This book aims to provide the scientific community with a novel and valuable approach based on fractal geometry concepts

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*Table of contents : Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems......Page 3Contents......Page 7About the Editors......Page 9List of Contributors......Page 11Series Preface......Page 13Preface......Page 151 Introduction to the Study of Environmental Fractals......Page 192 Introduction to Fractal Geometry, Fragmentation Processes and Multifractal Measures: Theory and Operational Aspects of their Application to Natural Systems......Page 293 Methods and Techniques for Fractal Analysis of Environmental Systems......Page 874 Fractal Structures and Mechanisms in Coagulation/Flocculation Processes in Environmental Systems: Theoretical Aspects......Page 1295 Fractal Mechanisms in Coagulation/Flocculation Processes in Environmental Systems......Page 1676 Fractal Approach to Adsorption/Desorption Processes on Environmental Surfaces......Page 1977 Applications of Fractals in the Study of Humic Materials......Page 2398 Fractal Geometry and Microorganisms in the Environment......Page 2579 Fractal Geometry of Aerosol Particles......Page 291Index......Page 335*

IUPAC Series on Analytical and Physical Chemistry of Environmental Systems. Volume 11

Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems Edited by PROFESSOR NICOLA SENESI Department of Biology and Chemistry, University of Bari, Italy PROFESSOR KEVIN J. WILKINSON Department of Chemistry, University of Montreal, Canada

A John Wiley and Sons, Ltd, Publication

Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems

IUPAC SERIES ON ANALYTICAL AND PHYSICAL CHEMISTRY OF ENVIRONMENTAL SYSTEMS

Series Editors Jacques Bufﬂe, University of Geneva, Geneva, Switzerland Herman P. van Leeuwen, Wageningen University, Wageningen, The Netherlands Series published within the framework of the activities of the IUPAC Commission on Fundamental Environmental Chemistry, Division of Chemistry and the Environment. INTERNATIONAL UNION OF PURE AND APPLIED CHEMISTRY (IUPAC) Secretariat, PO Box 13757, 104 T.W. Alexander Drive, Building 19, Research Triangle Park, NC 27709-3757, USA Previously published volumes (Lewis Publishers) Environmental Particles Vol. 1 (1992) ISBN 0-87371-589-6 Edited by Jacques Bufﬂe and Herman P. van Leeuwen Environmental Particles Vol. 2 (1993) ISBN 0-87371-895-X Edited by Jacques Bufﬂe and Herman P van Leeuwen Previously Published volumes (John Wiley & Sons, Ltd) Metal Speciation and Bioavailability in Aquatic Systems Vol. 3 (1995) ISBN 0-471-95830-1 Edited by Andre Tessier and David R Turner Structure and Surface Reactions of Soil Particles Vol. 4 (1998) ISBN 0-471-95936-7 Edited by Pan M. Huang, Nicola Senesi and Jacques Bufﬂe Atmospheric Particles Vol. 5 (1998) ISBN 0-471-95935-9 Edited by Roy M. Harrison and Rene E van Grieken In Situ Monitoring of Aquatic Systems Vol. 6 (2000) ISBN 0-471-48979-4 Edited by Jacques Bufﬂe and George Horvai The Biogeochemistry of Iron in Seawater Vol. 7 (2001) ISBN 0-471-49068-7 Edited by David R. Turner and Keith A. Hunter Interactions between Soil Particles and Microorganisms Vol. 8 (2002) ISBN 0-471-60790-8 Edited by Pan M. Huang, Jean-Marc Bollag and Nicola Senesi Physicochemical Kinetics and Transport at Biointerfaces Vol. 9 (2004) ISBN 0-471-49845-9 Edited by Herman P. van Leeuwen and Wolfgang Koster Environmental Colloids and Particles – Behaviour, Separation and Characterisation Vol. 10 (2006) ISBN 978-0470-024324 Edited by Kevin J. Wilkinson and Jamie Lead Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems Vol. 11 (2008) ISBN 978-0470-014745 Edited by Nicola Senesi and Kevin J. Wilkinson

IUPAC Series on Analytical and Physical Chemistry of Environmental Systems. Volume 11

Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems Edited by PROFESSOR NICOLA SENESI Department of Biology and Chemistry, University of Bari, Italy PROFESSOR KEVIN J. WILKINSON Department of Chemistry, University of Montreal, Canada

A John Wiley and Sons, Ltd, Publication

Copyright © 2008 IUPAC Published in 2008 by

John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone

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Email (for orders and customer service enquiries): [email protected] Visit our Home Page on www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to [email protected], or faxed to (+44) 1243 770620. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The Publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. The Publisher and the Author make no representations or warranties with respect to the accuracy or completeness of the contents of this work and speciﬁcally disclaim all warranties, including without limitation any implied warranties of ﬁtness for a particular purpose. The advice and strategies contained herein may not be suitable for every situation. In view of ongoing research, equipment modiﬁcations, changes in governmental regulations, and the constant ﬂow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. The fact that an organization or Website is referred to in this work as a citation and/or a potential source of further information does not mean that the author or the publisher endorses the information the organization or Website may provide or recommendations it may make. Further, readers should be aware that Internet Websites listed in this work may have changed or disappeared between when this work was written and when it is read. No warranty may be created or extended by any promotional statements for this work. Neither the Publisher nor the Author shall be liable for any damages arising herefrom. Other Wiley Editorial Ofﬁces John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 6045 Freemont Blvd, Mississauga, Ontario L5R 4J3, Canada Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Library of Congress Cataloging in Publication Data Senesi, N. (Nicola) Biophysical chemistry of fractal structures and processes in environmental systems / Nicola Senesi, Kevin Wilkinson. p. cm. — (Series on analytical and physical chemistry of environmental systems) Includes bibliographical references and index. ISBN 978-0-470-01474-5 (cloth : alk. paper) 1. Physical biochemistry. 2. Fractals. 3. Environmental chemistry. I. Wilkinson, Kevin J. II. Title. QD476.2.S36 2008 577 .14—dc22 2008011002 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 978-0-470-01474-5 Typeset in 10/12pt Times by Integra Software Services Pvt. Ltd, Pondicherry, India Printed and bound in Great Britain by TJ International Ltd, Padstow, Cornwall

Contents

About the Editors List of Contributors Series Preface Preface 1. 2.

3.

4.

5.

6.

7. 8. 9.

Introduction to the Study of Environmental Fractals Nicola Senesi and Kevin J. Wilkinson Introduction to Fractal Geometry, Fragmentation Processes and Multifractal Measures: Theory and Operational Aspects of their Application to Natural Systems Philippe Baveye, Charles W. Boast, Sarra Gaspard, Ana M. Tarquis and Humberto Millan Methods and Techniques for Fractal Analysis of Environmental Systems Graeme Bushell Fractal Structures and Mechanisms in Coagulation/Flocculation Processes in Environmental Systems: Theoretical Aspects Serge Stoll and Silvia Diez Fractal Mechanisms in Coagulation/Flocculation Processes in Environmental Systems Jean Yves Bottero, Armand Masion and Jérôme Rose Fractal Approach to Adsorption/Desorption Processes on Environmental Surfaces Zoﬁa Sokołowska and Stefan Sokołowski Applications of Fractals in the Study of Humic Materials James A. Rice Fractal Geometry and Microorganisms in the Environment Lynne Boddy and Damian P. Donnelly Fractal Geometry of Aerosol Particles Ian Colbeck

Index

vii ix xi xiii 1

11

69

111

149

179 221 239 273

317

About the Editors

Nicola Senesi is Professor of Soil Chemistry and Head of the Department of Agroforestal and Environmental Biology and Chemistry of the University of Bari, Italy, where he has been actively involved in research and teaching since 1969. He has taught courses in Soil Chemistry, Soil Science, Agricultural Chemistry, Wood Chemistry and Technology, Organic Chemistry, and General and Inorganic Chemistry. He has been a visiting professor and/or scientist for various periods at universities in Canada, USA, Somalia, Indonesia, Switzerland, Argentina, Brasil, Venezuela and Colombia. Dr. Senesi is a Fellow of the American Society of Agronomy (ASA) since 1995 and the Soil Science Society of America (SSSA) since 1996, and received the Gold Medal of the Polish Soil Science Society in 1994. He was conferred with a Doctorate Honoris Causa by the Institute National Polytechnique de Toulouse (INPT), France, in 2000. Dr. Senesi is currently President of Division VI: Chemistry and the Environment and Chair of the Subcommittee on Biophysico-Chemical Processes in Environmental Systems of IUPAC, President of MESAEP, and Vice-President of the Italian Soil Science Society (SISS), and was President of the International Humic Substances Society (IHSS) and Chairman of Division II-Soil Properties and Processes of the International Union of Soil Science (IUSS). He is currently an Associate Editor of Geoderma, Soil Science, Pedosphere, Pure and Applied Chemistry and CLEAN-Soil, Air, Water. Dr. Senesi’s research is focused on fundamental and applied aspects of the chemistry and biochemistry of organic matter from soils and other systems and materials, and its interactions with soil-applied organic chemicals and trace metals, by the use of advanced physico-chemical techniques and biochemical tools. Speciﬁc topics of research include the abiotic interactions of herbicides and endocrine disruptors with humic substances, the complexation mechanisms between trace metals of agricultural and environmental importance and natural and artiﬁcial humic materials, the physiological and antimutagenic effects of humic substances on plants, and the implications of recycling organic wastes on soil fertility and crop production. He is also pioneering the application of fractal geometry to the study of molecular conformation and aggregation processes of natural soil organic colloids. The results of his research are documented in about 300 scientiﬁc and technical papers and about 60 book chapters and invited reviews. Dr. Senesi has also co-edited 12 books and Proceedings Volumes.

viii

ABOUT THE EDITORS

Kevin J. Wilkinson received a PhD in Environmental Chemistry from the National Water Research Institute of the University of Quebec (INRS-Eau) in 1993. Following his PhD, he joined the research group of Professor Jacques Bufﬂe at the University of Geneva where he began to examine some of the important biophysical properties of environmental biopolymers and colloids. Following the establishment of his own research group in 1994, he focused his research onto relating the structure of environmental colloids and aggregates to their function in addition to initiating a research programme designed to develop a fundamental understanding of the chemical mechanisms of contaminant bioavailability. In 2005, he was appointed Associate Professor of Chemistry at the University of Montreal. His teaching includes (bio)analytical and environmental chemistry. His current research interests include: (i) improving our understanding of the role(s) of microorganisms on the biophysicochemistry of trace elements and colloids; (ii) development and optimisation of novel analytical techniques for quantifying bioavailability and colloidal/aggregate structure; (iii) characterising environmental biopolymers; and (iv) determining the role of diffusion in complex environmental media (bioﬁlms, ﬂocs, sediments). He is currently member of the editorial board of Environmental Chemistry and titular member of the IUPAC Chemistry and Environment Division. He also was editor for the previous volume in the IUPAC series on Analytical and Physical Chemistry of Environmental Systems (vol. 10): Environmental Colloids and Particles: Behaviour, Separation and Characterisation.

List of Contributors

Philippe Baveye SIMBIOS Centre, University of Abertay Dundee, Kidd Building, Bell Street, Dundee DD1 1HG, UK. Charles W. Boast Department of Natural Resources and Environmental Sciences, University of Illinois at Urbana-Champaign, 1102 South Goodwin Avenue, Urbana, II 61801, USA Lynne Boddy Cardiff School of Biosciences, Cardiff University Biomedical Building, Museum Avenue, Cardiff, CF10 3US, UK Jean Yves Bottero CEREGE UMR 6635 CNRS-UPCAM, Europole de l’Arbois, BP 80, 13545 Aix-enProvence, Cedex 04, France Graeme Bushell School of Chemical Sciences and Engineering, The University of New South Wales, UNSW Sydney NSW 2052, Australia Ian Colbeck Department of Biological Sciences, University of Essex, Colchester CO4 3SQ, UK Silvia Diez CIEMAT, Avda. Complutense 22, 28040, Madrid, Spain Damian P. Donnelly Cardiff School of Biosciences, Cardiff University Biomedical Building, Museum Avenue, Cardiff, CF10 3US, UK Sarra Gaspard Laboratoire COVACHIMM EA3592, Département de Chimie, UFR des Sciences Exactes et Naturelles, Université des Antilles et de la Guyane, BP 250, Campus Fouillole, Pointe-à-Pitre, 97159 GUADELOUPE, FWI (French West Indies) Armand Masion CEREGE UMR 6635 CNRS-UPCAM, Physical Chemistry, Europole de l’Arbois, BP 80, 13545 Aix-en-Provence, Cedex 04, France Humberto Millan Department of Basic Science, Universidad de Granma, Apdo 21, Bayamo, Granma 85100, Cuba

x

LIST OF CONTRIBUTORS

James A. Rice Department of Chemistry and Biochemistry, South Dakota State University, Box 2202, Brookings, SD 57007-0896 Jérôme Rose CEREGE UMR 6635 CNRS-UPCAM, Europole de l’Arbois, BP 80, 13545 Aix-enProvence, Cedex 04, France Nicola Senesi Dipartimento di Biologia e Chimica Agro-Forestale ed Ambientale, Università degli Studi di Bari, Via Amendola 165/A, 70126-Bari, Italy Zoﬁa Sokołowska Institute of Agrophysics, Polish Academy of Sciences, Do´swiadczalana 4, 20290 Lublin, Poland Stefan Sokołowski Department for the Modelling of Physico-Chemical Processes, Marie Curie– Skłodowska University, 20031 Lublin, Poland Serge Stoll Department of Inorganic, Analytical and Applied Chemistry CABE, University of Geneva, Sciences II, 30 Quai Ernest Ansermet, CH-1211 Geneva 4/Switzerland Ana M. Tarquis Department of Applied Mathematics to Agriculture Engineering, E.T.S. de Ingenieros Agrónomos, Universidad Politechnica de Madrid, Ciudad Universitaria s.n. 28040 Madrid, Spain Kevin J. Wilkinson Département de Chimie, Analitique et Biophysicochimie, Université de Montréal, C.P. 6128, Montréal, QC, H3C 3J7, Canada

Series Preface

The main purpose of the IUPAC Series on Analytical and Physical Chemistry of Environmental Systems is to make chemists, biologists, physicists and other scientists aware of the most important biophysicochemical conditions and processes that deﬁne the behaviour of environmental systems. The various volumes of the Series thus emphasise the fundamental concepts of environmental processes, taking into account speciﬁc aspects such as physical and chemical heterogeneity, and interaction with the biota. Another major goal of the series is to discuss the analytical tools that are available, or should be developed, to study these processes. Indeed, there still seems to be a great need for methodology developed speciﬁcally for the ﬁeld of analytical/physical chemistry of the environment. The present volume of the series focuses on the use of fractal geometry to provide a quantitative description of disordered systems. By their very nature, environmental systems are disordered and are thus obvious candidates for quantitative description using fractal dimensions. In addition, the physical, chemical and biological properties and processes of natural systems can be described using a fractal approach. This includes the quantiﬁcation of aggregate structures in air, water, soils and sediments; ﬂow through porous media; distributions of organisms, adsorption phenomena and reaction kinetics. This volume was realised within the framework of the activities of the IUPAC Division of Chemistry and the Environment. We thank the IUPAC ofﬁcers responsible, especially the executive director, Dr John Jost, for their support and assistance. Support from IUPAC provided the means to organize the discussion meeting of chapter authors (Bari, Italy, 2007), which formed such an essential step in the preparation and harmonisation of the various chapters of this book. Jacques Bufﬂe and Herman P. van Leeuwen Series Editors

Preface

Fractal geometry provides a powerful approach for the quantitative description of disordered systems. In addition, it is useful for describing the processes that lead to the formation of such complex, highly irregular and random systems and their physical behaviour.1 Fractals treat disorder as an intrinsic phenomenon that is described in terms of a nonintegral dimension with a degree of irregularity that is independent of scale. In weakly disordered systems, the disorder disappears as progressively smaller or larger length scales are probed, whereas there is repetition of the disorder at all length scales in strongly disordered systems. A large number of analytical techniques (light, X-ray and neutron scattering techniques; light, X-ray and electron microscopy; sedimentation and particle counting techniques; etc.) are now available to probe this repetitive disorder. In the natural environment, there is a great need to describe complex physicochemical systems and processes quantitatively. By their very nature, environmental systems are disordered and, thus, are perfect candidates for quantitative description using fractal dimensions. Indeed, many natural objects have been shown to be fractal, including star constellations, clouds, coastlines, trees, snowﬂakes, brain circumvolutions, proteins, cellulose, colloidal aggregates, several minerals and clays, limestones and sandstones, sediments, soils, and their organic, mineral and microbial components. In addition, the physical, chemical and biological properties and processes of natural systems may be described using a fractal approach. This includes the quantiﬁcation of aggregate structures in air, water, soils and sediments; ﬂow through porous media; distributions of organisms, adsorption phenomena and reaction kinetics. An ‘ideal’ or ‘regular’ fractal structure exhibits ‘self-similarity’ over all characterization length scales, i.e. the structure can be decomposed into smaller copies of itself, so that when any portion of the structure is magniﬁed it will appear identical to a larger part. Since natural structures tend to be self-similar over only a ﬁnite range of length scales,2 they are most often referred to as ‘random’ fractals. The notion of fractals can be used to describe very diverse objects according to the property of interest. In the natural sciences, a large number of physical properties and processes will depend upon the scaling behaviour of the mass, surface and pore spaces of the system (Figure 1). If the mass and the surface area scale in the same

1

Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman, New York. Pfeifer, P. and Obert, M. (1989). Fractals: basic concepts and terminology. In The Fractal Approach to Heterogeneous Chemistry, Avnir, D. (ed.). John Wiley and Sons, Ltd. 2

xiv

PREFACE

Figure 1 Different fractals considered in environmental sciences. (a) Surface fractal: colony of Bacillus subtilis on agar; see Figure 8.3 for more details. (b) Mass fractal: aggregate of hematite particles formed at pH 4, ionic strength 150 mM, in the presence of natural organic matter of [C] = 2.8 mg dm−3 ; see Chapters 4 and 5 for more details on aggregation processes. (c) Pore fractals: soil proﬁle in which a (darker) preferential pathway is visible; see Figure 2.21 for more details.

manner, then the system is considered a mass fractal; if the pore space and the surface follow the same scaling law, then the system is a pore fractal; and if only the surface is fractal, then the system is considered a surface fractal. Each system can be represented by a three-dimensional network that divides the system on the basis of its distribution of mass, surface or pore sites. Although the different fractal dimensions are not always distinguished in the literature (again contributing to some of the confusion), an attempt has been made in this volume to distinguish clearly the different measurements (even when it was not made clear in the original reference!). As a result of its practical utility for examining natural systems, fractal theory has developed in the geophysical, soil and atmospheric sciences, although little critical discussion has attempted to relate the different ﬁelds. The use of fractal concepts by scientists from very different disciplines and backgrounds has resulted in some confusion in the literature with respect to the meaning of the term fractal, nearly always resulting in unwarranted confusion, but also occasionally leading to unsound science. One of the objectives of this book is to reduce the confusion resulting from such a broad use of this important approach to the quantiﬁcation of disorder. Speciﬁcally, the book was written in order: (i) to provide an introduction to the theory of environmental fractals (Chapters 1 and 2); (ii) to summarize the available techniques for quantifying fractal structures in environmental systems (Chapter 3); (iii) to describe how the fractal approach can be employed to describe environmental processes such as coagulation, ﬂocculation, adsorption and desorption (Chapters 4–6); and (iv) and to describe critically a number of important environmental applications of fractal analysis (humic substances, Chapter 7; microorganisms, Chapter 8; aerosols, Chapter 9).

PREFACE

xv

In this volume, as for the other volumes in the series, the goal is to provide a critical review of the literature and a thorough explanation of the most important physicochemical processes. This book is the result of the efforts of a number of authors, collaborators and students. The International Union of Pure and Applied Chemistry (IUPAC) provided much of the structure and funding, through the Division of Chemistry and the Environment, which allowed this project come to fruition. The role of the series editors, Professor H.P. van Leeuwen and Professor J. Bufﬂe, is also greatly appreciated. N. Senesi Dipartimento di Biologia e Chimica Agro-Forestale ed Ambientale Università degli Studi di Bari K.J. Wilkinson Department of Chemistry, Université de Montréal

1 Introduction to the Study of Environmental Fractals Nicola Senesi1 and Kevin J. Wilkinson2 1

Dipartimento di Biologia e Chimica Agro-Forestale ed Ambientale, Università degli Studi di Bari, Via Amendola 165/A, 70126-Bari, Italy 2 Department of Chemistry, Université de Montréal, C.P. 6128, Montréal, QC, H3C 3J7, Canada

1.1 INTRODUCTION Fractal geometry provides a powerful approach for the quantitative description of complex, highly irregular and random, i.e. disordered systems [1]. Moreover, it can be used to describe the processes leading to the formation of such systems and their physical behavior. Fractal geometry relates to structures that cannot be described by Euclidean whole number dimensions of 1 (straight lines), 2 (ﬂat surfaces) or 3 (volumes), but instead have fractional dimensions. Surface structures can have fractal dimensions between 1 and 2 (a completely ﬁlled plane) and objects with volume can have fractal dimensions up to 3. The degree of irregularity of fractal structures is independent of scale and, thus, when they are examined under increasing magniﬁcation, more and more irregularities come into view. ‘Ideal’ or ‘regular’ fractal structures exhibit ‘self-similarity’, i.e. the structure can be broken down into smaller copies of itself. In other words, when any portion of the structure is magniﬁed, it will appear identical to a larger part over all characterization length scales. Natural structures tend to be self-similar only over a ﬁnite range of length scales [2]. Nonetheless, since they are more closely described by fractal rather than Euclidean geometry, they are designated ‘random’ fractals. Indeed, fractal concepts have been shown to be useful to describe a large number of natural objects/systems and their physical, chemical and biological properties, including constellations, clouds, coastlines, trees, snowﬂakes, brain circumvolutions, proteins, colloidal aggregates, cellulose, several minerals and clays, sediments, soils and their organic, mineral and microbial components [e.g. 3–13].

Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems c 2008, IUPAC and Kevin J. Wilkinson

Edited by Nicola Senesi

2

INTRODUCTION TO THE STUDY OF ENVIRONMENTAL FRACTALS

1.2 NATURAL FRACTALS Mathematical or nonrandom fractals are scale invariant, i.e. the pattern is the same at all scales (self-similar). Natural, real or random fractals are ‘quasi’ or ‘statistically’ self-similar over a ﬁnite length scale that is most often determined by the characterization technique that is employed. An object or process can be classiﬁed as fractal when the length scale of the property being measured covers at least one order of magnitude. Fractal structures obey a power law, allowing the fractal dimension D to be determined from experimental data: p ∝ vγ

(1.1)

where p is the property of interest, v is the variable being measured and γ is an exponent that can be related to D. For example, in Figure 1.1, the dark area is a two-dimensional representation of a natural structure, e.g. a soil particle or microbial colony, with an area A(s). The area can be estimated by counting the number of cells of size s 2 that are required to cover the object. For a fractal object A(s) ∝ s 2−D

(1.2)

D can be estimated by making measurements using different values of s.

Figure 1.1 A fractal structure overlaid by a grid of squares. In three dimensions the squares are replaced by boxes. Surface, mass and pore fractal dimensions can be estimated by counting the number of squares occupied by the object. Two-dimensional images of three-dimensional objects do not contain the full information about the object, so fractal dimensions of threedimensional objects need to be estimated in three dimensions [11]. Reproduced with permission of John Wiley & Sons, Ltd.

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3

1.3 MASS, SURFACE AND PORE FRACTALS Environmental particles, microbial colonies, and even patterns of movement of organisms can be characterized in terms of several different fractal dimensions [14] (Table 1.1). The fractal dimension of the surface DS (boundary/interface) of a solid structure is obviously an important characteristic, but many of the physical properties of solids also depend on the scaling behavior of the entire solid and/or of its pores. Systems where surface and mass scale similarly are termed ‘mass fractal’ systems, those where surface and the pore volumes scale similarly are described as ‘pore fractals’ and systems where only the surface is fractal are designated as ‘surface fractals’ (Table 1.1; Figure 1.2). Table 1.1 Classiﬁcation of fractal objects based on surface (DS ), mass (DM ) and pore (DP ) fractal dimensions [14]. Reproduced with permission of Academic Press, London Classiﬁcation

DS

DM

DP

Characteristics

Surface fractal

Da

db

d

Mass fractal Pore fractal

D D

D d

d D

The interface is fractal, but the actual object and the pore are not fractal Interface and the actual object are fractal Interface and pore structure are fractal

a

D: general fractal dimension. d: Euclidean dimension of embedding space (i.e. 1, 2 or 3 respectively in one-, two- or threedimensional systems).

b

As an example, consider the situation where a structure is overlaid by a grid. The underlying sites can be categorized as mass sites (occupied sites), surface sites (occupied sites with adjacent empty sites) or pore (empty) sites (Figure 1.2). In order to determine the fractal dimension, it is necessary to quantify the number of mass, surface or pore sites that fall within a distance R from a ﬁxed site (Figure 1.3) [15]. R must always be much greater than the distance between nearest neighbor sites. The quantity Msites (R) refers to the ‘mass’ in a sphere of radius R. It grows with increasing R [16]: Msites (R) ∝ R D

(1.3)

Each type of site gives rise to a mass–radius relationship of the following form: Mmass (R) ∝ R DM

(1.4)

Msurface (R) ∝ R

(1.5)

DS

Mpore (R) ∝ R DP

(1.6)

As D approaches Euclidean dimensions, the three types of fractal become indistinguishable.

4

INTRODUCTION TO THE STUDY OF ENVIRONMENTAL FRACTALS

Figure 1.2 Two-dimensional representations of mass fractal (a), surface fractal (b) and pore fractal (c) structures [14]. Reproduced with permission of John Wiley & Sons, Ltd.

Figure 1.3 Illustration of Msites (R). The sites are pictured as adsorbed molecules on a surface, but equally well could be monomers in a polymer chain, primary particles of a colloidal aggregate, etc. [15]. Reproduced with permission of John Wiley & Sons, Ltd.

1.4 METHODS FOR ESTIMATING THE FRACTAL DIMENSION Fractal geometry deﬁnes how objects occupy space. There are a variety of different methods that are employed for estimating D. For example, different approaches are used depending on whether biotic or abiotic components are measured or according to the size of the major components of the system. For example, an embedding,

NICOLA SENESI AND KEVIN J. WILKINSON

5

three-dimensional space has been used to describe both the inorganic and organic components of soils. In contrast, the in situ distribution of soil microorganisms cannot be described using similar techniques due to the inherent difﬁculties involved with recording their images in an opaque substratum. One means of avoiding this problem would be to grow the microorganisms on the surfaces of the soil. The most common experimental methods that have been proposed to estimate the fractal dimensions of environmental systems include: (a) scattering techniques; (b) turbidmetry; (c) vapor adsorption methods, (d) reaction rate methods; and (e) microscopy. Visible or laser light scattering, small-angle X-ray scattering (SAXS) and small-angle neutron scattering (SANS) (a) are based on the measurement of the fraction of incident beam intensity that is scattered by the sample as a function of the scattering angle. The fractal dimension is obtained from the experimentally measured density autocorrelation function [14, 17, 18]. Turbidimetry (b) is based on a measurement of the wavelength dependence of turbidity, i.e. determination of the scattered or nontransmitted light corresponding to a reduction in the intensity of the incident beam [19]. Static or dynamic vapor adsorption methods (c) are based on the measurement of fractal adsorption isotherms. They make use of various classical adsorption equations that have been rederived for fractal surfaces. Speciﬁcally, the number of adsorbed molecules per unit mass of sorbent, i.e. the monolayer value, is related to the size of adsorbed probe molecules [20, 21]. Measurements of reaction rates on a particle surfaces (d), i.e. catalysis, ion exchange, dissolution, etc., are performed as a function of the number of active sites available, provided that the reaction is not diffusion limited [22]. Finally, transmission (TEM) and scanning electron microscopy (SEM) and optical microscopy (e) are used to collect images that are analyzed using procedures based on the density–density autocorrelation function [23] or other image-analysis techniques [24]. The methods outlined above have been recently reviewed in relation to the fractal analysis of soil components and soil properties [6, 11]. They are also discussed in much greater detail in Chapter 3 [25]. Once data are acquired, the choice of an appropriate analysis technique is another highly critical step. For example, the box counting method is a common method for estimating the fractal dimensions of bacterial colonies and mycelial systems that is well suited to image analysis [13, 26–28]. This method has recently replaced the ‘concentric circles’ (two-point density–density correlation) method [29] that was used in several earlier studies of microbial colonies in agar and soils [e.g. 30, 31], but that was shown to exclude the extending colony margin from analysis. The technique is described in much greater detail in Chapter 3 [25], and applications are provided in Chapters 8 and 9 [32, 33].

1.5 APPLYING FRACTAL GEOMETRY TO QUANTIFY GROWTH, POLYMERIZATION AND AGGREGATION An example of how fractal geometry can be used to better understand complex environmental processes is brieﬂy described here and more thoroughly detailed in

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INTRODUCTION TO THE STUDY OF ENVIRONMENTAL FRACTALS

Chapters 4 and 5 [34, 35]. A number of computer simulations and statistical models have been proposed to explain growth, polymerization and aggregation processes that lead to the formation of fractal structures that closely resemble those found in nature. Attempts have been made to relate the fractal dimensions of aggregates to their formation mechanisms and aggregation kinetics. Two distinct models, particle–cluster aggregation and cluster–cluster aggregation [36], have been proposed to describe aggregation processes. The particle–cluster aggregation model is based on the sequential addition of particles to a growing cluster and is not believed to be relevant for the majority of colloidal aggregation processes that occur in nature. Aggregation in natural systems is best simulated using a cluster– cluster aggregation (CLA) model [36] in which a large number of initially dispersed and randomly moving particles collide and stick to form small clusters that grow following collision with other particles and clusters. CLA can be classiﬁed on the basis of its rate-limiting process [36]: (a) ballistic CLA, for which particles and clusters follow linear or ballistic trajectories and join at their point of ﬁrst contact; (b) diffusion-limited CLA (DLCA), where particles and clusters follow random walks (Brownian motion) and the fractal dimensions of the generated clusters range between 1.75 and 1.80; (c) a reaction-limited CLA (RLCA), where a short-range repulsive barrier must be overcome before direct contact and sticking can occur between particles and/or clusters. In this ﬁnal CLA class, on average, a large number of collisions are required before aggregation can take place. In RLCA, aggregates have fractal dimensions that range between ∼1.9 and 2.1 for d = 3 and ∼1.5 and 1.6 for d = 2 [36, 37]. Aggregation processes are time dependent; thus, in addition to the geometric aspects, kinetic aspects must also be considered. In real systems, two distinct regimes of colloidal aggregation are generally distinguished, i.e. fast and slow aggregation. Each regime has a different rate-limiting step, and the resulting aggregates have different fractal dimensions and size distributions. In order to make the models time dependent, a sticking coefﬁcient α, which represents the collision efﬁciency between two particles and/or clusters, is deﬁned. The sticking coefﬁcient speciﬁes the aggregation rate and depends on the value of the energy barrier that must be overcome to obtain aggregation. Values of α range between α = 0 and α = 1 for 0 % or 100 % collision efﬁciency respectively [38]. When α is large and close to 1, aggregation is fast, limited only by the diffusion of the particles and/or clusters. Under these conditions, loose aggregates, characterized by fractal dimensions between 1.75 and 1.80, are formed (Figure 1.4) [38], in agreement with the DLCA model. In contrast, when α is small (low collision efﬁciency), the aggregation rate is slow and the process is mainly controlled by the energy barrier that must be overcome for particle/cluster sticking to occur. In this case, dense aggregates are formed with high fractal dimensions (Figure 1.4) [38] and aggregation is best described by the RLCA model. The sticking probability and aggregation rate can be increased by a decrease of the repulsive component of the energy barrier. In practice, the repulsive barrier is decreased: (i) by decreasing the pH of the medium, with a resulting decrease in the negative

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rate

m n li

o usi diff rea ctio n li

mit

ed

rate

Figure 1.4 Schematic representation of a diffusion-limited and a reaction-limited cluster– cluster aggregation process leading to aggregates of either low fractal dimension (loose aggregate) or higher fractal dimension (denser aggregate). Reproduced with permission from [19], copyright American Chemical Society.

charge on the interacting particles and clusters; (ii) by increasing the ionic strength of the medium, resulting in an increased charge screening of the particles/clusters; or (iii) by increasing temperature and/or stirring to provide greater kinetic energy to the aggregating system. Most aggregation processes that occur in natural aquatic systems can be described adequately by the simple, limiting-case models outlined above. However, several natural aggregation processes are much more complex, in which case a variety of factors may affect the fractal dimension. For example, the fractal dimension can be signiﬁcantly reduced by the effects of long-range attractive or repulsive interactions. Furthermore, in the often fragile aggregates that form in aqueous media, a number of processes, including restructuring by bending, folding and twisting and even bond breaking, can occur, resulting in a fractal dimension that is somewhat larger than that expected from simple models [36]. For fragile aggregates, deﬂocculation and dispersion may also result from changing environmental conditions, such as pH, concentration, salinity, etc., further complicating the interpretation of the fractal dimension that is measured for these systems. Owing to all of the above factors, it is clear that much care is required when interpreting fractal measurements of aggregation in natural systems.

1.6 CONCLUSIONS AND RECOMMENDATIONS In the natural environment, there is a great need to describe complex physiochemical systems and processes quantitatively. By their very nature, environmental systems are disordered and are thus perfect candidates for their quantitative description using

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INTRODUCTION TO THE STUDY OF ENVIRONMENTAL FRACTALS

fractal dimensions. Since a large number of physical properties will depend upon the scaling behavior of the mass, surface and pore spaces of the system, the notion of fractals can be used to describe very diverse objects according to the propriety of interest. Indeed, fractal geometry has been successfully employed to describe aggregate structures in water, soils and sediments, ﬂow through porous media, organism distributions, adsorption and reaction kinetics, among other topics. Fractal analysis cannot be applied indiscriminately. For the values that are obtained to be mechanistically valid, comparisons must be made among similar measurements of fractal dimensions (e.g. mass, surface or pore fractal dimensions) that are made using a single given analytical technique or techniques based upon similar analytical principles. Future research is required to better relate measurements that are performed in different manners and indeed to better understand the limitations of the fractal measurements. Nonetheless, the majority of future research in the ﬁeld will necessarily be focused upon applying established determinations of fractal dimensions to understanding the physicochemistry of complex (i.e. natural) systems better (e.g. Chapters 4–9 [32–35, 39, 40]). LIST OF SYMBOLS AND ABBREVIATIONS D, DM , DS , DP R M CLA DLCA RLCA α

fractal dimension, mass fractal dimension, surface fractal dimension, pore fractal dimension distance from a ﬁxed site mass in a given distance cluster–cluster aggregation model diffusion-limited CLA model reaction-limited CLA model sticking coefﬁcient

REFERENCES [1] Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman & Co., New York. [2] Hastings, H.M. and Sugihara, G. (1993). Fractals. A User’s Guide for the Natural Sciences. Oxford University Press, Oxford. [3] Richardson, L.F. (1961). The problem of contiguity: an appendix of statistics of deadly quarrels. Gen. Syst. Yrbk., 6, 139. [4] Morse, D.R., Lawton, J.H., Dodson, M.M. and Williamson, M.H. (1985). Fractal dimension of vegetation and the distribution of arthropod body lengths. Nature, 314, 731. [5] Avnir, D. (1989). The Fractal Approach to Heterogeneous Chemistry: Surfaces, Colloids, Polymers. John Wiley & Sons, Ltd, New York. [6] Senesi, N. (1994). The fractal approach to the study of humic substances. In Humic Substances in the Global Environment and Implications on Human Health, Senesi, N. and Miano, T.M. (eds). Elsevier, Amsterdam, p. 3.

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[7] Senesi, N. (1996). Fractals in general soil science and in soil biology and biochemistry. In Soil Biochemistry, vol. 9, Stotzky, G. and Bollag, J.-M. (eds). Marcel Dekker, New York, p. 415. [8] Senesi, N. (1999). Aggregation patterns and macromolecular morphology of humic substances: a fractal approach. Soil Sci., 164, 841. [9] Anderson, A.N., McBratney, A.B. and Crawford, J.W. (1998). Applications of fractals to soil studies. Adv. Agron., 63, 1. [10] Baveye, P., Parlange, J.-Y. and Stewart, B.A. (1998). Fractals in Soil Science. CRC Press, Boca Raton, FL. [11] Okuda, I. and Senesi, N. (1998). Fractal principles and methods applied to the chemistry of sorption onto environmental particles. In Structure and Surface Reactions of Soil Particles, Huang, P.M., Senesi, N. and Bufﬂe, J. (eds). IUPAC Series. John Wiley & Sons, Ltd, Chichester, UK, p. 77. [12] Pachepsky, Ya. A., Crawford, J.W. and Rawis, W.J. (1999). Fractals in Soil Science. Geoderma, Special Issue, 88, 1. [13] Boddy, L., Wells, J.M., Culshaw, C. and Donnelly, D.P. (1999). Fractal analysis in studies of mycelium in soil. Geoderma, 88, 301. [14] Pfeifer, P. (1987). Characterization of surface irregularity. In Preparative Chemistry Using Supported Reagents, Laszlo, P. (ed.). Academic Press, London, p. 13. [15] Pfeifer, P. (1988). Fractals in surface science: scattering and thermodynamics of adsorbed ﬁlms. In Chemistry and Physics of Solid Surfaces, Vanselow, R. and Howe, R. (eds). Springer, Berlin, p. 283. [16] Pfeifer, P. and Obert, M. (1989). Fractal basic concepts and technology. In The Fractal Approach to Heterogeneous Chemistry: Surfaces, Colloids, Polymers, Avnir, D. (ed.). John Wiley & Sons, Ltd, New York, p. 11. [17] Martin, J.E. and Hurd, J. (1987). Scattering from fractal. J. Appl. Crystallogr., 20, 61. [18] Schmidt, P.W. (1989). Use of scattering to determine the fractal dimension. In The Fractal Approach to Heterogeneous Chemistry: Surfaces, Colloids, Polymers,Avnir, D. (ed.). John Wiley & Sons, Ltd, Chichester, England, p. 67. [19] Horne, D.S. (1987). Determination of the fractal dimension using turbidimetric techniques. Application to aggregation protein systems. Faraday Discuss. Chem. Soc., 83, 259. [20] Pfeifer, P. and Avnir, D. (1983). Chemistry in noninteger dimensions between two and three: I. Fractal theory of heterogeneous surfaces. J. Chem. Phys., 79, 3558. [21] Avnir, D., Farin, D. and Pfeifer, P. (1984). Molecular fractal surfaces. Nature, 308, 261. [22] Farin, D. and Avnir, D. (1989). The fractal nature of molecule–surface interactions and reactions. In The Fractal Approach to Heterogeneous Chemistry: Surfaces, Colloids, Polymers, Avnir, D. (ed.). John Wiley & Sons, Ltd, New York, p. 271. [23] Teixeira, J. (1986). Experimental methods for studying fractal aggregates. In On Growth and Form, Stanley, H.E. and Ostrowsky, N. (eds). Nijhoff, Dordrecht, The Netherlands, p. 145. [24] Kaye, B.H. (1989). Image analysis techniques for characterising fractal structure. In The Fractal Approach to Heterogenous Chemistry: Surfaces, Colloids, Polymers, Avnir, D. (ed.). John Wiley & Sons, Ltd, New York, p. 55. [25] Bushell, G. (2008). Methods and techniques for fractal analysis of environmental systems. In Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems, Senesi, N. and Wilkinson, K. (eds.), IUPAC Series, John Wiley & Sons Ltd, Chichester, UK, Chapter 3. [26] Obert, M., Pfeifer, P. and Sernetz, M. (1990). Microbial growth patterns described by fractal geometry. J. Bacteriol., 172, 1180.

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[27] Donnelly, D.P., Wilkins, M.F. and Boddy, L. (1995). An integrated image analysis approach for determining biomass, radial extent and box-count fractal dimension of macroscopic mycelial systems. Binary, 7, 19. [28] Donnelly, D P, Boddy, L. and Wilkins, M.F. (1999). Image analysis – a valuable tool for recording and analysing development of mycelial systems. Mycologist, 13, 120. [29] Witten, T.A. and Sander, L.M. (1981). Diffusion-limited aggregation, a kinetic critical phenomenon. Phys. Rev. Lett., 47, 1400. [30] Ritz, K. and Crawford J. (1990). Quantiﬁcation of the fractal nature of colonies of Trichoderma viride. Mycol. Res., 94, 1138. [31] Bolton, R.G. and Boddy, L. (1993). Characterisation of the spatial aspects of foraging mycelial cord systems using fractal geometry. Mycol. Res., 97, 762. [32] Boddy, L. and Donnelly, D.P. (2008). Fractal geometry and microorganisms in the environment. In Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems, Senesi, N. and Wilkinson, K. (eds.), IUPAC Series, John Wiley & Sons Ltd, Chichester, UK, Chapter 8. [33] Colbeck, I. (2008). Fractal geometry of aerosol particles. In Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems, Senesi, N. and Wilkinson, K. (eds.), IUPAC Series, John Wiley & Sons Ltd, Chichester, UK, Chapter 9. [34] Stoll, S. and Diez, S. (2008). Fractal structures and mechanisms in coagulation/ﬂocculation processes in environmental systems: Theoretical aspects. In Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems, Senesi, N. and Wilkinson, K. (eds.), IUPAC Series, John Wiley & Sons Ltd, Chichester, UK, Chapter 4. [35] Bottero, J.Y., Masion, A. and Rose. J. (2008). Fractal mechanisms in coagulation/ﬂocculation processes in environmental systems. Molecular aspects. In Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems, Senesi, N. and Wilkinson, K. (eds.), IUPAC Series, John Wiley & Sons Ltd, Chichester, UK, Chapter 5. [36] Meakin, P. (1991). Fractal aggregates in geophysics. Rev. Geophys., 29, 317. [37] Brown, W.D. and Ball, R.C. (1985). Computer simulation of chemically limited aggregation. J. Phys. A, 18, L517. [38] Bufﬂe, J. and Leppard, G.G. (1995). Characterization of aquatic colloids and macromolecules. 1. Structure and behavior of colloidal material. Env. Sci. Technol., 29, 2169. [39] Sokolowska, Z. and Sokolowski, S. (2008). Fractal approach to adsorption/desorption processes on environmental surfaces. In Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems, Senesi, N. and Wilkinson, K. (eds.), IUPAC Series, John Wiley & Sons Ltd, Chichester, UK, Chapter 6. [40] Rice, J.A. (2008). Applications of fractals in the study of humic materials. In Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems, Senesi, N. and Wilkinson, K. (eds.), IUPAC Series, John Wiley & Sons Ltd, Chichester, UK, Chapter 7.

2 Introduction to Fractal Geometry, Fragmentation Processes and Multifractal Measures: Theory and Operational Aspects of their Application to Natural Systems Philippe Baveye1 , Charles W. Boast2 , Sarra Gaspard3 , Ana M. Tarquis4 and Humberto Millan5 1

SIMBIOS Centre, University of Abertay Dundee, Kidd Building, Bell Street, Dundee DD1 1HG, UK 2 Department of Natural Resources and Environmental Sciences, University of Illinois at Urbana-Champaign, 1102 South Goodwin Avenue, Urbana, II 61801, USA 3 Laboratoire COVACHIMM EA3592, Département de Chimie, UFR des Sciences Exactes et Naturelles, Université des Antilles et de la Guyane, BP 250, Campus Fouillole, Pointe-à-Pitre, 97159 GUADELOUPE, FWI (French West Indies) 4 Department of Applied Mathematics to Agriculture Engineering, E.T.S. de Ingenieros Agrónomos, Universidad Politechnica de Madrid, Ciudad Universitaria s.n. 28040 Madrid, Spain 5 Department of Basic Science, Universidad de Granma, Apdo 21, Bayamo, Granma 85100, Cuba

2.1 INTRODUCTION Periodically, scientists uncover, in the treasure troves of mathematicians, a theory that allows the simple solution of a hitherto unresolved problem, or at least makes possible its formulation in a conceptual framework that eventually leads to an elegant solution. A typical example of this process is the adoption of tensor calculus by physicists in the early years of the 20th century. In the 1880s and 1890s, two Italian mathematicians, Gregorio Ricci-Curbastro (1853–1925) and Tullio Levi-Civita (1873–1941), spent years patiently elaborating a mathematical theory initially referred to as ‘absolute differential calculus’and later known as tensor calculus. This theory attracted virtually no attention outside of mathematical circles until Albert Einstein realized that it was precisely the tool he crucially needed to develop his general theory of relativity. He

Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems c 2008, IUPAC and Kevin J. Wilkinson

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spent time familiarizing himself with tensor calculus, apparently with some difﬁculty on account of the abstractness of the theory, and eventually used it successfully in his work. Thereafter, the use of tensor calculus widened and became routine in a broad range of disciplines. In due time, physicists and mechanicists developed a conceptual approach to tensor calculus that differs from that of mathematicians in a number of respects (e.g. is far less abstract), yet fundamentally preserves its intrinsic rigor. Compared with this common pattern of the adoption of a mathematical tool or theory by nonmathematicians, the case of fractal geometry is entirely atypical, and has been so from the start. Even though fractal geometry deals with mathematical objects, it was not at the onset deﬁned or developed as a mathematical theory. Mandelbrot [1], who coined the term ‘fractal’, did not feel that it was necessary to deﬁne precisely what the term referred to. In addition, of his own admission [e.g. 2], he was more attracted by the esthetic beauty of fractals, by the ‘hypnotically intricate visual patterns and images’ [3] that one may base on fractals, than by the mathematically rigorous study of their properties. His inﬂuential book on the Fractal geometry of nature [4] makes numerous statements about the features of fractals, few of which have been rigorously proven. For fear, apparently, of excluding some applications that could bring fame and glory to the theory (and its progenitor), Mandelbrot’s book [4] also presents several conﬂicting perspectives on the nature of fractals, often leaving readers somewhat bafﬂed as to exactly what the author had in mind. Before mathematicians had a chance to step in and develop fractal geometry as a truly rigorous mathematical theory,1 natural scientists became enthused by what they perceived as the enormous potential of fractals in their own ﬁelds, and ﬁlled gaps in the theory as needs arose. As a result of their research, fractal theory has developed in many different directions. In the resulting ﬂurry of activity, including in ﬁelds such as geophysics and soil science, rare have been the articles or books that have attempted to analyze carefully the connections existing among the different ‘natural’ fractals identiﬁed in the real world, or between them and the mathematical beings deﬁned and manipulated by the geometricians. Reluctance to embark on this analysis appears to have caused much confusion in the literature (e.g. see discussion in Baveye and Boast [6]); the term ‘fractal’, when applied to natural systems, often means very different things to different people, creating unnecessary difﬁculties in communication and occasionally leading to unsound conclusions. A ﬁrst objective of this chapter, therefore, is to attempt to ﬁll these gaps and, in particular, to make more explicit the connection between theoretical and natural fractals. A similar approach is followed in the last section of this chapter, which deals brieﬂy with the increasingly important multifractal measures. A second objective of this chapter is to point out that some uses of ‘fractals’ amount to little more than curveﬁtting exercises, and that any attempt to relate the resulting ‘fractal’ dimensions to geometrical features of natural systems should be approached with great caution.

1

They eventually succeeded in that endeavor, years later [e.g. 5].

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This chapter (which is an outgrowth and updated version of [7]) is not meant to be a treatise on fractal geometry. Its objective is to introduce the aspects of fractal theory that have found application in environmental science and which are used in later chapters of this book. The approach adopted, which we have found useful in our courses on the subject, is to follow, at least initially, a chronological description of the genesis of fractal geometry, starting with a number of strange-looking, even monstrous-looking, curves discovered by mathematicians in the 19th century. Only after these have been described in some detail do we attempt to present to the reader the commonalities among these curves and to draw from the resulting list of shared features a relatively clear picture of the concept of fractal. Experience has taught us that this chronological approach results in far fewer misconceptions about the true nature of fractals than where fractals are deﬁned from the onset. As much as possible, this chapter has been written in such a way as to be selfcontained and accessible to readers with little more than an elementary background in calculus. Nevertheless, numerous pointers to other sources of information are provided in case the incursion into set and measure theory found here is not sufﬁciently basic. In particular, Falconer [5 (pp. 3–16)] may be consulted in order to get up to speed; the mathematical concepts and the notation used throughout this chapter are introduce there. 2.2 A GALLERY OF MATHEMATICAL MONSTERS 2.2.1 The Precursors: Cantor’s Set, the Devil’s Staircase and the Peano–Hilbert Plane-Filling Curves In a landmark letter, sent on 20 June 1877, to his friend Richard Dedekind, the German mathematician Georg Cantor started a revolution in the ﬁeld of geometry [2]. He mentioned that he had doubts about the meaning, and even the validity, of the concept of dimension. He also proved that a surface limited by a square, which like any regular surface has a topological dimension2 equal to two (in Euclidian space), does not contain more points than any one of the square’s sides (of topological dimension equal to one)! Cantor’s derivation showed that there is a one-to-one correspondence between points on the sides and points on the surface, so that a single number sufﬁces to determine the position of any point in the square. A few years later, Cantor [8] gave the concept of dimension another serious jolt and created in the process the ﬁrst of what, after Henri Poincaré, came to be called the mathematical ‘monsters’. Known as the Cantor set, it is also commonly termed the ‘middle third’, the ‘ternary’ or the ‘triadic’ Cantor set. It is constructed by a sequence

2

The topological dimension DT of a set is always an integer and is 0 if the set is totally disconnected (i.e. for isolated points), 1 if each point of the set has arbitrarily small neighborhoods with boundary of dimension 0 (i.e. for lines), and so on for higher dimensions [5].

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INTRODUCTION TO FRACTAL GEOMETRY 0

1/3

2/3

1

I0 (‘initiator’)

I1 (‘generator’)

I2

I3

I4

Figure 2.1 Initial steps in the construction of the Cantor set by repeated removal of the middle third of intervals.

of removal operations from a unit-length interval, termed the initiator. This initiator is labeled I0 in Figure 2.1, where it corresponds to the interval [0, 1] (i.e. the set of numbers x such that 0 ≤ x ≤ 1). If one removes from this initiator the segment (1/3, 2/3) (containing the real numbers x such that 1/3 ≤ x ≤ 2/3), the set I1 results. This set is sometimes termed the generator and consists of the two intervals [0, 1/3] and [2/3, 1]. Removing the middle thirds of these intervals, i.e. applying the generator to each of them, yields I2 , which comprises four intervals of length 1/9. At the next stage (I3 ), there are 23 = 8 intervals of length (1/9)/3 = 3−3 . At the nth iteration of this deletion procedure, the set In consists of 2n intervals, each having a length 3−n , and is included in all the preceding sets I1 , I2 , . . . , In−1 in the sequence. The Cantor set P can be thought of as the limit of the sequence of sets In when n tends to inﬁnity. Mathematically [9], it is deﬁned as ∩∞ i = 1 In , the intersection of all the sets In , with n going from one to inﬁnity. It is obviously impossible to draw the set P itself, with its inﬁnitesimal detail, so ‘pictures of the Cantor set P ’ are in fact only illustrations of one of the sets In . It is apparent from Figure 2.1 that such representations are feasible only for relatively low values of n. At ﬁrst glance, it might appear that we have removed so much of the interval [0, 1], during the construction of the Cantor set, that nothing remains. In a sense, this is true. The (Lebesgue) measure m(In ) of the set In , i.e. practically, the total length of all the 2n intervals included in In , is given by (2/3)n . Since P = ∩∞ n = 1 In and P is included in In for every n, the measure m(P ) is equal to the limit as n → ∞ of (2/3)n , which is zero. In other words, the Cantor set has zero length and its topological dimension is zero. Yet it is an uncountable set, containing inﬁnitely many points in any neighborhood of each of its points. Furthermore, one can show that the points

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of the Cantor set can be put in one-to-one correspondence with those of the initiator [0, 1]; mathematically, P and [0, 1] have the same ‘cardinality’. This raises some very serious questions about the use of traditional dimensions to characterize these two sets, and justiﬁes the labels of ‘monstrous’ and ‘pathological’ that rapidly became associated with the Cantor set. The Cantor set P has a number of striking features [5]: 1. P contains exact copies of itself at many different scales. It is clear that the part of P in the interval [0, 1/3], or that in the interval [2/3, 1], are geometrically similar to P , except for the fact that they are scaled down by a factor 1/3. Again, the parts of P in each of the four intervals of I2 are geometrically similar to P , but scaled by a factor 1/9, and so on. Some authors [e.g. 5] call this feature of P ‘self-similarity’. Strictly speaking, however, as pointed out by Feder [10 (p. 63)], P is not entirely self-similar because of the ﬁnite length of its initiator. To obtain a self-similar set, one should enlarge P by an extrapolation procedure that covers the region [0, 3] by two Cantor sets deﬁned on the intervals [0, 1] and [2, 3]. Repetition of this procedure ad inﬁnitum generates a self-similar set on the half-line [0, ∞]. Every time the expression ‘self-similar’ is used in the remainder of this chapter, it will have to be interpreted within the context of Feder’s comment [10]. 2. The set P has a ‘ﬁne’ structure; it contains detail at arbitrary small scales. 3. Although P has an intricate and detailed structure, the actual deﬁnition of P is straightforward. 4. P is obtained by a recursive procedure. 5. The geometry of P is not easily described in classical terms; it is not the locus of the points that satisfy some simple geometric condition, nor is it the set of solutions of any simple equation. 6. It is awkward to describe the local geometry of P . Near each of its points are a large number of other points, separated by gaps of varying lengths. 7. Although P is in some ways quite a large set (it is uncountably inﬁnite), its size is not quantiﬁed by the usual measures such as length. By any reasonable deﬁnition, P has zero length. 8. The actual set P cannot be represented graphically. Many of these peculiar features of the Cantor set are shared by, or at least are similar to those of, the other mathematical monsters described later in this section. The above list will also be useful when we try to deﬁne as precisely as possible what is meant by the term ‘fractal’. Another mathematical ‘monster’, sometimes called the Cantor singular function [11], is closely related to the Cantor set. This singular function is constructed by integrating an appropriate distribution function deﬁned on the Cantor set. One such distribution function is constructed by ﬁrst considering the uniform distribution of mass on the interval [0, 1], with total mass equal to one (in some arbitrary units). For example, one could visualize the initiator I0 of Figure 2.1, not as a line segment, but as a bar of some material with unit mass density ρ0 and length l0 = 1. The operation

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that resulted in I1 in Figure 2.1 now consists of cutting the bar into two halves of equal mass (= 1/2) and then hammering them so that the length of each part becomes l1 = 1/3. The total mass is conserved in the operation, while the mass density in each part becomes ρ1 = 3/2. Repeating this process, one ﬁnds that, at the nth stage, the number of small bars is 2n , each of length ln = (1/3)n and of mass density ρn = (3/2)n . Integrating the mass density along x, we obtain the total mass contained between x = 0 and an arbitrary point x as x (2.1) Mn (x) = ρn (x) dx 0

where ρn = (3/2) on the 2 intervals of length ln = (1/3)n and is identically zero everywhere else. An example of such a function Mn (x) is shown in Figure 2.2. The n

n

2X

3X

Figure 2.2 Intermediate stage (n = 6) in the construction of the Cantor singular function or ‘Devil’s staircase’ (top) and schematic illustration of the self-afﬁnity of the Cantor singular function or ‘Devil’s staircase’ (bottom); the enlargement is identical to the original, but the enlargement (scaling) factors are different in the x and y directions.

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limit of Equation (2.1) as n → ∞ is a singular function, discovered by Cantor [11], and usually referred to as the ‘Devil’s staircase’. It is a continuous, nonconstant function that is (nearly) horizontal all over except on an uncountable set (the Cantor set), of Lebesgue measure equal to zero. At each point in this uncountable set, the derivative of the Cantor singular function is given by a Dirac δ distribution. The Cantor singular function has many of the features of the Cantor set (listed above), except (i), i.e. it does not contain exact copies of itself at different scales. Inspection of Figure 2.2 (top) shows that the shape of the function Mn (x) in the interval [0, 1/3] is similar to that of the whole function (in the interval [0, 1]), scaled down in the x direction (abscissa) by a factor of 1/3. However, it is also apparent that the scaling factor in the y direction (ordinate) in Figure 2.2 (top) is not 1/3 but 1/2 (see Figure 2.2, bottom). A set or function is said to be self-afﬁne when its scaling factors are different in different directions. Self-similarity requires these scaling factors to be identical in all directions. Therefore, the Cantor singular function or ‘Devil’s staircase’ is not self-similar, but self-afﬁne. Following in Cantor’s steps, another assault on the concept of dimension was made simultaneously by Peano [12] and Hilbert [13], in two short but inﬂuential articles. Both describe polygons that appear at ﬁrst glance to be perfectly innocent, but nevertheless happen to ﬁll a square more and more completely, so that, in the limit, they pass through every single point in the square. The construction of Peano’s original curve begins with a single line segment, the initiator (stage 0 in Figure 2.3). It is substituted by the generator (stage 1), which touches (but does not cross) itself at two points labeled A and B in Figure 2.3. If each straight line segment in stage 1 is replaced by a properly scaled-down generator,

A

B

Stage 0

Stage 1

Stage 2

Stage 3

Figure 2.3 First stages of the construction of Peano’s space-ﬁlling curve. At each step, each line segment is replaced by nine line segments scaled down by a factor of 1/3.

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INTRODUCTION TO FRACTAL GEOMETRY

one obtains stage 2. Independently of the direction (horizontal or vertical) of the line segments in stage 1, the scaling factor is equal to 1/3. At the next stage, the scaling factor is equal to 1/9, and so on. If one assumes that the length of the original line segment constituting the initiator is one, then it is easy to calculate the length of the curves at each stage. In stage 1, there are nine line segments of length 1/3 and the total length of the curve is three. In stage 2 there are 92 line segments of length 1/32 , amounting to a total length equal to nine. Expressed as a general rule, in each step of the construction, the resulting curve increases in length by a factor of three. In stage n, the length is thus 3n . If one pursues the above construction procedure to the limit as n → ∞, the number of straight-line segments and their total length tend to inﬁnity. The result is generally termed Peano’s curve and is of great interest mathematically because it is nowhere differentiable, i.e. it does not allow a tangent at any of its points (which are all ‘corners’). As with the Cantor set (but for different reasons), it is impossible to visualize the structure of this curve.All one can see, at any scale, is a completely ‘ﬁlled out’square, which does not look in the least similar to the early steps of the construction (see Figure 2.3). Nevertheless, the Peano curve, like P , contains exact copies of itself [e.g. 14, 15]. The same general characteristics are exhibited by Hilbert’s [13] plane-ﬁlling curve, except for the fact that it is self-avoiding, i.e. it never intersects or touches itself. 2.2.2 The Triadic Von Koch Curve The Swedish mathematician von Koch introduced in 1904 what is now called the (triadic) von Koch curve [e.g. 5, 14 (pp. 89–93)]. The construction of this curve, illustrated in Figure 2.4, starts with a line segment, I0 , of unit length. This initiator is replaced by the generator shown as the curve I1 , consisting of four segments of length 1/3. At the next stage, I2 is constructed by replacing each line segment in I1 by a properly scaled-down version of the generator. The scaling factor is equal to 1/3. The curve I2 consists of 42 = 16 segments each having a length of 1/32 = 1/9, so that the total length of I2 is equal to (4/3)2 = 16/9. At the nth stage, there are 4n segments of length δn = 3−n , with a total length of L(In ) = (4/3)n . This total length L(In ) may be expressed differently, in a way that makes explicit the dependence of L(In ) on δn . Taking the exponential of the natural logarithm of the total length L(In ) = (4/3)n and substituting the value of n (= − ln δn / ln 3) obtained by solving the equality ln δn = ln 3−n [e.g. 5 (p. 17)], one obtains L(In ) = δn1−ln 4/ ln 3

(2.2)

Equation (2.2) corresponds to a power-law (or ‘Paretian’) relationship between L and δn . Similarly, the number Nn of segments of length δn is also a power-law function of δn . Indeed, the equality L(In ) = Nn δn implies that Nn = δn− ln 4/ ln 3 . We shall return to these important power-law dependencies in later sections of this chapter. Even though we cannot represent it graphically, it is clear that, if we carry on the iterative procedure illustrated in Figure 2.4 to the limit where n → ∞, the resulting

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I0

Initiator

I1

Generator

I2

I3

I4

Figure 2.4 Construction of the triadic von Koch curve. At each step, the number of line segments increases by a factor of 4.

curve (the von Koch curve) has an inﬁnite number of vanishingly small segments and has a total length tending to inﬁnity. If, in Figure 2.4, one takes the part of I4 that corresponds to the interval [0, 1/3] of the initiator, and scales this part up horizontally and vertically by three, one obtains I3 . In the limit n → ∞, however, this same scaling up of any segment of In would reproduce the von Koch curve itself. Furthermore, like the Peano–Hilbert space-ﬁlling curves, the von Koch curve is nowhere differentiable. All of these curves have a topological dimension equal to unity; a simple stretching operation (or ‘rectiﬁcation’) transforms them into an inﬁnite straight line. At ﬁrst sight, the von Koch curve appears less monstrous than the Cantor set or Peano’s space-ﬁlling curve. Nevertheless, it has astonishing properties, which challenge the traditional concepts of dimension. Indeed, since the von Koch curve is inﬁnite in length and contains exact replicates of itself, any scaled-down sub-image is also of inﬁnite length [3]. This leads to the conclusion that for any two points on the curve, no matter how close they are, the curve between them is of inﬁnite length! In

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INTRODUCTION TO FRACTAL GEOMETRY

addition, if one joins together three initiators like I0 (in Figure 2.4) to form a triangle and one performs on each initiator the iteration procedure of Figure 2.4, the resulting geometrical construct, commonly known as the von Koch ‘island’ or ‘snowﬂake’, has inﬁnite length, even though it ﬁts within a ﬁnite area! 2.2.3 The Sierpinski Carpet and the Menger sponge Cantor [8] remarked that examples similar to the (Cantor) set P ‘can be easily constructed for higher dimensions’. One such example is the Sierspinski carpet. The iterative procedure leading to this carpet starts with a square initiator of side length l0 (Figure 2.5). In the next step, one divides the initiator in nine smaller squares of side length l0 /3 and removes the central square. When one applies the same procedure ad inﬁnitum to the remaining squares, a structure known as the Sierpinski carpet is obtained [e.g. 10 (p. 25)], which is Cantor-like in many respects, even though its initiator is two-dimensional. Like the Cantor set, the Sierpinski carpet is a self-similar, uncountable set with a topological dimension equal to zero. The two-dimensional Lebesgue measure of the Sierpinski carpet, i.e. practically, its area in the plane, is equal to zero, so the traditional concept of area does not provide a very useful description of the spatial coverage of the Sierpinski carpet. Some other type of dimension is needed.

I0

I1

I2

Figure 2.5 First stages of the construction of the Sierpinski carpet. I0 is the initiator. I1 is the generator.

When the iterative process used for the construction of the Sierpinski carpet is generalized to three dimensions and when the initiator is a cube, one obtains a geometrical structure of particular interest in geophysics and soil science, the Menger sponge (Figure 2.6). The construction of the Menger sponge is very similar to that of the Sierpinski carpet. Each square face of the cube is treated in exactly the same way as the square initiator of the Sierpinski carpet. This time, extracting a square shape involves punching a hole directly through the cube at right angles to the face concerned. Thus, at the ﬁrst stage I1 , three holes are punched through. This leaves 20 subcubes at one-third scale, each of which is repeatedly subdivided to create, ad inﬁnitum, the hollowed structure of the Menger sponge. Another way to describe the recursive construction process is to consider that, at the nth stage, each cube of size

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21

Figure 2.6 First three stages in the construction of the Menger sponge.

ln−1 is divided into 27 equal cubes of size ln = ln−1 /3, and that the central small cube is removed along with the six cubes with which it shares faces. This iterative construction may be envisaged from a physical standpoint. This produces mathematical relationships that will be useful later on. If the initiator is a cube of side length l0 , made of some material of uniform mass density ρ0 , the ﬁrstorder structure (obtained after one iteration) would have a porosity φ1 (void volume divided by total volume) equal to 7/27 and a mass density ρ1 equal to 20ρ0 /27. After the second iteration, the porosity φ2 would increase to 329/729 (≈0.45) and the density ρ2 would decrease to 400ρ0 /729 (≈0.55ρ0 ). At the nth iteration, porosity and density would be given by n 3−ln 20/ ln 3 20 ln φn = 1 − =1 − (2.3) 27 l0 and

n 3−ln 20/ ln 3 ρn 20 ln = = 27 l0 ρ0

(2.4)

The second of these two equations again gives rise to a power-law relationship (as in the case of the von Koch curve). Equations (2.3) and (2.4) will be mentioned again in a later section. For the time being, however, they can be used to illustrate some of the features of the Menger sponge, obtained when n → ∞. Since the exponent in Equations (2.3) and (2.4) is strictly positive (indeed, 3 − ln 20/ ln 3 ≈ 0.273), φn and ρn /ρ0 will tend to one and zero respectively as n → ∞. Mathematically, the three-dimensional Lebesgue measure of the Menger sponge, i.e. its volume practically, is zero. Like the Cantor set and the Sierpinski gasket, the Menger sponge is a self-similar, uncountable set of points with a topological dimension of zero. Therefore, totally against intuition, the Menger sponge is topologically equivalent to the Cantor set. Anecdotally, there is another connection between the Menger sponge and the Cantor set. The intersections of the Menger sponge with medians or diagonals of the initial cube are triadic Cantor sets [4 (p. 144)].

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INTRODUCTION TO FRACTAL GEOMETRY

2.2.4 Bolzano–Weierstrass-like Functions Our rapid overview of mathematical ‘monsters’would not be complete without a brief reference to an interesting family of continuous, nowhere differentiable functions. They are occasionally referred to as ‘Weierstrass-like’ but, for historical reasons, it seems more appropriate to call them ‘Bolzano–Weierstrass-like’. Indeed, Bernard Bolzano [16] appears to have been the ﬁrst (in a manuscript written around 1830 but published only in 1930) to provide an example of a continuous, nowhere differentiable function. A few years later, in 1872, Karl Weierstrass [17] showed that the function f (x) =

∞

bn cos(a n xπ )

(2.5)

n=0

has these properties, provided the product ab exceeds a certain limit. Hardy [18] showed that the conditions to be satisﬁed by a and b are 0 < b < 1, a > 1 and ab ≥ 1. Over the years, many examples of continuous, nowhere differentiable functions have been published; see Edgar [11 (pp. 7, 341)]. One of them, the so-called Weierstrass–Mandelbrot function, assumes a particular signiﬁcance in environmental science because it constitutes the theoretical basis of the ﬁrst article that used fractal geometry in connection with soils data. Burrough [19] used the Weierstrass– Mandelbrot function to describe the often erratic-looking spatial variation of soil properties along transects. The Weierstrass–Mandelbrot function w(t) is deﬁned in complex form by [10 (p. 27), 20] n ∞ (1 − eib t )eiϕn w(t) = (2.6) b(2−D)n n=−∞ where 1 < D < 2, b > 1 and the ϕn are arbitrary phase constants. Each choice of the phases ϕn deﬁnes a speciﬁc function w(t). For a given w(t), one may show that the variance of increments V (t) = [w(t0 + t) − w(t0 )]2 is a power function of time, t 4−2D , for t → 0 [19]. A simple function, the real part of w(t) with ϕn chosen as ϕn = 0, is the Weierstrass– Mandelbrot cosine function c(t): c(t) =

∞ 1 − cos bn t b(2−D)n n=−∞

(2.7)

The shape of this function is reasonably smooth for low values of D (see Figure 2.7). As D increases, however, c(t) begins to ﬂuctuate widely, as if larger and larger amounts of ‘noise’ were added to an underlying trend. Close inspection of Figure 2.7c reveals that c(t) is self-afﬁne, like Cantor’s singular function [10 (p. 29)]. Both w(t) and c(t) are functions of a single variable.

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4 a

c(t)

3

2 D = 1.2 b = 1.5

1

0 0.0

0.2

0.4

0.6

0.8

1.0

t 7 6

b

5 c(t)

4 3 2

D = 1.5 b = 1.5

1 0 0.0

0.2

0.4

0.6

0.8

1.0

t 15 c

c(t)

10

5 D = 1.8 b = 1.5 0 0.0

0.2

0.4

0.6

0.8

1.0

t

Figure 2.7 Graphical representations of the Weierstrass–Mandelbrot cosine function c(t) with b = 1.5 and (a) D = 1.2, (b) D = 1.5, and (c) D = 1.8. Modiﬁed from [10].

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INTRODUCTION TO FRACTAL GEOMETRY

Multivariate Weierstrass–Mandelbrot functions have, however, been deﬁned in the last decade [e.g. 21]. 2.2.5 Random Monsters and Fractional Brownian Motion The various mathematical monsters described above have been singled out for their historical signiﬁcance, not because they are somehow unique. For each of them, there are an inﬁnite number of variants. One could, for example, deﬁne the generator I1 of a Cantor-like set to be the union of two intervals [0, a] and [b, 1], for any a and b with 0 < a < b < 1. It is easy to imagine extending this deﬁnition to include three, four, or ﬁve intervals, and so on. Similarly, instead of replacing the middle-third segment in the initiator of the von Koch curve by the other two sides of an equilateral triangle, one could remove an interval of length a from the middle of the initiator, such that 0 < a ≤ 1/3. The interval could be replaced, for example, by the other three sides of a square or by the other two sides of a suitably short isosceles triangle. Obviously, there are an inﬁnite number of such sets. In addition to sharing all of the general features of the Cantor set (listed above), they are also deterministic, in the sense that their construction is based on the recursive application of an invariant and nonrandom generator. It is straightforward to enlarge this family of monsters to include random analogues. For example, in the von Koch construction, each time the middle third of an interval is replaced by the other sides of an equilateral triangle, one might toss a coin to determine whether to position the new part above or below the removed segment. After a few steps, one gets an irregular-looking curve (see Figure 2.8), which nevertheless retains much of the ‘look’ of the von Koch curve [5, 14]. It is no longer exactly self-similar, like its nonrandom counterpart, but it is said to be ‘statistically’self-similar. According to Feder [10 (p. 184)], a set S is statistically self-similar when S is the union of n distinct (nonoverlapping) subsets, each of which is scaled down by r from the original and is identical in all statistical respects to S. Put differently, enlargements of subsets of a statistically self-similar set have a statistical distribution identical to that of the whole set [5 (p. 225)].

Figure 2.8 Intermediate stage (I5 ) in the realization of a random version of the von Koch curve. Modiﬁed from [14].

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25

As with the deterministic monsters, it is possible to relax the requirement that the scaling factor r be identical in all directions. In this case, the random set or curve is said to be ‘statistically self-afﬁne’. Historically important examples of statistically self-afﬁne sets are the Brownian and fractional Brownian motions. Brownian motion, also referred to as a random walk or Wiener process, is named after the Scottish botanist Robert Brown, who in 1828 [22] described the erratic motion of pollen in aqueous suspensions observed with a light microscope. A particle undergoing Brownian motion seems to wander around without any distinct pattern (Figure 2.9). Some regions of the plane are ﬁlled densely by the particle’s trace. Increasing the resolution of the microscope and the time resolution produces a random walk that looks very much like that obtained at lower resolution.

Figure 2.9 Trace of the Brownian motion of a particle in a plane. The boxed detail of the trace (magniﬁed in the upper left portion of the ﬁgure) suggests an invariance of scale or self-similarity: the detail looks like the whole. Modiﬁed from [14].

The Brownian motion of a microscopic particle is due to its constant bombardment by the numerous smaller molecules of the medium (e.g. water) in which it is suspended [23]. A single molecule hardly ever has enough momentum for its effect on the suspended particle to become visible under a microscope. Nevertheless, when many molecules collide with the particle from nearly the same direction at nearly the same time, they noticeably deﬂect it. These events are separated in time by what is commonly referred to as an ‘atomistic’ or ‘collision’ time τ . The displacement ξ of the particle is not ﬁxed, since it depends on the number of molecules colliding with

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INTRODUCTION TO FRACTAL GEOMETRY

the microscopic particle and on their direction. Furthermore, the sizes and directions of successive displacements are uncorrelated. In 1923, Wiener [24] proposed a rigorous mathematical model that exhibits a behavior similar to that observed in random motion. In this model, the displacement ξ is governed by a Gaussian (bell-shaped) probability distribution, with zero mean and unit variance. Therefore, if one plots the successive displacements of a particle (in one dimension for simplicity) over a certain period of time, the result (Figure 2.10a) corresponds to Gaussian, or ‘white’, noise. The sum of successive steps ξ1 , ξ2 , . . ., ξn of the particle during n collision times is given by X(t = nτ ) =

n

ξi

(2.8)

i=1

where X is the position of the particle at time t, relative to its initial position at time t = t0 (Figure 2.10b). The function X(t) in Figure 2.10b is commonly referred to as the ‘graph’ or the ‘record’ of the Brownian motion and may be obtained by projection of the path of the Brownian motion onto one of the space axes. In the limit of vanishingly small time steps (τ → 0), the position X in Equation (2.8) becomes a random function, usually denoted by B(t). This function is such that for any two times t1 and t2 , with t2 ≥ t1 : |B(t2 ) − B(t1 )| ∝ (t2 − t1 )H

(2.9)

where ∝ denotes proportionality and H is the Hurst exponent. The variance of the increments is given by var(B(t2 ) − B(t1 )) ∝ (t2 − t1 )H

(2.10)

Early work concentrated on the ordinary ‘mathematical’ Brownian motion, with H = 1/2, and showed that this motion has a number of remarkable features. First, it cannot be represented graphically as can motion for ﬁnite τ (Figure 2.10b). In addition, Wiener [24] showed that B(t) is continuous, as one expects, but nowhere differentiable. Thus, a particle undergoing mathematical Brownian motion does not have a well-deﬁned velocity anywhere, and the curve y = B(t) does not have a welldeﬁned tangent anywhere either. A related fact that illustrates the extreme irregularity of B(t) is that in every time interval, no matter how small, a particle undergoing mathematical Brownian motion travels an inﬁnite distance! Figure 2.9 suggests that Brownian motion is statistically self-similar, and it is indeed so in R n for n ≥ 2. However, graphs of Brownian motion, such as that illustrated in Figure 2.10b, are statistically self-afﬁne rather than self-similar. The exponent H in Equations (2.9) and (2.10) does not have to be set equal to 1/2, but may instead vary arbitrarily. When 0 < H < 1, this generalization leads to the fractional Brownian motion (fBm), to which is associated the so-called fractional Gaussian noise (fGn), in the same manner that Gaussian noise produces the ordinary Brownian motion (see Figure 2.10).

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a 4

ξ

2

0

–2

–4 b

40

20

X

0

–20

–40

–60

0

500

1000

1500

2000

2500

t

Figure 2.10 A sequence of independent Gaussian random variables with zero mean and unit variance (Gaussian noise) ‘observed’ at every fourth time step, i.e. at intervals of length 4τ . (a) Independent random steps of the ‘particle’. (b) Position of the particle along the x axis. The time is in units of the atomistic time τ between steps. Modiﬁed from [10].

A remarkable feature of fBm is that it generates inﬁnitely long-run correlations (these correlations correspond, loosely, to the levels of dependence, in a probabilistic sense, of distinct spatial increments). The correlation function C(t) associated with fBm is equal to 22H −1 − 1 [e.g. 10 (p. 170)], i.e. it depends only on the value of H . When H = 1/2, fBm reduces to ordinary Brownian motion, in which successive displacements or increments are uncorrelated (C(t) = 0). For 0.5 < H < 1, there is a positive correlation between successive increments and fBm exhibits a ‘persistent’

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INTRODUCTION TO FRACTAL GEOMETRY

behavior, characterized by clear trends and relatively little noise [e.g. 10 (p. 181)]. On the other hand, for 0 < H < 0.5, fBm displays ‘anti-persistence’, the correlation is negative and, as a result, graphs of the fBm appear very noisy. As in the case of the deterministic sets and functions considered in previous sections, the topological dimension of a random monster sheds very little light on its intricate geometry. For example, both the deterministic and random triadic von Koch curves have the same topological dimension, i.e. equal to one and identical to that of a regular line segment. Fortunately, mathematicians have introduced a number of dimensions that can characterize even the most pathological sets or functions. These dimensions are deﬁned and analyzed in detail in the following section.

2.3 DIMENSIONS ASSOCIATED WITH THE MATHEMATICAL MONSTERS 2.3.1 Hausdorff Measure and Dimension Of all the dimensions of sets, the one introduced by Hausdorff [25] is undoubtedly the most useful for characterizing nowhere-differentiable sets. Familiarity with its deﬁnition, and with its limitations, is essential to understanding the concept of fractals. To understand the mathematical background of the Hausdorff dimension, it is useful to ﬁrst consider as an illustration the process of measuring the ‘size’ of a set of points deﬁning a surface in three-dimensional Euclidian space R 3 (Figure 2.11a). The customary measurement of this surface is its area. It can be approximated by the product A ≡ N (δ)δ 2 , where N (δ) is the number of squares of side length δ needed to

a b

c

Figure 2.11 Schematic illustration of various approaches to measuring the ‘size’ of a surface: (a) surface; (b) covering with squares; (c) covering with cubes. Modiﬁed from [10].

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tile or cover the surface (Figure 2.11b). For ‘ordinary’ surfaces, N (δ) tends to A0 /δ 2 in the limit of vanishing δ, so that A ≡ N (δ)δ 2 −−→ A0 δ 0 = A0 δ→0

(2.11)

where A0 is the area of the surface. We might also try to associate a volume with the surface by calculating the sum of the volumes of the cubes of side length δ needed to cover the surface (Figure 2.11c). Since N(δ) tends to A0 /δ 2 , in the limit of vanishing δ: V ≡ N (δ)δ 3 −−→ A0 δ 1 δ→0

(2.12)

which vanishes for δ → 0. Formally (even though it is clearly not feasible!), we might also try to approximate the surface by the total length L of a ﬁnite number of line segments of length δ. It is easy to see, after replacement of N (δ) by A0 /δ 2 , that L ≡ N (δ)δ −−→ A0 δ −1 δ→0

(2.13)

which diverges (tends to inﬁnity) when δ → 0. The conclusion to be drawn from this (simplistic) illustration is, of course, that the only useful measure of a set of points deﬁning an ordinary surface in R 3 is the area. Interestingly, the measure of this same set of points with a ‘yardstick’ that has a lower dimension than the area (i.e. a line segment) yields an inﬁnite (divergent) measure, while a ‘yardstick’ with a higher dimension than the area (i.e. a volume) leads to a zero measure. Therefore, the practically useful yardstick corresponds to a transition from divergent measures to zero measures. This same feature, as we shall see below, characterizes the Hausdorff dimension. The deﬁnition of the Hausdorff dimension is based on the concept of the Hausdorff measure, which itself makes use of the notion of the δ-cover of a set. If U denotes a nonempty subset of n-dimensional Euclidean space (i.e. U ⊂ R n ), the diameter of U , i.e. |U |, is the greatest distance separating any pair of points in U . Mathematically, |U | = sup {|x − y| : x, y ∈ U }, where sup is the conventional abbreviation of the supremum, the smallest upper bound of a set and |x − y| is the Euclidian distance between x and y. If F (shown in Figure 2.12a) is any subset of R n , then we can cover it with a (possibly large but countable) number of subsets Ui of R n , such that each of the Ui has a diameter at most equal to a given δ(0 < |Ui | ≤ δ, for each i). This covering is done schematically in two different ways in Figure 2.12b and c. In each case, F is included in the union of all of the values of Ui . A set {Ui } that has this property is said to be a δ-cover of F . Among all the δ-covers of F , we might be interested in determining which one is the ‘smallest’, in some sense. This δ-cover could be deﬁned as the one with the smallest sum of diameters. It would be equally acceptable, for example, to minimize the sums of the squares or of the third powers of the diameters. More generally, we

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INTRODUCTION TO FRACTAL GEOMETRY a

F b

c

δ Figure 2.12 A set F and two possible δ-covers of F . Modiﬁed from [5].

could raise the diameters |Ui | to any nonnegative number s. For δ > 0, we could then deﬁne [5 (p. 25)] H (F ) = inf s δ

∞

|Ui | : {Ui } is a δ-cover of F s

(2.14)

i=1

where inf represents the inﬁmum, the largest lower bound of a set. In Equation (2.14), we look at all δ-covers of F and we seek to minimize the sum of the sth power of the diameters. As δ decreases, the class of permissible covers of F in Equation (2.14) is reduced. Therefore, the inﬁmum cannot decrease as δ decreases. As δ → 0, the inﬁmum approaches a limit which may be inﬁnite or a real number greater than or equal to zero. This limit is deﬁned as H s (F ) = lim Hδs (F ) δ→0

(2.15)

and is termed the s-dimensional Hausdorff measure of F . For integral values of s, the Hausdorff measure is reduced to the traditional n-dimensional Lebesgue measure (i.e. the usual n-dimensional volume), multiplied by a constant. It is clear from Equation (2.14) that if δ < 1, for any given set F , the terms |Ui |s tend to decrease when s increases. Therefore, Hδs (F ) is nonincreasing with s for δ ≤ 1

31

Hausdorff measure Hs(F)

PHILIPPE BAVEYE ET AL.

0 0

n

DH (F) S

Figure 2.13 Graph of the Hausdorff measure against s for an arbitrary set F . The Hausdorff dimension is the value of s at which the Dedekind cut, or ‘jump’, from inﬁnity to zero occurs.

and, by Equation (2.15), H s (F ) has the same property. In fact, it is possible to show that, if t > s and {Ui } is a δ-cover of F : |Ui |t ≤ δ t−s |Ui |s (2.16) i

i

Taking inﬁma according to Equation (2.14), one ﬁnds that Hδt (F ) ≤ δ t−s Hδs (F ). Letting δ → 0, we see that if H s (F ) < ∞, then H t (F ) = 0 for t > s. Thus, a graph of H s (F ) against s (Figure 2.13) shows that there is a critical value of s at which H s (F ) ‘jumps’ from inﬁnity to zero. This critical value of s (at which H s (F ) may in general be undeﬁned) is DH , the Hausdorff dimension of F . The Hausdorff dimension of the triadic Cantor set is 0.631 and that of the von Koch curve equals 1.262. This latter number is entirely consistent with the von Koch curve being somehow ‘larger than one-dimensional’ (having inﬁnite length within a bounded domain) and ‘smaller than two-dimensional’ (having zero area). In similar fashion, it is not surprising that graphs of Brownian motions (see Section 2.2.5) have a Hausdorff dimension equal to 1.5. Not all Hausdorff dimensions of mathematical ‘monsters’ are fractional. For example, that of the Cantor singular function (Devil’s staircase) is DH = 1, whereas the Hausdorff dimensions of the Sierpinski tetrahedron, the Peano plane-ﬁlling curve and paths of the mathematical Brownian motion in R n (n ≥ 2) are all DH = 2. To say the least, the above description of the theory leading to the deﬁnition of the Hausdorff dimension does not suggest a simple and intuitive way of evaluating this dimension in practical cases. Indeed, the calculation of Hausdorff measures and dimensions is, in general, more than a little involved, even for simple sets [e.g. 26

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INTRODUCTION TO FRACTAL GEOMETRY

(pp. 156–157)]. Rigorous dimension calculations often involve pages of complicated manipulations. In some cases, e.g. the Weierstrass–Mandelbrot function, there remains some uncertainty on the correct outcome of these calculations [e.g. 10 (p. 27), 20]. Nevertheless, the Hausdorff dimension has the distinct advantage of being deﬁned for any set (cf. the handling of statistically self-similar sets in Hutchinson [27] and Graf et al. [28]). It can, therefore, provide a reference for comparison with values of dimensions computed via simpler, more straightforward methods (as described in the following subsections). 2.3.2 Similarity Dimension A heuristic method, which often results in a number equal to the Hausdorff dimension of sets of points, leads to the deﬁnition of another dimension. Following Falconer [5 (p. 31)], let us take the Cantor set P (see Figure 2.1) as an example. It can easily be split into a left part PL = P ∩ [0, 1/3] and a right part PR = P ∩ [2/3, 1]. Both parts are geometrically similar to P but scaled down by a ratio 1/3, and P is the (disjoint) union of PL and PR . Because the Hausdorff measure is additive and scales according to H s (λF ) = λs H s (F ), where F ⊂ R n and λ > 0, one ﬁnds that H s (P ) = H s (PL ) + H s (PR ) =

1 s 1 H (P ) + s H s (P ) s 3 3

(2.17)

Assuming that at the critical value s = DH (P ), we have 0 < H s (P ) < ∞ (note that this is an important assumption, but one that can be justiﬁed [5]), we may then divide Equation (2.17) by H s (P ) to get 1 = 2(1/3)s or s = ln 2/ ln 3 ≈ 0.631. In this estimate of the Hausdorff dimension of the Cantor set, we end up with a ratio of two natural logarithms. The denominator corresponds to the number of segments (three) into which the initiator is divided, while the numerator corresponds to the number of segments (two) that are retained in the generator. Alternatively, we could look at two as the number of copies of itself, scaled by a factor 1/3, which the Cantor set contains. One could extend this reasoning to other sets and consider that, in general, a set made up of m copies of itself scaled by a factor 1/r has a dimension, ln m/ ln r. The number obtained in this way is usually referred to as the similarity dimension DS of the set. It turns out that for virtually all the deterministic monsters introduced earlier, the similarity and Hausdorff dimensions coincide. For example, it is easy to see that the triadic von Koch curve has a similarity dimension DS = ln 4/ ln 3 ≈ 1.262, identical to its Hausdorff dimension (see previous section). For the Sierpinski gasket, DS = ln 3/ ln 2 ≈ 1.585, for the Sierpinski carpet, DS = ln 8/ ln 3 ≈ 1.89 and for the Menger sponge, DS = ln 20/ ln 3 ≈ 2.73; all three Ds values are identical to the corresponding Hausdorff dimensions. In the case of ‘nonmonstrous’ sets, the similarity dimension also behaves properly. For example, if we consider a straight-line segment, divide it into four sub-segments of length scaled by a factor 1/4, keep all four segments and iterate this procedure indeﬁnitely, the similarity dimension of the

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resulting set is DS = ln 4/ ln 4 = 1, which is the same as the Hausdorff dimension of the set. In Section 2.2.2, it was shown that, after n iterations in the construction of the triadic von Koch curve, the number Nn of segments of length δn is a power-law function of δn : Nn = δn− ln 4/ ln 3 . With the introduction of the similarity dimension, and by virtue of the equality between Hausdorff and similarity dimensions for the triadic von Koch curve, this power-law relationship may also be expressed as Nn = δn− ln 4/ ln 3 = δn−DS = δn−DH

(2.18)

from which it follows that DS = DH may be obtained graphically from the slope of Nn versus δn in a log–log graph. Even without having recourse to this graphical method, the similarity dimension is clearly very straightforward to compute. Unfortunately, it is meaningful only for a small class of strictly self-similar sets. It cannot be used to evaluate the dimension of self-afﬁne, statistically self-similar or statistically self-afﬁne sets. For these, other easily measurable dimensions are necessary, like the box-counting dimension. 2.3.3 Box-Counting Dimension In calculating the inﬁmum Hδs (F ) in Equation (2.14), on the route to deﬁning the Hausdorff dimension, the covering sets Ui were allowed to have various diameters, within the constraint |Ui | ≤ δ. One possible approach to a dimension that would be equally general, yet simpler to compute than the Hausdorff dimension, would be to require that the diameters of all the Ui equal δ, or, equivalently, to replace |Ui | by δ in Equation (2.14). Using this approach, one would obtain a number Nδ (F ), associated with a given nonempty bounded subset F of R n , such that s s δ : {Ui } is a (ﬁnite) δ-cover of F (2.19) Nδ (F )δ = inf i

In other words, Nδ (F ) is the smallest number of sets of diameter at most δ that can cover F . Based on the deﬁnition of Equation (2.19), it is convenient to deﬁne a new dimension, called the box-counting dimension, denoted here by DBC , as DBC (F ) = lim δ→0

ln Nδ (F ) − ln δ

(2.20)

The motivation for this (apparently arbitrary) deﬁnition is that, for many sets of points, there is a nearly linear relationship between ln Nδ (F ) and ln δ for small δ, and, as δ → 0, the slope of this relationship approaches DBC (F ). Nevertheless, DBC (F ) as deﬁned by Equation (2.20) has meaning only when the limit as δ → 0 exists (see discussion in [5]). The dimension DBC may be thought of as an indication of the efﬁciency with which a set may be covered by small sets of equal size, whereas the Hausdorff dimension

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DH involves coverings by sets of small but perhaps widely varying size. The boxcounting dimension is sometimes also referred to as the ‘Minkowski’ dimension [5], ‘Minkowski–Bouligand’ dimension [4, 29, 30], ‘Bouligand–Minkowski’ dimension [31], ‘capacity’ dimension [5, 32] or the ‘Kolmogorov entropy’ [5]. Up to this stage, the geometrical shape of Ui has not been speciﬁed in any way, which suggests that the deﬁnition of Equation (2.19) can accommodate a variety of geometries. This variety is quite large indeed. It is possible to show mathematically [5 (p. 41)] that Nδ (F ) in Equation (2.20) can be any of the following options, illustrated in Figure 2.14:

F

(i )

(ii )

δ

(iii )

(iv )

(v )

Figure 2.14 Five ways of ﬁnding the box dimension of F (see text for details on (i)–(v)). Modiﬁed from [5].

• The smallest number of closed balls of radius δ that cover F . (A closed ball Br (x) is a set deﬁned as Br (x) = {y : |y − x| ≤ r}, where x and r are called the center and the radius of the set.) • The smallest number of cubes of side δ that cover F . • The number of ‘δ-mesh’ cubes of side δ that cover F (this number has been used for decades by geographers and cartographers, and justiﬁes the ‘box-counting’ qualiﬁer). • The smallest number of sets of diameter at most δ that cover F . • The largest number of disjoint balls of radius δ with centers in F . It is not readily obvious that this number is mathematically equivalent to the others, yet such is the case [e.g. 5 (p. 41)]. Mathematically, the deﬁnition of Equation (2.20) implies that Nδ (F )δ s → ∞ if s < DBC (F ) and that Nδ (F )δ s → 0 if s > DBC (F ). In other words, the product Nδ (F )δ s

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has a (Dedekind) cut similar to that experienced by the Hausdorff measure (cf. Figure 2.13). The analogy does not extend further, however. The limit of Nδ (F )δ s → 0 as δ → 0 does not, unfortunately, correspond to a measure on subsets of R n . This has a number of unpleasant consequences, one of them being that countable sets of points can have nonzero box-counting dimensions. For example, the countable (compact) set {0, 1, 1/2, 1/3, . . .} has a Hausdorff dimension equal to zero but a box-counting dimension equal to 0.5. A number of alternative dimensions have been proposed to overcome the difﬁculties associated with the traditional box-counting dimension DBC . They include the lower and upper modiﬁed box-counting dimensions [5] and the packing or Tricot dimension [5, 33, 34]. Unfortunately, these dimensions reintroduce all the difﬁculties of calculation associated with DH , and in some cases are even more awkward to use! One key advantage of the box-counting dimension DBC over the similarity dimension DS is that DBC can be used to evaluate the dimension of self-afﬁne sets. In these sets, however, DBC is not uniquely deﬁned; instead, it assumes two different values: a local or small-scale value and a global or large-scale value [e.g. [10 (p. 187), 31 (p. 55), 35 (p. 8)]. In the case of the fractional Brownian motion (Section 2.2.5), the local DBC value is equal to the Hausdorff dimension and is given by 2 − H , where H is the Hurst exponent, whereas the global value of DBC = 1 [e.g. 10 (p. 189)].

2.3.4 Divider Dimension In the case of curves that do not intersect themselves, it is possible to deﬁne another dimension, denoted here by DD and generally referred to as the divider or compass dimension. If C is a curve in R n and δ > 0, we deﬁne Nδ (C) as the maximum number of points x0 , x1 , . . . , xm on the curve C, in that order, such that |xk − xk−1 | = δ for k = 1, 2, . . . , m. Thus, (Nδ (C) − 1)δ may be thought of as the length of the curve measured using a pair of dividers with points set a distance δ apart, i.e. with ‘yardsticks’ of length δ. Under these conditions, the divider dimension DD is deﬁned as [5] DD (C) = lim δ→0

ln Nδ (C) − ln δ

(2.21)

assuming that this limit exists. It is possible to show that DD ≥ DBC , when both these dimensions exist. In simple self-similar examples, such as the von Koch curve, DD and DBC are equal, but in other cases the inequality has been shown to apply. For example, fractional Brownian motion (Section 2.2.5) has a local DD value equal to 1/H , where H is again the Hurst exponent, which is always strictly larger than its local DBC value, equal to 2 − H [e.g. 10 (p. 189)]. Tricot [33 (p. 234)] expresses reservations about the use of the divider method to evaluate the dimension of curves that are not exactly or statistically self-similar (e.g. the west coast of Britain, which will be mentioned again in Section 2.4.1). Indeed, he argues that the divider dimension does not have a precise metric meaning in these cases.

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2.3.5 Pointwise Dimension and Hölder Exponent Another dimension that has interesting properties is called the ‘pointwise’ or ‘mass’ dimension [e.g. 14 (p. 736), 32 (p. 334)]. To deﬁne it, one considers a curve in R n and samples it uniformly at a large number N0 of points, i.e. one determines the spatial coordinates of N0 points uniformly distributed along the curve. Then, one centers a ball Br (x) of radius r at some location x on the curve, and one counts the number N(r, x) of sampled points within this ball. The probability of ﬁnding a point in this ball is given by the limit μ(Br (x)) = lim

N0 →∞

N (r, x) N0

(2.22)

This limit may also be viewed as the measure or ‘mass’ of the curve that is contained in the ball Br (x). By analogy with previous deﬁnitions of dimensions, one may deﬁne the pointwise dimension Dp at the location x as follows: ln μ(Br (x)) (2.23) r→0 ln r In some cases, the limit as r → 0 is not taken in Equation (2.23) and the ratio ln μ(Br (x))/ ln r is termed in this case the Hölder [36], coarse Hölder [37] or Lipschitz–Hölder exponent [34]. It is traditionally denoted by α and it may be evaluated for any measure, deﬁned or not by Equation (2.22). This exponent is useful to characterize ‘singular’ measures, which have no local densities (i.e. for which the limit in Equation (2.23) does not exist), and it plays an important role in the deﬁnition of multifractal measures (Section 2.6). Dp (x) = lim

2.3.6 An Arsenal of Dimensions The previous subsections indicate that several concepts of dimension have been proposed over the years to characterize the geometrical properties of sets of points. Many more dimensions than those presented above exist. They include, for example, the Ljapunov dimension [14, 26], the spectral dimension [34], the information dimension DI [14, 26, 32, 38] and the Fourier dimension [5]. Another dimension, occasionally used in environmental science, is referred to as the variogram dimension. In Section 2.2.4, a relationship was mentioned between the variance of increments V (t) and the parameter D appearing in the formulation of the Weierstrass–Mandelbrot function w(t). In the case of spatial functions, it is more common to compute the semi-variance (half the variance of increments), also termed the variogram. By analogy with the behavior of V (t) at the origin, for the variogram, one may postulate a power-law dependency on the spatial increment h (or ‘lag’) as h → 0 and deﬁne a variogram dimension [e.g. 19, 39]. How many dimensions are there? Their number is in principle inﬁnite, as was illustrated elegantly by Hentschel and Proccacia [40], who deﬁned a collection of

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dimensions Dq , for q ≥ 0, now often referred to as the Rényi dimensions [e.g. 14 (p. 736)]. The ﬁrst in this collection (q = 0) corresponds to the box-counting dimension DBC . The second (q = 1) is equivalent to the information dimension DI , and the third (q = 2) to the correlation dimension DC . It can be proven mathematically that the Rényi dimensions Dq are decreasing with q, i.e. Dp ≥ Dq if p < q. In particular, this implies that DC ≤ DI ≤ DBC , where equality occurs only in special cases. In general, these dimensions are not equal. 2.3.7 Lacunarity At the same time that he coined the term ‘fractal’, Mandelbrot [4] pointed out that fractal dimensions would not sufﬁce to provide a satisfactory description of the geometry of lacunar fractals, and that at least one other parameter, which he termed ‘lacunarity’, would be necessary. The key reason for this requirement is vividly illustrated by the fact that Sierpinski carpets (Figure 2.15) with greatly different appearances can have precisely the same fractal (similarity) dimension. Therefore, the fractal dimension alone is not a very reliable diagnostic of the geometry and properties of lacunar fractals. For physical objects, such as porous media, where the geometry of interstices and pores inﬂuences a wide range of properties, this means that any attempt to ﬁnd a unique relationship between the fractal dimension of these objects and, for example, their transport or dielectric properties is most probably doomed to failure, unless one also takes lacunarity explicitly into account. Mandelbrot and co-workers [4, 41, 42] introduced the concept of lacunarity as a measure of the distribution of gap sizes in a given geometric object. In other words, lacunarity can be thought of as a measure of the ‘gappiness’ or ‘hole-iness’ of a geometric structure [43]. Objects are more lacunar if their gap sizes are distributed over a wider range. In that sense, the Sierpinski carpet of Figure 2.15a is more lacunar than that of Figure 2.15c. A more precise deﬁnition of lacunarity was introduced by Gefen et al. [44], who viewed lacunarity as a measure of the deviation of a geometric object from translational invariance. In a low lacunarity object, where gap a

b

c

Figure 2.15 Examples of 2nd iterate prefractals of Sierpinski carpets having different appearances, but identical fractal (similarity) dimension. In all three cases, the iterative construction process consists of dividing the initiator in 7 × 7 = 49 squares, and removing 3 × 3 = 9 squares. This process is associated with a similarity dimension Ds = ln(49 − 9)/ ln 7 = 1.8957.

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sizes are relatively homogeneous, different regions of the object tend to be similar to each other (as in Figure 2.15c). In contrast, in a high lacunarity object (like that in Figure 2.15a), different regions may be very dissimilar and cannot be made to coincide by simple translation. This characterization of the translational invariance of geometric objects is highly scale dependent: objects that are heterogeneous at small scales can be homogeneous when examined at larger scales or vice versa [44]. Amethod for calculating the lacunarity was outlined in general terms by Mandelbrot [4]. Subsequently, progress toward a usable method was made by Gefen et al. [44]. Allain and Cloitre [45] developed a straightforward algorithm, based on ‘gliding boxes’ (or ‘moving windows’) of increasing sizes, to evaluate the lacunarity of both deterministic and random fractals. More recently, Chappard et al. [46] described a different method to calculate the lacunarity, based on the mass–radius or ‘sandbox’ algorithm used routinely to estimate fractal dimensions. Both of these algorithms, and variants thereof, have been used extensively in the literature [e.g. 47–49]. In the calculation method suggested by Allain and Cloitre [45], a square structuring element or moving window of side length l is placed in the upper left-hand corner of an image of a solid geometrical structure of side length T (such that l ≤ T ). The calculation algorithm records the number or ‘mass’ m of pixels that are associated with the solid underneath the moving window. The window is then translated by one pixel to the right and the underlying mass is again recorded. When the moving window reaches the right side of the image, it is moved back to its starting point at the left side of the image and is translated by one pixel downward. The calculation proceeds in this fashion until eventually the moving window reaches the lower right-hand corner of the image, at which point it has explored every one of its (T − l + 1)2 possible positions above the solid. At that juncture, one may easily compute the number n(m, l) of times a particular value of the mass m has been recorded with the moving window of side length l. Division of n(m, l) by the total number (T − l + 1)2 of possible positions of the moving window above the image yields the probability distribution function Q(m, l) = n(m, l)/(T − l + 1)2 . The statistical moments ZQ(q) (l) of this probability distribution function are deﬁned as ZQ(q) (l) =

mq Q(m, l)

(2.24)

m

where q = 1 and 2 for the ﬁrst and second moments respectively, and the summation extends over all possible values of m (i.e. from 0 to l 2 for a moving window of side length l). On the basis of these moments, Allain and Cloitre [45] deﬁne the lacunarity GB (l), measured with a moving window of side length l, as GB (l) =

ZQ(2) (l) (ZQ(1) (l))2

(2.25)

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Since the ﬁrst moment ZQ(1) (l) is equal to the mean μQ of the probability distribution function Q, and the second moment ZQ(2) (l) is equal to the sum of the variance σQ2 of Q and the square of the mean, i.e. ZQ(2) (l) = σQ2 + μ2 , one may rewrite Equation (2.25) as GB (l) =

σ2 +1 μ2

(2.26)

which shows, perhaps more clearly than Equation (2.25), that GB (l) is fundamentally a relative measure of the width of the distribution Q(m, l). For a translationally invariant set, it is straightforward to show that ZQ(2) (l) = [ZQ(1) (l)]2 . Under these conditions, (l) becomes identically equal to unity and is independent of l. The algorithm developed by Chappard et al. [46] is based on the sand-box method elaborated earlier by Tél et al. [50]. In this algorithm, a number s of points or ‘seeds’are selected randomly within the geometrical domain occupied by the object under study, with s taken either arbitrarily or equal to the number of seeds at which convergence is reached for a number of targeted summary statistics. Around each seed, k concentric circles are drawn and the mass m of the object that is contained in each circle is recorded as a function of the circle diameter l. Once this process has been carried out for all s seeds, one may compute a mass distribution n(m, l) that depends on the mass m and circle diameter l. This mass distribution can readily be transformed into a probability distribution function Q(m, l) via division by the number s of seeds, i.e. according to Q(m, l) = n(m, l)/s. Using Equation (2.24) once again to deﬁne the moments of the probability distribution function Q(m, l) and remembering that the variance of Q(m, l) is equal to the difference of ZQ(2) (l) and (ZQ(1) (l))2 , one ﬁnds easily that the coefﬁcient of variation cv(l) of Q(m, l) is given by ZQ(2) (l) − (ZQ(1) (l))2 σ cv(l) = = (2.27) μ ZQ(1) (l) √ This coefﬁcient of variation is related to GB (l) by the relation cv(l) = GB (l) − 1. Chappard et al. [46] use this coefﬁcient of variation to deﬁne their lacunarity LC as follows: LC =

1 cv(l) k l

(2.28)

where the summation extends over all k circle diameters considered around the seeds. Comparison of Equations (2.25) and (2.28) makes it readily apparent that GB (l) and LC are fundamentally different concepts, even though they are both referred to as lacunarities. Indeed, whereas GB (l) depends on the side length l of the moving windows or structuring element, Chappard et al.’s [46] lacunarity LC does not depend on l, as a result of the averaging of cv(l) over the number of circle radii k in

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Equation (2.28). Unless one keeps in mind that there is no general agreement on the precise meaning of the term lacunarity, nor on its scale dependence, there is a high risk of confusion. Some of this confusion is evident in the literature.

2.4 FROM MATHEMATICAL MONSTERS TO ‘FRACTALS’ 2.4.1 Physical Motivation

Aggregate bulk density * 10–3 (kg m–3)

Section 2.2 introduced a large array of ‘monstrous’ mathematical beings that exhibit pathological properties defying the traditional concept of dimension. In Section 2.3, we saw that various alternatives to the traditional topological dimension have been devised by mathematicians. In spite of their multiplicity, these dimensions have tended to make the mathematical monsters somewhat less terrifying. Observations made by scientists over the years provided additional momentum in the same direction. The physical process whose study was perhaps most inﬂuential in stimulating interest in nowhere-differentiable functions is Brownian motion, described in Section 2.2.5. Besides Brownian motion, several other physical processes contributed to foster interest in nowhere-differentiable functions. A few years after Wiener’s work, Dedebant and Wehrlé [51] studied a number of scale issues arising in meteorology and concluded that ‘many meteorological processes, observed at a very small scale, are like the continuous, nondifferentiable functions that seemed forever to belong to the realm of speculative mathematicians’ [51 (p. 83)]. More recently, Chepil [52] observed a marked and systematic decrease of the densities of soil aggregates as the size of these aggregates increased (Figure 2.16). 1.3 1.2 1.1

1.0

0.9 y = 0.93 * x –0.12, R = 0.95 0.8

0.1

1

10

Aggregate radius (mm)

Figure 2.16 Bulk density of soil aggregates as a function of their radius. (Data from table 4 in [52], ﬁne sandy loam.)

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This behavior, which is not intuitive from a traditional geometric standpoint, is similar to that predicted by Equation (2.4) for intermediate steps in the iterative procedure leading to the Menger sponge. Indeed, if large aggregates are viewed as containing a greater variety of pore sizes than small aggregates – corresponding to the complexity found in the later stages in the iterative construction of the Menger sponge – then in Equation (2.4) one could keep ln and ρ0 constant and observe that, as l0 increases, the mass density ρn must decrease, since the exponent 3 − ln 20/ ln 3 is strictly positive (≈0.273). Therefore, at least some of the features of the Menger sponge may be useful to describe the properties of soil aggregates. Perhaps the most acute realization that the mathematical monsters of the previous section have properties very close to those routinely found in nature was made by geographers. Richardson’s [53] work is particularly signiﬁcant in this respect. He measured a number of geographical curves (coastlines and country borders), using the divider or compass method, that led to the deﬁnition of DD in Section 2.3.4. For a circle, the total length tends to a limit as the segment length goes to zero (see Figure 2.17). In all other cases, it increases as the side becomes shorter. This behavior is identical to that described above for the triadic von Koch curve (Section 2.1.2). In this latter case, the length of the nth iteration step In has a power-law or Paretian relationship with the ‘yardstick’ length δn (see Equation (2.2)). In other words, L(In ) and δn are related via a straight line in a log–log plot. This is precisely what is observed for the various geographical curves shown in Figure 2.17.

Log10 (Total Length in Kilometers)

AUSTRA

LIAN CO

AST

4.0 CIRCLE SOUTH AFRICAN COAST GERMAN

3.5

WES

LAND-FR

T CO

ONTIER,

3.0

AST

OF B

LAND-FR

RITA

ONTIER

1.0

1.5

1900

IN

OF PORT

UGAL

2.0 2.5 3.0 Log10 (Length of Side in Kilometers)

3.5

Figure 2.17 The length of coastlines as a function of yardstick length. (Data from [53].)

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2.4.2 ‘Deﬁnition’ of Fractals As the examples of the previous section illustrate, many scientists observed over several decades that natural objects or processes often have features akin to those of the von Koch curve or Menger sponge. This greatly stimulated interest in these monstrous sets; in this case, as in many others, interest in geometry was driven by its applications to nature [5]. This movement led to the publication by Mandelbrot in 1975 [1] of an essay in which he highlighted the similarities among the then-known continuous, nowheredifferentiable sets. He coined for these sets the term ‘fractal’, to emphasize the fact that their Hausdorff dimensions are often fractional. In the words of Dyson [54], ‘fractal is a word invented by Mandelbrot to bring together under one heading a large class of objects that have [played] . . . an historical role . . . in the development of pure mathematics’. Mandelbrot believed initially that one would do better without a precise deﬁnition of fractals. His original essay [1] contains none. By 1977, however, he saw the need to produce at least a tentative deﬁnition. It is the now classical statement that ‘a fractal is a set for which the Hausdorff dimension strictly exceeds the topological dimension’ [4, 5, 10]. For example, the Cantor set is a fractal, according to this viewpoint, since DH = 0.631 > DT = 0. The above deﬁnition immediately proved unsatisfactory, in that it excluded a number of sets with properties very similar to those of sets that satisﬁed the deﬁnition and, therefore, which also ought to be regarded as fractals. Indeed, according to this deﬁnition, the Cantor singular function (DH = DT = 1) and the Peano planeﬁlling curve (DH = DT = 2) are not fractals. Various other deﬁnitions of fractals have been proposed [e.g. 4 (p. 362)], but they all seem to suffer from the same drawback. Perhaps by modifying slightly the list of attributes of the Cantor set (Section 2.2.1), and by relaxing or deleting some of them, one could approach, as closely as possible, to a deﬁnition of the concept of fractal. This was done by Falconer [5], in what appears to be the best approximation of this concept to date. According to this author, when one refers to a set F (of points) as a fractal, one typically has one or more of the following properties in mind: • F has a ﬁne structure, i.e. detail on arbitrarily small scales; • F is too irregular to be described in traditional geometrical language, both locally and globally; • F often has some form of self-similarity, perhaps approximate or statistical; • usually, the ‘fractal dimension’ of F (deﬁned in some way) is greater than its topological dimension; • in most cases of interest, F is deﬁned in a very simple way, perhaps recursively (in which case, the various stages of the iterative construction are usually referred to as prefractals).

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Falconer [5] has perhaps best captured the spirit with which the above deﬁnition of fractals needs to be undertaken: My personal feeling is that the deﬁnition of a ‘fractal’ should be regarded in the same way as the biologist regards the deﬁnition of ‘life’. There is no hard and fast deﬁnition, but just a list of properties characteristic of a living thing, such as the ability to reproduce or to move or to exist, to some extent, independently of the environment. Most living things have most of the characteristics on the list, though there are living objects that are exceptions to each of them.

While probably agreeing in principle with Falconer’s [5] perception, many authors (in the fractal geometry literature at least) tend to give preferential weight to the requirement that fractals exhibit some form of self-similarity (see above, third point in the list of attributes of fractals).3 Feder [10 (p. 11)], for example, deﬁnes a fractal as a ‘shape made of parts similar to the whole in some way’. In other words, a fractal appears the same regardless of the scale of observation; its ‘look’ is scale invariant.4 Perhaps the most important aspect of the various deﬁnitions above is that they all consider fractals to be sets of points in R n , i.e. geometric constructs. This feature will assume particular signiﬁcance in Section 2.5, where we shall discuss nongeometric fractals. As with the concept of fractals itself, a certain level of vagueness characterizes the deﬁnition of the fractal dimension. The approach advocated by Mandelbrot in 1975 [1], and reiterated in his 1982 book, is to use the expression ‘fractal dimension’ as a generic term applicable to all the variants described in Section 2.3, and to use in each speciﬁc case whichever deﬁnition is most appropriate. This suggestion is adopted by a number of authors [e.g. 56]. However, it could, potentially, lead to considerable confusion if it is followed inconsistently, particularly in cases where different dimensions assume different values (see Section 2.3 for examples). Therefore, many mathematicians consider it safer to refer to speciﬁc dimensions by name, such as the correlation dimension, instead of using the generic term ‘fractal dimension’ [e.g. 5]. 2.4.3 ‘Natural’ versus Mathematical Fractals Can fractals, deﬁned as in Section 2.4.2, serve as appropriate representations of natural objects or processes? The answer to this question is (surprisingly perhaps, yet uncompromisingly) no. Strictly speaking, there are no true (mathematical) fractals in

3

It should be kept in mind, however, that self-similarity cannot be the sole deﬁning characteristic of fractals. A straight line segment is exactly self-similar, yet it hardly qualiﬁes as a fractal! 4 However, graphs like that of Figure 2.17 indicate that speciﬁc geometric features of fractals, such as perimeter length or surface area, depend strongly on observation scale. Instead of ‘scale invariance’, therefore, one might consider, with Nottale [55], that the expression ‘scale-covariance’ captures the essence of fractals better. However, exploration of this notion of ‘scale covariance’is beyond the scope of the present chapter.

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nature [e.g. 5, 57 (p. 319)]. Nevertheless, under speciﬁc conditions, physical objects, like soil aggregates or clouds, may have features that are very accurately described by fractals or prefractals. A similar situation pertains with other geometrical structures; there are no true straight lines or circles in nature, yet one would hardly claim that these concepts have absolutely no use in describing nature! Two of the ways in which true fractals can fail to represent physical objects are illustrated by the attempt to use the Menger sponge as a model of soil aggregates. First, soil aggregates have a porosity strictly smaller than unity and a mass density different from zero (otherwise there would be no aggregates to speak of!), but the Menger sponge has both porosity equal to unity and zero density (Section 2.2.3). Second, even if soil aggregates exhibit self-similarity over a range of observation scales (e.g. in thin sections observed under the microscope at different magniﬁcations), this selfsimilarity eventually disappears if the aggregates are viewed at sufﬁciently small observation scales (e.g. in the extreme, at subatomic scales) or, at the other end of the spectrum, at observation scales commensurate with the size of the aggregate itself. By contrast, the Menger sponge has a ﬁne structure at arbitrarily small scales. For these two reasons, the Menger sponge cannot serve as a model of soil aggregates. In discussing Chepil’s [52] results, however, we have seen that the prefractals associated with the Menger sponge (i.e. the sets of points obtained at intermediate steps in the iterative procedure which leads ad inﬁnitum to the Menger sponge) have properties that are closely related to those of real aggregates (see Figure 2.16). These prefractals have a porosity 0. Analysis of other examples, such as coastlines, clouds or landscapes, can lead to the same conclusion: fractal geometry is never an exact description of nature and often the geometry of a prefractal is a closer approximation to a physical object than is its associated fractal. Although physical systems are commonly referred to as ‘natural fractals’ (and we shall uphold that usage in the following to conform to standard usage), the expression ‘natural prefractals’ would probably be far more appropriate. When describing the geometry of a given soil aggregate with a prefractal of the Menger sponge, one has to decide on the iteration step with which this prefractal is associated. To this iteration step corresponds a dimension rn (the individual subcube size), which represents the lower value of the range of scales at which the prefractal exhibits self-similarity. This length, when related to natural fractals, is usually referred to as the inner cutoff length [e.g. 57]. Physically, in the case of soil aggregates, this length is associated with the size of individual particles. On the other hand, since a soil aggregate is necessarily of ﬁnite size, there must be an upper limit to the range of scales at which it may be observed. For a given system, this upper limit, referred to as the upper or outer cutoff length, may be the actual size of the system itself. However, it may be more accurate to consider that it corresponds to the largest scale at which the system displays self-similarity. The existence of an inner cutoff length has important consequences with respect to the evaluation of the dimensions of natural fractals. All the dimensions described in Section 2.3, except the similarity dimension, require a passage to a limit (e.g. limit

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to vanishingly small ball radius, box side or divider length). For physical reasons, the use of a limit is precluded for natural fractals. In consequence, the deﬁnitions of these dimensions have to be slightly modiﬁed when one applies them to natural fractals. For example, the box-counting dimension DBC can no longer be deﬁned as in Equation (2.15). At best, one would be able to determine the number of boxes Nδ (F ) only in the range of δ values between the inner and outer cutoff lengths. In practice, the range of δ values accessible via measurement may be much narrower. Under these conditions, Equation (2.15) may best be replaced by deﬁning DBC (F ) as the slope of the graph of ln Nδ (F ) versus − ln δ over a sufﬁciently large range of δ or, equivalently, by ﬁnding a best-ﬁt value of DBC in the power-law relationship Nδ (F ) ∝ δ −DBC . A direct consequence of this necessary change in the deﬁnition of the various dimensions is the fact that equalities or inequalities between dimensions that have been proven mathematically in the limit of vanishingly small ball radius, grid size or divider length may no longer be valid when such limits are not taken. Furthermore, the ﬁnite length of the ‘yardsticks’ may create serious practical difﬁculties. One of the key ones associated with the divider method (see Section 2.3.4), for example, relates to the existence of a remainder. This remainder stems from the fact that, most often, a noninteger number of steps is required to cover a given curve. At present there is no general consensus on how to deal with this remainder [e.g. 39]. In addition to the above problems, the existence of an inner cutoff length raises the question of what is ‘fractal’ in natural fractals. To understand this point, it is useful to take once again the example of the Menger sponge. As deﬁned in Section 2.2.3, the Menger sponge is a set of points in R 3 with (Lebesgue) measure equal to zero. One may also consider that there is another set of points that is closely associated with this ﬁrst one: the points that were removed during the iterative attrition process leading to the Menger sponge. In other words, to the ‘solid’ structure of the Menger sponge is associated a volume of ‘voids’. If one decides to interrupt the iterative process leading to the Menger sponge, then the ‘solid’ structure (no longer of measure equal to zero!) and the voids now have an interface. This feature is common to many natural fractals. In general, if the interface is fractal and scales like the mass of solids (i.e. follows a power law, or Paretian relationship, and has the same power-law, or Pareto-, exponent), the system is termed a mass fractal [e.g. 58]. If void (or pore) space and surface happen to scale alike, then the system is called a pore fractal, whereas if only the surface is fractal, then the system is called a surface fractal, or boundary fractal. In each case, one may use these fractals to characterize the dimensions introduced in Section 2.3, leading, for example, to ‘box-counting pore fractal dimensions’ or ‘correlation surface fractal dimensions’. 2.4.4 Is ‘Power Law’ Equivalent to ‘Fractal’? Repeatedly in the preceding sections, the analysis of the geometrical properties of mathematical and ‘natural’fractals has resulted in power-law or Paretian relationships

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INTRODUCTION TO FRACTAL GEOMETRY

between selected parameters. At no point in the text has this existence of a Paretian relationship been presented as a deﬁning characteristic of fractals; it was instead obtained as a consequence of fractal geometry.Anumber of authors, however, consider that the essence of fractals is not the notion of an underlying geometry, as predicated above, but the evidence of Paretian behavior. From this viewpoint, a fractal is a set for which some statistical distribution function is a power law [e.g. 59]. Some authors [e.g. 60] label the latter fractals as probabilistic, to distinguish them from the geometric fractals, deﬁned in Section 2.4.2. To understand clearly the connections between these two types of fractals, it is worth reviewing brieﬂy the development of the Pareto distribution and its relation to other statistical distributions in common use. The Pareto distribution is named after the Italian-born Swiss professor of economics, Vilfredo Pareto (1848–1923). Originally, it dealt with the distribution of income over a population and may be stated as follows [61–66]: N = Ax −a

(2.29)

where N is the number of persons having income ≥x. A and a are positive, real parameters. The relation of Equation (2.29) is now usually referred to as the ‘Pareto distribution of the ﬁrst kind’, to distinguish it from alternative forms [e.g. 67 (p. 234)]. When applied to discrete data (e.g. the length of words), it is also often referred to as the Zipf distribution [60, 64]. In the following, we shall simply call it the Pareto distribution. When N is plotted as a function of x, for given values of A and a, the distribution of Equation (2.29) is characterized by a very long right tail. Over the years, this Paretian behavior has been observed in relation with many socio-economic and other naturally occurring quantities. Examples [e.g. 67, 68] are the distributions of city population sizes, insurance claims, occurrence of natural resources, stock price ﬂuctuations, size of ﬁrms, and of error clusters in communication circuits, to list only a few. Usage of the Pareto distribution to describe data exhibiting very long right tails has been criticized by many researchers on the grounds that the Pareto distribution is not the only distribution with a very long right tail and that, often, it does not convincingly outperform its competitors. Macaulay [69], in particular, argues that ‘the approximate linearity of the tail of a frequency distribution charted on a double logarithmic scale signiﬁes relatively little, because it is such a common characteristic of frequency distributions of many and various types’. Indeed, in many cases, the exponential [38, 70], Weibull [38, 59] and lognormal [60] distributions mimic the Pareto distribution over certain ranges, or even provide a statistically better ﬁt to data than the Pareto distribution [e.g. 65, 71]. In that context, whether or not one uses the Pareto distribution to ﬁt experimental data often appears linked to one’s belief in the universality of the Paretian behavior, and to the deviation from Paretian behavior that one is willing to tolerate. In some well-publicized cases of the use of the Pareto distribution in the literature [e.g. 60 (ﬁgure 4.7) and discussion in 38 (p. 205)], there is enough deviation from linearity

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in log–log plots, and the deviation is systematic enough over the whole range of experimental data, that skepticism in the applicability of the Pareto distribution is warranted. Yet the authors of the original articles consider a unique Pareto exponent appropriate, implying that the systems under study exhibit power-law behavior. A related issue concerns the method one uses to ﬁt the Pareto distribution to experimental data. There are an array of methods available, including weighted or unweighted linear regression of log-transformed data, and nonlinear regression. Each of these methods may be applied to the full data set, or to a portion thereof, corresponding to points that are selected because they appear to fall on a straight line on a log–log plot. When the latter approach is adopted, one is faced with the problem of determining how big a range this linear segment should span to provide any conﬁdence that the data indeed exhibit a power-law behavior. Brock [72] argues that data points showing linearity should span at least two or three orders of magnitude in the abscissa before one tries to ﬁt to them with the Pareto distribution. A similar recommendation is made by other authors [e.g. 38]. Figure 2.18 illustrates some of the practical difﬁculties frequently encountered when trying to determine if the Pareto distribution adequately describes experimental data, and when evaluating its parameters. Originally, Cargill et al. [73] ﬁtted a power law to their data via weighted linear regression, after elimination of the leftmost ﬁve data points (considered to be artifacts resulting from a less efﬁcient extraction of copper from the ore). The weighting is proportional to the amount of ore. The Cargill et al. [73] regression line (dashed line in Figure 2.18) has a slope of −0.386,

C, Cumulative Cu grade

0.04

0.01

0.007 1011

1012 M, Cumulative Cu ore (kg)

1013

Figure 2.18 Dependence of cumulative copper grade C on cumulative copper ore M mined in the United States (data from [73]). The meanings of the different lines are described in the text.

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INTRODUCTION TO FRACTAL GEOMETRY

which, according to a formula derived by Turcotte [59], corresponds to an unspeciﬁed ‘fractal’ dimension D = 1.16. If an unweighted nonlinear regression is carried out on the same truncated data set, then the slope of the resulting regression line (thin solid line in Figure 2.18) equals −0.406 (i.e. D = 1.22), with an R value of 0.994. Finally, when the Pareto distribution is ﬁtted via an unweighted nonlinear regression to the full data set (thick solid line in Figure 2.18), the slope equals −0.327 (i.e. D = 0.98). There is some waviness in the data, but the ﬁt is quite close (R = 0.995), and is comparable to that found in the literature in situations where power-law behavior has been postulated. A consequence of this good ﬁt is that the ﬁrst ﬁve data points no longer appear artifactual or deviatory, as assumed by Cargill et al. [73] and Turcotte [59]. However, the most interesting consequence is in terms of the fractal behavior of the cumulative copper grade C. If one uses the truncated data set to estimate D and one follows Brock’s [72] recommendation, then C is not a probabilistic fractal, since its linear segment extends over less than one order of magnitude in M. On the other hand, if one considers the full data set, the linear segment spans very nearly two orders of magnitude in M and one may at least consider making the case that C is a probabilistic fractal. A similar analysis could be performed on many data sets described in the literature [e.g. 59]. This clearly highlights the fact that the evaluation of fractal dimensions, whether of geometric or probabilistic fractals, is not straightforward. It also shows that there is often some subjectivity involved in determining whether or not a given system has a Paretian behavior. These observations affect the practical use of both the geometric and the probabilistic fractals. However, the subjectivity is particularly signiﬁcant for probabilistic fractals, since evidence of Paretian behavior is their sole deﬁning feature. Afurther difference between geometric and probabilistic fractals concerns the range of values that are acceptable for the Pareto exponent a in Equation (2.29). In the various illustrations of Paretian relationships associated with mathematical monsters and fractals in preceding sections of this chapter (e.g. Equations (2.2) and (2.4)), the value of the exponent was always constrained to be (inclusively) between the topological dimension of the set and the Euclidian dimension of the embedding space R n . No such restriction applies to the parameter a in Equation (2.29) when this equation is taken simply as the mathematical formulation of a particular statistical distribution. Clearly, when Equation (2.29) is applied to data related to the size of insurance claims or to stock price ﬂuctuations, one cannot expect a to be restricted to a certain range (e.g. 0 ≤ a ≤ 3) for geometrical or physical reasons. Even when Equation (2.29) is used to describe some physical attribute of a real system, it is not guaranteed that either A or a, or even x for that matter, will have a clear physical meaning and that the values of a will be geometrically constrained. Some of the most convincing illustrations of this fact are found in applications of dimensional analysis to equations describing dynamical systems. Indeed, the so-called pi-theorem, one of the cornerstones of dimensional analysis, states that if there is a relationship among N + 1 variables involving N independent dimensional units, then the relation can be expressed in

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terms of a single nondimensional parameter, giving a power-law dependence [74]. This single nondimensional parameter, and consequently the coefﬁcient and exponent of the power law, may not always have a clear physical interpretation, as is illustrated by Schmidt and Housen [74] in the case of the cratering efﬁciency of conventional and nuclear explosives. In all these situations, and indeed for all probabilistic fractals, the range of values that the exponent a can assume is not restricted. Cases of physical systems where this exponent (after transformation whenever necessary) leads to ‘fractal dimensions’ >3 are routinely reported in the literature [e.g. 59, 72, 75, 76]). Such values are entirely acceptable for probabilistic fractals [e.g. 72, 77], even though they would be meaningless for geometric fractals embedded in R 3 . This distinction appears to have been frequently overlooked in the literature, as demonstrated by the lasting, yet groundless, controversy about supposedly ‘unphysical’ fractal dimensions >3 for natural fractals [e.g. 78]. Because of the constraints imposed on the exponent a in the case of geometric fractals, it is clear from the above analysis that whereas a geometric fractal is automatically also a probabilistic fractal, the reverse is not true in general. In environmental science, this conclusion has particular signiﬁcance with respect to the use of fragmentation models to account for the size distribution of soil particles [e.g. 76] or fragmented rocks and other geological materials [e.g. 59]. These fragmentation models are described in the following subsection. 2.4.5 Fragmentation Fractals The earliest attempt to develop a fragmentation model that accounts for the Paretian distribution of the size of fractured or fragmented solids seems to have been Matsushita’s [77] fracture cascade model (see Figure 2.19). The simplest derivation of this model, in two-dimensional space, starts with a square with a unit side length. At the ﬁrst stage, the square is divided into four equal subsquares of side 1/2. One of these subsquares, chosen randomly, is shaded (hatched diagonally) to indicate that it will not be fractured or fragmented further. At the second stage, the remaining three subsquares are divided each into four equal sub-subsquares of side (1/2)2 , and one of them (chosen randomly) is again shaded. The same procedure is in turn applied to the remaining 32 unshaded sub-subsquares, and so on. At the nth stage, there are 3n−1 newly shaded squares of side rn = 2−n . For ﬁnite n >> 1, the cumulative number N (rn ) of shaded squares of side length greater than rn = 2−n may be expressed as follows [77]: −Dfragm

N (rn ) = 1 + 3 + 32 + . . . + 3n−1 ∝ 3n = rn

(2.30)

where Dfragm ≡ ln 3/ ln 2 ∼ = 1.585 Equation (2.30) amounts to a power-law relationship between N (rn ) and rn , so, as n → ∞, it deﬁnes a probabilistic fractal. Following Kaye [43], it is occasionally called a fragmentation fractal, and Dfragm is termed the fragmentation fractal dimension.

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INTRODUCTION TO FRACTAL GEOMETRY

Figure 2.19 Fifth iteration in Matsushita’s fracture cascade in two dimensions. The hatched squares are arranged regularly for convenience. (Modiﬁed from [77].)

There is an interesting connection between Matsushita’s [77] fracture cascade model and some of the mathematical monsters (or geometric fractals) of Section 2.2. For example, if instead of dividing the initial square in Figure 2.19 into four equal subsquares and shading one, we divide it into nine equal subsquares and shade one, it is easy to see that the collection of shaded squares that we are producing in this manner corresponds to the square holes that are punched during the attrition process leading to the Sierpinski carpet (see Section 2.2.3). The exponent Dfragm of Equation (2.30), under these conditions, is equal to ln 8/ ln 3 ∼ = 1.89, which is the value of the Hausdorff and similarity dimensions of the Sierpinski carpet itself (see Section 2.3.2). The fracture cascade has been considered so far in two-dimensional space. To be applicable to the fragmentation of solid bodies, the model has to be extended to R 3 . In this case (see Figure 2.20), we could start with a cube with unit side length, divide it into eight subcubes, hatch three, apply the same procedure to each of the remaining ﬁve subcubes, and so on. The resulting value of Dfragm would be ln 5/ ln 2 ∼ = 2.322. More generally [e.g. 38], one could consider that the initial cube is divided into b3 equal-sized subcubes of side length 1/b, and that i randomly chosen subcubes are hatched, where b and i are arbitrary integers satisfying 1 < b and 1 ≤ i < b3 . At the

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1

1/2

1/4

Figure 2.20 Schematic representation of a fracture cascade in three dimensions. Three cubes are hatched (i.e. not fragmented further) at each fragmentation step.

nth iteration of this procedure, the cumulative number of hatched cubes larger than rn = 1/bn is −Dfragm

N(rn ) = i[1 + (b3 − i) + (b3 − i)2 + . . . + (b3 − i)n−1 ] ∝ (b3 − i)n = rn

(2.31)

where Dfragm ≡ ln(b3 − i)/ ln b. The power-law relationship of Equation (2.31) may be expressed differently if one introduces a ‘probability of fragmentation’ pc ≡ (b3 −i)/b3 , representing the fraction of the nth-order cubes of size b−n that are further fragmented. Under these conditions, Dfragm in Equation (2.31) becomes Dfragm =

ln(b3 pc ) ln b

(2.32)

The inequality 1 ≤ i < b3 (see above) implies that 1/b3 ≤ pc < 1, i.e. that 0 ≤ Dfragm < 3. This restriction on the range of values taken by Dfragm is a deﬁnite sign that the model described above is not satisfactory as a ‘physical’ explanation of the general Pareto distribution, since values of a in Equation (2.29) as high as 4 or 5 are routinely measured in practice [e.g. 59, 76]. On the other hand, the inequality Dfragm < 3 has been considered by some to be an indication that geometric fractal concepts are appropriate to describe any fragmented material [59]. This viewpoint deserves further analysis, since it assumes that every fragmented material was created by a process similar to that shown in Figures 2.19 and 2.20. If fragments or particulate matter are characterized using Equation (2.29), then the exponent a characterizes only

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INTRODUCTION TO FRACTAL GEOMETRY

the size of the individual fragments. It provides no quantitative insight concerning the geometry of the arrangement of these same fragments. By piling up or aggregating the fragments in speciﬁc ways, it may be possible to create with them various pore, mass or surface fractals. There are probably inﬁnitely many ways to do this. However, there are also many ways to assemble the fragments to create nonfractal structures. One such structure is in fact illustrated in Figure 2.20. Indeed, it is always possible to assemble the fragments resulting from a given fracture cascade in such a way as to reconstitute the original cube, which is deﬁnitely not fractal! Arguments that the dimension found when the size distribution of a fragmented material is characterized using Equation (2.29) should obey the inequality a < 3 (because Dfragm < 3) are, thus, without basis. 2.5 OPERATIONAL ASPECTS OF THE EVALUATION OF THE FRACTAL CHARACTERISTICS OF NATURAL SYSTEMS In the application of fractal geometry to natural systems, researchers encounter a number of daunting operational issues, some of which have been addressed in the literature in recent years [e.g. 79] (also see Chapter 3). Dubuc and Dubuc [80] summarized some of these issues in the following manner: . . . different methods as well as different scale ranges and resolutions can lead to estimates of dimension that are drastically different. Estimates of fractal dimension will never be valuable until one fully understands the importance of the various error factors involved in the estimation process.

To illustrate one of these operational problems, associated with the resolutions of images of natural systems, we shall consider the case of the fractal analysis of a preferential pathway in an orchard soil (Figure 2.21). The image in Figure 2.21a depicts a section of a moderately well-drained soil located in the Cornell University orchard (see [81] for details). On 14 July 1995, a metal ring was pushed into the surface layer of the soil. A solution of blue food coloring was poured inside the cylinder and rapidly inﬁltrated into the soil. Fifteen minutes later, a 1.8 m deep trench was dug with a backhoe, followed by carefully removing soil with shovels, in order to obtain as vertical as possible a soil proﬁle. Color slides were taken of the exposed soil facies with a hand-held camera. The slides were scanned and the software Adobe Photoshop was used to manipulate and analyze the digitized soil images. Using features of this software, images of the soil proﬁle were coarsened at different resolutions. To ensure that all the digitized images would receive identical treatments, precisely the same ﬁeld of view was cropped, i.e. delineated and cut, in each case. In addition, to maximize the contrast between stained and background soil material, the storage format of the cropped images was changed from RGB (red–green–blue) to CYMK (cyan–yellow–magenta–black), and the cyan channel was retained for further analysis. This channel corresponds very closely with the color of the dye used in the

PHILIPPE BAVEYE ET AL. a

53 b

c

Figure 2.21 (a) Gray-scale, 1685 × 2627 pixel image of a soil proﬁle in which a (darker) preferential pathway is visible; (b) black-and-white (binary) image obtained by thresholding (a) with the intermeans algorithm; (c) same as (b), but after the starting gray-scale image was coarsened 16 times. (The frames around (b) and (c) have been added solely to indicate the limits of the digitized images.)

ﬁeld experiment, a feature that makes the stain patterns much more sharply contrasted than for any of the other channels available in Adobe Photoshop. Gray-scale versions of the cyan channels were thresholded in two different ways, using the intermeans and minimum error algorithms, to produce binary (black and white) images suitable for fractal analysis, with the preferential pathway appearing in black and the background soil in white (Figure 2.21b and c). The box-counting information and correlation mass fractal dimensions of the preferential pathway were calculated using a method proposed by Liebovitch and Toth [82]. The results of the fractal analysis of the preferential pathway (see Figure 2.22) suggest that the values found for the fractal dimension are inﬂuenced by a number of the steps involved in the analysis. The fractal dimensions determined with images thresholded with the intermeans algorithm are generally higher than those based on the minimum-error threshold (open symbols tend to be higher than full symbols in Figure 2.22). Nevertheless, quantitatively, the inﬂuence of the thresholding method on fractal dimensions remains somewhat small; the largest difference, 0.034, is found in the case of the information dimension, in the second set of points from the right in Figure 2.22. More signiﬁcant, quantitatively, is the inﬂuence of the choice of a speciﬁc deﬁnition for the fractal dimension. The absolute difference among dimensions can be somewhat large, amounting to as much as 0.138 in one of the images. Nevertheless, clearly the most important source of variability of the fractal dimension is related to the resolution of the images used in the analysis. There is a marked tendency for fractal dimensions to increase with resolution, i.e. to increase when the pixel size decreases. This variation of the fractal dimension amounts to up to 0.2, which may

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INTRODUCTION TO FRACTAL GEOMETRY 1.85

‘Fractal’ dimension

1.8 1.75 1.7 1.65 1.6 1.55

0

0.2

0.4

0.6

0.8

1

Actual Pixel Size (cm)

Figure 2.22 Inﬂuence of physical pixel size (image pixel size translated into actual physical length on the soil proﬁle) on a number of mass fractal dimensions of the preferential pathway in Figure 2.21. Open symbols correspond to images thresholded with the intermeans algorithm, whereas full symbols correspond to the minimum-error algorithm. Circles, squares, and diamonds are associated with the box-counting, information and correlation dimensions respectively. (Modiﬁed from [81].)

not appear to be large; however, one has to remember that the mass fractal dimension of the preferential pathway in Figure 2.21 can only vary between 1.0 and 2.0, by deﬁnition. In addition, calculations by Baveye et al. [81] on the basis of a textbook fractal, the von Koch island, suggest that the fractal dimensions of the preferential pathway might have been larger still, and might have tended toward 2.0, if images at ﬁner resolutions had been available.

2.6 MULTIFRACTAL MEASURES: HOPE FOR THE FUTURE? 2.6.1 Deﬁnition The introduction to fractal geometry in this chapter would not be complete without a short mention of an area that is conceptually challenging,5 yet is the object of considerable interest in the literature, i.e. multifractal measures. As with fractals, many authors try to get by without having to provide a precise deﬁnition of multifractal measures. Consequently, the term ‘multifractal measure’

5

Korvin [38] humorously comments that multifractal measures are not for the squeamish!

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often means different things to different people. At the same time, different terminologies are used to refer to the same concept; ‘multifractals’ is often used in lieu of, or interchangeably with, the expression ‘multifractal measures’. In the following, we shall consistently use the term ‘multifractal measures’ to underline the fact that, unlike geometric (mono)fractals, which are self-similar sets of points, multifractal measures are self-similar measures deﬁned on speciﬁc sets of points. This deﬁnition of multifractal measures follows the treatment of Mandelbrot [36], Falconer [5] and Evertsz and Mandelbrot [37]. Other deﬁnitions will be brieﬂy mentioned at the end of this section. 2.6.2 The Binomial Fractal Measure The self-similarity of measures is best illustrated with the binomial measure [e.g. 32, 36, 37], which is generated recursively via a process known as a multiplicative cascade. This process starts (at the stage k = 0) with a uniformly distributed unit of mass on the unit interval I = [0, 1] (see Figure 2.23a). At the next stage (k = 1), a fraction m0 of the mass is distributed uniformly on the left half (I0 = [0, 1/2]) of the unit interval, and the remaining fraction m1 = 1 − m0 is distributed uniformly on the right half (I1 = [1/2, 1]). At this stage, the left half thus carries the measure μ(I0 ) = m0 and the right half carries the measure μ(I1 ) = m1 . Of course, the total mass is conserved, so that μ(I0 ) + μ(I1 ) = μ(I ) = 1. For that reason, μ appears like probabilities and could be considered as a probability measures [e.g. 37]. At the next stage (k = 2) of the multiplicative cascade, the subintervals I0 and I1 receive the same treatment as the original unit interval. For example, I0 is split into two intervals I00 = [0, 1/4] and I01 = [1/4, 1/2] of size 2−k , and the mass is also fragmented as it was at stage k = 1. It is easy to see that, for example, μ01 ≡ μ(I01 ) = m0 m1 , with similar relations for the other measures μ00 , μ10 and μ11 . In other words, the measure associated with a given interval is obtained by multiplication of a

b 0.05

0

0.04

1 m1

μ (x)

m0

0.03 0.02 0.01

0 0 m 0

1 m1

m0

m1 1

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Figure 2.23 (a) First two stages (k = 0 and k = 1) of the multiplicative cascade leading to the binomial measure, with m0 = 0.25 and m1 = 0.75. (b) Resulting measure μ(x) for k = 11, as a function of the segment position x = i × 2−11 . (Modiﬁed from [10].)

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INTRODUCTION TO FRACTAL GEOMETRY

suitable combinations of the measures m0 and m1 , hence the adjective ‘multiplicative’ that characterizes the cascade process. At the kth stage, the measure of a given interval, say Iβ1 β2 . . . βk , is given by μβ1 β2 . . . βk = mβ1 mβ2 . . . mβk = m00 m11 n

n

(2.33)

where n0 is the number of βi equal to zero and n1 ≡ k − n0 . For example, the ‘peak’ just to the left of x = 0.5 (in fact, in the interval [0.5 − 2−11 , 0.5]) in Figure 2.23b corresponds to a measure μ(x) equal to (0.25)(0.75)10 ≈ 0.0141, three times smaller than that at x = 1 where μ(x) = (0.75)11 ≈ 0.0422 (the latter peak is hidden by the right ordinate axis). Even though it is not possible to count them all on Figure 2.23b, there are 11 peaks with a measure equal to that of the peak near x = 0.5. In general, if k we write ξ ≡ n0 /k, the number of peaks with measure μ = (mξ0 m1−ξ 1 ) is given by the number of ways one can distribute n0 = ξ k zeros among k positions in the addresses β1 β2 . . . βk of the intervals Iβ1 β2 . . . βk . This number of peaks is given by the binomial coefﬁcient k! k (2.34) ≡ Nk (ξ ) = ξk (ξ k)![(1 − ξ )k]! which explains why the term binomial is applied to the binomial measure [e.g. 37 (p. 929)]. The limit of this process, as k → ∞, called the binomial measure, is exactly selfsimilar. The mass of an arbitrary interval Iβ1 β2 . . . βk is μβ1 β2 . . . βk times smaller than the mass, equal to unity, of the entire unit interval I . But the distribution of mass within these intervals Iβ1 β2 ...βk and I is distributed in exactly the same way. Indeed, by spatially rescaling the subinterval Iβ1 β2 ...βk by a factor 2k and renormalizing its mass by a factor (μβ1 β2 ...βk )−1 , one recovers the mass distribution in the interval I . It is in this sense that the multiplicatively generated binomial measure is said to be (exactly) self-similar. Self-similar measures like the binomial measure are ‘singular’, i.e. their local density, deﬁned as limε→0 μ[x, x + ε]/ε, is (almost) everywhere undeﬁned. To illustrate this, let us consider the value of the binomial measure in the neighborhood of the point 0. Because of Equation (2.34), μ[0, 2−k ] = mk0 = (2−k )v0 with v0 = − log2 m0 . That is, the measure in the neighborhood of zero scales as μ[0, ε] ∝ ε α , where α = v0 is the coarse Hölder exponent when ε is of ﬁnite length (see Section 2.3.5). The (nonlocal) density μ/ε, consequently, scales like ε α−1 , and if α = 1 (i.e. if m0 = 1/2), then the local density (the limit of the μ/ε as ε → ∞) is degenerate, equal to either zero (if m0 < 1/2) or inﬁnity (if m0 > 1/2). This is true not only in the neighborhood of zero: the local density of the binomial measure is either zero or inﬁnity almost everywhere in the interval [0, 1]. 2.6.3 Parameterization of Multifractal Measurements For multifractal measures to be useful in practice, we need to ﬁnd ways to characterize and parameterize their geometrical properties. This is achieved in particular by the

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‘f (α) curve’, whose theoretical foundation is now described. Evertsz and Mandelbrot [37] and Lavallée et al. [83] describe various methods for its evaluation. By simple arithmetic [e.g. 37 (p. 932)], one may show that, at stage k of the construction of the binomial measure, the coarse Hölder exponent associated with a given interval Iβ1 β2 . . . βk is α = ξ v0 + (1 − ξ )v1

(2.35)

where v1 = − log2 m1 and, as above, v0 = − log2 m0 , ξ ≡ n0 /k, and n0 is the number of βi values that equal zero. If, as in Figure 2.23, m0 < m1 , then v1 < v0 and, by virtue of Equation (2.33), v1 ≤ α ≤ v0 . The extreme values of α are usually denoted by αmin and αmax , so that αmin≡ v1 ≤ α ≤ v0 ≡ αmax . The limits αmin and αmax are independent of the size 2−k of the subintervals at stage k, i.e. are independent of the scale at which the measure is probed. The fact that the value of α is bound within a ﬁxed range makes it an ideal index to associate with the intervals within the support of the binomial measure, i.e. within the geometrical domain in which the measure assumes nonzero values. Of course, given the one-to-one relationship between α and ξ in Equation (2.35) and the fact that the values of ξ are bound (within [0, 1]) like those of α, one could conceivably also use ξ as an index [e.g. 10]. However, ξ is strongly tied to the concept of multiplicative cascade, which is a given in the case of the binomial measure but does not necessarily exist for other measures [37]. By contrast, α may be evaluated for any measure. From that viewpoint, it is interesting to inquire about the values that the coarse Hölder exponent assumes over the support of the measure and about how these values are distributed. For a given k, this distribution is given by the binomial coefﬁcient Nk (ξ ) of Equation (2.34). Because of Equation (2.35), each ξ corresponds to a unique α. Therefore, the number of intervals Ik of size 2−k with coarse Hölder exponent α is given by [37] k Nk (α) = ∝ (2−k )−f (α) (2.36) ξ(α)k where f (α) = log2 [ξ −ξ (1 − ξ )−(1−ξ ) ] with, based on Equation (2.35), ξ = (α − αmin )/(αmax − αmin ). The proportionality relationship in Equation (2.36) is an approximation based, in part, on the application of Stirling’s formula to the factorials in the binomial coefﬁcient. Although the coarse Hölder exponent α has meaning only for ﬁnite k, as k becomes very large Nk (α) dα is approximately ‘the number of intervals with a coarse Hölder exponent between α and α + dα’ [37 (p. 935)]. Thus, the function f (α) provides the information needed to ﬁnd the frequency distribution of α for large k. The graph of the f (α) curve for the binomial multifractal measure is shown in Figure 2.24a. Several of the features of this particular curve (e.g. symmetry around its maximum, local quadratic behavior near its maximum) are not generally typical of self-similar measures. Nevertheless, the f (α) curve of Figure 2.24a shares

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INTRODUCTION TO FRACTAL GEOMETRY a

1.0 0.8

f (α)

0.6 0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

5

10

α b

20

τ (q)

15 10 5 0 –5 –10

–5

0 q

Figure 2.24 (a) f (α) curve for the binomial measure with m0 = 0.25 and m1 = 0.75. (b) Sequence of mass exponents τ (q) as a function of moment order q, for the binomial measure with m0 = 0.25 and m1 = 0.75. (Modiﬁed from [10].)

two characteristics with other f (α) curves: ﬁrst, f (α) ≤ α for all α; second, the maximum value of f (α) corresponds to the box-counting dimension DBC of the support of the measure (DBC = 1 for the binomial measure because its support is a line segment). The f (α) curve is not the only way to characterize the properties of multifractal measures. Another, widely used, approach to describe their features involves sequences of the mass exponents, τ (q). To deﬁne these exponents, it is convenient to consider a set s covered with δ-mesh ‘cubes’, as in Figure 2.14(iii). Within each one of these δ-mesh cubes, one may deﬁne a measure μ that is similar to that of Equation (2.22) and represents the probability of ﬁnding an element (a point) of set s within the δ-mesh cube. With this probability, one may construct, for any real number q, the measure

PHILIPPE BAVEYE ET AL.

Md (q, δ) =

N (δ)

59

μqi δ d ≡ N (q, δ)δ d −−→ δ→0

i=1

0, for d > τ (q) ∞, for d < τ (q)

(2.37)

where the summation limit N (δ) is the total number of δ-mesh cubes that cover the set s, q plays the role of a moment order, and N (q, δ) is the weighted number of δ-mesh cubes in the cover. In close analogy with the Dedekind cut in Figure 2.13, the measure Md (q, δ) as δ → 0 approaches zero when d is large and approaches inﬁnity when d is small. The value of d at the division point is called the mass exponent and is denoted τ (q) to indicate its dependency on the moment order q. The measure Md (q, δ) is characterized by a whole sequence of exponents τ (q) that controls how the moments of the probabilities {μi } scale with δ. It follows from Equation (2.37) that the weighted number of δ-mesh cubes N (q, δ) has the form N (q, δ) =

N

μqi ∝ δ −τ (q)

(2.38)

ln N (q, δ) ln δ

(2.39)

i=1

and the mass exponent τ (q) is given by τ (q) = − lim δ→0

This relationship between τ (q) and q is illustrated in Figure 2.24b for the binomial measure described previously. If the probabilities μi are normalized (i.e. the sum of the μI is unity), then, by Equations (2.38) and (2.39), τ (1) = 0 (see Figure 2.24b). Another interesting special case is q = 0, where all the μqi = 1 in Equation (2.38) and, therefore, N (q = 0, δ) = N (δ) is simply the number of δ-mesh cubes needed to cover the set, so τ (0) equals the box-counting dimension of the set. This suggests that the value q = 0 corresponds to the value of α for which f (α) is maximum, and that there is a connectivity between the two ways to characterize multifractal measures. In fact, one may show that τ (q) and f (α) are intimately related via the relationships α(q) = −

dτ (q) dq

(2.40)

and f (α(q)) = qα(q) + τ (q)

(2.41)

This pair of equations in effect constitutes a ‘Legendre’ transformation from the independent variables τ and q to the independent variables f and α and may be very valuable in practical situations, when one of these two sets of independent variables turns out not to be particularly easy to evaluate directly (see [84]). Another interesting relationship exists between the sequence of exponents τ (q) and the Rényi dimensions Dq mentioned in Section 2.3.7. One can indeed show [e.g. 10 (p. 87)] that τ (q) = (1 − q)Dq .

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2.6.4 Alternative Deﬁnitions of Multifractal Measurements At this juncture, one may wonder why, for example, the binomial measure is termed multifractal. Indeed, in our coverage of the theory in the preceding two subsections, we have not provided a rationale either for the ‘multi-’ or for the ‘-fractal’ parts of this qualiﬁer. Following Frisch and Parisi [85], who coined the term ‘multifractals’, and a number of other authors [e.g. 86, 87], we might look at the Paretian relationship of Equation (2.36) as a manifestation of fractal behavior. If the length δ ≡ 2−k of the intervals Ik is shrunk to zero, then indeed the exponent f (α) in Equation (2.36) is deﬁned by an expression that is formally similar to Equation (2.15), deﬁning the box-counting dimension DBC . On that basis, one could consider that the geometrical support s of a multifractal measure is the union of fractal sets Sα, each with a particular Hölder exponent equal to α and a ‘fractal’ dimension equal to f (α). In much of the literature on multifractal measures, this statement is reversed in a manner that is very similar to the one discussed in Section 2.5.1 in the case of fractals; multifractal measures, or ‘multifractals’, are deﬁned as measures whose geometric support is the union of one or more fractal sets. Yet another deﬁnition of multifractals, occasionally found in the literature, is even farther removed from the concept introduced in Section 2.6.1. It stems from the observation that the log–log plots of speciﬁc parameters exhibit several distinct linear segments. When there are two such segments, the system is sometimes called ‘bifractal’ [e.g. 38]. When there are more than two linear segments, or when the curve in the log–log plot is nonlinear, the system has been called ‘multifractal’, but the use of the word in that context has nothing to do with the multifractal measures described here. From a mathematical standpoint, this alternative deﬁnition of multifractal measures, due to Frisch and Parisi [85], is fraught with difﬁculties. Indeed, a closer look at f (α) shows that it cannot be looked at as a box-counting dimension, in particular because the intervals counted by Equation (2.36) when δ ≡ 2−k is small need not be contained in those counted when δ is much larger (since the Hölder exponent associated with an interval δ1 is not necessarily equal to that of a larger interval which contains δ1 !) [e.g. 5 (p. 255)]. In the case of the binomial measure, one ﬁnds [e.g. 37 (p. 935)] that the (true) box-counting dimension of each of the sets Sα is unity, whereas f (α) varies between zero and one (see Figure 2.24a). Interestingly, it turns out that for a special class of multifractal measures, including the binomial measure, the value of f (α), for each α, is equal to the Hausdorff dimension of Sα [37]. Therefore, some of the sets Sα may be geometric fractals, but this does not appear to be a general characteristic of multifractal measures. 2.6.5 Beyond the Binomial Measure In the multiplicative cascade used to generate the binomial measure, the number of intervals was increased by two at each step. One could easily lift that restriction and consider instead a ﬁxed number b of intervals, larger than two. The resulting, more

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general, measures are usually termed multinomial measures [e.g. 37 (p. 935)].Another way to expand the notion of multiplicative cascade is to use random multipliers. At each step in the cascade, the multipliers are obtained by using a random number generator, subject of course to the constraint that the total measure be conserved. This process leads to random multifractal measures, which are statistically self-similar. The various cases considered so far have all involved multifractal measures whose geometrical supports are not fractal. It is easy to imagine measures of practical interest in environmental science that might belong to this category. Evertsz and Mandelbrot [37 (p. 922)] provide the example of the ‘quantity of groundwater’ down to some prescribed depth in a given (in general nonfractal) geographical area, such as an agricultural ﬁeld, a watershed, state, country, continent or island. As one subdivides the support more and more ﬁnely, following the multiplicative pattern of Figure 2.23a, it may turn out that the distribution of the quantity of groundwater is self-similar at different stages of the cascade, i.e. at different scales of observation. In this case, the quantity of groundwater is a multifractal measure, according to the deﬁnition given above. There are also situations where the support of a multifractal measure is itself a fractal. It is very easy to conceive a multiplicative cascade that would generate such a multifractal measure. Indeed, if in Figure 2.23a one distributes the fractions m0 and m1 of the original mass not over the two intervals [0, 1/2] and [1/2, 1] but over the intervals [0, 1/3] and [2/3, 1] respectively, then it is clear that, as k → ∞, the support of the measure is the Cantor set. Clusters generated by a diffusion-limited aggregation (DLA) model [10, 88] can also serve as a basis for constructing multifractal measures with fractal support. In this case, one may deﬁne on the DLA cluster a measure associated with the growth probability of the cluster. It is evaluated as follows. One starts with a cluster obtained by running the DLA model a certain number of times (e.g. 104 or 105 ). The model is then run several thousand more times, but without growing the cluster further; instead, each random walker that reaches the cluster is recorded as a ‘hit’ on the element it reaches and is removed. For each site of the cluster, the sum of the number of hits may be divided by the total number of random walkers considered. In this process, a measure is deﬁned that quantiﬁes the probability for a given site to be contacted by a random walker. This measure assumes its highest values at the periphery of the cluster [e.g. 89], and in many cases is multifractal. This existence of multifractal measures with a fractal support raises an interesting question that has practical consequences. One may look at a DLA cluster, for example, as a fractal or as the support of a multifractal measure. Even though these two viewpoints are not exclusive, one of them might lead to a much deeper insight into the properties of the cluster. Which one is it? In other words, is it more fruitful to concentrate on characterizing the fractality of given physical objects, or to identify multifractal measures deﬁned on these objects and investigate their features in detail? Obviously, the answer to this question will vary from situation to situation. In the context of environmental science, it is not clear at this stage what the answer may be.

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2.7 CONCLUSIONS AND RECOMMENDATIONS The purpose of this introductory chapter was to present some of the key concepts of fractal geometry and multifractal measures, which will be put to use in later chapters of this book. One of the principal take-home messages of this introduction is that extreme caution needs to be exercised when applying ‘fractals’ to natural systems. Indeed, unlike with most other mathematical theories that have been used to describe natural systems, starting with Euclidian geometry, there has historically been a considerable level of vagueness and ambiguity associated with the deﬁnition of most fractal concepts. As yet, unresolved operational issues complicate the evaluation of fractal characteristics for many systems. These challenges lead to a high risk of using the same terms to mean different things, of using different terminologies to cover the same basic reality, or of having different observers end up with very different perspectives on the fractality of given systems. The literature is replete with examples of confusion that can be traced back to these inherent ambiguities in the theory. Only if one is very clear about precisely what is meant by each term, and about how given parameters are evaluated, will one be truly able to assess the usefulness of fractal geometry for the description of natural systems.

LIST OF SYMBOLS AND ABBREVIATIONS A a C c(t) C(t) DBC DC DD Dfragm DH DI Dp Dq DS DT DLA F fBm φ H

surface area Paretian exponent, Lipschitz–Hölder exponent curve Weierstrass–Mandelbrot cosine function correlation function box-counting dimension correlation dimension divider or compass dimension fragmentation fractal dimension Hausdorff dimension of F information dimension pointwise dimension Rényi dimensions similarity dimension topological dimension diffusion-limited aggregation set of points, any subset of R n fractional Brownian motion porosity Hurst exponent

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H s (F ) l, δ, a, b L LC GB (l) m Mn (x) μi n, N Nδ (F ) P pc ϕn Q(m, l) r R3 Rn ρ t T τ τ (q) U V V (t) ZQ(q) (l) X ξ w(t)

63

s-dimensional Hausdorff measure of F length, distance total length lacunarity lacunarity mass mass density function along x probability number number of boxes Cantor set probability of fragmentation arbitrary phase constant probability distribution function scaling factor, radius three-dimensional Euclidian space Euclidian dimension of the embedding space mass density time side length time steps mass exponent nonempty subset of n-dimensional Euclidean space volume variance of increments statistical moments of probability distribution function position of the particle particle displacement Weierstrass–Mandelbrot function

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[59] Turcotte, D.L. (1992). Fractals and Chaos in Geology and Geophysics. Cambridge University Press, Cambridge, UK. [60] Crovelli, R.A. and Barton, C.C. (1995). Fractals and the Pareto distribution applied to petroleum accumulation-size distributions. In Fractals in Petroleum Geology and Earth Processes, Barton, C.C. and La Pointe, P.R. (eds). Plenum Press, New York, p. 59–72. [61] Pareto, V. (1897). Cours d’Économie Politique, volume 2. F. Rouge, Lausanne, Switzerland. [62] Mandelbrot, B.B. (1960). The Pareto–Levy law and the distribution of income. Int. Econ. Rev., 1(2), 79–106. [63] Mandelbrot, B.B. (1963). New methods in statistical economics. J. Polit. Econ., 71(5), 421–440. [64] Arnold, B.C. (1983). Pareto Distributions. International Cooperative Publishing House, Fairland, MD. [65] Montroll, E.W. and Schlesinger, M.E. (1983). Maximum entropy formalism, fractals, scaling phenomena, and 1/f noise: a tale of tails. J. Stat. Phys., 32(2), 209–230. [66] Persky, J. (1992). Pareto’s law. J. Econ. Persp., 6(2), 181–192. [67] Johnson, N.L. and Kotz, S. (1970). Distributions in Statistics. Continuous Univariate Distributions – 1. John Wiley & Sons, Ltd, New York. [68] Zipf, G. (1949). Human Behavior and the Principle of Least Effort. Addison-Wesley, Reading, MA. [69] Macaulay, F. (1922). Pareto’s laws and the general problem of mathematically describing the frequency distribution of income. In Income in the United States, its Amount and Distribution, 1909–1919. National Bureau of Economic Research, New York, Chapter XXIII. [70] Brown, W.K., Karpp, R.R. and Grady, D.E. (1983). Fragmentation of the universe. Astrophys. Space Sci., 94, 401–412. [71] Turcotte, D.L. (1986). Fractals and fragmentation. J. Geophys. Res., 91(B2), 1921–1926. [72] Brock, J.R. (1971). On size distributions of atmospheric aerosols. Atm. Environ., 5, 833–841. [73] Cargill, S.M., Root, D.H. and Bailey, E.H. (1981). Estimating usable resources from historical industrial data. Econ. Geol., 76(5), 1081–1095. [74] Schmidt, R. and Housen, K. (1995). Problem solving with dimensional analysis. Ind. Phys.,1(1), 21–24. [75] Turcotte, D.L. and Huang, J. (1995). Fractal distributions in geology, scale invariance, and deterministic chaos. In Fractals in the Earth Sciences, Barton, C.C. and La Pointe, P.R. (eds). Plenum Press, New York, pp. 1–40. [76] Van Damme, H. (1995). Scale invariance and hydric behaviour of soils and clays, C. R. Acad. Sci. Paris, Sér. IIa, 320, 665–681. [77] Matsushita, M. (1985). Fractal viewpoint of fracture and accretion. J. Phys. Soc. Jpn., 54(3), 857–860. [78] Avnir, D., Farin, D. and Pfeifer, P. (1992). Adiscussion of some aspects of surface fractality and of its determination. New J. Chem., 16, 439–449. [79] Crawford, J., Baveye, P., Grindrod, P. and Rappoldt, C. (1999). Application of fractals to soil properties, landscape patterns and solute transport in porous media. In Assessment of Non-point Source Pollution in the Vadose Zone, Corwin, D.L., Loague, K. and Ellsworth, T.R. (eds). Geophysical Monograph 108, American Geophysical Union, Washington, D.C., pp. 151–164. [80] Dubuc, B., and Dubuc, S. (1996). Error bounds on the estimation of fractal dimension, SIAM J. Numer. Anal., 33(2), 602–626. [81] Baveye, P., Boast, C.W., Ogawa, S., Parlange, J.-Y. and Steenhuis, T. (1998). Inﬂuence of image resolution and thresholding on the apparent mass fractal characteristics of preferential ﬂow patterns in ﬁeld soils. Water Resour. Res., 34(11), 2783–2796.

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[82] Liebovitch, L.S. and Toth, T.I. (1989). A fast algorithm to determine fractal dimensions by box counting. Phys. Lett. A, 141, 386–390. [83] Lavallée, D., Lovejoy, S., Shertzer, D. and Ladoy, P. (1993). Nonlinear variability of landscape topography: multifractal analysis and simulation. In Fractals in Geography, Lam, N.S.-N. and De Cola, L. (eds). PTR Prentice Hall, Englewood Cliffs, NJ, pp. 158–192. [84] Folorunso, O.A., Puente, C.E., Rolston, D.E. and Pinzón, J.E. (1994). Statistical and fractal evaluation of the spatial characteristics of soil surface strength. Soil Sci. Soc. Am. J., 58(2), 284–294. [85] Frisch, U. and Parisi, G. (1985). Fully developed turbulence and intermittency. In Turbulence and Predictability of Geophysical Flows and Climate Dynamics, Ghil, M., Benzi, R. and Parisi, G. (eds). North Holland, New York, p. 84. [86] Halsey, T.C., Jensen, M.H., Kadanoff, L.P., Procaccia, I. and Shraiman, B.I. (1986). Fractal measures and their singularities: the characterization of strange sets. Phys. Rev. A, 33, 1141. [87] Aharony, A. (1990). Multifractals in physics: successes, dangers and challenges, Physica A, 168, 479–489. [88] Witten, T.A. and Sander, L.M. (1981). Diffusion-limited aggregation, a kinetic critical phenomenon, Phys. Rev. Lett., 47, 1400–1403. [89] Stanley, H.E. and Meakin, P. (1988). Multifractal phenomena in physics and chemistry. Nature, 335, 405–409.

3 Methods and Techniques for Fractal Analysis of Environmental Systems Graeme Bushell School of Chemical Sciences and Engineering, The University of New South Wales, UNSW Sydney NSW 2052, Australia

3.1 INTRODUCTION A wide range of techniques and methods have been used to measure fractal structure in environmental systems. Which approach is appropriate in given circumstances depends on whether an in-situ measurement is required or whether the samples may be taken for ex-situ analysis, on the size of the materials one is analysing and on their optical and mechanical properties. We shall make a distinction between technique, which will refer to the process of making a measurement, and method, which involves processing a measurement to extract information about the fractal dimension. Both are required. Most of the techniques described here are straightforward as long as some basic steps are taken not to damage the sample. It is often the assumptions made in applying the method that introduce the most uncertainty into the results. When measuring fractal structures in environmental systems, one is often dealing with complex assemblies of smaller subunits (‘particles’). In practice, these assemblies may be naturally occurring or artiﬁcially induced and can be composed of bacterial cells, colloidal mineral particles, natural organic matter, pollutants such as surfactants and polymers, or various combinations of these. The systems may be airborne assemblies of smoke particles, dust particle assemblies in space or somewhat larger mineral and organic matter assemblies in soils or sediments. All of these structures shall be referred to by the generic term ‘aggregates’. Although there are fractal systems in the environment which are not aggregates, such as the fractal surface roughness of fractures and wear particles, the texture of clouds and the distribution of stars in the night sky, these are less often of interest in the study of environmental systems. Surface fractal dimensions are also considered important in environmental systems and, although they are subject to many of the same principles discussed later, are not treated explicitly here.

Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems c 2008, IUPAC and Kevin J. Wilkinson

Edited by Nicola Senesi

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3.1.1 General Considerations in Measuring Systems with Fractal Structure The measurement of fractal structure in environmental systems is accomplished by the measurement of at least two properties of the system that are related to each other through a fractal scaling law. Generally speaking, environmental systems of interest exhibit power-law scaling of mass with linear size such that ma ∝ LDa m

(3.1)

where ma is the aggregate mass or, more generally, something that is linearly proportional to mass, such as solid volume or number of constituent particles. The number of constituent particles is usually the best deﬁned measure in this respect. La is a measure of aggregate length, such as radius or diameter, and Dm is the mass fractal dimension, sometimes referred to as D3 , the fractal dimension in three-dimensional space. Choosing different measures for mass and linear size will simply result in different constants of proportionality, leaving the power law scaling unaffected. In the rest of this chapter, use will be made of a constant k, most commonly known in the literature as the power law prefactor. k will take a subscript denoting the type of aggregate radius used. For instance if we use the aggregate radius of gyration as our length, Expression (1) will become: D Rg,a m N0,a = kg (3.2) R0 where N0,a is the number of primary particles in an aggregate, Rg,a is the aggregate radius of gyration, R0 is the primary particle radius and kg is the prefactor pertaining to size measured as radius of gyration and most probably has a value around 1.3 for particles in point contact [1–3], increasing signiﬁcantly with any sintering or shortrange restructuring [1, 2, 4]. Indeed, Oh and Sorensen [1] show that the exact value of the prefactor depends on a number of factors, such as aggregation mechanism, space dimensionality, mass fractal dimension and aggregate size, and thus represents another measure that may be usefully employed to characterize aggregates. The relationship embodied in the expression in Equation (3.1) is shown in Figure 3.1 for data obtained from a computer simulation algorithm comparable to that propounded by Thouy and Jullien [5] and has been veriﬁed countless times in the literature for widely varying systems. In this case the algorithm selected two aggregates at random, selected one particle at random from each of the two aggregates and then placed the two aggregates such that the chosen particles were next to each other. If the aggregate placement did not result in any overlapped particles and the centre-to-centre distance of the two aggregates was within a range determined by parameters in the simulation, then the aggregation event was accepted and the new aggregate returned to the list of all aggregates. Example structures generated using this algorithm are shown in Figure 3.2 – biasing the centre-to-centre criterion towards close placement resulted in higher mass fractal dimensions, whereas biasing towards distant placement resulted in lower mass fractal dimensions. Different mass

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4.5 4 slope = 2.41

3.5

log10 (N0,a)

3 2.5 2 1.5

slope = 1.74

1

slope = 2.11

0.5 0

0

0.5

1

1.5

2

2.5

log10 (Rg,a / R0)

Figure 3.1 A log–log plot of number of particles N0,a in an aggregate against radius of gyration Rg,a of the aggregate. The data are obtained from computer simulations and show three different mass fractal dimensions, of 1.74, 2.11 and 2.41.

fractal dimensions result from different aggregate formation processes, two common cases being the so called reaction-limited cluster aggregation (RLCA) case with Dm = 2.1 and the diffusion-limited cluster aggregation (DLCA) case with Dm = 1.8 (see Chapter 4 for more details). Fractal scaling (in a fairly loosely deﬁned sense) in aggregates of ﬁne particles is so common [6] that it is the rule rather than the exception. The problem of measuring the fractal dimension in practice is the problem of inferring values of m and L (or their equivalents). The ways in which this can be achieved are quite diverse and should be chosen with care, as techniques well suited to a particular system may perform badly with others. Gregory [7] has made an early review of some of the ways by which fractal dimension might be measured, followed later by Bushell et al. [8]. 3.1.2 Fractal and Nonfractal Systems True fractal scaling as embodied in the expression in Equation (3.1) and Equation (3.2) never occurs in nature, as natural systems always have some kind of upper and lower characteristic size beyond which the scaling breaks down. For example, many people would be comfortable with the concept that a sheet of paper is two-dimensional, but this is only really a reasonable approximation at scales much larger than the paper thickness and much smaller than the sheet’s width. If we examine the sheet of paper on scales smaller than the thickness then it has characteristics of a three-dimensional

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Dm = 1.74

Dm = 2.11

Dm = 2.41

Figure 3.2 Example aggregates drawn from classes having mass fractal dimension (from top to bottom) of 1.74, 2.11 and 2.41. Each aggregate contains 2000 primary particles. Note the increasing compactness as the mass fractal dimension increases.

body. Likewise, if we examine it on scales much larger than the length of the sheet then it begins to look like a zero-dimensional object, or a point. Fractal systems in the environment raise exactly the same questions of perspective. To the extent that

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an aggregate can exhibit fractal scaling, it only occurs at scales somewhat larger than the particles comprising the aggregate and somewhat smaller than the aggregate size. Equations (3.1) and (3.2) can also be seen in two different ways: as formulae that express the underlying physics of the system, in which the true fractal dimension is asymptotically approached as the aggregates become very large; and as purely correlative equations in which the fractal dimension is a ﬁtting parameter to describe the scaling of the system over some limited range of aggregate size. Both meanings of fractal dimension have been used in the literature. Taking the ﬁrst of the two meanings, we are presented with the problem of knowing how large the system needs to be before ‘true’ fractal scaling emerges. This only happens when the component particle size becomes so small compared with the aggregate as not to have any signiﬁcant effect on the property we are observing. Unfortunately, there is no general answer to this question, as what constitutes ‘large enough’ varies depending on which property of the system we are interested in correlating. If we are interested in the scaling of the density autocorrelation function within an aggregate (deﬁned later), then a distance of four primary particle radii appears to be sufﬁcient [9, 10]; if, however, we are interested in the projected area of fractal aggregates coming close to its asymptotic scaling, then Figure 3.3 (and the work of Meakin et al. [11]) shows that even aggregates of 10 000 particles cannot be considered ‘large enough’.

Figure 3.3 Obscuration as a function of number of particles in an aggregate, for a simulated experiment with Dm = 1.74. Inﬁnite aggregates with this structure are geometrically transparent, meaning that the obscuration (fractional area of particles in the aggregate projection) should be constant. The ﬁgure shows that even aggregates of 10 000 particles cannot really be considered large enough for this to be the case. This is the origin of the error for the Dm = 1.74 data shown in Figure 3.6.

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Fractal scaling in real systems is not only limited by upper and lower size limits imposed by the physical size of the constituent units and the whole aggregate, but there may be intermediate scales where the structure changes. This is usually a result of a change in the nature of the physical processes during growth, e.g. a change in shear intensity at different stages of a ﬂocculation process [12] or shear-induced restructuring and breakup [13]. This kind of change in scaling can be a quite distinct transition but is commonly a smooth and continuous change. If we were to plot the mass of such aggregates versus their radius on a log–log plot we would not ﬁnd the straight line characteristic of fractal systems but quite possibly a curve with regions that may be more or less straight. This means that the system cannot be considered to be self-similar and, thus, truly fractal over the full size range of the aggregate (i.e. from primary particle size to aggregate extremity), though some degree of self-similarity can be considered to hold over narrower size ranges. This presents a rather serious problem to anyone interested in measuring the ‘fractal dimension’ of these systems. Even if the deviations from power-law scaling are not large, we may get quite different answers for fractal dimension if we use different techniques to analyse the same system. Take the example of the settling technique and the angular dependence of small-angle light scattering. Light scattering, as we shall see later, is able to resolve aggregate structure at scales much smaller than the overall size of the aggregate, probably at less than a tenth of the overall aggregate size. The settling velocity method, on the other hand, is most sensitive to the largest pores in the aggregate, i.e. the large-scale features that are larger than a tenth of the aggregate size. Even if the methods applied to these measurements were free from modelling errors (which they are not), we would still get different answers for fractal dimension if the system displays multiple levels of structure. The reason is that the two techniques are essentially probing different length scales. 3.1.3 Sample Handling Considerations As far as possible, when trying to measure the fractal dimension of environmental systems we should attempt to do so in situ. Many of these fractal systems, particularly those with low fractal dimensions, are extremely sensitive to shear. Any attempt to move the assemblages from one place to another, such as from the environment into some sort of laboratory equipment, may result in a potentially signiﬁcant level of shear-induced restructuring. This can be overcome in laboratory-based studies by performing the experiment in the very instrument that will be used to make the measurements. Examples of this are aggregation experiments performed in the sample cell of a small-angle light-scattering instrument [14], in-situ microscopy [15, 16] and even in-situ ultra-small-angle X-ray scattering (USAXS) [17]. Small-angle lightscattering experiments for ﬂame-generated aerosols [18, 19] are always performed in situ. Sampling of systems that are sheared in their natural environment is much easier. In this case, all one needs to do is make sure that in sampling the system the shear

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we induce is less than that naturally experienced by the aggregates [20]. A common example of this is the kind of fundamental ﬂocculation studies where aggregation is performed in some kind of sheared vessel. Samples of the system have been obtained by pumping through an external sampling loop [20, 21], by withdrawing sample from the vessel using tubing and a syringe [21] or by using a wide-mouthed or inverted pipette [22–24]. Structures smaller than a micrometre in overall size may be examined by X-ray and neutron scattering; for these techniques, sample handling is not normally given much consideration because structures at such scales are not generally considered sensitive to common shear environments.

3.2 SCATTERING One of the more common ways of measuring the fractal dimension of aggregated matter in natural systems is through the use of small-angle scattering. X-rays, neutrons and light have all been used to probe the structure of aggregates. Although the wavelength of the radiation is different, all of these methods have a common basis, in that they rely on interference between waves scattered from individual particles within the aggregate structure. General principles will be discussed ﬁrst, followed by consideration of speciﬁc applications. 3.2.1 Principles and Analysis The analysis that is used to determine fractal dimension in all X-ray, neutron and most light-scattering experiments is based upon the Rayleigh–Gans–Debye (RGD) formalism. For a more detailed discussion of the development of the equations for RGD scattering, readers are referred to the book by Guinier and Fournet [25]. In addition, Sorensen [26] has written an excellent and highly readable review that gives a good qualitative description of RGD scattering and comprehensively explains the state of current knowledge on light scattering from fractal aggregates. 3.2.1.1 Principle In conducting static scattering experiments, we normally measure the scattered intensity at a number of different angles, although the quantity of physical signiﬁcance is the magnitude of q, the scattering wave vector: 4πz θ q = |q| = sin (3.3) λ 2 where λ is the wavelength of the radiation in a vacuum, θ is the scattering angle and z is the refractive index of the dispersing medium. The idea behind the RGD theory is that the scattering body can be considered as a set of independent, noninteracting Rayleigh scattering elements. That is, that each particle in an aggregate scatters the radiation as if it were alone in space. An

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interference pattern reﬂective of the spatial arrangement of the constituent particles of the aggregate results because the path length of the radiation from source to detector is different for light scattered from each particle in the aggregate. Thus, the scattered wave from each particle arrives at the detector with a different phase. The intensity pattern at any point in space can then be calculated as the sum of the scattered waves from each individual scatterer according to I (q) = S(q)P (q)

(3.4)

where I (q) is the scattered intensity pattern as a function of scattering angle through the variable q. P (q) and S(q) are known as the form and structure factors, representing the angular scattering dependence of the individual particles and the interparticle effects respectively. 3.2.1.2 Form and structure factors, and the density autocorrelation function The form and structure factors are mathematical transforms of the distribution of the scattering elements in space. If the vector rij represents the displacement between scattering elements (particles) i and j , then the scalar q ·rij is the phase difference between light scattered from elements i and j at an angle θ corresponding to vector q. If q·rij 1, then the light scattered from i and j will be essentially in phase,leading to constructive interference. This means that the amplitudes of the scattered waves add and the scattered intensity is thus proportional to the square of the sums of the masses of the two scattering elements i and j . If q·rij 1, then the phase will be more or less random and the scattering incoherent. In that case the intensities add, the resulting intensity being proportional to the sum of squares of the masses of scatterers i and j . The signiﬁcance of this is that scatterers that are separated by a distance smaller than 1/q will scatter in phase, essentially as a single body. Scatterers separated by distances larger than 1/q will scatter incoherently. So the inverse magnitude of q acts as a probe of length scales, and the scattered intensity as a function of q provides information on how much mass exists at scales larger and smaller than 1/q. To calculate the structure factor we need information about the distances between the particles i and j comprising the aggregate. This is encapsulated in the density autocorrelation function g (r), which describes the distribution of separations between pairs of particles. This fundamental descriptor of aggregate structure will also make an appearance in our later discussion of aggregate hydrodynamics. It represents the average density of particles at a distance from any other particle in the aggregate: g (r) = g (|rj − ri |) =

N N i=1 j =1

1 φ(ri )φ(rj ) 4π|(rj ) − (ri )|2

(3.5)

where ri is a vector describing the position of particle i from an arbitrary ﬁxed origin, φ = 1 if there is a particle at the position of its argument and φ = 0 otherwise, and vertical lines denote magnitude.

GRAEME BUSHELL

The structure factor is then

77

∞

S(q) = 1 + 4π 0

r 2 [g (r) − 1]

sin(qr) dr qr

(3.6)

For fractal aggregates, the structure factor is like that shown in Figure 3.4. The form factor, which is the scattered intensity pattern from monodisperse spheres, is also shown. The overall scattering is shown in Figure 3.5. The main features to note are the Guinier scattering regime at q < 1/Rg,a and the fractal scaling regime at 1/Rg,a q 1/R0 . At q around 1/2R0 , the structure factor depends predominantly on the details of the coordination behaviour of the individual particles [27]. At this scale, the structure factor is sensitive to features of the order of the diameter of the particles: when all particles have a similar size, this means particles in direct contact. A higher average coordination number means a higher number of particles in contact and a larger structure factor in that region, independent of the fractal dimension of the system.

Figure 3.4 A typical structure factor S(q) for a fractal aggregate of monodisperse spheres (solid line) and the form factor P (q) for a collection of unaggregated monodisperse spheres. In this example the spheres have unit radius, the mass fractal dimension of the aggregate is 1.8 and the radius of gyration of the aggregate is 1000R0 .

At q 1/R0 the structure factor approaches unity and the scattering depends almost entirely on the form factor. The RGD scattering pattern exempliﬁed in Figure 3.5 may be used to determine both the size and fractal dimension of an aggregate. The size may be determined from

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Figure 3.5 Scattered intensity from a fractal aggregate, S(q)P (q) from Figure 3.4. The main features to note are the Guinier scattering regime at q < 1/Rg,a and the fractal scaling regime at 1/Rg,a q 1/R0 . At q around 1/R0 , the structure factor depends predominantly on the details of the coordination behaviour of the individual particles and at high q it depends on the structure of the individual particles.

the intensity data at q < 1/Rg,a , the Guinier regime, because in that range the scattered intensity is only a function of the radius of gyration of the aggregate, vis-à-vis 1 2 I (q) ∝ 1 − qRg,a 3

(3.7)

In the fractal regime, the intensity function is characterized by the power-law relationship I (q) ∝ q −Dm

(3.8)

and the fractal dimension may be determined from the negative slope of the fractal region on a log–log plot. Clearly, the fractal region does not become fully developed until q is somewhat larger than the inverse radius of gyration. Care must be taken not to include any of the transition region in the measurement of fractal dimension. Sorensen [28] advocates that, in measuring the fractal dimension this way, q should be at least 5/Rg,a . The author has found that another effective method is numerical differentiation of the scattered intensity data with respect to q and to take the fractal dimension as the negative of the derivative when it becomes reasonably constant.

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3.2.1.3 Cut-off functions The shape of the transition between the Guinier regime and the fractal regime relates to the shape of the aggregate. Physically, the density autocorrelation function does not continue fractal scaling forever, but falls to zero at lengths greater than the size of the aggregate. There have been a number of attempts to describe this decay through a so-called cut-off function, which when transformed through the RGD scattering equation (Equation (3.6)), would describe the shape of the transition region of the structure factor. These attempts have ranged from the mathematically convenient [29], to purely empirical [30, 31] and those derived from physical reasoning [3, 32, 33]. The Gaussian cut-off function is one of the simplest to use and is adequate for most purposes [26]. 3.2.1.4 Aggregate polydispersity Experimentally observed scattering patterns generally exhibit a much broader transition region than that depicted in Figure 3.5. This can be understood quite easily as a consequence of polydispersity in aggregate size. Each aggregate alone scatters in the manner of Figure 3.5,1 smaller aggregates showing the transition from Guinier to fractal scattering at higher q, and larger aggregates at lower q. Consider a mixture of different-sized fractal aggregates. If the aggregates are dilute enough to move independently of each other then their scattered intensities simply add. At high enough q (greater than ﬁve times the inverse radius of gyration of the smallest aggregate), we will be examining scattering from the fractal regime for all of the different-sized aggregates and, added together, these simply yield the same power law. At very low q (less than the inverse radius of gyration of the largest aggregate) we see the sum of the Guinier scattering regimes, which yields a functionality exactly as that described in Equation (3.7). The meaning of Rg,a in Equation (3.7) would now be somewhat different, being an average aggregate size known as the z-average [26]. Clearly the region between the Guinier regime and the fractal regime would now be wider than is the case with monodisperse aggregates. Sorensen et al. [34] have shown how the width of this transition region may be exploited via a graphical method to determine the width of the cluster size distribution under the assumption of a mathematical form for the cluster size distribution and a particular cut-off function. Sorensen et al. have compared most of the commonly used cut-off functions and concluded that the Gaussian cut-off is adequate [35]. The author’s own unpublished

1 Actually this is not quite true. A single aggregate shows quite a lot of deviation from fractal scaling due to the inherent randomness in these structures. It would be more accurate to say that the scattering in Figure 3.5 describes the average scattering from a large number of aggregates having exactly the same radius of gyration.

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work suggests that the Zeng and Meriani [33] approach2 is probably the most accurate, but differences from the Gaussian are not large. In the author’s opinion, the Gaussian function is accurate enough for the analysis of most real scattering experiments. This approach to measuring cluster size distribution is analogous to (although less sophisticated than) the way that most commercial laser diffraction particle-sizing instruments extract the particle size distribution from measured scattering data – the only real difference being the model for a single scatterer that is used.3 3.2.2 Small-Angle Light Scattering Light interacts primarily with the peripheral electrons in atoms and molecules and for this reason the nature of the interaction depends to a large extent on the chemical nature of the system.4 The extent to which materials refract and absorb light determines which systems can be analysed using the approach described here. An excellent review of what is known about light scattering from fractal aggregates has been published [26], and the interested reader should refer there for a much more detailed discussion. The previously presented discussion on the use of the RGD scattering approximation as a tool of use in determinating aggregate size and fractal dimension has limitations when applied to light scattering. For the RGD approximation to be strictly valid, the following rather restrictive conditions must be met [36]: |ζ − 1| 1

(3.9)

2k R0 |ζ − 1| 1

(3.10)

where ζ is the complex refractive index of the particles relative to the medium and k is the wave number, 2π/λ. Taking the example of polystyrene latex particles in water, the refractive index is 1.6. The relative refractive index is 1.6/1.33 = 1.20, approximately satisfying the ﬁrst condition. Taking the wavelength of incident light as 0.6328 m, the wavenumber is 9.93 m−1 . For the second condition to hold, R0 must be smaller than about 0.05 m. Numerical exploration of the limits of validity of the RGD approximation to light scattering for fractal aggregates has been made by Farias et al. [37], who found that if the conditions of Equations (3.9) and (3.10) are replaced by |ζ − 1| < 1

2k R0 < 0.6

(3.11) (3.12)

2 This was the cut-off function used to calculate the artiﬁcial structure factor that we see in Figures 3.4 and 3.5. 3 Commercial instruments work on the basis of scattering from spheres calculated using Mie theory, Sorensen et al.’s graphical method works on the basis of scattering from fractal aggregates calculated using RGD theory. 4 This is why various types of light spectroscopy are so useful in chemical characterization.

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then the size of the errors in predicting the absorption and scattering cross-sections will be less than about 10 %. Let us assume that this means that the scattered intensity at any given angle will be at most 10 % different from that expected under the RGD approximation. Let us further assume a worst-case scenario that we have 10 % departures from RGD in opposite directions at opposite ends of a decade of q range over which we are trying to measure fractal dimension. If we have a fairly typical fractal dimension around 2, then the intensity at the lower end of the q range will be 100 times that at the upper end of the q range. With maximum 10 % errors, the intensity at the lower end of the range would be 110, and at the upper end it would be 0.9. This would lead us to measure a fractal dimension of log(110) − log(0.9) = 2.09 instead of log(100) − log(1.0) = 2.00. An error of this magnitude is unlikely, as it rests on the unlikely assumption that maximal errors occur in opposite directions. In fact, the consensus in the literature suggests that fractal scaling is maintained even with signiﬁcant multiple scattering within aggregates [26, 38, 39] as long as the scattering system is not close to optical resonance [39, 40]. This seems to be true for constituent particles with size of the order of the wavelength of illuminating light, but has not been tested for very large particles. 3.2.3 Volume Obscuration In some ways, a more robust method of using small-angle light scattering to measure the mass fractal dimension of particle aggregates is to rely on the changing projected area of an assembly of particles as they aggregate, an idea ﬁrst used by Volker Oles in 1992 [20]. 3.2.3.1 Principle If we assume that the individual particles in an aggregate are large enough such that the extinction efﬁciency Qext for these particles is not very sensitive to particle size, then the extinction caused by an aggregate will be proportional to the projected area of that aggregate. The value of the extinction coefﬁcient approaches two as the particle size becomes large, but for many particles does not depart very far from this value once the condition 2x|ζ − 1| > 5 (3.13) has been achieved, where x is the size parameter of the particle (2π R0 /λ) [36]. This is particularly the case for absorbing particles. For latex microspheres irradiated by a He-Ne laser, this is equivalent to a particle diameter larger than about ﬁve microns. If, in addition to this condition, the particle concentration is low enough that the fraction of the total projected area obscured by other particles is small, then the obscuration observed in a scattering instrument will be proportional to the total projected area of the particles and aggregates: obscuration ∝ Na Ap,a

(3.14)

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where Ap,a is the projected area of an aggregate. If we have an aggregating fractal system, the generalized radius of aggregates Ra is related to the number of particles in an aggregate N0,a by N0,a ∝ RaDm

(3.15)

If the total mass of the system is conserved, then the total number of aggregates in the system is related to N0,T , the total number of primary particles, by Na =

N0,T N0,a

(3.16)

If the fractal dimension is higher than two, the projected area of an aggregate is related to Ra by (3.17) Ap,a ∝ Ra2 Therefore, the total projected area seen by the instrument is Ap,T = Na Ap,a ∝

N0,T 2 R RaDm a

(3.18)

so that Ap,T ∝ Ra2−Dm

(3.19)

In other words, a plot of the log of obscuration versus log of aggregate size should show a straight line with slope of (2−Dm ), for Dm > 2. For systems with mass fractal dimension less than two we would expect the obscuration to be constant because of the so-called ‘geometric transparency’ of these systems that ensures that essentially all particles in the structure are visible and hence contribute to the projection [11]. Oles [20] used a different formulation based on the solid volume fraction reported by a light-scattering instrument, but the principle is exactly the same. 3.2.3.2 Limitations There are three major limitations of this approach: (i) it requires the constituent particles in an aggregate assembly to be large enough that the scattering efﬁciency is independent of size; (ii) it requires the condition of geometric opacity to be met (fractal dimension to be somewhat higher than two, aggregates to be large); (iii) it requires conservation of sample mass during a change of aggregation state. The ﬁrst of these limitations is a purely practical one, in that the method cannot be applied to very small particles. The second is rather more serious, in that it has not yet been clearly established in the literature exactly what this means. The third limitation is again a practical one that limits the situations in which the method can be applied; for example, removal of particles by sedimentation will invalidate the approach. The ﬁrst limitation relates to the assumption made in derivation, that the extinction efﬁciency is constant as the particle size changes due to aggregation. This is probably the case for particles that comply with the condition given in Equation (3.13), since the

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extinction efﬁciency for particles that exceed this condition commonly does not vary by more than ±50 %. This may sound like a lot, but remember that we are plotting extinction on a log plot to extract the fractal dimension. In practice, the extinction is likely to vary a lot less than this, since the aggregates are not solid particles composed of the same material as the primary particles but are composite structures where, to some extent, the scattering identity of the individual particles is retained. The extent to which this is the case is not known at this time. Figure 3.6 shows a log–log plot of obscuration (normalized to the obscuration of single particles) against aggregate radius of gyration for the same data shown in Figure 3.1. The obscuration in this instance was calculated directly from the simulated aggregate geometry by dividing the total projected area of an aggregate by the total projected area of all of the particles in the aggregate, giving a fractional projected area. Thus, a collection of single particles will have an ‘obscuration’ of unity and any aggregated system will have an obscuration less than that as particles become partly obscured by other particles. Recall that the mass fractal dimensions of these systems (as characterized by the scaling of number of particles with radius of gyration in Figure 3.1) were 1.74, 2.11 and 2.41. With no knowledge of the real fractal dimension of the systems, the slope of the data in Figure 3.6 at large Rg,a would indicate fractal dimensions of 2.06, 2.19 and 2.35 respectively. The agreement between these and the fractal dimension obtained from N0,a and Rg,a is only fair for the high fractal dimension cases, and says nothing about systems with low fractal –0.1

log10 (Obscuration)

–0.2

slope = –0.06

–0.3

slope = –0.19

–0.4

slope = –0.35

–0.5

–0.6 0

0.5

1 1.5 log10 (Rg / R0)

2

2.5

Figure 3.6 A log–log plot of relative obscuration against aggregate radius of gyration for the three different simulated aggregate systems shown in Figure 3.1. From top to bottom, the mass fractal dimensions of the systems are 1.74 (triangles), 2.11 (squares) and 2.41 (stars).

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dimensions. Given that the slope of log(obscuration) versus log(Rg,a ) is supposed to be zero for systems with Dm < 2, the errors in slope obtained here are +0.06, +0.08 and −0.06. These errors are probably due to the second of the two limitations listed earlier. Since the simulation data in terms of obscuration and size were determined directly from the projected area and geometry of simulated aggregates, the physics of light scattering does not yet enter into consideration. The problem arises because the derivation assumes that the aggregates are inﬁnite in extent, so that we can use Equation (3.17) and, hence, attribute all of the loss of projected area to aggregate structure, rather than to aggregate size effects. That this is not the case is clearly illustrated in Figure 3.3, which shows the obscuration (which in this work is determined as the fractional area of single particles accounting for the whole aggregate projected area) as a function of the number of particles in the aggregate for the simulation where Dm = 1.74. Even aggregates of 10 000 particles are not large enough for the obscuration to be independent of aggregate size. What is required here is a ﬁnite size correction to the expected cross-sectional area for aggregates of given structure. Such corrections have been derived for the problem of measuring fractal dimension from projected images of aggregates [41] but this remains to be done for projected area, relevant to the analysis presented here. Köylü et al. [42] cite several authors who have used an empirical relationship between fractional projected area and number of particles in an aggregate, although the relationship does not directly address the problem of partial particle obscuration as discussed here. Meakin et al. [11] have suggested that the projected area of ﬁnite aggregates follows the general form γ δ Ap,a = GN0,a + BN0,a

(3.20)

where G, B, γ and δ are functions of Dm . δ has a value between zero and one, and γ = 1 is supposed if Dm < 2 and a value of 2/Dm if Dm > 2. A useful approach might be to ﬁt Equation (3.20) to data from computer simulations so that we know G, B, γ and δ for a range of fractal dimensions. We could then use an iterative approach whereby we analyse the experimental data to obtain an estimate for the fractal dimension, use Equation (3.20) to correct the analysis and then repeat the process with a better estimate of fractal dimension. The third of the major limitations with this approach is the requirement that the total mass of the system in the scattering volume must be conserved so that Equation (3.16) is valid. This immediately means that most systems in natural environments cannot be analysed in situ using this approach, unless we have an independent way of measuring and thereby correcting for the solids concentration presented to the instrument. Most of the applications of this technique have been in the laboratory, where the solids concentration is known and constant. Even then problems may arise, as experienced by Oles [20] in his original publication. In Oles’ experiment, the aggregating material eventually started to stick to the optical windows of the scattering cell, with the result that the effective solids concentration increased within the scattering

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volume, leading to an increase in obscuration unrelated to aggregate structural and size effects. Oles analysed 2.17 m latex spheres aggregating in a Couette ﬂow system and found that the aggregates exhibited a mass fractal dimension of 2.1 early during the aggregation process, which increased to 2.5 in later stages. The change in structure was attributed to selective breakup and reaggregation to more shear-resistant conﬁgurations. Kusters et al. [22] analysed polystyrene latex particles of about 1 m diameter aggregated in salt solution in a stirred tank, with samples withdrawn from the tank by pipette for light-scattering measurements. They found mass fractal dimensions of around 2.5 using the volume obscuration method. Spicer et al. [21] analysed 0.87 m particles in a stirred tank system similar to that used by Kusters et. al. and found, using the volume obscuration method, mass fractal dimensions of 2.1 for small ﬂocs increasing to 2.5 for larger ﬂocs, consistent with the ﬁndings of Oles. The increase was attributed to shear-induced restructuring. 3.2.3.3 Small-angle multiple light scattering A related method has been demonstrated by Lambert et al. [43]. This relies on the idea that multiple scattering inside an aggregate reduces the magnitude of the form factor. By measuring the scattered light at a relatively large angle (q corresponding to 1/R0 ) we can measure the magnitude of the form factor and compare it with the magnitude that we would see if all the primary particles were free entities. The reduction in the magnitude of the form factor results from the reduction in effective optical contrast (refractive index of the particles divided by that of the suspending medium) caused by the presence of neighbouring particles that effectively raise the refractive index of the suspending medium. The authors present a calculation for determining the mass fractal dimension based on the known form factor for the primary particles and the measured form factor for aggregates at large angles. The technique is only useful where the primary particles are well deﬁned and optically identical [43]. This condition is likely to be true for bacterial aggregates, and so it should be a useful approach when examining the aggregation behaviour of individual bacterial strains. 3.2.3.4 Turbidity ﬂuctuation Another method in the volume obscuration family is the measurement of turbidity ﬂuctuations in a ﬂowing suspension [44]. The physical principle is the same, in that the mass fractal dimension is inferred from changes in obscuration that result from changes in aggregation state, but the implementation is rather different. In turbidity ﬂuctuation measurement, the aggregated suspension is made to ﬂow through a sample tube which passes between a laser diode and photodetector arrangement. Aggregate size and number are determined indirectly from the size and frequency of ﬂuctuations in the photodetector voltage.

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3.2.4 X-ray and Neutron Scattering The RGD formalism introduced earlier describes X-ray scattering very well, as X-rays in general do not interact strongly with matter. This is particularly true of biological and environmental samples. This means that the interpretation of log– log scattering plots is straightforward, with the negative slope of an I versus q plot unequivocally representing the mass scaling exponent at the particular length scale corresponding to q. X-rays have a much smaller wavelength than light which, through Equation (3.3), means that much smaller sample length scales are probed at a given angle. USAXS allows a typical range for q in experiments between 5 × 10−3 and 5 nm−1 [45], or from sub-nanometre length scales to scales of the order of 100 nm [17]. Although the relatively weak interaction of X-rays and matter makes the analysis of scattering data simpler, it makes the measurement more complex. Because the interaction occurs with all the electrons in the sample rather than just the peripheral ones, it is differences in overall electron density that provide contrast and result in scattering. The chemistry of the system is unimportant; it is the atoms and their density that matter. For environmental samples suspended in water, the aggregates must have an electron density quite different to water. Biological materials are usually around 95% water, with the remainder being carbon and nitrogen compounds. This means that the electron density of the biological sample will be very similar to water; hence, biological samples suspended in water will be effectively invisible to X-rays, since sample concentrations are often very low. For example, Boukari et al. [17] reported that it took 25 min to acquire data over the whole angular range of their instrument when analysing silica particles. They used USAXS to examine the growth kinetics and structure of Stöber silica particles, showing that the early stages of that sol–gel process produced fractal structures. Even when looking at samples with rather good contrast, such as mineral oxides in water, much longer exposure times (of the order of minutes) must be used with X-ray scattering than with light (of the order of seconds) and samples must, in general, be much more concentrated. Berthon et al. [45] used USAXS to measure fractal structure in resorcinol–formaldehyde gels. The same comments made for X-ray scattering with regard to interaction with matter may be made for neutrons, only even more so. Neutrons have similar wavelengths to X-rays and so probe the same very short length scales. The contrast for neutrons is determined by the directions of spins in atomic nuclei, which has one unique and very useful practical consequence. Hydrogen and deuterium exhibit a very, very large difference in contrast even though they are chemically identical. This has the wonderful result that we can tune the contrast in our systems by doping them with different levels of deuterium. For instance, we could suspend our sample in different mixtures of ordinary and heavy water to achieve the desired

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contrast or ‘refractive index’. Contrast between biological samples and water might be achieved by this means. The same comments made for X-ray scattering with regard to analysis speed and expense apply even more so to neutron scattering. The experiments are, in general, slow and expensive compared with light scattering. For these reasons, neutron scattering is not preferred for the analysis of environmental systems except when there are very speciﬁc features of the system that indicate they would be of beneﬁt. The ability to control contrast very selectively without changing the system chemistry is one such aspect. Schmidt has made a useful review of a range of studies in small-angle X-ray and neutron scattering [46].

3.3 IMAGE ANALYSIS Images provide a great deal of information about a particulate system. Details of particle size, shape and textural features are all apparent in an image. The use of images for quantitative analysis, including the measurement of fractal dimensions, requires the use of image-processing algorithms. Balancing the great strength of image analysis in providing a wealth of structural information and the great weakness of reducing the information about a three-dimensional system to a two-dimensional representation, as well as the poor measurement statistics, when compared with the scattering techniques, represents the greatest challenge in making effective use of image analysis for structure determination. 3.3.1 Principles and Analysis The principle of geometric transparency discussed earlier is critical to how fractal structure is encoded in images of fractal objects. For large fractal structures with a mass fractal dimension Dm < 2, the area of the projected image will scale with exactly the same dimension as the mass scales in the real structure in three-dimensional space. When Dm > 2 the structure is geometrically opaque, which means that the projection has no ‘holes’ in it and scales according to power 2 as the size of the projection increases. What this means is that if we measure the fractal dimension describing how the area of a projected fractal scales with size, and if the dimension so obtained is less than two, then the value obtained is also the mass fractal dimension of the structure in three-dimensional space. We can express this sense of fractal scaling in images as Ap,a ∝ R DA

(3.21)

where Ap,a is the projected area of an aggregate image and DA is the projected area fractal dimension, sometimes referred to as D2 . The geometric transparency argument gives us DA = Dm (3.22)

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when Dm < 2, and DA = 2

(3.23)

when Dm > 2. In reality this is not quite so straightforward, as the arguments around geometric transparency and opacity are only true for inﬁnite fractals. With aggregates of ﬁnite size, the projection will display rather more ‘holes’ than would a larger fractal aggregate with the same structure. This is reﬂected in the form for projected area scaling suggested by Meakin et al. [11], reproduced as Equation (3.20). Nelson et al. [41] have suggested a correction to obtain the true fractal dimension Dm of the three-dimensional structure from the projected area dimension DA and successfully applied it to images of soot aggregates. No matter what the fractal dimension of the structure in three-dimensional space, the perimeter of the projection of the fractal will have fractal characteristics even though the projection itself may be compact. The perimeter dimension represents the scaling of the length of the perimeter P with the resolution used to measure it. This phenomenon was pointed out early on with respect to the length of coastlines, where, if the length of a coastline is measured on a map by stepping along it with a set of dividers, the measured length was found to increase as a power law as the step length was decreased. Effectively, the smaller the step length, the smaller the features of the coastline that were able to be resolved [47]. The fractal dimension describing this scaling is called the perimeter, boundary or edge dimension: Le (ε) ∝ ε1−De

(3.24)

where Le is the measured edge length of the image, ε is the length of the ‘yardstick’ used to measure it and De is the edge dimension, also known as the boundary or perimeter dimension or simply D1 . For ordinary nonfractal images, such as that of a circle, the edge has unit dimension and the measured edge length will be independent of the step length ε, provided that ε is sufﬁciently small. The perimeter or edge dimension, the projected area dimension and the mass dimension are all related to each other, although the relationship, at least for images of real fractal aggregates, is not trivial. Jullien et al. [48] made a numerical investigation of the measurement of fractal dimension from projections of computer-simulated aggregates and found that asymptotically, for large aggregates, the fractal dimension of the projection of the aggregate DA and the perimeter dimension Dp are the same as the mass fractal dimension of the aggregate when Dm < 2. When Dm > 2, DA = 2 and De decreased nontrivially with increasing mass dimension. These conclusions, while interesting and illustrative of fundamental aspects of fractal aggregate structure, are not of any use in measuring the fractal dimensions of these structures from images because they only apply for inﬁnitely large aggregates. When aggregates are ﬁnite in size these conclusions only hold approximately. In general, the ﬁnite size of aggregates makes it easier to ‘see through’ them, which will result in an underestimation of the fractal dimension using techniques such as box

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counting [49]. A mathematical correction, such as that derived by Nelson et al. [41], should be applied when measuring the fractal dimension from projected images. The behaviour of De for geometrically opaque aggregates raises the tantalizing possibility that some judgements about mass scaling might be made with measurements of both the projected area dimension and the perimeter dimension, but the relationship between perimeter dimension and mass dimension is, at the time of writing, not well understood. In addition, suitable ﬁnite size corrections for the perimeter dimension might be required. Another dimension that may be measured from images is a texture dimension, which may be applied to greyscale images [50]. This has mostly found application in analysis of medical images. 3.3.1.1 Image preprocessing Before the application of most algorithms that are used to deduce fractal dimensions from images, the image must be segmented, meaning that each pixel is unambiguously decided as being part of the image of the fractal or part of the background.5 The simplest way of doing this is by thresholding, which takes a greyscale image that is the usual result of electronic acquisition of microscope images and assigns a particular shade of grey (threshold value) to represent the transition from black to white. Everything darker than that tone is set to black and everything lighter is set to white. More sophisticated methods for obtaining a segmented image exist, such as edge detection by gradient or Laplacian methods. Readers are referred to the book by Gonzales and Woods [51] for a detailed discussion of image processing including segmentation and edge detection. Whichever combination of methods is chosen for segmenting the image, the extent to which the preprocessing affects the fractal analysis that follows should be tested. Where thresholding has been used for segmentation, one approach is to perform the fractal analysis with different values chosen for the threshold in preprocessing and accept the results where the returned fractal dimension does not change much with the choice of threshold value [24]. Equivalent tests should be conducted for other segmentation algorithms and parameters for the segmentation chosen that do not affect the fractal dimension results much. Another test worth doing is to take a clean image of a fractal with a known fractal dimension, such as triadic Koch island or a Sierpinski gasket, and ‘contaminate’ the image to a similar extent that experimentally derived images are contaminated by adding noise, blur or uneven lighting. If possible, the contamination should be applied with the same process that is used to measure the fractal images in the experiment. Then, the chosen image segmentation algorithm(s) and the chosen fractal analysis should be applied. The measured fractal dimension can then be compared with the analytically known fractal dimension.

5 The algorithm explained by Cross [50] to measure the texture dimension is an exception to this, it works on a greyscale image.

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Ideally, the images should be acquired in the ﬁrst place in such a way as to make segmentation straightforward. One should seek the best possible image resolution and focus, and the best possible contrast between the image and the background. Failing to consider image quality in the ﬁrst instance will make segmentation difﬁcult. For example, lighting from the side should be avoided if one wants to use thresholding for image segmentation, as illustrated in Figure 3.7. In this case, some of the image is brighter than the background and some is darker, making it impossible to separate image from background on the basis of brightness. Figure 3.8 shows an image much more suited to thresholding, with white particles viewed against a black background and overhead lighting, giving excellent contrast.

Figure 3.7 Flocs of iron hydroxide. Not only is the image somewhat out of focus, but the sideways lighting means that part of the ﬂoc image is brighter than the background and part of it is darker. Clearly, thresholding is not going to be able to segment this image properly.

3.3.1.2 Image processing algorithms Applying the various algorithms for measuring fractal dimensions for images is much easier than image preprocessing. For measurement of the perimeter dimension, the options are variations on box counting, divider or stepping methods, and aggregate population methods. For the measurement of the projected area dimension, the options are box counting, point correlation, including the sandbox method, and aggregate population methods. The box counting method involves covering an aggregate image with square boxes of different sizes and counting the number of boxes Nbox required to do this for each different box length Lbox . If the boxes are made to cover the entire projection of the

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Figure 3.8 Flocs of silica viewed against a black background with overhead lighting. Excellent contrast is achieved between the image and the background, making thresholding a good choice for image segmentation in this case.

aggregate, then the fractal dimension measured in this way will be the projected area dimension measured by box counting. If the boxes are made to cover only the outer edge of the aggregate, then the dimension will be the perimeter dimension measured by box counting. In either case: DA or De = lim

L→0

− log(Nbox ) log(Lbox )

(3.25)

An easy, although tedious, way to implement this procedure is to start with a highresolution black image of the fractal in any common image manipulation software and use an image histogram function to count the number of black pixels. This represents the number of squares of size ‘1’ required to cover the image. Next, reduce the resolution of the image by a factor of 2, ensuring that an interpolating function is used in the resize, convert any grey pixels to black and again count the image pixels. This is the number of squares of size ‘2’ required for coverage. Repeat for progressively coarser resolutions for several different images and extract the fractal dimension through Equation (3.25). An example of this implementation of the box counting method for the measurement of the projected area dimension can be seen in Figures 3.9 and 3.10. This analysis was implemented using simple and commonly available image manipulation software,6 starting with the full-resolution image and progressively halving the resolution for

6

Paint Shop Pro 4.15, from Jasc Software: http://www.jasc.com.

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Figure 3.9 Box-counting analysis of a projection of a computer-generated diffusion-limited cluster aggregate of 10 000 particles with mass fractal dimension of 1.88. The images have box sizes L of (top left to bottom right) 1, 2, 4, 8, 16, 32 and 64 pixels and require 205 245, 59 519, 17 062, 4895, 1436, 462 and 135 squares respectively to cover the image.

Figure 3.10 The number of boxes versus box size from the ﬂoc shown in Figure 3.9. The slope of the curve as log L goes to zero returns the fractal dimension. DA = 1.79 for this case, which is somewhat lower than the mass fractal dimension of 1.88 due to the ﬁnite size effects analysed by Nelson et al. [41].

each increase of box size. At each resolution, the image statistics displaying the total number of black pixels was recorded, representing the number of boxes required to cover the image at that resolution. The recovered fractal dimension is somewhat lower than the true value because of ﬁnite size effects. The divider method was the original method used to measure the fractal dimension of coastlines. Manual implementation of this method may be attempted for images of

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fractal aggregates, but this quickly becomes tedious. Such an implementation would involve opening a set of dividers to a distance substantially smaller than the aggregate image being measured and stepping around the perimeter, counting the number of steps required. The number of steps multiplied by the step length ε is the estimate Le (ε) of the perimeter. The process is then repeated with ever smaller steps. Plotting log(Le ) versus log(ε) should (from Equation (3.24)) yield a straight line with slope 1 − De . A much more common implementation of the divider method for computer images is the serial dilation method, which, along with box counting method, is explained by Cross [50]. Cross also reviews the use of fractal dimensions in image analysis, predominantly in the medical imaging context. A number of computer algorithms for measuring perimeter dimension are reviewed by Allen et al. [52], including several computer-based implementations of the divider method. They conclude that Adler and Hancock’s [53] distance transform method is a more accurate and faster analysis than the popular dilation method approach to box counting. This is generally considered the preferred method to use for measuring perimeter dimension. For both perimeter and projected area dimensions, the aggregate population methods may be used. When an image or set of images contains projections for fractal aggregates of many different sizes but arising from the same fundamental process, the perimeter or projected area can be measured at a ﬁxed ﬁne resolution and then correlated to aggregate size. This assumes that all the aggregates in the population have the same fractal structure. Equation (3.21) may be used to extract DA (or De ) from a log–log plot of Ap (or Le ) versus La , where Ap is the projected area of an aggregate and La is a measure of the aggregate length. The sandbox method is another approach to the measurement of the projected area dimension. This was the method used by Forrest and Witten to point out for the ﬁrst time that aggregates of ﬁne particles can exhibit fractal scaling [54]. The technique involves selecting a point near the centre of the image and counting the amount of projected area Ap,box that falls within a box of given length Lbox centred on the chosen point. This analysis is performed for boxes of different sizes and for different chosen initial points. The projected area fractal dimension is then extracted as log(Ap,box ) L→0 log(Lbox )

DA = lim

(3.26)

A method that lends itself much better to implementation on a computer is a direct measurement of the correlation function of the image though a two-dimensional equivalent of Equation (3.5). First, the image data are transformed such that every pixel that is part of the fractal segment (i.e. not the background) is listed as an x, y coordinate. Next, for every pixel in the segment i, we loop over all other pixels in the segment j and calculate the Euclidean distance r between them; i.e.: rij = (xi − xj )2 + (yi − yj )2 (3.27)

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2 For an image of a fractal represented with Npix pixels, these Npix distances are distributed into bins of width r representing different distance ranges so that the total number of times Nr that a distance between r − r/2 and r + r/2 occurs in the image is recorded for different r. The projected area dimension may then be obtained from a plot of log(Nr ) versus log(r), according to

Nr (r) ∝ r DA −1

(3.28)

2 These Npix distance calculations can represent a substantial computation for highresolution images, but this should not cause a problem for most modern computers. If computation time becomes an issue, an easy way to reduce the time while sacriﬁcing very little in the way of accuracy is to perform the loop over i for a randomly chosen subset of pixels in the segment, rather than all pixels in the segment. The accuracy and speed of the correlation function for measuring fractal dimension may be improved by only including pixels j which are closer to i than the aggregate edge nearest to i. This will not remove effects due to ﬁnite size and consequent increased transparency, but will eliminate gross aggregate shape effects.

3.3.2 Single Image Microscopy There are many kinds of microscopy that may be used for structural analysis of fractal aggregated systems. Most microscopes give a single image of a sample and, as a result, three-dimensional information about the image is generally lost. In preparing samples exhibiting fractal properties for microscopic analysis, considerable care must be taken to ensure that the sample handling does not alter the structure one is trying to analyse. Transmission electron microscopy has been used to measure fractal dimensions, commonly from soot or smoke particles collected directly onto the microscope grid [e.g. 55]. Bower et al. measured the boundary fractal dimension of lactose aggregates in 1,1,2-trichlorotriﬂuoroethane using serial perimeter dilation and Adler and Hancock’s distance transform function [56]. They found that it decreased as shear rate and aggregate mass fractal dimension increased. 3.3.3 Confocal Microscopy Confocal scanning laser microscopy (CSLM) (sometimes called confocal laser scanning microscopy and often simply abbreviated to confocal microscopy) is a relatively new technique that lends itself to fractal analysis in interesting ways (see also Chapter 5). The technique was ﬁrst used to measure fractal structures by Bremer et al. in 1993 [57]. The most notable feature is that laser focusing allows the acquisition of an image which is a slice through the object being measured, rather than a projection of it. This allows for extremely sophisticated analysis. For example, it is possible to image slices at many different heights through a single ﬂoc and to reconstruct the exact position of every particle in three-dimensional space [58]. Having imaged and reconstructed a ﬂoc in this way, the measurement of any structural feature one

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desires may be carried out. Similar measurements could be made by microtome slicing and traditional forms of microscopy, but CSLM presents fewer risks of sample modiﬁcation, as the measurement can be made wet. One of the easiest ways to measure fractal dimension with this technique is to capture images of slices through the structure and measure the fractal scaling of the image as discussed above. The dimension measured in this way is not the projected area dimension DA discussed earlier, because the image is a slice not a projection. It turns out that the dimension measured in this way is numerically equal to the mass fractal dimension minus one, by virtue of the codimension rule [58]. The measurement of fractal dimensions by this technique is not subject to the restriction of geometric transparency, as is the case with the analysis of projected images, and so fractal dimensions well over two can be measured. Snidaro et al. [59] used CSLM to measure the positions of bacteria in aggregates in three-dimensional space. Once the data were obtained they used a three-dimensional sandbox approach to measure the fractal dimension, with the sandbox centred on the centre of mass of the aggregate. They also used a two-dimensional sandbox method to obtain the two-dimensional fractal dimension, related to the usual Dm through the codimension rule. Small aggregates were shown to have a compact structure using both the two- and three-dimensional approaches. Thill et al. [24] used the technique to measure the fractal dimension of Brownian and shear-aggregated latex particles, using thresholding at different levels and a two-dimensional box-counting approach. They noted that higher fractal dimensions result in an increased chance of particles obscuring other particles and thus not being included in the analysis. Dinsmore and Weitz [60] avoided this problem by refractive index matching the (transparent) particles and suspending liquid, but this is unlikely to be an available option with environmental systems. Thill et al. [58] reconstructed the full three-dimensional coordinate information for latex aggregates and showed that the two-dimensional sectioning approach does indeed return the correct fractal dimension for the aggregates as determined from a three-dimensional pair correlation approach. The authors’ density matched the solution and particles to avoid any sedimentation and allowed the aggregation to occur by Brownian diffusion. The aggregates were sampled using a broad (0.8 cm) pipette7 and gently deposited onto the microscope slide without using a cover slip. This illustrates the level of care that needs to be taken in order to avoid the restructuring of aggregates by shearing. Mellema et al. [61] used CSLM to examine the structure of casein gels and included a detailed discussion of the mechanics of the technique and how they applied it to their experiments. They used the density autocorrelation approach to measure the area dimension of the two-dimensional image slices. These authors used thresholding to

7 Presumably the pipette was inverted, so that the sample was not drawn through the narrow opening at the tip of the pipette.

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segment the image and found that the area dimension was very sensitive to the choice of threshold level used. Schmid et al. [62] used CSLM to characterize activated sludge ﬂocs. They also used the two-dimensional correlation function for images and ﬂuorescence in-situ hybridization (FISH) to identify the organisms within the ﬂocs during CSLM. Dinsmore and Weitz (2002) [60] examined a model system of polymethylmethacrylate particles dyed with rhodamine in a density- and refractive-index-matched solution of decalin and cyclohexylbromide. They used CLSM to follow the (very slow) aggregation in real time and determined the particle positions in full threedimensional detail. From this they performed a comprehensive structural analysis of the gels, including measurement of coordination numbers and backbone fractal dimension8 of the structures, as well as the much more commonly measured mass fractal dimension. Confocal microscopy has shown itself to be one of the most versatile techniques for structural analysis of biological and environmental systems, in some cases allowing computer reconstruction of the full three-dimensional detail of the scanned sample. Once this level of detail is obtained about a sample, any structural measure desired can be made upon the reconstruction using the methods described here and many others. It is well suited to large aggregates of materials whose density is not greatly different to the supporting liquid. Clearly, resolution is limited by the wavelength of light, as with any optical microscopy. 3.3.4 X-ray Microtomography Although X-ray microtomography is not strictly speaking an imaging technique, twoand three-dimensional images are normally reconstructed from the raw data as a starting point for analysis. The technique works by scanning a thin X-ray beam through a sample at a number of different angles and measuring the absorbance of the X-rays by the sample. The three-dimensional structure that accounts for the observed absorbance patterns is constructed in a computer using the standard techniques of tomography. Many of the comments made regarding confocal microscopy can also be made regarding X-ray microtomography, in that a full three-dimensional reconstruction of the scanned sample results from the analysis. Thus, fractal analysis can be made by point-to-point correlations in the three-dimensional structure or within two-dimensional slices and then applying the codimension rule. As with X-ray scattering, the technique is not well suited to biological samples as the X-ray contrast of these systems is exceedingly poor. It does work well for mineral samples and might be applied for sediments and clays. An overview of the technique and comparison of different methods as applied to soil samples is given

8 The backbone fractal dimension is the mass fractal dimension of the ‘backbone’, i.e. that chain of particles which connects two arbitrary particles in the structure.

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by Gibson et al. [63]. At the time of writing, the practical resolution of commercial instruments is limited to the order of a couple of micrometres.

3.4 MISCELLANEOUS TECHNIQUES 3.4.1 Settling Velocity Measurement The settling velocity of fractal aggregates is one of their most signiﬁcant features in many environmental systems. Settling velocities of fractal aggregates have been observed to be signiﬁcantly higher than would be expected from impermeable spheres with the same mass and spatial extent as the aggregates [23, 64, 65]. The reason for this behaviour is that the fractal structures are often highly permeable, so that the ﬂuid drag on the aggregates is reduced. This is clearly a function of the geometry of the aggregates and the relative velocity of the aggregates and the ﬂuid. The fact that the drag is geometry dependent is highly signiﬁcant to many applications involving settling of aggregates because it implies that transport properties of fractally aggregated matter should depend on the fractal dimension of the system. In other words, it is important in practice to have models of settling velocity that properly account for the effects of fractal dimension. The geometry dependence of drag is also signiﬁcant, in that it potentially gives us a measurement that can be used to infer structural information about aggregates. Using settling velocity to infer fractal dimension and modelling settling velocity accurately are really two different problems with a common underlying physical basis. The ﬁrst of these problems will be dealt with here; the second is beyond the scope of this chapter. 3.4.1.1 Principle When settling aggregates have reached their terminal velocity, the buoyancy force Fb and drag force Fd have reached an exact balance: Fb = Fd

(3.29)

Without yet introducing any assumptions, this can be written as follows, which serves as a deﬁnition of the drag coefﬁcient Cd : ρL 1 ma 1 − g = ρL vt2 Ap,a Cd (3.30) ρS 2 where ma is the mass of the aggregate, ρL is the density of the ﬂuid, ρS is the density of the solid particles comprising the aggregate, g is acceleration due to gravity, vt is the terminal velocity of the aggregate and Ap,a is the projected area of the aggregate in the direction of vt . In the literature, the drag coefﬁcient is often represented as the product of the drag coefﬁcient of an impermeable sphere having the same radius of collision as the

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aggregate and a correction factor given the symbol which accounts for the increase in permeability of the aggregate over the equivalent sphere. Cd = Cd∗

(3.31)

where Cd∗ is the coefﬁcient of drag of an impermeable sphere having the same radius of collision as the aggregate in question. For creeping ﬂow (which is true for most fractal aggregates in environmental systems): 24 Re

(3.32)

2Rc,a ρL vt μL

(3.33)

Cd∗ = where the Reynolds number Re is given by Re =

where μL is the viscosity of the ﬂuid and Rc,a is the radius of collision of the aggregate. The use of Ap,a is rather difﬁcult in practice because it is often hard to measure the actual projected area of aggregates due to (i) the geometry of typical settling experiments and (ii) limited image resolution typical in such experiments. Additionally, all of the uncertainties associated with aggregate projected area that were discussed earlier in the context of the volume obscuration method also apply here. The usual practice is to take Aa as the projected area of the sphere (or sometimes the ellipsoid) that encloses the aggregate being observed. This will overestimate the projected area because ‘holes’ in the projection of the aggregate will be included as part of the projected area. What this means in practice is that the correction factor is as much a correction for projected area as for the drag coefﬁcient. Adopting this practice, the projected area Ap,a of the aggregate will be 2 Ap,a = πRc,a

(3.34)

Making use of an equation equivalent to Equation (3.2) for the mass of the aggregate: D Rc,a m 4 ma = kc (3.35) πR03 ρS R0 3 we can substitute into Equation (3.30) to derive an expression that will tell us about the dependence of settling velocity on size and fractal dimension: D 24μL Rc,a m 4 ρL 1 3 2 kc g = ρL vt2 π Rc,a πR0 ρS 1 − (3.36) R0 3 ρS 2 ρL vt 2Rc,a Rewriting with vt as the subject: vt =

2kc R02 (ρS − ρL )g 9 c μL

Rc,a R0

Dm −1 (3.37)

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which implies that a log–log plot of settling velocity versus aggregate collision radius for a set of aggregates having a fractal dimension will yield a straight line with slope Dm − 1. Now, let us critically examine the assumptions which underlie this analysis. If we have managed to accurately separate all of the size dependence of the system into the factor (Rc,a /R0 )Dm −1 , then Equation (3.37) will be correct. The biggest problem with this is the parameter c . The ﬁrst problem, of using settling velocity to determine an accurate value for the fractal dimension, is that of accurately knowing the Rc,a dependence of c . The second problem, of modelling settling velocity accurately requires a quantitative model for c . Clearly, the ﬁrst of these two requirements is the less restrictive. As explained above, the parameter c is known as a permeability correction, but rightly accounts for two different effects: the extent to which the projected area of an aggregate is different to the projected area of a sphere having the same radius of collision, and the extent to which the drag coefﬁcient of an aggregate is different to the drag coefﬁcient of a sphere having the same projected area as the aggregate. Taking these two effects together, c represents the extent to which the settling velocity of an aggregate is different to that of an impermeable sphere having the same drag force and radius of collision. Looking at the meaning of c in another way, if the hydrodynamic radius of an aggregate is the radius of a sphere that has the same settling velocity and drag force as the aggregate in question, then c can be shown to relate the radius of collision to the hydrodynamic radius: (3.38) Rh,a = c Rc,a Is c in fact constant with aggregate size as implied by Equation (3.38)? If it is, then this implies that the hydrodynamic radius is linearly related to the radius of collision, a conclusion that has considerable support in the literature [10, 66]. This would also mean that the dimensionless permeability of the aggregates is also constant with respect to aggregate size, a proposition that has had support recently [67, 68]. What are the issues relating to the shape of the aggregate, choice of size measurement and to preferential orientations in settling? Suppose that instead of the radius of collision of the aggregate we had preferred to use as a size characterization the radius of a sphere having the same projected area as the aggregate, RA,a . The drag force of the system would then be expressed as Fd =

24μL 1 2 A ρL vt2 πRA,a 2 ρL vt 2RA,a

(3.39)

2 The factor π RA,a , which represents the projected area of the aggregate, is now strictly correct and the drag correction A now represents a combination of drag coefﬁcient correction and Reynolds number correction because we have used a different radius in the Reynolds number calculation. Since the different measures of aggregate radius are all linearly related to each other if the aggregates are fractal (it follows from

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self-similarity), the same conclusions regarding the scaling of settling velocity with aggregate size and the constancy of the factor A are reached. Indeed, exactly the same conclusions are reached no matter which kind of radius or diameter we choose to use as a basis for the analysis. The equations presented so far rely on the proposition that the projected area of the aggregate is proportional to the square of the radius we are choosing to characterize the aggregate size. As discussed in the previous section on the volume obscuration method, this is only the case for very large aggregates, and is never the case for aggregates with fractal dimension less than two [23]. Johnson et al. [23] point out that Equation (3.37) will have a different power-law dependence between settling velocity and aggregate size depending on whether the fractal dimension is less than or greater than two and additionally make a correction to the drag coefﬁcient for somewhat higher Reynolds numbers |0.1 ≤ Re ≤ 10|. The characteristic length for the Reynolds number is taken as being the diameter of collision of the aggregate, in the absence of any better information. As explained by Meakin et al. [11], the reality with respect to projected area dependence on aggregate size is rather more complex than two simple cases depending whether the fractal dimension is larger or smaller than two. Small aggregates tend to present more projected area than would be expected, because of the ﬁnite size of the primary particles. This would lead to lower settling velocities for small aggregates than might be expected from Equation (3.37), which without correction would lead to an overprediction of Dm . As with the volume obscuration analysis, the way to account for this may well be to have the projected area well characterized in terms of G, B, γ and δ from computer simulations for aggregates with different size and structure. 3.4.1.2 Experimental approaches These kinds of experiments are without exception carried out in a column of ﬂuid, usually of the same composition as that from which the aggregates were sampled. The aggregates are introduced into the top of the column and one or more microscopes are used to measure the settling velocity. Farrow and Warren describe a ﬂoc density analyser [69] which may be used to determine the fractal dimension. Nobbs et al. [70] review many of the practical aspects involved in performing this type of experimental investigation. There are a number of difﬁculties with the settling approach in practice. For large ﬂocs, the settling velocity and induced drag may be large enough to restructure the aggregates under investigation because of the extremely weak mechanical strength of the aggregates [24]. Conversely, the settling velocities usually have such a small absolute value that stray convection currents can render the measurement meaningless. For this reason, control of ﬂuid movement in the column is absolutely critical. Settling columns are usually temperature controlled by recirculated air [70] or water [71]. Particular care needs to be taken with illumination – continuous, high-intensity light

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sources can easily add heat to the system and induce currents. Strobe-type illumination avoids this problem and additionally can provide very clear images, as the high but brief intensity allows a narrow aperture (good depth of ﬁeld) and short exposure time (no motion blur). Introducing the ﬂocs to the top of the settling column can also induce ﬂuid motion in the column, both through mechanical disturbance and by introducing too much sample, which effectively creates a density gradient in the column. Using an inverted pipette to introduce aggregates one at a time without disturbing the liquid in the column is a difﬁcult skill to master, but is the best way to avoid this kind of problem. An alternate approach to the problem of induced currents might be to introduce small ‘tracer particles’, density matched to the suspending ﬂuid. This allows the background ﬂuid motion to be visualized and corrected for.

3.4.2 Chord Length Measurement Focused beam reﬂectance measurement (FBRM) is a relatively new on-line particle measurement technique that measures chord lengths using a scanning laser. The instrument is designed as a robust probe, tipped with a sapphire window through which a circularly scanning laser is focused. Particles that approach the window reﬂect the laser back through the instrument’s optics and the duration of the reﬂected signal is converted to a chord length using the known scanning speed of the laser. Although the technique has been used to monitor ﬂocculation [72, 73], it appears that it has not yet been used to determine the fractal dimension of these systems. This should be a relatively straightforward matter: if one assumes that the instrument samples the aggregates in an unbiased manner, then the same kind of analysis as was applied for the volume obscuration techniques can be used. If the suspension ﬂowing past the instrument can be assumed to have a constant solids loading and the chord length distribution can be assumed to be related to the aggregate size distribution in a direct way, then the frequency of chord acquisition and the length of the chords will effectively tell us the concentration of aggregates of different sizes. This gives us the apparent solid volume fraction, which, combined with the known true solids loading, tells us the aggregate density and hence the fractal dimension. Kovalsky and Bushell speculate that it might even be possible to measure the fractal dimension directly from the texture of the reﬂected laser [74].

3.4.3 Particle Counting Techniques A very direct way of measuring the fractal dimension is to have direct measurement of the size of individual aggregates and of the amount of solid that comprises them, a technique sometimes referred to as the particle concentration technique

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[75]. Stoll et al. [76] and Jackson et al. [77] report this kind of analysis using aperture impedance particle sizing. Aperture impedance particle sizing works by passing a suspension of particles in electrolyte through a small oriﬁce with a voltage applied across it. Solid particles passing through the oriﬁce reduce the amount of electrolyte available for current ﬂow and hence change the impedance of the oriﬁce. The change of impedance is more or less directly related to the solid volume of the particle passing through the oriﬁce with little dependence on particle morphology. Jackson et al. [77] used two aperture impedance methods to determine the solid volume distribution of aggregating algal particles and additionally used image analysis to measure the size distribution of the same particles. A mass fractal dimension was then assumed that led to the best match between the two methods of size analysis, returning a value of 2.3. Sterling et al. [78] used a similar approach to examine the structure of crude oil and clay aggregates, except that they used forward light scattering instead of image analysis to determine the size distribution.

3.5 CONCLUSIONS AND FUTURE PERSPECTIVES A wide range of techniques exist for measuring the fractal dimensions in environmental systems. Which technique should be chosen depends on the particles being measured and the environment in which they are being measured. Many of the methods outlined here, such as scattering and settling experiments, are only suitable for laboratory-based work. Others, such as image analysis and FBRM, have the potential to be used in the ﬁeld. Small particles with size of the order of the wavelength of light or less are good candidates for light scattering experiments, whereas larger particles are suitable for optical microscopy and settling experiments. Large particles of size several times the wavelength of light, but with low refractive index, could be analysed by confocal microscopy. A summary of the techniques and methods discussed here and the systems to which each is applicable is given in Table 3.1. This chapter has illustrated how measurements from a range of different particle characterization techniques have been used to determine values for fractal dimensions. Other approaches can also be imagined. Whichever technique is chosen or devised, it is important to examine the assumptions underlying the analysis critically and to check their validity in the context of the particular system being analysed. The state of the art in measurement of fractal dimension for environmental systems at this time is probably CSLM, because the analysis requires no models or assumptions, unlike most other approaches, and is free of the two-dimensional limitation of other imaging techniques. To the extent that the sample can be handled appropriately and the measurement can be made reliably, the analysis is reliable and unambiguous.

Table 3.1 A comparison of different techniques and methods for determining fractal dimensions in environmental systems. Technique

Method

Particle size (order of magnitude)

Contrast

Relative density

Fractal dimension

Comments

Light scattering

Angular dependence Apparent volume

1 m

Moderate to high

Low

High

Angular dependence Angular dependence

1 m

Low

Low

Any

Light scattering X-ray scattering Neutron scattering Single image microscopy Single image microscopy Confocal laser scanning microscopy

Contrast should be low enough that image slices can be obtained from the interior of the structure. Higher contrasts may be used when Dm is low. Density should be low enough that the aggregate structure does not deform signiﬁcantly due to gravity

Table 3.1 (Continued). Technique

Method

Settling velocity measurement Chord length measurement Particle counting

Particle size (order of magnitude)

Contrast

Relative density

Fractal dimension

Comments

Permeability >1 m models

Moderate to high

Size dependent

High

Apparent volume

>1 m

Moderate to high

Any

High

Contrast should be high enough to allow optical measurement of settling velocity. Uncertainty in permeability models is less for high fractal dimensions. Density should be in a range that corresponds to Stokes’ law settling Untested technique

Mass/size scaling

Instrument dependent

Instrument dependent

Any

Any

Particle counting measures the number of particles in each aggregate. It must be used in combination with another technique that measures size

GRAEME BUSHELL

LIST OF SYMBOLS AND ABBREVIATIONS Abbreviations CSLM confocal scanning laser microscopy FISH ﬂuorescence in-situ hybridization USAXS ultra-small-angle X-ray scattering Variables A area B an area parameter C coefﬁcient D dimension (fractal) G another area parameter I intensity g acceleration due to gravity g correlation function k power-law prefactor k wave number L a generalized length m mass N number P form factor Q extinction efﬁciency q change in wave number on scattering q magnitude of q r displacement vector r magnitude of r R radius Re Reynolds number S structure factor v velocity x size parameter of a particle xi Cartesian coordinate of element i Cartesian coordinate of element i yi z real refractive index δ area exponent ε step size around a perimeter φ function describing the presence of a particle γ area exponent λ wavelength μ viscosity π pi (mathematical constant) θ angle ρ density

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drag coefﬁcient correction factor ζ complex refractive index Subscripts 0 primary particle 1 perimeter 2 area 3 structure in three-dimensional space a aggregate A area b buoyancy box relating to a box, such as in the box-counting or sand box methods c pertaining to the radius of collision d drag e pertaining to an edge or perimeter h pertaining to the hydrodynamic radius i label denoting any particular element, such as a particle or pixel different than j j label denoting any particular element, such as a particle or pixel different than i g pertaining to the radius of gyration L ﬂuid m pertaining to mass p pertaining to a projection pix pertaining to a pixel r pertaining to a distance S solid T total in the system t terminal

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[30] Jullien, R. (1992). From Guinier to fractals. J. Phys. I, 2, 759–770. [31] Mountain, R. D. and Mulholland, G. W. (1988). Light-scattering from simulated smoke agglomerates. Langmuir, 4, 1321–1326. [32] Hurd, A. J. and Flower, W. L. (1988). In situ growth and structure of fractal silica aggregates in a ﬂame. J. Colloid Interface Sci., 122, 178–192. [33] Zeng, Y. W. and Meriani, S. (1994). Scaling functions for the ﬁnite-size effect in fractal aggregates. J. Appl. Crystallogr., 27, 782–790. [34] Sorensen, C. M., Lu, N. and Cai, J. (1995). Fractal cluster size distribution measurement using static light scattering. J. Colloid Interface Sci., 174, 456–460. [35] Sorensen, C. M., Cai, J. and Lu, N. (1992). Test of static structure factors for describing light scattering from fractal soot aggregates. Langmuir, 8, 2064–2069. [36] Van de Hulst, H. C. (1981). Light Scattering by Small Particles. Dover, New York. [37] Farias, T. L., Köylü, Ü. Ö. and Carvalho, M.G. (1996). Range of validity of the Rayleigh– Debye–Gans theory for optics of fractal aggregates. Appl. Optics, 35, 6560–6567. [38] Nelson, J. (1989). Test of a mean ﬁeld theory for the optics of fractal clusters. J. Mod. Optics, 36, 1031–1057. [39] Botet, R., Rannou, P. and Cabane, M. (1997). Mean-ﬁeld approximation of Mie scattering by fractal aggregates of identical spheres. Appl. Optics, 36, 8791–8797. [40] Ortiz, G. P. and Mocán, W.L. (2003). Scaling condition for multiple scattering in fractal aggregates. Pysica B, 338, 103–106. [41] Nelson, J. A., Crookes, R. J. and Simons, S. (1990). On obtaining the fractal dimension of a 3D cluster from its projection on a plane – application to smoke agglomerates. J. Phys. D: Appl. Phys., 23, 465–468. [42] Köylü, Ü. Ö., Faeth, G. M., Farias, T. L. and Carvalho, M. G. (1995). Fractal and projected structure properties of soot aggregates. Combust. Flame, 100, 621–633. [43] Lambert, S., Moustier, S., Dussouillez, P., Barakat, M., Bottero, J. Y., Le Petit, J. and Ginestet, P. (2003). Analysis of the structure of very large baterial aggregates by small angle multiple light scattering and confocal image analysis. J. Colloid Interface Sci., 262, 384–390. [44] Gregory, J. and Chung, H. (1995). Continuous monitoring of ﬂoc properties in stirred suspensions. J. Water Supply Res. Technol. – Aqua, 44, 125–1313. [45] Berthon, S., Barbieri, O., Ehrburger-Dolle, F., Geissler, E., Achard, P., Bley, F., Hecht, A-M., Livet, F., Pajonk, G. M., Pinto, N., Rigacci, A. and Rochas, C. (2001). DLS and SAXS investigations of organic gels and aerogels. J. Non-Cryst. Solids, 285, 154–161. [46] Schmidt, P. W. (1991). Small-angle scattering studies of disordered, porous and fractal systems. J. Appl. Crystallogr., 24, 414–435. [47] Kaye, B. H. (1994). A Random Walk Through Fractal Dimensions. Weinheim, New York. [48] Jullien, R., Thouy, R. and Ehrburger-Dolle, F. (1994). Numerical investigation of twodimensional projections of random fractal aggregates. Phys. Rev. E, 50, 3878–3885. [49] Tence, M., Chevalier, J. P. and Jullien, R. (1986). On the measurement of the fractal dimension of aggregated particles by electron-microscopy – experimental-method, corrections and comparison with numerical-models. J. Phys., 47, 1989–1998. [50] Cross, S. S. (1994). The application of fractal geometric analysis to microscopic images. Micron, 25, 101–113. [51] Gonzales, R. C. and Woods, R. E. (2002). Digital Image Processin , second edition. Prentice Hall, New York. [52] Allen, M., Brown, G. J. and Miles, N. J. (1995). Measurement of boundary fractal dimensions: review of current techniques. Powder Technol., 84, 1–14. [53] Adler, J. and Hancock, D. (1994). Advantages of using a discrete distance transform function in the measurement of fractal dimensions by the dilation method. Powder Technol., 78, 191–196.

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[54] Forrest, S. R. and Witten, T. A. (1979). Long-range correlations in smoke-particle aggregates, J. Phys. A, 12, L109-L117. [55] Park, K., Kittelson, D. B. and McMurry, P. H. (2004). Structural properties of diesel exhaust particles measured by transmission electron microscopy (TEM): relationships to particle mass and mobility, Aerosol Sci. Technol., 38, 881–889. [56] Bower, C., Washington, C. and Purewal, T. S. (1997). The use of image analysis to characterize aggregates in a shear ﬁeld. Colloids Surf. A: Physicochem. Eng. Aspects, 127, 105–112. [57] Bremer, L. G. B., Bijsterbosch, B. H., Walstra, P. and van Vliet, T. (1993). Formation, properties and fractal structure of particle gels. Adv. Colloid Interface Sci., 46, 117–128. [58] Thill, A., Wagner, M. and Bottero, J. Y. (1999). Confocal scanning laser microscopy as a tool for the determination of 3D ﬂoc structure. J. Colloid Interface Sci., 220, 465–467. [59] Snidaro, D., Zartarian, F., Jorand, F., Bottero, J. Y., Block, J. C. and Manem, J. (1997). Characterization of activated sludge ﬂocs structure. Water Sci. Technol., 36, 313–320. [60] Dinsmore, A. D. and Weitz, D. A. (2002). Direct imaging of three-dimensional structure and topology of colloidal gels. J. Phys.: Condens. Matter, 14, 7581–7597. [61] Mellema, M., Heesakkers, J. W. M., van Opheusden, J. H. J. and van Vliet, T. (2000). Structure and scaling behaviour of aging rennet-induced casein gels examined by confocal microscopy and permeametry. Langmuir, 16, 6847–3854. [62] Schmid, M., Thill, A., Purkhold, U., Walcher, M., Bottero, J. Y., Ginestet, P., Nielsen, P. J., Wertz, S. and Wagner, M.(2003). Characterisation of activated sludge ﬂocs by confocal scanning microscopy and image analysis. Water Res., 17, 2043–2052. [63] Gibson, J. R., Lin, H. and Bruns, M. A. (2006). A comparison of fractal analytical methods on 2- and 3-dimensional computed tomographic scans of soil aggregates. Geoderma, 134, 335–348. [64] Li, D. H. and Ganczarczyk, J. J. (1988). Flow through activated sludge ﬂocs. Water Res., 22, 789–792. [65] Serra, T. and Logan, B. E. (1999). Collision frequencies of fractal bacterial aggregates with small particles in a sheared ﬂuid. Environ. Sci. Technol., 33, 2247–2251. [66] Hess, W., Frisch, H. L. and Klein, R. (1986). On the hydrodynamic behaviour of colloidal aggregates. Z. Phys. B – Condens. Matter, 64, 65–67. [67] Li, X. Y. and Logan, B. E. (2001). Permeability of fractal aggregates. Water Res., 35, 3373–3380. [68] Woodﬁeld, D. and Bickert, G. (2001). An improved permeability model for fractal aggregates settling in creeping ﬂow. Water Res., 35, 3807–3806. [69] Farrow, J. B. and Warren, L. J. (1993). Measurement of the size of aggregates in suspension. In Coagulation and Flocculation – Theory and Applications, Dobiás B. (ed.). Marcel Dekker, New York. [70] Nobbs, D., Tang, P. and Raper, J.A. (2002). The design, construction and commissioning of a low-cost optical particle size analyser speciﬁcally for measurement of settling velocities and size of ﬂocs. Meas. Sci. Technol., 13, 297–302. [71] Glover, S. M., Yan, Y. -D., Jameson, G. J. and Biggs, S. (2000). Bridging ﬂocculation studies by light scattering and settling. Chem. Eng. J., 80, 3–12. [72] Alfano, J. C., Carter, P. W., Dunham, A. J., Nowak, M. J. and Tubergen, K. R. (2000). Polyelectrolyte-induced aggregation of microcrystalline cellulose: reversibility and shear effects. J. Colloid Interface Sci., 223, 244–254. [73] Owen, A. T., Fawell, P. D., Swift, J. D. and Farrow, J. B. (2002). The impact of polyacrylamide ﬂocculant solution age on ﬂocculation performance. Int. J. Miner. Process., 67, 123–144. [74] Kovalsky, P. and Bushell, G. (2005). In situ measurement of fractal dimension using focussed beam reﬂectance measurement. Chem. Eng. J., 111, 181–188.

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[75] Logan, B. E. and Wilkerson, D. B. (1991). Fractal dimensions and porosities of Zoogloea ramigera and Saccharomyces cerevisae aggregates. Biotechnol. Bioeng., 23, 389–396. [76] Stoll, S., Elaissari, A. and Pefferkorn, E. (1990). Fractal dimensions of latex aggregates: correlation between hydrodynamic radius and cluster size. J. Colloid Interface Sci., 140, 98–104. [77] Jackson, G. A., Logan, B. E., Alldredge, A. L. and Dam, H. G. (1995). Combining particle size spectra from a mesocosm experiment measured using photographic and aperture impedance (Coulter and Elzone) techniques. Deep-Sea Res. II, 42, 139–157. [78] Sterling, M. C. Jr, Bonner, J. S., Ernest, A. N. S., Page, C. A. and Autenrieth, R.L. (2004). Characterizing aquatic sediment–oil aggregates using in situ measurements. Mar. Pollut. Bull., 48, 533–542.

4 Fractal Structures and Mechanisms in Coagulation/Flocculation Processes in Environmental Systems: Theoretical Aspects Serge Stoll1 and Silvia Diez2 1

Department of Inorganic, Analytical and Applied Chemistry CABE, University of Geneva, Sciences II, 30 Quai Ernest Ansermet, CH-1211 Geneva 4/Switzerland 2 CIEMAT, Avda. Complutense 22, 28040, Madrid, Spain

4.1 INTRODUCTION Coagulation/ﬂocculation, and more generally speaking aggregation, represent physical processes in which initially dispersed colloidal units (solid inorganic particles, fulvic and humic acids and biopolymers) stick together, under the inﬂuence of various attractive forces, to build characteristic structures whose size increases with time. Aggregation phenomena are not only of great importance in environmental systems, but also in many areas of colloid science, chemistry and physics and industrial processes. In natural waters and wastewater treatment plants, the transport and fate of both nutrients and toxic compounds largely depend on their interactions with colloidal particles, biopolymers and the aggregates they form. The transport and elimination of vital or detrimental compounds then depends upon the kinetics of formation, structure, and sedimentation of these aggregates (Figure 4.1). Because of the complexity and large number of biophysical and chemical factors inﬂuencing these processes, as well as the fact that natural colloids include several components, no rigorous analytical theory or models have been derived and applied to describe aggregation in environmental systems. On the other hand, owing to the development of computers and the introduction of scaling concepts, numerical and theoretical models have recently been applied to investigate the behaviour of colloidal suspensions. These models have proven to be important and convenient tools for the systematic investigation of some of the physicochemical factors (pH, temperature, solution ionic strength, colloidal concentration and chemical surface properties) that inﬂuence the morphology of colloidal structures

Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems c 2008, IUPAC and Kevin J. Wilkinson

Edited by Nicola Senesi

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Heterocoagulation

X Trace Compound Adsorption on inorganic particles

X

Biopolymer Production (biomass)

X

X

Inorganic particles and biopolymer complex formation

Fractal Aggregate Formation

X Flocculation

Sedimentation

X

X

X

Figure 4.1 Key mechanisms controlling the fate and transport of colloidal matter and associated trace compounds in natural waters.

induced by perikinetic aggregation (bridging ﬂocculation by polymers, salt-induced coagulation, heteroaggregation, etc.), in addition to addressing fundamental issues such as fractal growth. Microscopic observations of natural colloids and model systems have concluded that the formation of aggregates in aquatic systems is mainly controlled by three types of colloid: compact inorganic colloids, large rigid biopolymers, and aquagenic refractory organic matter in variable proportions, sizes, and chemical properties [1]. This chapter describes some of the simulation and theoretical models used to investigate aggregation processes. The models depend upon whether the problem under consideration is deﬁned at a microscopic or mesoscopic level and on the appropriate degree of complexity and rigour that is required to solve real practical problems of interest. Several techniques and examples will be described that are applicable to coagulation/ﬂocculation processes in colloidal dispersions. Owing to the fact that such processes are now recognized to lead to the formation of fractal objects, fractal concepts will be discussed.

4.2 NATURE AND MORPHOLOGY OF AQUATIC PARTICLES/COLLOIDS: USE OF FRACTAL CONCEPTS Aggregates formed in natural waters (rivers, lakes and oceans) and in wastewater treatment systems, as well as in controlled laboratory experiments, yield geometrical properties that can be conveniently described using fractal geometry [1–12]. The fractal concept was introduced in the 1970s by Mandelbrot [13]. His approach, essentially based on geometry, quickly became popular and is now successfully used in many areas of science for the description of complex geometrical structures that cannot be described by the classical Euclidean geometry. Full lines, surfaces and

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volumes have integer linear dimensions of 1, 2 and 3, respectively, whereas open, porous, complicated structures have noninteger dimensions, i.e. fractal dimensions. The fractal dimension is thus a quantitative measure of the more or less compact nature of the structure under consideration. Fractal structures can be divided into two groups: regular and irregular. They all exhibit a remarkable property: a self-similarity or invariance with changes in scale or size. Self-similarity means that if we enlarge a given portion of a fractal object then the new enlarged object is identical to the initial one. Regular fractal objects exhibit a regular shape with well-deﬁned positions between branches (Figure 4.2a). They are built with deterministic and iterative rules. Irregular fractals, for which some randomness occurs during their growth, are more representative of natural aggregation processes. In that case, the self-similarity property is valid on average, i.e. the new picture obtained after enlarging a part of an irregular fractal is only statistically equivalent to the original form (Figure 4.2b).

a) Regular Fractal Object

b) Irregular Fractal Object

Figure 4.2 Illustration of the self-similarity property of fractal aggregates. (a) A regular fractal aggregate: when the central (grey) part of the aggregate is enlarged (as displayed on the right), the same ﬁgure as the original one is obtained and scale invariance is veriﬁed. (b) An irregular (disordered) fractal with some randomness included in its growth where the self-similarity property now is only valid on average.

Quantitatively, the fractal character of a single large aggregate is often demonstrated by a power-law behaviour of its mass distribution in space [14, 15]. If one measures the mass m contained in a sphere of radius r centred at a given point (usually the centre of mass) of an irregular fractal aggregate, the following scaling relationship is obtained: m(r) ∼ r Dm where Dm represents the mass aggregate fractal dimension (Figure 4.3).

(4.1)

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(a)

Aggregate Mass

104

103

102

101

Dm = 2.5 Mass fractal Dimension

100 1

10 Aggregate Radius r (b)

Figure 4.3 (a) Three-dimensional computer-generated aggregate (Witten and Sanders Diffusion Limited Aggregate) consisting of 10 000 particles. In the presence of random motion of the particles, single particles are irreversibly stuck to a growing germ. (b) Log–log variation of the aggregate mass versus the aggregate size. From the slope, one can already see the fractal character of the structure. From the slope, the aggregate fractal dimension was calculated as 2.5.

The mass fractal dimension gives a quantitative description of the spatial mass repartition within an aggregate, a higher fractal dimension being representative of more compact structures. Table 4.1 presents some fractal dimensions for some computer-generated and ‘natural’ aggregates. The mass fractal dimension Dm usually

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Table 4.1 Examples of various mass fractal dimension Dm for computer-simulated, biological and inorganic aggregates in two (2D) and three dimensions (3D). 2D

3D

Ref.

Computer-generated aggregates Witten–Sanders aggregates (DLA)a CCA aggregates (DLCA)a CCA aggregates (RLCA)a Regular fractal aggregate (Figure 4.2)

1.71 1.44 1.55 1.46

2.49 1.78 2.09 1.77

[21] [19, 20, 30] [31, 48] —

Natural aggregates Marine and oceanic snow Hematite particles Hematite particles with schizophyllan

— — —

1.26–2.14 1.8–2.1 1.9–2.0

[6–8] [9–11] [12]

Inorganic aggregates Gold colloids (DLCA) Ludox silica spheres Polystyrene spheres Latex spheres with sodium chloride

— — — —

1.86–2.14 1.75–2.08 1.42–2.5 1.8–2.3

[21] [50] [118, 144] [98]

a

See Section 4.3.

ranges between 1.4 and 2.5 for aggregates produced in natural waters, sediments, soils and in vitro. If a system containing a number of aggregates is considered, the aggregate mass distribution is also expected to follow a power-law relationship: m ∼ l Dm

(4.2)

where l represents a characteristic length. It is important to note that the characteristic length l for the determination of Dm is not necessarily the maximum length of the aggregate. For example, the aggregate radius of gyration Rg is deﬁned by Rg2 =

N 1 (Rn − RG )2 N n=1

(4.3)

where RG corresponds to the position of the centre of mass of the aggregate (Figure 4.4), N is the total number of particles within the aggregate and Rn is the position of the particle n. The minimum or maximum aggregate diameter and the aggregate equivalent sphere area diameter can also be used as characteristic lengths to derive the aggregate fractal dimensions [16–18]. Because of the inhomogeneous distribution of particles within a fractal, an important feature of fractal aggregates is the apparent decrease in the mean aggregate density ρ with the increase of the size R of aggregates: ρ ∼ R Dm −3

(4.4)

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FRACTAL STRUCTURES AND MECHANISMS 105 104 103 2

〈Rg 〉

3D Dm = 2.03 ± 0.03

102 101 100 100

101

102 Aggregate Mass

103

104

Figure 4.4 Dependence of the mean square radius of gyration on the aggregate mass for a collection of reaction-limited cluster aggregation mechanism (RLCA) aggregates. Here, the scaling relationship Rg2 ∼ m2/Dm is veriﬁed.

This property is at odds with the ordinary concept of homogeneous density expected from the properties of porous or impermeable spheres and has important consequences for the hydrodynamic properties of fractal aggregates. Indeed, many previous studies based upon Stokes’ law have incorrectly calculated aggregate densities and porosities by considering aggregates as an assembly of permeable or impermeable spheres with a homogeneous distribution within the aggregate.

4.3 CLUSTER–CLUSTER AGGREGATION PROCESSES The cluster–cluster aggregation (CCA) model, was introduced simultaneously, but independently, in 1983 by Jullien and co-workers [19] and Meakin [20] as a realistic model to describe aggregation, such as for gold particles [21]. The CCA model begins with a collection of elementary spherical particles which are randomly distributed in a box. Particles are moved randomly in all directions in order to mimic Brownian motion (random walk with periodic boundary conditions). When two particles come into contact, they are assumed to stick together irreversibly and form a new rigid aggregate (dimer) that diffuses according to a diffusion coefﬁcient that is related to its size and geometry (small aggregates move faster than large aggregates). With time, larger, rigid aggregates are formed by the irreversible reactivity of the small clusters (Figure 4.5). Model iterations are stopped when only a single aggregate remains in the simulation box. Different extensions of the model have been developed to include intrinsic anisotropy, readjusting effects and polarizability [22].

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Figure 4.5 Four stages of an off-lattice diffusion-limited CCA process using 10 000 elementary spherical particles. In this simulation, the aggregate diffusion coefﬁcient is assumed to be controlled by the aggregate mass using D ∼ D0 m−1 where m represents the aggregate mass and D0 the diffusion coefﬁcient of a single particle. The aggregates generated by this model are self-similar with a fractal dimension of 1.8.

4.3.1 Particle–Particle Interactions If only the attractive van der Waals forces were operating on suspended particles in water, then one might expect the particles to stick together immediately and coagulate. Nonetheless, particles suspended in water or in solutions of high dielectric constant are usually charged due to the ionization or dissociation of surface sites or the adsorption of charged entities (multivalent ions, polyelectrolytes, humic or fulvic acids). Hence, particles generally undergo repulsive electrostatic forces originating from chemical reactions occurring at the particle–solution interface. Particle aggregation is expected to occur when the attractive van der Waals forces (which are always present) exceed the repulsive electrostatic interactions between the particles, as shown in Figure 4.6. The Derjarguin, Landau, Verwey, Overbeck (DLVO) theory [23–25] has established the potential energy–distance relationship between two particles as a function of the characteristics of both the particles and the suspending solution. In natural systems, this approach requires compilation [26–28] of the major key physicochemical parameters that characterize the colloid material, including: (a) colloid shapes

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Total Interaction Energy [kT]

250 200

Increasing salt concentration or decreasing surface potential

150 100

(a)

50

Electrostatic Repulsion

(b) Secondary Minimum

0 –50 van der Waals Attraction

(c)

–100 –150 –200

Primary Minimum

–250 0

5 10 15 20 25 30 35 Distance between particle surfaces [nm]

40

Figure 4.6 Calculated energy proﬁles of DLVO interactions as a function of the distance between particle surfaces. (a) Surfaces repel strongly; colloidal particles remain stable. (b) Surfaces come into stable equilibrium if the secondary minimum is deep enough; colloids then aggregate reversibly. (c) The interaction curve approaches the pure van der Waals curve and colloids coagulate irreversibly.

and sizes; (b) colloid surface potentials and/or surface charge densities characterizing the electrical double-layer repulsion; (c) Hamaker constants, which are proportional to the electronic densities of the interacting media and control the van der Waals attraction forces; and (d) the inverse Debye length, which characterizes the diffuse double-layer thickness and the range of the repulsive double-layer forces (given by the ionic strength of the solution). The combination of van der Waals and electrostatic forces usually results in two minima (primary and secondary minima) where aggregates are more or less stable. To achieve adhesive contact between two charged surfaces, surface potentials or charge densities must be ‘lowered’ into a primary minimum by screening charges by ion ‘clouds’ attracted close to the surface of oppositely charged particle surfaces. The attraction of two particles thus depends on the electrolyte concentration. Some energy proﬁles have been calculated in Figure 4.6 using the corresponding mathematical expression εaψ 2 Aa V = ln[1 + exp(−κh)] − (4.5) 2 12h (where h represents the surface–surface distance between the particles) and the following conditions: particle diameter, 2a = 1 m; Hamaker constant, A = 1.5×10−13 ergs; salt (NaCl) concentration, 0.001 mol dm−3 (curve a), 0.03 mol dm−3 (curves b and c); κ −1 = 9.58 × 10−7 cm; dielectric constant, ε = 78.54; particle surface potential, ψ = 35 mV (curve a), 35/30/25 mV (curves b), and 10 mV (curve c). For highly charged surfaces in dilute electrolyte, there is a strong electrostatic longrange repulsion (Figure 4.6, curve a). The corresponding energy barrier of several

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kB T units (1kB T = 0.4 × 10−20 J = 0.04 × 10−12 ergs) is too high to be overcome by the particles; consequently, they remain totally dispersed in the (stable) solution. As a result, when two particles move into a neighbouring position, the probability to form a physical or chemical bond (i.e. sticking probability) is practically zero. As the colloidal surface charge approaches zero (this may be achieved by adjusting the pH or by modifying the acid–base properties of the particle surface groups), or in more concentrated electrolyte solutions where charge screening is promoted, the interaction curve approaches the pure van der Waals curve. In this case, two particles now attract strongly and stick together irreversibly (Figure 4.6, curve c) in the ﬁrst primary minimum. The exact form of the DLVO potential, the inﬂuence of the DLVO forces on aggregation mechanisms and the structure of the resulting aggregates is still a major ﬁeld under investigation in colloidal chemistry. It should be noted here that the DLVO validity is, in principle, limited to large particles with small curvatures and to small separation distances between the two particles. 4.3.2 RLCA and DLCA Concepts Computer simulations have shown that the value of fractal dimension largely depends on whether the aggregation process is controlled by the diffusion rate of the clusters and single particles or by their chemical reactivity at the time of collision, the latter being mainly controlled by the DLVO forces. This observation, in agreement with experimental work on aerosols and colloids, has led to a new classiﬁcation of aggregation processes: the reaction-limited and diffusion-limited cluster aggregation (RLCA and DLCA respectively) processes. Key relationships exist between the kinetics of aggregate formation and the degree to which the resulting aggregates occupy the embedded space [28–34]. Several factors, such as the surface characteristics of the particles and the solution chemistry (ionic salt, pH), affect the fractal dimensions of aggregated material. If the rate of coagulation is limited by how quickly aggregates diffuse into contact, then a DLCA process is achieved. Then loose aggregates are formed with small fractal dimensions and the aggregate size-distribution is bell shaped. These structures can be explained by the fact that particles cannot penetrate the aggregate since they quickly attach to the particles with which they collide ﬁrst, i.e. on the aggregate exterior. In such a case, the sticking probability P (i, j ) (or adhesiveness) between two aggregates of sizes i and j to form a permanent bond is close to one. If thermal motion is not sufﬁcient to overcome the electrostatic repulsive barrier (which is typically of a few kB T in the case of partial screening), then particles interact with each other many times before being able to overcome the electrostatic barrier (Figure 4.6, curve b) so as to stick together irreversibly. In that case, the sticking probability P (i, j ) is somewhere between zero and one and an RLCA process is achieved. In addition, the resulting aggregates are more compact (due to the increased penetration of particles) and they have larger fractal dimensions (Figure 4.7). Owing

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FRACTAL STRUCTURES AND MECHANISMS

to the important size polydispersity of such systems, the size distribution curves are continuously decreasing functions with time.

Figure 4.7 DLCA (left) and RLCA (right) aggregates. Owing to the decrease of the sticking probability and increase of the penetration rate of the particles, the RLCA aggregate exhibits a more compact structure.

In aggregating systems, the bond formation between the colloids is described by the sticking probability factor [35, 36], which is experimentally deﬁned as the quotient of the kinetic constant for any given conditions divided by the kinetic constant for a rapid (diffusion-limited) aggregation. In rapid coagulation, P (i, j ) = 1, whereas lower values of P (i, j ) are obtained under conditions of slow coagulation. For two approaching particles, Fuchs [37] derived theoretically an expression (which was later improved by Honig et al. [38]) for P (i, j ): ∞ P (i, j ) = 2a 0

1 V (H ) dH exp (H + 2a)2 kB T

(4.6)

where H is the surface to surface distance of two approaching particles of radius a and V(H ) is the interaction potential, T the temperature and kB the Boltzmann constant. The general expression for V(H ) includes three contributions: electrostatic, attractive, and steric interactions. The last term is included to take into account the excluded volume of the colloids. DLCA and RLCA processes represent universality classes for homoaggregation (aggregation involving similar particles), i.e. the aggregates display some characteristic features for which the kinetics of particle coagulation and the fractal structure are independent of the details of the system. One line of evidence that supports this hypothesis is the observation that the fractal dimensions of aggregates formed by RLCA are remarkably constant, as are those formed by DLCA, for many different particle types [39, 40].

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4.3.3 Scaling Laws of Aggregate Formation The existence of scaling laws for the aggregation kinetics of colloidal particles has attracted considerable interest. These concepts have become a part of the description of systems far from equilibrium, since they allow a quantitative evaluation of the role of a number of physicochemical factors on aggregation kinetics. The formation of colloidal aggregates over time due to Brownian collisions was originally established by von Smoluchowski by considering [41] K(g,n)

[g] + [n] −−−→ [g + n]

(4.7)

which describes the formation of aggregates composed of g and n elementary particles, where K(g, n) represents the rate constant of the reaction. Swift and Friedlander [42] and Lushnikov [43] subsequently used von Smoluchowski’s kinetic equation in its integro-differential form to derive the scaling laws of aggregate formation in the presence of an excess of electrolyte sufﬁcient to ensure a regime of rapid aggregation: ∂c(g, t) = ∂t

g

∞ K(g − n, n)c(g − n, t)c(n, t) dn − 2c(g, t)

0

c(n, t)K(g, n) dn 0

(4.8) where c(g, t) gives the number of aggregates containing g associated primary particles at time t. The ﬁrst term on the right-hand side represents the increase in aggregates of size g due to the collision between aggregates of sizes (g − n) and n respectively, while the second term represents the decrease of aggregates of size g due to collisions between g-fold aggregates and aggregates of any sizes. The kernel K represents the collision frequency. Scaling theory demonstrates that Equation (4.8) is invariant under the following scalar transformation: g c , t = g02 c(g, g0 1−λ t) (4.9) g0 provided that K is constant or a homogeneous function of degree λ (of the form g + n or gn). In that case, the aggregate mass distribution c(g, t) can be represented in terms of a universal, time-invariant function ψ: c(g, t) = t −2/(1−λ) ψ[gt −1/(1−λ) ]

(4.10)

N0 (t) ∼ t −1/(1−λ)

(4.11)

with where N0 (t) is the moment of order zero of the distribution, i.e. ∞ N0 (t) =

c(g, t) dg 0

(4.12)

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which is equal to the total number of aggregates and elementary particle in the solution at a given time t. The function ψ does not depend explicitly on time and the ﬁrst moment of the distribution scales as t −w . w represents here a scaling exponent which depends on the aggregation efﬁciency and is equal to unity when it is maximum (λ = 0). Scaling theory can also be derived to explain the coagulation kinetics of Brownian particles in the presence of a low electrolyte concentration (RLCA conditions). With decreasing aggregation efﬁciency, w was found to become negative. This point can be explained by taking into account the probability of efﬁcient collisions between g- and n-fold aggregates which is proportional to a Boltzmann factor:

∞ p∼

exp

− Ge (x) dx kB T

(4.13)

0

where Ge (x) is the electrostatic free-energy of two charged particles at distance x. Equation (4.8) has also been solved numerically, and it was shown that Equation (4.10), the ‘self-preserving’ distribution, applies well after a sufﬁcient time lag [43, 44]. Equations (4.10) and (4.11) were also derived using Monte Carlo simulations [45–47] with a collision frequency K(g, n) expressed as K(g, n) =

kB T (Rg (g) + Rg (n))(D0 (g) + D0 (n)) 3η

(4.14)

where Rg (g), Rg (n), and D0 (g), and D0 (n) are respectively the radii of gyration and diffusion coefﬁcients of aggregates composed of g and n particles, and η is the solvent viscosity. The long-time behaviour of c(g, t) was found to scale like g c(g, t) ∼ t −w g −τ f z (4.15) t For diffusion coefﬁcients of the form Dg ∼ g γ with γ < 0, τ = 0 [48]; hence, Equations (4.10) and (4.15) are similar, with w = 2z and w = 2(1 − λ)−1 [49]. 4.3.4 Reversible Aggregation Process: Effects of Fragmentation and Restructuration In the original aggregation model, aggregates remain rigid with respect to their diffusive motion and do not rearrange themselves after sticking. Although many experimental systems can be described by irreversible and rigid bonds [21], there is experimental evidence, at least on small length scales, that other phenomena are expected to play a role in nature. For example, under certain experimental conditions, aggregates obtained by a DLCA mechanism can restructure to more compact aggregates after a certain time. This behaviour has been observed in diluted colloidal systems with small gold or silica particles in which the value of the fractal dimension increases from 1.75 to 2.1 [50] or to 2.4 [51].

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In practice, restructuring [52, 53] or aggregate fragmentation [54–56] is always present to some degree [57]. Compact aggregates result from the existence of combined aggregation and fragmentation processes, followed by internal reorganization. If we take into account all the detailed features of the interaction forces between particles, such mechanisms are relatively difﬁcult to simulate in a simple way. Some critical questions then arise. How do the structures of the aggregates change with time? Do these aggregates remain fractal objects during restructuring? If so, how does the fractal dimension Dm change with time? In order to simulate such processes, several modiﬁcations have been proposed to the fundamental aggregation models. For example, in two-dimensional modelling, some readjustments can been made to allow restructuring of the initial particle aggregates [58]. Compactiﬁcation was clearly visible at small scales but surprisingly less visible when extrapolated to the inﬁnite aggregate sizes. The fractal dimension was estimated to be 1.48, i.e. only very slightly larger than the 1.44 that was observed without restructuring. For a three-dimensional aggregation model, Meakin and Jullien [59] introduced restructuring by taking into account rotations of the rigid clusters about their contacting particles. In that case, restructuring was more efﬁcient, increasing the DLCA fractal dimension to about 2.2. It should be noted here that such a structural readjustment only increases the number of contacts between aggregates. As a result, the probability of reorganization is limited due to the irreversible nature of the clusters that are formed. Fragmentation allows the particles, or aggregates, to escape after sticking. This effect is different from that obtained by simply considering a sticking probability. In this case, some reversibility is allowed, which can be chemically justiﬁed if the attractive secondary minimum of the DLVO potential is sufﬁciently weak. In von Smoluchowski’s equation, it was assumed that aggregation was irreversible. Von Smoluchowski’s equation can, however, be extended to include fragmentation processes: d 1 K(i, j )Ci Cj − K(g, j )Cj Cg + F (j, g)Cj +g (Cg ) = dt 2 i+j =g j =1 j =1 −

1 F (i, j )Cg 2 i+j =g

(4.16)

where K(i, j ) is the element of the reaction kernel that describes the aggregation process and F (i, j ) is an element of the fragmentation kernel [54] which depends on the cluster dissociation constant kd : F (i, j ) = kd φ[i, j ]

(4.17)

where φ is a function describing the dependence of the fragmentation rate on the aggregate sizes i and j .

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In order to simulate the restructuring observed by experimentation [60], Meakin [61] and Kolb [62] considered a reversible model by modifying the original DLCA model by including random bond breaking. Although in the irreversible model [59] the effect on the fractal dimension was quite small, the fractal dimension value in the three-dimensional and two-dimensional reversible models was found to increase to 2.03 and 1.57 respectively at dynamic equilibrium. No change of Dm was observed with time. To construct a more realistic reversible aggregation model involving the rearrangement of particles according to energetic considerations, the CCA model can be modiﬁed with a ﬁnite interparticle attraction energy E [63–66]. Shih, Aksay and Kikuchi (SAK) [63] built a lattice reversible growth model in which the nearest neighbours have a ﬁnite attraction energy. Here, the bond breaking is controlled by a Boltzmann factor rather than being random as in the previous model. From the SAK model, it was demonstrated that the aggregates can remain fractal during restructuring, with values ranging from 1.4 to 2.0 in d = 2, depending on the interparticle bonding energy and time. The interparticle interaction can be described by a square-well potential: ⎧ rδ where d is the diameter of the colloidal particle, δ the attraction range and ε the depth of the interaction potential. Any attempted colloidal movements give a variation of energy E(r), whose probability p is proportional to the exponential E(r) p = exp − (4.19) kB T where E is the energy change due to the movement. By changing ε, the aggregation behaviour can be changed over a wide range from ramiﬁed aggregates to compact ones. Any attempted movement consistent with the nonoverlapping restriction is accepted by considering the Metropolis rule [64]: γ < exp(−E(r)/kB T ), where γ is a random number uniformly distributed between zero and unity. According to this condition, the process evolves until a thermal equilibrium state is reached (see Figure 4.8). This model shows that interparticle attractions play an important role in the aggregation process. Indeed, allowing the particles to rearrange themselves within the aggregates or even break from the aggregates encourages compaction of aggregates, and may also determine the saturated aggregate size. To illustrate the effect of restructuring, Figure 4.9 shows images obtained at the same simulation time which correspond to off-lattice reversible aggregation processes at four different values of ε. In these simulations, a ramiﬁed DLCA aggregate is considered as the initial conﬁguration.

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0

ε = 0.5 kT

Total Energy

–500

ε = 1 kT

–1000 –1500 –2000 –2500

ε = 2 kT –3000 0

2000 4000 Monte Carlo Steps

6000

Figure 4.8 Variation of the total energy with time (considered as Monte Carlo steps) for three different values of ε = 0.5, 1 and 2kB T at φ = 0.01 (particle volume fraction). According to the metropolis condition, the system approaches an equilibrium state after a certain simulation time in all three cases. Diez-Orrite, S., Stoll, S. and Schurtenberger, P. (2005). Study of aggregate formation in colloidal systems: off-lattice Monte Carlo simulations, Soft Matter, 1, 364–371. Reproduced by permission of the Royal Society of Chemistry.

When ε = 0.5kB T , the initial DLCA aggregate is broken into very small aggregates (frequently dimers) as well as free colloidal particles. At this value of bond energy, large aggregates form rarely. These conditions correspond to experimental temperatures that are sufﬁciently high such that the colloidal particles have high probability to break from the aggregates due to thermal motion. An increase of the interparticle energy promotes aggregation between colloidal particles. Simultaneously, the fragmentation process continues to produce free particles in the liquid phase. The combination of both mechanisms makes the aggregates more compact than the initial DLCA aggregate (see the largest aggregate obtained at ε = 2.5kB T and 3kB T ), suggesting the presence of phase separation in the system [67–69]. At very high values of bond energy (ε = 5kB T ), aggregation occurs at a faster rate than fragmentation, which makes the aggregates more ramiﬁed than aggregates obtained with less bond energy at the same simulation times. However, owing to the ﬁnite scale of the attractive interaction, reorganization within the aggregate is possible, giving an aggregate structure that is more compact at short length scales than the initial DLCA one. This suggests a Dm value slightly higher than 1.75. At a given particle concentration, the aggregate size decreases with bond energy because fragmentation is promoted. In Figure 4.10, the equilibrium aggregate size is plotted as a function of ε. The exponential decrease of the maximum aggregate size with

126

FRACTAL STRUCTURES AND MECHANISMS Initial DLCA aggregate (starting configuration)

(a)

(b)

(c)

(d)

Equilibrated Conformations

Figure 4.9 Inﬂuence of ε, the interaction potential value between particles, in the aggregation process: (a) ε = 0.5kB T , (b) ε = 2.5kB T , (c) ε = 3kB T and (d) ε = 5kB T . The particle volume fraction is equal to φ = 0.01. The balance between aggregation, fragmentation and restructuration controls the particle dispersion and aggregate structure. Diez-Orrite, S., Stoll, S. and Schurtenberger, P. (2005). Study of aggregate formation in colloidal systems: off-lattice Monte Carlo simulations, Soft Matter, 1, 364–371. Reproduced by permission of the Royal Society of Chemistry.

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N max

100

10

0.5

1.0

ε

1.5

2.0

Figure 4.10 Average maximum mass aggregate N max as a function of the interparticle interaction ε at a volume concentration φ = 0.05.

the decrease of ε shows that aggregates cannot grow indeﬁnitely in the aggregation process, but instead they reach a steady-state distribution of aggregate structure (see Figure 4.11). This is the main difference with the aggregates obtained by the DLCA model, for which the simulation was stopped once a single aggregate was obtained. In order to investigate the increase of Dm with time, the aggregation process can be started with aggregates of various sizes formed by the DLCA mechanism at a 10000

1000

DLCA

ε = 10 kT

ε = 3 kT

100

ε = 2.25 kT ε = 2 kT

10

Saturated cluster size

1

ε = 0.5 kT 100

10000 MC steps

Figure 4.11 Variation of the mean aggregate size (nn : number of elementary units) with the simulation time (Monte Carlo

i steps) at φ = 0.05. The mean aggregate size is deﬁned as nn = M1 /M0 , where Mi = n Nn is the ith-order moment of the size distribution [69]. Diez-Orrite, S., Stoll, S. and Schurtenberger, P. (2005). Study of aggregate formation in colloidal systems: off-lattice Monte Carlo simulations, Soft Matter, 1, 364–371. Reproduced by permission of the Royal Society of Chemistry.

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given particle concentration. From Shih et al. [63], aggregates can be considered as fractal during the reversible aggregation process. Therefore, the fractal dimension corresponding to different times of the aggregation process can be calculated from a log–log plot of aggregate mass against the radius of gyration (see Figure 4.12).

t=0 t = 800 t = 3900

N

100

10

Figure 4.12 Plot of the aggregate size N versus the mean gyration radius Rg for three different moments of the aggregation process. System conditions: φ = 0.01 and ε = 3kB T . Each point is the result of averaging over 10 samples. Diez-Orrite, S., Stoll, S. and Schurtenberger, P. (2005). Study of aggregate formation in colloidal systems: off-lattice Monte Carlo simulations, Soft Matter, 1, 364–371. Reproduced by permission of the Royal Society of Chemistry.

The value of Dm increases from 1.77 at the beginning of the simulation process (t = 0, where t is measured in Monte Carlo steps) to 2.03 at t = 3900 during the aggregation process, clearly showing how the aggregate structure becomes more compact as aggregation proceeds. 4.3.5 Heterocoagulation Processes Almost all simulations of nonequilibrium growth and aggregation (as reported above) are carried out under the assumption that the elementary particles are identical (homoaggregation). However, heteroaggregation involving different constituents is a much more common situation in natural systems (multicompoment systems). Both attractive and repulsive interactions can be present during heteroaggregation [70], e.g. when oppositely charged particles such as clay mineral and iron oxide or silica and alumina mixtures are considered [71, 72]. Such systems are too complicated to allow for the development of exact analytical theories, but computer simulations based on the concepts presented in Sections 4.1–4.3 can help to understand their behaviour. For example, simulations of the coagulation of a system made up of dissimilar A and B particles have been reported [73–76] and compared with experiment. The structures

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and growth kinetics of clusters formed by both A and B particles having variable sticking probabilities, and the inﬂuence of their relative concentrations, have been investigated in order to determine the inﬂuence of the unsuccessful A–A and B–B collisions on the rate of A–B aggregation, and to determine optimal coagulation conditions. It has also been shown that heterocoagulation was well described by size distribution scaling laws indicating that dynamic scaling is applicable over a wide range of coagulation processes. Aggregates produced by the heteroaggregation of A and B particles show a dependence of the fractal structures on the relative concentrations of the components. Figure 4.13 presents a series of images of heteroaggregates formed at equilibrium with identical elementary units (A and B) but at different relative concentrations. The particle number ratio of A and B had a dramatic effect on the ﬁnal heteroaggregate

A

B

C

D

Figure 4.13 Simulations relative to the coagulation of a three-dimensional system made up of dissimilar A (dark spheres) and B (bright spheres) particles. The inﬂuence of the unsuccessful A–A and B–B collisions on the rate of A–B aggregation and aggregate structures is illustrated here. The number of A particles was set to a constant value (Na = 1000) whereas the number of B particles Nb was adjusted to (a) 100, (b) 500, (c) 1000, (d) 3500. The fast coagulation regime corresponds to Na = Nb and the formation of DLCA-type structures. Less reactive and more compact structures are obtained when Na = Nb .

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sizes and structures. At poor or high number ratios, aggregation is terminated at early stages. Initial aggregate sizes are small and the particles which are in excess rapidly saturate the aggregates. For optimal number ratios, heteroaggregates are large and rapidly formed. By considering all the particle positions (A and B), the fractal dimension is expected to decrease, reﬂecting a transition from the RLCA to the DLCA regime (Figure 4.14) and then increase again when the number ratio increases. This behaviour is mainly a geometric effect arising from the different initial particle concentrations. It should be noted that heteroaggregates, as well as the underlying structures composed only of A or only of B particles, exhibit self-similar fractal geometry.

Fractal dimension Dm

2.4 2.2

RLCA Regime

2.0 DLCA Regime 1.8 1.6 0.01

A

B 0.1

1 10 Particle Number Ratio

100

Figure 4.14 Heteroaggregate fractal dimensions versus the particle number ratio Nb /Na . On the one hand, the fractal dimension is calculated by considering only A particles, whereas on the other hand it is calculated by considering only B particles.

4.4 BRIDGING FLOCCULATION PROCESSES It is well known that polymers may serve as bridges between colloidal particles to form ﬂocs; nonetheless, very little quantitative information is available about their structure and formation, despite the fact that they play key roles in environmental systems [1]. Particles may not only be bridged by polymers, but may also facilitate the formation of larger aggregates due to the adsorption of several polymer segments on the same particles. This process can be seen as an example of the CCA model, where polymer conformation, reactivity and total length play important roles. Computer models once again constitute a valuable tool that allows for predictions of ﬂocculation processes. 4.4.1 Model This section describes a three-dimensional model to mimic bridging ﬂocculation in which the size of the polymer chains is much larger than that of the interacting particles (a situation often encountered in natural waters).

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Off-lattice simulations were carried out in which colloidal particles and polymer chains were conﬁned to a cubic cell with normal periodic boundary conditions allowing them to pass freely through the periodic walls. The chains were represented at a mesoscopic scale by jointed spherical segments (pearl necklace model), and the relative sizes of chains and particles were controlled by adjusting the diameter of the colloidal particles and the length of the polymer chains through the number and size of the segments. In order to take into account a wide range of structural conﬁgurations of the chains and to adjust the polymer persistence length, from coils to rigid rods, angular constraints were imposed between the connected segments. Particles, chains, aggregates and ﬂocs were allowed to move in space according to their respective diffusion coefﬁcients recomputed at each step from their masses and conformations. Brownian motion was represented by random walks. The number of random walks during a unit of (relative) time was proportional to the diffusion coefﬁcients of the moving entities. The link between the relative and physical time was made by correlating the mean displacement of a reference particle (not participating in the aggregation process) and its diffusion coefﬁcients.

4.4.2 Kinetics and the Scaling Laws of Bridging Flocculation Both computational [30, 31] and experimental [77–79] results indicate that the aggregation kinetics of colloids, in the absence of large biopolymer chains, can be described by power laws over long periods of time and over large aggregate size ranges. The power laws are also valid for heteroaggregation processes involving dissimilar particles [74]. Bridging ﬂocculation involves a much more complicated system than the homoaggregation of spherical nondeformable colloids. Nonetheless, the application of similar scaling laws to bridging ﬂocculation can be justiﬁed by the fact that, after an initial period, for which the duration is a function of the particle–biopolymer concentration ratio, the solution contains very few, if any, free particles and bioploymers (Figure 4.15). Only ﬂocs and biopolymer–particle complexes remain in the simulation box. Following aggregation, the system behaves as if it were formed of only a single component with a sticking probability that is a function of the particle/biopolymer concentration ratio. This is an important ﬁnding, since it helps to explain why scaling laws used for single-component systems are valid for systems containing two (or more) components. Scaling laws for the aggregation of a single component system are usually described by the analysis of the temporal variation of the number N (t) and weight S(t) average aggregate sizes, which are calculated from the size distributions. They are deﬁned as N (t) = n Cn (t) Cn (t) (4.20) n

S(t) =

n

n

n2 Cn (t)

n

n Cn (t)

(4.21)

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Figure 4.15 Transmission electron micrographs of hematite in the presence of schizophyllan under conditions of rapid kinetics. Schizophyllan/hematite ratio 3.79, pH 4.5, I = 1 mmol dm−3 . The three-dimensional ﬂoc obtained by computer simulation (optimum conditions) explains the presence of a local linear order observed in the hematite ﬂoc due to the presence of rigid chains.

where Cn (t) is the concentration of ﬂocs composed of n elementary biopolymer chains at time t and C1 (t) represents the total concentration of the single biopolymer chains. From the size distribution histograms, S(t) and N (t) values can be calculated for various situations and dynamic scaling laws can be obtained by using the following relationships: N (t) ∼ t w

(4.22)

S(t) ∼ t z

(4.23)

where z and w are scaling exponents. The variations of S(t) and N (t) versus time are usually presented as log–log plots. The z and w values give a quantitative description of the ﬂocculation processes under the conditions of the space dimension d, the biopolymer persistence length or rigidity lp , biopolymer size lc and particle/biopolymer concentration ratio x. Hence z(d, lp , lc , x) and w(d, lp , lc , x) values can be calculated for various systems [80], and comparison between experiments and computer models to isolate the key parameters controlling both structure and kinetics is possible upon parameterization of the model with experimental data. For example, the formation kinetics and structure of hematite ﬂocs formed in the presence of a large rigid extra-cellular polysaccharide, schizophyllan, was investigated both experimentally and theoretically. Transmission electron microscopy (TEM),

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atomic force microscopy (AFM) and photon correlation spectroscopy (PCS) were used to characterize hematite particles and schizophyllan biopolymers and mixed ﬂocs. The evolution of the ﬂoc sizes was followed and ﬂoc fractal dimensions were calculated. Interpretation of the experimental results provided information on the inﬂuence of various factors on the optimal concentration ratio of schizophyllan biopolymer/hematite for the rapid formation of ﬂocs. Both experiments and computer simulations demonstrated the fractal character of the mixed ﬂocs. The optimal schizophyllan biopolymer/hematite concentration ratio obtained by simulation was smaller than that observed in laboratory experiments. The shift in the optimal dose was mainly attributed to a higher than predicted afﬁnity of hematite for the schizophyllan aggregates present in the initial solution in addition to the presence of a large proportion of chains that did not participate in the ﬂocculation process [12]. 4.4.3 Fractal Dimensions of Flocs Physical properties, such as ﬂoc densities, settling velocities or the diffusion of compounds inside aggregates and ﬂocs, depend on their fractal dimensions, which have to be calculated. The ﬂoc fractal dimensions can be determined directly from simulation, since each monomer and particle position is repeatedly computed during a simulation, allowing the mathematical deﬁnition of a fractal to be applied directly. The results discussed here have been obtained using a given ‘sticking probability’ set to allow both homoaggregation among particles and the ﬂocculation between particles and polymers. The effect of the polymer/particle concentration ratio was veriﬁed [81]. High rigid polymer chain concentrations lead to the formation of a polymeric network in which the particles were regularly distributed. In the presence of coiled polymer chains, ﬂoc structure resulted primarily from the association of particle aggregates and coiled polymers, all interacting via a classical CCA process that considers spherical units. One important conclusion can be made: the spatial disposition of particles in ﬂocs results from the biopolymer/particle concentration ratio in addition to the biopolymer conformations. In particular, ﬂocculation processes with rigid biopolymers resulted in the formation of a regular network characterized by fractal dimensions that were higher than those obtained on the basis of the classical DLCA or RLCA models (Figure 4.16). Despite the highly loose structure of the aggregate that was formed, the increase in fractal dimension reﬂected the high order of particles in such networks. A more reﬁned Monte Carlo ﬂocculation model was considered, based on energetic considerations rather than sticking probability factors, and included charged polymers (polyelectrolytes), the effect of solution pH, and the rearrangement of biopolymers at the particle surface. This model is presented in the next section, where the formation of a simple polymer–particle entity is examined, due to the required computational time.

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Mean Square Radius of Gyration

107 106 105 104

Dm = 2.52 ± 0.04

103 102 Dm = 2.03 ± 0.04

101 100

101

102 Aggregate Particle Number

103

Figure 4.16 Mean square radius of gyration versus the aggregate particle number. Open squares represent aggregates formed without the presence of rigid polymer chains. Black squares correspond to aggregates formed with the presence of rigid polymer chains (particle chain concentration ratio equal to 12.5).

4.4.4 Monte Carlo Simulations: Charged Polymer Chain Adsorption on a Colloidal Particle Owing to their important potential modiﬁcations of solution properties, mixtures of charged biopolymers (such as DNA and polysaccharides) and oppositely charged colloids (such as inorganic particles, globular proteins and micelles) have been followed with great interest in the ﬁelds of biology [82–84], wastewater treatment and industrial applications [83]. For example, the long-range attractive and/or repulsive character of the electrostatic interactions between polyelectrolytes and colloids, the role of solution chemistry, the geometry and concentration of polyelectrolytes and colloids and charge inversion are speciﬁc properties of these systems that are only partially understood. In order to better understand how polyelectrolytes (ﬂexible, semi-ﬂexible and rigid) interact with charged colloidal particles, a Monte Carlo approach is brieﬂy described here [86–94]. Owing to screening effects, the role of ionic strength is discussed, since it will play a key role, in controlling both chain conformations, via the electrostatic persistence length, and the interaction energy of polyelectrolytes with the particles. The total energy Etot (kB T units) for a given conformation is described as the sum of (i) repulsive electrostatic interactions between monomers, (ii) attractive electrostatic interactions between the chain and the particle, Eel and Etor , and (iii) the chain stiffness or chain bending energy. Hard core interactions Eev are also considered to include both monomer and particle excluded volumes.

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All pairs of charged monomers within the polyelectrolyte interact via a screened Debye–Hückel long-range potential: uel (rij ) =

zi zj e2 exp(−κrij ) 4πεr ε0 rij

(4.24)

where zi represents the amount of charge on a unit i. Monomers interact with the particle according to the Verwey–Overbeek potential: u el (rij ) =

zi zj e2 exp[−κ(rij − σp )] 4πεr ε0 rij 1 + κσp

(4.25)

Free ions are usually not included explicitly in the simulations, but their overall effects on monomer–monomer and monomer–particle interactions are described via the dependence of the inverse Debye screening length κ 2 (m−2 ) on the electrolyte concentration according to z 2 Ci i (4.26) κ 2 = 1000e2 NA ε ε 0 r kB T i The intrinsic chain stiffness is adjusted by a square potential with variable amplitude in order to vary its strength. This gives the bending energy: Etor =

N

kang (αi − α0 )2

(4.27)

i=2

where α0 = 180◦ , αi represents the angle achieved by three consecutive monomers i − 1, i, and i + 1 and kang (kB T /deg2 ) deﬁnes the strength of the angular potential or chain stiffness. Monte Carlo simulations are usually performed according to the Metropolis algorithm in the canonical ensemble. In this method, successive ‘trial’chain conﬁgurations are generated to obtain a reasonable sampling of low energy conformations [95]. After applying elementary movements which are randomly selected, the Metropolis selection criterion is employed either to select or reject the move. If the change in energy E resulting from the move is negative, then the move is selected. If Etot is positive, the Boltzmann factor p −Etot (4.28) p = exp kB T is computed and a random number rand (with 0 ≤ rand ≤ 1) is generated. If rand ≤ p, then the movement is selected. When rand > p, the trial conﬁguration is rejected and the previous conﬁguration is retained and considered as a ‘new’ state in calculating ensemble averages. The monomer distribution at the particle surface is largely controlled by the value of kang . When the chain ﬂexibility is important (kang ≤ 0.001kB T /deg2 ), ‘tennis ball’ conformations are achieved, whereas when rigid chains are considered

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FRACTAL STRUCTURES AND MECHANISMS

(kang > 0.001kB T /deg2 ) the intrinsic ﬂexibility forces the polyelectrolyte to adopt solenoid conformations, as predicted by the analytical model of Nguyen and Shklovskii [96]. Both the strong electrostatic repulsions between the consecutive turns and intrinsic chain rigidity keep the turns parallel to each other and maintain a constant distance between them. When Ci > 0.01 mol dm−3 , large changes in the chain dimensions are observed with increasing the chain stiffness. As long as kang ≤ 0.001kB T /deg2 , loops and a tail are formed (Table 4.2; Ci = 0.1 mol dm−3 and kang = 0kB T /deg2 ), resulting in an increased thickness of the adsorption layer. When kang ≥ 0.01kB T /deg2 , and with increasing Ci , the polyelectrolyte starts to leave the surface by unwinding. Extended tails in solution are formed concomitantly with a decrease in the number of turns of the solenoid and monomers in trains (Table 4.2; Ci = 0.1 mol dm−3 and kang = 0.02kB T /deg2 ). By increasing further the ionic strength or the intrinsic ﬂexibility of the chains, the polyelectrolyte becomes tangent to the particle surface with dimensions close to its free unperturbed dimensions. It must be noted that the adsorption of the charged polymers is not only controlled by the ionic Table 4.2 Equilibrated conformations from Monte Carlo simulations of complexes composed of a semi-ﬂexible polyelectrolyte and a single colloidal particle as a function of the solution ionic strength I and polyelectrolyte intrinsic rigidity kang . By increasing the chain stiffness, solenoid conformations are progressively achieved at the particle surface, whereas an increase in ionic strength leads to the desorption of the polyelectrolyte.

kang (kBT/deg2)

I (mol dm−3) 0.01

0

0.001

0.02

0.1

1

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strength, but also by the particle diameter. Surface curvature effects will limit the amount of adsorbed monomers. Large particles allow the polyelectrolyte to be spread over the surface; small particles limit the number of adsorbed monomers. Monte Carlo simulations provide a rewarding and invaluable approach to solving these systems, and computer simulations and theory can isolate the molecular factors that control polyelectrolyte conformations in solution. Therefore, they are extremely useful to address the optimization of colloid–polymer mixtures and guide the design of new experiments. A simple model involving one chain interacting with one particle has been described, but the same model can be extended to more concentrated systems, e.g. involving several chains (and/or colloidal particles) with explicit counter ions, co-ions and solvent molecules.

4.5 SETTLING VELOCITIES OF FRACTAL OBJECTS The sedimentation of ﬂocculated material and aggregate formation are amongst the most important processes, not only for the rational design of water treatment processes, but also for prediction of the diffusion of suspended matter and particle residence times in aquatic systems. Nonetheless, most current mathematical models used to simulate the circulation of trace compounds either do not take into account coagulation–sedimentation processes or, when they are considered, the coagulating material is described in a simplistic manner, most often as impermeable spheres that obey von Smoluchowski’s theory and Stokes’ law [41, 97]. In contrast, experimental data for settling rates of aggregates 2R, i.e. the diameter of the aggregate. This chapter is aimed at illustrating the mechanisms of fractal aggregate formation in environmental systems, including rivers, lakes, soils and industrial systems (e.g. water and wastewater treatment plants). In these systems, the formation of fractal aggregates is controlled by different mechanisms that must be analyzed at different length scales that range from nano- to meso- and macro-scales. In addition, when the subunits are in the submicrometer size range, structure analysis must also take into account additional aspects such as multiple scattering. Two examples have been selected to illustrate these systems. The ﬁrst examines aggregate formation by hydrolysis of Fe(III) ions and the role of ‘strong’ and ‘weak’ ligands in orienting aggregate structure at microscales. A special investigation of the nanoscale–mesoscale relationship is also developed. The second example addresses the formation of very large aggregates that originate from micrometer-sized subunits. These aggregates are commonly formed in soils and natural water bodies, and in industrial water and wastewater facilities.

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5.2 TECHNIQUES 5.2.1 Small-Angle X-Ray Scattering Small-angle X-ray scattering (SAXS) has been shown to be the best tool to investigate aggregate size and structure parameters for suspensions of very small colloids. At the largest values of the scattering vector q, information on the shape, size and polydispersity of the subunits/nuclei can be obtained. In the case of fractal aggregates, Df can be easily derived from the scattering intensity I (q), according to I (q) = F (q)S(q)

(5.3)

where F (q), called the form factor, is the scattering by the subunits/nuclei, and S(q) is the interference function describing the arrangement of subunits within the aggregate. For a fractal aggregate, S(q) scales as q −Df . Thus, a rigorous determination of Df implies that the subunits are well characterized. Another parameter of structural interest that can be derived from the SAXS curve is the pair distribution function, P (r). It can be obtained from the correlation function γ (r), which is calculated from the Fourier transform of IN (q) of the normalized intensity: 2 sin(qr) γ (r) = dq (5.4) q 2 IN (q) qr 2 In this case, P (r) = r 2 γ (r). The form of P (r) provides information on the size and shape of the aggregate and the coordination number of the particles in the aggregates. 5.2.2 Light Scattering 5.2.2.1 Homogeneous particles Dynamic light scattering allows measurement of the translational diffusion coefﬁcient DT of particles. For noninteracting particles, the ﬁrst-order autocorrelation function g(r) is directly proportional to DT . Polydisperse systems are analyzed using the method of cumulants, in which a distribution of exponential terms C(r) is assumed: C(r) = F () exp(−r) d

(5.5)

where = DT q , q = 4πn sin(θ/2)/λ, θ is the scattering angle, n is the optical index and λ is the wavelength of the incident beam. The function F () is expanded in a power series of r. The ﬁrst moment (cumulant) in the expansion is the average decay constant avg , which deﬁnes an effective diffusion coefﬁcient DT for the particle size distribution. DT is converted to the effective hydrodynamic radius rH using the Stokes–Einstein relationship. 2

5.2.2.2 Fractal aggregates If the size of the subunits is less than λ/20, then the fractal dimension Df can be calculated using the same law as for SAXS. In this case, Df is deﬁned in an angular

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domain which varies from the size of the subunit to the size of the aggregate. However, if the size of the subunits is not very different from λ, then multiple scattering cannot be neglected. With respect to this point, Botet et al. [19] recently developed a new scattering model, called the mean-ﬁeld approximation, which accounts for multiple scattering where the size of the scatterer is close to the scattering wavelength. These authors analyzed the multiple scattering results of Xu [20] by mean-ﬁeld theory using Mie theory for fractal aggregates. Mean-ﬁeld scattering provides solutions that are able to describe multiple scattering far from the optical resonance where the Rayleigh approximation is not consistent. An extension of this approach takes multiple scattering into account by introducing a new parameter, the mean optical index n [21]. This term characterizes the mean optical environment of scatterers belonging to a fractal object. In fractal aggregates, the optical contrast is deﬁned as the optical contrast between scatterers ns and their mean environment n, with the scatterers’ mean optical contrast denoted as ns /n. The form factor F (q) is thus deﬁned by this new optical contrast. On the basis of Equation (5.3), the scattered intensity I (q) depends on the mean optical contrast of the scatterer. Therefore, I (q) contains structural information not only through S(q), but also through the mean optical contrast that depends on the aggregate structure. In this case, the form factor F (q) can be calculated using the approximation Iscatt (q) ∝ I rs ,ns /n (q)S(q)

(5.6)

where I rs ,ns /n (q) is the Mie angular scattered intensity of a primary particle belonging to a fractal geometry characterized by a mean optical index n which follows the mean-ﬁeld approximation. The initial optical contrast m = ns /nm is replaced by the scatterers’ mean optical contrast ns /n, deﬁned with respect to the mean environment of the scatterers inside the aggregate. (1) The electromagnetic mean-ﬁeld has been expressed by the coefﬁcients d 1,n and (2)

d 1,n , used as the renormalized Mie coefﬁcients an and bn [19–21]. These new coefﬁcients express the mean ﬁeld scattered by a particle within an aggregate. They are independent of the position of the particle inside the aggregate and take into account the mean environment of the scatterers in a fractal aggregate. Far from the aggregate, each particle of a given cluster is assumed to radiate the same electromagnetic ﬁeld, (1) (2) except for its phase. The coefﬁcients d 1,n and d 1,n are expressed in terms of fractal and optical characteristics (an , bn ) using the notations of Botet et al. [19]: (1)

d 1,n =

an ∞ 1 + 2L (2ν + 1)an

(5.7)

ν=1 (2)

d 1,n =

bn ∞ 1 + 2L (2ν + 1)bn ν=1

(5.8)

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where L=

153

c2/Df [1 − (2/Df )] N 4 (kRg )2

k is the wave number, ν is a positive integer, Rg , is the radius of gyration, N is the number of subunits inside the aggregate, and the fractal dimension of the aggregate is greater than two. The parameters Df , Rg , and N are directly related to the optical parameters.

5.3 NUCLEATION–PRECIPITATION OF FRESH IRON OXYHYDROXIDES Over the years, the precipitation of iron oxides and hydroxides from acidic solutions has received considerable attention in various ﬁelds, such as catalyst synthesis, environmental sciences and industrial processes [e.g. 22–29]. Freshly formed iron hydroxide particles help to control pollution in aquatic systems, e.g. by ﬁxation and transport of phosphates, heavy metals and other reactive inorganic and/or organic species [30–33]. The high reactivity of these iron phases is mainly due to their small size. The formation and aggregation of iron colloids, which occurs in continental and marine aquatic systems [22, 34–38], is also employed in water and wastewater treatments [35, 39, 40]. In aquatic systems, Fe(III)-aquo ions may polymerize by deprotonation to form sparingly soluble colloidal iron oxyhydroxides. Although the crystalline end-products of this reaction, such as goethite (-FeOOH) and hematite (-Fe2 O3 ), are well characterized, the structure and shape of the intermediate phases are far from being fully elucidated. The shape and ‘porosity’ of freshly formed iron colloids and aggregates control their capacity to bind and transport a high number of elements. In some cases, e.g. where the size of the particles is not too small, a fractal approach can be useful to determine the reactivity of the iron colloids and aggregates and to explain their high speciﬁc surface and roughness. This approach, however, must be combined with a molecular-scale characterization, since these objects are very small and are not always geometrically self-similar. General models of nucleation have been proposed [41, 42] based on the concept of similarity between the structure of polymers and the structure of the resulting crystalline phases. Nonetheless, these models have failed to predict the precipitation processes correctly. A generalized approach based on the electrostatic ﬁeld theory [25, 43] was able to provide hints for understanding the formation of small clusters but could not be applied to larger ones. 5.3.1 Nucleation Several pioneering studies [41, 42, 44, 45] have developed the general concepts of polymerization, growth and precipitation of Fe(III) ions, but the intermediate

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polymers and clusters have not been identiﬁed experimentally. Structural information was based on the modeling of potentiometric data, not on the observation of the objects, which became possible later by the use of X-ray absorption spectroscopy (XAS) and SAXS [39, 40, 46–51]. The structure and size of iron polymers depends on the hydrolysis ratio n = OH/Fe, time t, and nature of the anions. At low hydrolysis ratios, two Fe monomers react to form a dimer in which both octahedra share an edge. Mössbauer spectroscopy has shown that, at n < 0.5, hydrolysis of Fe(NO3 )3 at 92 ◦ C leads to the formation of Fe2 (OH)4+ 2 [52]. Extended edge X-ray absorption ﬁne structure (EXAFS) spectroscopy has revealed the presence of a dimer formed from FeCl3 at n = 1.5 and t = 10 min [50], with the two Fe at a distance of 3.01 Å, corresponding to Fe octahedral edge sharing [53]. The dimer could be stabilized by organic ligands such as carboxylates and proteins [45], whereas the growth of Fe species in the absence of ‘stabilizers’ continues with the formation of a trimer (Figure 5.1) [50].

A)

B)

Figure 5.1 Structure of a planar trimer (A) and a double corner trimer (B) of Fe3 (OH)5+ 4 .

In addition to the trimers, in Fe(NO3 )3 solution, for n = 1.5 and t > 1 h, small polymers are detected with Fe–Fe interatomic distances of 2.85 Å, 3.06 Å, 3.52 Å and 3.95 Å, which correspond respectively to face, edge, double corner and single corner sharing between Fe octahedra [51, 53]. The unusual Fe–Fe distance of 2.85 Å, which is characteristic of Fe octahedra with face sharing [47, 53], has also been observed in freshly formed two-line ferrihydrite [54]. When the Fe concentration is in the decimolar range, XAS and SAXS data can be reconciled with models taking into account a limited number of Fe clusters (Figure 5.2). In more dilute solutions ([Fe] < 10−4 m), the number of edge- and corner-sharing neighbors at pH ∼ 2.8 suggests the formation of small Fe polymers that most likely consist of a mixture of dimers, trimers and tetramers [28]. Tetrameric species with edge and double-corner linkages were also shown to form during Cr 3+ and Ga3+ hydrolysis [55]. The relevant differences measured in the very ﬁrst steps of FeCl3 and Fe(NO3 )3 hydrolysis are most likely due to the nature and complexing strength of anions. In the early stages of FeCl3 hydrolysis, one or two of the six coordination sites in the Fe octahedra are occupied by chloride anions, whereas only OH and OH2 ligands

JEAN YVES BOTTERO, ARMAND MASION AND JÉRÔME ROSE DimF + PentaE

DimF + PentaE–DC

155

TrimF + Trim DC

Single corner linkage 14% Dim E + Tetra SC

12%

24%

15%

Trim E + Trim DC

Fe23 Ferrihydrite 2L

26%

12%

Figure 5.2 Six possible clusters present in Fe(NO3 )3 solution at n = 1.5. Dim = dimer, trim = trimer, tetra = tetramer, penta = pentamer, F = Face, E = edge; DC = double corner; SC = single corner. All clusters correspond to substructures of larger minerals. The Fe23 ferrihydrite 2L substructure is described in [54].

are detected in Fe(NO3 )3 solutions. Consequently, Fe(NO3 )3 exhibits more binding possibilities than FeCl3 , which likely includes single-corner bonds whose free energy of formation of 105 kJ mol−1 is lower than that of a double-corner bond [43]. Marked differences between the polymerization mechanisms of Fe(NO3 )3 and FeCl3 have also been shown to occur at higher hydrolysis ratios (or times). In the case of Fe(NO3 )3 , the cluster radii increase with hydrolysis ratio [48]. In particular, some authors have suggested that the dodecamer Fe12 (OH)2+ 34 may form in solutions of Fe(NO3 )3 at n > 2.3 [56], whereas others have postulated the formation of the polycation [FeO4 Fe12 (OH)24 (H2 O)12 ]7+ with a structure similar to that of the Al13 polymer [57]. In contrast, the size of aggregate subunits formed from FeCl3 does not appear to vary with the hydrolysis ratio [58]. On the basis of SAXS, EXAFS and Ar adsorption data, it has been suggested that the polymer Fe24 O12 (OH)16+ 32 , which shows a local structure similar to akaganeite (-FeOOH), can form [50] (Figure 5.3).

5.3.2 Diffusion, Aggregation, Fractal and Nonfractal Growth In contrast to FeCl3 hydrolysis, the hydrolysis of Fe(NO3 )3 leads to a more chaotic evolution of Fe polymers and Fe nuclei. Two possible mechanisms have been suggested to explain this observation. The ﬁrst consists of crystal growth or precipitation based on ion diffusion and includes the following steps [59]: (i) diffusion of Fe(III) ions to the surface of the nucleus; (ii) dehydration of Fe at the surface; (iii) adsorption of dehydrated Fe; (iv) diffusion of Fe on the surface to a more energetically favorable position. In contrast to the ion diffusion mechanism, the growth of Fe species appears to be explained better by the aggregation of primary polymers. This second

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FRACTAL MECHANISMS IN COAGULATION/FLOCCULATION polymerization

Fe-O/CI

16 Å Fe-O-Fe Fe24 n = 2.5 n=2 n=1 n=0 1

2

3

4

5

R (Å)

Figure 5.3 XAS Fe K-edge radial distribution function R and structural evolution of Fe polycations during hydrolysis in FeCl3 solution.

mechanism, i.e. the formation of larger polymers or precipitates, which are generally amorphous at ambient temperatures, has been studied using a fractal approach [48, 58]. In the case of FeCl3 , particle sizes measured by photon correlation spectroscopy (PCS) and SAXS [58] decrease with time and increase with n at a constant time. For example, at t = 24 h, the hydrodynamic radius rH of the particles is 12 nm at n = 1.5 and 200 nm at n = 2.7. Owing to the small size of Fe aggregates, SAXS curves do not show any fractal behavior at n = 1, whereas suspensions of Fe24 subunits appear arranged in fractal aggregates at n = 2.0 and 2.5 [58]. With increasing n, the sticking of polycations follows the usual rules of clustering as controlled by van der Waals interactions. An increase to n = 3 causes the formation of precipitates that exhibit a characteristic scattering curve with an apparent fractal dimension of Df = 2. In solutions at n = 1, the polycations feature a highly positive charge and the aggregation mechanism depends on long-range dipolar magnetic interactions with each subunit acting as a dipole that can realign in prevalently linear aggregates. More subunits are formed when n increases, but the positive charge at the subunit surface decreases, especially at n > 2.5, i.e. just before the ﬂocculation threshold of the sols. For this

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condition, electrostatic repulsion decreases and electrostatic attractive interactions become responsible for the fractal arrangement of colloids. The various steps of FeCl3 polymerization are summarized in Figure 5.3. In the case of Fe(NO3 )3 , aggregates are thought to take up a linear or semi-linear shape at the local range-order that consists of 4–5, 3–4, 7 and 9 subunits for n = 1.5, 2.0, 2.2 and 2.5 respectively [49]. At t < 10 min, greater branching and polydispersity is observed for n = 2.2 and 2.5, whereas aggregates have a fractal geometry with an apparent fractal dimension of Df = 1.75 for n = 2.8, typical of a cluster–cluster aggregation mechanism. Although this value is lower than that measured for FeCl3 solutions at n = 2.7(Df = 2), some authors have suggested that less aggregated and less dense, small particles can form in Fe(NO3 )3 solutions compared with FeCl3 solutions [59]. 5.3.3 The Effect of Strong Competing Ligands: The Case of Phosphate

3

P/Fe = 0.2 P/Fe = 0.5

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5 0

P/Fe = 0.2 P/Fe = 0.5

1

1.5 n (OH/Fe)

2

0

3.4 3.35 3.3 3.25 3.2 3.15 3.1 3.05 3

P/Fe = 0.2 P/Fe = 0.5

Fe-P distance (Å)

3.5

Df

Number of Fe dimers around PO4

When in competition with O and OH ligands, the presence of species such as phosphate, silicate, and organic matter is thought to affect the composition, structure, morphology and reactivity of Fe hydrolysis products [60–67]. PO3− 4 ions hinder the hydrolysis of Fe3+ cations by affecting the size and crystallinity of the particles even at low [PO4 ]/[Fe] molar ratios [61]. For example, EXAFS spectra at the Fe K-edge have indicated that, for n = 1 and [P]/[Fe] ≈ 0.2, Fe nucleation is extensively blocked at the dimerization step (Figure 5.4) [68] and that Fe–O–P linkages are formed [69]. For n = 1.5, dimers are the major Fe species (Figure 5.4) with P in the second coordination sphere of Fe at a distance of 3.2–3.3 Å [65, 67]. Binding of phosphate to the Fe dimers inhibits the binding of a third Fe octahedron to the dimer, unlike what occurs in FeCl3 solutions. For n = 2.0, each phosphate is able to bind three Fe dimers [69] in order to form small aggregates with a size of approximately 100–120 Å [58].

1

1.5 n (OH/Fe)

2

1

1.5 n (OH/Fe)

2

Figure 5.4 Number of Fe dimers surrounding each phosphate at various n and P/Fe ratios.

The fractal dimensions measured for n = 1.0, 1.5 and 2.0 are respectively 2.3, 2.4 and 2.85, and do not correspond to any classical aggregation mechanism. Nonetheless, structures tend to become denser and denser as n increases. At the semi-local

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scale, the evolution of Df with increasing n is very well correlated to the structural evolution at the local scale (Figure 5.5). EXAFS analysis indicates that the P–Fe distances decreased progressively from 3.35 to 3.15 Å as n increased from 1.0 to 2.0. Furthermore, the number of Fe atoms bound to a phosphate tetrahedron also increased with n without modifying the overall size of the aggregates, thus causing an additional density increase. P/Fe ≤ 0.2

P/Fe > 0.2

n=1

n = 1.5

n=2 n = OH/Fe

Iron hydrolysis is blocked at the dimer step The structure at the molecular level reflects the structure of the aggregates Phosphate governs the structure of the aggregates

Figure 5.5 Evolution of the Fe–PO4 polymers during the hydrolysis of Fe–PO4 solutions at various P/Fe ratios.

The structure of the system is modiﬁed drastically by an increase in the P/Fe ratio to 0.5 (Figures 5.4 and 5.5) [70]. For n = 1.5, the size of the aggregates is too high (>350 Å) to be measured by SAXS and the fractal dimension is much lower than in the case of P/Fe = 0.2. The Df value corresponds to a cluster–cluster-type aggregation mechanism with a sticking probability lower than unity, which suggests that the aggregates have a branched structure formed by connected subunits of small Fe dimer–PO4 complexes. For n = 2.0, as to the opposed to the rapidly settling precipitate that is formed at lower P/Fe, a gel is generated with more than 80% of the Fe as dimers and the rest as isolated monomers. Similar to the n = 1.5 case, the structure of the n = 2.0 sample is not uniform over the whole q range. The value of Df = 1.85 that is measured at high q is slightly lower than that for n = 1.5 and characteristic of a gel formed by a cluster–cluster aggregation. For n = 2.0, Fe monomers are thought to form bridges that bind two aggregates together, thus inducing the gelation process and preventing the aggregates from moving freely or approaching each other, leading to a less dense structure. The observation that the structures formed at P/Fe = 0.5

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are much more open than those obtained at the same hydrolysis ratio at P/Fe = 0.2 (Figure 5.5) may be due to the difference in stoichiometry of the two sample series. The ﬁrst reactions that occur, i.e. the condensation of a ﬁrst and a second Fe dimer onto a phosphate tetrahedron, are the most energetically favorable. Owing to the lower partial charge on the phosphate and to the steric hindrance created by the presence of two dimers, which reduces considerably the accessibility of the unoccupied sites on phosphate tetrahedra, a higher energy barrier must be overcome at P/Fe = 0.2. At this P/Fe ratio, the condensation of a third Fe dimer would result in dense structures. At P/Fe = 0.5, i.e. at a 2.5 times higher amount of phosphate, the probability of forming three dimer–phosphate units becomes even lower, since they are energetically disadvantaged with respect to two dimer–phosphate clusters. Thus, more linear and less dense structures that are characterized by a lower fractal dimension are formed (Figure 5.5). 5.3.4 Aggregation/Precipitation in Fe–SiO4 Systems 5.3.4.1 Effect of Si on Fe crystallization As Si and Fe are two of the most abundant elements in the Earth’s crust, there is a long history of studies describing the interactions between these elements. For example, the ﬁrst quantitative report of the formation of an FeSiO(OH)2+ 3 complex in dilute solution dates to the 1960s [71]. Structural studies of the Fe–Si systems have received much attention and focused mainly on the role of Si on the crystallization of the Fe oxides and oxyhydroxides. In this context, many authors have found that even low levels of Si are likely to affect the size and shape of the Fe–Si precipitates signiﬁcantly. For example, the presence of Si is shown to hinder the formation of goethite, and lead to the preferential formation of metastable ferrihydrite [61, 72]. Under acidic conditions, SiO4 is assumed to participate in the aggregation of nanosized (10–20 nm) goethite crystals [73]. Si contents as low as 0.1 mol% are found to delay considerably the dissolution–reprecipitation processes that transform ferrihydrite into goethite [24]. The zero point charge (ZPC) of the surface is displaced towards higher pH in the presence of Si. This effect is consistent with the formation of an Fe–O–SiO3 complex at the ferrihydrite surface, thus impeding its dissolution and transformation into goethite [74–77]. More recent data obtained over extended pH and Fe/Si ranges suggest that polymeric Si species can form at the surface of ferrihydrite [78]. These Fe–O–Si complexes are also shown to prevent the formation of hematite from ferrihydrite, which typically is favored by heating [79]. Although the above investigations are fundamental in determining to what extent the presence of Si modiﬁes the chemistry of Fe, they generally do not attempt to describe the structure of ‘X-ray diffraction (XRD)-invisible’or amorphous Fe species. In addition, the samples generally undergo heating and/or aging processes that favor the formation of crystalline structures. As a consequence, the speciation and structure of fresh Fe–Si phases, for which no organized structure is expected, have long been neglected. Although synchrotron-based SAXS and especially XAS techniques have

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gained popularity since the mid 1980s, the structure of freshly prepared Fe–Si systems has been studied only recently [28, 80–82]. These studies cover Fe hydrolysis over a large range of pH, Fe concentrations and Si/Fe molar ratios. 5.3.4.2 Nucleation and growth of Fe–Si species in the ﬁrst step of hydrolysis Analyses by Fe K-edge EXAFS show that, at low pH and in large excess of Si (Si/Fe = 5), Fe essentially forms edge-sharing Fe dimers for which each Fe binds one SiO4 tetrahedron [28]. In these studies, samples are ﬁltered to eliminate solids that may have formed, and no visible aggregation occurs within the ﬁrst few hours of aging. The formation of solid phases is observed in more concentrated Fe solutions (e.g. [Fe] = 0.2 mol l−1 ) and at lower Si/Fe ratios. For Si/Fe = 1 and Si/Fe = 2, the analysis of the n = 0.5 sample reveals the presence of monomeric Fe [83]. The formation of edge-sharing dimers is detected at n = 1.0, but signiﬁcant amounts of monomeric Fe are still present within the precipitate. These results indicate that Fe speciation in the aggregates is very similar to that observed in solution [28]. However, contrary to the dissolved species, and similar to other systems [84], Si was not found in the local environment of Fe in the precipitates. A signiﬁcant Fe polymerization is observed by increasing the hydrolysis ratio from n = 1 to n = 1.5. The total number of neighbors Ntot detected by EXAFS is higher at Si/Fe = 1 (Ntot = 3.11) than at Si/Fe = 2(Ntot = 2.47). This result is mainly due to the higher number of double corner linkages at Si/Fe = 1(NDC = 1.63) compared with Si/Fe = 2(NDC = 1.18). Similar scattering curves are obtained for samples at n = 0.5 and n = 1.5 (Figure 5.6) [83]. Linear domains are present at high q. For Si/Fe = 1, the slopes of the curves measured between q = 0.04 Å−1 and q = 0.20 Å−1 are −1.6 at n = 0.5 and −2.3 at n = 1.5, whereas the slopes are −1.2 at n = 0.5 and −1.9 at n = 1.5 for Si/Fe = 2. Although fractal dimensions cannot be derived from these slopes due to the limited q range and varying I0 , the slopes can be used to deﬁne general trends in the structure of Fe–Si aggregates. Indeed, with an increasing hydrolysis ratio, denser structures are formed, while increasing the concentration of Si lead to more open aggregates. For q values lower than ∼ 0.04 Å−1 , the scattering curves showed a marked increase in intensity, especially for the systems at n = 0.5. This effect was also observed for other Fe systems (e.g. chloride, nitrate, phosphate) at n = 1.0 and n = 1.5 [48, 58, 70], which suggests that this feature is not speciﬁc to a particular Fe system but can be generally related to a low hydrolysis ratio (n < 1.5) [83]. Nonetheless, neither the exact cause(s) nor the inﬂuence of the actual structural features on the increased intensity has been elucidated. For this reason, the low q portion of the scattering curves is generally not considered in the computation of the pair distribution functions P (r) (Figure 5.7). At n = 0.5, the ﬁrst peak of the P (r) function, which corresponds to the radius of the subunit, is located at almost the same value, i.e. r = 6 Å for both Si/Fe ratios. The presence of 6 Å clusters can only be explained by formation of Fe monomer–SiO4 complexes, whose occurrence as dissolved species can be predicted

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Si/Fe = 1 Si/Fe = 2

log I (a.u.)

Si/Fe = 1 Si/Fe = 2 (a)

(b) –2

–2.5

–1

–1.5

0

–0.5

log Q (Å–1)

Figure 5.6 SAXS curves of Fe–SiO4 colloids obtained at n = 0.5 (a) and n = 1.5 (b). Curves are shifted in intensity for better clarity.

n = 0.5

Fe/Si = 1 P (r)

Fe/Si = 2 n = 1.5

Fe/Si = 1 Fe/Si = 2 0

20

40

60

80

100

120

r (Å)

Figure 5.7 Experimental (full lines) and simulated (circles) P (r) functions for Fe–SiO4 colloids obtained at n = 0.5 and n = 1.5

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by equilibrium calculations at acidic pH [28]. The presence of Fe–O–Si bonds at such a low n value (i.e. even before Fe nucleation) is indicative of a high afﬁnity between Fe and SiO4 ligands. At n = 1.5, the shift of the ﬁrst peak of P (r) towards larger r values, i.e. r = 7.2 Å (Si/Fe = 1) and r = 6.5 Å (Si/Fe = 2), is consistent with the increased Fe polymerization. The higher Ntot value for Si/Fe = 1 is correlated with the larger subunit size. The P (r) functions are ﬁtted with linear combinations of theoretical curves using roughly spherical subunits [83], the number of which per unit volume increases with increasing n. This result can be attributed to the presence of denser structures, as suggested from the slopes of the scattering curves. Furthermore, at a ﬁxed n value, the number of spherical subunits decreases with increasing Si concentration, also as suggested from the slopes. Nonetheless, extreme care should be taken in interpreting these results in terms of structure, because the models do not provide any indication of the nature of bonds between the subunits. Considering the pH conditions, the presence of polymerized Si species that are able to connect Fe subunits is very likely. However, unfortunately, the nature and structure of Si species cannot be determined from SAXS due to the low electron contrast of Si compared with the heavier Fe, making Si ‘invisible’ on the scattering curves. The inﬂuence of SiO4 and PO4 on the ﬁrst steps of hydrolysis (n ≤ 2) differs greatly, although both ligands have a strong afﬁnity for Fe. PO4 is shown to block Fe polymerization from the lowest P/Fe ratios and predominantly form Fe dimers in all Fe–P systems at n ≥ 1.5 [85, 86]. In contrast, in the presence of SiO4 , Fe speciation and probably Si speciation varies with n and with the Si/Fe ratio. Furthermore, larger Fe clusters are formed. Therefore, the similarities in the evolution of aggregate structure that are observed, i.e. a denser structure obtained with increasing n or a decreasing ratio of ligand/Fe) appear to be purely coincidental. 5.3.4.3 Nature and structure of Fe–Si species at pH ≥3 The speciation and structure of Fe–Si systems have also been examined between pH 3 and 10 at n > 3 and Si/Fe = 0 to 4 [80–82]. Based upon XRD results, the phases that are formed are generally amorphous with only small amounts of poorly crystallized akaganeite detected at pH 3. The level of Fe polymerization within the aggregates is indicated by Ntot , i.e. the total number of Fe neighbors around the central Fe atom (Figure 5.8). The Ntot values are higher in samples without SiO4 . At all pH values, the presence of SiO4 results in less polymerized Fe species, i.e. Ntot decreases markedly from Si/Fe = 0 to Si/Fe = 1, whereas a further excess of Si (up to Si/Fe = 4) leads to an increase of Ntot . The formation of single and double corner linkages (SC and DC respectively) between Fe ions decreases sharply with the increasing Si/Fe ratio, whereas the evolution of edge (Ed) linkages follows that of Ntot (Figure 5.8). These results suggest that the growth regime of the Fe species is dependent on the Si/Fe ratio. At low Si concentrations (Si/Fe ≤ 0.5), the presence of both edge and corner linkages is consistent with

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Number of neighbours

6 5 4 3 2 1 0

0

0.25

0.5

1

2

4

Tot Ed DC SC

[Si] / [Fe]

Figure 5.8 Number of Fe neighbors around the central Fe ion as a function of the Si/Fe ratio at pH 3. SC: single corner; DC: double corner; Ed: edge; Tot: total.

the formation of three-dimensional structures, whereas the predominance of edge linkages at Si/Fe > 1 suggests a two-dimensional growth. Some information on Si speciation can be provided by Fourier transform infrared (FTIR) analysis in the spectral region between 800 and 1300 cm−1 . A simultaneous increase of the absorption bands at 930 cm−1 (Si–O–Fe) and at 800 cm−1 and 1300 cm−1 (symmetric and asymmetric Si–O–Si stretching respectively) is observed. Si polymerization increases with increasing Si and decreases with increasing pH, due to the higher solubility of Si at pH ≥ 7 [81]. Despite the presence of Fe, for samples with the highest Si polymerization level, i.e. Si/Fe = 4 at pH 3 and pH 5 and Si/Fe = 2 at pH 3, detection of 29 Si by NMR at about −100 ppm indicated the presence of Fe-free domains within the samples [81]. Information on the spatial arrangement of Si within these silica pockets can be obtained by measuring the magnetization S as a function of time. Indeed, S(t) scales as (Ct)β/6 or t α , where C is a constant and β is the dimensionality of the spin system. The slopes of the log S(t) versus log t plots yielded an average value of β = 2.2. Si K-edge EXAFS data can provide additional details on Si speciation in these systems, but they are often difﬁcult to interpret. In agreement with XRD results, the absence of correlation peaks in SAXS curves (Figure 5.9) indicates the presence of amorphous and disordered Fe–SiO4 aggregates. All curves display a linear portion for q < 0.075 Å−1 (Figure 5.9), but since I0 is not determined, only apparent fractal dimensions can be derived from the slopes measured in this q range (Table 5.1). Nonetheless, given that the subunits are small, the inﬂuence of I0 on the low q part of the SAXS curves is expected to be minimal; therefore, the apparent fractal dimensions derived from the log I (q) versus log q plots are considered to be good estimates of the rigorous values of Df . Hereafter,

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pH = 10 pH = 7

log I (a.u.)

pH = 3

–2.5

–2

–1.5 log q

–1

0

–0.5

(Å–1)

Figure 5.9 SAXS curves of Fe–SiO4 aggregates at pH ≥3. Table 5.1 Fractal dimension of Fe–SiO4 aggregates at pH ≥ 3.

Si/Fe = 0.5 Si/Fe = 1.0 Si/Fe = 2.0

pH 3

pH 7

pH 10

pH 7 (aged)

1.98 2.70 2.36

2.36 2.56 2.53

2.44 2.35 2.62

2.19 2.39 2.36

fractal dimensions determined in this manner are referred to as apparent fractal dimensions. The high Df values of all Fe–Si samples (Table 5.1) suggests the presence of dense structures consistent with a ligand-driven aggregation mechanism, as is the case for similar Fe– orAl–ligand systems [14, 70, 86, 87]. Since the scattering of Fe–Si systems is dominated by the heavier Fe species, data for both the ‘Fe structure’(Df = 2.36) and the ‘Si structure’ (Df = 2.2) are only available for the sample with Si/Fe = 2 at pH 3. No correlation was found between the growth regime of Fe species and their semilocal-scale arrangement.Actually, the two-dimensional Fe species at Si/Fe = 2.0 form denser structures than the three-dimensional Fe clusters at Si/Fe = 0.5. For these two Si/Fe ratios, the increase of fractal dimension with pH was attributed to the denser

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packing of Fe species favored by the decrease of size measured for Si polymers. On the contrary, at Si/Fe = 1, the observed depolymerization of Si at high pH resulted in lower Df values. The evolution of Df is not only dependent on SiO4 speciation [82], but also on Fe speciation, i.e. Df decreases linearly with increasing Fe polymerization (Ntot ; Figure 5.10). The semi-local structure of the Fe–Si aggregates is controlled by Fe complexation (presumably on the corner sites) by SiO4 , which causes dense aggregation and results in almost all Df values being above 2.2. 2.8 Si/Fe = 0.5 Si/Fe = 1.0 Si/Fe = 2.0

2.7 2.6 2.5 Df

2.4 2.3 2.2 2.1 2 1.9 1.5

2

2.5

3

3.5

4

N Tot

Figure 5.10 Evolution of Df with Fe polymerization Ntot for Fe–SiO4 aggregates at pH ≥ 3.

5.3.5 General Remarks The examples of the Fe–PO4 and Fe–SiO4 systems illustrate well some general ﬁndings about the local and semi-local structure of metal–ligand aggregates. While the hydrolysis of metal cations yields phases of well-deﬁned structure at equilibrium, the same process in the presence of ligands only rarely results in the formation of organized solids, even after appropriate heating/aging. The presence of the ligands delayed or inhibited the crystallization process. On a local scale, the polymerization of the metal cation was hindered by ligand complexation at its growth sites. In addition, ligand-induced depolymerization of already-formed metal species could be observed [88]. The structure of aggregates at the semi-local scale was also very sensitive to the presence of ligands. While classical diffusion-limited cluster aggregation was observed in their absence, a dense aggregation of metal monomers and polymers by ligand-driven bridging and/or charge screening was observed in their presence.

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Although the measurement of the fractal dimension of an aggregate can be performed with various imaging, scattering and settling techniques [89], the options available for the characterization of the semi-local-scale order (i.e. up to approximately 1000 Å) are limited to a few microscopic and scattering methods. Although imaging techniques such as transmission electron microscopy (TEM), scanning electron microscopy (SEM), and atomic force microscopy (AFM) are very versatile, they suffer from poor statistics, sample opacity, and poor transfer of information obtained in two dimensions to three-dimensional aggregates. Thus, X-ray scattering represents the main analytical tool for determining Df in this size range. Although scattering in heterogeneous systems is dominated by the heavier elements and no direct information about the arrangement of the lighter elements can be derived from SAXS data, the structural information provided by SAXS in the case of metal–ligand aggregates remains very valuable. Aggregate fractal dimensions measured by SAXS are often signiﬁcantly larger than two. When interpreting the evolution of Df with pH and/or ligand concentration, two typical situations must be considered: (a) metal speciation remains relatively unaffected by modiﬁcations of pH and/or ligand concentrations (and typically consists of monomers and oligomers, e.g. Fe–PO4 ); (b) metal speciation evolves with experimental conditions (e.g. Fe–SiO4 ). In the ﬁrst case, the scattering centers remain constant and variations of the fractal dimension reﬂect changes in ligand chemistry and/or structure. In the second case, although the ligand chemistry and structure still affect the overall density of the aggregate and the corresponding fractal dimension, structural changes in the size and shape of metal species, to which scattering is very sensitive, will dominate the signal and control, to a large extent, the evolution of Df . These trends are obvious only where an adequate difference in atomic number exists between the metal cation and the ligand (e.g. Fe–P, Fe–Si, Al/Fe–C). Where the metal and ligand produce similar electron contrast (e.g. Al–Si) [13, 90], additional data from independent techniques may be required to interpret structural data correctly.

5.4 FRACTAL DIMENSIONS OF VERY LARGE FLOCS BUILT WITH MICROSIZED UNITS A rapid and accurate determination of aggregate structure is desirable for understanding particle aggregation processes that occur in freshwater and marine environments, in water and wastewater treatment plants and for modeling various environmental phenomena [9, 91–94]. Among the techniques used for the optical determination of structural parameters [95–99], light scattering is certainly one of the most powerful for studying the evolution of the structure of large aggregates that are formed of units in the submicrometre size range. The relatively few studies that use light scattering in this context have used both numerical simulation and an experimental validation of the optical properties of well-known colloids.

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5.4.1 Numerical Simulations Some authors [19, 100–102] have tentatively calculated the mean optical contrast of scatterers within different computer-generated aggregates. More recently [101], using the approach described in Section 5.2 [19], and the same deﬁnition for the mean optical contrast generated by the presence of neighboring particles around a central particle within a fractal aggregate, the inﬂuence of the physical parameters Rg and Df has been tested on computer-generated fractal aggregates with Df > 2.0. 5.4.1.1 The inﬂuence of Df on the m/m ratio For fractal dimensions close to two, an observed decrease of the scatterers’ high mean optical contrast m/m with increasing fractal dimension (Figure 5.11) can be explained by the increasing number of neighbors around the scatterer. For Df > 2, the scatterers’ mean optical contrast is constant and tends towards the value of 1.05, which suggests that the scatterers are embedded in a compact structure. For Df > 2.3, scattering is small and the mean optical contrast is constant and close to unity. In this range of Df values, the average number of close neighbors around the scatterers is nearly constant and the neighbors exert a dominant inﬂuence on the scattering behavior of a particle within an aggregate. 1.20 1.16

m/m

1.12 1.08 1.04 1.00 2.00

2.10

2.20

2.30

2.40

2.50

Df

Figure 5.11 Scatterers’ mean optical contrast m/m as a function of the fractal dimension Df of aggregates made of 250 primary particles: () extinction values; (×) scattering values; () mean scattering and extinction values; (–) mean values for aggregates that do not ﬁt accurately the fractal model. Reproduced by permission of Elsevier.

The slope of the plot of the scatterers’ mean optical contrast as a function of Df can be explained by the evolution of the neighbors’ local concentration around the scatterers. In an inﬁnite fractal aggregate, the number of neighbors ninf surrounding a particle at a distance rinf (below which multiple scattering is important) is proportional

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to (rinf /rs )Df . One can assume that m/m evolves with the possible multiple scattering paths pms : pms = 1 + ninf + n2inf + n3inf + · · · + nxinf

(5.9)

where x is the multiple scattering order. When pms increases, m/m decreases. According to Equation (5.9), 1/pms decreases less quickly at high fractal dimensions than it does at low fractal dimensions. Thus, even if it is impossible to explain the decrease of m/m with Df precisely, the trends are consistent with the theoretical expectations. 5.4.1.2 The inﬂuence of Rg on the m/m ratio The evolution of the m/m ratio as a function of Rg for ﬁve sets of aggregates with 50, 100, 150, 200 and 250 subunits is shown in Figure 5.12. Circles have diameters that are proportional to the fractal dimension. For each set of aggregates, m/m increases with Rg , which in turn decreases with increasing fractal dimension. The Rg value can thus be used to characterize the aggregate structure. It would appear that whatever the number of monomers in an aggregate, variations of the scatterers’ mean optical contrast depend on the mean environment of the scatterers. Although determination of the form factor I rs ,ns /n (q) depends on Rg and Df , an increase in the number of subunits N appears to cause variations in the mean optical 12

10

8

Rg

N = 250 N = 200 N = 150

6

N = 100 N = 50

4

2 1

1.05

1.1

1.15

1.2

m/m

Figure 5.12 Rg versus m/m for aggregates of N primary particles. The circle diameters are directly proportional to the aggregate fractal dimensions Df . Reproduced by permission of Elsevier.

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contrast when the fractal geometry is not strictly consistent, in contrast to theoretical predictions. The scatterers’ mean optical contrast increases slowly with N, independent of the fractal dimension [12, 103]. Structural changes and an evolution of the aggregate size do not inﬂuence the mean optical contrast in the same way, i.e. an increase in size slightly increases the mean optical contrast, whereas an increase in fractal dimension causes the optical contrast to decrease. The difference in the two behaviors is more pronounced when the fractal dimension is low. In the case of aggregation kinetics, both size and structure of the aggregates change during the reaction. In summary, the scatterers’ mean optical contrast can provide useful information on aggregate size and structure. 5.4.2 Experimental Validation The concepts developed above have been tested experimentally with suspensions of latex beads [103]. From light-scattering experiments, the experimental scattered intensity Iexp (θ ) has been normalized for the number of scatterers: Inorm (θ ) = Iexp (θ )

Vs Iexp (θ ) = φVcell Ns

(5.10)

where Vs is the volume of the primary scatterer, φ is the volume fraction of scatterers in the suspension, Vcell is the scattering volume and Ns = φVcell /Vs is the number of scatterers in the scattering volume. The magnitude of scattering vector q, is related to the maximum interference distance d by qd = π

(5.11)

At large q values, S(q) tends towards unity (1/q rs ) and variations of the scattering intensity Inorm (q) are essentially due to form factor changes. In order to follow these changes, the largest available q that is associated with an interference distance d that is smaller than the monomer diameter has been selected. The intensity at this particular angle is given by Iexp (θ ) Inorm (θs ) = (5.12) Ns A numerical model is used to compute the scattered intensity Inum (θ ) according to Mie theory. At the lowest volume fraction, Inum (θs ) ﬁts Inorm (θs ) and no multiple scattering arises, such that the evaluation of parameters for the numerical model is straightforward. The mean optical contrast is adjusted to compute the normalized theoretical values Inum (θs ) so as to ﬁt the variation of Inorm (θs ) as a function of φ. It is then possible to calculate the value of the mean optical contrast by ﬁtting the form factor. From a kinetic point of view, the value of the mean optical contrast can be obtained at each step of the aggregation process. In addition, from the correlation between the mean optical contrast and the aggregate structure that is established by numerical simulation, it is possible to determine aggregate structure at each aggregation stage.

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Figure 5.13 shows the evolution of the m/m ratio as a function of the volume fraction φ for 4 × 10−6 ≤ φ ≤ 1.4 × 10−5 and for an optical contrast ns /nm = 1.2. In this case, the theoretical result is in very good agreement with the normalized experimental scattered intensities. At φ ≤ 10−6 , Mie theory is in good agreement with the experimental data and it is thus reasonable to conclude that multiple scattering is insigniﬁcant. At φ > 1.4 × 10−5 , m/m decreases as φ increases. 1.2

m/m

1.15

1.1

1.05

1 10–7

10–6

10–5

10–4

10–3

10–2

φ Figure 5.13 Evolution of m/m with the volume fraction φ. Reproduced by permission of Elsevier.

For aggregates of latex beads, the normalized intensity Inorm (θ ) decreases as a function of the scattering angle in the high q range. Figure 5.14 shows the normalized scattering intensities for a velocity gradient G of 63 s−1 [98]. During aggregation, the local average number of neighbors around the scatterer increases. As previously observed (Figure 5.13), the increased concentration of local scatterers is associated with a decrease of normalized scattered intensity Inorm (θ ) at large angles. Using the mean ﬁeld approximation [19], it is possible to associate a mean optical contrast [101] with each aggregation step. Using the same data treatment procedure as was applied to latex beads in suspension, the normalized scattering intensity at the angle θs (or qs ) can be used to extract the value of the mean optical contrast m/m which best ﬁts Inorm (θs ) values. When the mean optical contrast m/m is known, it is possible to compute the form factor I rs ,ns /n (q) by taking into account the mean environment of scatterers in the aggregate. In such a case, fractal dimensions can be calculated from the established correlation [101] and an experimental structure factor Sexp (q) can be extracted. As aggregation proceeds, the variation of Df can be plotted as a function of the m/m ratio. This procedure allows the calculation of fractal dimension for a scattering-angle range at which the scattering intensity is not ‘polluted’ by the aggregate size distribution.

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10–4

10–5

Inorm

10–6

10–7

10–8

10–9 0.1

1

10

θ (degrees)

θs

100

Figure 5.14 Inorm (θ ) at different aggregation times: () t = 60 s, () t = 300 s, (•) t = 840 s, () t = 1440 s, () t = 2040 s, () t = 3600 s. Reproduced by permission of Elsevier.

In the intermediate q range, the Sexp (q) slope changes during aggregation as a result of the scatterers’ arrangement inside the aggregates and as a function of the scatterers’ mean optical contrast and aggregate size distribution. The detailed evolution of the slope is due to a combination of the size polydispersity and structure of the aggregates [103, 104].

5.5 CONCLUSIONS AND RECOMMENDATIONS The processes at the origin of aggregation in environmental media are complex. Hydrolysis, surface complexation and adsorption occur, sometimes in opposition. Complexation can limit hydrolysis and growth, producing subunits in the nano-size range, which often results in the formation of fractal aggregates. The mesoscale structure is largely dependent on molecular-scale arrangements, as was discussed for the Fe–PO4 or Fe–SiO4 systems. The structural characterization of such systems requires the use of different experimental approaches in the nanoscale and mesoscale size ranges. In particular, EXAFS and SAXS techniques have demonstrated a close relationship between the structure of complexes formed between PO4 and Fe atoms and the fractal dimension. In particular, when the number of PO4 units increases around the central Fe, the fractal dimension also increases. The structure of nano- and meso-scale systems is now well understood. The characterization of large (much greater than micrometer) aggregates formed of

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micrometer-sized subunits using spectroscopy (e.g. light scattering) is not trivial. The necessity to have fast and reliable measurements of the structure and/or size distribution is of high interest. Recent work has shown that data should be revised in the light of this new approach. The good agreement between the structural values calculated from surface quasi-elastic light scattering and classical light scattering demonstrates that the fractal dimension can be determined from large angle static light scattering measurements. Both techniques allow the non-destructive, rapid (less than 30 minutes per sample), in situ structural determination of fractal aggregates. Nevertheless, structural information can only be extracted in cases where the nature and shape of the subunits are well known.

LIST OF ABBREVIATIONS AND SYMBOLS EXAFS NMR PCS SAXS XAS XRD ZPC d Df DT F (q) g(r) I (q) IN (q) Inum (θ ) Iexp (θ ) m = ns /nm n nm ns /n Ntot Vs Vcell P (r) q r rH rs R

extended X-ray absorption ﬁne structure nuclear magnetic resonance photon correlation spectroscopy small-angle X-ray scattering X-ray absorption spectroscopy X-ray diffraction zero point of charge interference distance fractal dimension translational diffusion coefﬁcient form factor ﬁrst order of the auto-correlation function Scattering intensity versus the moment of the wave vector q normalized by N (number of dispersed particles) of I (q) numerical intensity experimental intensity I initial optical contrast mean optical index optical index of the medium scatterers’ mean optical contrast total number of Fe neighbors primary scatterer volume scattering volume autocorrelation function moment of the wave vector q = 4π sin θ/λ distance hydrodynamic radius radius of the subunit radius

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173

structure factor or interference function volume volume fraction of scatterers pair-correlation function wavelength of the incident beam scattering angle

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6 Fractal Approach to Adsorption/Desorption Processes on Environmental Surfaces Zoﬁa Sokołowska1 and Stefan Sokołowski2 1

Institute of Agrophysics, Polish Academy of Sciences, Do´swiadczalna 4, 20290 Lublin, Poland 2 Department for the Modelling of Physico-Chemical Processes, Marie Curie–Skłodowska University, 20031 Lublin, Poland

6.1 INTRODUCTION The term ‘fractal’ and the concept of fractal dimension were introduced by Mandelbrot [1]. Since Mandelbrot’s work, many scientists have used fractal geometry as a means of quantifying natural structures and as an aid in understanding physical processes occurring within these structures. Fractals are objects that appear to be scale invariant. Mandelbrot deﬁnes them as ‘shapes whose roughness and fragmentation neither tend to vanish, nor ﬂuctuate up and down, but remain essentially unchanged as one zooms in continually and examination is reﬁned’. The above property is called ‘scale invariance’. If the transformations are independent of direction, then the fractal is self-similar; if they are different in different directions, then the fractal is self-afﬁne (see Chapter 2). Natural objects are not fractals in the mathematical sense because they do not exhibit scaling over an inﬁnite range of scales, but their fractal character is demonstrated over a ﬁnite scale range from a lower limit rmin to an upper limit rmax . Nonetheless, fractals may provide a useful characterization of natural structures between the two limits [rmin , rmax ]. Fractal dimensions can be deﬁned in several different ways, depending on the object under study and the measurements being made (see Chapter 2). The fact that there are different fractal dimensions may, in some cases, lead to confusion with respect to which fractal dimension is being employed. In the case of adsorption studies, two types of fractal, i.e. the surface fractal and mass fractal, are important. Surface fractals have a fractal boundary but a compact, nonfractal bulk. Common to all methods of fractal analysis is that ‘yardsticks’ of different size probe the surface. Depending on the method, the probes may be molecules of different size,

Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems c 2008, IUPAC and Kevin J. Wilkinson

Edited by Nicola Senesi

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electromagnetic waves diffracted at different angles, liquid menisci with different radii of curvature, etc. In general, in the case of surface fractals in a three-dimensional space, the number of yardsticks N (r) of linear size r necessary to cover the surface scales as N (r) ∝ r Ds (6.1) and the exponent Ds by deﬁnition is the surface fractal dimension. The mathematical deﬁnition of the surface fractal dimension is Ds = limr→0 [ln N (r)/ ln r]

(6.2)

Equation (6.1) can be generalized to the case of objects in a d-dimensional space. If Ds coincides with Dtop , the topological dimension of the system, then the system is Euclidean. In general, for any system Dtop ≤ Ds ≤ d. Obviously, in the case of real surfaces, the mathematical deﬁnition cannot be applied because the size of the smallest yardstick is always greater than zero. Therefore, in the real world, Equation (6.1) is satisﬁed only within some ﬁnite range, r ∈ [rmin , rmax ], called the fractality range. Typically, irregularities ﬁll the entire fractal surface, but in some cases only a part of the surface is fractal [1]. For example, the set of irregularities may correspond to the active sites of a catalyst surface, to the sites of preferential adsorption, etc. [2]. In a mass fractal, the entire bulk of mass solid is not compact but porous. For mass fractal, the mass m inside a characteristic length scale r scales according to m ∝ r DM

(6.3)

where DM is the mass fractal dimension that is smaller or equal to the dimension of Euclidean space. Pfeifer et al. [3] introduced the term ‘pore fractal’ for which not the solid phase, but ‘an empty space’ scales according to Equation (6.3). If the ‘mass’ of pores scales according to Equation (6.3), then their volume (in three-dimensional space) scales as Vp ∝ r 3−DP

(6.4)

where DF is the pore fractal dimension whose value in a three-dimensional space is in the range 0 ≤ DP ≤ 3. For DP = 2, the distribution of pore sizes is uniform. For increasing values of DF , the relative number of pores whose size is close to the minimum pore size increases. An adsorbent with a pore fractal dimension DP = 3 should have pores of one size only. The question can be posed whether the exponent α in the power law y ∝ t α relating the quantities t and y can always be interpreted in terms of a fractal dimension. Clearly, if this power law is applied to any arbitrary data, then one cannot expect α to be restricted to a certain range (e.g. 0 ≤ α ≤ 3). Even if the power law is used to describe some physical attribute of a real system, it cannot be guaranteed that the exponent will have a clear physical meaning nor that its values will be geometrically constrained. For example, cases of physical systems where this exponent leads to ‘fractal dimensions’ that were greater than three have been reported in the literature

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[4, 5]. In some cases, such values can be considered to be ‘probabilistic’ fractals. For a discussion of this problem, see the review of Baveye and Boast [6] and the references quoted therein, in addition to Chapter 2 in this book. A vast majority of theories describing the physical adsorption of ﬂuids on solids employ models which are based on Euclidean geometry. Usually, the adsorbing surfaces have been modeled as planar surfaces, straight-line step edges, slit-like or cylindrical pores, etc. In contrast, many systems of practical importance, e.g. soils and soil components, have structures that cannot be described in such terms. In this respect, the network formed by the pores in soils apparently forms ‘an iterative structure’, i.e. smaller pores are sub-pores of larger pores, etc. Owing to this effect, models in which structural features span a wide range of length scales, i.e. treating surface irregularities as recurrent and nested rather than as isolated entities, are required. A fractal adsorbing surface has the same structural features at length scales that are comprised between the inner cut-off limit rmin and the upper cut-off limit rmax . Its irregularity is described by the surface fractal dimension Ds . The fact that the complexity of the adsorbing surface is captured by a single number is extremely convenient and appealing, and allows for a great simpliﬁcation of the theoretical description of adsorption phenomena. The purpose of this chapter is to describe the developments and various aspects of adsorption phenomena on fractal surfaces. The review is arranged as follows.After this introductory section, Section 6.2 describes the methods of evaluating fractal dimensions using the so-called thermodynamic approach developed by Neimark [7, 8]. Section 6.3 provides an overview of adsorption equations of gases on geometrically nonuniform, but energetically homogeneous, solid surfaces. The description of adsorption on both geometrically and energetically nonuniform surfaces is discussed in Section 6.4. Examples of applications of the approaches described in Sections 6.2–6.4 to the studies of soil materials are presented in Section 6.5.

6.2 EVALUATION OF FRACTAL DIMENSION 6.2.1 Monolayer Capacities of Molecules of Different Sizes The monolayer capacity Nm , i.e. the number of adsorbed particles that cover the adsorbing surface and form a monomolecular layer, can be determined from standard adsorption measurements. The monolayer capacity is then used to calculate the surface area S = Nm ω, where ω is the cross-sectional area of adsorbed molecules [9], which is proportional to the square of the molecular diameter σ . For fractal surfaces: Nm ∝ ω−Ds /2

(6.5)

where Nm is evaluated from adsorption data and Ds is the surface fractal dimension [10]. Numerous literature data [11–19] reveal that, in most cases, the fractal model is applicable, with Ds values falling in the theoretical range of 2 ≤ Ds ≤ 3.

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Nonetheless, an initial difﬁculty may arise with respect to the correct assessment of the cross-sectional surface area of the adsorbate molecule, which also depends on the adsorbate–adsorbent interaction since this can change the orientation of the adsorbed molecules. Often, the adsorbed molecule occupies not only the site on the surface, but also some neighboring volume, thus creating steric difﬁculties for the adsorption of other molecules and leading to a possible underestimation of the monolayer capacity [20]. Moreover, in the case of some speciﬁc interactions, e.g. associative interactions, the structure of the adsorbed layer may change as an effect of adsorbate–adsorbate interactions. Under these circumstances, it may be useful to refer to appropriate literature data for ω [9]. Although in some cases, the values of Ds obtained from Equation (6.5) are not bounded within the limits of 2 and 3, values of Ds ≤ 2 are still interpretable in terms of the fractal dimension, namely as a subset of all surface points. On the other hand, values of Ds > 3 cannot be interpreted in terms of geometrical dimension. Apart from inaccurate models that are applied to evaluate ω, there are several effects that can lead to apparent values of Ds > 3. These include superposition of the fractals, some sieving effects, chemisorption and the nonfractal distribution of adsorbing sites [10]. On some substrates, such as expanding clay minerals, the adsorption of different molecules may occur according to different mechanisms. For example, the adsorption of nonpolar molecules can occur only on external surfaces, whereas polar molecules can also enter interlayer spaces. Consequently, application of Equation (6.5) to polar and nonpolar molecules may lead to different results [16, 21]. The range of the power law applicability is limited to a given set of σ (or ω) values. In most cases, the smallest value of σ is for argon, water vapor or nitrogen; thus, these molecules are the most frequently used in adsorption experiments. Unless polymer molecules are used, the range of available molecular sizes is rather small. Furthermore, molecules in the gas phase may not be able to reach all the vacant adsorption sites. For example, if the path involves a linear extension smaller than the dimension of the molecule (bottle neck), then the measured Nm will be smaller than the value that would have been obtained if molecules were able to cross the bottle neck. In conclusion, the value of Ds reﬂects ‘an effective geometry’ as seen by the speciﬁc set of adsorbed molecules. The value of Ds is not related to sample porosity. For example, adsorption on either a nearly nonporous or a porous carbon adsorbent, e.g. carbon black or Vulcan 3D [11], gave similar Ds values, i.e. Ds = 2.07 and 2.04 respectively. Application of Equation (6.5), therefore, should always take into account the question about which surface’s fractal dimension is actually measured in adsorption experiments. According to the mathematical derivation of this equation, the centers of the ‘yardstick molecules’ are theoretically located at the geometrical surface of the underlying solid. However, each molecule that is adsorbed within the ﬁrst layer adjacent to the solid is located at a certain distance from the surface, which is determined not only by the geometry of the underlying solid, but also by the form

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of the gas–solid interaction potential. The relationship between the fractal dimension of the underlying solid and the fractal dimension of the surface at which adsorption actually occurs has been discussed in the literature [22–25]. However, in contrast to previous work [22–25], in simulations of the adsorption of spherical particles on deterministic fractal curves, each vertex of the generated fractal curve is treated as the center of the adsorbing potential [26]. The energy of interaction of a gas molecule of diameter σ with the whole curve was assumed to be equal to the sum of interactions of the molecule with all vertices, and the energy of interaction of the vertex i with the molecule can be described by a Lennard–Jones (12, 6) function [27]: 12 6 σs σs ui (r) = 4ε (6.6) − r r where the energy ε and size parameter σs depend on the kind of adsorbed molecules, and r is the distance between the vertex and the molecule. According to the Lorentz– Berthelott mixing rules, σs = (σa + σ )/2, where σa is the distance between the nearest vertices. The ﬁrst layer of adsorbed molecules is located at the curve M that passes through the minima of the adsorbing potential, and the geometry of this curve depends on the size of adsorbed molecules σ . Examples of the curves M evaluated for a set of adsorbate molecules of different diameters σ are shown in Figure 6.1a. The fractal dimension can be computed by evaluating the number of spheres of diameter σ that are necessary to cover the curve M . Obviously, a new curve M is generated for each value of σ . Figure 6.1b compares the fractal dimension determined according to the

b 1.4

σ = σa σ = 5 σa

Dsm

σ = 2.5σa

a

1.2

1 1

1.2

1.4 Dst

Figure 6.1 (a) Examples of three curves M passing through minima of adsorption potential. Curves have been generated for molecules of three different sizes σ . (b) Correlation between the theoretical fractal dimension Dst and a fractal dimension that has been evaluated for the curves passing through the minima of the adsorption potential Dsm . Results shown refer to diadic and triadic Koch curves. Reproduced by permission of the American Chemical Society [26].

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above procedure Dsm with the theoretical surface fractal dimension of the underlying substrate Dst . In the case of diadic and triadic Koch curves [26], the fractal dimension calculated by accounting for energetic adsorption conditions was always lower than the mathematical fractal dimension of the underlying substrate. The number of particles (or granules) Ng , of an adsorbent substrate in a constant volume V scales with the size of the particles rg according to Ng ∝ rg−3 . It follows that the number of molecules necessary to cover these particles with a monomolecular layer Nmg scales as Nmg ∝ rgDs −3 (6.7) Equation (6.7) offers an alternative method to Equation (6.5) for the evaluation of the surface fractal dimension, which relies on the evaluation of the monolayer capacity for a series of adsorbent samples that have a constant volume and contain adsorbent particles in the size interval rg,min , rg,max [12, 15, 28–30]. A method that can be considered as a combination of the two described above has been proposed by Van Damme and Fripiat [21], who have applied the concept of the surface fractal dimension to adsorption on the so-called pillared clays [31, 32]. If, prior to the adsorption of small molecules having a cross-section ω, np large molecules (the pillars) of cross-section ωp are deposited on the surface, then the number of small molecules necessary to reach monolayer coverage on the remaining ‘free’ surface is given by ω Ds /2 p Nm = Nm − np (6.8) ω where Nm is the monolayer capacity of the initial bare surface. By varying the number of deposited pillars and measuring the monolayer capacities of N2 , it is possible to obtain the value of Ds . This method, however, is rather hard to implement as a general method. 6.2.2 The Thermodynamic Method of Neimark The method of Neimark [7, 8, 33, 34] for the determination of the surface fractal dimension of a microporous solid is based on the adsorption isotherm equation that was developed by Kiselev [35]. This equation relates the surface area of pores ﬁlled by the adsorbate S(x) to the amount of adsorbed molecules N (x) at a given relative pressure x = p/p0 : Np kT − ln(x) dN (6.9) S(x)φ = Vm N(x)

where p is the pressure, p0 is the saturated vapor pressure, φ is the surface tension, Vm is the molar volume of the adsorbate and Np is the maximum adsorbed amount in the pores, i.e. the value of the adsorption when p → p0 . The product S(x)φ in Equation (6.9) represents the work connected with the formation of a phase boundary

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between the liquid-like adsorbed ﬁlm and the gas phase, which equals the work corresponding to the growth of the adsorbed ﬁlm as represented by the right-hand side of Equation (6.9). Physically, this situation corresponds to the representation of an adsorbate surface as an ensemble of surfaces, each with a constant curvature. The radius of curvature of the liquid meniscus σC plays the role of a yardstick. According to the Kelvin equation, σC ∝ φ/ ln(x) [27, 35], because S ∝ σC2−Ds , Equation (6.9) yields ⎤ ⎡ Np d ⎣ln − ln(x) dN ⎦ (6.10) Ds = 2 + d[ln(− ln(x)] N (x)

Equation (6.10) was derived based upon thermodynamics alone without invoking any structure mode [7, 8]. The range of validity of the Kelvin equation corresponds to the range of mesopores [27]. A similar method was suggested for the calculation of the surface fractal dimension based on measurements of mercury porosimetry. The surface area measured by covering the surface with gauges of size σg scales as S ∝ σg2−Ds . In the case of mercury porosimetry, the gauge corresponds to the average radius of curvature r of the meniscus at the mercury–solid sample interface. Assuming that the pore space is represented by a bundle of nonintersecting pores of the same geometry (e.g. cylindrical, slit-like, etc.), the radius r is related to the mercury pressure p by the Washburn relationship: C1 r

p=

(6.11)

where C1 is a constant. The Washburn relationship plays a similar role to the Kelvin equation in the case of adsorption experiments. Rootare and Prenzlow [36] derived the following relationship between the surface of pores S1 and the volume of the intruded mercury V , i.e. the volume of pores of radius r: V S1 = C2 p dV (6.12) 0

In Equation (6.12), C2 = −1/(φ cos α) is a constant that is related to the surface tension φ and the cosine of the contact angle of mercury at the mercury–solid interface α. This area is interpreted as the one that would be measured by spheres of radius r. The total surface area of an adsorbent S is the sum of the surface area of the sample–mercury interphase at the beginning of the intrusion process S0 and the inner pore surface S1 . If S0 is neglected, then [7, 8] ⎤ ⎡ V d ⎣ln p dV ⎦ Ds = 2 + (6.13) d(ln p) 0

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Equation (6.13) requires only the measurement of the dependence of the volume of the intruded mercury on the applied pressure, and can be used to investigate irregularities of the solids within the range of scales provided by mercury intrusion data.Application of the mercury porosimetry technique has been reported in several papers studying soils [14, 17, 33, 37–45]. Both Equations (6.10) and (6.13) require some comments. If Equation (6.10) is used to analyze experimental data, then a problem often encountered is related to adsorption hysteresis in the mesopores. The results of calculations may depend on which branch of the isotherm (adsorption or desorption) is used as the basis of calculations. In such cases, one should preferentially employ the desorption branch in order to conform to the Kelvin theory of pore ﬁlling [8, 27, 46]. Nonetheless, values of the fractal dimension that are calculated from the adsorption and desorption isotherms should be comparable for those samples showing fractal properties over a sufﬁciently wide range of pore sizes [7, 8]. Another problem is related to the fact that Np is ill deﬁned and it can be difﬁcult to evaluate from experimental data. In particular, if the liquid adsorbate wets the adsorbent, then adsorption diverges as the pressure approaches the saturated vapor pressure. This divergence is due to the growth of a ﬁlm at the solid surface. In the case of mesoporous adsorbents, the value of Np is usually assumed to be equal to the amount of adsorbate at the end of the capillary condensation loop [7, 8]. A similar problem is encountered for the analysis of mercury porosimetry data. The approach based on Equation (6.13) neglects the surface area of the sample–mercury interphase at the beginning of the intrusion process S0 . If Equation (6.10) is applied to data over a range of mercury pressures, then the effect of neglecting S0 may be signiﬁcant. Therefore, Equation (6.10) should be corrected by incorporating S0 , thus obtaining ⎞⎤ ⎡ ⎛ V d ⎣ln ⎝S0 + C2 p dV ⎠⎦ Ds = 2 + (6.14) d(ln p) 0

However, the porosimetric method does not directly measure the initial area S0 . To evaluate S0 , Kozak et al. [40] modiﬁed Neimark’s approach based on the application of Equation (6.14). In addition to Neimark’s approach, other alternative approaches have been developed that are aimed at evaluating the fractal dimension from mercury intrusion data [47].

6.3 ISOTHERMS DESCRIBING ADSORPTION ON FRACTAL SURFACES 6.3.1 Brunauer–Emmet–Teller Theory Applied to Fractal Surfaces The Brunauer–Emmet–Teller (BET) model of adsorption is one of the most common tools for surface analysis, and is also recommended by IUPAC [48] as a standard tool for surface area determination. According to the BET model, adsorption occurs

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on an ideally ﬂat (Ds = 2) and energetically homogeneous surface. The energy of an adsorbate with N1 molecules in the ﬁrst layer and N2 molecules in higher layers is N1 ε1 + N2 ε2 , where ε1 and ε2 are the energies of adsorption in the ﬁrst and in the higher layers respectively. The adsorption energy of molecules in the higher layers is assumed to be equal to the heat of condensation of the bulk adsorbate, ε2 = εc . This theory neglects the long-range interactions between the adsorbent and the adsorbate, and the ﬁlling of all higher layers is governed by adsorbate–adsorbate interactions. If there is no restriction imposed on the maximum number of adsorbed layers, then the number of adsorbed molecules N grows with increasing the gas pressure p: 1 g(x) N = Nm 1 − x 1 + g(x)C

(6.15)

where g(x) = x/(1 − x), x = p/p0 and C = exp[−(ε1 − ε2 )/kT ], x corresponds to the relative pressure, p0 is the saturated vapor pressure, k is the Boltzmann constant and T is the absolute temperature. The ratio N/Nm gives the relative coverage of the surface, which is also a measure (in units of the adsorbate diameter σ ) of the mean ﬁlm thickness. The success of Equation (6.15) in describing experimental isotherms for a variety of systems has likely made it the most widely used isotherm equation in surface sciences. Therefore, it is quite obvious to extend BET theory to the case of adsorption on a fractal surface. In this respect, two approaches have been reported. The ﬁrst [3, 22] is based on the idea that as the ﬁlm thickness grows it ﬁlls all the surface irregularities, and consequently the number of adsorption sites in consecutive layers N (n) decreases. This decrease is described by a power law: N (n) = n2−Ds N (0)

(6.16)

where n is the layer number and N (0) is the number of bare surface sites, N (0) = Nm . For a ﬁnite system, this scaling holds only up to nmax ≈ Rmax /σ , where σ is the diameter of the adsorbate molecule and Rmax is the outer cut-off of the fractal regime. The isotherm equation has the form ϑ(x) = where f (x, D) =

f (x, Ds ) N = Nm 1 + C g(x) ˜ nmax

n2−Ds

nmax

(6.18)

j =n

n=1

g(x) ˜ =

j Ds −2 x j ,

(6.17)

nmax

xi

(6.19)

n=1

˜ reduces to the and ϑ(x) is the surface coverage. For nmax → ∞, the function g(x) term x/(1 − x) in Equation (6.15). For a planar surface with Ds = 2, Equation (6.17)

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reduces to the usual BET equation, Equation (6.15). For Ds = 3, Equation (6.17) assumes the form [49] ϑ(x) =

N Cx = Nm [Cx + (1 − x)2 ]{1 + ln[1/(1 − x)]}

(6.20)

The low-pressure behavior (x → 0) of Equation (6.17) is given by N = C[x + (2D−2 + 1 − C)x 2 + O(x 3 )] Nm

(6.21)

where O(x 3 ) denotes the rest of the order of x 3 . Equation (6.21) indicates that, at relative pressures where the contributions of order x 2 become important, the isotherms on fractal and on planar surfaces begin to differ. This regime is important for the determination of the monolayer capacity Nm . For any value of the relative pressure x, the amount of adsorbed molecules per surface site, θ = N/Nm , decreases with increasing Ds . This occurs because, for a given number of surface sites, increasing Ds imposes increasing spatial restrictions on the growth of the ﬁlm thickness. When the pressure approaches the saturated vapor pressure, i.e. for p → p0 or for x → 1, the asymptotic behavior of the adsorption isotherms is described by the power law [3] N

(3 − Ds ) ≈ Nm (1 − x)3−D

for 2 ≤ Ds < 3

N = ln[1/(1 − x)] for Ds = 3 Nm (6.22)

and

where is the gamma function. The divergence of the isotherm for p → p0 means that the fractal surface, similarly to a geometrically ﬂat surface adsorbs inﬁnitely thick layers. For Ds = 3, the divergence is logarithmic. Equation (6.17) suffers from the deﬁciency that during its derivation, ‘multiplesurface’ effects were neglected. Nonetheless, for the ﬁlling of a slit-like pore, the ﬁlm grows from two opposite adsorbing surfaces and stops growing when the two ﬁlms meet. A BET-like theory that takes into account multiple-surface effects has been developed by Cole et al. [50]. Such an approach replaces the Ds -dimensional surface by an equivalent system of pores, each of which can be treated as independent. In each pore, coverage can be evaluated from standard BET theory for adsorption between two plates by assuming that the maximum number of adsorbed layers nmax and the pore width H are related via 2σ nmax = H . In other words, adsorption in a single pore is given by θ (x; nmax ) =

Cx 1 + (nmax + 1)x nmax + nmax x nmax +1 1−x 1 + (C − 1)x − Cx nmax +1

(6.23)

The total number of adsorbed molecules is calculated by summing up the contributions from individual pores: N ϑ(x) = = (Ds − 2) Nm

∞ n1−Ds θ(x; n) dn 1

(6.24)

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The surface fractal dimension enters Equation (6.24) through the pore size distribution [50]. Equation (6.24) holds for any surface with 2 ≤ Ds ≤ 3. Despite the quite different form of the isotherms in Equations (6.17) and (6.24), the numerical values of adsorption that are calculated from these equations agree well, up to x ≈ 0.7. The asymptotic behavior of Equation (6.24) can be described as follows: for a geometrically ﬂat surface, Ds = 2 and Equation (6.24) yields the classical BET isotherm [3]. For x → 1, the growth of the adsorbed ﬁlm scales as N (D) ∝ (1 − x)3−Ds Nm where

for 2 ≤ Ds < 3 ∞

(Ds ) =

and

t 1 − exp(t)

t 2−Ds

d dt

and

(Ds ) = 1

0

N ln[1/(1 − x)] ∝ Nm 2

for Ds = 3 (6.25)

dt

for 2 < Ds ≤ 3

for Ds = 2

(6.26)

The function (Ds ) is remarkably close to unity except when Ds values approach 3. From Equations (6.22) and (6.26), it follows that, for Ds = 3 and for relative pressures approaching unity, Equation (6.24) predicts a ﬁlm thickness that is two times lower than that predicted by Equation (6.17). This is due to the fact that Equation (6.17) tends to overestimate the surface coverage. The overestimation may be as large as a factor of two because, in Equation (6.24), up to one half of all adsorbing surfaces act as a ‘ceiling’ rather than as a ‘base’ for the ﬁlm growth. For very low pressures, Equation (6.24) reduces to N x2 2 (6.27) = C x + (2 − C)x + O Nm − ln(x) For x → 0, Equations (6.17) and (6.24) coincide and are independent of the surface fractal dimension. However, in contrast to Equation (6.17), the term x 2 resulting from Equation (6.24) is independent of the fractal dimension. The region x < 0.15 is of interest for the determination of the monolayer capacity. Thus, for any value of Ds , monolayer capacities Nm can be determined from the standard BET method as long as the experimental measurements are restricted to the range of low relative pressures. Both Equations (6.22) and (6.25) predict that, for x → 1, the growth of the ﬁlm is described by the power law N ln ∝ (Ds − 3) ln(1 − x) (6.28) Nm Film growth is independent of the substrate–ﬁlm interaction, i.e. the parameter C. There is a simple heuristic justiﬁcation of the exponent Ds − 3 in the power law given above [3]. On a planar surface, the mean ﬁlm thickness for x → 1 can be estimated as σ/(1−x). This suggests that, for an arbitrary surface, the ﬁlm volume is V [σ/(1−x)],

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where V (z) is the volume of all points located within the ﬁlm at distances lower than or equal to z. Since the derivative dV (z)/dz can be interpreted as the surface area measured by molecules of diameter z, dV (z)/dz|z = σ should be equal to the speciﬁc surface area Nm σ 2 . Therefore, from Equation (6.17), it is possible to deduce that for a fractal surface and for z σ : z 3−Ds (6.29) V (z) = Bσ 3 σ where B is a constant. After substitution of z = σ/(1 − x), the power law is recovered in Equation (6.25) from Equation (6.29). For z values down to σ , Equation (6.29) can be corrected by including an additional term that ensures correct behavior at z = σ : z 3−D + B σ 3 (6.30) V (z) = Bσ 3 σ The constants B and B can be evaluated when the value of the derivative of the function V (z) at z = σ is V (σ ) = Nm σ 2 and when V (σ ) = Nm σ 3 is the volume of the monolayer. These two conditions uniquely ﬁx the values of B and B . Consequently: V (z) =

Nm σ 3 (3 − Ds )[(z/σ )3−D − (Ds − 2)]

(6.31)

The dependence of the Equations (6.17) and (6.24) and Equations (6.22) and (6.25) on the surface fractal dimension Ds offers a method for measuring Ds from a single experimental isotherm. This may be done by ﬁtting Equations (6.22) or (6.25) to the data obtained at relative pressures close to unity. In such a case, it is necessary to determine how large x should be in order for the asymptotic Equations (6.17) and (6.25) to be valid. A simple estimate proposed by Pfeifer et al. [3] follows from the requirements that N/Nm > 2, since the adsorption energy of molecules adsorbed in the second and higher layers is independent of the energy of interaction with the solid substrate. This estimate yields x > 1 − 2−1/(3−Ds ) as a minimal condition for the relative pressure x to be in the asymptotic regime. For Ds = 2.5, this condition gives x > 0.75, whereas the asymptotic regime is shifted towards higher relative pressures for larger values of Ds . To make a deﬁnitive statement about the fractal dimension and the length range, it would be useful to carry out analyses of adsorption data evaluated using different adsorbates. Unfortunately, such analyses are not always possible. Furthermore, since different adsorbate molecules can only be speciﬁcally adsorbed by a given type of site, some parts of the adsorbent are left almost completely uncovered and the values of Ds obtained for different adsorbates may be different. Equations (6.13) and (6.16) were developed to describe adsorption on a surface of topological dimension Dtop = 2, in three-dimensional space d = 3. Equation (6.17) can also be generalized to the case of an arbitrary topological and space dimension [3]. This generalization may be important because, in the case of the adsorption on fractal soil materials including clusters of aggregated colloids, polymeric substances (e.g. humic substances), etc., Dtop = 1 rather than Dtop = 2. Finally, it should be noted

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that fractal analogues of the BET isotherm have been developed by some authors [49, 51–54]. 6.3.2 Frenke–Halsey–Hill Theory The Frenkel–Halsey–Hill (FHH) isotherm was originally developed to describe the growth of thick ﬁlms and wetting phenomena on a ﬂat surface and was later extended to studying adsorption on fractal surfaces [3, 55]. In contrast to BET theory, FHH theory applies to long-range adsorbate–absorbent interactions and its approach is closely related to the so-called potential theory of adsorption of Eucken and Polanyi (see Ref. [35]). A common method for deriving the FHH adsorption isotherm is to determine the chemical potential of an adsorbate at a distance z from the substrate with respect to that in the bulk state. If the adsorbing surface is ﬂat and if the adsorbed ﬁlm is structurally similar to the bulk liquid adsorbate at a given temperature, then the difference between the chemical potentials, is A (6.32) z3 where A is a (positive) constant. Equation (6.32) is based on the fact that, for a planar adsorbing surface, the molecule–substrate potential can be modeled by the Lennard–Jones (9, 3) function: 9 3 z0 z0 (6.33) − v(z) = ε z z μ = μﬁlm − μbulk = −

where ε and z0 are constant [27]. A ﬁnal important assumption is that the vapor coexisting with the adsorbed ﬁlm is ideal, i.e. p = kT ln(x) (6.34) μ = kT ln p0 Since at distances larger than the monolayer distance (z > σ ) the repulsive energy (z0 /z)9 can be neglected, from Equations (6.32) and (6.33) it is possible to derive A = kT ln(x) (6.35) z3 Equation (6.35) is the FHH isotherm that describes the continuous growth of ﬁlm thickness with increasing relative pressure. For a ﬂat surface, the ﬁlm thickness is related to the adsorbed amount N through z = N σ/Nm . Consequently, for a ﬂat surface, the multilayer ﬁlm growth is governed by 1 N ∝ − ln[− ln(x)] ln (6.36) Nm 3 −

Since Equation (6.35) also provides a characteristic distance scale for a fractal surface, it is possible to construct the surface at the distance z from the fractal

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solid, which is in turn an ‘equidistance surface’. According to the potential theory of adsorption, the adsorption energy on this surface can be approximated by −kT ln(x). The corresponding surface coverage can be found by calculating the volume within the bounds of the equidistance surface V (z). If one neglects the B σ 3 term in Equation (6.30) and normalizes the volume V (z) so that V (σ ) = Nm σ 3 , then (3−Ds )/3 N 1 A = Nm kT σ 3 − ln x

(6.37)

Note that the behavior predicted by FHH theory in the regime N/Nm 1 is physically meaningless because, at small values of z, the repulsive energy term in Equation (6.33) cannot be neglected. The log–log plot of Equation (6.37): ln

N Nm

∝

Ds − 3 ln[− ln(x)] 3

(6.38)

allows direct evaluation of the value of Ds as the slope of a straight line that approximates the experimental data. The basic assumption underlying the FHH approach is that the potential energy over a fractal surface should depend only on z and it is given by the ‘ﬂat surface expression’, i.e. by the relation −A/z3 . This assumption neglects effects due to the energetic heterogeneity of adsorbing surfaces. Since the FHH method is relatively simple, it was widely used for determining the surface fractal dimension of several solids, including active carbons [35, 56], aerogels [57], metal ﬁlms, oxides and related compounds [55]. In particular, Pfeifer and Lui [55] have provided an almost comprehensive review on this topic. Section 6.5 will brieﬂy review applications to soils. The potential energy in Equation (6.33) assumes that the contributions from several nearby adsorbing surfaces are rare, i.e. that multiple-surface effects are unimportant. This adsorption regime is usually called the ‘van der Waals regime’. In the case of adsorption inside a porous network, the surface–molecule attraction is stronger than that predicted by the z−3 power law, due to the ‘overlapping surface’ effect. In such cases, adsorption is governed by the so-called capillary condensation mechanism. A discussion of an FHH isotherm that obeys this mechanism was presented in [35, 55], whereas an outline of the derivation of this isotherm based on [58] is provided below. In the case of adsorption in porous networks, the FHH isotherm was shown to be a generalization of the well-known Dubinin–Radushkevich (DR) equation [59, 60]: θDR (x; R) = exp{−MDR R 2 [kT ln(x)]2 }

(6.39)

which describes the ﬁlling process of a pore whose diameter is R and where MDR is a positive constant. In order to derive the FHH equation, Avnir and Jaroniec [58] considered a network of structurally heterogeneous pores of sizes in the range [Rmin , Rmax ].

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If the pores are independently ﬁlled and the ﬁlling of each pore is described by the DR equation, then the overall adsorption isotherm is Rmax θDR (x; R)J (R) dR

N = ϑ(x) = Nm

(6.40)

Rmin

where J (R) is the pore size distribution. For fractal pores, the size distribution has the form [48, 61] (6.41) J (R) = J0 R 2−Ds where J0 =

3 − Ds 3−Ds R − Rmin

(6.42)

3−Ds max

is the normalization constant. The values Rmin and Rmax are the experimental bounds, i.e. Rmin is the ﬁnest resolution at which fractality prevails (Rmin greater than or equal to the size of the adsorbate molecules) and Rmax is either the upper limit of fractality of the object or the upper size limit of the pores for which Equation (6.39) is valid. Equation (6.40) can be rewritten as ∞ ϑ(x) =

Rmin θDR (x; R)J (R) dR −

0

∞ θDR (x; R)J (R) dR −

0

θDR (x; R)J (R) dR

Rmax

(6.43) where the last two terms can be neglected at relative pressures of x > 0.05 [58]. The ﬁrst integral can be evaluated analytically, giving ϑ(x) =

N = K[−kT ln(x)]−(3−Ds ) Nm

(6.44)

(Ds −3)/2 depends on the adsorbent– where the constant K = 0.5J0 [(3 − Ds )/2]MDR adsorbate interaction. In summary, the FHH approach yields a straight line:

ln N ∝ −

1 ln[− ln(x)] m

(6.45)

which allows determination of the fractal dimension from the slope (−1/m). If adsorption occurs according to the van der Waals mechanism, then the fractal dimension is 1 (6.46) Ds = 3 1 − m whereas if adsorption is governed by the capillary condensation mechanism, it is Ds = 3 −

1 m

(6.47)

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It has been shown [55] that it is possible to distinguish between the two adsorption regimes based upon the magnitude of the exponent (1/m). In particular, adsorption occurs according to the van der Waals mechanism if 1/m < 1/3, whereas it occurs by the capillary condensation (or capillary wetting) mechanism if 1/m > 1/3. 6.3.3 Fractal Analogue of the Dubinin–Astakhov Adsorption Isotherm The DR isotherm describing adsorption in a single pore is a special case of a more general isotherm known in the literature as the Dubinin–Astakhov (DA) isotherm [35, 62]: (6.48) θDA (x; R) = exp{−MDR R n [kT ln(x)]n } where n can assume values from 1 to 6 [59, 62]. In the case of microporous carbons with cylindrical pores, the values of n are usually in the range from 1 to 2. For molecular sieves and clay minerals (slit-like pores), they are close to 2, whereas for zeolites possessing pores in quite complex networks the value of n can be as high as 6. The isotherm described by Equation (6.48) can be extended to the case of porous adsorbents whose pore-size distribution is governed by Equation (6.41). In Equation (6.40), it is possible to replace the integrand θDR (x; R) of Equation (6.39) with θDA (x; R) of Equation (6.48) to obtain [35] N = J0 (Rmin , Rmax ){MDR [−kT ln(x)]n }(3−Ds )/n Nm 3 − Ds n 3 − Ds n γ , Rmax MDR (kT ln x)n − γ , Rmin MDR (kT ln x)n n n where

(6.49)

b γ (a, b) =

t a−1 exp(−t) dr

(6.50)

0

is the incomplete gamma function. The main properties of the isotherm described by Equation (6.49) and its application to experimental adsorption data and related thermodynamic formulae (e.g. adsorption enthalpies, adsorption entropies) are discussed in papers [35, 63], which are devoted mainly to porous carbon adsorbents. 6.3.4 A Generalized Form of the Pore Size Distribution The adsorption isotherm described by Equation (6.49) was derived by assuming that the distribution of the pore sizes is governed by Equation (6.41). However, for real adsorbents, several types of pore-size distribution have been observed, leading to the application of a gamma-type function [62–65]: J (R) = J0 (Rmin , Rmax )R n−1 (R n − R0n )v exp[−Q(R n − R0n )]

(6.51)

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where Q, R0 , ν and n are constants. The normalization constant J0 depends on the pore range [Rmin , Rmax ]. For Q = 0 and R0 = 0, Equation (6.51) reduces to the distribution function described in Equation (6.41) with Ds related to the parameters ν and n through Ds = 3 − n(ν + 1). If the pore-size distribution in Equation (6.51) is substituted into Equation (6.40) and the integration is carried out within the limits (Rmin = 0, Rmax = ∞), then the isotherm N = ϑ(x) = {MDR [−kT ln(x)]n /Q + 1}−(v+1) Nm

(6.52)

is obtained, which can reduce to the FHH isotherm under certain assumptions [65]. Recently, Terzyk et al. [35] have proposed a ‘hybrid’ model that describes adsorption on porous solids and which takes into account the possibility of adsorption in pores and on external surfaces that are characterized by different fractal dimensions. The resulting adsorption isotherm is the sum of two terms, each involving the relevant fractal exponent. The ﬁrst term describes the pore ﬁlling and the second term accounts for the adsorption on external surfaces. Nonetheless, to the best of our knowledge, the hybrid isotherm of Terzyk et al. [35] has not yet been applied to describe experimental data. 6.3.5 Fractal Analysis Applied to Adsorption Data Numerous publications are available on the determination of fractal dimensions from adsorption data and on the application of fractal analysis to studies of adsorption properties and changes occurring during the modiﬁcation of adsorbants. Section 6.5 provides a comprehensive review of studies devoted to soils, whereas this section outlines very brieﬂy studies not involving soils. Nevertheless, the methods, techniques and theoretical approaches that are described in this section are likely also useful for the investigation of soils. In most studies, the main objective was to calculate and compare the fractal dimensions that were obtained using different approaches. For example, Fadeev et al. [13] investigated different fractal regimes in silicas by comparing Ds values obtained from mercury intrusion, N2 and benzene adsorption and the chemisorption of chlorosilanes. Porosity, pore fractal dimensions and surface fractal dimensions were interrelated. The silicas that were investigated behaved as surface fractals in the range of rmin = 0.4 nm to rmax = 0.9 nm. Ds values obtained from adsorption and chemisorption measurements varied from 2.05 to 2.90. The surface fractal dimension of the silicas obtained from mercury intrusion data were higher and ranged from 2.6 to 3. Various techniques used for the determination of surface fractal dimensions were discussed by Park [42]. Mahnke and Mögel [49] applied a modiﬁed fractal BET isotherm to the analysis of N2 adsorption on a TiO2 – ceramic ultraﬁltration membrane. A value of Ds ≈ 2.4 was obtained by ﬁtting their experimental data to the theoretical isotherm of Equation (6.17), whereas a value of Ds = 2.49 was obtained when Neimark’s method was employed (Equation (6.13)). According to these authors, the observed difference in Ds was due to the different

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assumptions used by two methods. In particular, it was stressed that the fractal BET equation ignores totally the lateral attractive interactions occurring in the adsorbed phase. Ds values that were obtained by ﬁtting N2 adsorption data for three shales to the FHH isotherm equation were lower (2.39, 2.24 and 2.20) than those obtained by smallangle neutron scattering (SANS; 2.83, 2.75 and 2.59) [66]. The differences were explained by showing that the theoretical adsorption isotherms that were obtained using a power-law distribution of uncoupled spherical pores exhibited the same general features as the experimental isotherms [66]. The authors thus concluded that the origin of the discrepancies could be attributed to the fact that, at low relative pressure, adsorption is governed by capillary condensation, whereas the van der Waals mechanism prevails at high relative pressure. Since the two possible regimes are partially superimposed, the relationship between the surface fractal dimension and the coefﬁcient (1/m) of the FHH isotherm is not unique (cf. Equations (6.46) and (6.47)); thus, the reliability of the surface fractal dimensions evaluated from adsorption data may be questioned. The relatively large differences observed between Ds values of clay minerals that are calculated from monolayer capacities for various adsorbates and those obtained from SAXS and NMR experiments led Malekani et al. [67] to conclude that the three methods measured a Ds value that was ‘seeing’ the material from a different perspective. In particular, SAXS probed chemical heterogeneity through the contrast in electron density while the adsorption method probed physical surface irregularity and the NMR technique measured the mass distribution of structural sites [35]. Other researchers [68, 69] have reached similar conclusions. In particular, studies on freeze-dried organic gels led to Ds ≈ 2.55 when the FHH isotherm equation was applied to N2 adsorption data and Ds > 2.6 when SAXS was used [68]. For three shale samples, Ma et al. [69] found that the Ds values obtained from N2 adsorption data were signiﬁcantly lower than those obtained from SANS experiments. The authors [69] suggested that the discrepancies were due to the different properties of the dense liquid phase in the small and large pores and to the different volumes that were occupied by the adsorbed ﬁlm with respect to that probed by SANS. In conclusion, while SANS (and SAXS) ‘sees’ the total porosity of a system (including inaccessible pores), adsorbate molecules can only penetrate pores that are both larger than their molecular diameter and accessible. On the other hand, Weidler et al. [70] observed reasonable agreement between Ds values (2.55 ± 0.05) obtained from SAXS and N2 adsorption data for two synthetic goethites whose surface morphology was modiﬁed by HCl treatment in order to obtain different porosities. Micropores were formed by short acid treatments and longer treatments caused the removal of pore walls, leading to the release of Fe. The best agreement between Ds values from N2 adsorption and those from SAXS experiments was observed for treatments ranging from 2 to 5 h. Similar agreement was obtained by Diduszko et al. [71] for Ds values determined on activated carbons when comparing benzene adsorption data and SAXS experiments.

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Jaroniec et al. [72] extensively evaluated fractal dimensions of a model porous solid by simulating adsorption isotherms, then using a density functional approach, and then ﬁnally analyzing data obtained by different methods. The surface fractal dimension depended on the range of pore sizes analyzed, thus conﬁrming the validity of the FHH method. Fractal analysis was also applied to investigate the structural changes of a number of modiﬁed adsorbents. In particular, Weidler and Stanjek [73] studied N2 adsorption on synthetic ferrihydrite samples that were heated at different temperatures for different times. They showed that the fractal dimensions were helpful when interpreting the structural transformations that occurred during thermal treatment. Sonwane et al. [17] used Ds values determined from N2 adsorption measurements, mercury porosimetry, SAXS and SANS to investigate the structure of mesoporous crystalline MCM-4 molecular sieves of different pore sizes. Tatller and Erdem-Senatalar [74] showed that the values of fractal dimensions measured at zeolite A borders over a wide range of reagent concentrations, temperatures and equilibration times depended on the synthesis process used. Furthermore, values were useful to predict the type of zeolite that was formed. In a subsequent study, Erdem-Senatalar et al. [75] observed that the H-form of an NaY zeolite that was obtained by calcination of the NH4 -exchanged form featured a higher adsorption capacity and a higher surface fractal dimension (Ds = 2.43) than those of the original sample. The measurement of the fractal dimensions of the pore spaces in building materials has allowed the development of a mathematical model that relates the water content of the porous material to the relative humidity of the surrounding environment at a given temperature, enabling predictions of water suction and adsorption [76]. The fractal approach has also been suggested for use in predictions of other technologically important properties of building materials, including resistance to frost, thermal conductivity and speciﬁc heat. The Ds values of chars that were produced from three Chinese coals at different burn-off ratios were obtained by N2 adsorption/desorption using the fractal BET equation and by scanning electron microscopy analysis [77]. Ds values reached a maximum in the 35–45 wt% char burn-off interval and then decreased with increasing the carbon burn-off ratio. The fractal approach was also used to investigate adsorption and desorption mechanisms of water vapor on active carbons that were derived from coconut shell, coal, coke and pitch ﬁber featuring a wide range of BET speciﬁc surface areas [78]. Ds values were measured for the water clusters adsorbed on primary carbon centers. Values ranging from 1.64 to 1.67 implied a diffusion-limited aggregation model on a pore wall plane, whereas higher Ds values (up to 1.86), measured at a relative pressure x = 0.95, implied the formation of water clusters that were partly merged vertically to the walls. Rizkalla et al. [79] studied the inﬂuence of surface fractality on the solid–liquid interface reactivity of magnesium oxides, hydroxides and silicates by determining

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their fractal dimensions from mercury intrusion and N2 adsorption data by application of the FHH isotherm equation and the method of Neimark. For magnesium silicate samples, good agreement was found between Ds values obtained by the Neimark method (Ds = 2.85) and by the FHH isotherm equation (Ds = 2.87), whereas a less satisfactory agreement was obtained for magnesium oxides and hydroxides. The mercury porosimetry method usually yielded Ds values that were lower than those obtained by adsorption methods. Furthermore, a correlation between the acid neutralization activity and the fractal properties was shown to exist for these adsorbent surfaces [79]. Podkos´cielny et al. [80] and Kano et al. [81] extended the theories used for gas adsorption to adsorption from binary liquid mixtures. In particular, the former authors [80] investigated the adsorption of aliphatic alcohols, benzene and aliphatic ketones on silica gels having Ds = 2.61 and 2.69, whereas the latter authors [81] studied the adsorption of several organic compounds from water on activated carbons having high Ds values (Ds ≈ 3). Laszlo et al. [82] compared the fractal dimensions of carbonaceous composite materials evaluated by: (a) N2 adsorption data using the FHH equation; (b) adsorption data of gaseous N2 and iodine and aqueous solutions of methylene blue, 2,3,4trichlorophenol and naphthalene using Equation (6.5); and (c) SAXS experiments. Ds values obtained using (a) and (b) were very different. For example, fractal dimensions determined for a carbonaceous adsorbent derived from auto tires gave Ds = 2.56 when applying the FHH equation, as opposed to Ds = 2.96 when using Equation (6.5). The authors suggested that observed discrepancies were potentially due to inaccurate values of the cross-sectional area estimated for the adsorbed molecules and employed in Equation (6.5). For example, for the adsorption of polar molecules with different binding possibilities from aqueous solutions, their orientation at the surface and, consequently, the value of their cross-sectional area might be inﬂuenced by solute composition. Furthermore, carbonaceous samples with a high ash content have a large number of active mineral sites where the conﬁguration of adsorbed polar molecules is uncertain. Numerical modeling studies have also been employed to investigate the behavior of adsorbates by fractal surfaces and fractal pores. For example, Park et al. [83] developed a computer method for which ﬁve fractal surfaces with Ds values ranging from 2.262 to 2.771 were generated. The reaction of CO with O2 on the surfaces was simulated by a Monte Carlo method. The results showed that catalytic CO oxidation proceeded more efﬁciently on surfaces with higher fractal dimensions. In addition, as the surface fractal dimension became higher, the ﬁrst-order kinetic transition (i.e. transition from a highly reactive phase to a practically nonreactive phase) was shifted to a higher partial pressure of CO. This implied that the poisoning of the catalyst surface occurred for more complex surfaces at a higher CO partial pressure. Applications of a molecular dynamics simulations to investigations of the microstructure and dynamic properties of nanoporous silica gels yielded densities down to 0.7 g ml−1 and mass fractal dimensions of 2.6 or higher [84]. The speciﬁc surfaces that

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were calculated for the porous silicas were in good agreement with those obtained from adsorption experiments; thus, the authors suggested that the algorithm that was used could be applied to other computer simulations of adsorption. By measuring surface fractal dimensions using adsorption isotherms, Fuji et al. [85] studied the surface structure of nonporous silica powders that were modiﬁed with n- and t-butyl alcohol and the effects of surface structure on wettability. The monotonous increase of the fractal dimension with increasing density of t-butoxy groups suggested that this functional group was rigid and did not change its conformation. On the other hand, Ds variations for the surface that was modiﬁed with n-butyl alcohol (n-surface) indicated that the structure of n-butoxy groups was ﬂexible and that its conformation changed with the group density. At low densities, the n-butoxy groups tended to lie parallel to the surface, whereas at higher densities their orientation was perpendicular to the surface. By comparing molecular dynamics simulations with experimental results, it was shown that variations of the mobility of the methyl group in the n-butoxy group were closely related to changes of the Ds value of the n-surface. Furthermore, variations in the conformations of the n-butoxy groups as obtained from molecular dynamics simulations were in good agreement with the change in wettability of the n-modiﬁed surface.

6.4 ADSORPTION ON ENERGETICALLY AND GEOMETRICALLY NONUNIFORM SURFACES The description of adsorption on geometrically uniform but energetically heterogeneous surfaces has been frequently discussed in the literature [60, 86]. For planar and energetically homogeneous surfaces, adsorption equilibria have been often described in terms of Equation (6.15), the BET equation, with the constant C being a function of the adsorption energy. The assumption of energetic homogeneity can be removed by considering the total measured adsorption as the sum of ‘local’ processes, each occurring on sites of the same adsorption energy. If these processes are independent, then the total adsorption isotherm ϑt (x) for any assigned surface heterogeneity is (6.53) ϑt (x) = χ (ε)θl (x, ε)

where θl (x, ε) is the local isotherm that governs adsorption on sites of adsorption energy ε, χ(ε) is the energy distribution function and is the range of possible energies of adsorption. This range is usually assumed to be = [εm , ∞), where εm is the lowest value of the energy of adsorption, which is assumed to be equal to the energy of condensation of the adsorbate εc . A number of isotherm equations can be derived assuming different analytical forms of θl (x, ε) and χ (ε). On the other hand, if the function θl (x, ε) is known and ϑt (x) is experimentally determined, then

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Equation (6.53) is an integral equation for χ (ε). Numerous methods of solution of Equation (6.53) with respect to χ (ε) are available in the literature [60, 64, 86]. Although several aspects of adsorption on energetically heterogeneous surfaces have been studied, including that of phase transitions [60, 86], most emphasis was placed on the effect(s) of energetic heterogeneity. Energetically and geometrically, heterogeneities might also be related. The energetic heterogeneity is associated with the presence of adsorbing sites of different chemical properties and with pores of different geometries. Since the adsorbing potential depends on the geometric characteristics of the pores, energetic heterogeneity may also result from geometric nonuniformity. However, the considerations above do not imply that the fractal dimensions must be a function of ε, nor that the energy distribution must be a function of Ds . The simulations discussed in Section 6.2.1 were aimed at checking how the fractal dimension of the geometric adsorbent surface was related to the fractal dimension of a surface passing through the minima of the adsorbent–adsorbate molecular potential. A similar numerical experiment can also be used to search for the dependence of the fractal dimension on the energy of adsorption ε. The simulations in Section 6.2.2 can be repeated [16, 23] by assuming that the molecular energy can vary from site to site with sites distributed differently along the investigated fractal curve. Two topographic models of the surface have been used: (a) the patch model, where sites of equal energy are grouped into patches and (b) the random topographic model, where different sites are distributed randomly over the whole surface. After distributing the adsorbing sites of different energies on the fractal curve, the curve passing through the minima of the ﬂuid–solid potential could be evaluated. The curve was then used to evaluate the fractal dimension according to the method described in Section 6.2.1. The main result was that the fractal dimension Ds was, in general, only slightly dependent upon the energetic heterogeneity of adsorbing centers on the surface. Nonetheless, the numerical experiments that were performed should not be considered as being exhaustive of all of the possibilities of a distribution of energetic centers along the fractal curve. Because these experiments have a statistical character, it is impossible to ascertain whether the results could be applied to any fractal surface. Very probably, surfaces exist for which the inﬂuence of energetic heterogeneity on the fractal dimension is different from that described above. In particular, the simulation described above was performed for adsorption on external surfaces, i.e. where multiple surface effects were neglected. 6.4.1 Models Based on No Correlations between Energetic and Geometric Heterogeneities The function χ (ε) refers to the total energetic heterogeneity of an adsorbent, and provides no information about the topography of energetic centers on a surface. Similarly, Ds is a global characteristic of the adsorbent, i.e. it does not refer to any speciﬁc site on the surface. However, in the case of a patchwise topography, it is

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possible to imagine a model in which each patch is characterized by its own fractal dimension. On the contrary, no possibility exists to deﬁne the fractal dimension for a given kind of energetic center for the case of a random topography. Consequently, Equation (6.53) can be generalized as [15, 23, 30, 44] ϑt (x, Ds ) = χ (ε)θl (x, ε; Ds ) (6.54)

where θl (x, ε; Ds ) is the local adsorption isotherm on the surface as characterized by the Ds value of the entire surface. By integrating Equation (6.54) with different functions χ (ε), the corresponding isotherm equations can be developed. In particular, by assuming that the local adsorption isotherm can be described by Equation (6.17), the fractal BET equation, one obtains N f (x, Ds ) g(x) exp[(ε − εc )/kT ] ϑt (x, Ds ) = dε (6.55) = χ (ε) Nm g(x) 1 + g(x) exp[(ε − εc )/kT ]

where the functions f (x, Ds ) and g(x) are deﬁned in Section 6.3. The form of Equation (6.55) is identical to that developed for geometrically uniform surfaces, such that the mathematical methods described in [47, 60, 64] can be applied. In particular, using the so-called condensation approximation, where BDR is a positive constant and gm = exp[(εc − εm )/kT ], it can be shown that g(x) f (x, Ds ) ϑt (x, Ds ) = exp −BDR ln (6.56) g(x) gm is related to a quasi-Gaussian distribution of the form 2 ε − εm ε − εm χ(ε) = 2BDR exp −BDR kT kT

(6.57)

In the case of ﬂat surfaces, Equation (6.56) reduces to the DR isotherm in Equation (6.39). The function χ(ε) possesses a single maximum, but in many cases the extremely complex nature of the adsorbent requires a more general approach that can allow for several maxima in the distribution functions, each corresponding to adsorption centers. Jaroniec and Madey [60] proposed the use of the following function: ∂ 2 χ (ε) = exp (6.58) bi [(ε − εm )/kT ] ∂ε i=1,L where {b1 , b2 , . . . , bL } are the distribution parameters, which must be selected in such a way that χ (ε) ≥ 0. The distribution in Equation (6.58) is normalized: χ (ε) dε = 1 (6.59)

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From Equations (6.55) and (6.58) one obtains N f (x, Ds ) i ϑt (x; Ds ) = exp = bj ln [g(x)/gm ] Nm g(x) j =1,L

(6.60)

Any application of Equation (6.60) to experimental data requires knowledge of the surface fractal dimension of the adsorbent investigated so that the evaluation of the energy distribution function is reduced to an approximation of the experimental data using Equation (6.60). This calculation is most conveniently performed using a method quite similar to that described by Jaroniec and Madey [60] for geometrically uniform surfaces. The approach described above was applied to the analysis of N2 and H2 O adsorption data on different horizons of an Orthic Luvisol [15, 30]. Ds values of the soil samples were determined from adsorption and monolayer capacities of both adsorbates on grain samples of different sizes (see Section 6.2.1). Ds values calculated from water adsorption data ranged from 2.25 to 2.35, whereas those evaluated from N2 adsorption data were slightly higher, ranging from 2.38 to 2.40. Because of the different chemical character of N2 and water vapor, the adsorption centers for these substances are different. In particular, only external adsorbing surfaces are available to N2 molecules, whereas water molecules may also explore the internal spaces of soil particles. As a result, monolayer capacities of water vapor are higher than those resulting from N2 experimental data. Experimental isotherms were used to determine the coefﬁcients {bi } and compute the distribution functions χ (ε) for soil samples. Finally, adsorption on energetically and geometrically nonuniform surfaces has also been studied by Lefebvre et al. [87], Giona et al. [46] and Tsunoda et al. [88].

6.4.2 Generalized Fractal BET Equation for Adsorption on Geometrically and Energetically Nonuniform Porous Solids It has been stressed in this chapter that the statistical independence of the fractal dimension and the energy distribution function can be assumed only for adsorption on external surfaces or in mesopores. Adsorption by a fractal porous network that takes into account the possible correlations between Ds and adsorption energies has been described by Rudzin´ ski et al. [89, 90]. In the simple case of a porous network with ‘chemically uniform’ surfaces, the dependence of the adsorption energy on the pore size can be easily determined. In particular, if the network is composed of slit-like pores and the interaction of a molecule with a single pore surface is described by Equation (6.34), then the potential energy of a molecule inside the pore can be calculated by summing the potentials from the two surfaces. The minima on the potential curves are identiﬁed as the adsorption energies. If the distribution of pore sizes J (R) is fractal, then χ (ε) depends on the type of distribution, and in turn on the Ds value. Rudzin´ ski et al. [89, 90] postulated

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the following functional form for the dependence between the energy of adsorption and the pore-size parameter R: R = exp[−F (ε)]

(6.61)

where F (ε) is a function describing the individual features of the adsorption system. By assuming that Rmin = 0, the following isotherm equation could be derived: ϑt (x, Ds ) =

r0 Rmax

3−Ds

2 2 x 3 − Ds ∂ F (ε) kT ln exp − 2 ∂ε 2 x 0 ε=ε0 (6.62)

where x0 = exp[(ε0 −εc )/kT ], r0 = exp[−F (ε0 )] and ε0 is a constant. Equation (6.62) is equivalent to Equation (6.56), except that the coefﬁcient [kT ln(x/x0 )]2 depends linearly on the fractal dimension Ds . One of the assumptions underlying the theory described above was that the lower fractality range was zero. The problem of the limits [Rmin , Rmax ] has already been discussed in Section 6.3. While accepting the ‘classical’ pore size distribution, Equation (6.41), along with its limited range of applicability, [Rmin , Rmax ], one can describe only a part of the experimental isotherm corresponding to pressures in the relevant range [pmin , pmax ]. To derive an isotherm equation that could be valid over the entire range of pressures, Rudzin´ ski et al. [89, 90] considered a new model of geometrical heterogeneity, according to which real adsorbents are never totally irregular (fractal) and never perfectly regular, but instead they have a structure that is intermediate between the two extremes. The isotherm for such adsorbents, which were named ‘partially correlated’, has the form

N/[Mm f (x, Ds )] ln 1 + N/[Mm f (x, Ds )]

= A0 + A1 kT ln g(x) + A2 [kT ln g(x)]2

(6.63)

where {Ai } are the constants. Unfortunately, the evaluation of the parameters that are described by Equations (6.62) and (6.63) requires the use of rather sophisticated numerical procedures because the isotherms are in a nonlinear form relating the amount of adsorbed molecules to the relative pressure. To evaluate these parameters, Terzyk et al. [35] constructed a special algorithm that was able to ﬁt the theoretical isotherm described by Equation (6.63) to the experimental data. This algorithm was a hybrid of a random search method and an evolutionary searching procedure. Numerical calculations of the parameters in Equations (6.62) and (6.63) require special care, and large differences in values might be obtained using different strategies for the calculations.

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6.5 FRACTAL ANALYSIS OF ADSORPTION DATA OBTAINED ON SOILS AND SOIL MINERALS 6.5.1 Soil Minerals Soil solid phases consist of a very complex mixture of inorganic and organic components that determine the physical and physicochemical properties of the soils. Organic components, which are usually present in much smaller quantities than inorganic components, include plant and animal residues at various stages of decomposition, living cells and tissues of soil organisms, and substances synthesized in soil, the so-called humic substances. Inorganic components include primary and secondary minerals ranging in size from very ﬁne colloids to large rocks. In soils, the sand and silt fractions consist exclusively of primary minerals, whereas minerals in the clay fraction are predominantly secondary and include layered silicates, and various oxides, carbonates and sulfur minerals. The term ‘clay mineral’ refers to soil inorganic materials less than 2 m in effective diameter and comprises minerals with the sheet silicate structure of phyllosilicates. The secondary minerals that are dominant in the soil clay fraction are kaolinites, smectites, vermiculites, mica and especially montmorillonites. Soil clay minerals often differ in their properties from those of pure minerals. In particular, they are generally much less ordered and smaller in size than pure minerals. Often, neighboring particles or sheets overlap. Furthermore, they exhibit surface irregularities at the molecular scale and a nonuniform microporous structure. Determinations of the fractal dimensions of clay minerals have been reported in several papers [11, 21]. For montmorillonite, a value of Ds ≈ 2 has been measured [11, 21, 91, 92], whereas other authors [87, 93, 94] have obtained a higher value of Ds = 2.7, e.g. by measuring N2 isotherms. Sokołowska and co-workers [15, 16] analyzed the monolayer capacities of montmorillonite with nonpolar and polar adsorbates and found values of Ds = 2.1 and 2.3 respectively. Similar differences in Ds values were also observed for kaolinite [16, 21, 91, 93]. The origin of these differences is probably related to the fact that experiments were carried out on mineral samples of different origins. Small, but variable, amounts of strongly bound water could change surface geometries. One of the questions that can be answered with the help of adsorption measurements concerns the microtexture of natural clay minerals. Several idealized models for the texture of soil clays (see [5]) have been considered, but rather than assuming one model a priori, one should try to gain useful information from experimental relationships between the size of clay particles and apparent density or surface area and internal porosity, as described in Sections 6.1 and 6.2.1. Experiments aiming at the evaluation of the microtextures of clay minerals were carried out by Ben Ohoud and van Damme [95], who studied kaolinite, sepiolite, palygorskite and 20 monoionic montmorillonite samples. The accessible surface area S of consecutive fractions of size r was measured by N2 adsorption using the classical BET method, whereas the open porosity P was measured from the amounts of adsorbed N2 at a relative vapor

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pressure of x = 0.99. These authors [95] also determined mass (fragmentation) fractal dimensions by measuring the volume occupied by a mass of powdered particles of a given size. They found, within experimental error, a value of DM = 3 for all clays studied, i.e. the samples were not mass fractals. In contrast, the behavior of the accessible surface area and open porosity of samples was much more interesting [95]. For the ﬁbrous clays, kaolinite and La-saturated montmorillonite, the values of S and P were independent of the grain size, i.e. the surface fractal dimension Ds = 3, which suggested that the ideally porous model was applicable in these cases. All of the other montmorillonite samples were characterized by a linear dependence of ln(S) versus ln(r) over a quite wide range of r values. The slopes of these curves led to nontrivial values of Ds (2 < Ds < 3). Furthermore, the same power laws were obtained for porosity with scaling exponents that were in very good agreement with Ds values. In addition, pronounced correlations between Ds values and the coherence length, i.e. the thickness of the ordered stacks of elementary clay sheets determined from X-ray diffraction, were observed [95]. A question has arisen with regard to the possible existence of a relationship between the properties of exchangeable cations and Ds values, which implies the possible capacity of different cations to generate stacks of smectite lamellae. In this respect, highly charged and/or small cations with a large polarizing power would be expected to have a strong local ordering tendency and, thus, be able to form these stacks. The left panel of Figure 6.2 displays the values of Ds obtained from N2 adsorption data [30, 95], whereas the right panel shows the average adsorption energies of water vapor on different monoionic forms of montmorillonite [96]. The trends shown here

a

Pb Ba Cu

2.8

2.4

Na

Ds 2.6

Li K Bi

0

3000

La,Fe Mg,Co Ca Ni Fe Ca Co Zn

Cs

1

ε–εc, kcal/mol

3

b

Ba

Ca 2000

Sr K Na

Ti 2 Z

Mg

Li

3

4

1000

1

2 Z

Figure 6.2 (a) Values of surface fractal dimensions Ds obtained from N2 adsorption data from the authors (diamonds), from [30] (circles), and from [95] (stars). (b) Average adsorption energies of water vapor (data from [96]) on various monoionic forms of montmorillonite as a function of the cation charge Z. Reproduced by permission of the Polish Academy of Sciences.

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are consistent, i.e. highly charged trivalent ions yield very high Ds values, bivalentions yield intermediate values and monovalent ions yield generally lower Ds values. The same trend is valid for the average adsorption energies, although a closer look shows that the correlation between the polarizing power and Ds is not simple. For example, in the presence of the large and monovalent ion Cs+ , a very high Ds value is obtained. Thus, the ﬁne structure of the hydration shells in the interlayer space and type of the cation interacting with montmorillonite sheets should be taken into account. Recently, Huang et al. [97] provided a good illustration of the contribution of fractal analysis to the determination of the adsorption mechanism and the effects of cation exchange on the pore and surface structure of montmorillonite. Results of fractal analysis of N2 adsorption data were compared with X-ray diffraction and differential scanning calorimetry patterns. Subsequently, isotherms were measured for benzene, hexane and cyclohexane adsorption onto cation-exchanged montmorillonite. An important ﬁnding was that Ti cations produced structural modiﬁcations that caused the opening of pores, with an increase of pore size, pore volume and pore connectivity with the result of an instantaneous smoothing of the surface. Wang et al. [98] studied the effects of the adsorption of basic dyes such as basic green and basic violet on a Ti-exchanged calcium montmorillonite. The changes in surface and pore structure and the adsorption behavior of the titanium montmorillonites were evaluated using fractal analysis and the FHH equation applied to the experimental N2 adsorption isotherms. The decrease in the BET surface area of the montmorillonites that followed the adsorption of the dyes was interpreted in terms of a partial reduction in surface roughness (surface screening effect) and a partial inhibition of N2 displacements within the pores (pore blocking effect). Changes in the Ds value before and after dye adsorption were used to evaluate these two effects with respect to the changes in mean pore size. Bentonite, which usually contains more than 50 % montmorillonite, is often used as a back-ﬁlling buffer material to prevent groundwater contamination because of its low hydraulic conductivity and ability to expand and ﬁll up openings. In this context, Xu and co-workers [92, 94] proposed a method to determine the surface fractal dimensions of swelling materials. They derived a relationship between the normalized water volume and the vertical overburden pressure of the form Vw ∝ pDs −3 (6.64) Vm where Vw is the water volume conﬁned by the montmorillonite in bentonite, Vm is the volume of montmorillonite in the bentonite, and p is the overburden pressure. The fractal dimension of Wyoming bentonite obtained from the swelling test, Ds = 2.64, was nearly equal to that evaluated from N2 adsorption using the FHH equation, Ds = 2.65, and larger than that measured by mercury porosimetry, Ds = 2.44. The authors [94] interpreted this discrepancy as resulting from an ‘oversimpliﬁed

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assumption’ in the interpretation of mercury porosimetry data, which considers the pore space as bundles of nonintersecting tubes. The so-called pillared montmorillonities are prepared by intercalating oligomeric metal hydroxide ions into interlayers [31]. Pillared clays have high surface areas and the structural parameters of their micropores vary as function of the pillaring material and preparation conditions. These materials show promise for use as highperformance adsorbents and active catalysts. Yamazaki et al. [99] performed N2 adsorption experiments to investigate the structure of pores and sizes of pillars of montmorillonites pillared with Al and Cr. Szücz et al. [32] prepared a series of substrates with different Pd contents (Pd-nanophase reactors) by depositing Pd nanoparticles on the surface and in the micropores of a montmorillonite pillared with aluminum oxide. The N2 adsorption and SAXS data showed that the deposition of Pd nanoparticles inside the micropores decreased the speciﬁc surface area and the porosity of the substrate. Ds values determined from SAXS experiments increased with increasing amounts of deposited Pd. In particular, the value of Ds increased from 2.76 for nonmodiﬁed pillared montmorillonite to 2.83 for montmorillonite containing 2.09 % Pd and 2.9 for the substrate containing 2.53 % Pd. In general, the surfaces of soil clay minerals are covered by amorphous oxyhydroxides and organic matter [100]. Fine grains of soils are often composed of random aggregates of clays, oxides and organic matter, including organomineral complexes, which are very common in soils. The structure and reactivity of both the organic and the mineral components are modiﬁed drastically when combined. Tombácz et al. [101] carried out a systematic investigation on samples of montmorillonite and kaolinite covered with natural organic matter and synthetic humic acids in order to determine the role of organic coatings on clay interfacial properties. A rough polyionic layer was shown to form on the surface of clay particles, causing a decrease of in the pore volume and the speciﬁc surface area and a simultaneous increase of the surface fractal dimension from Ds = 2.25 (uncoated kaolinite) to Ds = 2.34 (coated kaolinite) and from Ds = 2.47 (uncoated montmorillonite) to Ds = 2.61 (coated montmorillonite). Similar studies were conducted by Sokołowska and Sokołowski [102] for evaluating changes to the surface properties of kaolin following interactions with a natural humic acid. Józefaciuk et al. [38] used water vapor adsorption/desorption and mercury porosimetry to investigate modiﬁcations occurring in bentonite, illite and kaolin samples that were treated with randomly methylated -cyclodextrin (RAMEB). With increasing amounts of RAMEB, the surface areas and micropore volumes of the minerals decreased, whereas their surface fractal dimensions increased. The observed changes suggested the occurrence of strong interactions between the RAMEB and the clay minerals, in contrast to previous ﬁndings [103] that implied that RAMEB and minerals could coexist as separate, nonreactive phases.

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6.5.2 Whole Soils Various processes and phenomena that occur in soils, e.g. the movement of gases, the movement of water due hydraulic gradients and the distribution of organic and inorganic constituents, are markedly inﬂuenced by the irregularities of both the external surfaces of soil particles and the inner surfaces of soil pores. The fractal approach offers the possibility of being able to quantify the intrinsic heterogeneity of soil and related soil properties at different scales. Surprisingly, except in the case of some soil components like carbonaceous adsorbents and clay minerals, adsorption methods have rarely been employed to investigate the geometric heterogeneity of soils. Adsorption studies of soil geometric heterogeneity which have used a fractal approach include: (i) the comparison of the fractal dimensions evaluated from adsorption data with values obtained by other methods [24, 33, 43, 102, 104–110]; (ii) the use of the fractal dimension, in some cases together with the energy distribution function χ(ε) to characterize soils [24, 43, 44, 107–114]; and (iii) the monitoring of changes to soils caused by degradation, organic matter removal, alkalinization, acidiﬁcation, cultivation, etc. [14, 33, 38, 39, 41, 43, 44, 102, 104, 105, 109, 114, 115]. Surface fractal dimensions of a number of Cambisols and Luvisols were determined using the FHH equation from data obtained from N2 and water vapor adsorption isotherms. Values were compared with those obtained from the mercury intrusion method and with mass fractal dimensions that were evaluated from particle-size distributions using a modiﬁed number-based method [108] (Figure 6.3). This method was proposed by Kozak et al. [116] in order to correct some inconsistencies of previous approaches 3 R = 0.946

N2 2.8

H2O Hg

Ds

2.6

2.4

2.2

2

2.4

2.6

2.8

DM

Figure 6.3 Correlations between the mass fractal dimension DM and the surface fractal dimensions Ds obtained from N2 and H2 O adsorption data and from mercury porosimetry data for a number of Cambisol and Luvisol samples [108]. The correlation coefﬁcient R refers to the latter data. Reproduced by permission of Wiley–VCH.

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in estimating the mass fractal dimensions of soils [e.g. 117, 118]. Ds values evaluated from H2 O adsorption data ranged from 2.02 to 2.25 for Luvisols and from 2.02 to 2.37 for Cambisols. Ds values obtained from N2 isotherms were generally higher, ranging from 2.14 to 2.41 for Luvisols and from 2.10 to 2.59 for Cambisols. These results suggested that different surface sites with different fractality were responsible for adsorption of H2 O and N2 molecules and that the gas–solid interface was smoother for H2 O adsorption. The capacity of H2 O molecules to form hydrogen bonds may be an important factor for smoothing the adsorbed layer. However, not only water adsorption, but also water absorption may occur in soils, potentially distorting evaluations of the fractal dimension. In contrast, N2 molecules are mainly adsorbed via van der Waals forces; thus, the geometry of the interface might be expected to resemble that of the underlying solid more closely. No signiﬁcant correlation related Ds values evaluated from N2 adsorption isotherms and mercury porosimetry data for the soils investigated, possibly because these two methods provide information on different geometrical scales. On the other hand, DM values of these soils were positively correlated with Ds values obtained from mercury porosimetry [108]. Sławin´ ski et al. [106] applied the so-called ‘pore–solid–fractal’ (PSF) model to describe water retention curves that were measured for six peat soils in order to obtain their DM values. Furthermore, Ds values were determined by applying the FHH equation to water vapor adsorption and desorption and N2 adsorption isotherms. The PSF model gave DM values ranging from 2.88 to 2.94. In contrast to the results described above for Cambisols and Luvisols [108], Ds values obtained from N2 adsorption isotherms (2.25 ≤ Ds ≤ 2.575) were lower than those evaluated from water adsorption (2.58 ≤ Ds ≤ 2.70) and desorption (2.63 ≤ Ds ≤ 2.75) isotherms. Removal of carbohydrates from soil samples that were collected from a longterm (12 years) managed grassland increased Ds values from 2.75 to 2.85 [33]. Furthermore, these values were close to those obtained from mercury intrusion experiments on the same soils (cultivated with barley) [14]. Based upon these experiments, Pachepsky et al. [14] concluded that long-term management affected the speciﬁc soil surface area, but not the Ds values. Although no differences in Ds values were measured in the speciﬁc pore dimension range from a few nanometers to about 100 nm, differences in surface roughness could exist in other ranges [42, 105, 119]. Ds values obtained by applying the FHH isotherm equation to water vapor adsorption data were used to characterize various Bulgarian soils, including leached chernozem, smolnitza, mountainous light-gray podzol and meadow zinnamonic soils [110] and to describe the changes in physicochemical properties with the proﬁle depth of solonetz soils [45]. Water vapor and N2 adsorption data were collected for several Polish alluvial soils [24, 109] with the aim of determining their surface and energetic heterogeneity using the theory outlined in Section 6.4. All samples investigated contained only small amounts of micropores and mesopores; therefore, the fractal dimensions were evaluated using the FHH equation and by assuming a van der Waals adsorption regime. The Ds values that were obtained were always smaller than 2.5, and those obtained

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from the water adsorption isotherm were generally smaller than those estimated from N2 adsorption data. Weak correlations were found between the Ds values and the clay fraction percentages (Figure 6.4) and the cation exchange capacity. Even weaker correlations were observed between the average adsorption energies and these parameters. Furthermore, no correlation was observed between Ds values obtained from H2 O and N2 adsorption isotherms or between Ds and average adsorption energy. N2

Ds

2.4

2.2

2.0

H2O

0

10

20 30 Clay fraction, %

40

Figure 6.4 Correlations between the fractional clay content and the surface fractal dimension Ds obtained from H2 O and N2 adsorption data of several alluvial soils [109]. Reproduced by permission of Elsevier.

The fractal approach provides a convenient means for monitoring structural changes that may occur during various types of soil alteration. For example, soil acidiﬁcation generally leads to a loosening of soil structure and an increase in the water permeability, whereas alkalinization leads to soil slacking and a loss of water permeability. These effects imply contrasting change of the pore system for acid and alkaline soils. In contrast, Józefaciuk et al. [39] observed that pore volumes and radii generally decreased both with increasing soil acidiﬁcation or alkalinization, whereas the fractal dimension of mesopores decreased after acidiﬁcation but increased slightly after alkalinization. The decreased fractal dimension with acidiﬁcation corresponds to a smoother porous soil surface. Model investigations performed on clays [101, 102, 120, 121] and whole soils [43, 111, 122] showed the important role of organic matter in determining the surface heterogeneity of these adsorbents. In particular, Sokołowska et al. [43] investigated in the laboratory how the leaching of dissolved organic carbon (DOC) by solutions of NaOH at various pH values could alter surface properties of an Anthropogenic Regosol. The experiment was conducted on two soil portions: one was unchanged (A), whereas the other (B) was subjected to an acid pretreatment of 0.01 m HCl for 2 days, wash with 0.01 m NaCl and distilled water and a ﬁnal drying. The soil portion B lost

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about 5% of its total organic carbon, while the remaining organic material was peptized and deﬂocculated and the inorganic soil components experienced surface changes. The pore size distributions of both soil samples were obtained with the FHH equation on data measured by mercury intrusion and water vapor adsorption. Measured soil pore radii did not change from pH 3 to about pH 7, whereas they increased linearly above pH 8 (Figure 6.5). On the other hand, Ds values decreased almost linearly from about 2.8 (pH 3) to about 2.65 (pH 12), corresponding to an increased leaching of organic matter over the pH range investigated. Similar investigations were carried out on other soils [114, 122]. 2.4

a

2.8

20

1.6 A B

1.2

Ds

DOC, mg/g

R, nm

2

b

2.7

10

0.8 8

4 pH

12

4

8

2.6 12

pH

Figure 6.5 (a) Amount of DOC (top circles) and average pore radius R (bottom circles) and (b) surface fractal dimension Ds as a function of pH for two soils (A and B) [43]. Reproduced by permission of World Scientiﬁc.

Peat soils are typically rich in organic matter and subject to drainage; thus, secondary transformations depend greatly on drainage intensity. As a result, peat soils may exhibit different physicochemical properties, including different amounts of bound water, different kinds and contents of humic substances, and different sorption and exchangeable properties. Transformations in peat soils are accompanied by (i) changes in the geometrical and energetic characteristics of the soil particle surface, which can be revealed by evaluating soil porosity from mercury porosimetry data, and (ii) energetic and geometric heterogeneity, which can be evaluated from water vapor and N2 adsorption isotherms [44, 106–108, 123]. Weakly transformed peats exhibited signiﬁcantly smaller Ds values (about 2.4) than extensively transformed ones (Ds > 2.6.). The Ds values obtained from mercury porosimetry data were usually smaller than those derived from N2 adsorption data ﬁtted in the FHH isotherm. These results indicated that secondary transformation of peat soils caused an increased surface roughness. Further studies carried out on peat samples that were thermally treated at 50, 100 and 150◦ C, showed that the surface roughness of most transformed

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samples diminished with an increase in temperature [44, 107]. For some samples, however, the temperature effect was opposite. In general, it was difﬁcult to relate these changes to other physicochemical characteristics of the peat soils investigated.

6.6 CONCLUSIONS AND RECOMMENDATIONS This review has illustrated various applications of fractal concepts to the description of adsorption/desorption processes on environmental materials. The surface fractal dimension Ds can be determined by several independent methods, among which those based on adsorption measurements are shown to be the most convenient and straightforward. However, a comparison of fractal dimensions calculated from different approaches was shown to be problematic; therefore, the need for further studies in this ﬁeld is evident. Particular attention has been paid to the discussion of the fractal approach to the adsorption of gases by clay minerals and soils. Further promising developments are expected when applying fractal concepts in new environmental ﬁelds. For example, promising research has been conducted very recently on the fractal description of ion adsorption and on the inﬂuence of surface roughness on the electric properties of the double layer [124, 125]. Processes and models leading to the preparation of adsorbents with fractal surfaces have also been investigated [126]. Adsorption processes are controlled by the energetic properties of the absorbent and absorbate, to the distribution of sites where adsorption occurs and to the geometric heterogeneity of the adsorbent. Discriminating the contribution of each of these factors to the adsorption process as a whole is particularly challenging and requires an appropriate combination of experiment and computer modeling. Despite the substantial progress in the latter ﬁeld, only rather simple models of selected soil components (e.g. clay minerals) have been used in computer simulations so far [127, 128]. Largescale off-lattice simulations of gas adsorption onto soils remain extremely difﬁcult. Therefore, the further development of methods for performing computer simulations to investigate the formation of natural adsorbents and the structure of adsorbed layers will be extremely helpful. In the past few years, the concept of ‘quenched–annealed’mixtures has been widely applied to the study of the structure and thermodynamic properties of ﬂuids adsorbed in disordered porous matrices [129]. According to this approach, computer simulations and integral-equation theories [129] describing gas adsorption have been proposed that are based on the equilibration of adsorbate molecules in a matrix of particles frozen in a disordered conﬁguration and sampled from a given probability distribution. The novelty of this approach is that the structure of the porous network can be implicitly incorporated into the description by using a static structure factor of the adsorbent. In particular, the porous adsorbent can be considered as being formed by processes leading to the formation of mass fractals, e.g. particle–cluster or cluster–cluster diffusion-limited aggregation processes [130]. Although the concept

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of ‘quenched–annealed’ mixtures appears to be very promising for modeling gas adsorption onto soils, to the best of our knowledge, only a single paper that describes preliminary attempts to apply such models to adsorption processes in soils [131] has been published on this topic. Developments in this direction are encouraged and expected.

LIST OF SYMBOLS AND ABBREVIATIONS BET DA DOC DR FHH NMR RAMEB SANS SAXS BDR C d DM DP Ds Dtop J (R) m N Nm Np p p0 rmin rmax R Rmin Rmax S u(r) v(z) V Vm

Brunauer, Emmet and Teller theory of adsorption and isotherm equation Dubinin–Astakhov isotherm equation dissolved organic carbon Dubinin–Raduskhevich isotherm equation Frenkel, Halsey and Hill theory of adsorption and isotherm equation nuclear magnetic resonance randomly methylated -cyclodextrin small-angle neutron scattering small-angle X-ray scattering constant of the Dubinin–Raduskhevich isotherm equation constant of the Brunauer, Emmet and Teller isotherm equation space dimensionality mass fractal dimension pore fractal dimension surface fractal dimension topological dimension pore-size distribution function constant of the Frenkel, Halsey and Hill isotherm equation amount of adsorbed molecules monolayer capacity maximum adsorbed amount in the pores pressure saturated vapor pressure lower range of fractality upper range of fractality pore size (diameter) minimum pore size maximum pore size surface area Lennard–Jones (12, 6) interaction potential molecule–surface potential energy volume molar volume

214

Vp x χ ε εi εm φ μ μ σ ϑ ϑi ϑt

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pore volume relative pressure adsorption energy distribution function range of adsorption energies adsorption energy adsorption energy in the ith adsorbed layer the lowest value of the adsorption energy surface tension chemical potential difference of chemical potentials molecular diameter adsorption isotherm, surface coverage local adsorption isotherm total adsorption isotherm

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7 Applications of Fractals in the Study of Humic Materials James A. Rice South Dakota State University, Department of Chemistry & Biochemistry, Box 2202, Brookings, SD 57007-0896

7.1 INTRODUCTION Natural organic matter (NOM) in the environment can be classiﬁed into two broad categories, humic and nonhumic substances [1–5]. Nonhumic substances are generally described as those NOM components that belong to recognized molecular compound classes, such as sugars, peptides, lipids, fatty acids, etc. Humic materials are NOM components that cannot be so conveniently compartmentalized. They are generally described as refractory, heterogeneous, polyelectrolytic organic substances that color waters, sediments, or soils brown or black. Humic substances are divided into three fractions based on an operational deﬁnition that traces its origin back into the soils literature. They are operationally deﬁned by their solubility in an aqueous solution as a function of the solution pH (Figure 7.1): humin, which is insoluble in an aqueous solution at any pH value; humic acid, which is soluble in an alkaline aqueous solution but precipitates in acidic conditions; and fulvic acid, which is soluble in an aqueous solution at any pH value.1 A variety of texts describe the chemical nature of each fraction in detail [1–5]. Typical chemical characteristics of each fraction are shown in Figure 7.2.

1 Humin’s insolubility obviates its presence in a dissolved organic matter aquatic sample. In aquatic systems, the operational deﬁnitions of humic and fulvic acids are stated somewhat differently [4]. Humic acid and fulvic acid are deﬁned by the aqueous solution pH necessary to desorb them from XAD-8 resin: fulvic acid can be desorbed with water; humic acid is desorbed with an alkaline aqueous solution. However, the deﬁning operational characteristic, solubility in an aqueous solution as a function of pH, is consistent in both operations. Thus, the use of the term fulvic acid in this paper refers to the humic substance fraction soluble in aqueous solution at any pH value, and the term humic acid refers to the humic substance fraction soluble in alkaline aqueous solution but insoluble under acidic conditions.

Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems c 2008, IUPAC and Kevin J. Wilkinson

Edited by Nicola Senesi

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Humic Materials Dissolve in Basic Solution & Centrifugation Residue

Supernatant

Acidification & Centrifugation Precipitate

Humin

Humic Acid

Supernatant

Fulvic Acid

Figure 7.1 Operational nature of the deﬁnitions of each of the three fractions of humic substances.

The characteristic that remains associated with each humic fraction after its separation from an NOM sample is an extreme degree of heterogeneity. The general consensus is that each humic fraction is a very complex mixture [8, 9] whose components represent a continuum of chemical and molecular properties. The literature is replete with references to the inability to subdivide humic and fulvic acid into components that represent a quantitatively signiﬁcant fraction of the whole sample [8]. It has even been suggested that ‘. . . no two molecules of a humic substance are exactly alike . . .’ [10]. There has been some success in further fractionating the components of humin [11]. But even this fractionation ends with a material that conforms to the operational deﬁnition of humic acid, though its chemical characteristics allow it to be distinguished from humic acid. MacCarthy and Rice [12] have proposed that not only is this heterogeneity the deﬁning characteristic of humic materials, but in fact it also represents an environmental feedback that has emerged to ensure that humic materials persist to perform their ecological functions. In any case, the heterogeneity of each humic fraction limits the application of conventional structural determination approaches. If the deﬁning chemical characteristic of humic materials is, in fact, the lack of a single, characteristic chemical structure, then a primary structure of humic or fulvic acid does not exist (though certainly the individual components that comprise a humic material have primary structures)! Even the determination of secondary structures becomes quite problematic from this perspective. What is left to explore are the tertiary and quaternary structures which result from molecule–molecule, aggregate–molecule, and aggregate–aggregate interactions. Conceptual models of humic materials acknowledging this realization have begun to appear [13–15]. These types of interaction, and these types of structure, are ideally suited to study by a fractal approach.

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Humic Acid Elemental Composition 55.1%C, 5.0%H, 3.5%N, 1.8%S, 35.6%O Acidic Functional Group Content Total Acidity: 5.0 meq /g Carboxyl Acidity: 2.5 meq /g 200 150 100 50

0

Fulvic Acid Elemental Composition 46.2%C, 4.9%H, 2.5%N, 1.2%S, 45.6%O Acidic Functional Group Content Total Acidity: 8.7 meq /g Carboxyl Acidity: 5.2 meq /g 200 150 100 50

0

Humin

Elemental Composition 56.1%C, 5.5%H, 3.7%N, 0.4%S, 34.7%O

0 200 150 100 50 Chemical Shift (ppm)

Acidic Functional Group Content Total Acidity: 6.4 meq /g Carboxyl Acidity: 1.3 meq /g

Figure 7.2 13 C DPMAS NMR spectra of the three humic fractions isolated from a peat soil using the procedure outlined in Figure 7.1. Carbon-type distributions generally used to describe samples are aliphatic (0–50 ppm), carbohydrate (50–110 ppm), aromatic (110–190 ppm) and carboxyl (190–220 ppm). Elemental data [6] are average elemental compositions for each humic fraction. Functional group analyses are for these speciﬁc samples [7]. There is considerable variation in these chemical characteristics among samples, and between environments, for each fraction. For descriptions of compositional variations in humic materials from different environments the reader is referred to the references [1–5].

7.2 METHODS FOR DETERMINING FRACTAL PROPERTIES AND BEHAVIOURS OF HUMIC MATERIALS The majority of the studies utilizing a fractal approach to study humic materials have employed some form of scattering: light scattering [16–19], small-angle x-ray scattering (SAXS) [18–25], small-angle neutron scattering (SANS) [26–32], or turbidimetry [33–38]. The SAXS and SANS techniques have the advantage of being able to be applied to solid- or solution-phase samples. Static light-scattering data and SAXS

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or SANS data can be combined to cover a larger characterization length-scale that allows particles of a broad size-distribution to be studied. The application of scattering methods to the study of the fractal properties of materials have been described in detail by Schmidt [39], Homer [32], Martin and Hurd [40], and Bushell in Chapter 3 of this book. Raper and Amal [17] give a concise description of the theory of light scattering from fractal objects. The reader is directed to these references for detailed discussion of the theory of these methods. A brief introduction follows. Most of the information in a scattering experiment will be found at scattering angles that satisfy the criterion 0.1 ≤ ql

(7.1)

where l is the diameter of the scattering particle. The scattering vector, q, is deﬁned as q=

4π θ sin 2 λ

(7.2)

where λ is the wavelength of the incident radiation and θ is the scattering angle. The fractal properties of a scatterer are obtained in the Porod scattering regime when Equation (7.1) is satisﬁed [39, 40]. It can be shown that the intensity of the radiation scattered by a particle consisting of mass fractal scatterers as function of q is approximated by π sin (Dm − 1) 2 I (q) ≈ Ns I0 (qa)Dm (Dm − 1) (7.3) (qa)Dm where Ns is the number of scatterers, I0 (qa) is the scattered intensity from a single scatterer and (Dm − 1) is a gamma function [39, 40]. This is a power law where I (q) is proportional to the power law exponent Dm . The power-law exponent is the slope of the linear portion of a log–log plot of I (q) versus q. If the material is a mass fractal, then Dm will satisfy the condition Dm ≤ 3 and the fractal dimension is the absolute value of Dm . For a surface fractal it can be shown that π(Ds − 1) −(6−Ds ) 2 I0 (qa) ≈ πN0 δ Ie (5 − Ds )x sin (7.4) q 2 where N0 is a constant, Ie is the scattered intensity from a single electron, and (5−Ds ) is a gamma function [39, 40]. Equation (7.4) also describes a power law where I (q) is proportional to a power law exponent represented by −(6 − Ds ). The exponent can again be obtained from the slope of a log–log plot of I (q) versus q. If the material is a surface fractal, then the power-law exponent satisﬁes the condition 3 ≤ (6−Ds ) ≤ 4. The surface fractal dimension Ds is readily obtained by subtraction. Several other methods, notably turbidimetry [33–38], have also been used to characterize humic materials. Turbidimetry is a variation on scattering where the transmitted

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light is observed as opposed to the scattered light. The turbidity τ of a particle suspension can be described by ⎡ 2 ⎤ dη 2 2 32π n τ =⎣ (7.5) dc ⎦cMw Q 3N λ4 where dη/dc is the refractive index increment, N is Avogadro’s number, c is the concentration of particles with molar mass Mw in a monodisperse assemblage, and Q is the turbidity dissipation function. The value of Q is a consequence of the internal interference of light scattered at all angles. For a fractal object: 3 π Q= (7.6) P (θ )S(q) sin θ (1 + cos2 θ) dθ 8 0 where P (θ) is the particle form factor and S(q) is the structure factor that describes the distribution of scatterers within the particle and is given by Equation (7.7) for intermediate values of q: 1 π const lim S(q) = 1 + D(D − 1) sin (D − 1) ≈ (qα)−D (7.7) = 1+ D ζ →∞ (qα) 2 (qα)D This expression is a power law that can be used to evaluate scattering by a mass fractal object [30]. If Equation (7.7) is substituted into Equation (7.6), which is then substituted into Equation (7.5), τ can be directly related to D for large scatters. Senesi et al. [33] describe the derivation of this relationship in detail. The use of scattering to measure fractal properties of humic materials is limited by what is often referred to as the ‘characteristic length scale’, which is a function of the wavelength used (Equation (7.2)). If a scattering particle is larger than this length, then the entire particle cannot be observed (see Chapter 3 for a detailed description of the application of scattering techniques for determining fractal dimensions). Thus, this length forms the upper limit for the fractal regime that can be observed under a particular set of conditions. It is possible to extend the particle or cluster size that is being observed using two different wavelengths, e.g. by combining SAXS and static light scattering, or using SAXS and ultra-small-angle X-ray scattering. However, the extension of the characteristic length scale is usually not more than one order of magnitude in q. Because aggregation phenomena in humic materials span particlesize ranges from the nanoscale to the macroscopic scale, the characteristic length scale does limit the applicability of scattering techniques in large-particle humic systems. Microscopy is a technique that can be used to study the fractal nature of larger particles, even to the point of ﬂocculation. Transmission electron micrographs reveal what appears to be a fractal structure [41] whose fractal dimension would be amenable to measurement by counting methods. In these methods, object images are digitized and superimposed on a grid. Pixels representing a cluster within each grid unit are counted using commercial software to generate the greatest distance between pixels attributed to a particular cluster. Equation (7.8) describes a power-law relationship

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between the particle mass M and length l where the power-law exponent is the mass fractal dimension: M ∝ l Dm (7.8) The mass fractal dimension is obtained from a log–log plot of M versus l. This technique is described in detail in Chapter 3. Other, nonhumic components of NOM have also been studied using a fractal approach. These studies and the methods they employ are reviewed in Chapter 4.

7.3 WHAT FRACTALS HAVE SHOWN US ABOUT HUMIC MATERIALS There have been approximately two dozen papers published in the peer-reviewed literature since 1992 that have used the concept of fractals to characterize humic materials [16–38, 42]. Most of the studies were done in a burst of interest during a 7-year period in the 1990s; there have been ﬁve papers published in the peerreviewed literature on the application of fractals to the study of humic materials other than reviews [15, 32, 43, 44] since 1999. Two common applications of fractal geometry have been to study the mass distribution of colloidal humic materials and the aggregation processes that produced them. There have been a number of studies reported that have examined the effect of the presence or absence of humic materials and NOM on the fractal properties of other colloidal materials such as hematite, clays, colloidal hydrous oxides and similar materials [e.g. 45–48]. However, since the focus of this chapter is on the use of a fractal approach to explore the unique nature of humic materials, these studies are not discussed in this chapter. 7.3.1 Particle Structure Studies of humic materials employing fractal geometry have generally focused on solution-state samples, and most of these studies have been directed towards humic acid (Table 7.1). Exceptions to this generalization are studies using turbidimetry that characterize aqueous suspensions of humic acid particles [33–38] and the application of fractals to describe the surface morphology of humin [23–25]. Humic and fulvic acid particles in solution that have been characterized to date have demonstrated mass fractal behavior (Table 7.1). The fractal dimensions reported for humic acid samples in alkaline solutions using scattering methods in the range 2 ≤ Dm ≤ 3. Particles with mass fractal dimensions in this range can be viewed as open, three-dimensional aggregates with structures that become more space ﬁlling as Dm approaches 3. Humic materials behave like anionic polyelectrolytes in solution (particles are smaller and more subject to phase separation at low pH and high ionic strengths [10, 49], and they aggregate as a function of concentration [13, 50]. Thus, it is difﬁcult to make direct comparisons between samples or sample source-environments

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Table 7.1 Fractal dimensions reported for humic materials from various environments. Samples that had been modiﬁed are not included in this table. Roman numerals are used for reference to discussion in text. Table adapted from Homer [32]. Sample source Solution state Humic acid Stream sediment Peat Soil I Unspeciﬁed IHSS HA

Technique

Notes

Fractal dimension

Ref.

SAXS

Basic solution, [HA] = 6.7 g m−3

Dm = 2.5

[20]

SAXS SANS SANS

Basic solution, [HA] = 2.0 g m−3 pH 5, 3.6 g m−3 pH 5, 2.9 g m−3

Dm = 2.1 Dm = 2.3 Dm = 2.3

[16] [26] [26]

II

Soil

SANS

pH 5, scattering from ﬁve dilutions normalized with respect to concentration collapse to master curve

Dm = 2.35

[30]

III

Soil

SANS

pH 5

[C] = 10 g m−3 [C] = 15 g m−3 [C] = 20 g m−3

Dm = 2.10 Dm = 2.14 Dm = 2.02

[32]

IV

Aldrich

SANS

[C] = 0.4 g m−3

pH 5 pH 2 pH 10 pH 11 pH 11, Ca2+ added

Dm = 2.41 Dm = 2.45 Dm = 2.56 Dm = 2.66 Dm = 2.66

[32]

pH 3, 10 h equilibration pH 6, 10 h equilibration

Dm = 1.75

[42]

pH 3, 10 h equilibration pH 6, 28 h equilibration

Dm = 1.72

pH 3, 10 h equilibration pH 6, 10 h equilibration

Dm = 1.86

pH 3, 4 h equilibration pH 6, 28 h equilibration

Dm = 1.73

[C] = 10 g m−3

V

V

V

V

IHSS peat (1S103H)

TEM

IHSS soil (1S102H)

TEM

IHSS soil (1S106H)

TEM

soil

TEM

Fulvic acid Groundwater

SAXS

1 g m−3

1 g m−3

1 g m−3

1 g m−3

Aquatic NOM sample, [NOM] = 45 mg m−3

Dm = 1.52 [42]

Dm = 1.80 [42]

Dm = 1.54 [42]

Dm = 1.52

Dm = 1.6

[20]

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Table 7.1 (Continued). Sample source Solid state Humic acid Stream sediment Lignite Fulvic acid Stream sediment Lignite Soil Groundwater Humin Soil Soil Soil Peat In suspension Humic acid VI IHSS soil (1S106H)

VI

VI

IHSS soil (1S102H)

IHSS peat (1S103H)

Technique

Notes

Fractal dimension

Ref.

SAXS

Ds = 2.2

[20]

SAXS

Ds = 2.3

[20]

SAXS

Ds = 2.3

[20]

SAXS SAXS SAXS

Ds = 2.8 Ds = 2.5 Dm = 2.5

[20] [20] [20]

Ds = 2.7 Ds = 2.4 Ds = 2.9 Dm = 2.9 Ds = 2.2 Dm = 2.2

[24] [24] [24]

Dm = 2.77 Dm = 2.46 Dm = 2.20 Dm = 2.11 Dm = 1.86 Dm = 2.57 Dm = 2.55 Dm = 2.31 Dm = 1.78

[35]

Nonfractal Nonfractal Dm = 2.55 Dm = 2.13 Dm = 1.71 Nonfractal Nonfractal Nonfractal Dm = 1.95

[35]

Ds = 2.79 Ds = 2.87 Ds = 2.87

[35]

Aquatic NOM sample

SAXS SAXS SAXS SAXS

Turbidity

Turbidity

Turbidity

pH 3 pH 4 pH 5 pH 6 pH 7 pH 3 pH 4 pH 5 pH 6

[HA] = 30 mg m−3

pH 3 pH 4 pH 5 pH 6 pH 7 pH 3 pH 4 pH 5 pH 6

[HA] = 30 mg m−3

pH 3 pH 4 pH 5

[HA] = 30 mg m−3

[HA] = 40 mg m−3

[HA] = 40 mg m−3

[24]

Ds : surface fractal dimension; Dm : mass fractal dimension; HA: humic acid; NOM: unfractionated natural organic matter sample. IHSS: International Humic Substances Society.

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because of variations in the pH, ionic strength, and humic material concentration used to prepare samples. Humic acids characterized by transmission electron microscopy (TEM) in acidic solutions (Table 7.1; V) display mass fractal dimensions Dm < 2, even after aging for 28 h. These values are considerably smaller than those reported for SAXS or SANS measurements of humic acids in solution, regardless of whether the solution was basic or acidic at the time of the measurements (Table 7.1). They are also smaller than the dimensions reported for the same samples obtained by turbidimetry (Table 7.1; VI). There are insufﬁcient data and dissimilar experimental conditions (pH, ionic strength, sample source, etc.) to allow a deﬁnitive explanation of the differences between the Dm values of humic acids characterized by SAXS, SANS and TEM. These observations, however, suggest at least two possibilities: that there are methodological differences that ‘see’ different components of the sample (here, it would be useful to know what fraction of the original sample’s organic carbon was actually present on the TEM grids imaged) or that humic acid particles go through a range of fractal structures as the concentration increases and/or the solution pH decreases. The fractal properties reported for suspensions of humic particles can be characteristic of either mass fractals or surface fractals (Table 7.1; VI), apparently depending on the nature of the sample and the medium. At around pH 5, an increase in humic acid concentration, either in solution (Table 7.1; IV) or in suspension (Table 7.1: VI), results in an increase in Dm ; this suggests that the aggregates are becoming more space ﬁlling (i.e. more tightly packed), consistent with the general view that humic acid aggregation increases with increasing concentration. But data also exist from solutions of varying humic acid concentrations at pH 5 (Table 7.1; III) which suggest that the fractal dimension of a humic acid at this pH is independent of concentration. This contradiction does not appear to have been resolved; but, based on its operational deﬁnition, pH 5 probably represents a lower limit for humic acid solubility, and aggregation may have already begun to occur in the samples, especially given the relatively high concentrations used (Table 7.1; III). This ﬁnal point could explain why Dm was reported to be independent of concentration. Studies where pH and/or ionic strength have been varied (Figure 7.3) are generally consistent with a polyelectrolyte nature for humic acids: fractal dimensions increase with decreasing pH and increasing (CaCl2 ) ionic strength, indicating the development of a more dense, space-ﬁlling aggregate structure. The increase in Dm is presumably the result of charge screening (increase of ionic strength) or proton binding and hydrogen bonding (decrease of pH) engendering intermolecular interactions leading to changes such as coiling, compaction, etc. within the aggregate structure. This is consistent with Ghosh and Schnitzer’s [49] original qualitative depiction of the variations in the structure of humic and fulvic acids with respect to pH and ionic strength changes. The fractal dimension of the one aquatic (groundwater) NOM sample in solution reported to date has a Dm = 1.6 (Table 7.1). This sample was characterized at a realistic environmental concentration of [NOM] = 45 mg dm−3 . A Dm = 1.6 is indicative of

230

APPLICATIONS OF FRACTALS IN THE STUDY OF HUMIC MATERIALS A.NaCl 3

Mass Fractal Dimension

0 mM 1 mM

2.5

5 mM 2 10 mM 1.5 1 0.5 0 3

4

5 pH

6

7

B.CaCl2 3

Mass Fractal Dimension

0 mM 1 mM

2.5

10 mM 2 1.5 1 0.5 0 3

4

5 pH

6

7

Figure 7.3 Effect of ionic strength (as NaCl or CaCl2 concentration) and pH on the mass fractal dimension of a soil humic acid. Fractal dimensions are taken from Senesi et al. [35], and were determined by turbidimetry (see text for details).

a more linear aggregate structure than that discussed above for humic acid in solution. A more linear structure is similar to the results obtained from TEM images of very dilute solutions of humic acids (Table 7.1; V). It is also consistent with the depiction of colloid/macromolecular interactions for NOM based on electron microscopy and computational studies [51].

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In contrast to their solution characteristics, humic and fulvic acids in the solid state are surface fractals. Values of Ds are in the range 2 ≤ Ds ≤ 3 (Table 7.1). To date, these values are observed even with the varying pH or ionic strength conditions that existed when the samples were isolated. Humin is an organo-mineral composite [11] that is the least studied of the humic materials [52], and this is reﬂected in the small number of studies examining its fractal characteristics. Humin isolated from three mineral soils and a peat soil (Table 7.1) displays Ds values ranging from 2.5 to 2.9 over length scales of 1 to 15 nm. Two of these samples also have components that exhibit mass fractal behavior (Table 7.1), an observation that, while repeated, has not been satisfactorily explained. Selective, sequential removal of the organic matter from the surface of humin particles results in a decrease in Ds and concomitant decrease in surface-pore size and increase in surface area [25]. These results suggest that the mineral components of humin have smoother surfaces over these length scales than the original humin organo-mineral composite, and that it is the organic matter coatings that are responsible for the particle surface roughness. It was also noted in these studies that the humic acid-like component of humin exhibited a fractal nature being a mass fractal in solution and a surface fractal in the solid state. 7.3.2 Aggregation Mechanisms In Chapter 4, the various aggregation mechanisms operant in fractal systems are reviewed, e.g. reaction-limited aggregation (RLA) and diffusion-limited aggregation (DLA). For a detailed discussion of these mechanisms the reader is referred to that chapter and the references it contains. In the late 1990s, Homer [32] and Senesi [44] reviewed studies of humic material aggregation using a fractal approach. Since then, little in the way of new insights into either the mechanism or the applications of fractal aggregation has been presented. The early fractal studies of humic materials alluded to the application of sample fractal dimensions for the description of aggregation mechanisms operant in these systems. In the ﬁrst paper reporting a fractal nature for humic materials in solution, Österberg noted that the observed fractal dimensions were indicative of an RLA process [26]. It is difﬁcult to make generalizations from such a limited body of work that has used different samples, pHs, concentrations, ionic strengths, etc. With that as a caveat, subsequent investigations have discussed their results in terms of either RLA or DLA mechanisms [16, 18, 27, 28, 32, 34, 35, 42, 44], depending on pH. An RLA mechanism is suggested as dominant at higher pH conditions presumably because ionized functional groups generate electrostatic repulsions that hinder aggregation [16, 26, 28, 34, 35]. At low pH values, a DLA mechanism seems to be dominant, probably because charge neutralization has minimized or eliminated electrostatic repulsions. It has been reported that as a humic aggregate ages it undergoes a restructuring that tends to produce a more space-ﬁlling particle [28, 29, 42] with

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APPLICATIONS OF FRACTALS IN THE STUDY OF HUMIC MATERIALS

a concomitant increase in the fractal dimension (Figure 7.4). This behavior is consistent with the polyelectrolytic characteristics of these materials. The restructuring phenomenon is characteristic of RLA, where components of a fractal aggregate can reorganize with time to form denser particles [53].

Mass Fractal Dimension

2.5

2

1.5

1

0.5

0 1

11 34 Time (hours)

60

Figure 7.4 Effect of time on the mass fractal dimension of a soil humic acid. Fractal dimensions are taken from Österberg et al. [28], and were determined by SANS (see text for details).

7.4 WHAT FRACTALS COULD TELL US ABOUT HUMIC MATERIALS A fractal approach to the study of the chemistry and geochemistry of humic materials has a number of possible applications that the limited number of papers published on this property have only begun to consider. In the paragraphs below, a few possible examples are given. To fully realize the insights that a fractal approach could provide will require systematic study of a suite of carefully isolated and well-characterized humic substances. The International Humic Substances Society (IHSS) has such a collection of standard and reference humic materials that would be an ideal sample suite for this purpose [54, 55]. 7.4.1 Controls on the Fractal Dimension In 1999, Homer [32] noted that ‘sparse’ data existed to correlate the chemical characteristics of a humic material with its fractal dimension or aggregation mechanism. For example, a preliminary study is described that may be the only attempt to correlate the carbon types present in a humic sample to its fractal properties. Humic acids with a greater proportion of aromatic carbon than carbohydrate carbon (determined from a 13 C NMR spectrum) had larger fractal dimensions than those with smaller

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proportions of aromatic carbon. In the same report, Homer [32] suggested that functional group concentrations might also play a role. Though they represent a very limited data set, the fractal dimensions presented in Rice and Lin [12] seem to suggest that humic particle morphologies may vary with source environment. While the correlations from these comparisons may only weakly infer a cause-and-effect relationship, a systematic study of the chemical properties of a suite of carefullycharacterized humic and fulvic acids could, at the very least, point the way towards more speciﬁc studies of the chemical interactions resulting in the fractal aggregate structures observed for humic materials. 7.4.2 Effect of Environmental Variables on Aggregate Structure While the solution parameters discussed certainly fall within the realm of environmental variables, their limits in most natural environments are relatively constrained; for example, pH usually varies only between ∼ 5 and 8 [56]. Humic materials are strong ligands for multivalent cations and these same cations readily ﬂocculate humic materials, for some cations at very low concentrations. The resulting aggregates and the aggregation process(es) that form them could be studied with a fractal approach. For example, Ca2+ [35] and Fe3+ [32] have been cursorily examined. Calcium ions produced a particle with surface fractal properties (as opposed to mass fractal behavior) at all pH values. Contrary to the expected increase in fractal dimension, Fe3+ did not change the observed value of Dm for the humic acid studied, but it did increase the intensity of scattered neutrons as a result of what was believed to be an increase in aggregate size. Homer [32] proposes that this could be the result of either an increase in humic aggregate cluster size or the formation of new clusters from previously unaggregated components in a humic sample. 7.4.3 Environmental Applications of the Fractal Nature for Humic Materials Studies carried out on humic materials that have utilized fractals have focused on characterizing their structural organization and/or their aggregation behavior. The real power of a fractal approach lies not in simply characterizing humic materials, but in applying it to modeling and predicting the chemical and geochemical behavior of these materials in the environment. For example, Dachs and coworkers [57, 58] have developed a fractal approach that takes the ‘geometric heterogeneity’ of humic materials into account when modeling their interaction with organic contaminants. This approach [57] utilizes the different interaction probabilities of the active contaminant binding sites on fractal surfaces to accommodate linear and nonlinear sorption isotherms (i.e. ‘dual-mode’ sorption) using coverage extent, physical and chemical properties of the adsorbate and the existence of equilibrium conditions. An extension of this approach [58] treats partitioning as sorption to the inner sites of a fractal sorbent, in much the same manner that a hydrophobic organic chemical is viewed as partitioning into a humic

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material. In this approach, molecules are adsorbed at the outer sites on a fractal aggregate in a relatively fast process. With time, molecules slowly diffuse into the fractal aggregate and bind to the inner sites. In this process, partitioning is a signiﬁcant contribution to the overall sorption process at long equilibration times. Interestingly, this sequence of events is what Kohl et al. [59] have observed when solid-state 19 F NMR is used to follow the interaction of hexaﬂuorobenzene with natural organic matter sorbents. 7.5 CONCLUSIONS AND RECOMMENDATIONS A fractal approach seems ideally suited to the study of higher order structural organization and interactions in humic materials. But in the 15 years since it has been ﬁrst applied to them, there has been surprisingly little interest in applying this tool to advancing the understanding of the unique nature of these ubiquitous materials by any other than those who initially adopted it. There have been surprisingly few papers exploring the fractal nature of humic materials published during this time. However, there have been some basic insights into the fractal nature of humic materials that have come from this early work. Humic materials have been shown to be mass fractal aggregates in solution or in suspension, and surface fractals in the solid state. The fractal dimensions of both fractal types are generally between 2 and 3. But there has been no systematic study of how the fractal dimension varies with environmental variables (pH, ionic strength, humic concentration, sample source, identity and concentration of multivalent cations, aggregate size, etc.). A basic recognition has emerged that the aggregation of humic material can be described using a fractal approach. DLA is generally operant under acidic conditions and RLA becomes more important as the pH increases. The real value of a fractal approach in the study of humic materials will not be realized until it can be applied in a manner that describes properties, reactions or geochemical behavior for which we previously had no vocabulary. LIST OF SYMBOLS AND ABBREVIATIONS DLA diffusion-limited aggregation HA humic acid IHSS International Humic Substances Society NOM natural organic matter NMR nuclear magnetic resonance RLA reaction-limited aggregation SAXS small-angle X-ray scattering SANS small-angle neutron scattering TEM transmission electron microscopy

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c concentration of particles with a particular molar mass D fractal dimension Dm mass fractal dimension surface fractal dimension Ds Ie scattered intensity from a single electron I0 incident intensity I0 (qa) scattered intensity from a single scatterer I (q) intensity of scattered radiation as a function of the scattering vector q l diameter of a scattering particle M particle mass Mw molar mass N Avogadro’s number Ns number of scattering particles P (θ) particle form factor q scattering vector Q turbidity dissipation function S(q) structure factor dη/dc refractive index increment λ wavelength of the incident radiation θ scattering angle gamma function τ turbidity REFERENCES [1] Stevenson, F.J. (1982). Humus Chemistry. John Wiley & Sons, Ltd, New York. [2] Kononova, M.M. (1966). Soil Organic Matter, Nowakowski, T.A. and Newman, A.C.D. (transl.). Pergamon Press, Oxford. [3] Orlov, D.S. (1985). Humus Acids of Soils. Kothekar, V. (transl.)., A.A. Balkema, Rotterdam. [4] Thurman, E.M. (1985). Organic Geochemistry of Natural Waters. Martinus Nijhoff/ Dr Junk, Dordrecht. [5] Aiken, G.R., McKnight, D.M., Wershaw, R.L. and MacCarthy, P. (eds). (1985). Humic Substances in Soil, Sediment, and Water. John Wiley & Sons, Ltd, New York. [6] Rice, J.A. and MacCarthy, P. (1991). Statistical evaluation of the elemental composition of humic substances. Org. Geochem., 17, 635–648. [7] Rice, J.A. (1988). Studies on humus. I. Statistical studies on the elemental composition of humus; II. The humin fraction. PhD dissertation, Colorado School of Mines, Golden, CO. [8] MacCarthy, P. and Rice, J.A. (1985). Spectroscopic methods (other than NMR) for determining functionality in humic substances. In Humic Substances in Soil, Sediment, and Water: Geochemistry, Isolation, and Characterization, Aiken, G.R., McKnight, D.M., Wershaw, R.L. and MacCarthy, P. (eds). John Wiley & Sons, Ltd, New York, pp. 527–559. [9] Hayes, M.H.B., MacCarthy, P., Malcolm, R.L. and Swift, R.S. (1989). Structures of humic substances: the emergence of ‘forms’. In Humic Substances II. In Search of Structure, Hayes, M.H.B., MacCarthy, P., Malcolm, R.L. and Swift, R.S. (eds). Wiley–Interscience, Chichester, pp. 689–733.

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[10] Swift, R.S. (1985). Fractionation of soil humic substances In Humic Substances in Soil, Sediment, and Water: Geochemistry, Isolation, and Characterization, Aiken, G.R., McKnight, D.M., Wershaw, R.L. and MacCarthy, P. (eds). John Wiley & Sons, Ltd, New York, pp. 387–408. [11] Rice, J.A. and MacCarthy, P. (1990). A model of humin. Environ. Sci. Technol., 24, 1875–1877. [12] MacCarthy, P. and Rice, J.A. (1991). An ecological rationale for the heterogeneous nature of humic substances. In Scientists on Gaia, Schneider S. and Boston, P.J. (eds). MIT Press, Cambridge, MA, pp. 339–345. [13] Wershaw, R.L. (1993). Model for humus in soils and sediments. Environ. Sci. Technol., 27, 814–816. [14] Piccolo, A. (2001). The supramolecular structure of humic substances. Soil Sci., 166, 810–832. [15] Tombácz, E. (2002). Interfacial and colloidal properties of humic substances. Magy. Kem. Foly. Kem. Kozl., 108, 435–443. [16] Ren, S.-Z., Tombácz, E. and Rice, J.A. (1996). Dynamic light scattering from fractals in solution: Application of dynamic scaling theory to humic acid. Phys. Rev. E, 53, 2980–2983. [17] Raper J.A. and Amal, R. (1993). Measurement of aggregate fractal dimension using static light scattering. Part. Part. Syst. Charact., 10, 239–245. [18] Tombácz, E., Rice, J.A. and Ren, S.-Z. (1997). Fractal structure of polydisperse humic acid particles in solution studied by scattering methods. ACH Models Chem., 134, 877–888. [19] Rice, J.A., Tombácz, E. and Malekani, K. (2000). Applications of light and X-ray scattering to characterize the fractal properties of soil organic matter. In Fractals in Soil Science, Developments in Soil Science 27, Pachepsky, Ya.A., Crawford, J.W. and Rawls, W.J. (eds). Elsevier, Amsterdam, pp. 161–174. [20] Rice, J.A. and Lin, J.S. (1993). Fractal nature of humic materials. Environ. Sci. Technol., 27, 413–414. [21] Rice, J.A. and Lin, J.S. (1994). Fractal dimensions of humic materials. In Humic Substances in the Global Environment and Implications on Human Health, Senesi, N. and Miano, T.M. (eds). Elsevier, Amsterdam, pp. 115–120. [22] Dékány, I., Szekeres, M., Marosi, Balázs, T.J. and Tombácz, E. (1994). Interaction between ionic surfactants and soil colloids: adsorption, wetting and structural properties. Prog. Colloid Polym. Sci., 95, 73–90. [23] Malekani, K., Lin, J.S. and Rice, J.A. (1997). Fractal characterization of the surface of the humin fraction of soil organic matter. In Fractal Frontiers, Novak, M.M. and Dewey, T.G. (eds). World Scientiﬁc, Singapore, pp. 367–381. [24] Malekani, K., and Rice, J.A. (1997). Fractal character of humin and its components. Fractals, 5, 83–100. [25] Malekani, K., Lin, J.S. and Rice, J.A. (1997). The effect of sequential organic matter removal on the surface morphology of humin. Soil Sci., 162, 333–342. [26] Österberg, R. (1992). Fractal dimensions of humic acids: a small angle neutron scattering study. Eur. Biophys. J., 21, 163–167. [27] Österberg, R. and Mortensen, K. (1994). The growth of fractal humic acids: cluster correlation and gel formation. Radiat. Environ. Biophys., 33, 269–276. [28] Österberg, R., Szajdak, L. and Mortensen, K. (1994). Temperature-dependent restructuring of fractal humic acids: a proton-dependent process. Environ. Int., 20, 77–80. [29] Österberg, R. and Mortensen, K. (1994). Fractal geometry of humic acids. Temperaturedependent restructuring studied by small-angle neutron scattering. In Humic Substances in the Global Environment and Implications on Human Health, Senesi, N. and Miano, T.M. (eds). Elsevier, Amsterdam, 127–132.

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[30] Österberg, R. and Mortensen, K. and Ikai, A. (1995). Direct observation of humic acid clusters, a nonequilibrium system with a fractal structure. Naturwissenschaften, 82, 137–139. [31] Österberg, R. (2002). Fractal structure and reactions of the two-phase humic acid system. Trends Geochem., 2, 95–99. [32] Homer, V.J. (1998). Fractal probes of humic aggregation: scattering techniques for fractal dimension determinations. In Fractals in Soil Science, Baveye, P., Parlange, J.-Y. and Stewart, B.A. (eds). CRC Press, Boca Raton, FL, pp. 75–112. [33] Senesi, N., Lorusso, G.F., Miano, T.M., Maggipinto, G., Rizzi, F.R. and Capozzi, V. (1994). The fractal dimension of humic substances as a function of pH by turbidity measurements. In Humic Substances in the Global Environment and Implications on Human Health, Senesi, N. and Miano, T.M. (eds). Elsevier, Amsterdam, pp. 121–126. [34] Senesi, N., Rizzi, F.R., Dellino, P. and Acquafredda, P. (1996). Fractal dimension of humic acids in aqueous suspension as a function of pH and time. Soil Sci. Soc. Am. J., 60, 1773–1780. [35] Senesi, N., Rizzi, F.R., Dellino, P. and Acquafredda, P. (1997) Fractal humic acids in aqueous suspensions at various concentrations, ionic strengths, and pH values. Colloids Surf. A, 127, 57–68. [36] Costa da Silva, A., Sa Mendonca, E. and Reis, C. (2003). Fractal dimension of humic acids in different experimental conditions. Quim. Nova, 26, 344–346. [37] Costa da Silva, A., Sa Mendonca, E. and Reis, C. (2000). Fractal dimension of humic acids. Rev. Brasil. Cienc. Solo, 24, 759–766. [38] Costa da Silva, A. (2000). Fractal dimension of humic acids by turbidimetry. Rev. Ceres, 47, 103–112. [39] Schmidt, P. (1989). Use of scattering to determine the fractal dimension. In The Fractal Approach to Heterogeneous Chemistry, Avnir, D. (ed.). John Wiley & Sons, Ltd, Chichester, pp. 67–79. [40] Martin, J.E. and Hurd, A.J. (1987). Scattering from fractals. J. Appl. Crystallogr., 20, 61–78. [41] Chen, Y. and Schnitzer, M. (1989). Sizes and shapes of humic substances by electron microscopy. In Humic Substances II. In Search of Structure, Hayes, M.H.B., MacCarthy, P., Malcolm, R.L. and Swift, R.S. (eds). Wiley–Interscience, Chichester, pp. 621–638. [42] Rizzi, F.R., Stoll, S., Senesi, N. and Bufﬂe, J. (2004). A transmission electron microscopy study of the fractal properties and aggregation processes of humic acids. Soil Sci., 169, 765–775. [43] Senesi, N. (1994). The fractal approach to the study of humic substances. In Humic Substances in the Global Environment and Implications on Human Health, Senesi, N. and Miano, T.M. (eds). Elsevier, Amsterdam, pp. 3–41. [44] Senesi, N. (1999). Aggregations patterns and macromolecular morphology of humic substances: a fractal approach. Soil Sci., 164, 841–856. [45] Amal, R., Raper, J.A. and Waite, T.D. (1992). Effect of fulvic acid adsorption on the aggregation kinetics and structure or hematite particles. J. Colloid Interface Sci., 151, 244–257. [46] Schmidt, C., Thieme, J., Neuhausler, U., Schulte-Ebbert, U., Abbt-Braun, G., Specht, G. and Jacobsen, C. (2000). Association of particles and structures in the presence of organic matter. In AIP Conference Proceedings, 507 (X-Ray Microscopy). American Institute of Physics, pp. 313–318. [47] Waite, T.D., Amal, R., Ngo, H. and Vigneswaran, S. (1993). Effects of adsorbed organic matter on size, structure and ﬁlterability of iron oxyhydroxide ﬂocs.Water Sci. Technol., 27, 133–142.

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[48] Sokolowska, Z. and Sokolowski, S. (1999). Inﬂuence of humic acid on surface fractal dimension of kaolin: analysis of mercury porosimetry and water vapour adsorption data. Geoderma, 88, 233–249. [49] Ghosh, K. and Schnitzer, M. (1980). Macromolecular structures of humic substances. Soil Sci., 129, 266–276. [50] Guetzloff, T.F. and Rice, J.A. (1994). Does humic acid form a micelle? Sci. Total Environ., 152, 31–35. [51] Rizzi, F.R., Stoll, S., Senesi, N. and Bufﬂe, J. (2004). A transmission electron microscopy study of the fractal properties and aggregation processes of humic acids. Soil Sci., 169, 765–775. [52] Rice, J.A. and MacCarthy, P. (1988). Comments on the literature of the humin fraction of humus. Geoderma, 43, 65–73. [53] Meakin, P. (1988). Fractal aggregates. Adv. Colloid Interface Sci., 28, 249–331. [54] Campbel, W.L. and Malcolm, R.J. (1985). Preparation, homogenization, and storage of earth standards used and distributed by the International Humic Substances Society. Org. Geochem., 8, 109. [55] Swift, R.S. (1996). Organic matter characterization. In Methods of Soil Analysis. Part 3. Chemical Methods. SSSA Book Series No. 5. Soil Science Society of America, Madison, WI, pp. 1011–1069. [56] Baas Becking, L.G.M., Kaplan, I.R. and Moore, D. (1960). Limits of the natural environment in terms of pH and oxidation–reduction potentials. J. Geol., 68, 243–284. [57] Dachs, J. and Bayona, J.M. (1997). Langmuir-derived model for diffusion- and reactionlimited adsorption of organic compounds on fractal aggregates. Environ. Sci. Technol., 31, 2754–2760. [58] Dachs, J. and Eisenreich, S.J. (2001). Adsorption and partitioning of organic compounds onto/into fractal sorbents. Langmuir, 17, 2553–2537. [59] Kohl, S.D., Toscano, P.J., Hou, W. and Rice, J.A. (2000). Solid-state 19 F NMR investigation of hexaﬂuorobenzene sorption to soil organic matter. Environ. Sci. Technol., 34, 204–210.

8 Fractal Geometry and Microorganisms in the Environment Lynne Boddy1 and Damian P. Donnelly2 Cardiff School of Biosciences, Cardiff University, Biomedical Building, Museum Avenue, Cardiff, CF10 3US, UK

8.1 INTRODUCTION During the last two or three decades, microbial ecologists have concentrated on quantifying numbers (individual cells or colony forming units) and/or biomass in natural environments, as a measure of ‘how much is there’. However, such measures provided little indication of what the microorganisms are doing [1], nor of how the biomass is spatially distributed. Microorganisms are able to deploy the same amount of biomass in different ways, depending on species, resources available and other external environmental factors (see Sections 8.3–8.5). Many microorganisms (including bacteria and fungi) have modular body forms. When new modules are formed, e.g. by ﬁssion or budding, the modules commonly remain attached to the body from which they were formed, but sometimes they become detached and are able to function as separate physiological units [2]. Effectively, they have iterative, indeterminate growth patterns, tend to branch, and are sessile except when in dispersal phases. Such modular organisms contrast with unitary organisms, such as higher animals that generally exhibit noniterative, determinate growth patterns, are nonbranched and are motile. Since modular organisms have simple organisation, tend to lack motility at some stages of their life, and often have external digestion, they are vulnerable to environmental changes. However, they commonly have considerable phenotypic plasticity, i.e. genotype can be expressed differently under different environmental conditions. Thus, microorganisms often exhibit different physiologies and/or morphologies under different environmental regimes. Fungi exhibit considerable morphological and physiological plasticity (Section 8.3.2 and 8.5). Bacteria, on the other hand, are physiologically very plastic, but less so morphologically, though colony characteristics can sometimes change

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(Sections 8.3.1 and 8.4). Arguments have been put forward that bacterial colonies are not self-similar [3], but, like many naturally irregular structures, many modular organisms, including fungi and some bacteria under some conditions, are often approximately fractal. Colonies are, however, self-similar only over a ﬁnite range of length scales. Fractal geometry has been used to describe quantitatively inter- and intra-speciﬁc differences, within-colony differences, and environmentally induced differences in the morphology of bacteria and fungi (Sections 8.3–8.5). Since microbial morphology is most easily seen on artiﬁcial media, many studies have been performed in solid agar culture (Section 8.3), but this reﬂects the real world poorly, and studies are increasingly being undertaken in more realistic microcosms and even ﬁeld situations (Sections 8.4 and 8.5). The vast majority of these studies have concerned fungi; hence, these are emphasised, though the work on bacteria is also summarised. Some natural environments are themselves fractal, e.g. soil, and this affects the diversity, density and distribution of the inhabitant microorganisms (Section 8.6). Further, a range of mathematical models have been developed to explain fractal growth of microbial colonies [e.g. 4–8], but consideration of this topic is beyond the scope of this chapter.

8.2 DETERMINING FRACTAL DIMENSION OF MICROORGANISMS The two fractal dimensions of surface and mass provide different measurements of the morphology of microbial colonies. The surface fractal dimension (a subset of the mass fractal dimension) only describes the morphology of the edges of the colony, i.e. where, in images (see below), white pixels from the colony occur adjacent to black pixels of the background, including exterior edges at the colony margins and interior edges, e.g. where internal gaps persist between cords1 and hyphae of fungi. In terms of more regular geometric shapes, this would equate to describing the morphology of the perimeter. For nonfractal shapes, perimeters are linear with a Euclidean dimension of one, fractal shapes have perimeter values greater than one but less than the Euclidean area dimension (length and breadth) of two. The mass fractal dimension, on the other hand, includes all colony pixels, including those at perimeters and those interior to the perimeter. Thus, mass fractal dimension describes the morphology of the entire colony area, which will have values between the nonfractal values of two for completely ﬁlled areas and one for mycelia in which all colony pixels are at a linear perimeter (in 2 dimensions).

1 Mycelial cords: linear organs formed from hyphae aggregating together, behind a mycelial growing front. They often have a thickened outer rind and are differentiated internally for rapid transport of water and nutrients.

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The colonies of many bacteria and some fungi, or at least some fungi during early stages of outgrowth from a resource, are entirely space-ﬁlled at the centre, being only fractal at their boundaries, i.e. they are surface fractal (Chapter 1). In contrast, many fungi growing in terrestrial environments have gaps within their colonies, the structures being mass fractal. Fractal dimension is an appropriate means of quantifying the way in which these microorganisms deploy their biomass, but it is obviously necessary to use both mass fractal DM and surface fractal DS dimensions. Fractal dimensions have most commonly been determined from images of microorganisms in sterile agar culture and nonsterile soil microcosms. The latter have been used to great effect to visualise and quantify the changes in morphology of macroscopic mycelial systems of saprotrophic, pathogenic and ectomycorrhizal fungi. Microcosms, ranging in size from 20 cm × 20 cm to 57 cm × 57 cm, and consisting of soil compacted to ensure mycelial growth at the soil surface, are inoculated with resources, e.g. wood, leaf litter or plant roots precolonised with fungi (e.g. Figures 8.1 and 8.2). As microbial systems develop, be they on agar, soil or other substrata, images are captured over time for morphological quantiﬁcation. Image analysis involves a series of predeﬁned steps to extract the image of the microbe from the rest of the microcosm. Mycelia (a radially extending, interconnecting series of tubes, e.g. Figure 8.1a) growing in soil are the most complex systems used so far, thus the steps in image analysis and determination of fractal dimension are outlined below for these systems. Mycelial areas are marked out by user deﬁned pixel lines (Figure 8.1b), and unwanted areas such as resources are masked by being set to pixel values of 0 (black) (Figure 8.1c). Image quality may also be improved by applying software ﬁlters to remove image ‘noise’. Finally, the image is ‘thresholded’, whereby the user deﬁnes a greyscale boundary above which all pixels are set to white (pixel value = 1) and below which all values are set to black (pixel value = 0) (Figure 8.1c). This produces a binary image typically with the mycelium as a white image on a black background. There will obviously be some user bias during manual threshholding. However, provided that illumination has been constant throughout an experiment, the same threshold value can be used on each image for consistency, and is unlikely to result in a change in fractal dimension of more than about ±0.01. A number of methods (Chapter 2) have been used for estimating D, including concentric rings, length-projected area, length number scaling and box counting, but the latter has been by far the most common [9]. 8.2.1 Determining Fractal Dimensions by the Concentric Ring Technique The mass fractal dimension can be estimated by overlaying an image with a series of concentric rings of different radius: M(r) = Cr DM

(8.1)

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Figure 8.1 Image preprocessing of mycelial systems in soil microcosms: (a) unprocessed image; (b) masking of central inoculum and uncolonised soil area; (c) thresholded binary image; (d) overlaid grid for box-count determination of fractal dimension; (e) close-up of mycelial colonisation of wood resources and localised mycelial patch production (encircled dotted lines).

where M(r) is the mass contained within a circle of radius r, C is a constant and DM is the mass fractal dimension [10]. It is advisable to use at least ﬁve different radii. Analysis should not be taken to the edge of colonies due to boundary effects [10]. Constructing double logarithmic (Robertson) plots of M against r allows determination of the slope, which is equivalent to 2DM − 1 [11]. 8.2.2 Determining Fractal Dimensions by the Box-Counting Technique Box counting involves overlaying a series of grids of square boxes (with sizes of 3–63 pixels being appropriate for fungal mycelia in small microcosms) onto an image of the colony (Figure 8.1d). The number of boxes intersecting pixels representing the

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organism (rather than the matrix upon which it is growing) is recorded. Fractal images obey the power law over a range of length scales [12]: N (s) ≈ cs −DBM

(8.2)

where N (s) is the total number of boxes of side length s, which intersects the image; c is a constant; and DBM is the box-count fractal dimension. The total number of boxes N intersected by the set comprises both interior boxes (i.e. boxes that contain white pixels only) and border boxes (i.e. boxes that contain at least one white pixel and which contain or adjoin at least one black pixel): N (s) = Nborder (s) + Ninterior (s)

(8.3)

Surface fractal dimensions DBS are estimated from the linear portion of a plot of log Nborder (s) against log s: log Nborder (s) = log c + (−DBS ) log s

(8.4)

Similarly, mass fractal dimension DBM is estimated by regression of the linear part of a plot of Ninterior (s) against log s. However, since natural structures are only self-similar over a ﬁnite range of length scales, departures from the power law occur at very large and very small box sizes [11, 13–15]. Upper limits of length scale range are determined by the largest gap within the image (in practice usually 25 % of the maximum width of the image set) and lower limits by the pixel size of the image, and in mycelia this will be limited by individual hyphae [11, 14]. Since border boxes are not entirely ﬁlled, the area is progressively overestimated by the use of increasingly larger box sizes. To avoid this overestimation a correction is needed [16]. Thus, DBM is estimated by regression analysis on the linear portion of a plot of log[N (s) − 1/2 Nborder (s)] against s [16]. Imaging software techniques enable analysis of both entire images and analysis of distinct local areas of interest (AOIs). AOIs are typically demarked from the rest of the image by user deﬁned pixel lines (Figure 8.1e), and either copied or removed from the main image for subsequent separate analysis. This approach is particularly useful in quantifying fungal mycelia, as these systems often display a range of morphologies during colonisation of heterogeneous substrates. Thus, a single mycelium may be composed of a number of regions of differing fractal dimension, e.g. mycelial patches colonising discrete resources within soil (Figure 8.1e), and production of overgrowth fans during interaction with other fungi (Figure 8.2). 8.2.3 Potential Sources of Error One potential source of error in determining the fractal dimension from images is the resolution of the image constraining the size of the boxes that may be used. Ideally, the image resolution should be such that the smallest picture element is the same size as the smallest hyphal width. In reality, the image may not be fractal at all length scales and the power-law behaviour not obeyed for all box sizes, departures occurring

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Figure 8.2 Digital images of mycelial systems interacting on nonsterile soil (a, b) and peat (c–e). (a, b) Interactions between Stropharia caerulea (top right) and Hypholoma fasciculare (bottom left) after (a) 35 days and (b) 63 days. Note dense fan production by Hypholoma fasciculare. (c) Interaction between Paxillus involutus mycorrhizal with Pinus sylvestris seedlings and the root pathogen Rhizina undulata on peat. (d) Interaction between Paxillus involutus mycorrhizal with Betula pendula seedlings and the decomposer Phanerochaete velutina in microcosms (30 cm × 25 cm) comprising a thin layer (0.2 cm deep) of nonsterile peat (c–e). (e) Detail of the microcosm shown in (d) showing truncation of Phanerochaete vetutina cords by interaction with Paxillus involutus.

at very small and very large box sizes. In practice, the box size upper limit has been approximated as 25 % of the maximum width of the image (feret diameter) [15]. The lower box size limit using a ﬁnite resolution imaging system is determined by the pixel size of the image, as detail below this resolution cannot be resolved. In practice, this has been determined experimentally (D.P. Donnelly, unpublished) and the lower limit to which the power law was maintained was at boxes sizes of 3 pixel side length for images of 512×512 pixels.

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One of the crucial steps in image processing is the production of a binary image that distinguishes regions of interest from other background regions, achieved through user-deﬁned threshold selection of pixel values at and above which all pixels are white and below which pixels are black. During the thresholding, a false-colour image of the thresholded pixel level is displayed on the image, enabling the user to determine the best possible level that distinguishes the mycelium. If the threshold is set too high, then background will be included as mycelium and an overestimate of mycelial DBM produced, and a possible underestimate of DBS through loss of perimeter pixels. The converse will be true if the threshold level is set too low: parts of the mycelium will be included as background and a low DBM will be artiﬁcially produced. This may also increase DBS , as artiﬁcial erosion of mycelium image could increase the irregularity of the perimeter and add perimeters to mycelial interiors where no gaps really exist. 8.2.4 Methods Used in Three Dimensions The fractal nature of bacterial aggregates (ﬂocs) has implications for transport processes, including movement of dissolved respiratory gases to and from the outside of the aggregate, and for movement of dissolved nutrient sources and metabolic products. Several methods have been employed to estimate fractal dimensions of ﬂocs in three dimensions. One approach is to measure light scattering of suspended ﬂocs [17, 18]. The two-slopes method calculates fractal dimensions D3 from the slope of the cumulative size distribution for maximum length l and the slope of the cumulative solid volume V [17]: S(l) D3 = (8.5) S(V ) where S(l) and S(V ) are the exponents of the cumulative size distributions. D3 can also be determined from the small-angle light scattering of laser light by ﬂocs [17]: I (Q) ∝ Q−D3

(8.6)

where I (Q) is the intensity of the scattered light. Q = (4π n/λ) sin(θ/2), where n is the refractive index of the medium, λ is the wavelength of laser light, and θ is the scattering angle. The slope of the plot of log(I (Q)) against log(Q) gives the fractal dimension D3 within certain limits (i.e. this is only valid for size ranges much greater than the monomer size and up to a limited upper size) [17]. Other methods are also being developed to determine fractal dimension from light scatter data [18]. Confocal optical microscopy can be used to take a sequence of randomly chosen images through a bacterial ﬂoc. Methodologies for calculation of three-dimensional fractal dimensions have been described for this approach [18–20]. One method determines the fractal dimension of each section Df using a two-point correlation function C(r) [20]: C(r) = p(r )p(r + r ) dr ∼ r Df −d (8.7)

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where d is the Euclidean space dimension, i.e. 2 in two-dimensional sections, and p(r) is the density at point r. Df is obtained from the slope of a plot of log(C(r)) versus log(r). Since ﬂocs are of limited size, corrections for border effects are needed: for each cell centre, only correlations with other cells closer than the nearest edge are included [20]. Provided that cells are visible in at least two consecutive sections, it is possible to deduce the position of cells in three-dimensional space. The two-point correlation function (Equation (8.7)) can then be computed for d = 3 [20]. Another approach to determining the fractal dimension of three-dimensional ﬂocs has been to view them individually by light microscopy to measure the largest ﬂoc length [21]. Fractal dimension can be determined from the scaling relationship [21]: N ∼ l D3

(8.8)

where N is the number of cells in an aggregate, D3 is the fractal dimension of the cluster, and l is the longest length of the cluster. This has been termed a lengthprojected scaling method. Equation (8.8) is nondimensionalised by dividing by the average cell size lc : D3 l N = a1 (8.9) lc where a1 is a constant. Fractal dimensions in two dimensions D2 can be determined from the ﬂocs’ projected surface area, since aggregate projected surface area A is related to aggregate length l [21]: A = a2 l D2 (8.10) where a2 is a proportionality constant. 8.3 FRACTAL DIMENSION OF MICROORGANISMS IN ARTIFICIAL MEDIA 8.3.1 Bacteria and Unicellular Fungi The majority of studies of fractal geometry of bacteria and unicellular fungi (yeasts) have been performed in agar culture, in which the solidity of the medium, nutrient concentration, inhibitory chemicals and incubation conditions (temperature) have been varied. With regard to bacterial pathogens,2 Escherichia coli, Citrobacter freundii, Klebsiella pneumoniae, Proteus mirabilis, Salmonella anatum, Salmonella typhimurium and Serratia marcescens produced colonies with DBM values between 1.7 and 1.8 [22, 23], whereas Klebsiella ozaenae had more open colonies, DBM = 1.6 [24]. Colony morphology is dramatically affected by nutrient supply [19, 20] and nonlethal concentrations of antibiotics [5]. For example, the fractal dimension of

2

Organisms which kill other organisms, tissues or cells.

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Paenibacillus dendritiformis decreased from 2 to about 1.8 on agar in the presence of 3 g ml−1 co-trimoxazole (Septrin) [25]. Changes in these systems were attributed to changes in chemotactic signalling between bacteria, resulting in a shift in colonial morphology. Studies of bacteria from the natural environment, e.g. on Bacillus subtilis, also revealed effects of growth medium on colony pattern formation: circular colonies, colonies with dense branching, spreading colonies without openings, colonies with concentric rings and fractal colonies were produced depending upon the agar and nutrient concentration [26]. Fractal colonies developed on medium with a low peptone concentration (Figure 8.3); indeed, low nutrient concentration seems to be a general requirement for production of fractal bacterial colonies [26, 27]. Various species of fungi which form yeast also appear to produce different growth patterns, including compact, dense-branching and fractal growth, depending on the nutrient concentration of agar [28].

Figure 8.3 (a) Bacillus subtilis colony on agar containing 1 g l−1 peptone (after [4]). (b) Effect of agar peptone concentration on the fractal dimension of Bacillus subtilis (data derived from [4]).

The ability of many bacteria to swarm and of cells to move within colonies [29] is crucial to pattern production in bacterial colonies. Intricate communication and cooperative behaviour between individual bacterial cells within colonies are undoubtedly essential for self-organization of colonies, though aspects of colony morphology can be at least partially accounted for by ideas of pattern formation from nonbiological systems [2, 5, 30]. Not surprisingly, actinomycetes, whose cells remain in chains forming mycelia, produce fractal structures. Indeed, the ﬁrst fractal analysis of microbial colonies was on Streptomyces griseus [14]. Fractal dimension increased with time with a DBM of up to 1.49 ± 0.01. Fractal dimensions of both bacteria and yeast ﬂocs have also been determined in liquid culture in the laboratory, providing a useful quantiﬁcation of ﬂuid dynamic environment effects upon microbial aggregates. The bacterium Zoogloea ramigera had a DS = 1.69 ± 0.11 (determined from projected area, Section 8.2.4)

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and a DM = 1.79 ± 0.28 (determined by length-number scaling, Section 8.2.4) when grown in rotating test tubes [21]. However, when grown in a bench-top fermentor, aggregates lost their mass fractal characteristics (DM almost 3), but had a DS = 1.78 ± 0.11. The yeast Saccharomyces cerevisae had higher fractal dimensions (DS = 1.92 ± 0.08, DM = 2.66 ± 0.34) than the bacterium when growing in rotating test tubes [21]. The structure of large bacterial ﬂocs, produced by adding cationic polymers to liquid culture medium, has also received attention. Interest arises from applications such as liquid–solid separation in bioprocessing industries and in drinking water and wastewater treatment. Fractal dimensions of Escherichia coli ﬂocs at different times determined by two methods, small-angle light scattering and confocal image analysis, were between 1.9 and 2.19 (Section 8.2.4 [18]). Escherichia coli ﬂocs, ﬂocculated with chitosan, had fractal dimensions of 1.99 ± 0.08 and 2.14 ± 0.04 (determined by the small-angle light scattering and two-slopes methods, Section 8.2.4) [17]. 8.3.2 Mycelial Fungi Growth of a range of mycelial fungi, including saprotrophs, plant pathogens and mycorrhizal formers, has been examined on solid agar, and to a lesser extent in liquid medium. The whole range of fractal dimensions has been found in two dimensions (Figures 8.4 and 8.5), from close to unity for species producing linear rhizomorphs3 (Figure 8.4d) to complete space ﬁlling (Table 8.1). Fractal dimensions vary between and among species, depending on abiotic conditions and on time. With regard to the time, the lowest fractal dimensions, which then rise, are often found immediately after the germination of spores. For example, DBM = 1.05–1.2, 72 h after germination and DBM = 1.43–1.57 by 116 h with Armillaria luteus [33]. Likewise, DBM increased from 1.4 to about 2 over 5 h with Trichoderma viride, implying denser packing of hyphae with time [10]. On the other hand, fractal dimensions may decrease later on, e.g. DBM dropped from 1.73 at 20 days to 1.65 at 40 days with Aspergillus oryzae [36]. The DBM of Armillaria rhizomorphs was not, however, correlated with time [32]. Increase in fractal dimension implies that the colony is ﬁlling space more, whereas a decrease implies that the mycelium is thinning out. However, global measures of DBM may mask changes occurring in different regions. Because of the dynamic and modular nature of mycelial systems, mycelia may undergo morphological changes in one area concomitant with changes elsewhere in the same mycelium. Image analysis allows discrete morphological regions to be extracted and analysed separately (e.g. Figure 8.1e), e.g. localised fractal dimensions of mycelial fans produced as a result of interspeciﬁc mycelial interactions (Figure 8.2). This highlights the requirement for appropriate use and interpretation of fractal dimension alongside visual examination of mycelial images.

3 Linear organs of mycelial fungi, which often have a thick rind, are differentiated internally and extend by apical growth of the whole organ.

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Figure 8.4 Digital images of ectomycorrhizal mycelial systems of: (a) Suillus bovinus and (b) Paxillus involutus in peat microcosms (in association with Pinus sylvestris); (c) the pine root pathogen Rhizina undulata extending from a 50 cm3 section of pine stem; (d) Armillaria ostoyae extending from a 8 cm3 pine wood resource across nonsterile soil compacted in 24 cm × 24 cm trays.

Carbon and nutrient content of the agar medium can have large effects on fractal dimension. For example, four ectomycorrhizal species had a fractal dimension of almost 2 on a basal medium designed to give good growth of these species, but between 1.2 and 1.79 when glucose content was reduced by a half or three-quarters [37]. The presence of heavy metals, added as chloride salt in concentrations of 0.005– 3 mmol l−1 , had differential effects on Achlya bisexualis: there was an increase in DBS with Cu, Co and Hg, but no effect with Zn and Cd [31]. With 0.5 mmol l−1 Cu, DBS increased from 1.3 to nearly 1.5, and DBS increased to 1.6 with 3 mmol l−1 Hg. It was suggested that the increase in fractal dimension may have arisen from disrupted tip growth and branching at the edge of the mycelium. In terms of mycelial pattern formation as a whole, it has been suggested that the wide range of morphologies that develop arise by mycelia operating as nonlinear (i.e. regulated by feedback), hydrodynamic systems with indeterminate (i.e. indeﬁnitely expandable) boundaries [39]. Fractal dimension has also been occasionally determined for fungi growing in liquid culture. When cultures are agitated, mycelia tend to form pellets. In one study,

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DBS = 1.865 DBM = 1.872

I

I

DBS = 1.735 DBM = 1.964

DBS = 1.807 DBM = 1.929

I

I

DBS = 1.545 DBM = 1.847

I

DBS = 1.829 DBM = 1.862

I

Figure 8.5 Digital images of mycelial systems of (a) Agrocybe gibberosa, (b) Phallus impudicus, (c) Stropharia caerulea, (d) Stropharia aeruginosa after 28 days, (e) Coprinus picaceus after 180 days (image courtesy of Alaa Alawi) and (f) Resinicium bicolour after 30 days (image courtesy of G.M. Tordoff), extending from 4 cm3 beech wood resources onto nonsterile soil which has been compacted in 24 cm × 24 cm trays. White circles (a–d) are inert plastic caps; I are inocula. Note the differences between the mass (DBM ) and surface (DBS ) fractal dimensions of the species.

Aspergillus niger had values of DBM = 1.3–1.47 [34]. Fractal dimension has also been used in analysis of fragment size distribution, when mycelia have been fragmented in a homogeniser to form inoculum [40]. With the wood decay basidiomycete Phanerochaete chrysosporium, DBM decreased with increasing shear force [40]. A major problem with all of these studies on agar and in liquid media is that nutrients are, at least initially, distributed homogeneously, which is far from the situation in the natural environment, where there is considerable spatial heterogeneity over a range of scales. To get slightly closer to the ﬁeld situation, sheets of cellophane

Table 8.1 Examples of fractal dimensions of mycelial fungi, from different ecological groups, growing on artiﬁcial media. Species

Ecological group

Taxonomic group

Growth conditions

Measurement technique

Fractal dimensiona

Ref.

Achlya bisexualis

Saprotroph

Oomycota

Agar media with added heavy metals

Box counting

DBS = 1.26–1.6

[31]

Armillaria calvescens

Saprotroph, rhizomorph former

Basidiomycota

Agar, at different times during rhizomorph development over many days

Box counting

DBM = 1.46 ± 0.01; 1.40 ± 0.07

[32]

Armillaria luteus ( = A. gallica)

Saprotroph, rhizomorph former

Basidiomycota

Agar, at different times during colony development to 74 h

Box counting

DBM = 1.05–1.20

[33]

Armillaria luteus ( = A. gallica)

Saprotroph, rhizomorph former

Basidiomycota

Agar, at different times during rhizomorph development over many days

Box counting

DBM = 1.47 ± 0.03; 1.45 ± 0.03

[32]

Armillaria ostoyae

Tree root pathogen, rhizomorph former

Basidiomycota

Agar, at different times during colony development to 116 h

Box counting

DBM = 1.43–1.57

[33]

Table 8.1 (Continued). Species

Ecological group

Taxonomic group

Growth conditions

Measurement technique

Fractal dimensiona

Ref.

Armillaria sinapina

Saprotroph, rhizomorph former

Basidiomycota

Agar, at different times during rhizomorph development over many days

Box counting

DBM = 1.54 ± 0.03; 1.40 ± 0.04

[32]

Ashbya gossypii

Saprotroph

Ascomycota

Agar

Box counting

DBM = 1.94 DBS = 1.45

[14]

Aspergillus niger

Saprotroph

Deuteromycota/ Ascomycota

Pellets formed in liquid culture

Box counting

DBM = 1.3–1.47

[34]

Aspergillus oryzae

Saprotroph

Deuteromycota/ Ascomycota

Agar, at different times during colony development

Box counting

DBM = 1.55–1.8

[35]

Aspergillus oryzae

Saprotroph

Deuteromycota/ Ascomycota

Liquid-like medium (0.15 %)

Box counting

DBM = 1.63–1.73

[36]

Coltricia perennis

Ectomycorrhiza former

Basidiomycota

Agar with different concentrations of inorganic nitrogen and glucose

Concentric rings

DM = 1.47–2.12

[37]

Trichoderma viride

Saprotroph

Deuteromycota/ Ascomycota

Agar, at different times during colony development

Concentric rings

DM = 1.4–2.0

[10]

Laccaria bicolor

Ectomycorrhiza former

Basidiomycota

Agar with different concentrations of inorganic nitrogen and glucose

Concentric rings

DM = 1.64–2.6

[37]

Lactarius hepaticus

Ectomycorrhiza former

Basidiomycota

Agar with different concentrations of inorganic nitrogen and glucose

Concentric rings

DM = 1.20–1.79

[37]

Macrophomina phaseolina

Pathogen of beans

Ascomycota

Agar at different times during colony development

Box counting

DBM = 1.21–1.84

[38]

Paxillus involutus

Ectomycorrhiza former

Basidiomycota

Agar with different concentrations of inorganic nitrogen and glucose

Concentric rings

DM = 1.70–2.18

[37]

a

All fractal values were estimated in two dimensions: fractal dimensions greater than 2 presumably reﬂect inaccuracies in the estimation procedure.

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were inoculated with single germinated spores of T. viride, that were placed in Petri dishes, to which a discrete nutrient source (cylinder of agar 5 are not available, then the fractal dimension measurement should only be considered as qualitative. Di Stasio et al. [46] used this technique to study the agglomeration kinetics of growing soot generated in a diffusion atmospheric ﬂame. The number of primary particles per aggregate and the number concentration of clusters were evaluated on the basis of the measured angular patterns of the scattered-light intensity. They found that the kinetics of the coagulation process that yields to the formation of chain-like aggregates by soot primary particles can be described with a constant coagulation kernel in accord with the Smoluchowski coagulation equation in the free molecular regime. Earlier, this group [44] investigated particle morphology in an ethylene– air diffusion ﬂame for different ﬂow rates of the fuel gas. It is reported that soot

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FRACTAL GEOMETRY OF AEROSOL PARTICLES

Scattered light intensity, (I(q))

Rayleigh regime Power Law regime

Slope = –Df

Monomer regime

Slope = –4

R g–1

a–1

Magnitude of scattering wave vector, (q )

Figure 9.4 Schematic representation of the scattered light intensity I (q) versus q = 4π λ−1 sin θ/2, where θ is the scattering angle, from an ensemble of fractal aggregates of dimension Df on a log–log plot [43]. Reprinted with permission from Aerosol Science & Technology, ‘Light scattering by fractal aggregates’, 35, 648–687, Copyright 2001. Mount Laurel, NJ.

aggregates, at the early stages of the agglomeration process, ﬁrst grow as elongated chain-like structures and then, at later stages, assume more compact morphologies with signiﬁcantly smaller sizes. Thus, branched elongated aggregates grow up at low heights-above-burner with a fractal dimension that is practically constant (∼1.3). At larger heights-above-burner they self-reorganise into much shorter clusters with a fractal dimension increasing up to 1.9. It is proposed that this rearrangement of soot fractal aggregates can be explained in terms of local heat transfer balance at the cluster surface, which could cause, in the case of chain-like aggregates, a supplementary oxidation inside the pores of soot with possible breaking of the graphitic layers that act as bridges between primary particles within an aggregate. By constraining soot to a cylindrical annular ﬂame front with a radial thickness that appears thin relative to the increasing dimensions of the soot aggregates, thus conﬁning them to a twodimensional space during aggregation, Sorensen and Hageman [47] were able to show that such aggregates have a fractal dimension of 1.40. This is consistent with simulations of two-dimensional DLCA. Wang and Sorensen [48] considered large values of qRg to differentiate between single-cluster and polydispersity effects and showed that the effects are signiﬁcant, enough even to allow a measurement of the polydispersity of the aerosol. Recently, Kim and Choi [49] have developed an in situ light-scattering method for online measurement of aggregate size and morphology. Planar multi-angular lightscattering measurements were interpreted by Rayleigh–Debye–Gans scattering theory

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for fractal aggregates in order to obtain the mean radius of gyration and the fractal dimension along the ﬂame axis simultaneously, which are the parameters characterizing size and morphology of aggregates. The system developed was applied to study the evolution of silica aggregates produced in a methane–air premixed ﬂat ﬂame. To conﬁrm the suitability of the method as an online measurement of the growth of aggregates, they compared the results with those obtained by conventional point measurements and found that the two results agreed well with each other. They found that the fractal dimension increased with height above the ﬂame but that its variation became very small above 15 cm, where the fractal dimension was approximately constant at 1.7. 9.2.3 Aerodynamic/Mobility Methods Schmidt-Ott [34, 50] reported on a method which involves comparing the mobility radii of the aggregated particle before and after tempering. He showed that temperature increase induced reconstruction of silver aggregates, as observed by a decrease in mobility diameter, until a close-packed form was produced, i.e. tempering causes the aggregate to collapse until Df ≈ 3. Kutz and Schmidt-Ott [35] have described a method which was based on a comparison of the mobility equivalent radius, determined by differential mobility analysis, with the ratio of mass and drag, measured by a low-pressure impactor. For silver aggregates produced by heating silver up to 1000◦ C, they found a fractal dimension of 2.06. Weber and Friedlander [51] introduced a model to describe this rearrangement process based on the change in free energy per primary particle during restructuring. The excess free energy compared with the ﬁnal state drives the aggregate to become more compact, which results in an increase in the coordination number, as estimated from the changing fractal dimension of the aerosol aggregates. This restructuring behaviour of aggregated nanoparticles has recently been reviewed [52]. They identiﬁed three steps in the restructuring process: compaction, internal rearrangement, and evaporation. A clear connection was observed between the particle size and the compacting temperature: the smaller the particle, the lower the temperature. The fractal dimension can be estimated via the mass mobility relationship [53, 54] based on the scaling laws developed by Schmidt-Ott et al. [55]. In this relationship, it is assumed that the number of the primary particles N is proportional to the particle mass Mp , which requires the primary particle size distribution to be constant for all values of number of primary particles per aggregate as well as the assumption that the primary particle density is constant (which may not be strictly true if the aggregate is coated by a second species). The mass mobility relationship is given by Df dm Mp = C (9.2) dp where C is a constant, dp is the diameter of the individual spherules comprising the aggregate and dm is the mobility diameter (a diameter relating to a particle’s mobility

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FRACTAL GEOMETRY OF AEROSOL PARTICLES

in an electric ﬁeld [56]). It follows that the particle volume (the volume of the particle is comprised of all material and all void spaces enclosed within the particle envelope) is directly proportional to N. Equation (9.2) can then be rewritten thus: Df dm dv3 = C (9.3) dp Here, C is a constant and dv the volume equivalent diameter. Since dm can be measured with instruments such as a differential mobility analyser (DMA) or scanning mobility particle sizer (SMPS) and dv can be estimated as a function of N (for a given dp ) as described above, then Equation (9.3) can be used to estimate a fractal dimension based on the mass–mobility relationship with known values of dp and N . The fractal dimension may be obtained from Equation (9.2) as the slope of log(Mp ) versus log(dm ). An alternative method for estimating Df (for values of 2 or larger) can be derived using the results of Rogak et al. [57] and Schmidt-Ott [50]. For Df ≥ 2.0: dva3 =

C D

dp f

dmDf −2

(9.4)

where C is a constant and dva is the particle diameter in the free molecular regime (i.e. the mean free path of the gas is greater than the particle diameter). Then: dva3 = CdmDf −2

(9.5)

Hence, if the fractal dimension Df = 2, then dva is a constant. By plotting log(dva ) versus log(dm ), values of Df > 2 can be determined. This relationship has been shown to hold for all particles in the continuum regime, and for particles with Df ≥ 2 in any ﬂow regime [54, 55, 58]. For fractal dimensions Df 2, the interior spherules are no longer shielded by spherules on the exterior and, consequently, drag and mass are simply functions of the number of primary particles and scale together. In this case dva and dm measurements would not contain the information needed to determine Df (i.e. when Df < 2). It is uncertain how far into the transition regime this relationship holds for particles with Df ≤ 2 [54]. Based on the above, DeCarlo et al. [54] estimated aerodynamic and mobility diameters as a function of number of primary particles. These calculations predicted two distinct values of the fractal dimension based on the mass–mobility relationship and depending on the size of the aggregate. For an aggregate with N ≤ 60, Df = 2.46, whereas for N ≥ 60, Df = 1.79. These values of the fractal dimension are consistent with those observed for some types of combustiongenerated soot [54, 59]. The decrease in Df occurring with the transition to the large N regime can be interpreted as being due to fractal aggregates with fewer primary particles ﬁlling a larger fraction of the volume around their centre of mass than aggregates with much larger N. Slowik et al. [59] used the mass–mobility relationship to investigate the impact of fuel equivalence ratio on soot morphology produced by a propane–oxygen ﬂame. Two types of soot particle were observed, depending on the fuel equivalence ratio. For < 4, the fractal dimension of the particles was approximately 1.7 ± 0.15. These

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particles were composed mostly of black carbon, with the organic carbon content increasing as increased. For > 4, the particles were nearly spherical, with a fractal dimension of 2.95 ± 0.10. These particles were composed of about 50 % polyaromatic hydrocarbons, 45 % black carbon and 5 % aliphatic hydrocarbons. By combining inductively coupled plasma optical emission spectrometry with an epiphaniometer and a differential mobility analyser (DMA), it is possible to determine, simultaneously, the mass, surface area and diameter of aggregated aerosols [39]. From these measurements, the fractal dimension and the monomer size may be determined. The results indicate that silver aggregates, produced by spark discharge in argon, possess various fractal dimensions. A substantial increase in the density of the aggregates occurred by heating, even in the temperature range where the primary particle size remained unchanged. This increase was due to internal rearrangement leading to a more compact packing. In the domain of fractal-like structures, heating resulted in a gradual increase in mass fractal dimension from ∼1.8 to ∼3. The restructuring rate was governed by the temperature and the heating time. The results show that the applied methods have the potential to tailor aggregate shape in order to match special requirements in the production of aggregated structures. The papers by Burtscher and co-workers [60, 61] compare the mobility diameter and the mass of the particle to give the fractal dimension of aggregates from various combustion systems. Since some volatile species are emitted in the gas phase and condense on the particles when the exhaust gas cools down, the output from the exhaust pipe was passed through a thermodesorber. A fractal dimension of 3 was found with the thermodesorber at ambient temperature, whilst if the volatile fraction was removed by setting the desorber temperature to 350 ◦ C, the particles became smaller and the fractal dimension decreased to 2.2. It was suggested that particles consist of an aggregated, nonvolatile core, covered by a volatile liquid material which forms a droplet. As soon as the liquid is removed, the aggregated structure of the core is exposed. It is also possible to use a combination of two steady-ﬂow instruments in series: one classifying the particles according to mobility diameter and the other according to aerodynamic diameter. In these studies, the density of DMA-classiﬁed particles is measured by an impactor. Schleicher et al. [62] and Skillas et al. [63] used a single-stage low-pressure impactor. McMurry et al. [64] studied the particle density dimension of atmospheric aerosols by classifying particles with a DMA and then measured the particle mass with an aerosol particle mass analyser [65]. They hypothesized that the less ‘massive’ particles consisted of chain-aggregated soot. Maricq et al. [66] studied the density and the fractal dimension of diesel particles by using a DMA and an electrical low-pressure impactor (ELPI). They classiﬁed particles by DMA and then measured the aerodynamic size of the classiﬁed particles with the ELPI. Although a large number of individual particles are measured rapidly, these methods are time-consuming because one particle size has to be measured at a time. This method can give an average fractal dimension, because an ensemble of fractallike aggregate particles is present in the aerosol, although no actual values were reported.

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FRACTAL GEOMETRY OF AEROSOL PARTICLES

All of the methods based on mass and mobility measurements introduced above require mobility size classiﬁcation. Often the aerosols need to be size fractionated for further analysis and a separate size distribution measurement is needed. Virtanen et al. [67] have used an on-line method to estimate the effective density, fractal dimension, and size distribution of particles based on parallel measurements made by an SMPS and ELPI. A ﬁtting procedure is used to ﬁnd the particle density as a function of particle size and the fractal dimension. The method has been tested by simulation and by experimental measurements of particles with varying morphology. For fractal dimension values between 2.2 and 3.0, fractal dimension was measured with an accuracy of 0.1, and effective density was measured with 15 % relative accuracy. Recall that this method is only valid for fractal dimension values >2. Van Gulijk et al. [58] also used a combination of SMPS and ELPI. For diesel soot, they found a fractal dimension of 2.42 ± 0.12, close to the value of 2.15 ± 0.10 for ﬂame-generated soot and 2.3 ± 0.1 for vehicle exhaust particles reported by Maricq and Xu [68]. These values may be also compared with those reported by Skillas et al. [63], who found values between 2 and 2.5 (except for idle engines, where they found a value of about 3), and Burtscher [61], who found 2.2 for carbon core soot particles of spark engines. One of the main conclusions of the Van Gulijk et al. [58] work is that the performance of both instruments is affected by the fractal-like structure of diesel soot. The ELPI, if it is set to measure the aerodynamic diameter, gives an underestimation of the apparent size of particles due to their fractal-like structure. As a result, the number of particles is overestimated. Electrical mobility analysers are normally calibrated for spherical particles. As a consequence, data obtained from the instrument cannot directly provide the surface area and volume distributions for nanoparticle aggregates, although a new theory enabling this has recently been published (see Section 9.3.4). The SMPS is affected by multiple charging of soot aggregates, larger than 1000 nm, which are abundantly present in diesel exhaust gas and are not removed by the impaction stage at the inlet due to their small effective density. Multiple charging leads to an underestimation of the size of the aggregates and hampers size classiﬁcation in the instrument. The SMPS capability to measure particles down to 3 nm is an advantage when compared with the ELPI. Ideally, both instruments should be used. It is evident that the above direct measurements of the fractal dimension can be performed in various ways, e.g. light scattering [41–45], combining a DMA and an impactor [69], and exposed surface and mobility of the aggregates using a DMA, an ephiphanometer and a condensation particle counter [39]. Other authors have developed methods and theories that describe the fractal structure as a function of the effective density of the aggregate [66–68]. The different approaches are not in contradiction, as Smallwood et al. [70] have shown that the effective density is linked to the fractal scaling laws. The coupling between the radius of gyration and the mobility diameter is nontrivial; the radius of gyration is a function of the geometric structure of the aggregate, whereas the mobility diameter is a function of the interaction between the aggregate and the carrier gas. The mechanical mobility of an aerosol particle is proportional to its radius

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of gyration [34, 50, 55]. It has been argued that the scaling law holds approximately for particles down to a single monomer [50], whereas others [48, 71] say that it does not hold for very small aggregates (consisting of a few primary particles) in the free molecule regime. Papers describing light-scattering methods have the advantage that the technique measures the radius of gyration rather that the mobility diameter and thereby circumvents the nontrivial coupling between them. A disadvantage of light scattering is that scattering models are complex and may introduce uncertainties, e.g. unknown refractive indexes for the particles, skewed particle size distributions, and aggregate restructuring can inﬂuence the outcome of the measurement. In contrast, the alternative derivation [34] is based on models that describe the friction force that gases exert on moving particles. These models are straightforward when compared with models for light scattering by aerosols. Unfortunately, a quantitative comparison between the two approaches is not straightforward, since the experimental methods are based on very different measuring principles. Comparison of the various values of the fractal dimension quoted in the literature is difﬁcult, as aggregates that were generated with different sources or measured with different techniques tend to have different values for the fractal dimension. Generally, soot formed by internal combustion has a fractal dimension different from that produced by burners. In the absence of scanning electron microscopy (SEM) and transmission electron microscopy (TEM) micrographs, comparison is difﬁcult. The different histories of the soot particles in internal combustion engines and burners may be responsible for the different fractal dimensions. It has already been mentioned that volatile compounds may condense and restructure the aggregates. Methane, propane and acetylene ﬂames produce drier soots, as no heavy hydrocarbons are present. Other factors which could be responsible include soot oxidation, sintering and adsorption of gases, liquids and solids. These differences in the history of the soot particle and combustion conditions, together with the various measurement techniques, could well explain the differences in the reported fractal dimensions. A fundamental point of concern is that the fractal dimension is a rather coarse characteristic for the fractal-like structure of soot particles. Modelling studies have shown that the fractal dimension is weakly dependent on the number of primary particles in the aggregate [19]. This, together with the different history of each individual soot aggregate in the combustion chamber, results in a distribution of fractal dimensions. The fractal dimension determined by all the direct techniques represents an average over the ensemble of aggregates. Nevertheless, studies have indicated that the fractal dimension is precise enough to distinguish between different engine loads: the lower the engine load, the higher the fractal dimension [63]. Hence, the fractal dimension is a useful tool to study the characteristics of diesel soot. Computer simulations have shed light on the aggregation processes [17, 19, 72]. Assumptions are required concerning the nature of the aggregate motion between collisions, as well as the nature of the collision process. Such assumptions may approximate the physics of actual collisions or may be a highly idealised version of the real process. Additionally, these models will give a too optimistic prediction

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FRACTAL GEOMETRY OF AEROSOL PARTICLES

for the fractal dimension of real-life soot aggregates, as they do not incorporate the history of the soot particles. 9.2.4 Other Techniques For one cluster falling with velocity v1 and colliding with another moving with velocity v2 , the velocity of the resultant particle v3 is given by v1γ + v2γ = v3γ , where γ = Df /(Df − 1) [36]. Hence, if measurements of v1 , v2 and v3 are made, the fractal dimension may be obtained. To observe the process of kinematic coagulation and to measure the settling velocities, a camera with output to a video-recorder can be used. The process of kinematic coagulation could then be traced and photographed from the recorded images. The fractal dimension for carbonaceous (butane) particles ranged from 1.61 to 2.38 with an average value of 1.97. For magnesium oxide, the fractal dimension ranged from 1.10 to 1.41 with an average value of 1.19. Colbeck and Wu [73] have developed a relationship between aerodynamic diameter da and volume equivalent diameter dv which enables the fractal dimension to be determined: [1.5(Df −1)]/Df da = Cdv (9.6) The calculation of average, volume-equivalent, diameters was based upon total suspended particulate mass and the total number of particles, the latter determined by a condensation particle counter. Aerodynamic size distributions were determined via sedimentation. The operational assumption in this technique is that an insigniﬁcant number of the total number of aerosol particles are in the range below 0.1 m. The above method was utilised to determine the fractal dimension of smoke produced from a butane ﬂame and from burning magnesium. For the former, a fractal dimension of 1.96 ± 0.10 was reported and for the latter a value of 1.08 ± 0.03 was obtained. The low value for the magnesium oxide aggregates can be explained by a tip-to-tip variation of the particle–cluster model. In the tip-to-tip model, two clusters with opposite charges on their tips collide. Electrostatic attraction affects their relative motion prior to collision, resulting in a more linear cluster and, thus, a lower fractal dimension. In this method, the deduced fractal dimension is an average value for the ensemble of aerosol particles. The fractal dimension has also been determined from measurements of the terminal velocity VTS of a cluster and the electric ﬁeld strength E required to balance the gravitational force [74]. This technique utilises the classical Millikan cell. A plot of log(VTS ) against log(E) enables the fractal dimension to be computed from the gradient: 1 log(VTS ) = C + 1 − log(E) (9.7) Df where C is a constant. This method assumes that the clusters are spherically symmetrical and they possess the same number of charges. These problems have been overcome by using a similar technique in conjunction with a photoemission method

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which enables the fractal dimension for single aggregates in three dimensions to be determined [37]. In this method, a modiﬁed Millikan cell is used to obtain in-situ measurements of mass fractal dimension for single butane aggregates in the transition/continuum regime [75]. The method relies on the measurement of the aerodynamic diameter and the absolute mass of the aggregate using a photoemission method. Hence, the volume-equivalent diameter and the measured aerodynamic diameter allow the dynamic shape factor and the mobility equivalent diameter to be calculated. The latter diameter is equivalent to the hydrodynamic diameter dh . The relationship between dh and the radius of gyration Rg has been investigated both theoretically and experimentally and may then be used to derive the fractal dimension [76]: 1/2 Df Rg = dh (9.8) Df + 2 The only parameters required for these determinations are an accurate estimation of the sedimentation velocity and various balance voltages in the cell. As a veriﬁcation of the technique and equations, polyvinyltoluene particles were used as calibration standards. Mass fractal dimensions vary from 2.96 to 2.99 and show excellent agreement to the theoretical value of 3, hence verifying the method. A further development of the modiﬁed Millikan cell has enabled the in-situ determination of the fractal dimension after which the aggregate may be ‘plucked’ from the cell by inserting a 3 mm diameter SEM aluminium stub [77]. Despite all reasonable measures to minimise the handling of the sample, it was concluded that most of the captured aggregates were lost from the stub during the SEM gold-coating evacuation cycles. However, successful fractal analysis was performed on the SEM images obtained for two aggregates. For one, the mass fractal dimension was 2.08 and the structural fractal dimension was 1.32, whilst for the other the respective values were 2.29 and 1.06. No ﬁrm conclusions could be drawn from these ﬁgures, although internal features may be lost to a greater extent for a larger mass fractal dimension and these could be omitted when calculating the structural fractal dimension. Several studies suggested the following relationship between the projected area and number of primary particles [9, 27, 78–80]: α Aa N =ε (9.9) Ap where Aa is the projected area of the aggregate, Ap is the mean projected area of primary particles, ε is an empirical constant, and α is an empirical projected area exponent. Koylu et al. [80] suggested α = 1.09 and ε = 1.15, whilst Oh and Sorensen [11] reported that α = 1.19 and ε = 1.81. The radius of gyration can also be extracted from projected properties. Studies [9, 27, 80] have suggested that the maximum projected length L can be used as a substitute of Rg (i.e. L/2Rg = constant). Based on numerical and experimental work, Koylu et al. [80] suggested L/2Rg = 1.49 for N > 100, whilst Oh and Sorensen [11] and Brasil et al. [27] found that L/2Rg = 1.45

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FRACTAL GEOMETRY OF AEROSOL PARTICLES

and 1.50 respectively using numerical simulations. Hence, the fractal dimension based on maximum length can be determined as follows: Df L L N = kL (9.10) 2a where kL is the prefactor and Df L is the fractal dimension when the maximum projected length is treated as the characteristic size instead of the radius of gyration. For small aggregates (N < 100), the ratio (L/2Rg ) is not constant [80], and its value is inﬂuenced by the degree of overlapping [11]. For aggregates with Df < 2, projected properties found using TEM can be used to determine the fractal dimension with good accuracy whilst for Df > 2 the projected properties fail to predict the actual properties accurately, most likely due to primary particle screening effects [80]. Using this method, Park et al. [81] analysed TEM images of diesel particles and showed that the projected area equivalent diameter nearly equals the mobility diameter in the mobility size range from 50 to 220 nm. Doubly charged particles and possible ‘fragments’ were observed for the DMA-classiﬁed particles. The fractal dimension calculated from the TEM images of mobility-classiﬁed aggregates Df L was 1.75. The fractal dimension was also determined using particle mass and mobility size [50, 53] and a value of 2.35 was reported. These authors showed that the relationship between the projected area diameter and the maximum projected length can be used to ﬁnd the relationship between Df L and fractal dimension determined from mobility–mass measurements. In this case: Df Df L = (9.11) 1.26 By using Equation (9.11) with the measured value of the fractal dimension determined from mobility–mass measurements one obtains Df L = 1.87. The difference between the two values can be accounted for by experimental error and the assumption that the projected area diameter nearly equals the mobility diameter. Wentzel et al. [28] also used the projected maximum length method in order to determine the morphology of various soots. The fractal properties of 37 diesel soot aggregates were determined from TEM images. The average fractal dimension Df L of diesel soot derived from TEM was 1.70 ± 0.13. A second independent approach to determine the fractal properties of soot, based on computer simulations of the aerosol dynamics, was also used [82]. A good reproduction of the time evolution of mass and number concentrations and of the mobility size distribution was achieved. The primary particle diameters obtained from the computer simulations (25 ± 3 nm for diesel soot) were in excellent agreement with the TEM results. The fractal dimension of diesel soot calculated by computer simulation was 1.9 ± 0.2, consistent with the value obtained from TEM image analysis. Fractional Brownian motion (FBM) theory can, because of its self-similarity and long correlation properties, provide both a description and a mathematical model for many highly complex natural shapes and textures. FBM analysis can derive the fractal parameter, the Hurst coefﬁcient H , to represent the characteristics of a nonstationary

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zero-mean Gaussian random function such as the ﬂuctuation of soot aggregate textures. The value of H ranges from zero to one and is related to the fractal dimension by Df = De + 1 − H. The parameter De represents the Euclidean dimension, which equals one for linear data. Luo et al. [83] ﬁrst extracted the multiscale H values to represent the fractal texture of airborne particles by applying FBM combined with the Fourier-domain maximum likelihood estimator. They then explored the application of FBM analysis to SEM micrographs of soot aggregates emitted by a dynamometer [84]. By directly quantifying the surface texture of fractal-like aggregates to extract their Hurst coefﬁcients, the fractal dimension of such particles was found to be in the range 1.6 to 1.7. The impact of image properties on Df measurements due to digital image processing and data recording was also investigated. A twofold change in SEM magniﬁcation size gave rise to a 7 % deviation in the fractal dimension, and scaling up from the original image increased the discrepancy compared with miniaturisation. Brightness was not a serious interference factor, as its variance did not exceed the grey level value of 80. These results give more conﬁdence to earlier measurements based on image analysis. Gwaze et al. [85] have recently shown that the fractal dimension of aggregates from biomass combustion is, on average, 1.83 and so consistent with aggregate formation by DLCA. Importantly, the fractal dimension was determined by three different techniques: Df = 1.84 ± 0.05 from projected surfaces in the SEM images; Df = 1.80 ± 0.13 from the relationship between mobility and number of primary particles; and Df = 1.83 ± 0.05 from the mass and radius of gyration relationship. Although each technique has its limitations, this paper does give conﬁdence in the different approaches to the determination of the fractal dimension of aggregates.

9.3 APPLICATIONS 9.3.1 Combustion Aerosols Diesel particulate emissions are a major source of ﬁne and ultraﬁne atmospheric particles. These particles are of current interest due to their suspected adverse health effects and their impacts on the Earth’s radiation balance, visibility impairment, and atmospheric chemistry. Diesel particles are typically aggregates of ﬁne primary particles coated with condensed organic ﬁlms. The most common technique to characterise the structure of the aggregates is TEM, which provides projected two-dimensional properties of aggregates. Thus, the actual structural properties of aggregates are inferred from the projected properties. However, few studies of the structural properties of aggregates produced from diesel engines have been reported. Comparison of fractal dimensions reported in the literature is difﬁcult, because soot that was generated with different sources or measured with different techniques tends to have different values for the fractal dimension. Cai and Sorensen [71] found

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a fractal dimension of 1.8 for soot that was generated with a methane burner and measured by light scattering, whilst Kütz [86] found 2.13 for spark-generated soot. Luo et al. [83] found values in the range 1.6–1.7. The difference with diesel soot suggests that soot particles formed by internal combustion have a different fractal dimension than soot generated from burners. Kütz and Schmidt-Ott [69] report that condensation of vapour on aggregates induces restructuring, leading to denser, higher Df , structures. Burtscher [61] and Lepperhof et al. [87] found evidence that adsorption of hydrocarbons on soot particles increases the fractal dimension. Methane- and acetylene-ﬂame-generated soots tend to be much drier because no heavy hydrocarbons are present in these ﬂames. Nevertheless, the studies by Skillas et al. [63] showed that the fractal dimension is precise enough to distinguish between different engine loads: the lower the engine load, the higher the fractal dimension. They suggested that as the load increases the air/fuel ratio decreases and more primary soot particles are produced. At low loads, the fractal dimension is close to 3 and this reduces to 2.1 at higher loads. For low engine loads, the agglomeration process is mainly governed by monomer–cluster aggregation, whereas cluster–cluster agglomeration signiﬁcantly competes with monomer–cluster aggregation at high loads, lowering the fractal dimension. In contrast to this, work by Park et al. [53] and Maricq and Xu [68] found a small systematic increase in fractal dimension with decreasing engine power. A closer examination of the Skillas et al. [63] data reveals that, except for a few points at low load, the majority of their data lie within Df = 2:4 ± 0:15. This general consistency in fractal dimension also extends to spark-generated soot, where Schleicher et al. [62] found 2.05 < Df < 2.35 from DMA/low-pressure impactor measurements. Colbeck et al. [88] analysed various smokes and found textural fractal dimensions in the range 1.12–1.27 for petrol, diesel, fuel, oil, parafﬁn, butane and wood (Table 9.2), whilst the mass fractal dimensions were 2.04, 1.88, 1.97 and 2.35 for diesel, fuel oil, butane and wood respectively. This wide range represents a

Table 9.2 Fractal dimensions of different smokes [88]. Reprinted from J. Aerosol Sci., Vol. 28, Colbeck, I.,Atkinson, B. and Johar, Y., The morphology and optical properties of soot produced by different fuels, 715–723. Copyright (1997), with permission from Elsevier. Fuel

Petrol Diesel Fuel oil Parafﬁn Butane Wood

Fractal dimension Textural

Mass

1.27 1.25 1.15 1.19 1.25 1.12

— 2.04 1.99 — 1.97 2.35

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correspondingly broad variation in structure and suggests that chemical composition might affect the ﬁnal structure. Chakrabarty et al. [89] also found a range of fractal dimensions for different wildland fuels (sagebrush, poplar wood, ponderosa pine wood, dambo grass, ponderosa pine needles and white pine needles). From SEM images, the fractal dimension was determined from the relationship between the number of primary particles per aggregate and the radius of gyration (Equation (9.10)), yielding values between 1.67 and 1.83. The textural fractal dimension was also obtained and reported to be in the range 1.10–1.19. It should be noted that the presence of collapsed semi-liquid aggregate chains in some of the samples suggests the presence of organic coatings on some of the black carbon cores. Liu et al. [90] studied the morphology of ﬂame-generated submicrometre aerosols by characterizing both ﬂy ash generated in a pilot-scale coal combustor and controlledcomposition inorganic aerosols. They found that the ﬂy ash was fractal-like with mass fractal dimensions of 1.9–2 and prefactors (see Equation (9.1)) of 1.1–1.5. The morphology of submicrometre, ﬂame-generated inorganic aerosols is known to be fractal-like. Flame-generated silica, magnesia, sodium-doped silica, and magnesiumdoped silica produced under identical conditions were also fractal-like with mass fractal dimensions 1.7–1.8 and prefactors of 1.6–1.8. No dependence of these structural parameters on chemical composition, ﬂame residence time, or particle number density was observed over the ranges considered. Changing chemical composition did, however, lead to order of magnitude changes in primary particle diameter without any corresponding change in aggregate structure. These ﬁndings are consistent with a growth process governed in the late stages by cluster–cluster aggregation. Further, they indicate that for ﬂame-synthesised materials produced in the overall decreasing temperature gradient characteristic of coal combustors and industrial ﬂame reactors, the aerosol aggregate structure will not be affected by changes in chemical composition under conditions of coalescence-limited growth. The various values of the fractal dimension for compounds associated with ash derived from coal combustion are given in Table 9.3. Most measurements are consistent with a value of approximately 1.8 reported in computer simulations of the coagulation of aggregates in three dimensions in both the free molecular and continuum regimes [17, 95]. However, fractal dimensions of approximately 2.5 are expected when growth is dominated by the diffusion of primary particles to an existing aggregate [62]. The values for MgO show considerable variation, and this is a consequence of the particle generation method. The combustion conditions were closely controlled in the work of Liu et al. [90], resulting in well-deﬁned, uncharged, primary particles with an average diameter of 5 nm. On the other hand, the combustion of magnesium ribbon produced polydisperse charged primary particles [36, 73]. The ability to characterise aggregate morphology quantitatively, through the fractal dimension, has motivated research on aggregate structure and led to the discovery of ‘superaggregates’, i.e. structures with fractal dimensions that differ from the substructures [96–98]. Using small-angle light scattering, in laminar diffusion ﬂames for a wide range of heavily sooting fuels, a fractal dimension of approximately

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Table 9.3 Fractal dimensions for compounds found in ﬂy ash [90]. Reproduced with permission of American Association of Aerosol Research ‘The effect of chemical composition on the fractal-like structure of combustion-generated inorganic aerosols, Aerosol Sci. Technol., 33, 459–469’. Copyright 2000. Mount Laurel, NJ. Material

Fractal dimension

Synthesis method

Measurement method

Reference

MgO MgO TiO2

1.08 ± 0.03 1.19 ± 0.02 1.66 ± 0.67

Settling, SEM Settling TEM

[73] [36] [57]

SiO2 SiO2 SiO2 SiO2 Al2 O3

1.7–1.9 1.49 ± 0.15 1.84 ± 0.08 2.52 ± 0.05 1.54

Mg ribbon combustion Mg ribbon combustion Alkoxide thermal decomposition Commercial fumed silica Seeded premixed ﬂame Commercial fumed silica

TEM Light scattering Light scattering Light scattering TEM

[4] [91] [92] [93] [94]

Bituminous coal SiO2

2.05 ± 0.06

TEM

[90]

TEM

[90]

SiO2 SiO2 MgO

1.71 ± 0.15 1.70 ± 0.03 1.81 ± 0.12

TEM TEM TEM

[90] [90] [90]

1.67 ± 0.07

Seeded counter diffusion ﬂame Pilot-scale combustion facility Nebulising precursor-containing liquid solutions in ﬂame As above As above As above

1.8 was obtained, which is a result of a three-dimensional DLCA process. With increasing height above the burner, which corresponds to increasing time for the soot to grow, a very intense, second phase appears. This scattering behaviour may be interpreted to indicate a phase of around 12 m soot with a fractal dimension of ∼2.6 for length scales of 1.2–12 m and ∼1.8 for scales smaller than 0.3 m down to the primary particle size (25 nm as determined by TEM). These results support a scenario in which supramicrometre aggregates are created by a percolation of the submicrometre aggregates with mass fractal dimensions similar or equal to 1.8. For less sooty ﬂames, only the submicrometre soot is seen. These measurements have been corroborated by computer simulation [99]. As agglomeration evolves, the largest cluster develops a hybrid structure with mass fractal dimension similar or equal to 2.6 over large length scales, whereas the early time dilute-limit fractal structure is frozen at smaller length scales. The largest cluster is thus an aggregate of smaller aggregates with a different fractal dimension (see Figure 9.5). Current theories for aggregation kinetics involving the Smoluchowski equation are not adequate to describe these dense systems. Hence, both experimental and theoretical research are needed for this cluster dense regime. While there has been some success in modelling the growth dynamics of the aggregates, quantitative results are lacking.

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Fractal dimension ~ 1.8 over this length scale

Fractal dimension ~ 2.6 over this length scale

Figure 9.5 Superaggregate: an aggregate of one morphology composed of smaller aggregates of different morphology [96]. Reproduced with permission of Elsevier.

9.3.2 Atmospheric Aerosols Ambient air contains varying fractions of carbonaceous particulates and aggregates, mixtures of carbonaceous particulates and inorganic crystalline particulates often composed of thousands of nanocrystals [100]. The morphology of atmospheric aggregates inﬂuences their effects on public health, water vapour nucleation, and absorption and scattering of light. For example, aggregate transport rates determine their atmospheric residence time and patterns of deposition in the lung, but mathematical models of aggregate transport and light scattering are usually based on the assumption that the primary particles that compose the aggregates are monodisperse [56]. Despite numerous algorithms and experimental techniques for determining the fractal dimension of aerosols, a quantitative description of atmospheric aerosols is rare. Those which have been reported have involved image analysis of ambient aerosols collected on a ﬁlter or TEM grid [23, 101–103]. Kindratenko et al. [102] concluded that fractal analysis allows the unequivocal identiﬁcation of particles source. For samples from Siberia they quote a textural fractal dimension of 1.09 ± 0.015 for ﬂy ash and 1.04 ± 0.015 for soil. However, the majority of the ﬂy ash particles examined were spherical and did not exhibit any fractality. This suggests that, in this case, ﬂy ash was formed by different mechanisms. Fractal analysis can also provide information on the formation processes of aerosols [23]. A range of mass fractal dimensions were reported for carbonaceous aerosols in Phoenix, Arizona, indicating variations of DLA. Aggregates were divided into groups: those with Df < 2 and those with Df > 2. For the former, a mean fractal dimension of 1.65 ± 0.15 was reported. For the other aggregates there was evidence that restructuring may have occurred [23]. The range of fractal dimensions found forms a continuum from 1.35 to 1.89. Cluster–cluster aggregation is the dominant

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formation process for 1.7 < Df < 1.9, whereas formation can be explained by the tip-to-tip model of cluster–cluster aggregation for aggregates with Df = 1.35 to 1.6. Fractal dimensions for the Phoenix aerosol appear to be independent of aggregate size, but morphology is affected by factors such as fuel composition and air-to-fuel ratios. Hence, it may be possible to use fractal analysis for source apportionment of aerosols. It should be noted that the aggregates in this study were relatively large in size (of the order of 1 m), whereas more recent studies revealed smaller aggregates (of the order 100 nm). These ultraﬁne aggregates, which make up the greatest proportion of the aerosol by number, have much less possibility for variation in space-ﬁlling structure than large aggregates. Dye et al. [104] determined the fractal dimension of aggregates at a roadside and background site in Plymouth, UK. The aerosol was classiﬁed into aggregate and nonaggregate groups. At the roadside and background site, aggregate particles accounted for 94 % and 89 % respectively of the particles analysed. Mass fractal dimensions of 1.56 ± 0.09 and 1.57 ± 0.09 and boundary fractal dimensions of 1.15 ± 0.09 and 1.13 ± 0.08 were measured for the roadside and background sites respectively. This morphology change may be due to the inclusion of particles from other nonaggregate sources that have a low fractal dimension or, more probably, it is indicative of a smoother, aged roadside aerosol at the background site. The results indicate an increase in boundary fractal dimension with aggregate size at both sites. The greatest difference between the sites occurs in the 120–220 nm size range, which suggests that the change from fresh roadside aggregates to aged, compacted and smoothed background aggregates is important in the size range above 120 nm. These conclusions were supported by Xiong and Friedlander [105], who analysed 102 particles collected in the Los Angeles area. This work showed that mass fractal dimension increased from near one to more than two as the number of primary particles making up the aggregates increased from 10 to 180. In one set of measurements, the fraction of the particles present as aggregates was about 60 % for particles with aerodynamic diameters between 50 and 75 nm and 34 % for the range 75 to 120 nm. The primary particles that make up atmospheric aggregates are more polydisperse than soot aggregates generated from a single laboratory source, an ethane–oxygen ﬂame. Most measurements were made in the Los Angeles area, where the aggregates may represent a signature for diesel emissions. Rural aggregate concentrations in the size range 50 to 120 nm were less than 1 % of the concentrations at urban sites. The increase in mass fractal dimension with N can be explained by the aggregation of short chains with low mass fractal dimension values to form larger aggregates with higher mass fractal dimension. Such an increase has been observed in computer simulations of the aggregation of an aerosol cloud of unitary particles in a ﬁxed volume of gas [106]. At rural sites, values of the mass fractal dimension showed no trend with N, and were in the range 1.5–2 as expected for cluster–cluster aggregation. This result may be due to the absence of fresh sources of aggregates containing short chains. Mullholland et al. [106] concluded that inferences can be drawn concerning atmospheric aggregation dynamics from measurements of aggregate fractal properties, but simulations

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of aggregate dynamics should take into account the distribution of the mass fractal dimension as well as aggregate size. TEM images were used to calculate the average surface area of the aggregates, which was estimated by summing the surface area of the primary particles composing the aggregates and dividing by the number of aggregates. Each stage on an impactor fractionates aerosols into a different size range based on aerodynamic diameter. For aggregates on stage 8 of the impactor (size range 50–75 nm) the average surface area was 0.188 m2 . A sphere of unit density depositing on the same stage would have a surface area of 0.031 m2 . Thus, the area of a spherical particle of equivalent aerodynamic diameter would underestimate the aggregate surface area by approximately a factor of six. For the largest aggregate investigated, the value of the surface area was 26 times that of the value corresponding to the aerodynamic diameter. These results have implications for human health, as the high surface area of the aggregates provides adsorption sites for toxic substances, such as polycyclic aromatic hydrocarbons. Various uncertainties are associated with these measurements. A low-pressure impactor was used to determine the aggregate concentration. This method assumes that the 50 % aerodynamic cut-off diameter of the impactor is a step function, above which the collection efﬁciency is 100 %. It was further assumed that the aerodynamic cut-off diameter is equivalent to the electrical mobility diameter, which, on theoretical grounds, is not expected to be true for aggregates. There is also a small uncertainty in the concentration measured with the DMA, associated with the adjustments for charging inefﬁciencies and sampling losses. It is also possible that aggregates break up in the impactor, giving a high count compared with the atmospheric concentration. The above studies used collection methods that may have changed the morphology of the aerosol during collection. McDonald and Biswas [107] captured particles via size-independent electrostatic precipitation at two sites in St Louis. For the aggregates examined, the mass fractal dimension ranged from 1.39 to 1.89 with a mean value of 1.69. The temporal variation in mass fractal dimension was also determined and found to be 1.65 in the morning, 1.49 around midday and 1.87 in the evening. It is suggested that variation in vehicle type may be responsible, with cars dominating trafﬁc ﬂow in the early morning and evening and trucks more abundant during the day. McDonald et al. [108] determined the fractal dimension of atmospheric aerosol from four sites in Greater Cincinnati. At sites in the vicinity of major highways the mass fractal dimension was of the order 1.60 ± 009, whilst at a site not inﬂuenced by trafﬁc-type combustion emissions the mass fractal dimension was 1.41 ± 009. At this latter site the aggregate particles mostly result from aggregation of freshly nucleated particles. It appears likely that fractal analysis will continue to be developed and will provide an invaluable tool for the identiﬁcation and apportionment of particles. Johnson et al. [109] sampled atmospheric particles of diameter 0.2–2 m in the Mexico City MetropolitanArea.Although they didn’t determine the fractal dimension, they noted that the aerosols collected in city trafﬁc were almost entirely carbonaceous, consisting of soot aggregates with liquid coatings suggestive of unburned lubricating

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oil and water. Processing of the soot by ammonium sulphate occurs rapidly, resulting in more compact, internally mixed particles. This has important implications, as the internally mixed nature of processed soot particles is expected to affect the heterogeneous chemistry of the soot surface, including the interaction with water during wet removal. Additionally, the observation of an oil coating on fresh soot particles should be examined more closely, both in terms of reactivity and consequences for human health. The studies to date provide only snapshots for a few sites. Much more data will be needed to characterise atmospheric aggregates sufﬁciently to permit reliable estimates of their large-scale effects. However, the suggestion that the fractal dimension varies from urban to rural locations has public health implications. Urban aggregates have a large surface area and a small aerodynamic diameter [13]. Particles with the same size in background air have a lower fractal dimension and, hence, a smaller surface area and larger aerodynamic diameter. Penetration of particles into the respiratory system is governed by the aerodynamic diameter, and so urban aerosols will penetrate more deeply, with a larger surface area, than more aged aerosols at background sites. 9.3.3 Restructuring Despite an immediate need to quantify the lifetime of soot in the atmosphere better, relatively little is currently understood regarding the processing of soot in the atmosphere, particularly in urban environments. Aggregate restructuring has been attributed to a number of mechanisms, including humidity, tempering, electrostatic processes, condensation, evaporation, compaction, internal rearrangement, coagulation, pressure and wind shear. 9.3.3.1 Humidity-Induced Restructuring Freshly emitted soot is hydrophobic. However, aging processes such as coagulation, condensation, and chemical reactions can occur. Once sufﬁcient water-soluble material has coated the soot’s surface, the particles become globally hydrophilic. The atmospheric lifetime of hydrophilic aerosols is considerably shorter than that of hydrophobic aerosols. Hence, the aging process of soot is one of the key uncertainties governing its lifetime, burden, and radiative forcing. Laboratory studies demonstrate that aged soot attracts and retains water, and can be efﬁciently removed from the troposphere by entrapment in existing liquid cloud droplets or by activation as cloud condensation nuclei [110]. Some measurements have indicated that there may be modiﬁcation of fractal morphology by atmospheric processes [111, 112]. As a consequence of humidity cycling, the morphology of the tenuous aggregates becomes more compact and ‘spheroidal’, resulting in an increase of Df . Jullien and Meakin [113] concluded that such a phenomenon would be important in the consideration of subsequent aggregate atmospheric lifetimes and optical properties. Various laboratory experiments have demonstrated the existence of a number of different mechanisms by which aerosols

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can be restructured. Some mechanisms investigated are: (1) temperature gradient and mechanical shear in colloids [114]; (2) the tempering of silver aerosols (Df ≈ 2.18) at 80 to 270 ◦ C into coalesced spheres for which Df ≈ 3 [50]; (3) electrostatic-induced restructuring of single in situ butane smoke aggregates in discrete steps to values as high as Df ≈ 2.2 [38]; and (4) compaction/internal rearrangement [52]. Another mechanism that may be of importance for larger aggregates in the supermicrometre range is the effect of wind shear. However, such investigations under atmospheric conditions are currently unknown to the author. Humidity-induced restructuring under high humidities or low supersaturations, however, remains more thoroughly investigated than other mechanisms to date. An investigation by Hallett et al. [112] noted a moderate restructuring effect on acetylene smoke, while a type of aviation fuel exhibited virtually no effect. Compaction was attributed to capillary forces acting upon branched chains, facilitated by the condensation of humidity on hydrophilic components of the aggregate. Later studies used fractal analysis to quantify the degree of restructuring as a function of relative humidity. Under humid conditions, Colbeck et al. [115] observed an increase in the fractal dimension from Df ≈ 1.8 to 2.0–2.5 with time for butane smoke aggregates. A curve-ﬁt analysis of their data (Figure 9.6) suggests an exponential increase of Df with relative humidity. For typical cloud supersaturations of 1 %, a value Df ≈ 2.0 is 3

Fractal Dimension

y = 1.93 + 2 × 10–6 exp(x – 90.5)

2.5

2

1.5 96

98 100 102 Relative Humidity (%)

104

Figure 9.6 Mean values of the fractal dimension for butane smoke aged for 4 h and then subject to various relative humidities and supersaturations (data from [115]).

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FRACTAL GEOMETRY OF AEROSOL PARTICLES

implied for up to ﬁve humidity cycles. A further humidity cycling study by Huang et al. [21] investigated the restructuring of diesel aggregates in an environmental SEM. Increases in the boundary fractal dimension for three different sulphur-content fuels were observed, where the high-sulphur case (0.084 wt % S) surprisingly exhibited no signiﬁcant increase and was attributed to possible pre-sampling restructuring. Ramachandran and Reist [116] found that, when an aerosol composed of irregularly shaped aggregates is subject to condensation followed by evaporation, the aggregates change from a branched, chainlike structure to a more regular, near-spherical or clumplike structure reﬂected by a decrease in the structural fractal dimension. The textural fractal dimension remains constant because the primary particles, of which the aggregates are composed, do not change in shape. The degree of supersaturation and the number of condensation–evaporation cycles that the aerosol undergoes are major factors that inﬂuence morphological change. Even at low supersaturations, increasing the number of condensation–evaporation cycles makes the aggregates more regular and thus decreases the structural fractal dimension. To investigate the inﬂuence of the hygroscopicity, comparative measurements between pure soot aerosol and soot particles pretreated with n-butanol vapour were carried out [117]. Electron microscope analysis showed that the size distribution of initially pure soot aerosol did not shift signiﬁcantly after the interaction with water droplets, whereas the size distribution of n-butanol-processed soot aerosol shifted to smaller sizes with the mean size reduced by a factor of three. This size redistribution could be explained by the assumption that wettable particles penetrate inside a water droplet during the coagulation, and, by being uniformly squeezed by capillary forces and subsequent water evaporation, become more compact and, consequently, smaller. The structure analysis data have shown that the fractal dimension of soot aggregates increased from 1.72 ± 0.15 to 1.87 ± 0.15 as a result of such densiﬁcation of the microstructure [117]. The transformation will be more pronounced in clouds and fogs, as the characteristic size of atmospheric water droplets is substantially larger than the droplets used in this study. These studies illustrate the ease with which aggregates can be restructured by humidity cycling. The tenuous nature of supermicrometre aggregates has been further demonstrated in the restructuring of individual particles suspended in a modiﬁed Millikan cell. Various degrees of compaction could be induced with limiting values of Df as high as 2.2. As each aggregate exhibited different limiting values, it may be suggested that restructuring mechanisms are not only dependent on external inﬂuences (i.e. humidity cycling, etc.), but also on individual aggregate morphology. To what extent humidity-induced restructuring is relevant to atmospheric processes would require further investigation to measure Df under realistic conditions. Direct in-situ atmospheric observations of humidity-induced restructuring are difﬁcult to obtain and remain sparse. Most evidence for humidity processing has come from laboratory and ground-based measurements. Trials on the humidity processing of atmospheric samples attributed a decrease in size to the restructuring of freshly emitted smoke [69]. In the same trials, laboratory experiments indicated that aggregate

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size may be important in the degree of ﬁnal restructuring obtained. While size was not directly measured, the degree of restructuring increased for larger smoke aggregates, a result supported in later experiments [38]. A humidity processing study by Weingartner et al. [118] on petrol- and diesel-derived smoke demonstrated a small degree of restructuring for aggregates with a mobility diameter of ∼100 nm at humidities as low as 50–60 %. This may give an indication of the initial restructuring possible under typical humidity conditions encountered in the boundary layer. Restructuring at < 95 % relative humidity is also implied for the results of Colbeck et al. [115] in Figure 9.5, where it is suggested that Df increases from an initial value of 1.8–1.9 to ∼1.9–1.95. Such conclusions, however, need more veriﬁcation due to the overall sparseness of data and diverse experimental conditions. The possible degree of cloud-processing-induced restructuring has been hinted at by the absence of tenuous combustion aggregates in atmospheric samples [119, 120]. Katrinak et al. [23] reported many restructured aggregates. Various restructuring mechanisms were postulated, including humidity processing, aging, and combustion conditions. Some aggregates exhibited coatings, determined by electron energyloss spectroscopy to contain sulphate and nitrate compounds, which may have been present during formation or formed subsequently through coagulation/photochemical activity. The most intermediate effect of restructuring would be to alter the aggregate radiative properties. Whether this is signiﬁcant on a global basis remains to be assessed. Traditionally, Mie theory, which is directly applicable to spherical particles and may be extended to other regular shapes (e.g. cylinders, ellipsoids), has been used to predict aerosol optical properties. However, it is not directly applicable to fractal clusters or other complicated particle shapes. The erroneous application of Mie theory has led to predictions of optical properties that are strongly dependent on the extent of particle aggregation. However, because the coagulating carbonaceous particles are not spherical, they maintain a larger effective surface area than spheres would and, thus, the optical properties remain largely unaffected by the increasing particle size. There are several techniques available for calculating the optical properties of aggregates. A rigorous discussion of the techniques is beyond the scope of this chapter, and the reader is referred to either Chapter 2 or Sorensen [43] for further details. As shown by Zuberi et al. [110], chemically aged soot particles can become signiﬁcantly hydrophilic and have a strong attraction for water. Increased hydrophilicity has strong implications on its lifetime in the atmosphere. Hydrophilic soot particles can be entrained inside existing liquid cloud droplets, leading to efﬁcient removal by wet scavenging. At high supersaturations, these particles may also act as cloud condensation nuclei and, consequently, be removed by rainout. If cloud condensation nuclei concentrations rise, then competition for the total available liquid water may result in smaller cloud particles and, hence, increased scattering of radiation and overall cooling effects. The entrainment of highly absorbing black soot cores within liquid droplets may lead to a local warming and evaporation of the clouds, known as the cloud-burning effect [2].

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In summary, the typical ﬁnal restructured value Df ≈ 2.0–2.2 for laboratory results may be compared with a value Df ≈ 2.2 for simulations [72, 113]. In this model, diffusion-limited cluster–cluster aggregates restructured in discrete steps to Df ≈ 2.2, after which further increase was not possible. The situation in the atmosphere is more complex, and there probably exists a spectrum of Df values in the range 1.8 to 2.0–2.2, dependent mainly on aggregate age and meteorological conditions encountered. 9.3.3.2 Thermal Restructuring As was mentioned brieﬂy in Section 9.2.3, aggregates can adopt more compact conformations when heated, a phenomenon known as thermal restructuring. Weber and Friedlander [51] have shown that, for silver aggregates, the mobility diameter begins to decrease at about 100◦ C and levels off around 350◦ C when the aggregates have reached a close-packed state with a fractal dimension of approximately 3. A similar behaviour, although at different temperatures, has been observed for other materials [52]. Experiments have shown that the smaller the particle, the lower the compacting temperature. This is a consequence of the well-known phenomenon of melting-point depression, which states that the melting point is lowered as the diameter of the particle is decreased. Restructuring is driven by the deviation of the free energy of the low fractal dimension aggregate from the value for the compact state. The rate of restructuring is an activated process, and the restructuring energy can be found from an Arrhenius plot [51]. The activation energy for silver was similar to the bond energy calculated from the bulk Hamaker constants. This indicates that restructuring most likely occurs when aggregate branches rotate around the weakest bonds, and not from single primary particles detaching and relocating. 9.3.3.3 Elastic Properties Another aspect of chain aggregate behaviour is their elastic properties. This phenomenon was ﬁrst reported by Friedlander et al. [121], who observed that titania chain aggregates stretch under tension and contract when the tension is relaxed. They generated titania chain aggregates by thermal decomposition and deposited them on a transmission electron microscope grid. A typical aggregate selected for study with a fractal dimension of approximately 2.3, was about 250 nm in length and composed of a few hundred individual primary particles about 7 nm in diameter. After 10 to 20 min in the electron beam, a hole developed in the ﬁlm near the aggregate. As the hole grew, the aggregate which bridged the gap began to stretch but remained connected to both sides of the receding ﬁlm. After stretching up to 90 % the aggregate broke loose from one side and contracted to a folded chain on the other side of the hole. Chain aggregates made of other transition metal oxides showed similar behaviour [122, 123]. However, aggregates composed of relatively large primary particles, about 36 nm in diameter, did not exhibit the phenomenon, suggesting that it may be limited to chains composed of primary particles probably less than 10–20 nm. Suh and Friedlander [123]

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examined aggregated composed of carbon. Initially, the stretching of the chains took place by the unravelling of kinks along the aggregate. Reversing the system led to folding of the extended chain without showing elastic behaviour. When stretching was resumed, the chain eventually became taut and then broke at the site of the smallest primary particle. The elastic behaviour is probably associated with local folding of chain segments due to van der Waals forces. Under tension, folded chains straighten; but when the tension is relaxed the folds tend to reform, but not reversibly. Rotation and sliding probably occur at the boundaries between particles during stretching. These phenomena may have signiﬁcant potential for the development of ductile or elastic ceramics and could play a role in the action of nanoparticle additives such as fumed silica and carbon black, used to improve the properties of rubber. 9.3.4 Heterogeneous Reactions Heterogeneous reactions on combustion-generated fractals may well be of importance for the transformation of atmospheric pollutants.According to their surface properties, such particles are suspected, and in a number of cases have been shown, to interact with gaseous environmental chemicals, radicals, and other reactive intermediates. Of particular interest are the exhaust emissions from aircraft, where measurements have shown that not only carbonaceous particles contribute to contrail formation, but also that the aerosol surface areas are of central importance for ozone changes caused by heterogeneous chemical reactions [124, 125]. Aircraft impact studies have motivated laboratory investigations of heterogeneous soot reactions. In particular, heterogeneous reactions of nitrogen oxides on soot surfaces have attracted considerable attention during recent years since they may have an inﬂuence on the NOx /HNO3 ratio and on the HOx balance of the atmosphere [126–128]. Reactive uptake of nitrogen species and ozone on soot has been proposed to explain discrepancies between observations and model calculations [126, 127, 129]. Recent work by Kleffmann and Wiesen [130] suggests that the reactions of HNO3 and HNO3 +NO on soot surfaces are unimportant for a ‘renoxiﬁcation’ of the atmosphere and do not represent an atmospheric HO/NO source, as earlier laboratory data had been determined under experimental conditions not prevailing in the atmosphere, i.e. very high reactant concentrations, thus leading to unrealistic conclusions. For in-depth studies, knowledge of the transport properties and information on the surface area of fractal particles is required. Often, the surface area of soot aggregates is based on the sum of the surfaces of the monomers comprising the aggregate, or aggregates are assumed to be spherical. This issue has been addressed by Coelho et al. [131], who developed a detailed framework for calculating uptake rates of gases on fractal particles. They validated their model against laboratory data and found that model-calculated surface/mass ratios of soot aggregates were consistent with laboratory measurements. By not taking the fractal nature into account, derivations of the aggregate mass and the surface area enhancement factor were in error by at least an order of magnitude.

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For a single aggregate, the mass ﬂux Jf of a diffusing species may be given by −1 1 Jf = 5.65DC∞ Df Rg 1 + (9.12) PeDa where PeDa =

Ks Rg 2.55 D Df

Rg a

Df −2 (9.13)

and D is the gas diffusion coefﬁcient of the species, Ks is the surface reaction rate constant, PeDa is the Peclet–Dahmköhler number and C∞ is the concentration away from the aggregate. By incorporating a factor for the geometric standard deviations, these equations may be extended to a lognormal distribution of aggregates [131]. This ﬂux may be compared with that for a sphere of radius Rg . In the diffusionlimited regime (PeDa 1): Jf = 0.45Df (9.14) Jsphere and the aggregate effectively acts as a sphere of radius equal to the radius of gyration multiplied by roughly half of the fractal dimension. In the reaction-limited regime (PeDa 1) the ratio is Df −2 Jf Rg = 1.15 (9.15) Jsphere a For a fractal dimension of two, aggregates act approximately as spheres with a radius equal to the radius of gyration. However, for values that are different from two, Jf and Jsphere differ signiﬁcantly, as shown in Figure 9.7. Bekki et al. [132] used the above theory to investigate the uptake of chemical species on atmospheric soot in the lower stratosphere and the boundary layer. The uptake was found to be mostly reaction limited, justifying the assumption of proportionality between the soot surface area and the uptake rate. However, the uptake occurs in the transition regime for the relatively compact aggregates. The mass of a fractal mf may be approximated by Df −3 mf Rg ≈ (9.16) m a where m is the mass of a sphere of radius Rg . For soot aggregates found in the upper troposphere/lower stratosphere, the average radius is about 100 nm, whilst the monomers are of the order of 20 nm [133]. For studies which have ignored the fractal morphology and assumed spherical particles, the masses of individual aggregates could have been overestimated by a factor of 5–10 (Figure 9.8). The surface area enhancement Es is a measure of the enhanced area of an aerosol when its geometry deviates from a sphere. It can be approximated by Df /3 Rg Es ≈ 1.2 (9.17) a

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40 0.5

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1.8 2.0 2.2 Fractal Dimension

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Figure 9.7 Ratio of the uptake ﬂuxes to a fractal and a spherical particle as a function of the fractal dimension and Rg /a.

For typical soot aggregates in the upper troposphere/lower stratosphere the enhancement factor is in the range 2.7 to 3.5 (Figure 9.9). This is an order of magnitude lower that that used in modelling studies [126]. It should be borne in mind that there are limitations in the theory developed by Coelho et al. [131] and Bekki et al. [132]. The uptake on the soot surface is assumed to be a pseudo-ﬁrst-order process in the framework, but uptake can be a complex multistep process [134, 135], depending on the environmental conditions. Also, carbonaceous particles can contain a very large organic fraction that is mixed with an insoluble fraction composed predominantly of elemental carbon. The organic fraction is also strongly reactive [136]. Surface area distributions are also of great interest in applications to possible adverse health effects of ultraﬁne atmospheric aerosols and to the monitoring of the products of nanoparticle aerosol reactors. In the past, such calculations have been

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Figure 9.8 Ratio of the mass of a fractal to that of a spherical particle as a function of the fractal dimension and Rg /a.

based on time-consuming image analysis of electron micrographs of individual chain aggregates. However, recently, a method has been described which makes it possible to relate aggregate surface area and volume distributions to the electrical mobility diameter [137, 138]. This method is only applicable for aggregates composed of uniform primary particles, with all surfaces directly exposed to collisions with molecules from the surrounding gas, i.e. a mass fractal dimension less than two. The analysis takes into account the friction coefﬁcient and charging efﬁciency of chain aggregates, under the assumption that the primary particles composing the aggregates are at least one order of magnitude smaller than the mean free path of the surrounding gas. Lall and Friedlander [138] compared the surface area and the volume of aggregates with those of a sphere with an equivalent mobility diameter. Their results indicate that the surface area distributions are somewhat underpredicted if the calculations are based on the assumption of spherical particles. However, the volume distributions are greatly overpredicted, by an order of magnitude in some cases. Figure 9.10 shows the difference in the surface area distribution for spherical particles and idealised aggregates. For a primary particle of 18.5 nm diameter the difference in peak heights of the distributions was 17 % and the total aggregate surface area based on mobility diameter is underpredicted by about 15 %. For smaller primary particles, the ratio of

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Figure 9.9 Fractal surface area enhancement as a function of the fractal dimension and Rg /a. Based on surface area of spheres with diameter dm

dA/dlog(dm) (nm2/cm3)

8E + 10

Based on theory for idealised aggregates

6E + 10

4E + 10

2E + 10

0E + 00 10

100 Mobility diameter (nm)

1000

Figure 9.10 Surface area distribution based on spherical particles compared with the surface area distribution for silver aggregates using the theory of Lall et al. [138] for idealized aggregates. Primary particle size: 18.5 ± 3.5 nm. Reproduced with permission of Elsevier.

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the surface area of the aggregates to that of spheres with the same mobility diameter is expected to be much higher. Aggregate volumes calculated from the theory agreed well with those measured by electron microscopy. The precision of the technique is limited by the accuracy of estimation of charge distribution on aggregates, which is about 10 %. Hence, this method appears to give realistic information on aggregate volumes and surface areas without the use of electron microscopy. Therefore, it should be possible to determine aggregate surface area and volumes in real time without time-consuming image analysis. Section 9.3.3 reported that the fractal morphology may be altered in the atmosphere. This restructuring modiﬁes the surface area available for heterogeneous reactions. Adsorption and exchange of molecules at the surface also lead to partial or complete saturation of the chemically active sites. However, the results demonstrate that accounting for the fractal morphology of carbonaceous aggregates can substantially improve calculations of their properties or dynamics. Errors in deriving soot surface areas or uptake rates may prevent quantitative analysis of the impact of soot on atmospheric chemistry.

9.4 CONCLUSIONS AND RECOMMENDATIONS Many physical characteristics of aerosols are determined by combinations of their size and morphology. However, morphology is often ignored, despite its inﬂuence on the drag force and, hence, transport properties. Transport properties determine how particles deposit, and how they are size-classiﬁed by aerosol instrumentation. There is evidence that aggregate morphology inﬂuences injury to lung cells [139]. Aggregate surface areas depend on their morphology and may be much larger than that of compact particles. Larger surface areas may result in the adsorption of increased amounts of biochemically active chemical components within the lungs. Morphology is also central to light scattering by aerosols and so is crucial to the climate change debate. Atmospheric processing may also lead to changes in the fractal dimensions of aggregate particles, as they undergo cloud processing and/or grow by condensation. Recent measurements have shown that optical absorption by black carbon plays an important role in direct radiative effects of aerosols. In polluted atmospheres, the forcing effects of aerosol absorption can be equal in magnitude to scattering. Therefore, the optical properties of coated and uncoated aggregate soot particles need to be well understood. Also, hygroscopicity inﬂuences the tendency of particles to form cloud droplets, and may inﬂuence deposition in the lung or elsewhere. It is very likely that the fractal properties of coated hygroscopic particles would change during cloud processing, and such changes would lead to changes in optical and transport properties. Morphology also plays an important role in source apportionment, where it can be used to differentiate between chemically similar particles created by different processes. Collision

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rates among fractal structures are higher than those for spherical particles and this probably reduces atmospheric residence times. The characterisation of particle shape has been an active research area and source of contention for over 50 years. The application of fractal geometry to aerosol science lies in the necessity to parameterise complex aerosol shapes, such as those produced by combustion processes. Both computer simulation and experiments have been important in establishing our knowledge of fractal aggregates. Various in-situ techniques have been presented allowing the fractal dimension of both a single aggregate and an ensemble to be calculated. In many industrial applications, aerosol particles with a well-deﬁned structure are needed. Therefore, methods are required to determine the geometrical structure of aggregated aerosol particles. Moreover, these methods should be able to monitor small and fast structural changes of the aggregates. Then, one would be able to investigate the inﬂuences of the production conditions upon the resulting aggregate structure. The determination of structure after deposition of the aggregates on, for example, a ﬁlter can lead to wrong results and needs much preparation and evaluation time. In contrast to the sampling methods, in-situ methods are fast and do not suffer from large inﬂuences on the particle structure. With both the boundary and mass fractal dimension, some information is lost. However, each measurement method yields different details and there is diagnostic information on formation dynamics of the aerosol embedded in the fractal structure. The fractal characterization of aerosols has received increasing attention in recent years. Progress has been made in the study of such aspects as the hydrodynamic, diffusional, optical and elastic properties of fractal objects. However, little is known of the electrical properties of fractal aerosols. It is important both from basic and technological points of view to extend theories and experiments to fractals. The charging of fractal materials can give rise to interesting new phenomena which are not usually encountered in compact materials. The charging of low-density structures (e.g. smokes) could lead to their break-up; the possibility of restructuring them into more compact clusters could lead to a dramatic enhancement in their electrical mobilities. This result may have industrial applications in electrostatic precipitators for the removal of fractal clusters. There are indications that the fractal dimension of atmospheric aggregates are dependent on the time of day, suggesting that this may be used as an indicator of aerosol sources, or the history of aggregates prior to sampling. On-line methods for measuring particle morphology in the ultraﬁne region would contribute to an understanding of the nature and signiﬁcance of shape in engineered nanoparticles. These methods would further advance the ability to characterise aerosol exposure and ﬂux measurements in terms of particle number, surface area and mass, and would contribute to reconciling disparate measures of aerosol properties that are dependent on particle morphology. From the engineering view point, such measurements could provide rapid feedback on particle morphology and structure to monitor and control generation processes.

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LIST OF SYMBOLS AND ABBREVIATIONS DLA DLCA DMA ELPI FBM RLCA SEM SMPS TEM a Aa Ap C C∞ da D De Df Df L dh dm dp dv dva DS DT E Es H I (q) Jf Jsphere Ks L m mf Mp N PeDa q

diffusion-limited aggregation diffusion-limited cluster aggregation differential mobility analyser electrical low-pressure impactor fractional Brownian motion reaction-limited cluster aggregation scanning electron microscopy scanning mobility particle sizer transmission electron microscopy monomer radius projected area of the aggregate mean projected area of primary particles arbitrary constant concentration away from the aggregate aerodynamic diameter gas diffusion coefﬁcient Euclidean dimension fractal dimension fractal dimension when the maximum projected length is treated as the characteristic size hydrodynamic diameter mobility diameter diameter of the individual spherules volume equivalent diameter particle diameter in the free molecular regime structural fractal dimension textural fractal dimension electric ﬁeld strength surface area enhancement Hurst coefﬁcient scattered intensity mass ﬂux to fractal mass ﬂux to sphere surface reaction rate constant maximum projected length mass of a sphere mass of a fractal particle mass number of primary particles Peclet–Dahmköhler number scattered wave vector

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radius of gyration velocity of cluster i terminal velocity empirical projected area exponent fractal prefactor scattering angle incremental length wavelength fuel equivalence ratio

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Index

Page references in bold type refer to tables; those in italics to ﬁgures. Achlya bisexualis 251 Acidiﬁcation 210 Actinomycetes 247 Activated sludge 254–5 Adsorption Brunauer–Emmet–Teller (BET) modelling 186–91 dyes 206 inside porous networks 192–4 kinetics 183–4 from liquid mixtures 198 modelling 181–2 monolayer capacity 181–4 on nonuniform surfaces 199–200 no-correlation models 200–2 of polymers onto colloid surfaces 134–7 soils 208–12 Aerosols atmospheric 293–6 restructuring 296–301 Agar 249 Aggregates aerosol 276, 279–80 chain 273, 296–301 deﬁnition 69 mass fractal dimension 72 mass scaling 70 Aggregation 6–7 aerosols 285–6 superaggregates 292 cluster–cluster, see Cluster–cluster aggregation determining factor and fractal dimension 119–20 diffusion-limited 61, 122–3, 233, 276

humic materials 231–2 iron compounds 155–7 with phosphate 157–9 iron oxyhydroxides 153–66 iron–silica complexes 164–5 kinetics 121–2 overview 111–12 reaction-limited 6, 119–20, 231 see also Coagulation Algorithms, image analysis 90–4 Alkalinization 210 Analytic techniques 5, 103–4, 150, 151–3 aerosols 277–89 mobility methods 281–6 other 286–7 ﬂocculation 132–3 humic materials 223–6 scattering, see Scattering techniques settling velocity measurement 97–101 X-ray microtomography 96–7 Aperture impedance particle sizing 101–2 Applications aerosol analysis atmospheric aerosols 293–6 soots 290–3 humic materials 233–4 Aquatic particles 112–16 Aquatic samples, humic compounds 229–30 Armillaria luteus 248, 251 Aspergillus niger 250, 251 Atmospheric samples, heterogeneous reactions 302–4

Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems c 2008, IUPAC and Kevin J. Wilkinson

Edited by Nicola Senesi

318 Bacillus subtilis 247 Bacteria 85, 95 in artiﬁcial media 246–8 in environment 254–5 phenotypic plasticity 239–40 Bentonite 206 Binomial fractal measure 55–6, 57–9 Biological samples 86 Biomass soot 289, 291 Bolzano–Weierstrass functions 22–4 Boundary fractal dimension, see surface fractal, dimension, aerosols Box-counting 33–5, 34, 90–2 microorganisms 242–4 Bridging ﬂocculation 130–7 Brinkman’s law 139–40 Brownian motion 26–8, 288–9 Brunauer–Emmet–Teller (BET) adsorption modelling 186–91, 197 on heterogeneous surfaces 202–3 Cambisols 309–10 Cantor set 13–15, 21, 31 Cantor singular function 15–17, 16, 42 Capillary condensation 192–3 Carbohydrates 209 Carbonaceous materials 198, 273 Casein gels 95–6 Cation exchange, and fractal dimension 205–6 Chain aggregates 273 restructuring 296–301 Characteristic length scale 225 Charcoal 197 Chord length measurement 101, 105 Citrobacter freundii 246 Clays 204–5 adsorption behaviour 196 adsorption characteristics 182 Cloud-burning effect 299 Cluster–cluster aggregation (CCA) 6 fragmentation 122–8 heterogeneous particles 128–30 particle interactions 117–19 reaction- and diffusion-limited mechanisms 119–20 scaling laws 121 Coagulants 149 Coagulation 111–12 in environment 149–50 see also Aggregation

INDEX Coal ash 291, 292 Coastlines 35, 41 Coltricia perennis 252 Confocal microscopy 94–6, 104 Contrast and aggregate radius 168–9 and fractal dimension 167–8 neutrons 86–7 optical and form factor 170–1 and lighting 90, 91 X-rays 86 Cutoff functions 79 Cutoff length 44–5 Debye–Hückel potential 135 Density autocorrelation function 76–8 Density fractal dimension, see Mass fractal, dimension Derjarguin, Landau, Verwey, Overbeek (DLVO) theory 117–19, 149–50 Devil’s staircase 13–18 Diesel particulates 289–90 Differential mobility analyser (DMA) 283 Diffusion iron compounds 155–6 nutrients 266–7 Diffusion-limited aggregation (DLA) 61, 122–3, 233 aerosols 276 and multifractal measures 61 Diffusion-limited cluster aggregation (DLCA) 6, 119–20, 276 humic materials 231 restructuring 122–3 Dissolved organic carbon (DOC) 210–211 Divider 92–3 DLA, see Diffusion-limited aggregation Drag 97–8 Dubinin–Astakhov (DA) isotherm 194 Dubinin–Radushkevich (DR) isotherm 192–3 Dyes 206 adsorption 206 Ectomycorrhizal fungi 255, 256 Elastic properties, chain aggregates 300–1 Electrical low-pressure impactor (ELPI) 283–4 Electron microscopy 5 humic acid 229

INDEX Elemental composition, humic fractions 223 Engine load, and fractal dimension (of particle) 290–1 Eschericia coli 246, 248, 266 Exponential distributions 46–7 Extended edge X-ray absorption ﬁne structure (EXAFS) spectroscopy 154 Flocs 133, 150 bacterial 245, 248 in sludge 254–5 large 166–71 structural parameters 150 Fly ash 292 Focused beam reﬂectance measurement (FBRM) 101 Form factor 76–8 and contrast 169–71 Fractal dimension 1, 2, 4–5, 42, 275 and adsorption energy 202–3 aerosols, limitations of characterization 285 and aggregate radius 114 bacterial colonies 246–7 in environment 254–5 and cation exchange 205–6 deﬁnition 43 determination 4–5, 52–4 from adsorption data experiments 195–9 monolayer capacity 181–4 thermodynamic method 184–6 adsorption methods monolayer capacity 181–4 thermodynamic 184–6 aerodynamic methods 281–6 aerosols 277 algorithms 90–4 box-counting 33–5, 91–2, 242–4 by optical scattering 77–87 divider 35, 93 Hausdorff 28–32 microorganisms, box-counting method 242–5 other methods 36–7 similarity 32–3 ﬂocs 133, 150 large 166–71 fungi change with time 257–8 in environment 255–6

319 environmental factors 262–3 mycelial, in culture 251–3 and nutrient status 259–60 humic substances 227–8, 232–3 impermeable aggregates, and settling velocity 138–9 and inner cutoff length 44–5 microorganisms concentric ring technique 241–2 see also Fractal dimension, fungi and optical contrast 167–8 and permeability 139 pore 3, 180 soils 210 superaggregates 293 surface and adsorption 180–1 measurement 183–4 thermodynamic method 184–6 peat soils 209 and topological dimension 275 and uptake ﬂux 303 variation with scale 73–4 Fractals classiﬁcation 3 deﬁnition 2, 42–3 natural vs. theoretical 43–4 Fractal scaling, in natural systems 71–4 Fractal structure, measurement 70–1 Fractional Brownian motion 26–8, 288–9 Fracture cascade 49–52, 51, 52 Fragmentation fractals 49–52 Fragmentation processes 122–8 Frenke–Halsey–Hill (FHH) isotherm 191–4, 196 Fuels 290–1 Fulvic acid 21, 222, 223, 227–8 Fungi incubation 255–6 mycelial 242, 248–54 growth patterns, invertebrate grazing 264–5 phenotypic plasticity 239 unicellular 246–8 Gaussian noise 26, 27 Geometric fractals 48 Geometric transparency 87–8 Gliding box algorithm 38–9 Goethites 196–7 Grazing 264–5

320 Haematite, ﬂocculation 132–3 Hausdorff dimension 28–32, 42 Health hazard assessment 273, 303–4 Heavy metals, and fungal growth 249 Heteroaggregation 128–30 Heterogeneous reactions, aerosols 301–6 Hilbert’s curve 18 Hölder exponent 36, 57 Honey fungi 257 Humic acid 221, 222, 223, 227, 228 Humic materials analytic techniques 223–5 deﬁnition 221 particle structure 226–31 Humidity 296–301 Humin 221, 222, 223, 228 Hydrolysis ratio and iron polymer size 154 and scattering intensity, iron complexes 160 Hypholoma fasciculare 259, 261 Image analysis algorithms 90–4 microscopy 94–6 preprocessing 89–90 and fractal dimension estimation 52–4 fungi 242 principles 87–9 resolution (of image) 243–5 Inductively coupled plasma optical emission spectrometry 283 Inner cutoff length 4, 45 Inoculum 258–61 International Humic Substances Society (IHSS) 232 Invertebrate grazing 264–5 Iron oxides and hydroxides 153–66 Iron–silica complexes 159–66 Iron trichloride 154–5, 156 Iron trinitrate 154 Irregular fractal aggregates 113 Isotherms adsorption, heterogeneous surfaces 199 Brunauer–Emmet–Teller (BET) 186–91 Dubinin–Radushkevich (DR) 192–3 Frenke–Halsey–Hill (FHH) 191–4, 196 Kiselev 184–5

INDEX Kaolinite 204, 205, 207 Kinematic coagulation, aerosols 286 Kinetics adsorption 183–4 aggregation 6, 119–20, 121–2 bridging ﬂocculation 131–3 Kirkwood–Riseman theory 140 Kiselev isotherm 184–5 Klebsiella bacteria 246 Laccaria bicolor 253 Lactarius hepaticus 253 Lacunarity 37–40 and nutrient diffusion 266–7 Latex 95, 169–71 Lennard–Jones function 191 Ligands, competition 157–9 Light microscopy lighting and contrast 90, 91 and microbial ﬂocs 246 Light scattering techniques, aerosols 279–81 Luvisols 208–9 Macrophomina phaseolina 253 Magnesium compounds 197–8 Mass exponent 58–9, 58 Mass fractal 3, 4, 180 deﬁnition 45 dimension 3, 4, 115 humic materials 226 Mass-mobility relationship 281–3 Mass, scaling with size 70, 114–16 Matushita fracture cascade 49–50 Mean-ﬁeld approximation 152 Menger sponge 20–1 and soil aggregates 44, 45 Mercury porosimetry 185–6 Metropolis algorithm 124–5, 135 Microbes fractal dimension 240–6 interspeciﬁc interactions 263–4, 264 see also Bacteria; Fungi Microscopy 94–6, 104 humic materials 225–6 Mie theory 299 Millikan cell 286, 287 Minkowski dimension, see Box-counting Mobility diameter and radius of gyration 284–5 and surface area 304–6, 305

INDEX Modelling, heterogeneous surfaces 200 Molecules, diameter, and monolayer capacity 181–3 Monolayer capacity 181–4 Monte Carlo simulations adsorption 198 polymers 134–7 Montmorillonite 204, 205 pillared 207 Multifractal measures 54–61 deﬁnition 60 parameters 56–9 Multiplicative cascade 55–6, 60–1 Mycelial structures 242, 244, 248–54, 255 Navier–Stokes equations 140 Necrotrophic fungi 256 Nematodes 266 Neutron scattering 86–7 Nitrogen oxides, and soot 301 Nucleation iron oxides and hydroxides 153–5 iron–silica complexes 160–2 Nutrients availability, and fungal growth 258, 259–60, 261 diffusion in soils 266–7 Optical microscopy 94–6 confocal 94–6, 104 Optical scattering 80–1, 151–3, 166, 245–6 aerosols 279 Organic matter, and soil surface heterogeneity 210–211 Paenibacillus dendritiformis 246–7 Palladium 207 Pareto distribution 45–9 Particle counting 101–2, 105 Particle number, and projected area 287 Particle–particle forces, and aggregate rearrangement 124–5 Particle size, and analytic technique choice 102 P axillus involutus 253 Peano’s curve 17–18, 42 Peat soils 209, 211–12 Perikinetic aggregation 112 Perimeter dimension, projected images 88 Permeability, and settling velocity 139–40 pH, humic acids 229

321 P hallus impudicus 259, 261, 264 P hanerochaete velutina 259, 261, 264 Phosphates 157–9 Pointwise dimension, see Mass fractal, dimension Polydispersity 79–80 Polymers adsorption onto particle 134–7 ‘bridging’ in ﬂocculation 130 iron, nucleation 154 Pore fractal 3, 180, 210 deﬁnition 45 Pores adsorption 188, 194–5 networks, adsorption energies 202–3 Power-law parametric relationships, see Scaling laws Predators 266 Prefractals 44–5 Preprocessing, images 89–90 Pressure, and adsorption 186–91 Probabilistic fractals 48 Projected area dimension algorithms 93–4 and mass fractal dimension 87–8 and particle number 287 P roteus mirabilis bacteria 246 P ycnoporus cinnabarinus 254 Quenched–annealed mixtures

212–13

Radius of gyration, and mobility diameter 284–5 Random fractals 1, 24 Randomly methylated -cyclodextrin (RAMEB) 207 Rayleigh–Gans–Debye (RGD) formalism 75–6 Reaction-limited aggregation (RLA), humic materials 231 Reaction-limited cluster aggregation (RLCA) 6, 119–20 Rearrangement (of aggregate particles) 123–8, 296–301, 306 Regular fractal aggregates 113 Rényi dimension 36–7, 59 Resinicium bicolor 258 Restructuring 95, 123–8, 296–301, 306 Reynolds number 98 Ricci-Curbastro, Gregorio 11 Richardson plot 277, 278

322 Salmonella bacteria 246 Sample handling 95 Samples handling 74–5 lighting 90 in water, and X-ray scattering 86 Sandbox algorithm 39, 93 and confocal microscopy 95 SANS (small-angle neutron scattering) 5, 86–7 Saprotrophic fungi 255, 256 SAXS (small-angle X-ray scattering) 5, 151, 166 Scale invariance, aerosols 274 Scaling laws 45–9 aggregation 121–2 bridging ﬂocculation 131–3 mass 70, 114–16 radius of gyration 115–16 Scanning electron microscopy (SEM) 5 Scattering techniques 75–87, 104 and aerosols 279–81 compared with adsorption analysis 196–7 form and structure factors 76–8 humic materials 223–4, 223–6 neutron 5, 86–7 optical 80–5, 151–3, 166, 245–6 intensity 169–71 mean-ﬁeld approximation 152 turbidimetry 5, 85, 224–5 principles 75–6 X-ray 151, 166, 223–4 Segmentation (of images) 89 Self-afﬁnity 16 Self-similarity 1, 42, 43, 113 multifractal measures 55–6 statistical 24 Serratia marcescens 246 Settling velocity aggregates divergence from models 137 impermeable 137–9 permeable 139–40 measurement 97–101, 105 Shales 196 Shearing, and fractal dimension 74–5 Sierpinski carpet 20, 32, 37, 50 Silicas 157, 159–65 adsorption behaviour 195–6 powders 199

INDEX Silver aggregates 281, 283, 300 Similarity dimension 32–3 Simulations, optical scattering 167–9 Single image microscopy 94 Sludge 254–5 Small-angle light scattering (SALS) 80–1 volume obscuration 81–5 Small-angle multiple light scattering 85 Small-angle neutron scattering (SANS) 5, 86–7 Small-angle X-ray scattering (SAXS) 5, 151, 166 humic materials 223–4 Smectite lamellae 205 Smokes, fractal dimension 290 Smoluchowski equation 121, 123 Soil 40–1, 44, 208–12 as habitat 265–7 soil minerals 204–7 Solids, humic particles 228 Soot 273, 279–80, 282–3 fractal dimension, comparison 289–90 humidity-induced restructuring 296–300 Stokes law, and settling velocity 138 Stropharia caerulea 258–61, 259, 261, 263, 264 Structure factor 76–8, 77 Superaggregates 291–2, 293 Surface area, and uptake, aerosols 303–4 Surface forces 117–19 and restructuring 124–8 Surface fractal 3, 4 deﬁnition 45 dimension, aerosols 277 Surfaces, mathematical, characterization 28–30 Suspensions, humic particles 228, 229 Temperature, and fungal growth 262 Tensor calculus 11–12 Terminal velocity, aerosol clusters 286–7 Thermal restructuring 300 Thresholding 89–90 Tomography 245–6 Transmission electron microscopy (TEM) 5 humic acid 229 Trichoderma viride 248, 252, 254 Turbidimetry 5, 85, 224–5

INDEX Universality classes 120 Uptake, onto aerosol particles 301–3 van der Waals forces 117–19 Vapor adsorption 5 Variogram dimension 36–7 Verwey–Overbeek potential 135 Volume obscuration 81–5 von Koch curve 18–20, 19, 24, 183

323 Washburn relationship 185 Weierstrass–Mandelbrot function

22–4, 23

X-ray microtomography 96–7 X-ray scattering 86, 151, 166, 223–4 Yeasts

246, 248

Zeolites 197 Zipf distribution 46 Zoogloea ramigera 247–8