Interdisciplinary Applied Mathematics

Volume 16

Interdisciplinary Applied Mathematics

1. Gutzwiller: Chaos in Classical and Quantum Mechanics 2. Wiggins: Chaotic Transport in Dynamical Systems 3. Joseph/Renardy: Fundamentals of Two-Fluid Dynamics:

Part 1: Mathematical Theory and Applications 4. Joseph/Renardy: Fundamentals of Two-Fluid Dynamics:

Part II: Lubricated Transport, Drops and Miscible Liquids 5. Seydel: Practical Bifurcation and Stability Analysis:

From Equilibrium to Chaos 6. Hornung: Homogenization and Porous Media 7. Simo/Hughes: Computational Inelasticity 8. Keener/Sneyd: Mathematical Physiology 9. Han!Reddy: Plasticity: Mathematical Theory and Numerical Analysis

10. Sastry: Nonlinear Systems: Analysis, Stability, and Control 11. McCarthy: Geometric Design of Linkages 12. Winfree: The Geometry of Biological Time (Second Edition) 13. Bleistein/Cohen/Stockwell: Mathematics of Multidimensional

Seismic Imaging, Migration, and Inversion 14. Okubo!Levin: Diffusion and Ecological Problems: Modern Perspectives

(Second Edition) 15. Logan: Transport Modeling in Hydrogeochemical Systems 16. Torquato: Random Heterogeneous Materials: Microstructure and

Macroscopic Properties 17. Murray: An Introduction to Mathematical Biology 18. Murray: Mathematical Biology: Spatial Models and Biomedical

Applications 19. Kimmel/Axelrod: Branching Processes in Biology

Salvatore Torquato

Random Heterogeneous Materials Microstructure and Macroscopic Properties

With 218 Illustrations

~Springer

Salvatore Torquato Department of Chemistry and Princeton Materials Institute Princeton University Princeton, NJ 08544 USA torquato@electron. princeton.edu

Editors S.S. Antman Department of Mathematics and Institute for Physical Science and Technology University of Maryland College Park, MD 20742-4015 USA

L. Sirovich Division of Applied Mathematics Brown University Providence, RI 02912 USA

J.E. Marsden Control and Dynamical Systems Mail Code 107-81 California Institute of Technology Pasadena, CA 91125 USA

S. Wiggins Control and Dynamical Systems Mail Code 107-81 <:::alifomia Institute of Technology Pasadena, CA 91125 USA

Mathematics Subject Classification(2000): 82Bxx, 78 02, 73-02, 60D05

Library of Congress Cataloging-in-Publication Data Torquato, S.

Random heterogeneous materials: microstructure and macroscopic properties/Salvatore Torquato. p. cm.-(Interdisciplinary applied mathematics; v. 16)

Includes bibliographical references and index. ISBN 978-1-4757-6357-7 ISBN 978-1-4757-6355-3 (eBook) DOI 10.1007/978-1-4757-6355-3

1. Inhomogeneous materials. 2. Microstructure. I. Title. II. Series. TA418.9.153 2001 620.1'1-dc21

Printed on acid-free paper.

© 2002 Springer Science+Business Media New York Originally published by Springer Science+Business Media, Inc. in 2002 Softcover reprint of the hardcover 1st edition 2002

2001020203

All rights reserved. This work may not be translated or copied in whole or in part without written permission of the publisher Springer Science+Business Media, LLC. except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieyal, electronic adaptation, computer software, or by similar or dissimilar methodology now known or here­after developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production managed by Frank McGuckin; manufacturing supervised by Jerome Basma. Typeset by the Bartlett Press, Inc., Marietta, GA.

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springeronline.com

To My Wife,

KIM

My Daughters,

MICHELLE and LISA

and My Parents,

PALMA and VINCENT

"The fairest thing we can experience is the mysterious. It is the funda­mental emotion which stands at the cradle of true art and true science. He who does not know it and can no longer wonder, no longer feel amazement, is as good as dead, a snuffed-out candle."

-Albert Einstein, Forum and Century (1930)

"How novel and original must be each new man's view of the universe­for though the world is so old-and so many books have been written­each object appears wholly undescribed to our experience-each field of thought wholly unexplored .... "

-Henry David Thoreau, Journal4:421 (1852)

Preface

The interdisciplinary subject of random heterogeneous materials has experienced remarkable growth since the publication of the well-known monograph Statistical Con­tinuum Theories by Beran ( 1968). Many of these advances, especially those concerning the statistical characterization of the microstructure and its effect on the physical prop­erties of the material, have not been treated fully in any book. One of the intents of the present book is to fill this gap. This book also distinguishes itself in that it provides a unified rigorous framework to characterize the microstructures and macroscopic properties of the widely diverse types of heterogeneous materials found in nature and synthetic products. Emphasis is placed on providing foundational theoretical methods that can simultaneously yield results of practical utility.

This book treats a wide breadth of topics, but the choice of subjects naturally reflects my own interests. The sheer enormity of the field has prevented me from covering many important topics. I apologize to those colleagues, known and unknown, who may not find enough of their own work cited in the ensuing pages.

This book is intended for graduate students and researchers from various walks of scientific life, including applied mathematicians, physicists, chemists, materials sci­entists, engineers, geologists, and biologists. In order to reach this broad audience, I have attempted to make the book as self-contained as possible, assuming only a rudi­mentary knowledge of probability theory, statistical mechanics, advanced calculus, and continuum mechanics. In cases where I have fallen short in this regard, the numerous references provided should satiate the voracious appetite for knowledge of the most curious minded among us. The book contains as many proofs and derivations of key results as can be accommodated within the aforementioned constraints. All of these features and an attempt to avoid technical jargon should make the book accessible to

viii PREFACE

the nonspecialist. Indeed, it is my hope that motivated experimentalists will find this book useful.

The book is divided into two parts. Part I describes basic concepts and recent ad­vances in quantitatively characterizing the microstructure of random heterogeneous materials. Topics covered include the statistical mechanics of many-particle systems, the canonical n-point correlation function, lattice and continuum percolation theory, local volume-fraction fluctuations, computer-simulation methods, image analyses and reconstructions of real materials, and models of microstructures.

Part II treats a wide variety of macroscopic transport, electromagnetic, mechani­cal, and chemical properties of heterogeneous materials and describes how they are linked to the microstructure of model and real materials. Topics covered include ho­mogenization theory, variational principles and rigorous bounds, phase-interchange relations, exact results, effective-medium approximations, cluster expansions, contrast expansions, and cross-property relations.

A brief description of the topics covered in each chapter of the present book is given towards the end of Chapter 1, which provides the motivation for the book and an overview of its contents. It is unique among the chapters because it is purposely written in the nontechnical style of Scientific American in order to introduce the key ideas to the interdisciplinary audience for which the book is intended. The book is an outgrowth of a graduate course that I teach at Princeton University. The course begins with Chapter 1 and then immediately skips to Part II and covers in varying depths Chapters 13-21. I then cover most of the material contained Chapters 2-12 of Part I and subsequently return to Part II to cover Chapters 22 and 23. I follow this sequence because the property/microstructure connection (Part II) provides a motivation for the reasons why we will ultimately need to quantify the microstructure (Part I). However, this sequence certainly does not need to be followed in a course, especially if one is primarily interested in microstructural analysis. Although there is substantial cross referencing between Parts I and II, each part has been designed to be relatively inde­pendent of the other. Nonetheless, I believe that Parts I and II make a cohesive unit, and ideally, they should be read together. Those interested in further reading on the theme of Part II of this book are referred to the recent books by Cherkaev (2000) and Milton (2001), which emphasize the general theory of composites.

One of the most enjoyable parts of writing this book is thanking the many people without whose support it would have never been written. The contributions of my collaborators over the years, many of whom are cited in the text, have enriched my scientific experience. I thank George Stell, my former advisor at the State University of New York at Stony Brook, who instilled in me a love for research and introduced me to statistical mechanics and composite media. Hajime Sakai, John Quintanilla, Louis Bouchard, Juan Eroles, Sangil Hyun, Edward Garboczi, Konstantin Markov, Leonid Gibiansky, Tony Roberts, Thomas Truskett, Frank Stillinger, and Leonid Berlyand care­fully read various portions of the manuscript and provided valuable criticisms and suggestions. I am deeply indebted to all of them. I gratefully acknowledge fruitful and illuminating discussions with Erhan Cinlar, Thomas Spencer, Marco Avellaneda, and

PREFACE ix

Robert Kohn on certain aspects of the book. Many thanks are due to Christopher Yeong, Anuraag Kansal, and Juan Eroles, who produced the majority of the numerous figures that have greatly enhanced the book. Thomas Truskett wrote the first version of the Monte Carlo program that appears in one of the appendices. George Dvorak, Nick Martys, and Paul Stutzman each generously supplied a figure from their respective work.

I thank the Department of Energy and Air Force Office of Scientific Research for supporting much of my work. My year-long sabbatical in the School of Mathematics at the Institute for Advanced Study in Princeton, New Jersey as a Guggenheim Fellow during 1999-2000 enabled me to focus my efforts on the final product before you. I am especially grateful to Princeton University for their support and encouragement of my scholarship over the years.

On the home front, I want to express my deep thanks to my wife, Kim, and daughters, Michelle and Lisa, for their love, understanding, perseverance, and patience. Their unfailing support has made this book possible. Finally, I acknowledge the sacrifices made by my parents, Palma and Vincent, that enabled me to pursue my dreams.

The author would be grateful for reports of typographical and other errors to be sent electronically via the following webpage for the book:

http://cherrypit. princeton.edulbook.html,

where an up-to-date errata list will be maintained.

Princeton, New Jersey June 2001

Salvatore Torquato

Contents

Preface

1 Motivation and Overview 1.1 What Is a Heterogeneous Material? ...... . 1.2 Effective Properties and Applications . . . . . .

1.2.1 Conductivity and Analogous Properties 1.2.2 Elastic Moduli . . . . . . . . . . . . . 1.2.3 Survival Time or Trapping Constant .. 1.2.4 Fluid Permeability . . . . . . . . . . . . . 1.2.5 Diffusion and Viscous Relaxation Times 1.2.6 Definitions of Effective Properties .

1.3 Importance of Microstructure . . . . 1.4 Development of a Systematic Theory . .

1.4.1 Microstructural Details . . . . . . 1.4.2 Multidisciplinary Research Area .

1.5 Overview of the Book 1.5.1 Part I . 1.5.2 Part II 1.5.3 Scope .

I Microstructure Characterization

2 Microstructural Descriptors 2.1 Preliminaries . . . . . .

vii

1 1 3 6 7 8 8 9 9

10 12 12 14 17 17 18 19

21

23 24

xii

2.2 n-Point Probability Functions .. . 2.2.1 Definitions .......... . 2.2.2 Symmetries and Ergodicity . 2.2.3 Geometrical Probability Interpretation . 2.2.4 Asymptotic Properties and Bounds 2.2.5 Two-Point Probability Function

2.3 Surface Correlation Functions . 2.4 Lineal-Path Function ...... ·. 2.5 Chord-Length Density Function 2.6 Pore-Size Functions ...... . 2. 7 Percolation and Cluster Functions . 2.8 Nearest-Neighbor Functions .... 2.9 Point/q-Particle Correlation Functions 2.10 Surface/Particle Correlation Function .

3 Statistical Mechanics of Many-Particle Systems 3.1 Many-Particle Statistics ........ .

3.1.1 n-Particle Probability Densities 3.1.2 Pair Potentials ....... .

3.2 Omstein-Zemike Formalism ... . 3.3 Equilibrium Hard-Sphere Systems

3.3.1 Low-Density Expansions .. 3.3.2 Arbitrary Fluid Densities ..

3.4 Random Sequential Addition Processes 3.4.1 One-Dimensional Identical Hard Rods 3.4.2 Identical Hard Spheres in Higher Dimensions 3.4.3 General Hard-Particle Systems .. .

3.5 Maximally Random Jammed State .......... . 3.5.1 Random Close Packing Is Ill-Defined ..... . 3.5.2 Definition of Maximally Random Jammed State 3.5.3 Order Metrics ........... . 3.5.4 Molecular Dynamics Simulations 3.5.5 Concluding Remarks ....... .

4 Unified Approach to Characterize Microstructure 4.1 Volume Fraction and Specific Surface

4.1.1 Bounding Properties . . . . . 4.1.2 Example Calculations . . . .

4.2 Canonical Correlation Function Hn

4.2.1 Definitions ....... . 4.2.2 Asymptotic Properties .

4.3 Series Representations of Hn .

4.3.1 Mayer Representation .

CONTENTS

25 25 28 32 33 34 43 44 45 48 50 50 57 58

59 60 60 65 72 75 79 81 83 85 87 88 88 89 90 92 93 95

96 97

100 102 104 105 109 109 110

CONTENTS xiii

4.3.2 Kirkwood-Salsburg Representation 111 4.3.3 Bounding Properties . 112

4.4 Special Cases of Hn ....... 114 4.5 Polydispersivity . . . . . . . . . 116 4.6 Other Model Microstructures . 118

5 Monodisperse Spheres 119 5.1 Fully Penetrable Spheres ........ 120

5.1.1 n-Point Probability Functions 122 5.1.2 Surface Correlation Functions . 124 5.1.3 Lineal-Path Function ...... 125 5.1.4 Chord-Length Density Function . 127 5.1.5 Nearest-Neighbor Functions ... 128 5.1.6 Pore-Size Functions ........ 128 5.1.7 Point/q-Particle Correlation Functions 129

5.2 Totally Impenetrable Spheres . . . . . . 129 5.2.1 n-Point Probability Functions 130 5.2.2 Surface Correlation Functions . 134 5.2.3 Lineal-Path Function ...... 136 5.2.4 Chord-Length Density Function 137 5.2.5 Nearest-Neighbor Functions .. 139 5.2.6 Pore-Size Functions ....... 151 5.2.7 Point/q-Particle Correlation Functions 152

5.3 Interpenetrable Spheres . . . . . . . 153 5.3.1 Nearest-Neighbor Functions 154 5.3.2 Volume Fraction .. 155 5.3.3 Specific Surface . . . . . . . . 155 5.3.4 Pore-Size Functions ..... 157 5.3.5 Other Statistical Descriptors 157

5.4 Statistically Inhomogeneous Systems . 158

6 Polydisperse Spheres 160 6.1 Fully Penetrable Spheres ........ 161

6.1.1 n-Point Probability Functions 163 6.1.2 Surface Correlation Functions . 164 6.1.3 Lineal-Path Function ...... 165 6.1.4 Chord-Length Density Function 166 6.1.5 Nearest-Surface Functions ... 166 6.1.6 Pore-Size Functions ....... 167 6.1.7 Point/q-Particle Correlation Functions 167

6.2 Totally Impenetrable Spheres . . . . . . 167 6.2.1 n-Point Probability Functions 169 6.2.2 Surface Correlation Functions . 170

xiv

6.2.3 Lineal-Path Function ..... . 6.2.4 Chord-Length Density Function 6.2.5 Nearest-Surface Functions .. . 6.2.6 Pore-Size Functions ...... . 6.2.7 Point/q-Particle Correlation Functions

7 Anisotropic Media 7.1 General Considerations . . . . . . . . 7.2 Fully Penetrable Oriented Inclusions 7.3 Impenetrable Oriented Inclusions 7.4 · Hierarchical Laminates . .

8 Cell and Random-Field Models 8.1 Cell Models . . . . . . . . . . . . . . . . . . . .

8.1.1 Voronoi and Delaunay Tessellations 8.1.2 Cell Statistics ....... . 8.1.3 Symmetric-Cell Materials . 8.1.4 Random Checkerboard 8.1.5 Ising Model ...... .

8.2 Random-Field Models .... . 8.2.1 General Considerations 8.2.2 Gaussian Convolved Intensities

9 Percolation and Clustering 9.1 Lattice Percolation ........ .

9.1.1 Bond and Site Percolation 9 .1.2 Percolation Properties . . . 9.1.3 Scaling and Critical Exponents 9.1.4 Infinite Cluster and Fractality 9.1.5 Finite-Size Scaling ...

9.2 Continuum Percolation . . . . . . . 9.2.1 Percolation Properties .... 9.2.2 Two-Point Cluster Function 9.2.3 Critical Exponents ..... .

10 Some Continuum Percolation Results 10.1 Exact Results for Overlapping Spheres

1 0.1.1 One Dimension . . . . . . . . . . 1 0.1.2 Higher Dimensions . . . . . . . . 10.1.3 Low-Density Expansions of Cluster Statistics .

10.2 Omstein-Zemike Formalism .. 10.3 Percus-Yevick Approximations .

10.3.1 Permeable-Sphere Model

CONTENTS

171 171 172 176 176

177 177 179 181 183

188 188 189 192 194 199 201 203 203 207

210 211 211 215 217 222 223 224 227 230 231

234 234 235 240 242 243 245 246

CONTENTS

10.3.2 Cherry-Pit Model ......... . 10.3.3 Sticky Hard-Sphere Model ... .

10.4 Beyond Percus-Yevick Approximations. 10.5 Two-Point Cluster Function ...... . 10.6 Percolation Threshold Estimates .... .

10.6.1 Overlapping Disks and Spheres . 1 0.6.2 Nonspherical Overlapping Particles .. 1 0.6.3 Interacting Particle Systems . . . . ..

11 Local Volume Fraction Fluctuations 11.1 Definitions . . . . . . . . 11.2 Coarseness . . . . . . . . . . .

11.2.1 General Formula ... 11.2.2 Asymptotic Formula . 11.2.3 Calculations . . . . . .

11.3 Moments of Local Volume Fraction 11.4 Evaluations of Full Distribution . .

12 Computer Simulations, Image Analyses, and Reconstructions 12.1 Monte Carlo Simulations ..

12.1.1 Introduction ............. . 12.1.2 Importance Sampling ....... .

12.2 Metropolis Method for Gibbs Ensembles. 12.2.1 Markov Chain ....... . 12.2.2 Algorithm . . . . . . . . . . 12.2.3 Practical Implementation . 12.2.4 Hard Spheres ..... . 12.2.5 Other Particle Systems .. 12.2.6 Cell Models ........ .

12.3 Methods for Generating Nonequilibrium Ensembles. 12.4 Sampling in Particle Systems .....

12.4.1 Radial Distribution Function .. 12.4.2 n-point Probability Functions . 12.4.3 Surface Correlation Functions . 12.4.4 Cluster-Type Functions ..... 12.4.5 Other Correlation Functions ..

12.5 Sampling Images and Digitized Media 12.5.1 Two-Point Probability Function 12.5.2 Lineal-Path Function ..... . 12.5.3 Chord-Length Density Function 12.5.4 Pore-Size Functions ..... 12.5.5 Two-Point Cluster Function ..

XV

248 249 250 250 251 252 254 255

257 258 260 260 261 262 264 265

269 270 270 271 273 273 275 275 277 278 279 279 281 281 283 285 285 286 287 289 291 292 292 293

xvi

12.6 Reconstructing Heterogeneous Materials 12.6.1 Reconstruction Procedure 12.6.2 Illustrative Examples ....... .

CONTENTS

294 295 297

II Microstructure/Property Connection 303

13 Local and Homogenized Equations 13.1 Preliminaries .... . 13.2 Conduction Problem ..... .

13.2.1 Local Relations .... .

305 306 308 308

13.2.2 Conduction Symmetry 311 13.2.3 Model One-Dimensional Problem . 313 13.2.4 Homogenization of Periodic Problem in ffid 315 13.2.5 Homogenization of Random Problem in rna 318 13.2.6 Frequency-Dependent Conductivity. 321

13.3 Elastic Problem . . . . . . 321 13.3.1 Local Relations . . . . . . . . . . . . . 321 13.3.2 Elastic Symmetry. . . . . . . . . . . . 324 13.3.3 Homogenization of Random Problem in ffid • 332 13.3.4 Heterogeneous Materials . . . . . . . . . . . . 334 13.3.5 Relationship Between Elasticity and Viscous Fluid Theory . 337 13.3.6 Viscosity of a Suspension . 338 13.3.7 Viscoelasticity. . . . . . . 339

13.4 Steady-State Trapping Problem . 339 13.4.1 Local Relations . . . . . . . 341 13.4.2 Homogenization of Random Problem in rna . 341

13.5 Steady-State Fluid Permeability Problem . . . . . . 344 13.5.1 Local Relations . . . . . . . . . . . . . . . . . . 345 13.5.2 Homogenization of Random Problem in md . 346 13.5.3 Relationship to Sedimentation Rate 348

13.6 Classification of Steady-State Problems. 349 13.7 Time-Dependent Trapping Problem . . . . . 350

13.7.1 Basic Equations . . . . . . . . . . . . 350 13.7.2 Relationship Between Survival and Relaxation Times . 353

13.8 Time-Dependent Flow Problem . . . . . . . . . . . . . . . . . . 354 13.8.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . 354 13.8.2 Relationship Between Permeability and Relaxation Times 356

14 Variational Principles 14.1 Conductivity .......... .

14.1.1 Field Fluctuations .. . 14.1.2 Energy Representation

357 359 359 361

CONTENTS xvii

14.1.3 Minimum Energy Principles ... . 14.1.4 Hashin-Shtrikman Principle ... .

14.2 Elastic Moduli . . . . . . . . . . . ... .

363 367 368 369 370 373 377 379 379 380 383 383 385

14.3

14.4

14.2.1 Field Fluctuations ...... . 14.2.2 Energy Representation ... . 14.2.3 Minimum Energy Principles . 14.2.4 Hashin-Shtrikman Principle . Trapping Constant . . . . . . . . . . 14.3.1 Energy Representation ... 14.3.2 Minimum Energy Principles Fluid Permeability . . . . . . . . . . 14.4.1 Energy Representation . . . . 14.4.2 Minimum Energy Principl~s

15 Phase-Interchange Relations 390 15.1 Conductivity . . . . . . . . . . . . . . . . . . . . 390

15.1.1 Duality for Two-Dimensional Media . 390 15.1.2 Three-Dimensional Media . . 397

15.2 Elastic Moduli . . . . . . . . . . . . . . 398 15.2.1 Two-Dimensional Media . . . 398 15.2.2 Three-Dimensiqn.al Media . . 401

15.3 Trapping Constant and Fluid Permeability . . 402

16 Exact Results 403 16.1 Conductivity . . . . . . . . . . . . . . . . . . . 404

16.1.1 Coated-Spheres Model. . . . . . . . . 404 16.1.2 Simple Laminates . . . . . . . . . . . 407 16.1.3 Higher-Order Laminates and Attainability. 410 16.1.4 Fiber-Reinforced Materials . . . . . . . . 413 16.1.5 Periodic Arrays oflnclusions. . . . . . . 413 16.1.6 Low-Density Cellular Solids 415 16.1. 7 Field Fluctuations . . . . . . . . . . . 416

16.2 Elastic Moduli . . . . . . . . . . . . . . . . . . 417 16.2.1 Coated-Spheres Model . . . . . . . . 417 16.2.2 Simple Laminates . . . . . . . . . . . . . 419 16.2.3 Higher-Order Laminates and Attainability . 424 16.2.4 Periodic Arrays of Inclusions . . . . . . 426 16.2.5 Low-Density Cellular Solids. . . . 428 16.2.6 Equal Phase Shear Moduli . . . . . . . 429 16.2.7 Sheets with Holes . . . . . . . . . . . . . 429 16.2.8 Dispersions of Particles in a Liquid . . 429 16.2.9 Cavities (Bubbles) in an Incompressible Matrix (Liquid) 429 16.2.10 Field Fluctuations . . . . . . . . . . . . . . . . . . . . . . . 430

xviii

16.2.11 Link to Two-Dimensional Conductivity . 16.2.12 Link to Thermoelastic Constants.

16.3 Trapping Constant ........... . 16.3.1 Diffusion Inside Hyperspheres . 16.3.2 Periodic Arrays of Traps .... .

16.4 Fluid Permeability ........... . 16.4.1 Flow Between Plates and Inside Tubes 16.4.2 Periodic Arrays of Obstacles ..... .

17 Single-Inclusion Solutions 1 7.1 Conduction Problem . . . . . . . . . . . .

17 .1.1 Spherical Inclusion . . . . . . . . 17 .1.2 Polarization Within an Ellipsoid .

17.2 Elasticity Problem . . . . . . . . . . . . . 17 .2.1 Spherical Inclusion . . . . . . . . 17.2.2 Polarization Within an Ellipsoid.

1 7.3 Trapping Problem . . . . . . . . . . 17 .3.1 Spherical Trap . 17 .3.2 Spheroidal Trap .

1 7.4 Flow Problem . . . . . . 17 .4.1 Spherical Obstacle 17.4.2 Spheroidal Obstacle

18 Effective-Medium Approximations 18.1 Conductivity ................ .

18.1.1 Maxwell Approximations .... . 18.1.2 Self-Consistent Approximations . 18.1.3 Differential Etifective-Medium"Approximations

18.2 Elastic Moduli . . .. .. . . . . . . . . . . . . . 18.2.1 Maxwell Approximations .............. . 18.2.2 Self-Consistc:mt'A.pproximanons ........ . 18.2.3 Differential Effective-Medium Approximations

18.3 Trapping Constant 18.4 Fluid Permeability

19 Cluster Expansions 19.1 Conductivity . . . . . . . . . . . . . . . .

19.1.1 Dilute Dispersions of Spheres . 19 .1.2 Dilute Dispersions of Ellipsoids 19.1.3 Nondilute Concentrations ...

19.2 Elastic Moduli . . . . . . . . . . . . . . . 19.2.1 Dilute Dispersions of Spheres . 19.2.2 Dilute Dispersions of Ellipsoids

CONTENTS

430 431 432 432 433 434 434 436

437 437 437 441 442 442 448 451 451 453 455 455 457

459 459 460 462 467 470 470 474 477 479 481

485 486 488 490 491 496 497 500

CONTENTS

19.3

19.4

19.2.3 Nondilute Concentrations ...... . Trapping Constant . . . . . . . . . . . . . . . . 19.3.1 Dilute Dispersions of Spherical Traps 19.3.2 Dilute Dispersions of Spheroidal Traps. 19.3.3 Nondilute Concentrations . Fluid Permeability . . . . . . . . . . 19.4.1 Dilute Beds of Spheres ... 19.4.2 Dilute Beds of Spheroids . 19.4.3 Nondilute Concentrations

20 Exact Contrast Expansions 20.1 Conductivity Tensor . . . . . . . . . . . . . . . . . .

20.1.1 Integral Equation for Cavity Electric Field 20.1.2 Strong-Contrast Expansions 20.1.3 Some Tensor Properties ........ . 20.1.4 Weak-Contrast Expansions . . . . . . . 20.1.5 Expansion of Local Electric Field ... 20.1.6 Isotropic Media . . . . . . . . . . . . ..

20.2 Stiffness Tensor . . . . . . . . . . . . . . . . . . 20.2.1 Integral Equation for the Cavity Strain Field 20.2.2 Strong-Contrast Expansions . . . . 20.2.3 Weak-Contrast Expansions ..... 20.2.4 Expansion of Local Strain Field . . 20.2.5 Isotropic Media . . . . . . . . . . . .

21 Rigorous Bounds 21.1 Conductivity . . . . . . . . . . .

21.1.1 General Considerations 21.1.2 Contrast Bounds . . . . . 21.1.3 Cluster Bounds . . . . . . 21.1.4 Security-Spheres Bounds

21.2 Elastic Moduli . . . . . . . . . . 21.2.1 General Considerations 21.2.2 Contrast Bounds . . . . 21.2.3 Cluster Bounds . . . . . 21.2.4 Security-Spheres Bounds- . . . .

21.3 Trapping Constant . . . . . . . . . . . · . 21.3.1 Interfacial-Surface Lower Bound 21.3.2 Void Lower Bound ....... . 21.3.3 Cluster Lower Bounds ..... . 21.3.4 Security-Spheres Upper Bound 21.3.5 Pore-Size Upper Bound .....

xix

501 502 502 503 504 505 505 506 507

509 510 511 514 519 520 521 521 530 530 534 539 540 541

552 554 554 555 563 564 566 566 568 576 577 578 579 580 581 582 584

XX

21.4 Fluid Permeability . . . . . . . . . . . . . 21.4.1 Interfacial-Surface Upper Bound 21.4.2 Void Upper Bound . . . . . . . . 21.4.3 Cluster Upper Bounds ..... . 21.4.4 Security-Spheres Lower Bound

21.5 Structural Optimization 21.6 Utility of Bounds

22 Evaluation of Bounds 22.1 Conductivity ...

22.1.1 Contrast Bounds 22.1.2 Cluster Bounds . 22.1.3 Security-Spheres Bounds

22.2 Elastic Moduli ..... . 22.2.1 Contrast Bounds .... . 22.2.2 Cluster Bounds ..... . 22.2.3 Security-Spheres Bounds

22.3 Trapping Constant . . . . . . . . 22.3.1 Interfacial-Surface Lower Bound 22.3.2 Void Lower Bound ....... . 22.3.3 Cluster Lower Bounds ..... . 22.3.4 Security-Spheres Upper Bound 22.3.5 Pore-Size Upper Bound .... .

22.4 Fluid Permeability ........... . 22.4.1 Interfacial-Surface Upper Bound 22.4.2 Void Upper Bound ....... . 22.4.3 Cluster Upper Bounds ..... . 22.4.4 Security-Spheres Lower Bound

23 Cross-Property Relations 23.1 Conductivity and Elastic Moduli .

23.1.1 Elementary Bounds .... 23.1.2 Translation Bounds ford = 2 . 23.1.3 Translation Bounds ford= 3.

23.2 Flow and Diffusion Parameters . . . 23.2.1 Permeability and Survival Time 23.2.2 Permeability, Formation Factor,

and Viscous Relaxation Times . 23.2.3 Viscous and Diffusion Relaxation Times

A Equilibrium Hard-Disk Program

B Interrelations Among Two- and Three-Dimensional Moduli

CONTENTS

585 585 586 587 589 590 592

593 594 594 609 610 611 611 620 620 621 621 623 624 625 625 627 627 629 630 631

632 633 633 636 642 647 647

650 654

656

661

CONTENTS

References

Index

xxi

663

693