Book: Random Heterogeneous Materials Random Heterogeneous Materials: Microstructure and Macroscopic Properties By Salvatore TorquatoPublisher: Springer-Verlag (2002) Information on ordering through Springer-Verlag Information through Amazon.com Information through Barnes and Noble Book IntroExcerpts from Chapter 1: The determination of the transport, electromagnetic and mechanical properties of heterogeneous materials has a long and venerable history, attracting the attention of some of the luminaries of science, including Maxwell (1873), Lord Rayleigh (1892) and Einstein (1906). In his Treatise on Electricity and Magnetism, Maxwell derived an expression for the effective conductivity of a dispersion of spheres that is exact for dilute sphere concentrations. Lord Rayleigh developed a formalism to compute the effective conductivity of regular arrays of spheres that is used to this day. Work on the mechanical properties of heterogeneous materials began with the famous paper by Einstein in which he determined the effective viscosity of a dilute suspension of spheres. Since the early work on the physical properties of heterogeneous materials, there has been an explosion in the literature on this subject because of the rich and challenging fundamental problems it offers and its manifest technological importance. This book is divided into two parts. Part I deals with the quantitative characterization of the microstructure of heterogeneous materials via theoretical, computer-simulation and imaging techniques. Emphasis is placed on theoretical methods. Part II treats a wide variety of effective properties of heterogeneous materials and how they are linked to the microstructure. This is accomplished using rigorous methods. (Readers interested in property prediction can immediately skip to Part II.) Whenever possible, theoretical predictions for the effective properties are compared to available experimental and computer-simulation data. The overall goal of the book is to provide a rigorous means of characterizing the microstructure and properties of heterogeneous materials that can simultaneously yield results of practical utility. A unified treatment of both microstructure and properties is emphasized. In Chapter 2, the various microstructural functions that are essential in determining the effective properties of random heterogeneous materials are defined. Chapter 3 provides a review of the statistical mechanics of particle systems that is particularly germane to the study of random heterogeneous materials. In Chapter 4, a unified approach to characterize the microstructure of a large class of media is developed. This is accomplished via a canonical n-point function H_n from which one can derive exact analytical expressions for any microstructural function of interest. Chapters 5, 6 and 7 apply the formalism of Chapter 4 to the case of identical systems of spheres, spheres with a polydispersivity in size, and anisotropic particle systems (including laminates), respectively. In Chapter 8, the methods of Chapter 4 are extended to quantify the microstructure of cell models. Here the random-field approach is also discussed. Chapter 9 reviews the study of percolation and clustering on a lattice and introduces continuum percolation. Chapter 10 describes specific developments continuum percolation theory. Chapter 11 describes a means to study microstructural fluctuations that occur on local length scales. Finally, Chapter 12 discusses computer-simulation techniques (primarily Monte Carlo methods) to quantify microstructure. Moreover, it is shown how to apply the same methods to compute relevant microstructural functions from two- and three-dimensional images of the material. In Chapter 13, the local governing equations for the relevant field quantities and the method of homogenization leading to the averaged equations for the effective properties are described. The aforementioned four different classes of problems are studied. In Chapter 14, minimum energy principles are derived that lead to variational bounds on all of the effective properties in terms of trial fields. Chapter 15 proves and discusses certain phase-interchange relations for the effective conductivity and elastic moduli. Chapter 16 derives and describes some exact results for each of the effective properties. In Chapter 17, we derive the local fields associated with a single spherical or ellipsoidal inclusion in an infinite medium for all problem classes. Chapter 18 presents derivations of popular effective-medium approximations for all four effective properties. In Chapter 19, cluster expansions of the effective properties of dispersions are described. Chapter 20 presents derivations of so-called strong-contrast expansions for the effective conductivity and elastic moduli of generally anisotropic media of arbitrary microstructure. In Chapter 21, rigorous bounds on the all of the effective properties are derived using the variational principles of Chapter 14 and specific trial fields. Chapter 22 describes the evaluation of the bounds found in Chapter 21 for certain theoretical model microstructures as well as experimental systems using the results of Part I. Finally, cross-property relations between the seemingly different effective properties considered here are discussed and derived in Chapter 23. Table of Contents Table of Contents Preface vii 1 Motivation and Overview 1 1.1 What Is a Heterogeneous Material? . . . . . . . . . . . . . . . . . . . . . 1 1.2 Effective Properties and Applications . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Conductivity and Analogous Properties . . . . . . . . . . . . . . 6 1.2.2 Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Survival Time or Trapping Constant . . . . . . . . . . . . . . . . 8 1.2.4 Fluid Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.5 Diffusion and Viscous Relaxation Times . . . . . . . . . . . . . . 9 1.2.6 Definitions of Effective Properties . . . . . . . . . . . . . . . . . . 9 1.3 Importance of Microstructure . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Development of a Systematic Theory . . . . . . . . . . . . . . . . . . . . 12 1.4.1 Microstructural Details . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.2 Multidisciplinary Research Area . . . . . . . . . . . . . . . . . . . 14 1.5 Overview of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.5.1 Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.5.2 Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.5.3 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 I Microstructure Characterization 21 2 Microstructural Descriptors 23 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2 n-Point Probability Functions . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.2 Symmetries and Ergodicity . . . . . . . . . . . . . . . . . . . . . . 28 2.2.3 Geometrical Probability Interpretation . . . . . . . . . . . . . . . 32 2.2.4 Asymptotic Properties and Bounds . . . . . . . . . . . . . . . . . 33 2.2.5 Two-Point Probability Function . . . . . . . . . . . . . . . . . . . 34 2.3 Surface Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . 43 2.4 Lineal-Path Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.5 Chord-Length Density Function . . . . . . . . . . . . . . . . . . . . . . . 45 2.6 Pore-Size Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.7 Percolation and Cluster Functions . . . . . . . . . . . . . . . . . . . . . . 50 2.8 Nearest-Neighbor Functions . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.9 Point/q-Particle Correlation Functions . . . . . . . . . . . . . . . . . . . 57 2.10 Surface/Particle Correlation Function . . . . . . . . . . . . . . . . . . . . 58 3 Statistical Mechanics of Many-Particle Systems 59 3.1 Many-Particle Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.1.1 n-Particle Probability Densities . . . . . . . . . . . . . . . . . . . 60 3.1.2 Pair Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2 OrnsteinZernike Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.3 Equilibrium Hard-Sphere Systems . . . . . . . . . . . . . . . . . . . . . 75 3.3.1 Low-Density Expansions . . . . . . . . . . . . . . . . . . . . . . . 79 3.3.2 Arbitrary Fluid Densities . . . . . . . . . . . . . . . . . . . . . . . 81 3.4 Random Sequential Addition Processes . . . . . . . . . . . . . . . . . . 83 3.4.1 One-Dimensional Identical Hard Rods . . . . . . . . . . . . . . . 85 3.4.2 Identical Hard Spheres in Higher Dimensions . . . . . . . . . . 87 3.4.3 General Hard-Particle Systems . . . . . . . . . . . . . . . . . . . 88 3.5 Maximally Random Jammed State . . . . . . . . . . . . . . . . . . . . . 88 3.5.1 Random Close Packing Is Ill-Defined . . . . . . . . . . . . . . . . 89 3.5.2 Definition of Maximally Random Jammed State . . . . . . . . . 90 3.5.3 Order Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.5.4 Molecular Dynamics Simulations . . . . . . . . . . . . . . . . . . 93 3.5.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4 United Approach to Characterize Microstructure 96 4.1 Volume Fraction and Specific Surface . . . . . . . . . . . . . . . . . . . 97 4.1.1 Bounding Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.1.2 Example Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.2 Canonical Correlation Function Hn . . . . . . . . . . . . . . . . . . . . . 104 4.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.2.2 Asymptotic Properties . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.3 Series Representations of Hn . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.3.1 Mayer Representation . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.3.2 KirkwoodSalsburg 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 6.2.3 Lineal-Path Function . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.2.4 Chord-Length Density Function . . . . . . . . . . . . . . . . . . . 171 6.2.5 Nearest-Surface Functions . . . . . . . . . . . . . . . . . . . . . . 172 6.2.6 Pore-Size Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6.2.7 Point/q-Particle Correlation Functions . . . . . . . . . . . . . . . 176 7 Anisotropic Media 177 7.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.2 Fully Penetrable Oriented Inclusions . . . . . . . . . . . . . . . . . . . . 179 7.3 Impenetrable Oriented Inclusions . . . . . . . . . . . . . . . . . . . . . . 181 7.4 Hierarchical Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 8 Cell and Random-Field Models 188 8.1 Cell Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 8.1.1 Voronoi and Delaunay Tessellations . . . . . . . . . . . . . . . . 189 8.1.2 Cell Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 8.1.3 Symmetric-Cell Materials . . . . . . . . . . . . . . . . . . . . . . . 194 8.1.4 Random Checkerboard . . . . . . . . . . . . . . . . . . . . . . . . 199 8.1.5 Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 8.2 Random-Field Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 8.2.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . 203 8.2.2 Gaussian Convolved Intensities . . . . . . . . . . . . . . . . . . . 207 9 Percolation and Clustering 210 9.1 Lattice Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 9.1.1 Bond and Site Percolation . . . . . . . . . . . . . . . . . . . . . . 211 9.1.2 Percolation Properties . . . . . . . . . . . . . . . . . . . . . . . . . 215 9.1.3 Scaling and Critical Exponents . . . . . . . . . . . . . . . . . . . 217 9.1.4 Infinite Cluster and Fractality . . . . . . . . . . . . . . . . . . . . 222 9.1.5 Finite-Size Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 9.2 Continuum Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 9.2.1 Percolation Properties . . . . . . . . . . . . . . . . . . . . . . . . . 227 9.2.2 Two-Point Cluster Function . . . . . . . . . . . . . . . . . . . . . 230 9.2.3 Critical Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 10 Some Continuum Percolation Results 234 10.1 Exact Results for Overlapping Spheres . . . . . . . . . . . . . . . . . . . 234 10.1.1 One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 10.1.2 Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 10.1.3 Low-Density Expansions of Cluster Statistics . . . . . . . . . . . 242 10.2 OrnsteinZernike Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 243 10.3 PercusYevick Approximations . . . . . . . . . . . . . . . . . . . . . . . . 245 10.3.1 Permeable-Sphere Model . . . . . . . . . . . . . . . . . . . . . . . 246 10.3.2 Cherry-Pit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 10.3.3 Sticky Hard-Sphere Model . . . . . . . . . . . . . . . . . . . . . . 249 10.4 Beyond PercusYevick Approximations . . . . . . . . . . . . . . . . . . . 250 10.5 Two-Point Cluster Function . . . . . . . . . . . . . . . . . . . . . . . . . . 250 10.6 Percolation Threshold Estimates . . . . . . . . . . . . . . . . . . . . . . . 251 10.6.1 Overlapping Disks and Spheres . . . . . . . . . . . . . . . . . . . 252 10.6.2 Nonspherical Overlapping Particles . . . . . . . . . . . . . . . . . 254 10.6.3 Interacting Particle Systems . . . . . . . . . . . . . . . . . . . . . 255 11 Local Volume Fraction Fluctuations 257 11.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 11.2 Coarseness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 11.2.1 General Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 11.2.2 Asymptotic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 261 11.2.3 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 11.3 Moments of Local Volume Fraction . . . . . . . . . . . . . . . . . . . . . 264 11.4 Evaluations of Full Distribution . . . . . . . . . . . . . . . . . . . . . . . 265 12 Computer Simulations, Image Analyses, and Reconstructions 269 12.1 Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 12.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 12.1.2 Importance Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 271 12.2 Metropolis Method for Gibbs Ensembles . . . . . . . . . . . . . . . . . . 273 12.2.1 Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 12.2.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 12.2.3 Practical Implementation . . . . . . . . . . . . . . . . . . . . . . . 275 12.2.4 Hard Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 12.2.5 Other Particle Systems . . . . . . . . . . . . . . . . . . . . . . . . 278 12.2.6 Cell Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 12.3 Methods for Generating Nonequilibrium Ensembles . . . . . . . . . . . 279 12.4 Sampling in Particle Systems . . . . . . . . . . . . . . . . . . . . . . . . . 281 12.4.1 Radial Distribution Function . . . . . . . . . . . . . . . . . . . . . 281 12.4.2 n-point Probability Functions . . . . . . . . . . . . . . . . . . . . 283 12.4.3 Surface Correlation Functions . . . . . . . . . . . . . . . . . . . . 285 12.4.4 Cluster-Type Functions . . . . . . . . . . . . . . . . . . . . . . . . 285 12.4.5 Other Correlation Functions . . . . . . . . . . . . . . . . . . . . . 286 12.5 Sampling Images and Digitized Media . . . . . . . . . . . . . . . . . . . 287 12.5.1 Two-Point Probability Function . . . . . . . . . . . . . . . . . . . 289 12.5.2 Lineal-Path Function . . . . . . . . . . . . . . . . . . . . . . . . . 291 12.5.3 Chord-Length Density Function . . . . . . . . . . . . . . . . . . . 292 12.5.4 Pore-Size Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 292 12.5.5 Two-Point Cluster Function . . . . . . . . . . . . . . . . . . . . . 293 12.6 Reconstructing Heterogeneous Materials . . . . . . . . . . . . . . . . . 294 12.6.1 Reconstruction Procedure . . . . . . . . . . . . . . . . . . . . . . 295 12.6.2 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . 297 II Microstructure/Property Connection 303 13 Local and Homogenized Equations 305 13.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 13.2 Conduction Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 13.2.1 Local Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 13.2.2 Conduction Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 311 13.2.3 Model One-Dimensional Problem . . . . . . . . . . . . . . . . . . 313 13.2.4 Homogenization of Periodic Problem in _d . . . . . . . . . . . . 315 13.2.5 Homogenization of Random Problem in _d . . . . . . . . . . . 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 _d . . . . . . . . . . . . 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 _d . . . . . . . . . . . . 341 13.5 Steady-State Fluid Permeability Problem . . . . . . . . . . . . . . . . . 344 13.5.1 Local Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 13.5.2 Homogenization of Random Problem in _d . . . . . . . . . . . . 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 357 14.1 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 14.1.1 Field Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 14.1.2 Energy Representation . . . . . . . . . . . . . . . . . . . . . . . . 361 14.1.3 Minimum Energy Principles . . . . . . . . . . . . . . . . . . . . . 363 14.1.4 HashinShtrikman Principle . . . . . . . . . . . . . . . . . . . . . 367 14.2 Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 14.2.1 Field Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 14.2.2 Energy Representation . . . . . . . . . . . . . . . . . . . . . . . . 370 14.2.3 Minimum Energy Principles . . . . . . . . . . . . . . . . . . . . . 373 14.2.4 HashinShtrikman Principle . . . . . . . . . . . . . . . . . . . . . 377 14.3 Trapping Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 14.3.1 Energy Representation . . . . . . . . . . . . . . . . . . . . . . . . 379 14.3.2 Minimum Energy Principles . . . . . . . . . . . . . . . . . . . . . 380 14.4 Fluid Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 14.4.1 Energy Representation . . . . . . . . . . . . . . . . . . . . . . . . 383 14.4.2 Minimum Energy Principles . . . . . . . . . . . . . . . . . . . . . 385 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-Dimensional 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 of Inclusions . . . . . . . . . . . . . . . . . . . . . 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 16.2.11 Link to Two-Dimensional Conductivity . . . . . . . . . . . . . . 430 16.2.12 Link to Thermoelastic Constants . . . . . . . . . . . . . . . . . . 431 16.3 Trapping Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 16.3.1 Diffusion Inside Hyperspheres . . . . . . . . . . . . . . . . . . . . 432 16.3.2 Periodic Arrays of Traps . . . . . . . . . . . . . . . . . . . . . . . . 433 16.4 Fluid Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 16.4.1 Flow Between Plates and Inside Tubes . . . . . . . . . . . . . . . 434 16.4.2 Periodic Arrays of Obstacles . . . . . . . . . . . . . . . . . . . . . 436 17 Single-Inclusion Solutions 437 17.1 Conduction Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 17.1.1 Spherical Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 437 17.1.2 Polarization Within an Ellipsoid . . . . . . . . . . . . . . . . . . . 441 17.2 Elasticity Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 17.2.1 Spherical Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 442 17.2.2 Polarization Within an Ellipsoid . . . . . . . . . . . . . . . . . . . 448 17.3 Trapping Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 17.3.1 Spherical Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 17.3.2 Spheroidal Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 17.4 Flow Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 17.4.1 Spherical Obstacle . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 17.4.2 Spheroidal Obstacle . . . . . . . . . . . . . . . . . . . . . . . . . . 457 18 Effective-Medium Approximations 459 18.1 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 18.1.1 Maxwell Approximations . . . . . . . . . . . . . . . . . . . . . . . 460 18.1.2 Self-Consistent Approximations . . . . . . . . . . . . . . . . . . . 462 18.1.3 Differential Effective-Medium Approximations . . . . . . . . . 467 18.2 Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 18.2.1 Maxwell Approximations . . . . . . . . . . . . . . . . . . . . . . . 470 18.2.2 Self-Consistent Approximations . . . . . . . . . . . . . . . . . . 474 18.2.3 Differential Effective-Medium Approximations . . . . . . . . . 477 18.3 Trapping Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 18.4 Fluid Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 19 Cluster Expansions 485 19.1 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 19.1.1 Dilute Dispersions of Spheres . . . . . . . . . . . . . . . . . . . . 488 19.1.2 Dilute Dispersions of Ellipsoids . . . . . . . . . . . . . . . . . . . 490 19.1.3 Nondilute Concentrations . . . . . . . . . . . . . . . . . . . . . . 491 19.2 Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 19.2.1 Dilute Dispersions of Spheres . . . . . . . . . . . . . . . . . . . . 497 19.2.2 Dilute Dispersions of Ellipsoids . . . . . . . . . . . . . . . . . . . 500 19.2.3 Nondilute Concentrations . . . . . . . . . . . . . . . . . . . . . . 501 19.3 Trapping Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 19.3.1 Dilute Dispersions of Spherical Traps . . . . . . . . . . . . . . . 502 19.3.2 Dilute Dispersions of Spheroidal Traps . . . . . . . . . . . . . . . 503 19.3.3 Nondilute Concentrations . . . . . . . . . . . . . . . . . . . . . . 504 19.4 Fluid Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 19.4.1 Dilute Beds of Spheres . . . . . . . . . . . . . . . . . . . . . . . . 505 19.4.2 Dilute Beds of Spheroids . . . . . . . . . . . . . . . . . . . . . . . 506 19.4.3 Nondilute Concentrations . . . . . . . . . . . . . . . . . . . . . . 507 20 Exact Contrast Expansions 509 20.1 Conductivity Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 20.1.1 Integral Equation for Cavity Electric Field . . . . . . . . . . . . 511 20.1.2 Strong-Contrast Expansions . . . . . . . . . . . . . . . . . . . . . 514 20.1.3 Some Tensor Properties . . . . . . . . . . . . . . . . . . . . . . . . 519 20.1.4 Weak-Contrast Expansions . . . . . . . . . . . . . . . . . . . . . . 520 20.1.5 Expansion of Local Electric Field . . . . . . . . . . . . . . . . . . 521 20.1.6 Isotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 20.2 Stiffness Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 20.2.1 Integral Equation for the Cavity Strain Field . . . . . . . . . . . 530 20.2.2 Strong-Contrast Expansions . . . . . . . . . . . . . . . . . . . . . 534 20.2.3 Weak-Contrast Expansions . . . . . . . . . . . . . . . . . . . . . . 539 20.2.4 Expansion of Local Strain Field . . . . . . . . . . . . . . . . . . . 540 20.2.5 Isotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 21 Rigorous Bounds 552 21.1 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 21.1.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . 554 21.1.2 Contrast Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 21.1.3 Cluster Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 21.1.4 Security-Spheres Bounds . . . . . . . . . . . . . . . . . . . . . . . 564 21.2 Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 21.2.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . 566 21.2.2 Contrast Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 21.2.3 Cluster Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 21.2.4 Security-Spheres Bounds . . . . . . . . . . . . . . . . . . . . . . . 577 21.3 Trapping Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 21.3.1 Interfacial-Surface Lower Bound . . . . . . . . . . . . . . . . . . 579 21.3.2 Void Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 21.3.3 Cluster Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 581 21.3.4 Security-Spheres Upper Bound . . . . . . . . . . . . . . . . . . . 582 21.3.5 Pore-Size Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . 584 21.4 Fluid Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 21.4.1 Interfacial-Surface Upper Bound . . . . . . . . . . . . . . . . . . 585 21.4.2 Void Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 21.4.3 Cluster Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 587 21.4.4 Security-Spheres Lower Bound . . . . . . . . . . . . . . . . . . . 589 21.5 Structural Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 21.6 Utility of Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 22 Evaluation of Bounds 593 22.1 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 22.1.1 Contrast Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 22.1.2 Cluster Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 22.1.3 Security-Spheres Bounds . . . . . . . . . . . . . . . . . . . . . . . 610 22.2 Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 22.2.1 Contrast Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 22.2.2 Cluster Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620 22.2.3 Security-Spheres Bounds . . . . . . . . . . . . . . . . . . . . . . . 620 22.3 Trapping Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 22.3.1 Interfacial-Surface Lower Bound . . . . . . . . . . . . . . . . . . 621 22.3.2 Void Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 22.3.3 Cluster Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 624 22.3.4 Security-Spheres Upper Bound . . . . . . . . . . . . . . . . . . . 625 22.3.5 Pore-Size Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . 625 22.4 Fluid Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 22.4.1 Interfacial-Surface Upper Bound . . . . . . . . . . . . . . . . . . 627 22.4.2 Void Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 22.4.3 Cluster Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 630 22.4.4 Security-Spheres Lower Bound . . . . . . . . . . . . . . . . . . . 631 23 Cross-Property Relations 632 23.1 Conductivity and Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . 633 23.1.1 Elementary Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 633 23.1.2 Translation Bounds for d _ 2 . . . . . . . . . . . . . . . . . . . . . 636 23.1.3 Translation Bounds for d _ 3 . . . . . . . . . . . . . . . . . . . . . 642 23.2 Flow and Diffusion Parameters . . . . . . . . . . . . . . . . . . . . . . . 647 23.2.1 Permeability and Survival Time . . . . . . . . . . . . . . . . . . . 647 23.2.2 Permeability, Formation Factor, and Viscous Relaxation Times . . . . . . . . . . . . . . . . . . . . 650 23.2.3 Viscous and Diffusion Relaxation Times . . . . . . . . . . . . . . 654 A Equilibrium Hard-Disk Program 656 B Interrelations Among Two- and Three-Dimensional Moduli 661 References 663 Index 693 Errata for First Printing See below for the PDF version, or the corrected pages in the 2nd printing. Errata for Second Printing See below for the PDF version, or the corrected pages in the 2nd printing. Description The study of random heterogeneous materials is an exciting and rapidly growing multidisciplinary endeavor. This field demands a unified rigorous means of characterizing the microstructures and macroscopic properties of the widely diverse types of heterogeneous materials that abound in nature and synthetic products. This book is the first of its kind to provide such an approach. Emphasis is placed on foundational theoretical methods that can simultaneously yield results of practical utility. The first part of the book deals with the quantitative characterization of the microstructure of heterogeneous materials. The second part of the book 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. Contemporary topics covered include the statistical mechanics of many-particle systems, the canonical n-point correlation function, percolation theory, computer-simulation methods, image analyses and reconstructions of real materials, homogenization theory, exact property predictions, variational bounds, expansion techniques, and cross-property relations. This clear and authoritative volume will be of particular interest to graduate students and researchers in applied mathematics, physics, chemistry, materials sciences, engineering, geophysics, and biology. Moreover, the book is self-contained and approachable by the nonspecialist. Salvatore Torquato is a Professor in the Department of Chemistry and the Materials Institute at Princeton University. He also holds affiliated appointments at Princeton University in the Applied and Computational Mathematics Program and in Chemical Engineering. Among other honors, he was a John Simon Guggenheim Fellow in 1998. He has published over two hundred journal articles across a variety of scientific disciplines. The book is 701 pages, containing 218 illustrations and over 725 references, and is now available.