Random Heterogeneous Materials

by Torquato, Salvatore

• ISBN13: 9780387951676
• ISBN10: 0387951679
• Format: Hardcover
• Copyright: 2001-10-01
• Publisher: Springer Verlag

Summary

The overall aim of this book is to provide a rigorous means to characterize 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. The book is divided into two parts. Part I deals with the quantitative characterization of the microstructure of heterogeneous 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. Wherever possible, theoretical predictions for the effective properties are compared to available, experimental and computer-simulation data. This book will be of interest to graduate students and researchers in applied mathematics, physics, chemistry, materials science and engineering.FROM THE REVIEWS:¿This is an important book, which is recommended to all stochastic geometers as well as spatial statisticians, at least to learn what a leading physicist has to say about stochastic geometry . . . [Many] of the formulas in Part II are perfectly general and make no assumption about the type of microstructure, i.e., they are valid for foams, heterogeneous polymer phases, granular media, colloids, composites, etc. . . . [This] is a book which is very important and which merits great attention in the mathematical-statistical community.¿¿METRIKA"'Random Hetergeneous Materials' should become a standard reference for those beginning a study of random hetergeneous media."¿APPLIED MECHANICS REVIEWS¿The book at hand has at least two aims: to bring to the literature the current state of the analytic theory of random heterogeneous structures and to expose the tools that have led to a resurgence of interest in these and related materials. The author is in a very good position to achieve these goals, since he is responsible for much of the recent progress in this subject area, and because of his interest in the study of complex systems. Furthermore, much of the material has not appeared in any previous book on the subject . . . This is a well-written monograph with a high degree of originality. The pedagogy is strong, with most of the mathematical derivations elaborated in detail and without excessive verbiage. Much of the book is written so as to be accessible to specialists and advanced students. The mathematical background required of the reader is not extensive. In fact, a number of open research questions are outlined in the course of presentation. Torquato¿s book does not contain any exercises, but on the other hand there are many carefully worked out examples and meticulous notes about the history and literature in the field . . . [This book] answers a demand for a long awaited textbook for those who wish to learn about this fascinating and still poorly understood area of condensed matter physics which is sure to remain a focus of intense research activity in the years to come. Torquato¿s monograph should make an excellent text for self-study or for a graduate course in disordered materials. It will be welcomed by the physical and materials science communities.¿¿OPTICS & PHOTONICS NEWS

Author Biography

Salvatore Torquato is a Professor in the Department of Chemistry and in the Materials Institute at Princeton University.

Table of Contents

Preface	vii
Motivation and Overview	1	(20)
What Is a Heterogeneous Material?	1	(2)
Effective Properties and Applications	3	(7)
Conductivity and Analogous Properties	6	(1)
Elastic Moduli	7	(1)
Survival Time or Trapping Constant	8	(1)
Fluid Permeability	8	(1)
Diffusion and Viscous Relaxation Times	9	(1)
Definitions of Effective Properties	9	(1)
Importance of Microstructure	10	(2)
Development of a Systematic Theory	12	(5)
Microstructural Details	12	(2)
Multidisciplinary Research Area	14	(3)
Overview of the Book	17	(4)
Part I	17	(1)
Part II	18	(1)
Scope	19	(2)
I Microstructure Characterization	21	(282)
Microstructural Descriptors	23	(36)
Preliminaries	24	(1)
n-Point Probability Functions	25	(18)
Definitions	25	(3)
Symmetries and Ergodicity	28	(4)
Geometrical Probability Interpretation	32	(1)
Asymptotic Properties and Bounds	33	(1)
Two-Point Probability Function	34	(9)
Surface Correlation Functions	43	(1)
Lineal-Path Function	44	(1)
Chord-Length Density Function	45	(3)
Pore-Size Functions	48	(2)
Percolation and Cluster Functions	50	(1)
Nearest-Neighbor Functions	50	(7)
Point/q-Particle Correlation Functions	57	(1)
Surface/Particle Correlation Function	58	(1)
Statistical Mechanics of Many-Particle Systems	59	(37)
Many-Particle Statistics	60	(12)
n-Particle Probability Densities	60	(5)
Pair Potentials	65	(7)
Ornstein-Zernike Formalism	72	(3)
Equilibrium Hard-Sphere Systems	75	(8)
Low-Density Expansions	79	(2)
Arbitrary Fluid Densities	81	(2)
Random Sequential Addition Processes	83	(5)
One-Dimensional Identical Hard Rods	85	(2)
Identical Hard Spheres in Higher Dimensions	87	(1)
General Hard-Particle Systems	88	(1)
Maximally Random Jammed State	88	(8)
Random Close Packing Is Ill-Defined	89	(1)
Definition of Maximally Random Jammed State	90	(2)
Order Metrics	92	(1)
Molecular Dynamics Simulations	93	(2)
Concluding Remarks	95	(1)
Unified Approach to Characterize Microstructure	96	(23)
Volume Fraction and Specific Surface	97	(7)
Bounding Properties	100	(2)
Example Calculations	102	(2)
Canonical Correlation Function Hn	104	(5)
Definitions	105	(4)
Asymptotic Properties	109	(1)
Series Representations of Hn	109	(5)
Mayer Representation	110	(1)
Kirkwood-Salsburg Representation	111	(1)
Bounding Properties	112	(2)
Special Cases of Hn	114	(2)
Polydispersivity	116	(2)
Other Model Microsturctures	118	(1)
Monodisperse Spheres	119	(41)
Fully Penetrable Spheres	120	(9)
n-Point Probability Functions	122	(2)
Surface Correlation Functions	124	(1)
Lineal-Path Function	125	(2)
Chord-Length Density Function	127	(1)
Nearest-Neighbor Functions	128	(1)
Pore-Size Functions	128	(1)
Point/q-Particle Correlation Functions	129	(1)
Totally Impenetrable Spheres	129	(24)
n-Point Probability Functions	130	(4)
Surface Correlation Functions	134	(2)
Lineal-Path Function	136	(1)
Chord-Length Density Function	137	(2)
Nearest-Neighbor Functions	139	(12)
Pore-Size Functions	151	(1)
Point/q-Particle Correlation Functions	152	(1)
Interpenetrable Spheres	153	(5)
Nearest-Neighbor Functions	154	(1)
Volume Fraction	155	(1)
Specific Surface	155	(2)
Pore-Size Functions	157	(1)
Other Statistical Descriptors	157	(1)
Statistically Inhomogeneous Systems	158	(2)
Polydisperse Spheres	160	(17)
Fully Penetrable Spheres	161	(6)
n-Point Probability Functions	163	(1)
Surface Correlation Functions	164	(1)
Lineal-Path Function	165	(1)
Chord-Length Density Function	166	(1)
Nearest-Surface Functions	166	(1)
Pore-Size Functions	167	(1)
Point/q-Particle Correlation Functions	167	(1)
Totally Impenetrable Spheres	167	(10)
n-Point Probability Functions	169	(1)
Surface Correlation Functions	170	(1)
Lineal-Path Function	171	(1)
Chord-Length Density Function	171	(1)
Nearest-Surface Functions	172	(4)
Pore-Size Functions	176	(1)
Point/q-Particle Correlation Functions	176	(1)
Anisotropic Media	177	(11)
General Considerations	177	(2)
Fully Penetrable Oriented Inclusions	179	(2)
Impenetrable Oriented Inclusions	181	(2)
Hierarchical Laminates	183	(5)
Cell and Random-Field Models	188	(22)
Cell Models	188	(15)
Voronoi and Delaunay Tessellations	189	(3)
Cell Statistics	192	(2)
Symmetric-Cell Materials	194	(5)
Random Checkerboard	199	(2)
Ising Model	201	(2)
Random-Field Models	203	(7)
General Considerations	203	(4)
Gaussian Convolved Intensities	207	(3)
Percolation and Clustering	210	(24)
Lattice Percolation	211	(13)
Bond and Site Percolation	211	(4)
Percolation Properties	215	(2)
Scaling and Critical Exponents	217	(5)
Infinite Cluster and Fractality	222	(1)
Finite-Size Scaling	223	(1)
Continuum Percolation	224	(10)
Percolation Properties	227	(3)
Two-Point Cluster Function	230	(1)
Critical Exponents	231	(3)
Some Continuum Percolation Results	234	(23)
Exact Results for Overlapping Spheres	234	(9)
One Dimension	235	(5)
Higher Dimensions	240	(2)
Low-Density Expansions of Cluster Statistics	242	(1)
Ornstein-Zernike Formalism	243	(2)
Percus-Yevick Approximations	245	(5)
Permeable-Sphere Model	246	(2)
Cherry-Pit Model	248	(1)
Sticky Hard-Sphere Model	249	(1)
Beyond Percus-Yevick Approximations	250	(1)
Two-Point Cluster Function	250	(1)
Percolation Threshold Estimates	251	(6)
Overlapping Disks and Spheres	252	(2)
Nonspherical Overlapping Particles	254	(1)
Interacting Particle Systems	255	(2)
Local Volume Fraction Fluctuations	257	(12)
Definitions	258	(2)
Coarseness	260	(4)
General Formula	260	(1)
Asymptotic Formula	261	(1)
Calculations	262	(2)
Moments of Local Volume Fraction	264	(1)
Evaluations of Full Distribution	265	(4)
Computer Simulations, Image Analyses, and Reconstrctions	269	(34)
Monte Carlo Simulations	270	(3)
Introduction	270	(1)
Importance Sampling	271	(2)
Metropolis Method for Gibbs Ensembles	273	(6)
Markov Chain	273	(2)
Algorithm	275	(1)
Practical Implementation	275	(2)
Hard Spheres	277	(1)
Other Particle Systems	278	(1)
Cell Models	279	(1)
Methods for Generating Nonequilibrium Ensembles	279	(2)
Sampling in Particle Systems	281	(6)
Radial Distribution Funtion	281	(2)
n-point Probability Functions	283	(2)
Surface Correlation Functions	285	(1)
Cluster-Type Functions	285	(1)
Other Correlation Functions	286	(1)
Sampling Images and Digitized Media	287	(7)
Two-Point Probability Function	289	(2)
Lineal-Path Function	291	(1)
Chord-Length Density Function	292	(1)
Pore-Size Functions	292	(1)
Two-Point Cluster Function	293	(1)
Reconstructing Heterogeneous Materials	294	(9)
Reconstruction Procedure	295	(2)
Illustrative Examples	297	(6)
II Microstructure/Property Connection	303	(353)
Local and Homogenized Equations	305	(52)
Preliminaries	306	(2)
Conduction Problem	308	(13)
Local Relations	308	(3)
Conduction Symmetry	311	(2)
Model One-Dimensional Problem	313	(2)
Homogenization of Periodic Problem in Rd	315	(3)
Homogenization of Random Problem in Rd	318	(3)
Frequency-Dependent Conductivity	321	(1)
Elastic Problem	321	(18)
Local Relations	321	(3)
Elastic Symmetry	324	(8)
Homogenization of Random Problem in Rd	332	(2)
Heterogeneous Materials	334	(3)
Relationship Between Elasticity and Viscous Fluid Theory	337	(1)
Viscosity of a Suspension	338	(1)
Viscoelasticity	339	(1)
Steady-State Trapping Problem	339	(5)
Local Relations	341	(1)
Homogenization of Random Problem in Rd	341	(3)
Steady-State Fluid Permeability Problem	344	(5)
Local Relations	345	(1)
Homogenization of Random Problem in Rd	346	(2)
Relationship to Sedimentation Rate	348	(1)
Classification of Steady-State Problems	349	(1)
Time-Dependent Trapping Problem	350	(4)
Basic Equations	350	(3)
Relationship Between Survival and Relaxation Times	353	(1)
Time-Dependent Flow Problem	354	(3)
Basic Equations	354	(2)
Relationship Between Permeability and Relaxation Times	356	(1)
Variational Principles	357	(33)
Conductivity	359	(9)
Field Fluctuations	359	(2)
Energy Representation	361	(2)
Minimum Energy Principles	363	(4)
Hashin-Shtrikman Principle	367	(1)
Elastic Moduli	368	(11)
Field Fluctuations	369	(1)
Energy Representation	370	(3)
Minimum Energy Principles	373	(4)
Hashin-Shtrikman Principle	377	(2)
Trapping Constant	379	(4)
Energy Representation	379	(1)
Minimum Energy Principles	380	(3)
Fluid Permeability	383	(7)
Energy Representation	383	(2)
Minimum Energy Principles	385	(5)
Phase-Interchange Relations	390	(13)
Conductivity	390	(8)
Duality for Two-Dimensional Media	390	(7)
Three-Dimensional Media	397	(1)
Elastic Moduli	398	(4)
Two-Dimensional Media	398	(3)
Three-Dimensional Media	401	(1)
Trapping Constant and Fluid Permeability	402	(1)
Exact Results	403	(34)
Conductivity	404	(13)
Coated-Spheres Model	404	(3)
Simple Laminates	407	(3)
Higher-Order Laminates and Attainability	410	(3)
Fiber-Reinforced Materials	413	(1)
Periodic Arrays of Inclusions	413	(2)
Low-Density Cellular Solids	415	(1)
Field Fluctuations	416	(1)
Elastic Moduli	417	(15)
Coated-Spheres Model	417	(2)
Simple Laminates	419	(5)
Higher-Order Laminates and Attainability	424	(2)
Periodic Arrays of Inclusions	426	(2)
Low-Density Cellular Solids	428	(1)
Equal Phase Shear Moduli	429	(1)
Sheets with Holes	429	(1)
Dispersions of Particles in a Liquid	429	(1)
Cavities (Bubbles) in an Incompressible Matrix (Liquid)	429	(1)
Field Fluctuations	430	(1)
Link to Two-Dimensional Conductivity	430	(1)
Link to Thermoelastic Constants	431	(1)
Trapping Constant	432	(2)
Diffusion Inside Hyperspheres	432	(1)
Periodic Arrays of Traps	433	(1)
Fluid Permeability	434	(3)
Flow Between Plates and Inside Tubes	434	(2)
Periodic Arrays of Obstacles	436	(1)
Single-Inclusion Solutions	437	(22)
Conduction Problem	437	(5)
Spherical Inclusion	437	(4)
Polarization Within an Ellipsoid	441	(1)
Elasticity Problem	442	(9)
Spherical Inclusion	442	(6)
Polarization Within an Ellipsoid	448	(3)
Trapping Problem	451	(4)
Spherical Trap	451	(2)
Spheroidal Trap	453	(2)
Flow Problem	455	(4)
Spherical Obstacle	455	(2)
Spheroidal Obstacle	457	(2)
Effective-Medium Approximations	459	(26)
Conductivity	459	(11)
Maxwell Approximations	460	(2)
Self-Consistent Approximations	462	(5)
Differential Effective-Medium Approximations	467	(3)
Elastic Moduli	470	(9)
Maxwell Approximations	470	(4)
Self-Consistent Approximations	474	(3)
Differential Effective-Medium Approximations	477	(2)
Trapping Constant	479	(2)
Fluid Permeability	481	(4)
Cluster Expansions	485	(24)
Conductivity	486	(10)
Dilute Dispersions of Spheres	488	(2)
Dilute Dispersions of Ellipsoids	490	(1)
Nondilute Concentrations	491	(5)
Elastic Moduli	496	(6)
Dilute Dispersions of Spheres	497	(3)
Dilute Dispersions of Ellipsoids	500	(1)
Nondilute Concentrations	501	(1)
Trapping Constant	502	(3)
Dilute Dispersions of Spherical Traps	502	(1)
Dilute Dispersions of Spheroidal Traps	503	(1)
Nondilute Concentrations	504	(1)
Fluid Permeability	505	(4)
Dilute Beds of Spheres	505	(1)
Dilute Beds of Spheroids	506	(1)
Nondilute Concentrations	507	(2)
Exact Contrast Expansions	509	(43)
Conductivity Tensor	510	(20)
Integral Equation for Cavity Electric Field	511	(3)
Strong-Contrast Expansions	514	(5)
Some Tensor Properties	519	(1)
Weak-Contrast Expansions	520	(1)
Expansion of Local Electric Field	521	(1)
Isotropic Media	521	(9)
Stiffness Tensor	530	(22)
Integral Equation for the Cavity Strain Field	530	(4)
Strong-Contrast Expansions	534	(5)
Weak-Contrast Expansions	539	(1)
Expansion of Local Strain Field	540	(1)
Isotropic Media	541	(11)
Rigorous Bounds	552	(41)
Conductivity	554	(12)
General Considerations	554	(1)
Contrast Bounds	555	(8)
Cluster Bounds	563	(1)
Security-Spheres Bounds	564	(2)
Elastic Moduli	566	(12)
General Considerations	566	(2)
Contrast Bounds	568	(8)
Cluster Bounds	576	(1)
Security-Spheres Bounds	577	(1)
Trapping Constant	578	(7)
Interfacial-Surface Lower Bound	579	(1)
Void Lower Bound	580	(1)
Cluster Lower Bounds	581	(1)
Security-Spheres Upper Bound	582	(2)
Pore-Size Upper Bound	584	(1)
Fluid Permeability	585	(5)
Interfacial-Surface Upper Bound	585	(1)
Void Upper Bound	586	(1)
Cluster Upper Bounds	587	(2)
Security-Spheres Lower Bound	589	(1)
Structural Optimization	590	(2)
Utility of Bounds	592	(1)
Evaluation of Bounds	593	(39)
Conductivity	594	(17)
Contrast Bounds	594	(15)
Cluster Bounds	609	(1)
Security-Spheres Bounds	610	(1)
Elastic Moduli	611	(10)
Contrast Bounds	611	(9)
Cluster Bounds	620	(1)
Security-Spheres Bounds	620	(1)
Trapping Constant	621	(6)
Interfacial-Surface Lower Bound	621	(2)
Void Lower Bound	623	(1)
Cluster Lower Bounds	624	(1)
Security-Spheres Upper Bound	625	(1)
Pore-Size Upper Bound	625	(2)
Fluid Permeability	627	(5)
Interfacial-Surface Upper Bound	627	(2)
Void Upper Bound	629	(1)
Cluster Upper Bounds	630	(1)
Security-Spheres Lower Bound	631	(1)
Cross-Property Relations	632	(24)
Conductivity and Elastic Moduli	633	(14)
Elemetary Bounds	633	(3)
Translation Bounds for d = 2	636	(6)
Translation Bounds for d = 3	642	(5)
Flow and Diffusion Parameters	647	(9)
Permeability and Survival Time	647	(3)
Permeability, Formation Factor, and Viscous Relaxation Times	650	(4)
Viscous and Diffusion Relaxation Times	654	(2)
A Equilibrium Hard-Disk Program	656	(5)
B Interrelations Among Two- and Three-Dimensional Moduli	661	(2)
References	663	(30)
Index	693