Random Heterogeneous Materials: Microstructure and Macroscopic Properties

719 Pages·2002·22.18 MB·English

by  Salvatore Torquato (auth.)

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 J.E. Marsden
Department of Mathematics Control and Dynamical Systems
and Mail Code 107-81
Institute for Physical Science and Technology California Institute of Technology
University of Maryland Pasadena, CA 91125
College Park, MD 20742-4015 USA
USA
L. Sirovich S. Wiggins
Division of Applied Mathematics Control and Dynamical Systems
Brown University Mail Code 107-81
Providence, RI 02912 <:::alifomia Institute of Technology
USA 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 2001020203
Printed on acid-free paper.
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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 funda
mental 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 universe
for though the world is so old-and so many books have been written
each 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 Con
tinuum Theories by Beran ( 1968). Many of these advances, especially those concerning
the statistical characterization of the microstructure and its effect on the physical prop
erties 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 sci
entists, 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 rudi
mentary 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 ad
vances 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, mechani
cal, and chemical properties of heterogeneous materials and describes how they are
linked to the microstructure of model and real materials. Topics covered include ho
mogenization 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 inde
pendent 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 care
fully 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.p rinceton.edulbook.html,
where an up-to-date errata list will be maintained.
Princeton, New Jersey Salvatore Torquato
June 2001
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