Random Heterogeneous Materials

We have seen that random heterogeneous materials exhibit a remarkably broad spectrum of rich and complex microstructures. Our focus in Part I of this book is to develop a machinery to characterize statistically this broad class of microstructures, i.e., to develop a statistical, or stochastic, geometry of heterogeneous materials. How or where does one begin to address this challenging task? The answer, of course, depends on what is the goal of the statistical characterization. Our goal is ultimately the prediction of the macroscopic or effective physical properties of the random heterogeneous material, and thus this determines our starting point. The diverse effective properties that we are concerned with in this book naturally and necessarily lead to a wide variety of microstructural descriptors, generically referred to as microstructural correlation functions. As we noted in Chapter 1, such descriptors have applicability in other seemingly disparate fields, such as cosmology (Peebles 1993, Saslaw 2000) and ecology (Pielou 1977, Diggle 1983, Durrett and Levin 1994).

Statistical mechanics is the branch of theoretical physics that attempts to predict, by starting at the level of atoms, molecules, spins, or other small “particles,” the bulk properties of systems in which a large number of these particles interact with one another. In other words, it links the microscopic properties of matter (molecular interactions and structure), as determined from the laws of quantum or classical mechanics, to its macroscopic properties (e.g., pressure of a liquid). The province of statistical physics is more general, extending to any situation in which one is interested in the collective behavior of interacting entities, from population dynamics through solids, liquids, and gases to cosmology as well as random heterogeneous materials. Systems composed of many interacting particles (albeit much larger than molecular dimensions) are often useful models of random heterogeneous materials, and thus one can exploit the powerful machinery of statistical mechanics to study such materials. Moreover, as we will see in subsequent chapters, the formalism of statistical mechanics can be extended to nonparticulate systems.

A unified and general formalism has been developed by Torquato (1986c) to characterize statistically the microstructure of heterogeneous materials composed of d-dimensional spherical inclusions distributed throughout a matrix phase. He accomplished this by introducing a general n-point correlation function H _n and by deriving two different series representations for it in terms of the probability density functions that characterize the particle configurations (quantities that are known, in principle). This methodology provides a means of calculating any of the various types of microstructural correlation functions that have arisen in rigorous expressions for the transport, electromagnetic, and transport properties (see Chapter 2 and Part II) for such nontrivial models of two-phase disordered media as well as generalizations of these correlation functions. For this reason, H _n is referred to as the canonical n-point correlation function.).

In this chapter we will evaluate the series representations of the canonical function H _n , derived in the previous chapter, for various assemblies of identical (i.e., monodisperse) spheres of radius R. Such models are not as restrictive as one might initially surmise. For example, one can vary the connectedness of the particle phase (and therefore its percolation threshold) by allowing the spheres to interpenetrate one another in varying degrees. We saw in Chapter 3 that one extreme of this interpenetrable-sphere model is the case of spatially uncorrelated (i.e., Poisson distributed) spheres that we call fully penetrable (or overlapping) spheres. Overlapping spheres, at low sphere densities, are useful models of nonpercolating dispersions (see Figure 3.4). At high densities, overlapping spheres can be used to model consolidated media such as sandstones and sintered materials (Torquato 1986b). Figure 5.1 shows a distribution of identical overlapping disks at a very high density that resembles the sandstone depicted in Figure 1.3.

Polydispersivity in the size of the particles constitutes a fundamental feature of the microstructure of a wide class of dispersions of technological importance, including composite solid propellant combustion (Kerstein 1987), sintering of powders (Rahaman 1995), colloids (Russel et al. 1989), transport and mechanical properties of particulate composite materials (Christensen 1979), and flow in packed beds (Scheidegger 1974). The effect of particle-size distribution on the microstructure and effective properties of dispersions can be dramatic and therefore is of great interest.

There is a vast class of random heterogeneous materials whose microstructures cannot be modeled as a distribution of inclusions or cavities of well-defined shape in a matrix. This classification includes animal and plant tissue (which have a cellular structure), bone, foams, froths, polycrystals, block copolymers, and microemulsions, to mention but a few examples. In this chapter we will consider analytical methods to characterize the microstructure of two different nonparticulate models: cell and random-field models. For simplicity, we focus primarily on the determination of the n-point probability functions for these models.

Percolation theory deals with the effects of varying the connectivity of elements (e.g., particles, sites, or bonds) in a random system. A cluster is simply a connected group of elements. Roughly speaking, the percolation transition, or threshold, of the system is the point at which a cluster first spans the system, i.e., the first appearance of long-range connectivity. In the thermodynamic limit, the percolation threshold is the point at which a cluster becomes infinite in size. The percolation transition is a wonderful example of a second-order phase transition and critical phenomenon.

The intent of the present chapter is to derive and discuss some basic results and specific developments in continuum percolation theory. We will begin with a discussion of exact results for cluster statistics and other percolation descriptors for a prototypical model of continuum percolation, namely, identical overlapping spheres in d dimensions. Subsequently, we will describe an Ornstein-Zernike formalism to find the pair-connectedness function P ₂(r) for general isotropic models of continuum percolation. The reader should note the beautiful correspondence of this theory to the Ornstein-Zernike formalism for the total correlation function h(r) of equilibrium (or thermal) systems discussed in Chapter 3. This will be followed by a discussion of various approximation schemes to close the resulting integral equation, including the Percus-Yevick approximation. The next topic will be the two-point cluster function C ₂(r). First we will present an exact series representation of C ₂(r) for dispersions and then discuss its analytical evaluation for certain models. The chapter will conclude with a presentation of percolation thresholds for overlapping sphere systems, overlapping particles of nonspherical shape, and interacting particle systems. The reader is referred to Meester and Roy (1996) for a more mathematical treatment of continuum percolation.

One of the most important morphological descriptors of heterogeneous materials is the volume fraction of the phases or, in the case of porous media, the porosity (i.e., the volume fraction of the fluid phase). Although the volume fraction is constant for statistically homogeneous media, on a spatially local level it fluctuates. An interesting question is the following: How does the “local” volume fraction fluctuate about its average value? The answer to this query has relevance to a number of problems, including scattering by heterogeneous media (Debye et al. 1957), the study of noise and granularity of photographic images (O’Neill 1963, Bayer 1964, Lu and Torquato 1990c), transport through porous media (Hilfer 1991, Hilfer 1996), mechanical properties of composites (Ostoja-Starzewski 1993), the properties of organic coatings (Fishman, Kurtze and Bierwagen 1992), and the fracture of composites (Botsis, Beldica and Zhao 1994, Torquato 2000a). It is actually in the context of photographic science that this question of local volume fraction fluctuations was first probed, and here primarily for simple two-dimensional models of photographic emulsions that do not account for impenetrability of the grains (O’Neill 1963, Bayer 1964).

The evaluation of microstructural correlation functions via computer simulations is a two-step process. First, one must generate realizations of the random medium using a particular simulation method. Second, one must sample each realization for the correlation functions of interest and then average over all realizations to get the ensemble-averaged correlation functions. Similar sampling techniques can be used to extract correlation functions from images of real heterogeneous materials.

Homogenization theory is concerned with finding the appropriate homogenized (or averaged, or macroscopic) governing partial differential equations describing physical processes occurring in heterogeneous materials when the length scale of the heterogeneities tends to zero. In such instances it is desired that the effects of the microstructure reside wholly in the macroscopic or effective properties via certain weighted averages of the microstructure. In its simplest form, the method is based on the consideration of two length scales: the macroscopic scale L,characterizing the extent of the system, and the microscopic scale ℓ, associated with the heterogeneities. Moreover, it is supposed that some external field is applied that varies on a characteristic length scale Λ. If ℓ is comparable in magnitude to Λ or L,then one must employ a microscopic description, i.e., one cannot homogenize the equations.

For random media of arbitrary microstructure, exact analytical solutions of the effective properties are unattainable, and so any rigorous statement about the effective properties must be in the form of rigorous bounds. To get variational bounds on effective properties, one must first express the effective parameter in terms of some functional and then formulate an appropriate variational (extremum) principle for the functional. We shall primarily deal with “energy” functionals. Once the variational principle is established, then specific bounds on the property of interest are obtained by constructing trial, or admissible, fields that conform with the variational principle. Specific bounds derived from trial fields are the subject of Chapter 21. In this chapter we will derive variational principles for the effective conductivity, effective elastic moduli, trapping constant, and fluid permeability.

We refer to expressions that link the effective properties of a two-phase heterogeneous material to the effective properties of the same microstructure but with the phases interchanged as phase-interchange relations. When such rigorous relations apply to a wide class of heterogeneous materials, they can provide useful tests of analytical and numerical estimates of the effective properties. We focus here on the effective conductivity and elastic moduli of two-phase composites and, to a lesser extent, of polycrystals. The two-dimensional relations are shown to have interesting implications for percolation behavior. Finally, we remark on phase-interchange relations for the trapping constant and fluid permeability.

Due to the complexity of the microstructure, there are relatively few situations in which one can evaluate the effective properties of heterogeneous materials exactly. Such rare results are nonetheless quite valuable as benchmarks to test theories and computer simulations. Exact results also provide useful insights into the mechanisms responsible for the effective behavior, insights that extend beyond the specific microstructures for which they are derived. In Chapter 15 we discussed exact phase-interchange relations. In the ensuing sections we will derive and discuss some other known exact results that we will make use of in subsequent chapters.

A variety of estimates of the effective properties of heterogeneous media utilize the solution of the boundary value problem of the relevant field for a single inclusion of one material in a matrix of another material. Such estimates include effective properties of dispersions in the dilute-concentration limit (Chapter 19), a variety of effective-medium type approximations (Chapter 18), and rigorous bounds on the effective properties (Chapter 21). In this chapter we will derive single-inclusion solutions for all four classes of problems, focusing on spherical and ellipsoidal inclusions.

Heterogeneous materials composed of well-defined inclusions (e.g., spheres, cylinders, ellipsoids) distributed randomly throughout a matrix material have served as excellent starting points for modeling the complex field interactions in random composites. The celebrated formulas of Maxwell (1873) and Einstein (1906) for the effective conductivity and effective viscosity of dispersions of spheres, respectively, assume that the particles do not interact with another and therefore are valid through first order in the sphere volume fraction ø ₂. Similar formulas for the trapping constant of dilute distributions of traps or the fluid permeability of dilute beds of spheres can easily be obtained from the classical results of Smoluchowski (1917) and Stokes (1851), respectively, given in Chapter 17.

For two-phase media in which variations in the phase properties are small, formally exact perturbation series for both the effective conductivity (Beran 1968, Phan-Thien and Milton 1982) and effective elastic moduli (Beran 1968, Dederichs and Zeller 1973, Willis 1981) have been developed. Such weak-contrast expansions are found by first obtaining corresponding expansions of either the local electric field E(x) or the local strain field ε(x) via integral equations. For specificity, it is useful to state the weak-contrast form of the effective conductivity σ_e of a macroscopically isotropic medium, keeping in mind that analogous results exist for anisotropic media and for the effective elastic tensor.

Since it is generally impossible to determine exactly the effective properties of random heterogeneous media, any rigorous statements about effective properties must take the form of bounds. In Chapter 14 we formulated variational (extremum) principles for the effective properties that are given in terms of trial fields. Here we will derive specific existing bounds on the effective conductivity, effective elastic moduli, trapping constant, and fluid permeability by constructing specific trial fields that conform with the variational principles.

Although some improved bounds have been in existence for nearly four decades, they have, until the recent past, lain dormant and untested because of the difficulty associated with ascertaining the various types of correlation functions involved, even for simple models (e.g., random arrays of spherical particles). Advances in the quantitative characterization of the microstructure, described in Part I of this book, have paved the way for the computation of improved bounds on the effective conductivity, effective elastic moduli, trapping constant, and fluid permeability of nontrivial models of two-phase random heterogeneous materials.

An intriguing fundamental as well as practical question in the study of heterogeneous materials is the following: Can different properties of the medium be rigorously linked to one another? Such cross-property relations become especially useful if one property is more easily measured than another property. Since the effective properties of random media reflect certain morphological information about the medium, one might expect that one could extract useful information about one effective property given an accurate (experimental or theoretical) determination of a different effective property, even when their respective governing equations are uncoupled. Cross-property relations provide a means of ascertaining the possible range of values that different effective properties can possess (i.e., the allowable region in multidimensional property space) and thus have important implications for the design of multifunctional composites.

Here we provide a Fortran 77 program to generate equilibrium configurations of equal-sized hard disks via the Metropolis algorithm. It also computes the radial distribution function g ₂(r) by averaging over the configurations. The extension of the program to d-dimensional hard spheres is obvious. For simplicity, it does not use a “neighbor list,” as described in Chapter 12.

Consider a d-dimensional linear isotropic homogeneous material with bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio denoted by K ^(d), G ^(d),E ^(d), and v ^(d),respectively. These are precisely the moduli contained in the stress-strain relation (13.89) and strain-stress relation (13.90), except that here we explicitly indicate the dependence on dimensionality with the superscript (d).