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Introduction

There are numerous contributions to the problem of predicting the effective properties of inhomogeneous materials [1,2,3] and reviews appear regularly on this topic. Here we focus the discussion on transport properties [1] such as the viscosity of suspensions, dielectric constant, refractive index, thermal conductivity and related physical properties of mixtures. Much of the research has been limited to the classical case of spherical particle suspensions and composites and a few other particle shapes that allow analytical treatment [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. Even for suspensions of hard spheres in fluids, rigorous results are limited to the first few virial coefficients [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20] and analytical bounds on the effective properties at higher volume fractions [21,22]. Recent progress has been made in numerical calculation of transport properties by finite element [23] and Brownian dynamics methods [24], which is associated with the advent of sufficiently powerful computational resources and faster computational algorithms like the conjugate gradient method [25]. These new numerical results are very helpful in testing theoretical ideas about the effective properties of mixtures.

Much of the previous research, even the more recent numerical work, has focused on mixtures involving simple shapes such as spheres, since the overall goal has usually been the development of a theory applicable at high volume fractions of suspended matter. The choice of simple particle shapes is motivated by the existence of the relatively few exact analytical results at lower concentrations, which provide a benchmark test for the numerical calculations.

However, many real particles in fluid-solid suspensions and in solid composites are not well-represented by these simple shapes. The present work allows for general centro-symmetric particle shape, while focusing solely on the dilute limit. This is a natural first step towards treating complex-shaped particle mixtures at higher concentrations. Originally, the research was motivated by theoretical arguments, described below, that suggested a relation between the leading order virial coefficients for the viscosity and conductivity of suspensions of conducting particles having arbitrary shape. We have gathered analytical results for these properties, which are scattered widely throughout the mathematical and technical literature and have calculated new results as necessary (analytically and by finite element methods) to obtain a wide range of shapes with which to check the conjectured relation. The results obtained should have independent interest in the problem of developing a more realistic description of the properties of mixtures in terms of a more faithful description of the mixture components.

In Sec. 2 we summarize classical results for the conductivity virial expansion and discuss the relation between the leading virial coefficient, called the intrinsic conductivity [σ] and certain functionals of particle shape that arise in other physical contexts--the electric polarizability, magnetic polarizability, and virtual mass. All these functionals of shape involve solving Navier-Stokes or the Laplace equation on the exterior of a body with various boundary conditions [26]. Exact results for these functionals are summarized for simple particle shapes to illustrate the general effect of particle anisotropy. Sec. 3 summarizes classical results for the viscosity virial expansion of suspensions of particles and the virial expansion for the shear modulus of an elastic material with inclusions. An angular preaveraging approximation is invoked, as in previous calculations for the translational friction of a Brownian particle [27], to relate the intrinsic viscosity [η] to the intrinsic conductivity [σ]_∞ for conducting inclusions. The intrinsic viscosity is the leading order virial coefficient for the viscosity of dilute mixtures. Examination of exact results for [η] and [σ]_∞ show that the conjectured relation is not exact, but rather a good approximation. An extensive range of particle shapes is considered in this comparison. In Sect. 4 we pursue the universality of the [η] -- [σ]_∞ relation for a variety of complicated shapes using finite element methods. The conjectured relation is found to hold to a good approximation (±5%) for all shapes considered.

More general results are possible in d = 2 because of powerful conformal mapping methods. We exploit a mathematical identity of Polya [26, 28], which implies that [σ] for conducting and non-conducting inclusions is exactly related to the 'transfinite diameter' C_L of the inclusions. This relation is useful since this quantity is fundamental in classical conformal mapping theory and, consequently, this property has been extensively investigated [26,29,30]. All previously known exact results for [σ]_∞ in d = 2, plus many new results, are obtained from our new relation expressing [σ]_∞ purely in terms of the geometrical quantities, C_L and the particle area. Numerical results for [η] in d = 2 are obtained using finite element methods. Again the predicted approximate relation between [η] and [σ]_∞ is well confirmed. Analytical results for ellipses show that the conjectured relation is actually exact at all aspect ratios and we conjecture that [η] equals [σ]_∞ for all shapes in two dimensions.

Sec. 6 considers the case of flat 'plate-like' objects, which requires special analytic treatment. Plate-like particles are intermediate in their properties between the three and two dimensional cases and these problems tend to be especially difficult analytically. Numerical finite element calculations, however, can be carried out using methods similar those used for other shapes (see Appendix E). This case has important applications in the scattering of sound waves and electromagnetic radiation through apertures and many numerical results have accumulated in the technical literature. We summarize these connections since they provide the source of predictions for [η] through our proposed relation between [η] and [σ]_∞.