Bulk properties of two-phase disordered media. III. New bounds on the effective conductivity of dispersions of penetrable spheres

Abstract

Rigorous upper and lower bounds on the effective electrical conductivity σ * of a two-phase material composed of equi-sized spheres distributed with an arbitrary degree of impenetrability in a matrix are obtained and studied. In general, the bounds depend upon, among other quantities, the point/n-particle distribution functions G(i)n, which are probability density functions associated with finding a point in phase i and a particular configuration of n spheres. The G(i)n are shown to be related to the ρn, the probability density functions associated with finding a particular configuration of n partially penetrable spheres in a matrix. General asymptotic and bounding properties of the G(i)n are given. New results for the G(i)n are presented for totally impenetrable spheres, fully penetrable spheres (i.e., randomly centered spheres), and sphere distributions between these latter two extremes. The so-called first-order cluster bounds on σ * derived here are given exactly through second order in the sphere volume fraction for arbitrary λ (where λ is the impenetrability or hardness parameter) for two different interpenetrable-sphere models. Comparison of these low-density bounds on σ * to an approximate low-density expansion of σ * derived here for interpenetrable-sphere models, reveals that the bounds can provide accurate estimates of the second-order coefficient for a fairly wide range of λ and phase conductivities. The results of this study suggest that general bounds derived by Beran, for dispersions of spheres distributed with arbitrary λ and through all orders in φ2, are more restrictive than the first-order cluster bounds for 0≤λ<1; with the two sets of bounds being identical for the case of totally impenetrable spheres (λ=1). For most values of λ in the range 0≤λ<1, however, the numerical differences between the Beran and cluster bounds should be small; the greatest difference occurring when λ=0. The analysis also indicates that the cluster bounds will be easier to compute than the Beran bounds for dispersions of partially penetrable spheres.