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

Rigorous upper and lower bounds on the effective electrical conductivity (T * 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), which are probability density functions associated with finding a point in phase i and a particular configuration of n spheres. The G ~i) are shown to be related to thepn, 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) are given. New results for the G ~i) 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 (T * derived here are given exactly through second order in the sphere volume fraction for arbitrary A (where A is the impenetrability or hardness parameter) for two different interpenetrable-sphere models. Comparison of these low-density bounds on (T * to an approximate low-density expansion of (T * 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 A and phase conductivities. The results of this study suggest that general bounds derived by Beran, for dispersions of spheres distributed with arbitrary A and through all orders in ¢2' are more restrictive than the first-order cluster bounds for O':;;A < 1; with the two sets of bounds being identical for the case of totally impenetrable spheres (A = 1). For most values orA in therangeO':;;A < 1, however, the numerical differences between the Beran and cluster bounds should be small; the greatest difference occurring when A = O. The analysis also indicates that the cluster bounds will be easier to compute than the Beran bounds for dispersions of partially penetrable spheres.