Microstructure of two-phase random media. III. The n-point matrix probability functions for fully penetrable spheres

Publisher Website
Google Scholar
Abstract
We examine the n-point matrix probability functions Sn (which give the probability of finding n points in the matrix phase of a two-phase random medium), for a model in which the included material consists of fully penetrable spheres of equal diameter (i.e., a system of identical spheres such that their centers are randomly distributed in the matrix). Exploiting the special simplicity of the model we give an explicit closed-form expression for S3 as well as sharp bounds on S3 and S4. Our best lower bound on S3 and our corresponding upper bound on S4 satisfy certain asymptotic forms (for both small and large separation of points) that are satisfied by the exact S3 and S4 for impenetrable as well as penetrable spheres, even though the bounding properties of our expressions can only be guaranteed for penetrable spheres. These expressions (and the resulting approximation for S4 in terms of S1 and S2 obtained from them) are thus highly appropriate approximants for both systems to be used in composite-media transport-coefficient expressions that involve integrals over the Sn. The S3 expression has in fact been suggested some time ago by Weissberg and Prager; our methods here provide further justification for this expression as well as one means of systematically generalizing it to Sn for higher n.