Porosity and specific surface for interpenetrable-sphere models of two-phase random media

Abstract

We derive and numerically evaluate expressions for the porosity (matrix volume fraction) φ and the specific surface (interface area per unit volume) s for two specific models of a random two-phase medium in which the medium is considered as a suspension of interpenetrable spheres of radius R, embedded in a uniform matrix. The models and quantities considered have applications to a wide range of problems concerning transport, mechanical, and chemical properties of composite media. Both models contain a continuously variable hardness parameter ε, such that for ε=1 they reduce to mutually impenetrable spheres, and for ε=0 they reduce to fully penetrable spheres. The first of these models, the permeable-sphere model, has been defined only in the context of the Percus–Yevick approximation, which yields for it a unique pair distribution function g2(r1, r2). To find the associated gn(r1,...,rn) for n≳2, which are needed to evaluate φ and s we use the generalized superposition approximation. The second model, the concentric-shell model, can be fully defined without recourse to any particular approximation and proves to be isomorphic to the picture described by the scaled-particle theory of Reiss, Frisch, and Lebowitz. In this case we evaluate φ and s in the scaled-particle approximation introduced by those authors to implement the scaled-particle theory. For both models, we present numerical plots of φ vs dimensionless density, and of s vs φ. We also briefly discuss the relations between the results obtained in the two cases. Finally, we consider generalizations of φ and s that define the volume and surface available to a particle of finite size, rather than a point.