Bulk properties of two-phase disordered media. II. Effective conductivity of a dilute dispersion of penetrable spheres

The calculation of the effective electrical conductivity σ* of a dilute dispersion of equisized spheres of radius R distributed with arbitrary degree of penetrability is considered. It is demonstrated that σ*, through second order in the inclusion volume fraction φ₂, can be written in terms of the zero-density limits of the pair-connectedness and pair-locking functions, and certain polarizability tensors which involve one and two inclusions. Rigorous upper and lower bounds on σ*, through order φ₂², are shown to depend upon, among other quantities, the aforementioned pair distribution functions and are evaluated for two models: an interpenetrable-sphere model and a certain sphere distribution in which the minimum distance between sphere centers is greater than or equal to 2R. An approximate expression obtained for the low-density expansion of σ* for dispersions of penetrable spheres always lies between the derived bounds on σ*. The study demonstrates that the effect of connectivity of the inclusion phase on σ*, through second order in φ₂, can be substantial relative to the conductivity of dispersions of spheres characterized by a pair-connectedness function that is zero for all sphere separations.