Effective electrical conductivity of two-phase disordered composite media

TLDR

The relationships between the conductivities of certain models that are exactly realized by some of these bounds and the perturbation expansions are discussed. The study derives a new expression for the conductivity of a broad class of three‑dimensional dispersions of inclusions. Perturbation expansions of the effective electrical conductivity σe of any two‑phase isotropic composite medium of arbitrary dimensionality d (where d=2,3) are derived. Certain Padé approximants of a particular series representation of σe yield known rigorous bounds, and the derived formula accurately predicts σe for both periodic and random arrays of impenetrable spheres across a wide range of phase conductivities and inclusion volume fractions.

Abstract

Perturbation expansions of the effective electrical conductivity σe of any two-phase isotropic composite medium of arbitrary dimensionality d (where d=2,3) are derived. It is shown that certain Padé approximants of a particular series representation of σe yield known rigorous bounds on the conductivity of the composite. The relationships between the conductivities of certain models that are exactly realized by some of these bounds and the perturbation expansions are discussed. A new expression for the conductivity of a broad class of three-dimensional dispersions of inclusions is derived. The formula for σe, which depends upon, among other quantities, a certain three-point probability function of the composite medium, is shown to accurately predict σe of both periodic and random arrays of impenetrable spheres, for a wide range of phase conductivities and inclusion volume fractions.