Effective electrical conductivity of two-phase disordered composite media

Perturbation expansions of the effective electrical conductivity σ_e 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 σ_e 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 σ_e, which depends upon, among other quantities, a certain three-point probability function of the composite medium, is shown to accurately predict σ_e of both periodic and random arrays of impenetrable spheres, for a wide range of phase conductivities and inclusion volume fractions.