Interchangeability and bounds on the effective conductivity of the square lattice

The effective conductivity σ* of an infinitely interchangeable two-component random medium is considered. This class of media includes cell materials in the continuum and the bond lattice on ℤ^d, where the cells or bonds are randomly assigned the conductivities σ_1 and σ_2 (σ_1, σ_2 ne0) with probabilities p_1 and p_2 = 1−p_1. A rigorous basis for the very old and widely used low volume fraction expansion of σ* is established, by proving that σ* is an analytic function of p_2 in a suitable domain containing [0, 1]. In the case of the bond lattice in d = 2, rigorous fourth-order upper and lower bounds on σ* valid for all p_2, σ_1, and σ_2 are derived. The four perturbation coefficients entering into the bounds are obtained from the first-order volume fraction coefficient using the method of infinite interchangeability.