The coherent potential approximation is a realizable effective medium scheme

The effective electrical conductivity of an aggregate, composed of grains of various conductivities, is frequently estimated by the coherent potential approximation, which embodies a local effective medium concept. It is proved rigorously that this approximation is exact for a wide class of hierarchical model composites made of spherical grains: the starting material 0 in the hierarchy is chosen arbitrarily, otherwise, materialj=1, 2, ... consists of equisized spheres, sayj-spheres, of arbitrary conductivities embedded in materialj — 1. The spatial distribution of thej-spheres must satisfy a mild homogeneity condition and their radiusr _j must, asymptotically, increase faster than exponentially withj. Furthermore, the minimum spacing, 2s _j , between thej-spheres is such that the ratios _j /r _j diverges. On the basis of these and some further ancillary conditions it is established that the coherent potential approximation becomes asymptotically exact for the effective conductivity of materialj. The results extend to other effective parameters of the composites, including the thermal conductivity, dielectric constant and magnetic permeability. In addition, the model composites and the proof of realizability may be generalized to allow non-spherical grains.