Statistical theory of effective electrical, thermal, and magnetic properties of random heterogeneous materials. IV. Effective−medium theory and cumulant expansion method

Abstract

Several perturbation solutions for the effective permittivity in a completely random medium are evaluated and the validity of the approximations is discussed. It is shown through a diagram technique that the effective−medium theory in a macroscopically inhomogeneous material is equivalent to the coherent−potential approximation in a discordered binary alloy not only in its physical concept but also in its mathematical structure. A cumulant expansion method which substantially takes into account the clusteringeffects is proposed while the effective−medium theory or the coherent−potential approximation is, by its nature, a single−site approximation and neglects the clusteringeffects. The numerical results obtained by the effective−medium theory and by the cumulant method for a binary mixture of a conducting material and an insulating material are compared with the computer simulation data on the effective conductivity of a three−dimensional random network. The solution of the cumulant method gives a remarkably good agreement with the computer simulation for the whole range of parameters. An important point is that the cumulant expansion theory holds excellently even near the critical percolation concentration where the clustering plays an essential role and where the effective−medium theory fails to work.