On the effective conductivity of flat random two-phase models

TL;DR

This paper derives an approximate and exact solution for the effective conductivity of two-phase inhomogeneous systems, highlighting the influence of structural details and discussing the percolation limit.

Contribution

It introduces a hierarchical model and mean-field approximation to analyze effective conductivity, revealing nonuniversality and structural dependence.

Findings

Derived an approximate equation for sigma_eff with finite inhomogeneity scale.

Found an exact solution and discussed its physical meaning.

Showed sigma_eff can depend on structural details and is not universally fixed.

Abstract

An approximate equation for the effective conductivity sigma_eff of systems with a finite maximal scale of inhomogeneities is deduced. An exact solution of this equation is found and its physical meaning is discussed. A two-phase randomly inhomogeneous model is constructed by a hierarchical method and its effective conductivity at arbitrary phase concentrations is found in the mean-field-like approximation. These expressions satisfy all the necessary symmetries, reproduce the known formulas for sigma_eff in the weakly inhomogeneous case and coincide with two recently found partial solutions of the duality relation. It means that sigma_eff even of two-phase randomly inhomogeneous system may be a nonuniversal function and can depend on some details of the structure of the inhomogeneous regions. The percolation limit is briefly discussed.