Duality and Effective Conductivity of Random Two-Phase Flat Systems

TL;DR

This paper explores the effective electrical conductivity of random two-phase systems, proposing new approximate formulas based on duality relations and series expansions, which differ from traditional models and suggest nonuniversality.

Contribution

It introduces two explicit approximate expressions for effective conductivity using duality and series expansion, expanding understanding beyond the effective medium approximation.

Findings

Proposed formulas satisfy all necessary physical requirements.

Formulas reproduce known results in weak inhomogeneity limit.

Effective conductivity may depend on specific inhomogeneity structures.

Abstract

The possible functional forms of the effective conductivity sigma_e of the randomly inhomogeneous two-phase systems at arbitrary values of concentrations are discussed. Two explicit approximate expressions for effective conductivity are found using a duality relation, a series expansion of sigma_e in the inhomogeneity parameter z and some additional conjectures about functional form of sigma_e. They differ from the effective medium approximation, satisfy all necessary requirements and reproduce the known formulas for sigma_e in weakly inhomogeneous case. This can signify also that sigma_e of the two-phase randomly inhomogeneous systems may be a nonuniversal function, depending on some details of the structure of the random inhomogeneities.