Duality and Effective Conductivity of Two-dimensional Two-phase Systems

TL;DR

This paper explores the effective electrical conductivity of two-dimensional two-phase systems, introducing a new duality-based functional equation, a hierarchical modeling approach, and mean-field solutions that account for inhomogeneity and percolation effects.

Contribution

It presents a novel functional equation generalizing duality relations and a hierarchical model for calculating effective conductivity in inhomogeneous systems.

Findings

Derived formulas satisfy necessary inequalities and symmetries.

Formulas reproduce known results in weakly inhomogeneous cases.

Effective conductivity may depend on specific structural details of the inhomogeneities.

Abstract

The possible functional forms of the effective conductivity sigma_{eff} of the randomly inhomogeneous two-phase system at arbitrary values of concentrations are discussed. A new functional equation, generalizing the duality relation, is deduced for systems with a finite maximal characteristical scale of the inhomogeneties and its solution is found. A hierarchical method of the construction of the model random inhomogeneous medium is proposed and one such simple model is constructed. Its effective conductivity at arbitrary phase concentrations is found in mean field like approximation. The derived formulas for the effective conductivity are different and also (1) satisfy all necessary inequalities and symmetries, including a dual symmetry; (2) reproduce the known formulas for sigma_{eff} in weakly inhomogeneous case. It means that in general sigma_{eff} of the two-phase randomly inhomogeneous systems may be a nonuniversal function and can depend on some details of the structure of the randomly inhomogeneous regions. The percolation limit is briefly discussed.