On applicability of differential mixing rules for statistically homogeneous and isotropic dispersions

TL;DR

This paper critically examines the applicability of differential mixing rules, especially the asymmetrical Bruggeman model, for homogeneous and isotropic dielectric dispersions, revealing physical inconsistencies and limitations in their use.

Contribution

The study demonstrates the inconsistency of the ABM and proposes generalized mixing rules, clarifying their valid concentration ranges using the compact group approach.

Findings

ABM is physically inconsistent for certain systems.

Generalized mixing rules depend on system structure and concentration.

Validity of mixing rules is limited by Hashin-Shtrikman bounds.

Abstract

The classical differential mixing rules are assumed to be independent effective-medium approaches, applicable to certain classes of systems. In the present work, the inconsistency of differential models for macroscopically homogeneous and isotropic systems is illustrated with a model for the effective permittivity of simple dielectric systems of impenetrable balls. The analysis is carried out in terms of the compact group approach reformulated in a way that allows one to analyze the role of different contributions to the permittivity distribution in the system. It is shown that the asymmetrical Bruggeman model (ABM) is physically inconsistent since the electromagnetic interaction between previously added constituents and those being added is replaced by the interaction of the latter with recursively formed effective medium. The overall changes in the effective permittivity due to addition of one constituent include the contributions from both constituents and depend on the system structure before the addition. Ignoring the contribution from one of the constituents, we obtain generalized versions of the original ABM mixing rules. They still remain applicable only in a certain concentration ranges, as is shown with the Hashin-Shtrikman bounds. The results obtained can be generalized to macroscopically homogeneous and isotropic systems with complex permittivities of constituents.