Effective permittivity of mixtures of anisotropic particles

TL;DR

This paper introduces a new theoretical approach to derive dielectric mixing rules for mixtures of anisotropic, inhomogeneous particles, accounting for shape and dielectric anisotropy, and provides explicit formulas for effective permittivity.

Contribution

A novel method based on macroscopic compact groups to derive dielectric mixing rules for anisotropic particle mixtures, including explicit formulas and special cases like ellipsoidal particles.

Findings

Derived Lorentz-Lorenz type formulas for effective permittivity.

Explicit integral expressions for particle polarizabilities.

Extended to Bruggeman-type formulas and discussed validity ranges.

Abstract

We use a new approach to derive dielectric mixing rules for macroscopically homogeneous and isotropic multicomponent mixtures of anisotropic inhomogeneous dielectric particles. Two factors of anisotropy are taken into account, the shape of the particles and anisotropy of the dielectric parameters of the particles' substances. Our approach is based upon the notion of macroscopic compact groups of particles and the procedure of averaging of the fields over volumes much greater than the typical scales of these groups. It enables us to effectively sum up the contributions from multiple interparticle reemission and short-range correlation effects, represented by all terms in the infinite iterative series for the electric field strength and induction. The expression for the effective permittivity can be given the form of the Lorentz-Lorenz type, which allows us to determine the effective polarizabilities of the particles in the mixture. These polarizabilities are found as integrals over the regions occupied by the particles and taken of explicit functions of the principal components of the permittivity tensors of the particles' substances and the permittivity of the host medium. The case of a mixture of particles of the ellipsoidal shape is considered in detail to exemplify the use of general formulas. As another example, Bruggeman-type formulas are derived under pertinent model assumptions. The ranges of validity of the results obtained are discussed as well.