Effective-medium theory for weakly nonlinear composites

Abstract

We propose an approximate general method for calculating the effective dielectric function of a random composite in which there is a weakly nonlinear relation between electric displacement and electric field of the form $\mathbf{D}=\ensuremath{\epsilon}\mathbf{E}+\ensuremath{\chi}{|\mathbf{E}|}^{2}\mathbf{E}$, where $\ensuremath{\epsilon}$ and $\ensuremath{\chi}$ are position dependent. In a two-phase composite, to first order in the nonlinear coefficients ${\ensuremath{\chi}}_{1}$ and ${\ensuremath{\chi}}_{2}$, the effective nonlinear dielectric susceptibility is found to be ${\ensuremath{\chi}}_{e}=\ensuremath{\Sigma}{i=1,2}^{}(\frac{{\ensuremath{\chi}}_{i}}{{p}_{i}}){(\frac{\ensuremath{\partial}{\ensuremath{\epsilon}}_{e}}{\ensuremath{\partial}{\ensuremath{\epsilon}}_{i}})}_{0}{|\frac{\ensuremath{\partial}{\ensuremath{\epsilon}}_{e}}{\ensuremath{\partial}{\ensuremath{\epsilon}}_{i}}|}_{0}$, where ${\ensuremath{\epsilon}}_{e}^{(0)}$ is the effective dielectric constant in the linear limit (${\ensuremath{\chi}}_{i}=0,i=1,2$) and ${\ensuremath{\epsilon}}_{i}$ and ${p}_{i}$ are the dielectric function and volume fraction of the ith component. The approximation is applied to a calculation of ${\ensuremath{\chi}}_{e}$ in the Maxwell-Garnett approximation (MGA) and the effective-medium approximation. For low concentrations of nonlinear inclusions in a linear host medium, our MGA reduces to the results of Stroud and Hui. An exact calculation of ${\ensuremath{\chi}}_{e}$ is carried out for the Hashin-Shtrikman microgeometry and compared to our MG approximation.