Effective-medium theory for weakly nonlinear composites

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 $\mathbf{D}=\epsilon \mathbf{E}+\chi {\left|\mathbf{E}\right|}^2\mathbf{E}$, where $\epsilon$ and $\chi$ are position dependent. In a two-phase composite, to first order in the nonlinear coefficients ${\chi }_1$ and ${\chi }_2$, the effective nonlinear dielectric susceptibility is found to be ${\chi }_e=\Sigma {i=1,2\mathrm{}}^{}\left(\frac{{\chi }_i}{p_i}\right){\left(\frac{\partial {\epsilon }_e}{\partial {\epsilon }_i}\right)}_0{\left|\frac{\partial {\epsilon }_e}{\partial {\epsilon }_i}\right|}_0$, where ${\epsilon }_e^{\mathrm{}\left(0\right)\mathrm{}}$ is the effective dielectric constant in the linear limit (${\chi }_i=0,i=1,2\mathrm{}$) and ${\epsilon }_i$ and $p_i$ are the dielectric function and volume fraction of the ith component. The approximation is applied to a calculation of ${\chi }_e$ 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 ${\chi }_e$ is carried out for the Hashin-Shtrikman microgeometry and compared to our MG approximation.