Effective-medium theory for two-component nonlinear composites

The effective-medium approximation (EMA) is employed to study the effective nonlinear response of a two-component composite. The first component, of fraction p, is nonlinear and obeys a current-field (J-E) characteristic of the form J=${\mathrm{\sigma }}_1$ E+${\mathrm{\chi }}_1$‖E ${\mathrm{\Vert }}^2$ E while the second component, of fraction q, is linear with J=${\mathrm{\sigma }}_2$ E. Near the percolation threshold (${\mathit{p}}_{\mathit{c}}$ or ${\mathit{q}}_{\mathit{c}}$), we examine the conductor-insulator (C/I) limit (${\mathrm{\sigma }}_2$=0) and the superconductor-conductor (S/C) limit (${\mathrm{\sigma }}_2$=∞). For the C/I limit and p>${\mathit{p}}_{\mathit{c}}$, the effective linear- and nonlinear-response functions behave as ${\mathrm{\sigma }}_{\mathit{e}}$≊(p-${\mathit{p}}_{\mathit{c}}$ ${)}^{\mathit{t}}$ and ${\mathrm{\chi }}_{\mathit{e}}$≊(p-${\mathit{p}}_{\mathit{c}}$ ${)}_2^{\mathit{t}}$, respectively. For the S/C limit and q<${\mathit{q}}_{\mathit{c}}$, ${\mathrm{\sigma }}_{\mathit{e}}$ and ${\mathrm{\chi }}_{\mathit{e}}$ are found to diverge as ${\mathrm{\sigma }}_{\mathit{e}}$≊(${\mathit{q}}_{\mathit{c}}$-q${)}^{\mathrm{-}\mathit{s}}$ and ${\mathrm{\chi }}_{\mathit{e}}$≊(${\mathit{q}}_{\mathit{c}}$-q${)}_2^{\mathrm{-}\mathit{s}}$. Explicit calculations are done in two dimensions and generalized to d dimensions. The exponents are found to be s=t=1 and ${\mathit{s}}_2$=${\mathit{t}}_2$=2; ${\mathit{p}}_{\mathit{c}}$=1/d and ${\mathit{q}}_{\mathit{c}}$=(d-1)/d within EMA. For a finite-conductivity ratio h and at percolation, ${\mathrm{\sigma }}_{\mathit{e}}$ and ${\mathrm{\chi }}_{\mathit{e}}$ are found to cross over from the fractal (h=0) to homogeneous (h=1) behavior. In the limits of small h and p-${\mathit{p}}_{\mathit{c}}$ (or ${\mathit{q}}_{\mathit{c}}$-q), the EMA results can be rescaled to collapse onto a universal curve. The scaling function is extracted and compared to a general scaling theory and an excellent agreement is found.