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=σ₁E+χ₁{}E²E while the second component, of fraction q, is linear with J=σ₂E. Near the percolation threshold (pc or qc), we examine the conductor-insulator (C/I) limit (σ₂=0) and the superconductor-conductor (S/C) limit (σ₂={∞}). For the C/I limit and p>pc, the effective linear- and nonlinear-response functions behave as σₑ{}(p-pc{)}ᵗ$ and ${{{χ}}}ₑ(p-{p}c)₂ᵗ, respectively. For the S/C limit and qqc, σₑ and χₑ are found to diverge as σₑ{}(qc-q)^-s and χₑ{}(qc-q)₂^-s. Explicit calculations are done in two dimensions and generalized to d dimensions. The exponents are found to be s=t=1 and s₂=t₂=2; pc=1/d and qc=(d-1)/d within EMA. For a finite-conductivity ratio h and at percolation, σₑ and χₑ are found to cross over from the fractal (h=0) to homogeneous (h=1) behavior. In the limits of small h and p-pc (or qc-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.