Percolation effects in two-component strongly nonlinear composites: Universal scaling behavior

We have studied the effective nonlinear response of a class of strongly nonlinear conducting composite media which obey a current-field (J-E) relation of the form J={χ}{}E^2βE, where {χ} is the nonlinear coefficient and {β}>0. We consider a d-dimensional, two-component strongly nonlinear composite in which a volume fraction p of nonlinear material of coefficient χᵢ is randomly embedded in a host medium of volume fraction q of χₘ (p+q=1). We examine the normal conductor-insulator (N/I) composite (χₘ=0) and the superconductor--normal conductor (S/N) composite (χₘ={∞}). Near the percolation threshold (pc or qc), the effective nonlinear response behaves as χₑ{}(p-pc{)}ᵗ$ for the N/I limit while ${{{χ}}}ₑ({q}c$-q${)}^{{{-}}s}$ for the S/N limit. We summarize various estimates of the critical exponents t({β}) and s({β}) and discuss a more general duality relation in two dimensions. For a finite χₘ/χᵢ, we propose various scaling relations for χₑ. The effective nonlinear response is calculated in the effective-medium approximation and numerical simulations. The scaling functions are extracted and the exponents are determined and compared to proposed scaling forms; excellent agreement is found.