Renormalized effective-medium theory for weakly nonlinear composites.

Abstract

The effective response is calculated in random nonlinear conductance networks consisting of two kinds of conductors obeying a current-voltage ($I\ensuremath{-}V$) characteristic of the form $I={\ensuremath{\sigma}}_{i}V+{\ensuremath{\chi}}_{i}{V}^{3}$, ($i=1, 2$). The effective linear response ${\ensuremath{\sigma}}_{e}$ and nonlinear response ${\ensuremath{\chi}}_{e}$ are calculated in a simple effective-medium approximation (EMA); the results are compared with numerical simulations in nonlinear conductance networks. It is found that while the simple EMA fits the simulation data of ${\ensuremath{\sigma}}_{e}$ reasonably well over the entire range of volume fraction, strong deviations are obtained for ${\ensuremath{\chi}}_{e}$. In an attempt to improve the EMA, we invoke a real-space renormalization-group (RG) analysis to rescale the network by small-cell transformations and bring it farther away from percolation, where EMA is applicable. The renormalized EMA (REMA) formalism, which couples RG and EMA, is able to extend the range of validity of EMA substantially. Results from the REMA are compared with numerical simulations. We find a remarkable improvement over the simple EMA in the entire range of volume fraction, especially at high contrast between the components.