The bulk effective response of non-linear random resistor networks: Numerical study and analytic approximations

The effective response of nonlinear random resistor networks consisting of two different types of resistor is studied numerically and compared with some simple approximations. One type of resistor is assumed to be ohmic, while the other is assumed to have a non-linear I-V response of the form i= chi ^{upsilon beta +1}. The effective response is calculated by solving Kirchhoff's equations for the voltages at each node of the network. Numerical results, for beta =2 and beta =4, are compared to theoretical predictions of a recently derived Clausius-Mossotti approximation for such networks. The Clausius-Mossotti results are found to provide a good description of the results of the simulations in cases of low contrast between the two components or a small fraction of the non-linear component in the network. It is also found that the range of validity of this non-linear Clausius-Mossotti approximation is larger than that of the classical Clausius-Mossotti approximation for linear two-component random resistor networks. An extension of Bruggeman's effective-medium approximation to this case is found to give somewhat better agreement with the numerical results.