Percolative conduction in two-component strongly nonlinear composites

Abstract

The effective-medium approximation (EMA) is developed to study the bulk effective response of a class of strongly nonlinear conducting composite media which obey a current-field ( J-E) relation of the form J=y~E~eE, where y is the nonlinear coefficient of the medium and p) 0. First, a variational approach is invoked for solving the electrostatic boundary-value problem of a spherical inclusion embedded in a host medium. Explicit asymptotic behaviors of the field distribution are derived at both large and small ratios between the coefficients of the constituents. Second, in an attempt to extend the approximation to the vicinity of a percolation threshold, we formulate the EMA for a two-component strongly nonlinear composite from the asymptotic behaviors of the field distribution. Scaling forms of the effective response are extracted and compared to a general scaling theory and an excellent agreement is found.

Path-integral approach to strongly nonlinear composites

“…1, we compare for d = 3 this exact expression for p c to numerical results [25,33,35]. It thus seems that the variational method used in [25,33] does not lead to the correct values of the electric field around the impurity. The fact that the percolation threshold depends on κ is the bad feature of this approximation.…”

“…The obtained formula is relatively simple and satisfies conditions (i) and (ii). We give the analytical expressions for numerically estimated quantities [25,33,35]. The only drawback of our result is that the percolation threshold depends on κ (in the same way as in [35]).…”

“…The second approach is in the Bruggeman spirit [32] and consists essentially in considering an impurity in an effective host. Bruggeman's theory was reformulated in order to apply to this problem [22] and it was further investigated by different authors [25,[33][34][35]. This approach predicts the effective medium values s = 1 and t = 1 + κ and reproduces the weak contrast expansion to second order.…”

“…In addition to allow calculations, this scheme ensures that the weak-contrast expansion will be recovered. The numerical study of this problem can be found in [25,33,35].…”

“…)] (the function f is defined up to a constant factor, and there is a global additional factor 1/(κ + 2) in our result). These four different coefficients were estimated numerically [25,33,35]. The percolation threshold is given by…”