Effective complex permittivity of two-phase random composite media:A test of the two exponent phenomenological percolation equation

The nature of percolation in continuum inhomogeneous media is a current topic of debate. In this work, Monte Carlo and finite element simulations of the effective complex permittivity, ε = ε ′ − i ε ″ , of two-phase random composite media are analyzed by using the two exponent phenomenological percolation equation (TEPPE) by McLachlan [ J. Am. Ceram. Soc. 73 , 2187 ( 1990 ) ; Phys. Rev. B 56 , 1236 ( 1987 ) ; Phys. Rev. B 58 , 14880 ( 1998 ) ; Phys. Rev. B 58 , 13558 ( 1998 ) ; Phys. Rev. B 67 , 024206 ( 2003 )] . The continuum-percolation system consists of two-dimensional equilibrium distributions of randomly distributed monodisperse circular and partially penetrable disks (or parallel, infinitely long, identical, partially penetrable circular cylinders) throughout a host matrix. The study is performed on a set of calculations, covering wide ranges of various parameters, including the intrinsic constituent permittivity, the surface fraction, and the degree of impenetrability. In our analysis, we first determine the parameters that characterize the critical behavior at the percolation threshold. Our data suggest that the TEPPE does not fit the simulation data well over the entire range of surface fraction and whatever is the degree of impenetrability considered. This is attributed, in part, to the fact that the effective medium approximation (restricted to dipolar interactions only) explicitly ignores the local-field fluctuations. Moreover, the mixtures exhibit clustering in equilibrium, which is not conceptually incorporated in the TEPPE, i.e., the inclusions form a cluster with a percolating spongelike topology accompanied by a strongly dependent shape of the radial distribution function on the degree of impenetrability. It is argued that further efforts are still needed to fully grasp the numerically (and experimentally) observed features of the effective properties of dielectric heterostructures.