Finite-element modeling method for the prediction of the complex effective permittivity of two-phase random statistically isotropic heterostructures

This article is devoted to the study of the complex permittivity of two-dimensional two-phase statistically isotropic heterostructures on a small scale such that the quasistatic limit is applicable. Even though several analytical approximation techniques have been developed in the past, today it is desirable to be able to simulate these media via computer, which necessitates the development of efficient numerical techniques for the solution of the resulting equations. The simulation data concern the effective permittivity of continuum composites consisting of distributions of hard disks of a dielectric phase randomly dispersed in a continuous matrix of another dielectric phase. The three-dimensional equivalent of this model would be a composite with cylindrical symmetry, i.e., all interfaces are parallel to a fixed direction. The two constituents are assumed to be isotropic and homogeneous materials with scalar permittivities. Ab initio calculations are accomplished self-consistently with a computer code. The distribution of monodisperse inclusions is equilibrated by the Monte Carlo method and the dielectric study was carried out using the finite element method. Results are first presented documenting the effects of the surface fraction of the disks and the permittivity contrast between the two phases on the complex effective permittivity of the composite material. The numerical results are then compared with available effective medium approaches and bounds methods. The effective complex permittivity is found to lie within the four-point bounds in the complex plane and between the curves corresponding to Maxwell Garnett and asymmetric Bruggeman formulas. The scaling analysis reported here highlights a number of complexities not previously noted for this system. The exponents governing the critical behavior of the real and imaginary parts of the complex permittivity can be quite different from those characterizing the continuum percolation phenomena in a statistically inhomogeneous system of partially penetrable disks. (C) 2005 American Institute of Physics.