Complex effective permittivity of a lossy composite material

Abstract

In recent work, boundary integral equations and finite elements were used to study the (real) effective permittivity for two-component dense composite materials in the quasistatic limit. In the present work, this approach is extended to investigate in detail the role of losses. We consider the special but important case of the axisymmetric configuration consisting of infinite circular cylinders (assumed to be parallel to the z axis and of permittivity ε1) organized into a crystalline arrangement (simple square lattice) within a homogeneous background medium of permittivity ε2=1. The intersections of the cylinders with the x–y plane form a periodic two-dimensional structure. We carried out simulations taking ε1=3−0.03i or ε1=30−0.3i and ε2=1. The concentration dependence of the loss tangent of the composite material can be fitted very well, at low and intermediate concentrations of inhomogeneities, with a power law. In the case at hand, it is found that the exponent parameter depends significantly on the ratio of the real part of the permittivity of the components. We argue, moreover, that the numerical results discussed here are consistent with the Bergman and Milton theory [D. J. Bergman, Phys. Rep. 43, 377 (1978) and G. W. Milton, J. Appl. Phys. 52, 5286 (1981)].