On the Range of Complex Effective Permittivities of Isotropic Two-Phase Composites and Related Problems

Abstract

Almost four decades ago, Bergman and Milton independently showed that the isotropic effective electric permittivity of a two-phase composite material with a given volume fraction is constrained to lie within lens-shaped regions in the complex plane that are bounded by two circular arcs. An implication of particular significance is a set of limits to the maximum and minimum absorption of an isotropic composite material at a given frequency. Here, after giving a short summary of the underlying theory, we show that the bound corresponding to one of the circular arcs is at least almost optimal by introducing a new class of hierarchical laminates. In regard to the second arc, we show that a tighter bound can be derived using variational methods. This new bound is optimal as it corresponds to assemblages of doubly-coated spheres, which can be easily approximated by more realistic microstructures. We briefly discuss the implications for related problems including bounds on the complex polarizability.