Tight Bounds on the Effective Complex Permittivity of Isotropic Composites and Related Problems

TL;DR

This paper refines the bounds on the complex permittivity of isotropic composites, demonstrating near-optimal limits using hierarchical laminates and variational methods, with implications for absorption and polarizability constraints.

Contribution

It introduces a class of hierarchical laminates and variational bounds that improve existing limits on effective permittivity in isotropic composites.

Findings

Hierarchical laminates nearly achieve the circular arc bounds.

Variational methods yield tighter bounds for the second arc.

Optimal bounds correspond to assemblages of coated spheres.

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 certain class of hierarchical laminates. In regard to the second arc, we show that a tighter bound can be derived using variational methods. This tighter 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.