Analytic framework for the modeling of effective media

Synthetic materials in which new electromagnetic properties are obtained from the combination of two or more materials have been of theoretical and practical interest for nearly a century. The ability to explain and predict the properties of these materials has traditionally relied on combining physicomathematical models of the effective environment seen by the various constituents of the mixture with some assumptions about the way these microscopic properties should translate into macroscopic homogeneous parameters. Thus, even in the simplest case of the binary mixture, with every new set of assumptions, a new effective medium theory (EMT) results, and, with each new theory, stronger claims of correctness and applicability are made. This issue of correctness becomes critical when the properties of one of the constituents is unknown a priori and the claim is made that by inverting a fit of experimental results to the EMT model those properties can be ascertained. For this inverse procedure to be possible, the EMT theory should not only be correct, it should be unique in the analytic sense. In this article, a generalized framework is developed through which the analytic properties of all binary mixture EMTs can be deduced and compared. In the process it is shown that in the complex plane of the variable u = i/(∈_eff-1), it is straightforward to separate the morphology dependent properties of the EMT from its dependence on the susceptibilities of the components. The frequency dependence of the EMT model as a function of the arbitrary complex properties of the filler is easily summarized as a compact sum of the poles of a complex function. This process is demonstrated for a number of common EMTs.