Analysis and Computation of Electromagnetic Transport in Composite Materials

University Of Utah, Salt Lake City UT

Investigators

Abstract

The interaction of an electromagnetic field with an inhomogeneous material is a pervasive problem arising in a broad array of applications. Often a wave interacts with a random or structured medium, and one is interested in the effective behavior as the wave propagates through or is localized by the medium. Moreover, systems as varied as photonic crystals and band gap structures, random resistor netorks, sea ice and other porous media, and electrorheological fluids exhibit critical behavior in their effective electromagnetic properties as some parameter is varied. The investigators conduct fundamental, mathematical studies of the interaction of electromagnetic fields with such composites, in conjunction with state of the art numerical experiments on model problems. In particular, a method based on spectral theory and analytic continuation has been developed to obtain rigorous bounds on the effective complex permittivity in the quasistatic limit. More recently, this approach has led to the finding that the effective transport properties of two-component media share the same analytic properties as the order parameters in statistical mechanics, such as the magnetization in an Ising model. These powerful relations have only been exploited in the static case and raise many important questions, yet the ideas of statistical mechanics form a natural framework for analyzing critical behavior of wave phenomena as well. Much of the analytic structure displayed in the static case carries over to the Helmholtz equation in a composite. This observation, as well as its implications for using statistical mechanics in this context, is investigated analytically and numerically. These investigations may yield fundamental advances in the mathematics, computation, and physics of electromagnetic fields and their interactions with composite systems. Graduate and undergraduate students are involved in key projects combining mathematical and computational analysis. In a broad range of problems across many disciplines, electromagnetic fields such as light, radar, or microwaves interact with inhomogeneous materials such as semiconductors, oil-filled rocks, bone or heart tissue, radar absorbing coatings, or shipping containers. Examples arise in physics, materials science, electrical and bio-engineering, chemistry, biology, geophysics, and astrophysics, and are central to communications and medical technologies. The investigators conduct fundamental mathematical studies of electromagnetic fields interacting with composite media, in conjunction with state of the art numerical experiments on model problems. In many important examples, the effective electromagnetic properties depend critically on some parameter in the system. For example, whether or not a wave can propagate through some types of structured media, called photonic crystals, depends critically on the ratio of the wavelength to the scale of the structure or other properties of the medium. Recently it was found that the mathematics underlying electrical transport in composites is almost identical to the mathematics underlying statistical mechanics. Statistical mechanics deals with phase transitions such as the freezing of water at the critical temperature of zero degrees Centigrade, and provides a natural framework for posing key questions about these systems. The investigators develop and apply the ideas of statistical mechanics to electromagnetic systems using mathematical analysis and sophisticated computations. They seek fundamental new insights, as well as novel ways of understanding and predicting how electromagnetic fields interact with composites. The project actively involves graduate and undergraduate students engaged in analytical and computational projects that are central to the overall goals. Results, particularly concerning critical properties, could potentially affect a broad range of application areas, including media structured on the nanometer scale.