Spectral densities of embedded interfaces in composite materials

The effective resistivity and conductivity of two media that meet at a randomly rough interface are computed in the quasistatic limit. The results are presented in the spectral density representations of the Bergman-Milton formulation for the properties of two-component composite materials. The spectral densities are extracted from computer simulations of resistor networks in which the random interface separates two regions containing different types of resistors. In the limit that the bond lengths in the resistor network are small compared to parameters characterizing the surface roughness, the resistor network simulation approximates the continuum limit of the two-component composite. The Bergman-Milton formulation is used to obtain a set of exact sum rules in the continuum limit for the spectral densities in terms of parameters describing the surface roughness and the simulation results are found to agree with these limiting forms. Perturbation theory results of the composite in the continuum limit for weakly rough random interfaces are also presented. An expansion of the spectral density is determined to second order in the surface profile function of the random interface and compared with the Bergman-Milton sum rules and computer simulation results. The formalism is applied to surface plasmons, electron energy loss, and light scattering from rough surfaces. Layered structures are discussed briefly.