Dielectric constant of two-component, two-dimensional mixtures in terms of Bergman–Milton simple poles

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Abstract
According to the Bergman–Milton theory [D. J. Bergman, Phys. Rep. 43, (No. 9), 377 (1978), and G. W. Milton, J. Appl. Phys. 52, 5286 (1981)], the effective dielectric constant or conductivity of a two-component mixture is a function of the ratio of the dielectric constants (or conductivities) of those components. The function has simple poles only at some negative values of the ratio, and positions of the poles and residues are dependent only on the microgeometry of the mixture. In this study, the location of the poles and the values of the residues are found for two-component, two-dimensional composite materials with periodic, simple-square lattice structures using the Fourier series expansion technique. With the locations of the poles and residues determined, the effective dielectric constants or conductivities of two-component composite materials may be expressed in practical forms and may be predicted for composite mixtures even when the dielectric constants of the component materials are complex.