Transport properties of a three-phase composite material: the square array of coated cylinders

Abstract

The analytic properties of the effective dielectric constant of a class of three-phase composite materials are studied. Specifically, we investigate the effective dielectric constant of a periodic array of coated cylinders, as a function of the core dielectric constant ($\epsilon _{\text{c}}$) and the shell dielectric constant ($\epsilon _{\text{s}}$), while keeping the matrix dielectric constant ($\epsilon _{\text{b}}$) fixed. We show that when $\epsilon _{\text{s}}=-\epsilon _{\text{c}}$, the composite has exactly the same effective dielectric constant as a periodic array of solid cylinders with dielectric constant $\epsilon _{\text{c}}$ and radius equal to the outer radius of the original coated cylinder. We also show that when $\epsilon _{\text{s}}$ = -1, the composite has exactly the same effective dielectric constant as a periodic array of solid cylinders with dielectric constant $\epsilon _{\text{c}}$, and radius exceeding the shell radius. We explore the location of poles and zeros of the three-phase effective dielectric constant in the ($\epsilon _{\text{s}},\epsilon _{\text{c}}$) plane. The lines $\epsilon _{\text{s}}$ = -1 and $\epsilon _{\text{s}}+\epsilon _{\text{c}}$ = 0 are loci of essential singularities. We also comment on the behaviour of the effective dielectric constant in the neighbourhood of the two special points ($\epsilon _{\text{s}},\epsilon _{\text{c}}$) = (0,0) and ($\epsilon _{\text{s}},\epsilon _{\text{c}}$) = (-1, +1).