Asymptotic behaviour of the effective dielectric constants of composite materials

Asymptotic formulae are derived for the effective dielectric constants of composites containing densely packed spherical inclusions with dielectric constants differing greatly from that of the background. The asymptotic results are obtained by treating the influence of nearest-neighbouring inclusions analytically. All the results obtained are in agreement with results obtained using other methods and in some cases provide an additional term not given before. Uniform asymptotic results are obtained as a function of both the spacing of the spheres and their dielectric constants simultaneously, whereas previous analyses treat these separately. Results are obtained for a pair of spheres, for an infinite chain of spheres and for a lattice of spheres having the sodium chloride structure. Results are also presented for some Bravais lattices and a large class of structures having specified nearest-neighbour configurations. The results may also be of use in studying percolating, structures.