The effective conductivity of a periodic array of spheres

Abstract

We consider the problem of determining the effective conductivity k * of a composite material consisting of equal-sized spheres of conductivity α arranged in a cubic array within a homogeneous matrix of unit conduc­tivity. We modify Zuzovski & Brenner’s (1977) method and thereby obtain a set of infinite linear equations for the coefficients of the formal solution equivalent to that derived by McKenzie et al . (1978) using a method originally devised by Rayleigh (1892). On solving these equations we derive expressions for k * to O ( c 9 ) ─ c being the volume fraction of the spheres ─ for simple, body-centred and face-centred cubic arrays, and also obtain numerical values for k * over the whole range of α and c . We show that these results for cubic arrays can be used to estimate k * for random arrays of identical spheres. For arrays of highly conducting and nearly touching spheres, Batchelor & O’Brien (1977) showed that k * ∼ {─ K 1 In (1 ─ χ ) ─ K 2 ( α = ∞, χ = ( c / c max ) ⅓ → 1), 2 K 1 In α ─ K ʹ 2 ( χ = 1, α → ∞), where c max corresponds to the volume fraction when the spheres are actually touching each other, and determined K 1 for the three cubic arrays. Our numerical results are consistent with the above asymptotic expressions except for the fact that the numerical values for the constants K 2 and Kʹ 2 thereby obtained do not quite satisfy the relation Kʹ 2 = K 1 (3.9 ─ In 2) + K 2 given by Batchelor & O’Brien. We have been unable to find the reason for this slight discrepancy.