The effective conductivity of a periodic array of spheres

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(c9) ─ 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* ∼ {─ K1 In (1 ─ χ) ─ K2 (α = ∞, χ = (c/cmax)⅓ → 1), 2K1 In α ─ Kʹ2 (χ = 1, α → ∞), where cmax corresponds to the volume fraction when the spheres are actually touching each other, and determined K1 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 K2 and Kʹ2 thereby obtained do not quite satisfy the relation Kʹ2 = K1(3.9 ─ In 2) + K2 given by Batchelor & O’Brien. We have been unable to find the reason for this slight discrepancy.

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