## Abstract

We consider the problem of determining the effective conductivity k$^*$ of a composite material consisting of equal-sized spheres of conductivity $\alpha$ 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 $k^{\ast}$ 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 $\alpha$ and $c$. We show that these results for cubic arrays can be used to estimate $^{\ast}$ for random arrays of identical spheres. For arrays of highly conducting and nearly touching spheres, Batchelor & O'Brien (1977) showed that \begin{equation*} k^{\ast} \sim \begin{cases} -K_1\ln (1-\chi)-K_2 (\alpha = \infty, \chi=(c/c_{\max})^{\frac{1}{3}}\rightarrow 1), \\ 2K_1\ln \alpha-K'_2\quad (\chi = 1, \alpha \rightarrow \infty),\end{cases} \end{equation*} 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 $K'_2$ thereby obtained do not quite satisfy the relation $K'_2 = K_1(3.9 - \ln 2) + K_2$ given by Batchelor & O'Brien. We have been unable to find the reason for this slight discrepancy.

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