Effective conductivity of hard-sphere dispersions

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Abstract
Three-point bounds on the effective conductivity σe of isotropic two-phase composites, that improve upon the well-known two-point Hashin–Shtrikman bounds [J. Appl. Phys. 23, 779 (1962)], depend upon a key microstructural parameter ζ2. A highly accurate approximation for σe developed by Torquato [J. Appl. Phys. 58, 3790 (1985)] also depends upon ζ2. This paper reports a new and accurate algorithm to compute the three-point parameter ζ2 for dispersions of hard spheres by Monte Carlo simulation. Data are reported up to values of the sphere volume fraction φ2 near random close-packing and are used to assess the accuracy of previous analytical calculations of ζ2. A major finding is that the exact expansion of ζ2 through second order in φ2 provides excellent agreement with the simulation data for the range 0≤φ2 ≤0.5, i.e., this low-volume-fraction expansion is virtually exact, even in the high-density region. For φ2 >0.5, this simple quadratic formula is still more accurate than other more sophisticated calculations of ζ2. The linear term of the quadratic formula is the dominant one. Using our simulation data for ζ2, we compute three-point bounds on the conductivity σe and Torquato’s approximation for σe .