Effective conductivity of suspensions of overlapping spheres

An accurate first-passage simulation technique formulated by the authors [J. Appl. Phys. 68, 3892 (1990)] is employed to compute the effective conductivity σ_e of distributions of penetrable (or overlapping) spheres of conductivity σ₂ in a matrix of conductivity σ₁. Clustering of particles in this model results in a generally intricate topology for virtually the entire range of sphere volume fractions φ₂ (i.e., 0≤φ₂≤1). Results for the effective conductivity σ_e are presented for several values of the conductivity ratio α=σ₂/σ₁, including superconducting spheres (α=∞) and perfectly insulating spheres (α=0), and for a wide range of volume fractions. The data are shown to lie between rigorous three-point bounds on σ_e for the same model. Consistent with the general observations of Torquato [J. Appl. Phys. 58, 3790 (1985)] regarding the utility of rigorous bounds, one of the bounds provides a good estimate of the effective conductivity, even in the extreme contrast cases (α≫1 or α≅0), depending upon whether the system is below or above the percolation threshold.