Improved bounds on the effective conductivity of high-contrast suspensions

Conventional upper and lower bounds on the effective conductivity σ_e of two-phase composite media diverge from one another in the infinite-contrast limits (α=∞ or 0). We have derived a generally nontrivial upper bound on σ_e for suspensions of identical spheres when the spheres are superconducting, i.e., the upper bound does not necessarily become infinite in the limit α→∞. Similarly, a generally nontrivial lower bound on σ_e is derived for the aforementioned suspension when the spheres are perfect insulators, i.e., the lower bound does not necessarily vanish in the limit α→0. The bounds are computed for two models: simple cubic arrays and random arrays of spheres.