Bounds on the effective thermal conductivity of a dispersion of fully penetrable spheres

We evaluate upper and lower bounds on the effective thermal conductivity K_e of a model of two-phase composite materials in which one of the phases consists of spherical inclusions (or voids) of conductivity K₂ and volume fraction φ₂, dispersed randomly throughout a matrix phase of conductivity K₁ and volume fraction φ₁. Our evaluations compare third-order bounds of Beran and of Brown, which utilize the three-point matrix probability function of the model, with bounds of De Vera and Strieder (which apply only to the aforementioned "fully penetrable sphere" model) and of Hashin and Shtrikman. The comparisons are made over extended ranges of values of both φ₂ and α = K₂ K₁ and reveal that the best bounds among those we have tested (generally those of Beran) are sharp enough to give quantitatively useful estimates of K_e for 0.1 ≤ K₂ K₁ ≤ 10 over a wide range of φ₂ values. They are sharp at high φ₂ values (i.e., φ₂ = 0.9) and very sharp at low φ₂ values (e.g. φ₂ = 0.1) where they remain useful for K₂ K₁ ≈ 100. They are less sharp at intermediate values (e.g. φ₂ = 0.5). As is well known, such results immediately translate into equivalent results for the electrical conductivity, dielectric constant, or magnetic permeability of composites.