Bounds on the effective properties of polydispersed suspensions of spheres: An evaluation of two relevant morphological parameters

Expressions for the two microstructural parameters that appear in the variational third-order bounds [G. W. Milton, Phys. Rev. Lett. 46, 542 (1981)] for the effective conductivity and elastic moduli of composite media are derived analytically to first order in the sphere concentration c for random well-mixed dispersions of impenetrable spheres with an arbitrary size distribution. These relations lead to rigorous bounds on the effective properties which are exactly valid to order c² for such models. The apparent linear behavior of the microstructural parameters up to moderately high c enables one to apply the bounds beyond second-order in c, however. Employing these results, the effect of polydispersivity on the effective properties is examined. It is worth noting that, under some conditions, polydispersivity can actually lead to a slight decrease of the shear modulus, whereas, for highly conducting particles, polydispersivity always increases the effective conductivity.