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

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Abstract
Expressions for the two microstructural parameters that appear in the variational third‐order bounds [G. W. Milton, Phys. Rev. Lett. 4 6, 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.