Effective conductivity of random composites with shape-distributed near-spherical inclusions

While traditional and widely accepted mixing rules are based on the concept of a fixed inclusion shape, real particle composites are usually shape-distributed. For such composite materials, however, the analytical homogenization techniques are not well developed, while their numerical analysis becomes particularly complicated. We introduce and analyze two analytical models for the effective conductivity of a particular class of two-phase composites, namely those containing shape-distributed rigid inclusions. The models are derived with the use of the Bergman-Milton spectral density function approach. As we show, one of them, which involves the Gauss hypergeometric function, provides reasonable agreement with experimental data within the wide range of filling factors for ion-exchange resin beads in aqueous solutions. Another model, which is based on the uniform spectral density function, gives rise to a logarithmic singularity for the effective conductivity at the percolation threshold.