Variational Approach to Percolation Threshold of Nanocomposites Considering Clustering Effect

Abstract

To mathematically model the phenomenon of complete dispersion, a crucial step is taken in a recent paper on isotropic formulation of the homogenized Eshelby’s tensor, leading to derivation of the ellipsoidal bound. In this paper, the writers show that in the same variational framework, the effect of clustering can be mathematically modeled as anisotropy of the homogenized Eshelby’s tensor. The degree of clustering or dispersion is accordingly represented by the aspect ratio of the ellipsoidal shape associated with the anisotropic Eshelby’s tensor. The asymptotic results and calculation demonstrate that such a new anisotropic formulation consistently leads to an increase of the percolation threshold due to the clustering effect.