2D vs 3D clustering of the elliptic particulates: The correlation with the percolation thresholds

Abstract

We develop a continuum percolation procedure for the aggregation of the elliptic fillers in the 2 and 3-dimensional media. Given random distributions for the locus and rotations of the elements with a specified original density p, each medium achieves chains of elements through overlapping to achieve a connection density of ρ. In this regard, typically 3D aggregation is more efficient than 2D due to the possibility of additional connectivity from the in/out (i.e. depth) directions. Hence, when increasing the number of fillers the 3D percolation system experiences an early increase in the connection density ρ, which typically occurs in the neighborhood of the percolation threshold p_c. We initially develop a new iterative method to compute the percolation threshold p_c in finite systems. Subsequently, we show that such early divergence between 2D-3D percolation systems is followed by a later convergence stage, as the number of fillers progressively increases. Consequently, we show, conceptually and computationally, that the maximum 2D-3D difference in the connections density Δ_ρ_max correlates directly with the respective 2D-3D difference in the percolation thresholds Δ_p_c , where a large pool of computational samples were generated by varying the aspect ratio as well as the relative scale of the particles. The results and respective analyses could be useful for the design of binary composite membranes of a specified thickness (i.e. thin → 2D, thick → 3D) for achieving the desired homogenized physical property.

Group Members

William A. Goddard III

Charles and Mary Ferkel Professor