Continuum percolation of polydisperse hyperspheres in infinite dimensions

Connectedness percolation of fractal liquids

We apply connectedness percolation theory to fractal liquids of hard particles, and make use of a Percus-Yevick liquid state theory combined with a geometric connectivity criterion. We find that in fractal dimensions the percolation threshold interpolates continuously between integer-dimensional values, and that it decreases monotonically with increasing (fractal) dimension. The influence of hard-core interactions is significant only for dimensions below three. Finally, our theory incorrectly suggests that a percolation threshold is absent below about two dimensions, which we attribute to the breakdown of the connectedness Percus-Yevick closure.

Role of the particle size polydispersity in the electrical conductivity of carbon nanotube-epoxy composites

Carbon nanotubes (CTNs) with large aspect-ratios are extensively used to establish electrical connectedness in polymer melts at very low CNT loadings. However, the CNT size polydispersity and the quality of the dispersion are still not fully understood factors that can substantially alter the desired characteristics of CNT nanocomposites. Here we demonstrate that the electrical conductivity of polydisperse CNT-epoxy composites with purposely-tailored distributions of the nanotube length L is a quasiuniversal function of the first moment of L. This finding challenges the current understanding that the conductivity depends upon higher moments of the CNT length. We explain the observed quasiuniversality by a combined effect between the particle size polydispersity and clustering. This mechanism can be exploited to achieve controlled tuning of the electrical transport in general CNT nanocomposites.

Random geometric graph description of connectedness percolation in rod systems

The problem of continuum percolation in dispersions of rods is reformulated in terms of weighted random geometric graphs. Nodes (or sites or vertices) in the graph represent spatial locations occupied by the centers of the rods. The probability that an edge (or link) connects any randomly selected pair of nodes depends upon the rod volume fraction as well as the distribution over their sizes and shapes, and also upon quantities that characterize their state of dispersion (such as the orientational distribution function). We employ the observation that contributions from closed loops of connected rods are negligible in the limit of large aspect ratios to obtain percolation thresholds that are fully equivalent to those calculated within the second-virial approximation of the connectedness Ornstein-Zernike equation. Our formulation can account for effects due to interactions between the rods, and many-body features can be partially addressed by suitable choices for the edge probabilities.