Extending the excluded volume for percolation threshold estimates in polydisperse systems: The binary disk system

For dispersions containing a single type of particle, it has been observed that the onset of percolation coincides with a critical value of volume fraction. When the volume fraction is calculated based on excluded volume, this critical percolation threshold is nearly invariant to particle shape. The critical threshold has been calculated to high precision for simple geometries using Monte Carlo simulations, but this method is slow at best, and infeasible for complex geometries. This paper explores an analytical approach to the prediction of percolation threshold in polydisperse mixtures. Specifically, this paper suggests an extension of the concept of excluded volume, and applies that extension to the 2D binary disk system. The simple analytical expression obtained is compared to Monte Carlo results from the literature. The result may be computed extremely rapidly and matches key parameters closely enough to be useful for composite material design.