Geometric percolation thresholds of interpenetrating plates in three-dimensional space

The geometric percolation thresholds for circular, elliptical, square, and triangular plates in the three-dimensional space are determined precisely by Monte Carlo simulations. These geometries represent oblate particles in the limit of zero thickness. The normalized percolation points, which are estimated by extrapolating the data to zero radius, are η_c=0.9614±0.0005 , 0.8647±0.0006 , and 0.7295±0.0006 for circles, squares, and equilateral triangles, respectively. These results show that the noncircular shapes and corner angles in the plate geometry tend to increase the interparticle connectivity and therefore reduce the percolation point. For elliptical plates, the percolation threshold is found to decrease moderately, when the aspect ratio ɛ is between 1 and 1.5, but decrease rapidly for ɛ greater than 1.5. For the binary dispersion of circular plates with two different radii, η_c is consistently larger than that of equisized plates, with the maximum value located at around r₁/r₂=0.5 .