Percolation Threshold of Clustered, Oriented and Polydisperse Sticks in a Plane

Monte-Carlo methods are used to study the influence of stick polydispersity, orientation and clustering on the percolation of sticks in the plane. The results show that for polydisperse stick lengths, the percolation threshold exhibits a linear dependence on the reciprocal of the length-weighted mean stick length. Reinterpretation of data from the literature confirms this dependence and shows it to persist for oriented networks. Also, the number of intersections per stick at percolation, Bc=3.6$B_c = 3.6$ is independent of orientation for percolation perpendicular to the direction of preferential orientation; in the direction of preferential orientation, it is a linear function of an orientation parameter. For networks of sticks clustered according to a compound Poisson process, a cluster intensity, psi is introduced; when this is less than the percolation threshold of a random network of sticks, the percolation threshold increases linearly with cluster intensity and is independent of cluster size. Above this, non-linear size-dependent relationships are observed.