Lattice model for percolation on a plane of partially aligned sticks with length dispersity

A lattice-based model for continuum percolation is applied to the case of randomly located, partially aligned sticks with unequal lengths in 2D which are allowed to cross each other. Results are obtained for the critical number of sticks per unit area at the percolation threshold in terms of the distributions over length and orientational angle and are compared with findings from computer simulations. Consistent with findings from computer simulations, our model shows that the percolation threshold is (i) elevated by increasing degrees of alignment for a fixed length distribution, and (ii) lowered by increasing degrees of length dispersity for a fixed orientational distribution. The impact of length dispersity is predicted to be governed entirely by the first and second moments of the stick length distribution, and the threshold is shown to be quite sensitive to particulars of the orientational distribution function.