Percolation of linear k-mers on a square lattice: From isotropic through partially ordered to completely aligned states

Numerical simulations by means of Monte Carlo method and finite-size scaling analysis have been performed to study the percolation behavior of linear k-mers (also denoted in publications as rigid rods, needles, sticks) on two-dimensional square lattices L x L with periodic boundary conditions. Percolation phenomena are investigated for anisotropic relaxation random sequential adsorption of linear k-mers. Especially, effect of anisotropic placement of the objects on the percolation threshold has been investigated. A detailed study of the behavior of percolation probability R-L(p) that a lattice of size L percolates at concentration p in dependence on k, anisotropy, and lattice size L has been performed. A nonmonotonic size dependence for the percolation threshold has been confirmed in the isotropic case. We propose a fitting formula for percolation threshold, p(c) = a/k(alpha) + b log(10) k + c, where a, b, c, and alpha are the fitting parameters depending on anisotropy. We predict that for large k-mers (k (sic) 1.2 x 10(4)) isotropically placed at the lattice, percolation cannot occur, even at jamming concentration. DOI: 10.1103/PhysRevE.86.061116