Percolation of linear $k$-mers on square lattice: from isotropic through partially ordered to completely aligned state

TL;DR

This study uses Monte Carlo simulations to analyze how the orientation and size of linear $k$-mers affect percolation thresholds on square lattices, revealing nonmonotonic size dependence and predicting no percolation for very large $k$-mers.

Contribution

It introduces a fitting formula for the percolation threshold considering anisotropy and large $k$-mers, expanding understanding of percolation behavior in anisotropic systems.

Findings

Percolation threshold exhibits nonmonotonic size dependence in isotropic cases.

A new fitting formula for $p_c$ as a function of $k$ and anisotropy is proposed.

Percolation does not occur for large $k$-mers ($k oughly 1.2×10^4$) at jamming concentration.

Abstract

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 the literature as rigid rods, needles, sticks) on two-dimensional square lattices $L \times L$ 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. Moreover, the behavior of percolation probability $R_L(p)$ that a lattice of size $L$ percolates at concentration $p$ has been studied in details in dependence on $k$, anisotropy and lattice size $L$. A nonmonotonic size dependence for the percolation threshold has been confirmed in isotropic case. We propose a fitting formula for percolation threshold $p_c = a/k^{\alpha}+b\log_{10} k+ c$, where $a$, $b$, $c$, $\alpha$ are the fitting parameters varying with anisotropy. We predict that for large $k$-mers ($k\gtrapprox 1.2\times10^4$) isotropic placed at the lattice, percolation cannot occur even at jamming concentration.