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

Yuri Yu. Tarasevich; Nikolai I. Lebovka; Valeri V. Laptev

doi:10.1103/physreve.86.061116

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arXiv · 1208.3602

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

Yuri Yu. Tarasevich·Nikolai I. Lebovka·Valeri V. Laptev

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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 $p_c = a/k^α+b\log_{10} k+ c$, where $a$, $b$, $c$, $α$ 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.

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Yuri Yu. Tarasevich, Nikolai I. Lebovka, Valeri V. Laptev. 2012-08-17. Percolation of linear $k$-mers on square lattice: from isotropic through partially ordered to completely aligned state. https://doi.org/10.1103/physreve.86.061116

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