Jamming and percolation in random sequential adsorption of straight rigid rods on a two-dimensional triangular lattice

TL;DR

This study uses Monte Carlo simulations to analyze how linear rods on a triangular lattice percolate and jam, revealing a nonmonotonic size dependence of the percolation threshold and confirming the universality class of the transition.

Contribution

It extends previous research by exploring larger system sizes and longer rods, identifying a maximum length beyond which percolation ceases, and confirming the universality class of the phase transition.

Findings

Percolation threshold varies nonmonotonically with rod length.

A maximum rod length exists beyond which percolation does not occur.

The critical exponents match those of ordinary percolation.

Abstract

Monte Carlo simulations and finite-size scaling analysis have been performed to study the jamming and percolation behavior of linear $k$-mers (also known as rods or needles) on the two-dimensional triangular lattice, considering an isotropic RSA process on a lattice of linear dimension $L$ and periodic boundary conditions. Extensive numerical work has been done to extend previous studies to larger system sizes and longer $k$-mers, which enables the confirmation of a nonmonotonic size dependence of the percolation threshold and the estimation of a maximum value of $k$ from which percolation would no longer occurs. Finally, a complete analysis of critical exponents and universality have been done, showing that the percolation phase transition involved in the system is not affected, having the same universality class of the ordinary random percolation.

Full text

Supplementary Material for:

Jamming and percolation in random sequential adsorption of straight rigid rods on a two-dimensional triangular lattice

Data collapsing

The scaling behavior can be further tested by plotting $R^{X}_{L,k}(\theta)$ vs $\left(\theta-\theta_{p,k}\right)L^{1/\nu}$, $PL^{\beta/\nu}$ vs $|\theta-\theta_{p,k}|L^{1/\nu}$ and $\chi L^{-\gamma/\nu}$ vs $\left(\theta-\theta_{p,k}\right)L^{1/\nu}$ and looking for data collapsing Stauffersup . Figs. 1 and 2 show the excellent collapse of curves of $R^{A}_{L,k}$ (a), $P$ (b), $\chi$ (c) and the cumulant $U_{L}$ (d), for two typical cases ($k=8$ and $k=32$) and different lattice sizes, as indicated. The plots were made using the value of $\theta_{p,k=8}=0.4118$ and $\theta_{p,k=32}=0.4303$ calculated above and the exact values of the critical exponents of the ordinary percolation $\nu=4/3$, $\beta=5/36$ and $\gamma=43/18$. This leads to independent controls and consistency checks of the values of all the critical exponents.