Nonmonotonic size dependence of the critical concentration in 2D percolation of straight rigid rods under equilibrium conditions

TL;DR

This study investigates how the critical concentration for percolation of straight rigid rods on 2D lattices varies with rod length, revealing a nonmonotonic pattern linked to phase transitions, and confirms the universality class matches random percolation.

Contribution

It uncovers the nonmonotonic size dependence of the percolation threshold and connects it to the isotropic-nematic phase transition in 2D rod systems.

Findings

Percolation threshold decreases then increases with rod length.

The percolation universality class matches that of random percolation.

A nonmonotonic size dependence is observed for the critical concentration.

Abstract

Numerical simulations and finite-size scaling analysis have been carried out to study the percolation behavior of straight rigid rods of length $k$ ($k$-mers) on two-dimensional square lattices. The $k$-mers, containing $k$ identical units (each one occupying a lattice site), were adsorbed at equilibrium on the lattice. The process was monitored by following the probability $R_{L,k}(\theta)$ that a lattice composed of $L \times L$ sites percolates at a concentration $\theta$ of sites occupied by particles of size $k$. A nonmonotonic size dependence was observed for the percolation threshold, which decreases for small particles sizes, goes through a minimum, and finally asymptotically converges towards a definite value for large segments. This striking behavior has been interpreted as a consequence of the isotropic-nematic phase transition occurring in the system for large values of $k$. Finally, the universality class of the model was found to be the same as for the random percolation model.