Critical behavior of long straight rigid rods on two-dimensional lattices: Theory and Monte Carlo simulations

TL;DR

This study investigates the phase transition behavior of long rigid rods on 2D lattices, revealing a nematic phase and deriving how the critical density depends on rod length through theory and simulations.

Contribution

It combines geometric and entropy-based analytical methods with Monte Carlo simulations to predict the critical density dependence on rod length and identifies the minimum length for nematic phase formation.

Findings

Nematic phase exists for rods of length ≥7 on triangular lattices.

Critical density scales inversely with rod length, θ_c(k) ∝ k^{-1}.

Continuous phase transition occurs at intermediate densities.

Abstract

The critical behavior of long straight rigid rods of length $k$ ($k$-mers) on square and triangular lattices at intermediate density has been studied. A nematic phase, characterized by a big domain of parallel $k$-mers, was found. This ordered phase is separated from the isotropic state by a continuous transition occurring at a intermediate density $\theta_c$. Two analytical techniques were combined with Monte Carlo simulations to predict the dependence of $\theta_c$ on $k$, being $\theta_c(k) \propto k^{-1}$. The first involves simple geometrical arguments, while the second is based on entropy considerations. Our analysis allowed us also to determine the minimum value of $k$ ($k_{min}=7$), which allows the formation of a nematic phase on a triangular lattice.