The phase transition from nematic to high-density disordered phase in a system of hard rods on a lattice

TL;DR

This paper investigates the phase transition behavior of hard rods on lattices, revealing a first order transition at high densities for large rod lengths and providing both theoretical and simulation evidence.

Contribution

It demonstrates that the second phase transition in such systems is first order for large rod lengths and offers asymptotic estimates for transition parameters across dimensions.

Findings

Transition is first order for large k in all dimensions > 1

Chemical potential at transition scales as k ln(k/ln k)

Density drops sharply from (ln k)/k^2 to exp(-a k)

Abstract

A system of hard rigid rods of length $k$ on hypercubic lattices is known to undergo two phases transitions when chemical potential is increased: from a low density isotropic phase to an intermediate density nematic phase, and on further increase to a high-density phase with no orientational order. In this paper, we argue that, for large $k$, the second phase transition is a first order transition with a discontinuity in density in all dimensions greater than $1$. We show the chemical potential at the transition is $\approx k \ln [k /\ln k]$ for large $k$, and that the density of uncovered sites drops from a value $ \approx (\ln k)/k^2$ to a value of order $\exp(-ak)$, where $a$ is some constant, across the transition. We conjecture that these results are asymptotically exact, in all dimensions $d\geq 2$. We also present evidence of coexistence of nematic and disordered phases from Monte Carlo simulations for rods of length $9$ on the square lattice.