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

A system of hard rigid rods of length k on hypercubic lattices is known to undergo two phase 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 that the chemical potential at the transition is ≈kln[k/lnk] for large k, and that the density of uncovered sites drops from a value ≈(lnk)/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≥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.