Phase diagram of a bidispersed hard rod lattice gas in two dimensions

We obtain, using extensive Monte Carlo simulations, virial expansion and a high-density perturbation expansion about the fully packed monodispersed phase, the phase diagram of a system of bidispersed hard rods on a square lattice. We show numerically that when the length of the longer rods is $7$, two continuous transitions may exist as the density of the longer rods in increased, keeping the density of shorter rods fixed: first from a low-density isotropic phase to a nematic phase, and second from the nematic to a high-density isotropic phase. The difference between the critical densities of the two transitions decreases to zero at a critical density of the shorter rods such that the fully packed phase is disordered for any composition. When both the rod lengths are larger than $6$, we observe the existence of two transitions along the fully packed line as the composition is varied. Low-density virial expansion, truncated at second virial coefficient, reproduces features of the first transition. By developing a high-density perturbation expansion, we show that when one of the rods is long enough, there will be at least two isotropic-nematic transitions along the fully packed line as the composition is varied.