A Monte Carlo study of the thermodynamic properties of “hard hexagons” on the triangular lattice

Abstract

A Monte Carlo method, developed originally to calculate thermodynamic properties in a grand canonical ensemble of lattice systems containing molecules which have finite mutual attractions, has been modified for studying lattice systems in which the intermolecular potential is an infinite repulsion. The grand ensemble formulation is reviewed briefly, and equations are given for a specific choice of transition probabilities defining a Markov process with a stationary distribution which is identical to the probability distribution of states in the grand ensemble. Results are presented for “hard hexagons” on the two-dimensional triangular lattice. For hexagons of a size that prevents simultaneous occupation of a pair of nearest-neighbor sites by two molecules, the results indicate a possible second-order phase transition for μ/ kT ≈ 2.3 at a density around 82% of the close-packed density. The existence, nature, and location of this transition have been more firmly established by Runnels and Combs, and by Gaunt, using other techniques. For larger hexagons excluding simultaneous second-neighbor occupancy, an apparent first-order transition occurs at μ f kT ≈ 1.68, with “fluid” and “solid” densities of about 69 % and 77 % of the close-packed density. These results agree, except for the effects of finite lattice size, with those extrapolated by Orban and Bellemans from exact calculations for semi-infinite lattices. When third neighbors are excluded also, the Monte Carlo results give no indication of a transition. However, Orban and Bellemans have estimated that the transition in this case occurs at μ/ kT ≈ 4.6, well above the value (about 3.0) where convergence of this particular Monte Carlo method becomes too slow for practical use.