A Simple Theory of Melting and Condensation in Two-Dimensional System

Abstract

Solid-liquid-gas phase transition is studied, using a simple two-dimensional model proposed by Collins. It is a simplified Bernal lattice which contains equilateral triangles and squares. Some lattice sites would be vacant, representing holes. An interatomic potential has a hard core repulsion and a short range attraction. Two kinds of order parameters are introduced to describe disorder of the system. One is related to the Bernal structure and plays a role at the solid-liquid transition. The other is related to the hole density and plays a role at the liquid-gas transition. We get a phase diagram which has the three phases including a triple point and a critical point. The temperature of the triple point and the critical point is located at kBT,jE-::::. 0.81 and kBT,jE-::::.1.27, respectively, where E is the depth of an attractive potential. The density change and the latent heat are calculated along each transition line. The obtained results are quite analogous to the behavior of usual three-dimensional substances. Ordinary substances have three phases, that is, solid, liquid and gas. At the triple point, the three phases coexist in equilibrium. At pressure lower than the triple point pressure, the liquid phase cannot exist and the transition takes place only between the solid and the gas phases. At pressure higher than the triple point pressure, the three phases can exist. Melting or freezing transition occurs between the solid and the liquid phases. Condensation or evaporation occurs be­ tween the liquid and the gas phases. Condensation curve has a critical point, whereas melting curve has no end. The liquid state can be changed into the gas state continuously around the critical point. At pressure higher than the critical point pressure, or temperature higher than the critical point temperature, there is no distinction between the liquid and the gas phases. This state is often called a fluid phase. Although these features of a typical phase diagram are quite well known, our understanding is still far from satisfaction for them. Such a phase diagram can be investigated by means of a computer experi­ ment.ll Monte Carlo calculations were carried out for a classical three-dimensional system of particles interacting through a Lennard-Jones potential. The results of these computer experiments could reproduce the phase diagram of rare gas like Ar fairly well, though there remained a little discrepancy between the results of two computer experiments.2),a) Some computer experiments were carried out for a