Interfacial Tension Effects in Finite, Periodic, Two-Dimensional Systems

We consider a two-dimensional system whose pressure specific area (p—v) isotherm (the temperature T being held fixed throughout our discussion) displays a first-order phase transition in the thermodynamic limit, i.e., as the number of molecules N and the total area V both tend to infinity for various fixed specific volumes v=V/N. When N is large but finite we adopt a macroscopic description of the system in which the total Helmholtz free energy A is given by additive contributions from the bulk properties of the two coexisting phases and an interfacial term appropriate to periodic boundary conditions. At fixed N and V we minimize A with respect to variations in the specific areas and amounts of the two phases and with respect to the shape of the interface, making a number of simplifying assumptions. The resulting p—v isotherm is found to have ``loops'' involving jump discontinuities, with three different forms appearing, depending upon the value of a dimensionless parameter involving the number of molecules N and the interfacial tension, and with coexistent phases found to exist only in the two cases involving larger values of N. The amplitude of the loop obtained by Alder and Wainwright for a system of 870 hard disks is used to make a crude estimate of the interfacial tension between two phases.