In this work we concentrate on phase equilibria in two-dimensional condensed systems of particles where both translational and internal degrees of freedom are present and coupled through microscopic interactions, with a focus on the manner of the macroscopic coupling between the two types of degrees of freedom. First, an unconventional description of the translational degrees of freedom is developed, in which the randomly varying spatial connectivity of the particles is represented by a random lattice whose dynamic structure is given by triangulating the spatial configurations. Based on this random-lattice description, a series of three statistical-mechanical models are then constructed. All of the three models are in essence spin-1/2 Ising models where the spins, representing internal degrees of freedom, are associated with hard-disk particles and nearest-neighbor particles interact through spin-spin interactions that may have spatial dependence. The fluctuating number of nearest neighbors and the possible spatial dependence of the spin-spin interactions couple microscopically the spin degrees of freedom to the translational degrees of freedom. The first model (I) is a random-lattice Ising model with conventional nearest-neighbor spin-spin interactions. The second model (II) is an extension of this model to include a spatial dependence of the nearest-neighbor spin-spin interactions. The third model (III) is a modification of the second model that accounts for spin states with different internal degeneracy. Monte Carlo simulation techniques, including both a special algorithm for the random-lattice description and histogram and finite-size scaling analysis, are used to investigate the phase behavior of all three models. It is shown that the order-disorder spin transition in model I is decoupled from a first-order singularity-lattice melting-associated with the translational degrees of freedom and remains critical and falls in the universality class of the standard two-dimensional Ising model on regular lattices. Model II is shown to exhibit a phase diagram that has a region where the spin degrees of freedom are slaved by the translational degrees of freedom and develop a first-order singularity in the order-disorder transition that accompanies the lattice-melting transition. The internal degeneracy of the spin states in model III implies that the spin order-disorder singularity can be of first order throughout the phase diagram. It is found that this first-order singularity can be either coupled to or decoupled from the lattice-melting singularity, depending on the strength of the microscopic coupling. The calculated phase diagram and the associated thermodynamic transitional properties for model III are discussed in relation to experiments on planar bilayers of lipid-chain molecules whose properties are determined by a subtle coupling between the translational variables and the intrachain conformational states.
Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 1996, Vol 54, Issue 6, p. 6889-6905