RANDOM-LATTICE MODELS AND SIMULATION ALGORITHMS FOR THE PHASE-EQUILIBRIA IN 2-DIMENSIONAL CONDENSED SYSTEMS OF PARTICLES WITH COUPLED INTERNAL AND TRANSLATIONAL DEGREES OF FREEDOM

In this work we concentrate on phase equilibria in two-dimensional con densed systems of particles where both translational and internal degr ees of freedom are present and coupled through microscopic interaction s, 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 ran domly 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 s eries of three statistical-mechanical models are then constructed. All of the three models are in essence spin-1/2 Ising models where the sp ins, representing internal degrees of freedom, are associated with har d-disk particles and nearest-neighbor particles interact through spin- spin interactions that may have spatial dependence. The fluctuating nu mber of nearest neighbors and the possible spatial dependence of the s pin-spin interactions couple microscopically the spin degrees of freed om to the translational degrees of freedom. The first model (I) is a r andom-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 interac tions. The third model (III) is a modification of the second model tha t accounts for spin states with different internal degeneracy. Monte C arlo simulation techniques, including both a special algorithm for the random-lattice description and histogram and finite-size scaling anal ysis, are used to investigate the phase behavior of all three models. It is shown that the order-disorder spin transition in model I is deco upled from a first-order singularity-lattice melting-associated with t he 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 ha s a region where the spin degrees of freedom are slaved by the transla tional degrees of freedom and develop a first-order singularity in the order-disorder transition that accompanies the lattice-melting transi tion. The internal degeneracy of the spin states in model III implies that the spin order-disorder singularity can be of first order through out the phase diagram. It is found that this first-order singularity c an be either coupled to or decoupled from the lattice-melting singular ity, depending on the strength of the microscopic coupling. The calcul ated phase diagram and the associated thermodynamic transitional prope rties 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 intracha in conformational states.