We use several Monte Carlo computer (MC) simulation techniques to calculate theudphase diagram of a system of hard disks interacting through a discrete square-shoulderudsquare-well potential. The phase diagram shows the gas, liquid and five crystal phases,udand we find that all the melting lines are first-order phase transitions, despite theudsystem being two dimensional. The melting line of the square crystal exhibits audtemperature maximum, meaning that above a certain pressure P the density of liquidudbecomes higher than that of a crystal. The same melting line also exhibits a pressureudmaximum that implies inverse melting, meaning that at constant pressure the liquidudcrystallizes by heating.udTo increase the range of pressure over which inverse melting occurs, we vary theudpotential parameters systematically and determine that the extent of the shoulder isudthe parameter that has the greatest impact. We calculate the new melting curve forudthe new potential parameter set, and we check the accuracy of the calculations byudseveral methods including the calculation of the Gibbs free energy as a function ofuddensity at conditions of constant P and temperature T. The melting transition is firstudorder and to a liquid rather than to a hexatic or to a quasicrystal.udFinally, we perform MC simulations at constant P, T and number of particlesudN, to study the high pressure phase behaviour of a model with parameters thatudproduce pronounced inverse melting. We detect three fascinating behaviours. First, the high pressure triple point present in the original model disappears, leaving behinduda “liquid corridor” in the phase diagram for which the liquid appears to retain itsudposition as the thermodynamically stable phase down to low temperature. Howeverudwe find a new crystal that likely usurps the liquid as the stable phase. Second, we finduda particular state point, which we name the “funny point”, at which the free energyudbarrier between the liquid and the high density triangular crystal vanishes along theirudcoexistence line. Although the explanation of this funny point remains a mystery, itudappears to be connected to the third discovery: a transition between low and highudtemperature forms of the high density triangular crystal.udThe potential studied in this thesis was previously developed to help understandudanomalous behaviour in systems such as water and liquid metals. Moreover similarudpotentials have been used to model lipids interacting within bilayer membranes. Thus,udit is possible that some of the phenomenology we observe for the model is relevant inudthese or related real systems.
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