Title:

Structural phase behaviour via Monte Carlo techniques

There are few reliable computational techniques applicable to the problem of structural phase behaviour. This is starkly emphasised by the fact that there are still a number of unanswered questions concerning the solid state of some of the simplest models of matter. To determine the phase behaviour of a given system we invoke the machinery of statistical physics, which identifies the equilibrium phase as that which minimises the freeenergy. This type of problem can only be dealt with fully via numerical simulation, as any less direct approach will involve making some uncontrolled approximation. In particular, a numerical simulation can be used to evaluate the freeenergy difference between two phases if the simulation is free to visit them both. However, it has proven very difficult to find an algorithm which is capable of efficiently exploring two different phases, particularly when one or both of them is a crystalline solid. This thesis builds on previous work (Physical Review Letters 79 p.3002), exploring a new Monte Carlo approach to this class of problem. This new simulation technique uses a global coordinate transformation to switch between two different crystalline structures. Generally, this `lattice switch' is found to be extremely unlikely to succeed in a normal Monte Carlo simulation. To overcome this, extendedsampling techniques are used to encourage the simulation to visit `gateway' microstates where the switch will be successful. After compensating for this bias in the sampling, the freeenergy difference between the two structures can be evaluated directly from their relative probabilities. As concrete examples on which to base the research, the latticeswitch Monte Carlo method is used to determine the freeenergy difference between the facecentred cubic (fcc) and hexagonal closepacked (hcp) phases of two generic model systems  the hardsphere and LennardJones potentials. The structural phase behaviour of the hardsphere solid is determined at densities near melting and in the closepacked limit. The factors controlling the efficiency of the latticeswitch approach are explored, as is the character of the `gateway' microstates. The facecentred cubic structure is identified as the thermodynamically stable phase, and the freeenergy difference between the two structures is determined with high precision. These results are shown to be in complete agreement with the results of other authors in the field (published during the course of this work), some of whom adopted the latticeswitch method for their calculations. Also, the results are favourably compared against the experimentally observed structural phase behaviour of stericallystabilised colloidal dispersions, which are believed to behave like systems of hard spheres. The logical extension of the hard sphere work is to generalise the latticeswitch technique to deal with `softer' systems, such as the LennardJones solid. The results in the literature for the structural phase behaviour of this relatively simple system are found to be completely inconsistent. A number of different approaches to this problem are explored, leading to the conclusion that these inconsistencies arise from the way in which the potential is truncated. Using results for the groundstate energies and from the harmonic approximation, we develop a new truncation scheme which allows this system to be simulated accurately and efficiently. Latticeswitch Monte Carlo is then used to determine the fcchcp phase boundary of the LennardJones solid in its entirety. These results are compared against the experimental results for the LennardJones potential's closest physical analogue, the raregas solids. While some of the published raregas observations are in approximate agreement with the latticeswitch results, these findings contradict the widely held belief that fcc is the equilibrium structure of the heavier raregas solids for all pressures and temperatures. The possible reasons for this disagreement are discussed. Finally, we examine the pros and cons of the latticeswitch technique, and explore ways in which it can be extended to cover an even wider range of structures and interactions.
