Some problems in the theory of equilibrium crystal shapes

Recently, great progress has been made in the theory of equilibrium crystal shapes, and a general thermodynamic framework of the problem has been developed. Chapter 1 serves as a brief survey of the theory, in order to provide basic background material of this thesis. It has recently been observed that the shape of N aCl crystals in stable coexistence with the vapor, while strictly cubical at low temperature, undergoes a corner-rounding transition at T0 ~ 650°C. The theory developed in Chapter 2 takes proper account of the statistical mechanics of the atomic-scale interface fluctuations by adopting a restricted solid- on- solid model of the interface. This model is equivalent to an antiferromagnetic Ising model on a triangular lattice, and its phase diagram provides a framework for understanding the qualitative differences observed between the thermal evolution of NaCl and that of metal crystals. Corner rounding occurs for N aCl via an energy- entropy crossover mechanism. The role of the step-like excitations in building vicinal surfaces is particularly emphasized. Estimates of To depend directly and sensitively on the energy per unit length of steps. A detailed calculation of the atomic relaxation effects on the edge and step energies per unit length is performed, which leads to a value of T0 ~ 1000K, in good agreement with the experiments. Gas adsorption at a crystal surface changes the interfacial free energy and, therefore, modifies the corresponding equilibrium crystal shape. In Chapter 3, the effects of adsorption on equilibrium crystal shapes are studied by two approaches. First, a simple lattice-gas model of the adsorption process, including both adatom- adatom and adatom-crystal-atom interactions, is solved exactly at zero temperature. The dependence of the crystal shape on the adgas chemical potential is calculated for the cubic crystals with a variety of crystal-atomcrystal- atom interactions, and the shape- change systematics is given in terms of crystal shape "phase" diagram. Secondly, a phenomenological theory of the problem at nonzero temperature is developed. By using available experiment data of the adsorption process, the theory is applied to the crystals of the practicely important material Pt, and good correspondence between the theory and experiments is obtained. Finally, a possible application of the theory to the designing of more efficient metal catalysts is presented.U of I OnlyThesi