PHASE FIELD MODELS FOR HYPERCOOLED SOLIDIFICATION

Properties of the solidification front in a hypercooled liquid, so cal led because the temperature of the resulting solid is below the meltin g temperature, are derived using a phase field (diffuse interface) mod el. Certain known properties for hypercooled fronts in specific materi als are reflected within our theories, such as the presence of thin th ermal layers and the trend towards smoother fronts (with less pronounc ed dendrites) when the undercooling is increased within the hypercoole d regime. Both an asymptotic analysis, to derive the relevant free bou ndary problems, and a rigorous determination of the inner profile of t he diffusive interface are given. Of particular interest is the incorp oration of anisotropy and general microscale interactions leading to h igher order differential operators. These features necessitate a much richer mathematical analysis than previous theories, Anisotropic free boundary problems are derived from our models, the simplest of which i nvolves determining the evolution of a set (a solid particle) whose bo undary moves with velocity depending on its normal vector. Considerabl e attention is given to the identification of surface tension, to comp arison with previous theories and to questions of stability.