Phase field models for hypercooled solidification

Properties of the solidification front in a hypercooled liquid, so called because the temperature of the resulting solid is below the melting temperature, are derived using a phase field (diffuse interface) model. Certain known properties for hypercooled fronts in specific materials are reflected within our theories, such as the presence of thin thermal layers and the trend towards smoother fronts (with less pronounced dendrites) when the undercooling is increased within the hypercooled regime. Both an asymptotic analysis, to derive the relevant free boundary problems, and a rigorous determination of the inner profile of the diffusive interface are given. Of particular interest is the incorporation of anisotropy and general microscale interactions leading to higher 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 involves determining the evolution of a set (a solid particle) whose boundary moves with velocity depending on its normal vector. Considerable attention is given to the identification of surface tension, to comparison with previous theories and to questions of stability.