Intrinsic Equations of Motion for Interfaces in Models of Solidification

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A methodology is proposed for deriving equations of motion for interfaces in transition between two phases, such as the solidification of a liquid into a solid, in the asymptotic limit of small diffusion. The investigator will show that the velocity of the interface is a function of its local geometry, described by so-called intrinsic quantities such as the curvature. This model allows the computation of the radius at which a growing spherical particle becomes unstable and may allow a deeper understanding of the complex dendritic structures seen in solidification. The problem of the solidification of a liquid has many industrial applications, such as the growing of a quartz crystal from a melt for use in semiconductor devices like computer chips. For these applications it is desirable to grow smooth, regular crystals. Unfortunately, many substances tend to grow into irregular snowflake-like dendrites. In this study the investigator proposes a methodology for reducing the complex equations governing solidification to a relatively simple geometric description. This reduction should allow the prediction of the transition from regular to dendritic solidification. It may also give some insight into the process by which irregular dendritic structures form and grow.