Thermodynamic basis for a variational model for crystal growth

Variational models provide an alternative approach to standard sharp interface models for calculating the motion of phase boundaries during solidification. We present a correspondence between objective functions used in variational simulations and specific thermodynamic functions. We demonstrate that variational models with the proposed identification of variables are consistent with nonequilibrium thermodynamics. Variational models are derived for solidification of a pure material and then generalized to obtain a model for solidification of a binary alloy. Conservation laws for internal energy and chemical species and the law of local entropy production are expressed in integral form and used to develop variational principles in which a "free energy," which includes an interfacial contribution, is shown to be a decreasing function of time. This free energy takes on its minimum value over any short time interval, subject to the laws of conservation of internal energy and chemical species. A variational simulation based on this model is described, and shown for small time intervals to provide the Gibbs-Thomson boundary condition at the solid-liquid interface.