 
Summary: Thermodynamic basis for a variational model for crystal growth
Bayard K. Johnson* and Robert F. Sekerka
Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Robert Almgren
University of Chicago, Chicago, Illinois 60637
Received 19 October 1998
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 GibbsThomson boundary condition at the solidliquid interface. S1063651X 99 023077
PACS number s : 81.10.Aj, 05.70.Ln, 02.70. c, 64.70.Dv
I. INTRODUCTION
In computational modeling of crystal growth, standard
