Computation of Dendrites Using a Phase-Field Model of Solidification

Richard J. Braun, University of Delaware

Bruce T. Murray, NIST

Juan Soto, now at Lockheed-Martin-Lorale

The solidification of a pure material may be modeled in a number of ways. One method is to describe the boundary between the frozen crystal and its melt as a free surface of zero thickness, where boundary conditions are applied and which must be determined as part of the solution to the problem. Another approach is to use a diffuse-interface method where the interface is postulated to be a thin layer varying smoothly from one phase to another. In the phase-field model we use here, an additional field variable describes the phase and it evolves according to a nonlinear reaction-diffusion equation that is coupled to a nonlinear reaction-diffusion equation for the temperature field.

We have solved this coupled set of pde's using the package VLUGR2 developed by J.G. Blom, R.A. Trompert and J.G. Verwer (ACM TOMS vol 22 no 3, Algorithm 758: VLUGR2, A Vectorizable Adaptive Grid Solver for PDEs in 2D, pp. 302-328) of CWI in the Netherlands. The code uses a finite difference mesh that is spatially adapted using local uniform grid refinement and a second order BDF method in time with a variable time step. The graphics in this note were made via modification of postprocessing routines available from J.G. Blom.

Inherent in solving these equations are severe differences in the length scale associated with the interface thickness, the radius of curvature of the features of the growing dendrite, and the length scale of the decay of the temperature field to its far-field value. The wide range of length scales involved prompted us to use and adaptive method in the hopes that we could compute at smaller values of undercooling (how much the temperature of the melt away from the dendrite is lower than the melting point). We show some results for the a dimensionless undercooling of 0.25. In the upper portions of the plots, the blue area is melt and the red is the solidified crystal. In the lower portion of the plots, the stairstepped lines separate the regions of finest mesh (nearest crystal-melt interface) from coarser regions (the coarsest regions are farthest from the interface). The finest mesh has a grid spacing 8 times smaller than the coarsest spacing.

t=1.0

t=2.5

t=5.0

This work has resulted in a paper that was given as an invited presentation at the Tenth American Conference on Crystal Growth held in Vail, CO in the summer of 1996. The paper will appear in the Journal of Crystal Growth vol 174 (1997) 41-43. (A gnu-zipped postscript version of the submitted paper is available). It has also resulted in a second paper (postscript version here) that covers a wider range of dimensionless undercooling (from 0.8 to 0.1); that paper appeared in Modelling and Simulation in Materials Science and Engineering vol 5 (1997) 365-380.