Wheeler, A.A., Murray, B. T. and Schaefer, R. J.
Computation of dendrites using a phase field model
Physica D, 66, (1-2), . (doi:10.1016/0167-2789(93)90242-S).
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A phase field model is used to numerically simulate the solidification of a pure material. We employ it to compute growth into an undercooled liquid for a one-dimensional spherically symmetric geometry and a planar two-dimensional rectangular region. The phase field model equation are solved using finite difference techniques on a uniform mesh. For the growth of a sphere, the solutions from the phase field equations for sufficiently small interface widths are in good agreement with a numerical solution to the classical sharp interface model obtained using a Green's function approach. In two dimensions, we simulate dendritic growth of nickel with four-fold anisotropy and investigate the effect of the level of anisotropy on the growth of a dendrite. The quantitative behavior of the phase field model is evaluated for varying interface thickness and spatial and temporal resolution. We find quantitatively that the results depend on the interface thickness and with the simple numerical scheme employed it is not practical to do computations with an interface that is sufficiently thin for the numerical solution to accurately represent a sharp interface model. However, even with a relatively thick interface the results from the phase field model show many of the features of dendritic growth and they are in surprisingly good quantitative agreement with the Ivantsov solution and microscopic solvability theory.
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