Quantitative phase-field modeling of dendritic growth in two and three dimensions

We report the results of quantitative phase-field simulations of the dendritic crystallization of a pure melt in two and three dimensions. These simulations exploit a recently developed thin-interface limit of the phase-field model [A. Karma and W.-J. Rappel, Phys. Rev. E 53, R3017 (1996)], which is given here a detailed exposition. This limit makes it possible to perform efficient computations with a smaller ratio of capillary length to interface thickness and with an arbitrary interface kinetic coefficient. Simulations in one and two dimensions are first carried out to test the accuracy of phase-field computations performed within this limit. Dendrite tip velocities and tip shapes are found to be in excellent quantitative agreement with exact numerical benchmarks of solvability theory obtained by a boundary integral method, both with and without interface kinetics. Simulations in three dimensions exploit, in addition to the asymptotics, a methodology to calculate grid corrections due to the surface tension and kinetic anisotropies. They are used to test basic aspects of dendritic growth theory that pertain to the selection of the operating state of the tip and to the three-dimensional morphology of needle crystals without sidebranches. For small crystalline anisotropy, simulated values of ${\ensuremath{\sigma}}^{*}$ are slightly larger than solvability theory predictions computed by the boundary integral method assuming an axisymmetric shape, and agree relatively well with experiments for succinonitrile given the uncertainty in the measured anisotropy. In contrast, for large anisotropy, simulated ${\ensuremath{\sigma}}^{*}$ values are significantly larger than the predicted values. This disagreement, however, does not signal a breakdown of solvability theory. It is consistent with the finding that the amplitude of the $\mathrm{cos}4\ensuremath{\varphi}$ mode, which measures the departure of the tip morphology from a shape of revolution, increases with anisotropy. This departure can therefore influence the tip selection in a way that is not accurately captured by the axisymmetric approximation for large anisotropy. Finally, the tail shape at a distance behind the tip that is large compared to the diffusion length is described by a linear law $r\ensuremath{\sim}z$ with a slope $dr/dz$ that is nearly equal to the ratio of the two-dimensional and three-dimensional steady-state tip velocities. Furthermore, the evolution of the cross section of a three-dimensional needle crystal with increasing distance behind the tip is nearly identical to the evolution of a two-dimensional growth shape in time, in accord with the current theory of the three-dimensional needle crystal shape.