We compute time-dependent solutions of the sharp-interface model of dendritic solidification in two dimensions by using a level set method. The steady-state results are in agreement with solvability theory. Solutions obtained from the level set algorithm are compared with dendritic growth simulations performed using a phase-field model and the two methods are found to give equivalent results. Furthermore, we perform simulations with unequal diffusivities in the solid and liquid phases and find reasonable agreement with the available theory.

We compare time-dependent solutions of different phase-field models for dendritic solidification in two dimensions, including a thermodynamically consistent model and several ad hoc models. The results are identical when the phase-field equations are operating in their appropriate sharp interface limit. The long time steady state results are all in agreement with solvability theory. No computational advantage accrues from using a thermodynamically consistent phase-field model. PACS Numbers: 81.10. Aj, 05.70.Ln, 64.70.Dv, 81.30.Fb Dendrites are the most commonly observed solidification microstructures in metals. The free growth of a single dendrite is a prototype for problems of pattern selection in materials science [1][2][3][4][5] and has been extensively studied experimentally and theoretically. It is still not possible to compare theory with experiment because of the difficulties in computing three dimensional microstructures in the temperature and material's parameter range of the experiments.Recently, a significant step forward was taken by Karma and Rappel [6,7], who not only showed how to compute accurately two dimensional dendritic growth but also were able to compare their results with theoretical predictions. Their calculations used the so-called phase-field formulation of solidification, in which a mathematically sharp solid-liquid interface is smeared out or regularized and treated as a boundary layer, with its own equation of motion. The resulting formulation, described in more detail below, no longer requires front tracking and the imposition of boundary conditions, but must be related to the sharp interface model by an asymptotic analysis. In fact, there are many ways to prescribe a smoothing and dynamics of a sharp interface consistent with the original sharp interface model, so that there is no unique phase-field model, but rather a family of related models. An important part of Karma and Rappel's work was an improved asymptotic analysis which allows coarser spatial grids to be used in the numerical computations than was previously possible.Although the phase-field method has gained acceptance as a useful way to study solidification problems, a debate still exists over the interpretation and validity of the phase-field models themselves. Each model includes a double-well potential field which enforces the above properties of the phase-field. Some models can be shown rigorously to satisfy entropy inequality [8,9]. These are sometimes called "thermodynamically consistent" models. On the other hand, it has been argued that the precise form of the phase-field equations should be irrelevant so long as the computations are performed at the asymptotic limit where the phase-field model converges to the sharp interface limit [10].The purpose of this Letter is to compare the dynamics of the different phase-field models proposed. To this end, we have performed accurate and extensive computations using a specially developed adaptive mesh refinement algorithm [11,12]. We find that when properly used, all ...