Modeling of dendritic growth using a quantitative nondiagonal phase field model

Abstract

The phase field method has emerged as the tool of choice to simulate complex pattern formation processes in various domains of materials sciences. For the phase field model to faithfully reproduce the dynamics of a prescribed free-boundary problem with transport equations in the bulk and boundary conditions at the interfaces, the so-called thin-interface limit should be performed. For a phase transformation driven by diffusion, the kinetic cross-coupling between the phase field and the diffusion field has recently been introduced, allowing a control on interface boundary conditions in the general case where the diffusivity in the growing phase ${D}_{S}$ neither vanishes (one-sided model) nor equals the one of the disappearing phase ${D}_{L}$ (symmetric model). Here, we investigate the capabilities of this nondiagonal phase field model in the case of two-dimensional dendritic growth. We benchmark our model with Green's function calculations (sharp-interface model) for the symmetric and one-sided cases, and our results for arbitrary ${D}_{S}/{D}_{L}$ allow us to propose a generalization of the theory by Barbieri and Langer [Phys. Rev. A 39, 5314 (1989)] for finite anisotropy of interface energy. We also perform simulations that evidence the necessity of introducing the kinetic cross-coupling and of eliminating surface diffusion. Our work opens up the way for quantitative phase field simulations of phase transformations with diffusion in the growing phases playing an important role in the pattern and velocity selections.