A fully quantitative mesh-independent cellular automaton model for dendrite growth

While a number of cellular automaton (CA) based models for dendrite growth were proposed, none so far have been validated, which casts doubt on their quantitative capabilities. Furthermore, these models are mesh-dependent and cannot correctly describe crystallographic orientation. In this paper, we propose a CA-based model for dendritic growth controlled by solutal and thermal effects in the low Peclet number regime. The model does not use an analytical solution to determine the velocity of the solid-liquid (SL) interface as function of undercooling, as common in other models. Instead, it solves the solute and heat conservation equations subjected to the boundary conditions at the interface. Using this approach the model does not need to use the concept of marginal stability and stability constant to uniquely define the steady state velocity and radius of the dendrite tip. The model contains an expression for the stability parameter but the process determines its value. It is found that the stability parameter is not a constant, but rather changes with time and angular position during dendrite formation. The model proposes solutions for the evaluation of local curvature, solid fraction, trapping rules and anisotropy of the mesh, which eliminates the mesh dependency of calculations, common in other models. The model is able to reproduce most of the dendritic features observed experimentally, such as secondary and tertiary branching, parabolic tip, arms generation and selection, etc. Computation results are validated in two ways. First, the simulated secondary dendrite arm spacing (SDAS) is compared with literature values for an Al-4wt% Cu alloy. Second, the predictions of the classic Lipton-Glicksman-Kurz (LGK) theory for steady state tip velocity and radius, are compared with simulated values as function of melt undercooling for an Al-18wt% Mg alloy. Model calculations were found to be in very good agreement with both the analytical model and the experimental data.