Growth competition between columnar dendritic grains – Cellular automaton versus phase field modeling

Abstract

Cellular Automaton (CA) simulations of two-dimensional growth competition among columnar dendritic grains are carried out for a succinonitrile - 0.4 wt% acetone alloy. This is achieved by computing the Grain Boundary (GB) orientation during directional solidification of a bi-crystal in a frozen temperature gradient approximation, each crystal being defined by its own orientation. Comparisons are subsequently conducted with recent Phase Field (PF) results derived under the same conditions as well as with the Geometrical Limit (GL) criterion and the Favorably Orientated Grain (FOG) criterion. The GL criterion is defined mathematically considering infinitely small branching within each grain in directions perpendicular to the main dendrite arms growth directions. The FOG criterion states survival of the grain having the growth direction best aligned with the temperature gradient. The GB orientation is investigated by CA simulations as a function of the cell size, the cell neighborhood and the position used to compute the growth velocity. Results reveal that sufficiently small cells lead to the convergence of the GB orientation towards the GL criterion, while sufficiently large cells lead to the FOG criterion. Within a range of intermediate cell size, excellent agreement is found with a revised version of the FOG criterion (rev-FOG) extracted from PF simulations over a wide range of grain orientations. The cell size needs to be of the order of the maximum step between primary stationary dendrite tips of the two competing grains. The Moore neighborhood provides better results than the von Neumann neighborhood. Noticeable improvement is also observed when computing the growth velocity at the leading dendrite tip positions compared to using the cell center approximation. With computational times several orders of magnitudes lower than PF, the CA method offers a realistic and useful alternative for direct simulations of solidification grain structures in casting processes. This work is also an example of upscaling between models, showing how PF dedicated to model phenomena at the scale of the solid-liquid interface and sidebranching competition can be used to evaluate and calibrate CA developed for large scale simulations.