Growth competition between columnar dendritic grains – The role of microstructural length scales

The growth competition between columnar dendritic grains is investigated by both phase-field (PF) and cellular automaton (CA) models in a growth regime where the primary dendrite spacing is much larger than the solutal diffusion length. This growth regime favors the formation of highly branched hierarchical dendritic microstructures prevalent in castings. Previous PF and CA studies have shed light on the complex relationship between GB orientation selection and GB bi-crystallography. They showed that, in the CA model, the orientation is governed by the favorably oriented grain (FOG) criterion and the geometrical limit (GL) in the limit of large and small cell size, respectively. The present study focuses on exploring how to quantitatively bridge length scales between PF simulations that resolve the whole solid-liquid interface dynamics and the CA model that resolves the dynamics of the grain envelope under a certain set of assumptions. For this purpose, we study grain boundary (GB) orientation selection as a function of the imposed temperature gradient G under the frozen-temperature approximation, which allows us to vary G-dependent microstructural length scales with a fixed GB bi-crystallography. PF simulations reveal the existence of a transition from FOG to GL dominance with decreasing G at non-degenerate bi-crystallography. Simulations further reveal that this transition can be quantitatively reproduced by the CA model with a choice of the cell size that corresponds, in PF simulations, to the active secondary dendrite arm spacing of the favorably oriented grain preceding the tertiary branching event that gives birth to a new stable primary dendrite. PF simulations are also used to obtain a detailed quantitative characterization of the dynamics of the grain envelope and its internal length scales, thereby providing a quantitative test of the inherent approximations of the existing CA approach and paving the way for its future development.