Axisymmetric front tracking model for the investigation of grain structure evolution during directional solidification

A recent numerical investigation by the present authors Battaglioli et al. (2017) had shown that the heat and mass transfer associated with steady-state Bridgman furnace solidification is dependent on advection of sensible and latent heat, and also on the axisymmetric geometry of the crucible. The present work extends the previous one by considering more complex non-steady solidification scenarios, where transient (limited-duration) Bridgman solidification is subsequently combined with a controlled power-down cooling process. A significant advancement with respect to the previous model is the inclusion of a front tracking method for the simulation of columnar growth. In the front tracking method the columnar mush region is demarcated by a series of markers that advance at a growth rate governed by the solutal undercooling at the dendrite tips. A classic micro-segregation law is used to govern the evolution of solid fraction in the columnar mush and the release of latent heat. The inclusion of the front tracking method has provided greater insights into how heat fluxes and thermal conditions can influence the final grain structure. Firstly, we show that by analysing the trajectories of the markers, it is possible to predict the transition from axial columnar growth (directional solidification) to unwanted radial columnar growth due to significant radial heat fluxes in the sample. Secondly, we show that model can simulate an undercooled liquid region ahead of the columnar front where equiaxed grains could possibly nucleate and grow, leading to a columnar to equiaxed transition (CET). Two indirect CET prediction methods from literature have been included to the model to assess the likelihood of a CET occurring in the as-cast grain structure of the alloy. The model was employed to simulate experimental scenarios from literature involving a γ-TiAl alloy where the samples were subjected to different transient cooling conditions in a Bridgman furnace. Examples of axial columnar growth, radial columnar growth, and CET are discussed in detail with elucidation from the model.