| Form casting, precision casting, soldering, welding are all manufacturing processes that involve solidification. In casting, solid nuclei appear in the liquid, leading to the formation of columnar or dendritic (tree-like structures) crystal structures. The type of microstructures that forms, whether dendritic or columnar, will influence such important material properties as strength and toughness. In this paper a computer simulation of a dendrite in three dimensions (3-D) based on the cellular model is described. Our model assumes constant temperature and the dendrite develops due to concentration gradients. The simulation is carried out for Al-Si-Cu alloys. Concentration profiles in three dimensions are obtained from the transport equation of solute diffusion and changes in concentration due to phase field changes. Concentration at each cell with a liquid fraction is calculated for every time step. The interface velocity is calculated from the curvature at the interface, concentrations of each alloying component, undercooling and a kinetic coefficient constant. The time step and the velocity provide a new position of the solid-liquid interface for each iteration. In this model, curvature at each cell is calculated using an averaged phase field. The average phase field in 3-D is obtained by multiplying the solid fraction of each cell and its neighbors by weight factors. This model takes anisotropy effects into account in order to avoid splitting of the dendrite tips. The data obtained from the model allow us to analyze the dendrite morphology and parameters such as tip radius, spacing between secondary arms, growth velocity and their dependence on undercooling. It is expected that the dendrite tip will grow with a stable parabola-like shape while the growth at the sides of the dendrite will be unstable. This instability at the solid liquid interface can result in the growth of secondary arms. The results are compared with analytical model data. |
