Numerical simulation of dendritic growth of binary alloys
A two-dimensional finite element model for simulation of dendritic solidification of binary alloys is developed. The model solves the coupled time-dependent temperature and solute concentration equations on two independent meshes: a fixed mesh for the temperature and an adaptive interface-conforming mesh for the concentration. The temperature is solved on the whole domain while the concentration is solved only on the liquid region because diffusion in the solid is much smaller than that in the liquid and can be neglected. Temperature and concentration are coupled at the interface through the generalized Gibbs-Thompson relation. The solid-liquid interface is explicitly tracked with a set of marker points that defines its position at all times. Latent heat of fusion, interfacial energy, kinetic mobility and crystalline anisotropy are taken into account. The adaptive mesh is generated at every time step as the interface position changes. The model is easy to use in the sense that it works with physical variables as opposed to those based on the phase-field variable and level-set method. The model is very accurate as demonstrated by a series of calculations that compare to exact solutions or predictions by solidification theories. In simulations of solidification of pure materials, where only the energy equation is solved, the model produces very complicated dendritic structures that are in close agreement with experimental observations, and the computation is very efficient. Calculations under a variety of conditions show that the undercooling and surface tension are the main factors that determine the final dendritic structures. In simulations of alloy solidification, where both the energy and solutal concentration equations are solved, the results for the onset of interface instability and the prediction of the most unstable wavelength are in good agreement with linear stability theory. When applied to simulations of directional solidification of Pb-Sb alloys, the model generates dendritic structures and solute segregation similar to those observed experimentally, and the interface-development from a planar form to cells to dendritic structures is clearly demonstrated. These simulations are the first of their kind.
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