Solidification of silicon in a one-dimensional slab and a two-dimensional wedge

We have developed mathematical models in both one and two spatial dimensions for the solidification of silicon. The one-dimensional model describes slab casting related to a set of thin-casting experiments. The model is fitted to thermocouple data and accounts for various heat transfer mechanisms as well as the latent heat. The model can be used to predict the time taken for the material to completely solidify and the solidification distance (the point where solidification fronts meet which can be observed as a discontinuity in the grain microstructure). Simple approximate analytical results, which agree very well with the full-scale numerical solutions on Matlab and COMSOL, are provided. The two-dimensional model relates to a wedge casting experiment where, again, various heat transfer mechanisms and latent heat need to be accounted for. Experimental data from thermocouples is used to quantify the heat transfer coefficients by fitting to two-dimensional COMSOL simulations. A very simple analytical “Triangle model” is derived by assuming that the solidification fronts move as flat surfaces from each of the two wedge walls and the air surface, independently of each other, as three separate one-dimensional quasi-steady approximations. This model predicts that the area of liquid silicon will diminish as shrinking self-similar triangles. This simplified model provides analytical results for the solidification time and distances which agree very well with the COMSOL simulations.