Three-dimensional adaptive finite volume scheme for transport phenomena in materials processing: Application to Czochralski crystal growth

A large class of materials processes involve the presence of multiple phases, moving interfaces and a variety of driving forces in addition to three dimensional behavior. Numerical manifestation of such processes demands generalized solution schemes in conjunction with adaptive grid generation methodology. A three dimensional curvilinear finite-volume based numerical algorithm is developed to model such processes. A body-fitted transformation based approach is adopted in conjunction with a multizone adaptive grid generation (MAGG) technique to accurately handle the three-dimensional problems of phase-change in irregular geometries with free and moving surfaces. The numerical model has been extensively validated by solving a variety of problems ranging from natural convection in a cuboid to fluid-superposed porous layer in a cylindrical system (hydrothermal growth) as well as to non-uniform solidification in a cavity.; The multizone adaptive model is then used to perform three-dimensional simulation of Czochralski growth silicon single crystals. The demand for large diameter, defect-free, single crystal has rendered accurate full three-dimensional modeling of this process absolutely necessary. It is found that the flow and temperature fields in the melt become asymmetric for various combinations of governing parameters even in the presence of symmetric external conditions. The melt/crystal interface also becomes asymmetric in shape and can oscillate under these conditions. This prediction is very important from crystal quality point of view as thermal stresses in the crystal and dopant distribution are strongly coupled to the flow and temperature fields. Although computing power remains a major hindering factor, the present algorithm can successfully model three-dimensional multiphase, multicomponent systems.