Three dimensional modelling of grain boundary interaction and evolution during directional solidification of multi-crystalline silicon

•Three dimensional phase filed model for grain boundary interaction and evolution.•Anisotropic energy and mobility for CSL and coherent grain boundaries.•Describing the grain boundaries using the normal vectors.•Twist and tilt sigma twin boundaries are considered.•Good agreement with theory and experiments. The development of grain structures during directional solidification of multi-crystalline silicon (mc-Si) plays a crucial role in the materials quality for silicon solar cells. Three dimensional (3D) modelling of the grain boundary (GB) interaction and evolution based on phase fields by considering anisotropic GB energy and mobility for mc-Si is carried out for the first time to elucidate the process. The energy and mobility of GBs are allowed to depend on misorientation and the GB plane. To examine the correctness of our method, the known the coincident site lattice (CSL) combinations such as (∑a+∑b→∑a×b) or (∑a+∑b→∑a/b) are verified. We frther discuss how to use the GB normal to characterize a ∑3 twin GB into a tilt or a twist one, and show the interaction between tilt and twist ∑3 twin GBs. Two experimental scenarios are considered for comparison and the results are in good agreement with the experiments as well as the theoretical predictions.