## 3 Results and Discussion
### Characteristics of Silicon Crystal Growth  
Based on above analysis, we know that the shape of the crystal-melt interface mainly depends on the boundary conditions. In this part, we keep adiabatic conditions on the top and side boundaries while the heat transfer coefficient of 1000 W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\) is used for the bottom boundary.  
Figure 7 shows the process of microstructure of silicon grain growth. Blue represents the melt, and other different colors are used to represent different crystal grains. Thus, the interface between blue and other colors is the crystal-melt interface. The nuclei usually appear at the bottom boundary. At the beginning, most nuclei grow gradually with different sizes. The shape of the crystal-melt interface is zigzag faceted in the preliminary stage and the heat of crystallization is taken away by the melt and mc-Si ingot. The heat flux transports slowly and the zigzag faceted interface propels into silicon melt with lower velocity during the crystal growth, which is in line with references [24; 25].  
Figure 7: The crystal growth at different times: (**a**) \(t=5\) s; (**b**) \(t=50\) s; (**c**) \(t=100\) s; (**d**) \(t=150\) s; (**e**) \(t=200\) s; and (**f**) \(t=250\) s.  
During crystal growth, the sharp corners in front of interfaces disappear gradually and the shape of the crystal interface becomes smoother. The crystal-melt interface is convex to silicon melt and the convexity decreases with the increase of silicon crystal length. Some grain boundaries disappeared because the array of atoms is similar among neighboring columnar crystal zones with similar phase, and the silicon grains with the higher energy may be merged with the lower energy ones. In addition, structure and energy fluctuations exist in the crystal-melt interface during crystal growth. It will release the latent heat in front of the crystal-melt interface during crystallization, resulting in some silicon grains with high energy which will be fused and stop the growth.  
The undercooling is large at the bottom of the computational area because heat transfer is intense between the melt and bottom boundary. Therefore, many fine grains are formed at the bottom of silicon melts. The silicon grains grow quickly due to the higher transfer coefficient. Being influenced by the thermal field, the nuclei appear firstly at the center of the bottom and then the nucleation is expanded to the side boundaries until the process of nucleation is finished at the bottom of the silicon melt.  
As shown in Figures 3, 4 and 7, in conditions of top adiabatic and bottom constant heat flux, the shape of the crystal-melt interface changes from concave to convex with the decrease of the heat transfer coefficient on the side boundaries. In the industrial production of polycrystalline silicon, the concave crystal-melt interface easily leads to many stresses and dislocations, which reduce the crystal quality. Conversely, the flat or a slightly convex crystal-melt interface can reduce stresses and dislocations [26], so reducing the heat transfer coefficient on side boundaries is necessary to produce high-quality polycrystalline silicon crystal.