## 3 Results and Discussion  
### Temperature Distribution  
The temperature distribution has a strong influence on crystal growth velocity, interface shape and grain boundaries during the crystal growth process. A suitable temperature field may decrease the defects in silicon ingot and increase the quality of silicon wafers. Therefore, it is necessary to discuss the temperature field during the process of silicon crystal growth. According to the Procast software, the range of the heat transfer coefficient is about 0-2000 W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\) between inorganic nonmetallic materials and air. In this part, we analyzed the effect of the side boundary conditions on temperature field within the computational area. The heat transfer coefficients were 0 and 1000 W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\) at the top and bottom, respectively. The temperature distribution is shown in Figures 3 and 4 with different heat transfer coefficients on the side boundaries, where the temperature isotherms of the melting point represent the crystal-melt interface at different times. The color of the isotherms reflects the temperature value, with red representing the maximum temperature and deep blue representing the minimum temperature. As clearly shown in Figures 3a and 4a, the computational area is full of silicon melt at an early stage of crystal growth. The temperature isotherms are straight and the maximum temperature gradient appears at the bottom boundary, which is attributed to the fact that heat transfer between environment and silicon melt is more intense. It is beneficial for crystal growth in that it releases the latent heat of crystallization with a high temperature gradient. In the columnar crystal zone, heat \begin{table}
\begin{tabular}{c c} \hline
**Property** & **Value** \\ \hline Melting temperature, \(T_{mel}\) (K) & 1687 \\ Initial temperature, \(T_{ini}\) (K) & 1750 \\ Density of silicon solid, \(\rho_{s}\) (kg\(\cdot\)m\({}^{-3}\)) & 2330 \\ Density of silicon melt, \(\rho_{m}\) (kg\(\cdot\)m\({}^{-3}\)) & \(-0.206\times T\) + 2930.94 \\ Thermal conductivity of silicon solid, \(\kappa_{s}\) (W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\)) & 22 \\ Thermal conductivity of silicon melt, \(\kappa_{m}\) (W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\)) & 66.5 \\ Heat capacity of silicon solid, \(C_{s}\) (J\(\cdot\)kg\(\cdot\)1\(\cdot\)K\({}^{-1}\)) & \(-1.354\times 10^{-4}\times T^{2}+0.4845\times T\) + 584.3 \\ Heat capacity of silicon melt, \(C_{m}\) (J\(\cdot\)kg\({}^{-1}\cdot\)K\({}^{-1}\)) & 950 \\ Latent heat, \(Q\) (J\(\cdot\)kg\({}^{-1}\)) & \(1.587\times 10^{6}\) \\ Microcosmic cell size, \(\Delta x_{mic}\) (m) & \(1\times 10^{-4}\) \\ Macrocosmic cell size, \(\Delta x_{mac}\) (m) & \(1\times 10^{-2}\) \\ Standard deviation, \(\Delta T\sigma\) (K) & 0.1 \\ Mean undercooling, \(\Delta T_{ma}\) (K) & 1.0 \\ Maximum nucleation density, \(n_{\max}\) (m\({}^{2}\)) & \(1\times 10^{4}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Physical properties and calculation parameters used in the present model. transfer is the dominant factor for crystal growth and the isothermal surface is likely to reflect the shape of the crystal-melt interface. As shown in Figure 3, when the heat transfer coefficient is 100 W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\) on the side boundaries, the shape of the crystal-melt interface changes from straight to slightly curved because the heat transport near the side boundaries is faster than that in the bottom boundary with the increase of crystal length. It can be seen that the temperature distribution at the center point of silicon melt is more uniform when the process of solidification is finished. In this case, the quality of the mc-Si ingot in the center position is much better than that near the side and bottom boundaries where the silicon ingot having higher temperature gradient. In order to get more straight crystal-melt interface, The temperature field was optimized by means of adjusting the heat transfer coefficient of the side boundaries.