## 2 Description of the Model and Numerical Calculation Method
### LBM Model  
A single relaxation time lattice Bhatnagar-Gross-Krook (LBGK) method, the D2Q9 model, were employed [17]. In consideration of latent heat source term and concentration source term, the particles' space-time evolution equations used to calculate velocity, temperature and solute content field's distribution function are given as follows [18]. \[f_{i}(\mathbf{r}+\mathbf{e}_{i}\Delta t,t+\Delta t)=f_{i}(\mathbf{r},t)+\frac{1}{\tau_{f} }\Big{(}f_{i}^{eq}(\mathbf{r},t)-f_{i}(\mathbf{r},t)\Big{)}+F_{i} \tag{1}\]\[h_{i}(\mathbf{r}+\mathbf{e}_{i}\Delta t,t+\Delta t) =h_{i}(\mathbf{r},t)+\frac{1}{\tau_{t}}\Big{(}h_{i}^{eq}(\mathbf{r},t)-h_{i }(\mathbf{r},t)\Big{)}+H_{i} \tag{2}\] \[g_{i}(\mathbf{r}+\mathbf{e}_{i}\Delta t,t+\Delta t) =g_{i}(\mathbf{r},t)+\frac{1}{\tau_{c}}\Big{(}g_{i}^{eq}(\mathbf{r},t)-g_{ i}(\mathbf{r},t)\Big{)}+G_{i} \tag{3}\]  
In above equation, \(\mathbf{r}\) and \(t\) represent position and time, respectively. \(\Delta t\) is time step. \(\mathbf{e}_{i}\) is the speed of \(i\) direction. \(\tau_{f}\), \(\tau_{t}\) and \(\tau_{c}\) are the relaxation time of velocity, temperature and solute content field respectively, then the relaxation time shows the speed level of distribution functions of each particles (\(f_{i}\), \(h_{i}\) and \(g_{i}\)) close to equilibrium distribution functions of particles (\(f_{i}^{eq}\), \(h_{i}^{eq}\) and \(g_{i}^{eq}\)). \(f_{i}\), \(h_{i}\) and \(g_{i}\) are the distribution functions of particles of velocity, temperature and solute content field respectively. \(f_{i}^{eq}\), \(h_{i}^{eq}\) and \(g_{i}^{eq}\) are the equilibrium distribution functions of velocity, temperature and solute content field, respectively. \(F_{i}\), \(H_{i}\) and \(G_{i}\) are the source terms of external force, heat and concentration, respectively. The source term of external force \(F_{i}\) shows interaction force on particles. The heat source term \(H_{i}\) and concentration source term \(G_{i}\) are the latent heat release and solute redistribution during the growth of multi-crystalline silicon. In the D2Q9 format, the discrete particle velocity \(\mathbf{e}_{i}\) (\(i=0\sim 8\)) in lattice point can be expressed as  
\[\mathbf{e}_{i}=\left\{ \begin{array}{cc}(0,0)&i=0\\ (\cos[(i-1)\pi/2],\sin[(i-1)\pi/2])c&i=1,2,3,4\\ (\cos[(2i-9)\pi/4],\sin[(2i-9)\pi/4])c&i=5,6,7,8\end{array} \right. \tag{4}\]  
In above equation, \(c=\Delta x/\Delta t\) is the grid velocity, \(\Delta x\) is the space step. The macroscopic physical quantity of the fluid, such as fluid density \(\mathbf{\rho}\), speed \(\mathbf{u}\), concentration \(C_{I}\) and temperature \(T\) can be obtained by the corresponding distribution functions of particles (Equation (5)-(8))  
\[\mathbf{\rho}=\sum_{i=0}^{8}f_{i} \tag{5}\]  
\[\mathbf{\rho}\mathbf{u}=\sum_{i=0}^{8}f_{i}\mathbf{e}_{i}+\frac{\Delta t}{2}\mathbf{F} \tag{6}\]  
\[C_{I}=\sum_{i=0}^{8}g_{i} \tag{7}\]  
\[T=\sum_{i=0}^{8}h_{i} \tag{8}\]  
where \(\mathbf{F}\) is the buoyancy of liquid. According to the approximation of [PERSON], the density in the buoyancy term is the linear function of the temperature gradient and the concentration gradient. Hence, the buoyancy term approximates to the equation of (9)  
\[\mathbf{F}=\mathbf{g}\rho_{0}[\beta_{T}(T-T_{0})+\beta_{C}(C-C_{0})] \tag{9}\]  
where \(\mathbf{\rho}_{0}\) is the density when the temperature is \(T_{0}\) and the concentration is \(C_{0}\). \(\mathbf{g}\) is the acceleration of gravity. \(\beta_{T}\) and \(\beta_{C}\) are the expansion coefficients of temperature and concentration respectively. In Equation (1)-(3), the external force term \  
where \(w_{i}\) is the weight coefficient, \(\Delta f_{s}\) is the increment of solid fraction, \(\Delta H_{\text{f}}\) is the molar latent heat, \(C_{\text{p}}\) is the molar liquid heat capacity, \(C_{\text{f}}\) is the liquid phase solute component in the unit, \(k\) is the solute partition coefficient. The value of \(w_{i}\) is the following Equation (13). \[w_{i}=\left\{ \begin{array}{ll}4/9&i=0\\ 1/9&i=1,2,3,4\\ 1/36&i=5,6,7,8\end{array} \right.