## 2 Description of the Model and Numerical Calculation Method
### CA Model
#### 2.2.1 Nucleation  
This work uses a continuous nucleation model based on Gaussian distribution [19]. The density of nucleation \(n(\Delta T)\) can be expressed by  
\[n(\Delta T)=\int_{0}^{\Delta T}\frac{dn}{d(\Delta T)}\cdot d(\Delta T) \tag{20}\]  
In the equation, \(\Delta T\) is undercooling, \(dn/d(\Delta T)\) represents an increase in nucleation density \(dn\) when the undercooling \(d(\Delta T)\) is increased, and the calculation of \(dn/d(\Delta T)\) under the Gaussian normal distribution is given by  
\[\frac{dn}{d(\Delta T)}=\frac{n_{\text{max}}}{\sqrt{2\pi}\Delta T_{\sigma}}\exp[ -\frac{1}{2}(\frac{\Delta T-\Delta T_{meta}}{\Delta T_{\sigma}})^{2}] \tag{21}\]where \(n_{\text{max}}\) is the maximum nucleation density, \(\Delta T_{\sigma}\) is the standard deviation and \(\Delta T_{mea}\) is the mean undercooling. In addition, then the above function is neatened as follows:  
\[n(\Delta T)=\frac{n_{\text{max}}}{2}\bigg{[}\text{tanh}\bigg{(}\frac{\Delta T- \Delta T_{\text{max}}}{1.25\Delta T_{\sigma}}\bigg{)}+\text{tanh}\bigg{(} \frac{\Delta T_{\text{max}}}{1.25\Delta T_{\sigma}}\bigg{)}\bigg{]} \tag{22}\]  
where \(\Delta T_{\text{max}}\) is maximum nucleation undercooling. We have considered two different models to find nucleation points in our system. One is nucleation at silicon-crucible interface as a result of the interfacial interaction between the molten silicon and crucible. The other is nucleation on SiC particles as shown in Figure 1. In fact, during the solidification process, carbon rejection at the solid-liquid interface leads to an increase of the carbon concentration in the silicon liquid. When carbon concentration reaches the maximum solubility limit, the silicon reacts with carbon to form SiC, allowing SiC particles to precipitate into the Si-melt (i.e., the supersaturation with respect to SiC) [20]. The solubility limit of carbon in Si-melt \(C_{L}(T)\) is expressed by a polynomial function [21]:  
\[C_{L}(T)=8.6250\times 10^{-4}T^{2}-2.7643T+2222.9 \tag{23}\]  
\(n_{\text{max}}\) is generally given in a three-dimensional form, therefore in this paper we convert it into two-dimensional:  
\[n_{S}^{*}=\sqrt{\frac{4n_{S}}{\pi}} \tag{24}\]  
\[n_{V}^{*}=\sqrt[3]{\frac{6n_{V}^{2}}{\pi}} \tag{25}\]  
where \(n_{S}^{*}\) and \(n_{S}\) are the maximum nucleation density of silicon-crucible interface for the two-dimensional and three-dimensional, respectively, \(n_{V}^{*}\) and \(n_{V}\) are the maximum nucleation density of impurities sites for the two-dimensional and three-dimensional, respectively. Consequently, the probability of nucleation is written as:  
\[P=\frac{\delta N}{N}=\delta n\cdot V_{C} \tag{26}\]  
where \(\delta N\) is the total number of new grains, \(N\) is the total number of cells, \(V_{C}\) is a cell volume. The nucleation probability \(P\) of any cell cannot be too large, its range is \([0,1)\). Compared probability nucleation \(P\) with a random number \(r(0<r<1)\), and if \(P>r\), the cell will be nucleated, whereas the cell cannot be nucleated [19].  
Figure 1: Nucleation at silicon-crucible interface and on SiC particles.