## 2 Description of the Model and Numerical Calculation Method
### CA Model
#### 2.2.2 Model for the Growth of the Columnar Faceted Interface  
The undercooling of the front of the solid-liquid interface consists of solute diffusion undercooling, thermodynamic undercooling and curvature undercooling. \[\Delta T=m(C_{l}-C_{0})+T_{L}-T+\Gamma Kf(\theta,\phi) \tag{27}\]  
In above equation, \(m\) is the liquidus slope, \(C_{l}\) is the actual solute concentration of the liquid phase in the cell, \(C_{0}\) is the initial concentration, \(T_{L}\) is the liquidus temperature, \(T\) is the node temperature of the cell, \(\Gamma\) is Gibbs-Thomson coefficient given by \(\frac{\gamma_{S}l}{\Delta S_{l}}\), \(K\) is the interface curvature given by the following Equation (28) [22]. \[K=\left[\left(\frac{\partial f_{s}}{\partial x}\right)^{2}+\left(\frac{ \partial f_{s}}{\partial y}\right)^{2}\right]^{\frac{-3}{2}}\left[2\frac{ \partial f_{s}}{\partial x}\frac{\partial f_{s}}{\partial y}\frac{\partial^{2 }f_{s}}{\partial x\partial y}-\left(\frac{\partial f_{s}}{\partial x}\right)^ {2}\frac{\partial^{2}f_{s}}{\partial y^{2}}-\left(\frac{\partial f_{s}}{ \partial y}\right)^{2}\frac{\partial^{2}f_{s}}{\partial x^{2}}\right] \tag{28}\]  
\(f(\theta,\phi)\) is the interfacial anisotropy function given by the following Equation (29) [23]. \[f(\theta,\phi)=1+\varepsilon\cos(4(\phi-\theta)) \tag{29}\]  
In above equation, \(\varepsilon\) is the anisotropy strength, \(\phi\) is the angle between the local growth direction and the X axis direction, \(\theta\) is the crystallographic orientation. The relationship between the growth rate and the undercooling can be written as  
\[v_{\rm e}=\mu(\phi)\Delta T^{2} \tag{30}\]  
where \(v_{\rm e}\) is the growth rate, \(\Delta T\) is the kinetic undercooling. \(\mu(\phi)\) is the kinetic coefficient equation obtained by the equation of (31) [24]  
\[\mu(\phi)=\overline{\mu}q(\phi) \tag{31}\]  
where \(\overline{\mu}\) is the angular average of the kinetic coefficient, \(q(\phi)\) is the anisotropy function obtained by the equation of (32). \[q(\phi)=1-\delta+\delta\text{tanh}\frac{\lambda}{|\tan 2\phi|} \tag{32}\]  
Here, \(\delta\) and \(\lambda\) are parameters. In the case of the presence of undercooling, once the cell has nucleated, it will grow with a preferential direction relying on its crystallographic orientation. The growth rate is calculated by Equation (30), and the horizontal projection length of crystal growth \(l(t)\) can be given by (33) [25]  
\[l(t)=\frac{\int_{t}^{t+\delta t}v_{e}(\Delta T(t))dt}{\cos\theta+\sin\theta} \tag{33}\]  
where \(\delta t\) is time step. The solid fraction \(f_{s}(t)\) can be obtained with  
\[f_{s}(t)=\frac{l(t)}{\delta_{x}} \tag{34}\]  
where \(\delta_{x}\) is the distance between two adjacent cells. When \(f_{s}(t)\geq 1\), which means the growth front of the solid cell touches the center of its neighboring liquid cell, the neighboring cell transforms its state from liquid to solid and gets the same orientation index as this solid cell [26]. [PERSON] [27] proposed that the undercooling at the growth front is the key parameter to divide growth behaviors. The grain growth behavior is expected to be changed between \(\Delta T=3.8\) and \(6.3\) K. When the moving velocity of solid-liquid growth interface is slow and the interfacial morphology is flat, the \((111)\) grain extended to lateral direction during growth. In this simulation, the solid-liquid interfacefront-side undercooling fluctuates between 1.5 and 0.4 K. Its value is less than 3.8 K and the moving velocity of solid-liquid growth interface is slow and the interfacial morphology is flat. Therefore, this work mainly simulates the growth morphology of the dominant grain orientation \(<111>\). An integer in the range of \([1-q]\) is randomly selected and assigned to each cell after nucleation, which indicates the crystallographic index (i.e., grain orientation) of that point. Sixty crystallographic indexes \((q=60)\) have been considered in our model.