*2.1. CA Model*

The solute equilibrium model (ZS model) proposed by Zhu [14] is used in this paper. The growth driving force is the difference between the equilibrium crystallization concentration and the actual liquid concentration at the interface. The actual liquid concentration *CL* can be calculated by LBM, and the equilibrium crystallization concentration *Ceq <sup>L</sup>* can be calculated by the following formula according to the equilibrium crystallization theory:

$$\mathbf{C}\_{L}^{eq} = \mathbf{C}\_{0} + \left[T\_{L} - T\_{L}^{eq} + \Gamma \mathbf{K} f(\phi\_{\prime}, \theta\_{0})\right] / m \tag{1}$$

where *C*<sup>0</sup> is the initial concentration of the alloy; m is the slope of the liquidus; *TL* is the actual temperature of the interface; *Teq <sup>L</sup>* is the liquidus temperature at the initial concentration of *C*0, Γ is the Gibbs Thomson coefficient; K is the average curvature at the solid/liquid interface, *f*(φ, θ0) is the anisotropic function of the interface energy. K can be calculated from the spatial distribution of the interface solid phase ratio.

$$\mathcal{K} = \left[ \left( \frac{\partial f\_{\mathcal{S}}}{\partial \mathbf{x}} \right)^{2} + \left( \frac{\partial f\_{\mathcal{S}}}{\partial \mathbf{y}} \right)^{2} \right]^{-3/2} \cdot \left[ 2 \frac{\partial f\_{\mathcal{S}}}{\partial \mathbf{x}} \frac{\partial f\_{\mathcal{S}}}{\partial \mathbf{y}} \frac{\partial^{2} f\_{\mathcal{S}}}{\partial \mathbf{x} \partial \mathbf{y}} - \left( \frac{\partial f\_{\mathcal{S}}}{\partial \mathbf{x}} \right)^{2} \frac{\partial^{2} f\_{\mathcal{S}}}{\partial \mathbf{y}^{2}} - \left( \frac{\partial f\_{\mathcal{S}}}{\partial \mathbf{y}} \right)^{2} \frac{\partial^{2} f\_{\mathcal{S}}}{\partial \mathbf{x}^{2}} \right] \tag{2}$$

According to the Gibbs-Thomson formula, the interface energy anisotropy function in Equation (1) can be expressed as:

$$f(\phi\_\prime \theta\_0) = \Psi(\phi\_\prime \theta\_0) + \frac{\partial^2}{\partial \phi^2} \Psi(\phi\_\prime \theta\_0) = 1 - \delta \cos[4(\phi - \theta\_0)] \tag{3}$$

where δ = 15ε is the anisotropy coefficient (ε is the anisotropic strength of the interface energy), ψ(φ, θ0) is the anisotropy function of the interface energy, φ is the angle between the normal direction of the solid-liquid interface and the horizontal direction, and θ<sup>0</sup> is the preferred growth direction. The anisotropy function ψ(φ, θ0) and the growth angle φ can be calculated from Equations (4) and (5).

$$\psi(\phi, \theta 0) = 1 + \varepsilon \cos[4(\phi - \theta 0)] \tag{4}$$

$$\phi = \begin{cases} \begin{array}{cc} \cos^{-1}\left[\frac{\partial f\_S}{\partial x}\left(\left(\frac{\partial f\_S}{\partial x}\right)^2 + \left(\frac{\partial f\_S}{\partial y}\right)^2\right)^{-1/2}\right] & \frac{\partial f\_S}{\partial y} \ge \mathbf{0} \\\ 2\pi - \cos^{-1}\left[\frac{\partial f\_S}{\partial x}\left[\left(\frac{\partial f\_S}{\partial x}\right)^2 + \left(\frac{\partial f\_S}{\partial y}\right)^2\right]^{-1/2}\right] & \frac{\partial f\_S}{\partial y} < \mathbf{0} \end{array} \tag{5}$$

When*CL* <sup>&</sup>lt; *Ceq <sup>L</sup>* , the solid fraction increment Δ*fs* in a time step is calculated by the following formula:

$$
\Delta f\_{\mathbb{S}} = \frac{(\mathbb{C}\_L^{eq} - \mathbb{C}\_L)}{\mathbb{C}\_L^{eq} (1 - k)} \tag{6}
$$

In order to partially eliminate anisotropy, this paper uses an improved eight-neighbor capture method proposed by Zhu [15].
