*2.2. LBM Model*

According to Boussinesq's approximation, the effect of latent heat and solutes on the density during the solidification process can be expressed by the following formula:

$$\rho = \rho\_0 \left[ 1 - \beta\_T (T - T\_0) - \beta\_C (\mathbb{C} - \mathbb{C}\_0) \right] \tag{7}$$

where ρ0, *T*0, and *C*<sup>0</sup> represent the initial density, temperature, and concentration of the liquid phase, respectively, and *T* and *C* are the temperature and concentration of the liquid phase at the current

moment. β*<sup>T</sup>* and β*<sup>C</sup>* are the volume expansion coefficients of temperature and concentration changes, respectively. The resultant force of the fluid particles is:

$$\begin{array}{lcl} \mathbf{F} &= \mathbf{g}\rho\_0 \left[ 1 - \beta\_T (T - T\_0) - \beta\_\mathbf{C} (\mathbf{C} - \mathbf{C}\_0) \right] + (-\rho\_0 \mathbf{g}) \\ &= \mathbf{g}\rho \mathbf{o} \left[ -\beta\_T (T - T\_0) - \beta\_\mathbf{C} (\mathbf{C} - \mathbf{C}\_0) \right] \end{array} \tag{8}$$

The distribution function of the flow field can be expressed as:

$$f\_i(\mathbf{x} + \mathbf{e}\_i \Delta t, t + \Delta t) = f\_i(\mathbf{x}, t) + \frac{1}{\tau\_f} (f\_i^{eq}(\mathbf{x}, t) - f\_i(\mathbf{x}, t)) + F\_i \tag{9}$$

where *Fi* is the component force of the particle under the external force field in the i direction, and its magnitude is expressed as:

$$F\_i = (1 - \frac{1}{2\pi\_f})\omega\_i[3\frac{\mathbf{e}\_i - \mathbf{u}}{c^2} + 9\frac{\mathbf{e}\_i \cdot \mathbf{u}}{c^4}]\Delta t \cdot \mathbf{F} \tag{10}$$

where ω*<sup>i</sup>* is the weight coefficient in each direction, which represents the probability of particles moving in different directions, which can be expressed as:

$$\omega\_{i} = \begin{cases} 4/9 & i = 0 \\ 1/9 & i = 1, 2, 3, 4 \\ 1/36 & i = 5, 6, 7, 8 \end{cases} \tag{11}$$

The equilibrium distribution function and relaxation time of the flow field in Equation (9) are respectively expressed as:

$$f\_i^{eq}(\mathbf{x}, t) = \omega\_i \rho (1 + 3\frac{\mathbf{e}\_i \cdot \mathbf{u}}{c^2} + \frac{9}{2} \frac{(\mathbf{e}\_i \cdot \mathbf{u})^2}{c^4} - \frac{3}{2} \frac{\mathbf{u} \cdot \mathbf{u}}{c^2}) \tag{12}$$

$$
\pi\_f = 3\nu/(c^2\Delta t) + 0.5\tag{13}
$$

where ν is the dynamic viscosity of the fluid. The macroscopic density ρ and velocity *u* are obtained by adding the distribution function.

$$\rho = \sum\_{i=0}^{8} f\_i \tag{14}$$

$$\mu = \left(\sum\_{i=0}^{8} \mathbf{e}\_i f\_i + \mathbf{F} \cdot \Delta \mathbf{t} / 2\right) / \rho \tag{15}$$

The distribution functions of temperature field and solute field are similar to those of flow field:

$$h\_i(\mathbf{x} + \mathbf{e}\_i \Delta t, t + \Delta t) = h\_i(\mathbf{x}, t) + \frac{1}{\tau\_\alpha} (h\_i^{eq}(\mathbf{x}, t) - h\_i(\mathbf{x}, t)) + H\_i \tag{16}$$

$$\mathcal{g}\_i(\mathbf{x} + \mathbf{e}\_i \Delta t, t + \Delta t) = \mathcal{g}\_i(\mathbf{x}, t) + \frac{1}{\tau\_D} (\mathcal{g}\_i^{eq}(\mathbf{x}, t) - \mathcal{g}\_i(\mathbf{x}, t)) + G\_i \tag{17}$$

where *hi*(*x*,*t*) and *gi*(*x*,*t*) are the distribution functions of temperature field and solute field at the position x at time t, respectively, and *hi*(*x*,*t*) and *gi*(*x*,*t*) represent the temperature field and the equilibrium distribution function of the solute field, respectively defined as:

$$h\_i^{eq}(\mathbf{x}, t) = \alpha\_i T(1 + 3\frac{\mathbf{e}\_i \cdot \mathbf{u}}{c^2} + \frac{9}{2}\frac{(\mathbf{e}\_i \cdot \mathbf{u})^2}{c^4} - \frac{3}{2}\frac{\mathbf{u} \cdot \mathbf{u}}{c^2}) \tag{18}$$

$$\mathcal{g}\_i^{eq}(\mathbf{x}, t) = \omega\_i \mathbb{C} (1 + 3 \frac{\mathbf{e}\_i \cdot \mathbf{u}}{c^2} + \frac{9}{2} \frac{(\mathbf{e}\_i \cdot \mathbf{u})^2}{c^4} - \frac{3}{2} \frac{\mathbf{u} \cdot \mathbf{u}}{c^2}) \tag{19}$$

The relaxation time τα in the temperature field and the relaxation time τ*<sup>D</sup>* in the solute field can be obtained by using the corresponding diffusion coefficients:

$$
\tau\_a = 3\alpha / (c^2 \Delta t) + 0.5\tag{20}
$$

$$
\pi\_D = 3D/(c^2 \Delta t) + 0.5\tag{21}
$$

where α is a temperature diffusion coefficient, and *D* is a concentration diffusion coefficient. Macro temperature and concentration are:

$$T = \sum\_{i=0}^{8} h\_i(\mathbf{x}, t); \mathbf{C} = \sum\_{i=0}^{8} g\_i(r, t) \tag{22}$$

The source term *Hi* of the temperature field and the source term *Gi* of the concentration field can be expressed as:

$$H\_{\rm i} = \omega\_{\rm i} \Delta T; G\_{\rm i} = \omega\_{\rm i} \Delta \mathcal{C} \tag{23}$$

In the formula, Δ*T* and Δ*H* respectively represent the latent heat released by the solidification of the alloy and the excluded solutes.

#### *2.3. Ladd Method to Calculate the Solid-liquid Interface Interaction Force*

In Figure 1, *xb* is the particle boundary point, *xl* is the liquid phase lattice point of the boundary node along the *c*−*<sup>i</sup>* direction, *xs* is the solid phase lattice point of the boundary node along the *ci* direction, and *ub* is the particle velocity. The calculation formulas of *xs* and *xb* are:

$$\mathbf{x}\_s = \mathbf{x}\_l + \Delta t \cdot \mathbf{c}\_i \tag{24}$$

$$
\mu\_b = V\_b + \mathcal{W}\_b(\mathbf{x}\_b - \mathbf{x}\_c) \tag{25}
$$

where Δ*t* is the time step, *Vb* and *Wb* are the translational and rotational speeds, respectively, and *xc* is the center of mass of the solid particles. The solid-liquid distribution functions at time t are *fi*(*xl*,*t*) and *f*−*i*(*xs*,*t*). After Δ*t*/2 time, the two particles move to the boundary and collide. The distribution function at this time is:

$$f\_{-i}(\mathbf{x}\_b, t + \Delta t/\mathbf{2}) = f\_i(\mathbf{x}\_l, t) + \mathbf{2}\omega\_{-i}\rho \frac{c\_{-i}\mu\_b}{c\_s^2} \tag{26}$$

$$f\_i(\mathbf{x}\_b, t + \Delta t/\mathbf{2}) = f\_{-i}(\mathbf{x}\_b, t) + \mathbf{2}a\_i \rho \frac{c\_i u\_b}{c\_s^2} \tag{27}$$

**Figure 1.** Sketch map of half step rebound format.

After the Δ*t*/2 time, the fluid particles bounce to the corresponding lattice points respectively. At this time, the distribution functions of the liquid phase and solid phase lattice points are:

$$f\_{-i}(\mathbf{x}\_{l\prime}t + \Delta t) = f\_{-i}(\mathbf{x}\_{b\prime}t + \Delta t/\mathbf{2})\tag{28}$$

*Crystals* **2020**, *10*, 70

$$f\_i(\mathbf{x}\_{\nu}, t + \Delta t) = f\_i(\mathbf{x}\_{\nu}, t + \Delta t/\mathbf{2}) \tag{29}$$

Force exerted by fluid particles on solid particles:

$$F\_i = \frac{\Delta \mathbf{x}^2}{\Delta t} \Big[ f\_{-i} \Big( \mathbf{x}\_{f'}, t + \Delta t \Big) + f\_i \Big( \mathbf{x}\_{f'}, t \Big) - f\_i (\mathbf{x}\_{s\prime}, t + \Delta t) - f\_{-i} (\mathbf{x}\_{s\prime}, t) \Big] \mathbf{c}\_i \tag{30}$$

The total force *F* on the solid particles is:

$$F = \frac{\Delta x^2}{\Delta t} \sum\_{\mathbf{x}\_b} \sum\_i \left[ f\_{-i}(\mathbf{x}\_{f'}, t + \Delta t) + f\_i(\mathbf{x}\_{f'}, t) - f\_i(\mathbf{x}\_{s'}, t + \Delta t) - f\_{-i}(\mathbf{x}\_{s'}, t) \right] \mathbf{c}\_i \tag{31}$$

The force moment on the solid particles is:

$$T\_{\mathbf{t}} = \sum\_{\mathbf{x}\_{\mathbf{b}}} F\_{\mathbf{x}\_{\mathbf{b}}} (\mathbf{x}\_{\mathbf{b}} - \mathbf{x}\_{\mathbf{c}}) \tag{32}$$

According to Newton's second law, the translation speed and rotation speed of the grains can be calculated respectively as:

$$V = \frac{F + G}{M\_S} dt\\\mathcal{W} = \frac{T\_\text{t}}{I\_S} dt\tag{33}$$

where *MS* and *IS* are mass and moment of inertia, respectively, and *G* is the combined force of gravity and buoyancy.
