*3.3. Undercooling*

Kurz et al. [24] developed the KGT (Kurz-Giovanola-Trivedi) model. It was introduced to simulate and calculate the growth process of the dendrite tip. De-Chang et al. [25] proposed that the undercooling of the solid–liquid interface mainly includes temperature, concentration, and curvature in the CA model. The anisotropy of the interface energy has a great effect on the curvature undercooling, so the model must take the interface anisotropy into consideration. At the same time, the degree of undercooling in the solid–liquid interface prerequisite is calculated as [26]:

$$
\Delta T(t\_{\rm h}) = T' - T\_{\rm i,j} + m(\rm C\_0 - \rm C\_l^\*) + \Gamma \mathcal{K}(t\_{\rm h}) \left\{ 1 - 15 G\_{\rm x} \cos \left[ 4(\theta - \theta\_0) \right] \right\} \tag{15}
$$

where *T* is the temperature at the interface, *T*i,j is the temperature of the undercooled melt, *C*<sup>0</sup> is the initial solute concentration, *C*<sup>l</sup> <sup>∗</sup> is the solute concentration of the liquid at the interface, and *m*<sup>l</sup> is the slope of the liquidus. Besides, Γ is the Gibbs–Thompson coefficient and *K*(*t*n) is the interface curvature calculated from Equation (12). *G*x is the anisotropy intensity of the liquid–solid interface.

$$\theta\_0 = \cos^{-1}\left(\frac{\partial f\_\text{s}/\partial \mathbf{x}}{\left(\left(\partial f\_\text{s}/\partial \mathbf{x}\right)^2 + \left(\partial f\_\text{s}/\partial \mathbf{y}\right)^2\right)^{1/2}}\right) \tag{16}$$

$$\theta = \arctan(\frac{\partial f\_{\text{s}}/\partial y}{\partial f\_{\text{s}}/\partial \mathbf{x}}) \tag{17}$$

Then, *θ*<sup>0</sup> represents the angle between the growth direction of the dendrites and the positive direction of the coordinate axis, and it can be calculated from Equation (16). Likewise, *θ* is also the angle between the normal of the solid–liquid interface and the positive direction of the coordinate axis, which is derived from Equation (17). As the heat undercooling is relatively small, the effect of dynamics on high dendritic growth is not taken into account and the solute diffusion in the solid phase can be neglected.
