*3.2. Solute Diffusion Model*

For binary alloys or multicomponent alloys, the solute diffusion also plays an important role on dendrite growth in the weld molten pool and the solute concentration gradient is the driving force for solute diffusion in the solid and liquid phases. Then, the solute concentration in the solid and liquid phases is determined by solving the governing equation for each phase separately, as follows:

$$\frac{\partial \mathbf{C}\_{\mathbf{i}}}{\partial t} = D\_{\mathbf{i}} \nabla^{2} \mathbf{C}\_{\mathbf{i}} + \mathbf{C}\_{\mathbf{i}} \cdot (1 - k) \frac{\partial f\_{\mathbf{s}}}{\partial t} \tag{10}$$

where *C* is the solute concentration, *D* represents the solute diffusivity, the subscript *i* denotes a solid or liquid, and *k* is the partition coefficient.

At the solid–liquid interface, the solute partition between the liquid and solid is given by:

$$
\mathbb{C}\_s{}^\* = k \cdot \mathbb{C}\_l{}^\* \tag{11}
$$

where *C*s ∗ and *C*∗ <sup>l</sup> denote the interface solute concentrations in the solid and liquid phases, respectively.

Based on the counting method advanced by Nastac [22], the interface curvature of a cell with a solid fraction can be derived by calculating the nearest and second-nearest neighboring cells:

$$K = \frac{1}{l\_{\rm CA}} (1 - 2\frac{f\_{\rm s} + \sum\_{j=1}^{N} f\_{\rm s}(j)}{N + 1}) \tag{12}$$

where *l*CA represents the length of the CA cell side, *N* is the number of the nearest and the second-nearest neighboring cells, and *f*s(*j*) is the solid fraction of neighboring cells.

The liquid concentration in the interface cell is given as [23]:

$$\mathbf{C}\_{\mathrm{l}} = \mathbf{C}\_{\mathrm{l}}^{\*} - \frac{1 - f\_{\mathrm{s}}}{2} \mathbf{l}\_{\mathrm{CA}} \cdot \mathbf{G}\_{\mathrm{c}} \tag{13}$$

where *G*<sup>c</sup> is the concentration gradient in front of the solid-liquid interface, and the interface equilibrium composition *C*<sup>l</sup> ∗ is obtained by:

$$\mathcal{C}\_{\rm I}{}^{\*} = \mathcal{C}\_{0} + \frac{1}{m\_{\rm l}} [T^{\*} - T^{\rm eq}{}\_{\rm l} + \Gamma \mathcal{K}] \tag{14}$$

where *C*<sup>0</sup> indicates the initial solute concentration, *T*∗ is the interface equilibrium temperature calculated by Equation (1), *T*eq <sup>1</sup> is the equilibrium liquidus temperature at *C*0, *m*<sup>l</sup> is the liquidus slope, Γ is the Gibbs-Thomson coefficient, and *K* is the average curvature of the liquid-solid interface.
