Variational formulation of a quantitative phase-field model for nonisothermal solidification in a multicomponent alloy

A variational formulation of a quantitative phase-field model is presented for nonisothermal solidification in a multicomponent alloy with two-sided asymmetric diffusion. The essential ingredient of this formulation is that the diffusion fluxes for conserved variables in both the liquid and solid are separately derived from functional derivatives of the total entropy and then these fluxes are related to each other on the basis of the local equilibrium conditions. In the present formulation, the cross-coupling terms between the phase-field and conserved variables naturally arise in the phase-field equation and diffusion equations, one of which corresponds to the antitrapping current, the phenomenological correction term in early nonvariational models. In addition, this formulation results in diffusivities of tensor form inside the interface. Asymptotic analysis demonstrates that this model can exactly reproduce the free-boundary problem in the thin-interface limit. The present model is widely applicable because approximations and simplifications are not formally introduced into the bulk's free energy densities and because off-diagonal elements of the diffusivity matrix are explicitly taken into account. Furthermore, we propose a nonvariational form of the present model to achieve high numerical performance. A numerical test of the nonvariational model is carried out for nonisothermal solidification in a binary alloy. It shows fast convergence of the results with decreasing interface thickness.

Figure 1

(a) Dependence of $a_4$ and $a_I$ on $q_a$. (b) Dependence of diffusivity $q_i^n$ on $\varphi$ calculated for $q_{s,i}={10}^{-2}$, ${10}^{-3}$, and ${10}^{-4}$. $q_a={10}^{-3}$ and $a_4=0.5478$ were employed in (b).

Figure 2

(a) Convergence behavior of (a) the interface velocity (open plots) and the tip radius (filled plots), (b) the compositions in the solid near the dendrite tip, $c_{s,1}^{*}$ and $c_{s,2}^{*}$ ($\varphi \approx 0.99$), and (c) error in the Gibbs-Thomson relation, calculated by the present model (squares), the present model without cross-coupling terms in $\varphi$ equation (circles), and the standard model (triangles).