Grand-potential formulation for multicomponent phase transformations combined with thin-interface asymptotics of the double-obstacle potential

In this paper, we describe the derivation of a model for the simulation of phase transformations in multicomponent real alloys starting from a grand-potential functional. We first point out the limitations of a phase-field model when evolution equations for the concentration and the phase-field variables are derived from a free energy functional. These limitations are mainly attributed to the contribution of the grand-chemical-potential excess to the interface energy. For a range of applications, the magnitude of this excess becomes large and its influence on interface profiles and dynamics is not negligible. The related constraint regarding the choice of the interface thickness limits the size of the domain that can be simulated and, hence, the effect of larger scales on microstructure evolution can not be observed. We propose a modification to the model in order to decouple the bulk and interface contributions. Following this, we perform the thin-interface asymptotic analysis of the phase-field model. Through this, we determine the thin-interface kinetic coefficient and the antitrapping current to remove the chemical potential jump at the interface. We limit our analysis to the Stefan condition at lowest order in $\epsilon$ (parameter related to the interface width) and apply results from previous literature that the corrections to the Stefan condition (surface diffusion and interface stretching) at higher orders are removed when antisymmetric interpolation functions are used for interpolating the grand-potential densities and the diffusion mobilities.

Figure 1

The grand-chemical-potential difference varies across the interface and has a form similar to that of a potential. At equilibrium, the two phases are at the same grand chemical potential, which is seen qualitatively from the graph. Notice also the asymmetry of the potential around ${\phi }_{\alpha }=0.5$, which is inherited from the asymmetry in the chemical free energy states of the two phases.

Figure 2

Illustration of the driving force for phase transformation between two phases.

Figure 3

The phase diagram of the Al-Cu system fitted on the Al-rich side using an ideal solution type of model.

Figure 4

Plot of the dendrite tip velocities simulated at a temperature of $T=0.9843$. We selectively plot points corresponding to a simulation to show the convergence of the velocities. We span a range where $\epsilon$ varies by a factor 4 and achieve convergence in the velocities. The simulation with the $\epsilon =112.5$ has run the least in nondimensional time ($1.5\times {10}^8$), but long enough to confirm convergence of the velocities.

Figure 5

Chemical potential plot along a linear section at the dendrite tip in the growing direction, superimposed with the lines showing the equilibrium chemical potential and the theoretically predicted chemical potential obtained by considering the shift because of Gibbs-Thomson effect due to curvature. The curvature used in the calculation is measured at the dendrite tip from the simulation. ${\sigma }_{\theta \theta }$ represent the second derivative of the surface tension as a function of the polar angle, and the sum $\sigma +{\sigma }_{\theta \theta }$ represents the stiffness of the interface.

Figure 6

Isolevel ${\phi }_{\alpha }=0.5$ denoting the binary interface between the solid and the liquid at various times is shown in (a), while the contours of the chemical potential at a particular instant during evolution is displayed in (b). Values of some of the contours of the respective nondimensionalized chemical potential are superimposed on the plot. The simulations correspond to the case when $\epsilon =300$.

Figure 7

Defined contour level of the surface energy $\sigma$ and the interface width $\lambda$ plotted as a function of the simulation parameters ${\gamma }_{\alpha \beta }$ and $\epsilon$. In (a), the contours are calculated for the temperature $T=0.9843$, while in (b) they are for a temperature of $T=0.988$. The contours of the interface width are calculated from the defined level for the surface energy and the value of the $\epsilon$ used in the simulation. All terms are dimensionless in the graphs.

Figure 8

Phase-field contours of a free growing equiaxed dendrite, with $\epsilon =1688$ at a temperature of $T=0.988$ ($T_m=0.99$) with the grand-potential formulation.

Figure 9

Comparison of the leading order solution of the phase field in the case of the grand-potential functional and the free energy functional. The profiles have been superimposed, and the vertical line representing the position of the binary interface is drawn for comparison.