Grand-potential-based phase-field model for multiple phases, grains, and chemical components

Grand-potential-based phase-field model for multiple phases, grains, and chemical components is derived from a grand-potential functional. Due to the grand-potential formulation, the chemical energy does not contribute to the interfacial energy between phases, simplifying parametrization and decoupling interface thickness from interfacial energy, which can potentially allow increased interface thicknesses and therefore improved computational efficiency. Two-phase interfaces are stable with respect to the formation of additional phases, simplifying implementation and allowing the variational form of the evolution equations to be used. Additionally, we show that grand-potential-based phase-field models are capable of simulating phase separation, and we derive conditions under which this is possible.

Figure 1

Geometrical analysis of a grain boundary allotriomorph, with length $L$, thickness $S$, and dihedral angle $\varphi$ indicated. ${\gamma }_{\alpha \beta }=4.5$, ${\gamma }_{\beta \beta }=1.5$, ${\varphi }^{\mathrm{an}}={135}^{\circ }$. The color bar represents the value of ${\sum }_i{\eta }_{\beta i}^2$ and is used to provide a visualization of the microstructure.

Figure 2

Snapshots of growth and coarsening of $\beta$-phase particles from a supersaturated ($c=0.15$) polycrystalline matrix. Simulation times are (a) $t=50$, (b) $t=100$, (c) $t=150$. Corner (triple-junction) particles grow at the expense of edge (grain boundary) particles because of the effect of curvature. The color bar represents the value of ${\sum }_i{\eta }_{\alpha i}^2$ and is used to provide a visualization of the microstructure. The color bar is applicable to each subfigure.

Figure 3

Simulation of the evolution of a nonequilibrium $\alpha \text{−}\beta$ interface with ${\eta }_{\delta 1}=0.1$ in the initial conditions. Simulation times are (a) $t=0$, (b) $t=1$, (c) $t=2$, (d) $t=5$. ${\eta }_{\delta 1}$ decreases to 0 and the $\alpha \text{−}\beta$ interface evolves to the equilbrium interfacial profile, demonstrating that for the materials parameters considered here, the $\alpha \text{−}\beta$ interface is stable with respect to formation of a third phase.

Figure 4

Simulations of growth of $\beta$ phase from supersaturated $\alpha$ phase (parameters given in Table 1, with $c_A=0.9$ in the $\beta$ phase and $c_A=0.15$ in the $\alpha$ phase). (a) Growth of a plate of $\beta$ phase (1D geometry). (b) Growth of a spherical precipitate of $\beta$ phase (3D geometry). The fit is to the linear portion of the $\mathrm{\Delta }x=1$ results, and in each case the slope of the fit line is in good agreement with the analytical prediction. The inset shows the effect of decreasing mesh resolution to the point where the interface is no longer adequately resolved.

Figure 5

Simulations of phase separation in a two-phase binary system using a grand-potential-based phase-field model with a single order parameter. Simulation times are (a),(c) $t=0$; (b),(d) $t=200$. The left column represents the initial configurations of the supersaturated matrix (see text), and the right column shows the concentration map after phase separation is complete. The upper row shows the morphology of second phase developed during separation for the case of high spinodal concentration ($c^s=0.5$), and the lower row captures the morphology developed for the case of low spinodal concentration ($c^s=0.167$).

Figure 6

Simulations of phase separation in a two-phase binary system using a grand-potential-based phase-field model with ${\eta }_{\alpha }=1,{\eta }_{\beta }=0$ representing the $\alpha$ phase, and ${\eta }_{\alpha }=0,{\eta }_{\beta }=1$ representing the $\beta$ phase. (a) The initial condition of $c$ ($t=0$); (b) $c$ after phase separation is complete ($t=600$). The average concentration $c^{\mathrm{avg}}=0.5$ is above $c^s=0.4$, so the system phase separates and forms a lamellar microstructure.