Quantitative phase-field modeling of two-phase growth

A phase-field model that allows for quantitative simulations of low-speed eutectic and peritectic solidification under typical experimental conditions is developed. Its cornerstone is a smooth free-energy functional, specifically designed so that the stable solutions that connect any two phases are completely free of the third phase. For the simplest choice for this functional, the equations of motion for each of the two solid-liquid interfaces can be mapped to the standard phase-field model of single-phase solidification with its quartic double-well potential. By applying the thin-interface asymptotics and by extending the antitrapping current previously developed for this model, all spurious corrections to the dynamics of the solid-liquid interfaces linear in the interface thickness $W$ can be eliminated. This means that, for small enough values of $W$, simulation results become independent of it. As a consequence, accurate results can be obtained using values of $W$ much larger than the physical interface thickness, which yields a tremendous gain in computational power and makes simulations for realistic experimental parameters feasible. Convergence of the simulation outcome with decreasing $W$ is explicitly demonstrated. Furthermore, the results are compared to a boundary-integral formulation of the corresponding free-boundary problem. Excellent agreement is found, except in the immediate vicinity of bifurcation points, a very sensitive situation where noticeable differences arise. These differences reveal that, in contrast to the standard assumptions of the free-boundary problem, out of equilibrium the diffuse trijunction region of the phase-field model can (i) slightly deviate from Young’s law for the contact angles, and (ii) advance in a direction that forms a finite angle with the solid-solid interface at each instant. While the deviation (i) extrapolates to zero in the limit of vanishing interface thickness, the small angle in (ii) remains roughly constant, which indicates that it might be a genuine physical effect, present even for an atomic-scale interface thickness.

Figure 1

Experimental phase diagram of the transparent organic eutectic alloy $\mathrm{C}{\mathrm{Br}}_4-{\mathrm{C}}_2{\mathrm{Cl}}_6$ (modified from Mergy, Faivre, and Akamatsu). $C$ is the concentration of ${\mathrm{C}}_2{\mathrm{Cl}}_6$, and we have indicated the eutectic temperature $T_{\mathrm{E}}$, the equilibrium concentrations of each phase at the eutectic temperature, $C_{\alpha }$, $C_{\beta }$, and $C_{\mathrm{E}}$, and the width $\Delta C$ of the eutectic plateau.

Figure 2

Sketch of the lamellar geometry with the definitions of the contact angles ${\theta }_{\alpha }$ and ${\theta }_{\beta }$; $\lambda$ is the lamellar spacing.

Figure 3

(Color online) Triple-well potential, visualized as a surface over the Gibbs simplex. The black lines mark the borders of the Gibbs simplex, drawn on the surface; for this function, they also coincide with the trajectory of the equilibrium interface solutions.

Figure 4

(Color online) Elementary tilting function $g_i$, visualized over the Gibbs simplex. The $i$ vertex is to the right.

Figure 5

The $f_c$ part of the free-energy density is an interpolation between three parabolic bulk phase free energies; the equilibrium compositions and chemical potentials can be obtained by the common tangent construction illustrated by the straight, dashed lines for both solid-liquid equilibria. This construction corresponds to the conditions of equal chemical and grand potential in the two coexisting phases.

Figure 6

(Color online) Function used to lift the saddle between two phases and hence alter the surface tension. It vanishes on the other two interfaces. The position and strength of the hill in the middle are controlled by the parameter $b$ in Eq. (3.56) (here, $b=12$).

Figure 7

(Color online) Illustrative surface plot of the two phase fields $p_{\alpha }$ and $p_{\beta }$, for two half lamellae as described in the text. $40\times 40$ grid points shown; the system is actually larger in the $z$ direction.

Figure 8

Configuration of the multigrid scheme. In front of the solids that grow upward, a fixed number $n_f$ of rows is treated on the fine grid with spacing $\Delta \tilde{x}$, which corresponds to a distance $d=n_f\Delta \tilde{x}$. Two coarser grids of spacing $2\Delta \tilde{x}$ and $4\Delta \tilde{x}$ have again $n_f$ rows each, which corresponds to distances $2d$ and $4d$, respectively. Beyond the coarsest grids, the solution is one dimensional.

Figure 9

Lamellar states, simulated with a constant temperature that yields almost zero growth speed, and comparison to the analytical critical stationary solution that consists of circular arcs. Two different values of $b$ are used together with $a_{\mathrm{L}}=12$ (predicted angle $\theta \approx 66°$). Since, at constant temperature, the origin of the $z$ axis is arbitrary, the solutions have been shifted in order for lamella tips to coincide. The curves shown are the isocontours $p_{\alpha }=p_{\mathrm{L}}$ and $p_{\beta }=p_{\mathrm{L}}$, to be compared with the theoretical circle arcs with the predicted contact angle $\theta$ (solid curve). The isocontour $p_{\alpha }=p_{\beta }$, exactly vertical and located at $x∕W=8$, has been omitted for clarity.

Figure 10

Convergence test for lamellar shapes in the symmetric model alloy at the eutectic composition. The solid-liquid interfaces are plotted for (a) the complete model, (b) without the antitrapping current, and (c) a model with several thin-interface corrections taken from Ref. 26. The parameters for all runs are given in Table . Thin solid lines, $\lambda ∕W=32$; dotted lines, $\lambda ∕W=64$; dashed lines, $\lambda ∕W=96$; dashed–dotted lines, $\lambda ∕W=128$; thick solid line, result of the boundary integral code of Ref. 17.

Figure 11

Difference between the contact angle and the equilibrium value 30° as a function of $W∕\lambda$ for the same series of runs as Fig. 10a.

Figure 12

Dimensionless average undercooling of the front versus dimensionless spacing ($l_D=D∕v_p$ is the diffusion length) for the first two parameter sets of Table . The curve is a best fit to the Jackson-Hunt law (see text for details).

Figure 13

Convergence test for lamellar shapes in $\mathrm{C}{\mathrm{Br}}_4-{\mathrm{C}}_2{\mathrm{Cl}}_6$. The parameters of all runs are given in Table . Thin solid lines, $\lambda ∕W=32$; dotted lines, $\lambda ∕W=64$; dashed lines, $\lambda ∕W=96$; dashed–dotted lines, $\lambda ∕W=128$; thick solid line, result of the boundary integral code of Ref. 17.

Figure 14

Trijunction rotation angle $\varphi$ (see text for definition) versus $W∕\lambda$ for the same series of runs as Fig. 13.

Figure 15

Oscillatory limit cycle in the symmetric model alloy for $\lambda =2.2{\lambda }_{\mathrm{JH}}$. (a) Snapshots of the two solid-solid interfaces (thick lines) and successive solid-liquid interfaces (thin lines). (b) Oscillation amplitude $A∕\lambda$ vs inverse interface thickness, $\lambda ∕W$. (c) Blowup of the trijunction region. Solid lines, instantaneous interfaces; dashed line, final solid-solid interface; arrow, direction of the trijunction motion.

Figure 16

Amplitude of saturated oscillatory limit cycles versus lamellar spacing, for parameters corresponding to the first (circles) and third (squares) lowest resolution in Fig. 15b ($\lambda ∕W=64$ and 128 there, respectively). Inset, same data, where the squares have been shifted along the $\lambda$ axis by 0.07.

Figure 17

Sketch of the diffusion field around a trijunction in the sharp-interface picture. Lines with arrows indicate the flux of solute around the trijunction.

Figure 18

(Color online) Sketch of the phase-field state space. The representative point of the system is confined to the plane $p_1+p_2+p_3=1$, whose intersection with the axes defines the vertices of the shadowed triangle, the so-called Gibbs simplex.