Multiphase-field model for surface diffusion and attachment kinetics in the grand-potential framework

Grand-potential based multiphase-field model is extended to include surface diffusion. Diffusion is elevated in the interface through a scalar degenerate term. In contrast to the classical Cahn-Hilliard-based formulations, the present model circumvents the related difficulties in restricting diffusion solely to the interface by combining two second-order equations, an Allen-Cahn-type equation for the phase field supplemented with an obstacle-type potential and a conservative diffusion equation for the chemical potential or composition evolution. The sharp interface limiting behavior of the model is deduced by means of asymptotic analysis. A combination of surface diffusion and finite attachment kinetics is retrieved as the governing law. Infinite attachment kinetics can be achieved through a minor modification of the model, and with a slight change in the interpretation, the same model handles the cases of pure substances and alloys. Relations between model parameters and physical properties are obtained which allow one to quantitatively interpret simulation results. An extensive study of thermal grooving is conducted to validate the model based on existing theories. The results show good agreement with the theoretical sharp-interface solutions. The obviation of fourth-order derivatives and the usage of the obstacle potential make the model computationally cost-effective.

Figure 1

Equilibrium phase field ${\varphi }_{\alpha }$, for a cylindrical particle of radius $r$, with a relatively large interfacial width ($\epsilon /r=1/2$) in the current model. The solid lines indicate the isolines of the phase field, and the dashed line represents a radial cross section of the interface.

Figure 2

Simulated equilibrium phase-field profile ${\varphi }_{\alpha }$ along the radial direction (dashed line in Fig. 1), rescaled in terms of $\eta$ (solid symbols). The profile of ${\varphi }_{\alpha }^0$ is shown as a solid line.

Figure 3

Simulated equilibrium phase-field profile ${\varphi }_{\alpha }$, along the radial direction (dashed line in Fig. 1), rescaled in terms of $\eta$ (points). The profiles derived as in the asymptotic analysis are additionally shown as lines. The simulation result almost perfectly matches the derived profile, including the second-order corrections in $\epsilon$. The value of ${\kappa }_0=1/r$ is calculated from the isoline ${\varphi }_{\alpha }=0.5$ of the simulation data (see Fig. 1).

Figure 4

Numerically solved groove profiles, $z\left(u\right)$, under surface diffusion. Seven profiles are shown for different values of $m=tan\theta$, calculated in a finite difference scheme, according to [20]. An additional curve of the analytical solution by Mullins [19] is shown, which perfectly coincides with the numerical solution for $\theta =0^{\circ }$.

Figure 5

Analytical groove profile for volume diffusion-governed grooving in the small-slope framework [105].

Figure 6

Initial setup used for the simulations performed in this work.

Figure 7

Measurement of the groove dimensions and equilibrium angle $\theta$ at the groove root, by sixth-order polynomial fits (solid lines) to the isoline ${\varphi }_{\delta }=0.5$ (closed symbols). Points shown as transparent symbols are excluded from the fits.

Figure 8

Simulational groove surface (${\varphi }_{\delta }=0.5$ isoline) for ${\gamma }_{\mathrm{GB}}/{\gamma }_s=1$, at different times. Early, intermediate, and late times correspond to $t=1.5\times {10}^4{u}_t$, $t=9\times {10}^4{u}_t$, and $t=2\times {10}^5{u}_t$, respectively.

Figure 9

Evolution of the groove surface for ${\gamma }_{\mathrm{GB}}/{\gamma }_s=1$, performed at the same times as in Fig. 8, in rescaled coordinates for the proposed multiphase-field model (MPF) and according to Mullins [19] and Robertson [20]. The rescaling is applied, with the value of $B$ being calculated according to Eq. (99).

Figure 10

Characteristic groove dimensions in the final state, obtained from the multiphase-field simulations (closed symbols) and according to the theory in [19, 20], as the grain boundary energy is varied. The theoretical triple-junction angle $\theta$ is expressed through Eq. (126).

Figure 11

Groove width $w$, as a function of time, (to the power of $1/4$) from a multiphase-field simulation at ${\gamma }_{\mathrm{GB}}/{\gamma }_s=1.5$ (solid symbols) and according to theory [19, 20] (dashed and solid lines, respectively). For the theoretical curves, the value of $B$ has been calculated according to the derived formula [Eq. (99)].

Figure 12

The chemical-potential field $\mu \left(\mathit{x}\right)$ (the unit $u_E/u_l^3$ is omitted) at an early time $t=1.5\times {10}^4{u}_t$, in a region close to the triple junction ($x=0$), for equal interfacial energies ${\gamma }_{\mathrm{GB}}={\gamma }_s$. The isolines of the chemical potential and the phase field of the vapor phase ${\varphi }_{\delta }$ are superimposed. The arrows indicate the normal vectors on the isoline ${\varphi }_{\delta }=0.5$ and the dashed line represents a normal cross section of the interface.

Figure 13

Chemical potential $\mu$ [purple (black)] and phase field ${\varphi }_{\delta }$ [green (light gray)] across the interface (along the dashed line in Fig. 12). $r$ represents the signed distance from the center of the interface (${\varphi }_{\delta }=0.5$). The horizontally dashed line is the theoretical chemical potential, according to the Gibbs-Thomson effect, calculated from the numerical sharp-interface solution [20], at the same position ($x$ taken from ${\varphi }_{\delta }=0.5$ intersection) and time.

Figure 14

Scale-invariant chemical potential $\mu {\left(Bt\right)}^{1/4}$, as a function of the rescaled $x$ coordinate $u=x/{\left(Bt\right)}^{1/4}$. The solid lines show the chemical potential in the center of the interface (along the isoline ${\varphi }_{\delta }=0.5$) for phase-field simulations of varying ${\gamma }_{\mathrm{GB}}/{\gamma }_s$ ratios, at early [purple (black), $t=1.5\times {10}^4{u}_t$], intermediate [teal (gray), $t=9\times {10}^4{u}_t$] and late [green (light gray), $t=2\times {10}^5{u}_t$] times. The dashed and dash-dotted lines are the theoretical curves, according to the relation obtained in the asymptotics without and with small-slope approximation, respectively.

Figure 15

Rescaled groove profiles from two different representations of the groove surface, at a high resolution ($\epsilon =2u_l$, $\mathrm{\Delta }x=0.125u_l$) and at a late time $t=1.95\times {10}^5{u}_t$. Note the blunt in the isoline ${\varphi }_{\delta }=0.5$, which is absent in the curve ${\varphi }_{\delta }={\varphi }_{\beta }$. The sharp interface solution from [20] is shown additionally.

Figure 16

The relative error between the phase field and sharp interface solutions for simulations at small interface widths (values tabulated in Table 4). The symbols represent the errors obtained from the phase-field simulations, whereas the dashed line shows a polynomial fit to the closed data points. For the fit, the open symbols are not taken into account. The colored (grayish) symbols refer to simulations at a higher resolution.