Phase-field crystal study of grain-boundary premelting

We study the phenomenon of grain-boundary premelting for temperatures below the melting point in the phase-field crystal model of a pure material with hexagonal ordering in two dimensions. We investigate the structures of symmetric tilt boundaries as a function of misorientation $\theta$ for two different inclinations and compute in the grand canonical ensemble the “disjoining potential” $V\left(w\right)$ that describes the fundamental interaction between crystal-melt interfaces as a function of the premelted layer width $w$, which is defined here in terms of the excess mass of the grain boundary via a Gibbs construction. The results reveal qualitatively different behaviors for high-angle grain boundaries that are uniformly wetted, with $w$ diverging logarithmically as the melting point is approached from below, and low-angle boundaries that are punctuated by liquid pools surrounding dislocations, separated by solid bridges. The latter persist over a superheated range of temperature. This qualitative difference between high- and low-angle boundaries is reflected in the $w$ dependence of the disjoining potential that is purely repulsive [$V^{\prime }\left(w\right)<0$ for all $w$] for misorientations larger than a critical angle ${\theta }_c$, but switches from repulsive at small $w$ to attractive at large $w$ for $\theta <{\theta }_c$. In the latter case, $V\left(w\right)$ has a minimum that corresponds to a premelted boundary of finite width at the melting point. Furthermore, we find that the standard wetting condition ${\gamma }_{\text{gb}}\left({\theta }_c\right)=2{\gamma }_{\text{sl}}$ gives a much too low estimate of ${\theta }_c$ when a low-temperature value of the grain-boundary energy ${\gamma }_{\text{gb}}$ is used. In contrast, a reasonable lower-bound estimate can be obtained if ${\gamma }_{\text{gb}}$ is extrapolated to the melting point, taking into account both the elastic softening of the material at high homologous temperature and local melting around dislocations.

Figure 1

(Color online) Density profiles of solid-liquid interface and grain boundaries. The complete two-dimensional density field $\psi \left(x,y\right)$ is shown. The superimposed line gives, for any point along the direction normal to the grain boundary, the value of the density averaged over the direction parallel to the grain boundary $\left[{\overline{\psi }}_x\left(y\right)=\left(1/L_x\right)\int \psi \left(x,y\right)dx\right]$. Top: Solid-liquid interface, $\theta =21.8°$. Middle: Solid-solid interface close to the melting point, $\overline{\psi }=-0.1980$, $\theta =32.2°$. Bottom: Solid-solid interface far from the melting point, $\overline{\psi }=0.180$, $\theta =32.2°$.

Figure 2

Symbols: Free-energy densities $f$ of systems with the same misorientation and average densities $\overline{\psi }$ but different lengths $L_y$ perpendicular to the grain boundary, plotted versus $1/L_y$. Line: Linear fit to the data. The slope gives twice the grain-boundary energy, where the factor of 2 is due to the fact that in periodic systems there are always two grain boundaries. In this example, $ϵ=0.1$ and $\overline{\psi }=-0.1$ and the misorientation is $\theta =6°$.

Figure 3

(Color online) Snapshots of a high-angle grain boundary with $\theta =32.2°$ for different values of $u$, which increase from bottom right to top left (see text and Fig. 5 for details). The “liquid” forms a rather homogeneous film. Only part of the simulation box is shown.

Figure 4

(Color online) Snapshots of a low-angle grain boundary with $\theta =9.4°$ for different values of $u$, which increases from bottom right to top left (see text and Fig. 5 for details). The grain boundary consists of individual dislocations and undergoes a structural transition. Only part of the simulation box is shown.

Figure 5

Ratio of film thickness $w$ to lattice spacing $a$ as a function of scaled chemical potential $u$ for two different grain boundaries. The inset shows a blowup of the vicinity of the melting point. The symbols mark the states that are depicted in the snapshot pictures in Figs. 3, 4.

Figure 6

(Color online) Film thickness $w$ as a function of $u$ for various misorientations, for symmetric tilt grain boundaries of inclination $\varphi =0°$ (top) and $\varphi =30°$ (bottom) close to the melting point. All angles are given in degrees, and a vertical line has been drawn at the melting point $u=0$. Inset: Film thickness of the three largest angles versus $-\text{ln}\left(-u\right)$; the divergence of the film thickness is logarithmic.

Figure 7

(Color online) Various grain-boundary states for inclination $\varphi =0°$ and misorientation $\theta =9.4°$. The curve $w$ versus $u$ exhibits a hysteresis: filled symbols are calculated with increasing $u$ (same curve as shown in Fig. 6), and open symbols with decreasing $u$. The snapshot pictures show different grain-boundary states, all at $u=-0.072$. Top: single-dislocation state; middle: dislocation pair calculated with a single-dislocation state as initial condition; bottom: dislocation pair calculated with the initial conditions described in Sec. 2B.

Figure 8

(Color online) Ratio of grain-boundary energy to twice the solid-liquid free energy for grain boundaries of inclination $\varphi =0°$ as a function of the misorientation $\theta$, shown for different chemical potentials $u$. The lowest curve, $u=-0.006$, is very close to the melting point, while the upper curve is far inside the solid region. The lines are fits to the Read-Shockley law [Eq. (36)], using the values of ${\gamma }_{\text{gb}}$ for $\theta <15°$. The shear modulus has been fixed to the theoretical value at the corresponding chemical potential, and the only fit parameter is the dislocation core radius $r_0$. Inset: The core radius $r_0$ obtained from the fits as a function of chemical potential. For comparison, the lattice constant is $a\approx 7.255$ and the value estimated by Elder and Grant (Ref. 20) is $r_0\approx 4.4$.

Figure 9

(Color online) Symbols: grain-boundary energy determined from direct simulations; lines: grain-boundary energy calculated by thermodynamic integration; dash-dotted lines: prediction of the Read-Shockley law with $r_0=3.75$ and elastic constants depending on $u$. See text for details.

Figure 10

(Color online) Disjoining potential for inclination $\varphi =0°$ and various misorientations (inset: detailed view of the region around the minimum). The angles are given in degrees.

Figure 11

(Color online) Ratio of the grain-boundary energy and the melting point ${\gamma }_g^m$ and twice the solid-liquid free energy as a function of misorientation for inclination $\varphi =0°$. The line is a fit to the Read-Shockley law for the points with $\theta <10°$.

Figure 12

(Color online) Critical value $u^{\ast }$ that corresponds to the limit of superheated states associated with the breaking of solid bridges. Symbols: simulation results; line: sharp-interface prediction according to Eq. (40).

Figure 13

Critical value $u^{\ast }$ versus ${\text{sin}}^2\theta$ for low-angle grain boundaries of inclination $\varphi =30°$. Inset: Ratio of the liquid-pool radius $r^{\ast }$ where the breakoff occurs and the dislocation spacing $d$ versus misorientation.