Role of the mesoscale in migration kinetics of flat grain boundaries

Classical molecular dynamics simulations of bicrystalline systems are a commonly used tool for exploring the migration of grain boundaries. Most simulation work to date has focused on measuring the mobility of grain boundaries, assuming it to be an intrinsic property of a boundary of a given geometry. Here we present results from simulations of the migration of a typical high-angle grain boundary that show that the concept of intrinsic mobility fails for defect-free, flat boundaries of the type frequently simulated and that key assumptions often made in analyzing the kinetics of migration do not hold. Our dynamical simulations of grain boundary migration show that the grain boundary velocity is not simply proportional to the driving force for grain boundary motion, as commonly assumed, and shows a strong and complex dependence on the system size. By analyzing the migration mechanism at the larger mesoscale we show that defect-free, flat boundaries must migrate via the homogeneous nucleation and growth of islands of transformed crystal volume on the grain boundary surface. We present a detailed analysis of the kinetics of this process, which only emerges in simulations of large grain boundary areas. An island-based mesoscale mechanism implies an energy barrier for migration that is inversely proportional to the driving force for migration—in the experimental (zero-force) limit such boundaries must be immobile. This calls into question the concept of an intrinsic mobility for defect-free, flat grain boundaries and suggests that mobility of real boundaries at low temperatures is rather a function of their morphology and defect content and at high temperatures is a result of thermal roughening.

Figure 1

A schematic illustration of a simple barrier hopping model of grain boundary migration. The labeled energies are discussed in detail in Sec. 2a.

Figure 2

Results of grain boundary migration simulations showing velocity as a function of driving force over a range of temperatures. The error bars show an estimate of one standard deviation in the measured velocity [derived using bootstrap resampling (Ref. [42])]. The horizontal arrows indicate the adjustments to the nominal driving force strength required to account for thermal fluctuations. The inset shows the nominal mobility (see main text for a definition) as a function of driving force (in $\text{eV}/\text{atom}$) up to very high driving forces.

Figure 3

Results derived from fitting to Arrhenius plots of the data in Fig. 2 as a function of driving force. Main figure: activation energy—the solid (red) line is a guide for the eye. Inset: the compensation effect—the dashed (blue) line is a linear fit to the data as a guide for the eye.

Figure 4

The velocity of a grain boundary in MD simulations at $600\text{K}$ as a function of system size (length of simulation cell in the dimensions in the grain boundary plane) and driving force. The velocities are normalized to the value in the largest system at each value of the driving force. Representative error bars [one standard deviation in the estimated velocity (Ref. [42])] are shown for a single system size. The colored arrows mark the approximate minimum system size at which convergence in the velocity occurs. Corresponding points are marked in Fig. 9. The inset shows a schematic of the observed trends and is discussed in Sec. 3b.

Figure 5

Snapshots of a migrating grain boundary surface. The contours show the edges of (white) islands of migrated grain boundary area at different times for a large system ($340\times 420{\text{Å}}^2$, left) and a smaller system ($170\times 210{\text{Å}}^2\approx 4r_{\text{crit}}$, where $r_{\text{crit}}$ is the radius of a critically stable island nucleus), both with a driving force of $5\text{meV}/\text{atom}$ at $600\text{K}$.

Figure 6

A schematic illustration of the island nucleation and growth model of grain boundary migration. The solid volume represents crystal volume in the favored orientation. The upper half of the bicrystal, in the disfavored orientation, is not shown. (a) Islands (blue) form on the surface of the flat boundary (red) due to thermal fluctuations. (b) Double kinks (yellow) form in the island edges and (c) migrate and coalesce to give growth of the islands. (d) The islands grow until they coalesce, at which point the grain boundary has migrated by the edge height.

Figure 7

The excess energies of circular islands in a grain boundary surface. The symbols show the results of static atomistic calculations (see inset) using an EAM potential. The curves are predictions of the analytical model in Eq. (7) (see main text).

Figure 8

Snapshots of islands of migrated grain boundary area at an early stage in the migration process (approximately the critically stable island population) as a function of driving force in a system $340\text{Å}\times 420\text{Å}$.

Figure 9

A map of the grain boundary migration mechanism as a function of driving force and system size. The solid red line shows the critical radius $r_{\text{crit}}$ from the analytical model in Eq. (7). Typical parameters [10, 13, 14, 48] of simulations in the literature are marked. The symbols (and dashed line) mark the threshold for size convergence of the grain boundary velocity from Fig. 4.

Figure 10

Arrhenius plots of the size-converged velocity of grain boundaries. The lines show the results of fitting the analytical model of island-based kinetics in Eq. (12), as discussed in the text. Note that the roughening transition temperature is $\sim 700\text{K}$.

Figure 11

(a) A comparison of (points) the internal energy barriers extracted from dynamical simulations and (line) the analytical predictions of Eq. (14). (b) The compensation effect (a proportionality between the logarithm of the kinetic prefactor and the internal energy barrier) arises as a natural consequence of the island-based migration mechanism. The points are extracted from dynamical simulations and the line is the prediction of the analytical model.

Figure 12

The energy barrier for grain boundary migration as a function of driving force. The points are the internal energy barrier $U_0$ [equivalently $F_{\text{A}}(P,T=0)]$ extracted from the dynamical simulations by fitting the data in Fig. 10. The solid (red) line is the prediction of Eq. (14) [or of Eq. (9) at $T=0\text{K}$]. The dashed (blue) line is the free energy barrier $F_{\text{A}}$ at $600\text{K}$ from Eq. (9).