Atomistic migration mechanisms of atomically flat, stepped, and kinked grain boundaries

We studied the migration behavior of mixed tilt and twist grain boundaries in the vicinity of a symmetric tilt $\langle 111\rangle \mathrm{\Sigma }7$ grain boundary in aluminum. We show that these grain boundaries fall into two main categories of stepped and kinked grain boundaries around the atomically flat symmetric tilt boundary. Using these structures together with size converged molecular dynamics simulations and investigating snapshots of the boundaries during migration, we obtain an intuitive and quantitative description of the kinetic and atomistic mechanisms of the migration of general mixed grain boundaries. This description is closely related to well-known concepts in surface growth such as step and kink-flow mechanisms and allows us to derive analytical kinetic models that explain the dependence of the migration barrier on the driving force. Using this insight we are able to extract energy barrier data for the experimentally relevant case of vanishing driving forces that are not accessible from direct molecular dynamics simulations and to classify arbitrary boundaries based on their mesoscopic structures.

Figure 1

Schematics of the investigated grain boundaries [(a) to (c)] and their static structures after $T=0$ K relaxation [(d) to (h)]. (a) Pure symmetric $\mathrm{\Sigma }7$ tilt boundary with a misorientation angle of $38.{21}^{\circ }$ around the $\left[1\overline{1}1\right]$ axis, relaxing to an atomically flat structure as displayed in (d). (b) Mixed boundary obtained by introducing a small rotation angle around the $\left[12\overline{1}\right]$ direction in the grain boundary plane, relaxing to a structure with straight steps as shown in (e) and (g). (c) Mixed boundary obtained by introducing a small inclination angle to the symmetric boundary, around an arbitrary low symmetry rotation axis in the grain boundary plane as indicated by the black arrow. As a specific representation the $\left[54\overline{1}\right]$ rotation axis has been chosen. This boundary relaxes to a structure with kinked steps as shown in (f) and (h). In (d) through (h) perfect bulk atoms have been removed for clarity, and the color indicates the position of the grain boundary atoms along the boundary plane normal as indicated by the color bar.

Figure 2

Position versus time dependence of the three representative grain boundaries at (a) 500 K, (b) 600 K, and (c) 700 K subjected to a driving force of 5 meV per atom. At 700 K all three boundaries have similar kinetics and the curves overlap.

Figure 3

Arrhenius representation of the grain boundary velocity as a function of inverse temperature at various driving forces for (a) flat, (b) stepped, and (c) kinked boundaries. The squares indicate the calculated data and the dashed lines are fits, quadratic for the flat boundary according to the island nucleation model [14] and linear for the stepped and kinked boundaries. The dotted black line indicates the roughening transition temperature of the grain boundaries, and the gray area marks the region above the transition temperature.

Figure 4

Energy barriers extracted from the Arrhenius representation of Fig. 3 shown at (a) a small energy scale to capture all grain boundaries and (b) a larger scale to emphasize the dependence for the stepped and kinked boundaries. For the flat boundary, the fitted curve (red dashed line) was obtained using the island nucleation model [14]. The fit for the stepped boundary (green dashed line) is based on the model developed in Sec. 4a, and for the kinked boundary the fit is a constant (blue dashed line). The range of experimentally observed energy barriers [25] is indicated by the black bar at low driving forces.

Figure 5

(a), (b), and (c) are snapshots of migration in the three grain boundaries [(a) flat, (b) stepped, and (c) kinked] at 500 ps, 3 meV, and 450 K where the bulk atoms have been removed and the atoms belonging to the shrinking grain have been made invisible to help resolve the atomistic details on the growing side of the boundaries. (d), (e), and (f) are the magnified views of the areas indicated in the snapshots. The black arrows indicate the $\left[12\overline{1}\right]$ direction in all grain boundaries.

Figure 6

Geometry and energy barrier of the double kink nucleation process. (a) A double kink of length $d$ on a stepped grain boundary and with a kink area $A$. (b) The formation free energy of the double kink as a function of kink separation as derived with the nucleation model [Eq. (8)] at different driving forces. The black dotted line shows the dependence in the absence of any driving force. (c) Compares the activation barrier as derived from the MD simulations at different driving forces (filled green squares) and the analytic nucleation model (green dashed line). The arrows mark the driving forces for which the explicit energy curves are shown in (b).

Figure 7

(a), (b), and (c) are surface growth parallels to the grain boundaries of Figs. 5, 5, and 5, respectively. (a) represents a layer-by-layer growth and (b) and (c) are vicinal surfaces that grow via a step flow mechanism.