Understanding long-time vacancy aggregation in iron: A kinetic activation-relaxation technique study

Vacancy diffusion and clustering processes in body-centered-cubic (bcc) Fe are studied using the kinetic activation-relaxation technique (k-ART), an off-lattice kinetic Monte Carlo method with on-the-fly catalog building capabilities. For monovacancies and divacancies, k-ART recovers previously published results while clustering in a 50-vacancy simulation box agrees with experimental estimates. Applying k-ART to the study of clustering pathways for systems containing from one to six vacancies, we find a rich set of diffusion mechanisms. In particular, we show that the path followed to reach a hexavacancy cluster influences greatly the associated mean-square displacement. Aggregation in a 50-vacancy box also shows a notable dispersion in relaxation time associated with effective barriers varying from 0.84 to 1.1 eV depending on the exact pathway selected. We isolate the effects of long-range elastic interactions between defects by comparing to simulations where those effects are deliberately suppressed. This allows us to demonstrate that in bcc Fe, suppressing long-range interactions mainly influences kinetics in the first 0.3 ms, slowing down quick energy release cascades seen more frequently in full simulations, whereas long-term behavior and final state are not significantly affected.

Figure 1

Energy landscape of the divacancy states and saddle points according to the A04 and M07 force fields. The energy zero corresponds to two isolated vacancies ($\infty$).

Figure 2

Three-vacancy cluster diffusion. Small light spheres are occupied lattice sites, large dark spheres are vacancies.

Figure 3

Binding energy of minima and saddle points along the $a-b-c-d$ trajectory (cf. Fig. 2). Note that k-ART does not provide any energy information for intermediate points.

Figure 4

Trajectory of four independent runs with six randomly placed vacancies. The bottom plot shows the cohesive energy of the system over time. The central plot shows the total squared displacement. The top plot shows the cluster-size distribution, where the linewidth is proportional to the number of vacancies in clusters of the respective size. High atomic mobility appears to be correlated with clusters of size three.

Figure 5

Flow chart of vacancy cluster formation in the six-vacancy system. Clusters with $n$ vacancies are labeled ${\mathrm{v}}_n$, monovacancies are not considered except for the initial state.

Figure 6

Top: diffusion rate of various cluster combinations in six-vacancy simulations. Clusters of $n$ vacancies are denoted as ${\mathrm{v}}_n$. Remaining vacancies are monovacancies. There are two distinct configurations of three-vacancy clusters: mobile (${\mathrm{v}}_3^m$) and immobile (${\mathrm{v}}_3^i$) (see text).

Figure 7

K-ART simulation results of four independent runs with randomly placed vacancies. Bottom plot: energy of the four runs. Most energy is released within the first 100 $\mu \mathrm{s}$. Circles mark the highest effective barrier of the runs (see Fig. 8 for magnifications). Center plot: fraction of monovacancies. After about 1 ms, all vacancies are clustered. Top plot: average cluster size. The simulations are terminated when seven (run 3) or five (other runs) clusters remain and no further progress is made.

Figure 8

Magnifications from the four runs showing the trajectory through minima and saddle points. The highest effective barrier (i.e., difference between saddle-point energy and previous lowest energy) is highlighted. In all four plots, the abscissas are KMC steps, while the ordinates are energy in eV. For runs 1 and 2, an energetically favorable configuration is obtained within a few KMC steps. For runs 3 and 4, the energy does not drop below the former optimal energy right away. This is also an example of the “replenish and relax” mechanism described in Ref. [17].

Figure 9

Details of run 2: Energy, average cluster size, and monovacancy fraction are plotted over simulation time. For further details, see text.

Figure 10

Basin threshold control and relaxation progression of run 2. The crosses show the height of the saddle point above the lower of the two states that it links. Red symbols are regular barriers, gray symbols represent barriers included in the basin. The smaller blue symbols show the effective barrier, i.e. the height of the saddle point above the minimal state obtained so far. The basin threshold is shown by the solid black line.

Figure 11

GE k-ART simulation results of four independent runs with randomly placed vacancies. Bottom plot: energy. Circles highlight the highest effective barrier of each run (0.858/0.897/0.823/0.903 eV). Center plot: fraction of monovacancies. Top plot: average cluster size.

Figure 12

Comparison between standard and generic embedding k-ART runs from same initial configurations. GE runs are shown in dark colors and bold lines, whereas the corresponding standard runs use thinner lines and lighter colors. The time axis is nonlogarithmic.