Stability of $\langle 100\rangle$ Dislocations formed in W Collision Cascades

The $\langle 100\rangle$ dislocations explored consist of 34 defects formed in 230 collision cascades. The procedure used for the collision cascade simulations and the identification of defect morphologies is discussed in detail in [6, 7]. The database consists of collision cascades using three interatomic potentials: Finnis-Sinclair potential [20] as modified by Juslin et al. (JW) [21], the potential by Derlet et al. [22] with the repulsive part fitted by Björkas et al. [23] (DND-BN), and the potential by Marinica et al. [24], stiffened for cascade simulations by Sand et al. (M-S) [25]. Collision cascades from multiple potentials help in analysing the effect of choice of potential on the stability of $\langle 100\rangle$ dislocations loops. We selected these three potentials owing to the large collision cascade database that is available with us for these three potentials.

The stability analysis and transition energy calculation is carried out on the simulation results of several annealing MD simulations. The simulations were performed for each $\langle 100\rangle$ dislocation loop in isolation at different temperatures. We first extract the desired defect along with five extra unit cells around it from the simulation box of the collision cascade. We then add more unit cells of W on each side of the extracted volume to form a simulation box of at least five times the extracted number of unit cells in each direction to take care of finite size effects. The Large-scale Atomic Molecular Massively Parallel Simulator (LAMMPS) [26] code is used to carry out MD simulations to relax the system using an NPT ensemble at 300 K with periodic boundary conditions (PBC) and zero pressure. This relaxation step is essential given that we have placed the defects and their nearby unit-cells into a new crystal. We analyse the dislocation after the relaxation for any change in structure or inconsistency.

For each relaxed system we carry out a temperature ramping simulation from 300K to 2000K. The temperature at which the defect transitions to $\langle 111\rangle$ dislocation ($T_{r}$) in a ramping simulation is noted. We calculate 12 temperatures ($T_{1}$ to $T_{12}$) around $T_{r}$ with a difference of 25K i.e. $(T_{2}-T_{1})=(T_{3}-T_{2})=$ … $=(T_{12}-T_{11})$ = 25K and $T_{7}=T_{r}$. After that, sixteen different NPT runs are carried out at each of these temperatures with different random number seeds for initializing the temperature. Periodic boundaries are set at zero pressure. These simulations are stopped either at 50 ns or if the defect transitions to $\langle 111\rangle$ dislocation. For cases where most of the sample runs transition while a few do not transition at a certain temperature, the non-transitioning simulations are extended for 100 ns. The temperatures where transitions occur are included for the transition energy calculation. We do not calculate the transition energy of a defect if it does not transition for the complete 100 ns run even at a high temperature of 2000 K. We use an output frequency of 2 ps for checking transitioning and configuration changes.

For transition energy calculation, the temperature (T) dependence of time for transition ($\tau$) is taken to be described by an Arrhenius expression:

where $\tau_{0}$ is a pre-exponential factor, $E_{t}$ is the activation energy for transition to $\langle 111\rangle$ loop and $k_{b}$ is the Boltzmann constant. A linear regression fit for the equation is found for different runs of each defect. Figure 1 shows a sample fit for a defect of size 11. The plot shows the time of transition of the $\langle 100\rangle$ loop to $\langle 111\rangle$ loop for different runs at eleven different temperatures. Out of the total twelve temperatures considered initially, for the lowest temperature a few of the runs did not transition within the 100 ns time limit. For this reason only eleven temperatures are shown in the plot. The temperature range is from 550K to 800 K with an interval of 25K. At each temperature there are sixteen different runs having different transition times. All these data-points from the eleven temperatures are included for the transition energy calculation. The transition time is governed by a Poisson distribution which results in a spread of transition times at each temperature. Moreover, different transition pathways and changes in configurations (as discussed in Section 3.1 and Section 3.3) further add to the variation in the transition times of different runs at a specific temperature.

We track morphological changes using the SaVi algorithm [6] and use CSaransh [27] to visualize the defects. In addition to identifying when the morphological transition occurs, SaVi also outputs various parameters such as the number of dumbbells/crowdions in specific orientations at each step and the number of neighbors of each dumbbell/crowdion. These parameters define the internal morphology and help understand the relationship between stability and internal configuration.

SaVi algorithm uses computational geometry and graph data-structure to find the various morphological components of a defect and identify the overall morphology based on the components. The algorithm finds lines across all the dumbbells and crowdions. The direction of the lines decide the orientation of a dumbbell or a crowdion. The lines that share a specific angle and distance form a morphological component. For instance, a dislocation is formed by lines that are approximately parallel to each other, while a hexagonal ring is formed by lines that have 60 degree angle. The direction of the Burgers vector for a dislocation loop is found by the orientation of the constituent lines while the magnitude is decided by the number of net defects in each line. The algorithm involves several details that make it robust to noise such as finding cycles for asserting rings. The algorithm can be used to find if a dumbbell is at the center of the defect or on the surface by counting its number of nearest neighbouring parallel lines.

The arrangement of dumbbells and crowdions in a dislocation decides the energy density and stability[19, 18]. This forms the basis of examining relationship between stability and internal configuration. For the discussions of internal configurations, it is important to express it in a way that conveys the shape of the defect especially the factors that are observed to affect the stability.

The $\langle 100\rangle$ dumbbells in a $\langle 100\rangle$ loop arrange in rectilinear form [18]. More specifically, in a parallelogram or rhombus shape [10] along with a few residual dumbbells outside the full parallelogram [6]. We utilize the length, breadth and residual number of $\langle 100\rangle$ dumbbells to name the configuration of $\langle 100\rangle$ dumbbells. The naming scheme also signifies the degree to which dumbbells are packed at the center in a configuration and can help in understanding stability differences for same sized defects. The scheme used to name a configuration of size $s$ is as follows:

where, $n$ is the number of $\langle 100\rangle$ dumbbells while $m$ is the number of non-$\langle 100\rangle$ dumbbells on the fringes. The configuration of $\langle 100\rangle$ dumbbells is further divided into $l$, $b$ and $r$ where $l$ and $b$ are the length and breadth of the number of completely filled rows and columns (sides of parallelogram) formed by the $\langle 100\rangle$ dumbbells. $r$ is the remaining or residual number of $\langle 100\rangle$ dumbbells. The values of $l$ and $b$ are to be chosen such that the residual $r$ has a minimum value. Figure 2 shows a schematic representation of a few defect configurations and their corresponding parameters for naming.

We say that two defect configurations are different if any of the values of $l$, $b$, $r$ or $m$ are different. Another feature that we will note in a defect configuration is the ratio of $l$ and $b$, which we refer to the aspect ratio of the $\langle 100\rangle$ component. This ratio together with number of residual dumbbells gives an idea about the number of dumbbells that will be on the edges. As shown in [18], the $\langle 100\rangle$ loops tend to have lower aspect ratio.

In place of current naming scheme, one can find the number of $\langle 100\rangle$ neighbors for each $\langle 100\rangle$ dumbbell and use this neighbour count for comparing the stability of defects where sizes are same. A dumbbell in the center would have four $\langle 100\rangle$ nearest neighbors while the one on the interface will have less than four. This scheme might be more appropriate for quantitative comparison or for loops that are more often circular than rectilinear. One drawback of the neighbor count based scheme is that it does not give a picture of the arrangement for qualitative discussion. In Section 3.3, we will discuss the relationship of the stability of a configuration with the different factors of the notation that we have used.

We find the changes in configurations and transition mechanisms in the simulations by tracing the movements and rotations of dumbbells that are in the center and on the edges. Although in some cases the configuration changes can be too quick and non-recurring, most of the times the transition pathways and configuration changes fall clearly in one of the two categories shown in Figure 5. These two paths are representative of the reoccurring patterns that we observe for all the transitioning defects. The figure traces the significant changes in defect configurations for two separate runs of a size-12 defect as it transitions to $\langle 111\rangle$ dislocation. The initial configuration of the defect is $(3\times 3+2)+1$ (shown left of the center in Figure 5).

Path-A is the preferred path of configuration changes and transition when $r=0$. In Figure 5 path-A starts when the defect transitions from initial configuration to $3\times 3+0$ configuration. The non-$\langle 100\rangle$ dumbbells move around the fringes from one side to the other while the $\langle 100\rangle$ dumbbells remain stable in the same configuration. This movement can result in a climb of the dislocation loop, which is observed to occur rarely. However, glide in these dislocations is much more common. As the non-$\langle 100\rangle$ dumbbells move around, they may meet and cluster, making the defect slightly unstable, inducing rotation in adjacent $\langle 100\rangle$ dumbbells to non-$\langle 100\rangle$. In our example, this is shown by Figure 5 (i) to (v) of Path A. The rotation of the adjacent dumbbells in the $\langle 100\rangle$ bunch may go back and forth for some time. At some point, a good majority of the dumbbells in the main bunch rotate, resulting in the transition to highly glissile $\langle 111\rangle$ dislocation. In a bigger $\langle 100\rangle$ dislocation, the number of $\langle 100\rangle$ dumbbells is more, and it becomes difficult to reach an instance when a majority of main $\langle 100\rangle$ dumbbells are in the non-$\langle 100\rangle$ direction. A bigger defect may either stay as a multi-component mixed loop or may keep looping between stages similar to (iv) and (v), thus taking longer to eventually transition.

Path-B is typical when $r>0$ or multiple non-$\langle 100\rangle$ dumbbells are distributed unevenly around the $\langle 100\rangle$ component. The exact arrangement of the $\langle 100\rangle$ dumbbells keeps changing very often as the $\langle 100\rangle$ dumbbells on the edges keep rotating from $\langle 100\rangle$ to non-$\langle 100\rangle$ as shown in Figure 5 (ii) to (v). At a certain moment, the rotation may travel from the edge towards the center of the dislocation. If a majority of dumbbells get rotated away from $\langle 100\rangle$, the dislocation rather than coming back to $\langle 100\rangle$ sways entirely to $\langle 111\rangle$ dislocation (Figure 5 (vi)). Again, if the value of $n$ is greater with more $\langle 100\rangle$ dumbbells packed inside, it is more likely for the rotating non-$\langle 100\rangle$ dumbbells to sway back rather than the disturbance penetrating deep resulting in the rotation of all $\langle 100\rangle$ dumbbells.

It must be noted that at any time the further dynamics of defect configurations may change from one path to another if the defect configuration changes in such a way. For instance, if an incomplete parallelogram ($r>0$) transitions to a smaller but complete parallelogram ($r=0$) then the pattern of further configuration changes will be based on path-A.

For both paths, the transition starts from the edges and moves towards the central part. Figure 6 shows transition of a size twenty-one defect. The rotation of dumbbells can be clearly seen to be initiated at the edges and then moving towards the center. For a bigger defect the half rotated configurations in first row are more stable. The defect may stay as a mixed loop for longer time. The configuration then may sway back to $\langle 100\rangle$ again or may trickle down to $\langle 111\rangle$ loop as in the second row of the Figure 6. The final transition stages are quick and same in both the paths.The relative instability of dumbbells/crowdions at the edges compared to central ones can be understood by their higher energy density as shown in [19, 6]. This difference in stability is also observed in our further examination of the relationship between internal morphology and stability in Section 3.3.

Figure 7 shows the transition energy of various $\langle 100\rangle$ dislocations for transitioning to $\langle 111\rangle$ dislocations as a function of defect size. The configuration of these defects are shown in Figure 4. In case there are multiple defects of the same size, the defects in Figure 4 are labeled by alphabets (a), (b) and so on, starting from defect with lower transition energy in Figure 7. The overall trend in Figure 7 shows an increase in defect stability with an increase in size.

The majority of $\langle 100\rangle$ dislocation loops that get formed in a collision cascade are of the size ranges that will transition to $\langle 111\rangle$ dislocation loops at elevated temperatures. This is in agreement with the experimental findings that the number of $\langle 100\rangle$ loops reduce at elevated temperatures [28].

The transition energy does not increase linearly with size but rather in steps. It remains comparable for a few similar sizes and then increases sharply. For example, in Figure 7 we see that sizes 7 to 11 have comparable transition energy, and then for size 12, it increases sharply. Again, the transition energy values are similar for sizes 19, 21, and 24, whereas size 29 is too stable to transition within the MD simulation time limits, even at a temperature of 2000 K. We also observe that defects of the same size may have different transition energy. For example, out of the three defects of size six, two have similar transition energy, whereas the third has higher transition energy (Figure 7) though all three defects are created in collision cascades using the JW interatomic potential. The internal morphology as discussed in next subsection reveals the reason behind these observations.

The transition energies of $\langle 100\rangle$ dislocations of different sizes show similar values and trends for all three potentials. However, there are a few differences, especially in the M-S potential.

The size-9 defect in M-S potential sometimes transitions to a mixed morphology defect having a component of $\langle 111\rangle$ dislocation and a hexagonal ring as shown in Figure 9. Although the transition to mixed morphology is rare, it is significant because once formed, this mixed configuration is very stable in M-S and does not transition even at very high temperatures like 2000 K. Also, such ring-like defects are formed relatively more often in collision cascades with M-S potential than with the other two potentials [7]. The M-S potential has been fitted using liquid configurations in addition to perfect crystal and point defects. For this reason, it might be more accurate about the stability of rings in W.

Defects of size 19 onward in M-S do not transition within the limits of our simulations, while for DND-BN, the defects up to size 24 transition. The two defects with size 12 have same configurations but different interatomic potentials. The one that belongs to M-S potential has higher transition energy than the other one that belongs to the DND-BN potential (Figure 4). In [17], it has been shown that the big multi-component dislocations are highly stable in M-S, while it is not the case with other potentials that they use. It has also been shown that the M-S potential stabilizes $\langle 100\rangle$ dislocation loops over 1/2 $\langle 111\rangle$ [29]. Our observation also indicate that the transition energy for $\langle 100\rangle$ dislocations is higher in the M-S potential compared to the other two potentials especially for bigger sizes in our dataset.