Probing grain boundary sink strength at the nanoscale: Energetics and length scales of vacancy and interstitial absorption by grain boundaries in $\alpha$-Fe

The energetics and length scales associated with the interaction between point defects (vacancies and self-interstitial atoms) and grain boundaries in bcc Fe was explored. Molecular statics simulations were used to generate a grain boundary structure database that contained $\approx$170 grain boundaries with varying tilt and twist character. Then, vacancy and self-interstitial atom formation energies were calculated at all potential grain boundary sites within 15 Å of the boundary. The present results provide detailed information about the interaction energies of vacancies and self-interstitial atoms with symmetric tilt grain boundaries in iron and the length scales involved with absorption of these point defects by grain boundaries. Both low- and high-angle grain boundaries were effective sinks for point defects, with a few low-$\Sigma$ grain boundaries (e.g., the $\Sigma 3$$\left\{112\right\}$ twin boundary) that have properties different from the rest. The formation energies depend on both the local atomic structure and the distance from the boundary center. Additionally, the effect of grain boundary energy, disorientation angle, and $\Sigma$ designation on the boundary sink strength was explored; the strongest correlation occurred between the grain boundary energy and the mean point defect formation energies. Based on point defect binding energies, interstitials have $\approx$80$\%$ more grain boundary sites per area and $\approx$300$\%$ greater site strength than vacancies. Last, the absorption length scale of point defects by grain boundaries is over a full lattice unit larger for interstitials than for vacancies (mean of 6–7 Å versus 10–11 Å for vacancies and interstitials, respectively).

Figure 1

Schematic of the process used to initialize, test, and analyze point defects in grain boundaries. The example to the right shows a grain boundary system from which a single grain boundary is selected and then the point defect formation energy of every potential grain boundary site is subsequently tested to build a database of formation energies.

Figure 2

Symmetric tilt grain boundary misorientation-energy relationship for (a) the $\langle 100\rangle$ tilt axis, (b) the $\langle 110\rangle$ tilt axis, and (c) the $\langle 111\rangle$ tilt axis. The low $\Sigma$ grain boundaries ($\Sigma ⩽13$) in each system are identified. The strongest cusps are the $\Sigma 3$$\left(112\right)$, the $\Sigma 11$$\left(332\right)$, and the $\Sigma 5$$\left(310\right)$ boundaries.

Figure 3

$\langle 100\rangle$ symmetric tilt grain boundary structures with structural units outlined for the $\Sigma 17$$\left(410\right)$, $\Sigma 5$$\left(210\right)$, $\Sigma 29$$\left(730\right)$, $\Sigma 5$$\left(310\right)$, and $\Sigma 17$$\left(530\right)$ boundaries. Black and white denote atoms on different $\left\{100\right\}$ planes. The different structural units are labeled A, B, C, and A'. The $\Sigma 29$$\left(730\right)$ is composed of structural units from the two favored $\Sigma 5$ boundaries in a ratio determined by the structural unit model.[51, 90, 91]

Figure 4

$\langle 110\rangle$ symmetric tilt grain boundary structures with structural units outlined for numerous boundaries, including the “favored” $\Sigma 3$$\left(112\right)$ and $\Sigma 3$$\left(111\right)$ boundaries. Black and white denote atoms on different $\left\{110\right\}$ planes. The different structural units are labeled as in Fig. 3. Similar to Fig. 3, the structural units of intermediate misorientations can be determined from the structural units of the favored boundaries.

Figure 5

Vacancy and interstitial formation energies as a function of spatial position for three $\langle 100\rangle$ symmetric tilt grain boundaries: $\Sigma 5$$\left(210\right)$, $\Sigma 29$$\left(730\right)$, and $\Sigma 5$$\left(310\right)$ boundaries. The upper (lower) images show the distribution of formation energies lower (higher) than bulk values. The point defect formation energies are shown in the unrelaxed position; while the interstitial atoms are placed 0.5 Å away from the vacancy sites, the atoms are shown in the same location due to the grain boundary periodicity. The structural units are outlined and correspond to those shown in Fig. 3.

Figure 6

Vacancy and interstitial formation energies for two low-angle boundaries in the $\langle 100\rangle$ symmetric tilt grain boundary system: the $\Sigma 41\left(910\right)\theta =12.{68}^{\circ }$ and $\Sigma 85\left(760\right)\theta =81.{20}^{\circ }$ grain boundaries. As in Fig. 5, the point defect formation energies are shown in the unrelaxed position with formation energies for interstitial atoms and vacancies shown in the same location due to periodicity. The affected region of lower defect formation energies surrounding the dislocations is noticeably larger for interstitials than for vacancies.

Figure 7

(a) Vacancy and (b) interstitial formation energies as a function of distance from the grain boundary for all $\langle 100\rangle$ symmetric tilt grain boundaries. The vacancy and interstitial distances are the unrelaxed distances to facilitate comparison. The point defect formation energies are lower in the grain boundary region, which extends $\approx$5 Å from the center of the boundary for the $\langle 100\rangle$ tilt system.

Figure 8

Vacancy (a) and (b) and interstitial (c) and (d) formation energies as a function of the disorientation angle for $\langle 100\rangle$, $\langle 110\rangle$, and $\langle 111\rangle$ symmetric tilt grain boundaries. (a) and (c) The distribution of formation energies is plotted (black dots) as well as the minimum formation energy for each grain boundary (symbol). (b) and (d) The mean formation energy for each grain boundary is plotted against disorientation angle; additional twist and asymmetric grain boundaries are added to show the trend.

Figure 9

Vacancy (a) and (b) and interstitial (c) and (d) formation energies as a function of the grain boundary energy for $\langle 100\rangle$, $\langle 110\rangle$, and $\langle 111\rangle$ symmetric tilt grain boundaries. The symbols denote the minimum (a) and (c) and mean formation energy (b) and (d) for each grain boundary, as in Fig. 8. The strongest trend occurs between the grain boundary energy and the mean defect formation energies (c) and (d).

Figure 10

Vacancy (a) and (b) and interstitial (c) and (d) formation energies as a function of the $\Sigma$ value for $\langle 100\rangle$, $\langle 110\rangle$, and $\langle 111\rangle$ symmetric tilt grain boundaries. The symbols denote the minimum (a) and (c) and (mean formation energy (b) and (d) for each grain boundary, as in Fig. 8. Aside from a few boundaries, very little trend is observed with respect to $\Sigma$ value.

Figure 11

Grain boundary site preference is shown by plotting the vacancy binding energy against the corresponding interstitial binding energy for each site in this study. The line represents equal binding energies for vacancies and interstitial atoms. Similar to prior results in the $\langle 100\rangle$ system,[79] albeit with a larger sampling of grain boundaries, there is an energetic preference for binding of interstitials to grain boundaries over vacancies.

Figure 12

The percentage of grain boundary sites that prefer an interstitial to a vacancy as a function of the binding energy threshold given. While there is a difference between the different grain boundary systems, overall grain boundary sites have a preference for interstitial atoms over vacancies in terms of binding energies.

Figure 13

Histograms of the grain boundary (a) and (b) site density and (c) and (d) site strength for (a) and (c) vacancies and (b) and (d) interstitials. Grain boundary sites were classified according to a criterion that required the binding energy to be greater than 0.05 eV, distinguishing atoms in the bulk lattice from those in the grain boundary.

Figure 14

Evolution of the vacancy and interstitial mean binding energies as a function of the distance from the grain boundary center. The mean binding energies for each 2 Å bin are calculated from the binding energy distributions for the sampled grain boundaries (inset histogram). Grain boundaries tend to have both a larger binding energy and absorption length scale for interstitial defects over vacancies.

Figure 15

Evolution of the (a) vacancy and (b) interstitial mean binding energies as a function of the distance from the grain boundary center. The mean binding energies for each 2 Å bin are calculated from the binding energy distributions for the different grain boundary systems displayed. The binding energy curves and absorption length scales for both interstitial defects and vacancies are sensitive to the type of grain boundary system.