# Energetic driving force for preferential binding of self-interstitial atoms to Fe grain boundaries over vacancies

### Abstract

Molecular dynamics simulations of 50 Fe grain boundaries were used to understand their interaction with vacancies and selfinterstitial atoms, which is important for designing radiation-resistant polycrystalline materials. Site-to-site variation of formation energies within the boundary is substantial, with the majority of sites having lower formation energies than in the bulk. Comparing the vacancy and self-interstitial atom binding energies for each site shows that there is an energetic driving force for interstitials to preferentially bind to grain boundary sites over vacancies.

Author(s): Mark A. Tschopp, Mark F. Horstemeyer, F. Gao, X. Sun and M. Khaleel

Corresponding Author: Mark Tschopp

Figure 1. (a) ⟨1 0 0⟩ symmetric tilt grain boundary energy as a function of misorientation angle. The low-R grain boundaries in each system are identified. (b) ⟨1 0 0⟩ symmetric tilt grain boundary structures with structural units outlined for the Σ5(2 1 0), Σ29(7 3 0) and Σ5(3 1 0) STGBs. Black and white denote atoms on different {1 0 0} planes. The different structural units are labeled B and C. (click on the image to enlarge).
Figure 2.. The vacancy and self-interstitial atom formation energies as a function of location for the Σ5(2 1 0), Σ29(7 3 0) and Σ5(3 1 0) STGBs (click on the image to enlarge).
Figure 3. Evolution of (a) vacancy and (b) interstitial formation energies as a function of distance from the grain boundary for all 50 ⟨1 0 0⟩ symmetric tilt grain boundaries (STGBs). Low and high angle boundaries are colored differently. The inset image is an example of a low angle boundary. (click on the image to enlarge).

### Methodology

The formation energies of vacancies and self-interstitial atoms (SIAs) as a function of location within/around the GB is calculated for 50 ⟨1 0 0⟩ symmetric tilt grain boundaries in body-centered cubic (bcc) Fe to energetically assess the GB sink strength. Nanoscale simulations were required to capture the physics of vacancy and interstitial formation energies at the GB interface. A parallel MD code, LAMMPS [1] , was used to run all simulations in this work. First, a GB database consisting of 50 ⟨1 0 0⟩ symmetric tilt grain boundaries was generated using bicrystal simulation cells with three-dimensional periodic boundary conditions [2][3][4]. A minimum distance of 12 nm between the two grain boundaries was used during generation to eliminate any potential interaction between the two boundaries. As with previous work [5][6], multiple initial configurations with different in-plane rigid body translations and an atom deletion criterion were used to properly access an optimal minimum energy GB structure via the Polak–Ribie`re conjugate gradient energy minimization. For an initial generation of the structures, the updated version of the Mendelev et al. [7] interatomic potential for Fe was used. This embedded-atom method [8] potential has been shown to perform well in nanoscale simulations for nuclear applications [9]. A large number of grain boundaries were used to sample the range of GB structures and energies that might be observed in polycrystalline materials. The ⟨1 0 0⟩ symmetric tilt grain boundary (STGB) system chosen has several low order coincident site lattice (CSL) grain boundaries (the Σ5 and Σ13 boundaries), as well as both general high angle boundaries and low angle grain boundaries (615).Figure1a shows the GB energy as a function of misorientation angle for the ⟨1 0 0⟩ symmetric tilt grain boundary system, similar to that found previously in Fe–Cr simulations [10]. The low-order CSL grain boundaries are also shown on this figure. For the ⟨1 0 0⟩ tilt axis, only minor cusps were observed in the energy relationship, most noticeably at the R5(3 1 0) boundary. In addition to many general high angle boundaries, several low angle boundaries (615) are also plotted. The range of GB energies sampled was 500 mJ m^{-2}. The GB structure plays an important role on the GB properties [11]. For low angle boundaries, the grain boundary is best represented by an array of discrete dislocations spaced a certain distance apart. However, at higher misorientation angles the spacing between dislocations is small enough that dislocation cores overlap and dislocations rearrange to minimize the energy of the boundary. The resulting GB structures are often characterized by structural units [12]. Grain boundaries with certain misorientation angles (and typically a low Σ value) correspond to “favored” structural units, while all other boundaries are characterized by structural units from the two neighboring favored boundaries. Figure 1b shows an example for the ⟨1 0 0⟩ STGB system, where the two Σ5 boundaries are favored STGBs, and the Σ29(7 3 0) boundary is a combination of structural units from the two Σ5 boundaries. The structural units for the Σ5(2 1 0) and Σ5(3 1 0) STGBs are labeled B and C,respectively, in a convention similar to that used for face-centered cubic metals [13]. Also notice that the ratio of structural units in the Σ29 GB can be determined by the crystallographic relationship of the two favored boundaries. All boundaries between the favored structural units of the R5 boundaries and the “structural units” of the 0 single crystals have a similar makeup. The formation energies of vacancies and self-interstitial atoms at specific locations within/around the grain boundary were calculated for each grain boundary. The approach here is similar to that used for modeling point defects in Cu [14]. First, all atoms within 20 A ˚ of each grain boundary were identified as potential sites. Then, for vacancy formation energy simulations, an atom at a particular site a was removed and the simulation cell was relaxed through an energy minimization. The vacancy formation energy for that particular site a was then calculated by:

$E^{\alpha}_{f}=E^{\alpha}_{GB}-E_{GB}+E_{coh}$
where $E_{coh}$ is the cohesive energy /atom of a perfect bcc lattice, and $E^{alpha}_{GB}$ are the total energies of the simulation cell with and without the vacancy. On the other hand, for self-interstitial formation energy simulations, an atom was inserted approximately 0.5 A ˚ away at site a, and the simulation cell was again relaxed through an energy minimization. The self-interstitial atom formation energy for site a was calculated by:
$E^{\alpha}_{f}=E^{\alpha}_{GB}-E_{GB}-E_{coh}$
The only difference is that $E_{coh}$ is subtracted instead of addition to reflect the added energy due to placing that atom in an interstitial site. This procedure was then performed for all 50 grain boundaries shown in Figure 1a and the vacancy and interstitial formation energies were calculated as a function of location for each boundary.

### Results

Figure 4. Evolution of (a) vacancy and (b) interstitial formation energies as a function of distance from the grain boundary for all 50 ⟨1 0 0⟩ symmetric tilt grain boundaries (STGBs). Low and high angle boundaries are colored differently. The inset image is an example of a low angle boundary. (click on the image to enlarge).

Figure 2 shows the vacancy and interstitial formation energies that correspond to atomic positions in the three GB structures shown in Figure 1b. The formation energy depicted for each location corresponds to a simulation. In this graph, the color bar is normalized such that the high value corresponds to the formation energy in the bulk, so that vacancy and interstitial energies can be easily compared. For all three GBs in Figure 2, there are atoms lying symmetrically along the GB plane that have vacancy formation energies slightly higher than in the bulk (E_f = 1.72 eV). However, there are several sites in each grain boundary that have vacancy formation energies lower than in the bulk, suggesting an energetic driving force for vacancy diffusion from the single crystal to the grain boundary. As the vacancy site is shifted away from the boundary, the formation energies approach that of the bulk. For interstitials, most of the GB sites depicted here have a lower formation energy than in the bulk (E_f = 3.52 eV). In the figure, the 29 GB has formation energies similar to general high angle GBs with higher Sigma values, while the Σ5 GBs have formation energies that differ from most GBs.
AtomEye is used to visualize the simulation results [15]. The evolution of the (a) vacancy and (b) interstitial formation energies as a function of distance from the grain boundary is shown in Figure 3 for all 50 grain boundaries. Interestingly, most of the formation energies that differ from the bulk occur within 5–8 A ˚ from the boundary center. The majority of formation energies within this GB-affected region are lower than in the bulk. However, for both vacancy and self-interstitial atoms, there are locations with formation energies higher than in the bulk, with the highest near the boundary center. Additionally, this plot separates the high angle boundaries from the low angle boundaries, which is motivated by previous studies (e.g. [16]). For interstitial formation energies, most high angle grain boundaries have much lower formation energies than in the bulk.

The majority of interstitial formation energies near bulk values within 5 A ˚ of the boundary occur in the single crystal regions between dislocations for low angle boundaries. For example, the inset image shows an example of a low angle boundary. The visible dislocations in the low angle boundary have a local effect on formation energies, and increasing the dislocation spacing (i.e. lower misorientation angle) merely results in shifting these localized regions further apart. For low angle boundaries, the formation energies trend to that of an isolated dislocation in a single crystal. The GB binding energy for vacancies and self-interstitial atoms are plotted against each other for each site in Figure 4. The GB binding energy for a particular site α is calculated by subtracting the formation energy from the bulk formation energy, $E^{\alpha}_{binding}= E_{f,bulk}-E^{\alpha}_{f}$ . The line delineates sites where interstitial binding energy is greater than vacancy binding energies (above the line). The large amount of binding energies above this line indicates that the system energy is decreased more through interstitials occupying GB sites, rather than vacancies. Hence, this letter shows that there is an energetic driving force for interstitials to segregate to GB sites over vacancies. This study also supports the interstitial loading–unloading mechanism [17] by showing that it is energetically favorable for interstitials to initially “load” grain boundaries for a wide range of GB types. This is significant for nuclear applications where radiation damage generates lattice defects and grain boundaries act as sinks for both vacancies and interstitial atoms. In summary, this letter shows that self-interstitial atoms have a larger energetic driving force for binding to grain boundaries than vacancies, based purely on formation energies of these point defects in thousands of GB sites. Fifty ⟨1 0 0⟩ symmetric tilt grain boundaries were utilized here to sample some of the variability in GB degrees of freedom that is observed in experimental polycrystalline materials, including general low and high angle grain boundaries as well as some low-R boundaries. By iteratively calculating the formation energies of both vacancies and self-interstitial atoms at every site within 20 A ˚ of the boundary, this study methodically examined the binding energy associated with GB segregation of point defects and calculated the energetic driving force for point defects to grain boundaries in Fe.

### Acknowledgments

M.A. Tschopp would like to acknowledge funding provided by the US Department of Energy’s Nuclear Energy Advanced Modeling and Simulation (NEAMS) program at Pacific Northwest National Laboratory. PNNL is operated by Battelle Memorial Institute for the U.S. Department of Energy under Contract No. DE-AC05-76RL01830.

### References

The initial methodology was used in the following papers:

1. S. Plimpton, J. Comp. Phys., 117, 1-19 (1995).
2. J.D. Rittner, D.N. Seidman, Phys. Rev. B 54 (1996) 6999.
3. M.A. Tschopp, D.L. McDowell, Philos. Mag. 87 (2007) 3147
4. M.A. Tschopp, D.L. McDowell, Philos. Mag. 87 (2007) 3871.
5. M.A. Tschopp, D.L. McDowell, Philos. Mag. 87 (2007) 3147
6. M.A. Tschopp, D.L. McDowell, Philos. Mag. 87 (2007) 3871.
7. M.A. Tschopp, D.L. McDowell, Philos. Mag. 87 (2007) 3147
8. M.S. Daw, M.I. Baskes, Phys. Rev. B 29 (1984) 6443.
9. L. Malerba et al., J. Nucl. Mater. 406 (2010) 19.
10. Y. Shibuta, S. Takamoto, T. Suzuki, ISIJ Int. 48 (2008) 1582.
11. Y. Mishin, M. Asta, J. Li, Acta Mater. 58 (2010) 1117.
12. A.P. Sutton, V. Vitek, Philos. Trans. R. Soc. Lond. A 309 (1983) 1.
13. J.D. Rittner, D.N. Seidman, Phys. Rev. B 54 (1996) 6999.
14. A. Suzuki, Y. Mishin, Interface Sci. 11 (2003) 131.
15. J. Li, Modell. Simul. Mater. Sci. Eng. 11 (2003) 173.
16. F. Gao, H.L. Heinisch, R.J. Kurtz, J. Nucl. Mater. 386–388 (2009) 390.
17. X.-M. Bai, A.F. Voter, R.G. Hoagland, M. Nastasi, B.P. Uberuaga, Science 327 (2010) 1631.