Ab initio solute–interstitial impurity interactions in vanadium alloys: the roles of vacancy

Abstract

This study aims to characterize the interactions between substitutional solutes (3d, 4d and 5d transition metals) and interstitial impurities (C and O) in vanadium alloys, with or without the presence of an adjacent vacancy. For this purpose, the binding energies for solute–impurity and vacancy–impurity pairs, as well as solute–vacancy–impurity complexes are investigated by means of first-principles calculations, with or without the elastic correction. The vacancy–impurity binding energies suggest that it is energetically favorable to form stable 1nn vacancy–impurity pairs. For large-sized solutes, the solute–impurity interactions present strong repulsive interactions when a vacancy is absent, while showing strong attractive ones in the presence of a vacancy. Furthermore, a comprehensive study on the binding energy of defects revealed a positive correlation between the elastic correction energies and solute volumes, indicating that the elastic correction for the binding energies needs to be considered when a vacancy is absent in the vicinity of defects. Based on the binding preference, we can infer that a vacancy prefers to bond with large solutes adjacent to it and thus the resulting solute–vacancy pair can serve as a strong impurity trapper to form a defect complex, enhancing the nucleation and growth of precipitates in V alloys.

---

1. Introduction

Vanadium (V) alloys are considered as candidates for blanket structural materials for fusion reactor systems due to their low neutron irradiation-induced activation characteristics, and remarkable elevated temperature mechanical and thermal properties in the fusion environment.^1–3 At present, V–4Cr–4Ti is regarded as the leading candidate due to its superior properties.³ With systematic efforts, studies showed that the mechanical strength of V alloys can be improved by a high number density of tiny precipitates dispersed in the matrixes.^4,5 Meanwhile, these nanoscale precipitates in V–4Cr–4Ti alloys are Ti-rich and most likely to be Ti–(O, N, C).^6,7 Especially, Zhu et al.⁸ using high-resolution electron microscopy (HREM) reveals that the precipitates are platelet-like, with NaCl structure and preferentially distribute within the grains rather than at the grain boundaries in the V–4Cr–4Ti alloys. Therefore, the optimization of size and distribution of Ti–(O, N, C) precipitates are crucial for good mechanical properties of the V–4Cr–4Ti alloys. Recently, Miyazawa et al.⁹ also reported that Y addition enhanced the formation of precipitates in V–4Cr–4Ti alloy, which can improve their high temperature strength. Thus it is beneficial to make clear the formation process of nanoclusters or precipitates, especially by understanding the interactions of substitutional solutes with interstitial impurities. However, our understandings of those interactions are far from completed.

Effects of interstitial impurities (C, O, and N) on mechanical property of V alloys are long-standing research subject. Even though the estimated containing for these impurities are low than 1000 wppm, the influence on the mechanical properties are strong enough. In the past, the impurities were usually considered to take the responsibility for the loss of ductility, and were removed as much as possible. Possible embrittling mechanisms were presented by DiStefano et al.¹⁰ where O enriched in boundary and have affinity to solute Ti in near-boundary regions, resulting in the grain boundary weakening. However, deeper analysis of the experimental results revealed that interstitial impurities such as C, O and N can effectively harden alloys due to the larger lattice distortion.^11,12 Even these impurities can also be considered as alloying elements in V, analogous to C in steel. The effect of O and N levels on hardness and precipitation behavior was investigated for V–4Cr–4Ti alloys by Heo et al.⁴ They pointed out that the hardness and the density of precipitates in alloys are increasing with O and N level.

Ab initio calculations based on the density functional theory (DFT) have been the most powerful tool for evaluating the atomic interactions and understanding the basic atomic phenomena involved in the V bulk materials. For instance, Zhang et al.^13,14 investigated the behaviors of H and He impurities in V and found that single H or He atom prefers to stay at the tetrahedral interstitial site rather than the octahedral site, and exhibits strong attractions with vacancy. Li et al.¹⁵ discussed the stability and diffusion behaviors of C, O and N impurities in V. They concluded that two same impurities reveal repulsive interactions when the distances between them are within about 3 Å. Furthermore, Gong et al.¹⁶ even studied the diffusivity of C, O and N atoms on the V surface. Recently, our group investigated the interactions of solute–solute and solute–vacancy in V.¹⁷ It is interesting to note that larger solutes, i.e., Sc, Y, Zr, La, and Hf, have larger binding energies for bonding with vacancies, and those values are by far larger than corresponding solute–solute binding energies, suggesting that larger solutes are more favorable to bind with vacancies than solute atoms. Therefore, large solutes can be considered as vacancy trappers in the crystal. Despite of the above mentioned efforts, there is no study reported for the interactions of solutes with interstitial impurities in V, which is of great importance for understanding the formation mechanisms of precipitates.

In this paper, a thorough study of the interactions of solutes with interstitial impurities (C and O) is carried out to uncover the possible formation mechanisms of precipitate in V alloys. The main objective of this work is using DFT calculations to study the solute–impurity binding with different neighboring sites in the BCC V lattice. Meanwhile, we also assess how the vacancy affects the binding preferences and the formation of nanoclusters or precipitates.

2. Methodology

All the energies are computed within the Vienna Ab Initio Simulation Package (VASP)^18,19 with the projector augmented wave (PAW) method^20,21 and generalized gradient approximation Perdew–Burke–Ernzerhof functional (GGA-PBE).²² The computations perform within a 54-atom periodic simulation cell. The binding energies are obtained with 400 eV plane-wave cutoff and 9 × 9 × 9 k-point meshes. Once the Hellmann–Feynman force acting on atoms is less than 0.01 eV Å⁻¹, the atoms are regard as being fully relaxed. The climbing-image nudged elastic band (CI-NEB) method²³ is used to estimate the migration barriers and determine the transition state. The required force convergence for all atoms on the CI-NEB technique is set to 0.04 eV Å⁻¹. The present equilibrium lattice constant for bulk V is 2.99 Å, which is consistent with theoretical results^16,24 and the experimental value of 3.03 Å.²⁵ All calculations performed here are not spin polarized since the magnetic interactions are very small in V.

The binding energy E^A−B_bind between two point defects (A, B) in the supercell is calculated by:^26,27

E^A−B_bind = E^A_tot + E^B_tot − E^A+B_tot − E^bulk_tot (1)

where E^A_tot, E^B_tot, and E^A+B_tot stand for the total energies of the supercell contained defect A, B, and two defects with the first nearest-neighbor (1nn) to fourth nearest-neighbor (4nn) site distance, respectively. E^bulk_tot is the total energy of perfect V bulk. In eqn (1), A can represent the solute (Sol) atom or the vacancy (Vac), and B represents the impurity (Imp) atom. If A is the solute atom, the E^A−B_bind indicates the Sol–Imp binding energy; if the latter, the E^A−B_bind means Vac–Imp binding energy. In Fig. 1, the model describes the binding configurations of Sol (or Vac)–Imp at different nn distances.

Fig. 1 Configurations of the solute (or vacancy) binding with impurity atoms with the different distance. The value i represents the first- to the fourth-nearest-neighbor (1nn to 4nn) octahedral site relative to the solute (or vacancy). For the triple defects configurations, the solute atom locate at ‘X’ labeled site and impurity atom locate at the i = 1 labeled site.

The binding energy of the Sol–Vac–Imp complex in the supercell, representing the stability of the complex with respect to the isolated defect, is defined as the energy difference between the three supercells with individual point defect and the supercell with the triple defects.²⁸ It can be calculated by

E^A−B−C_bind = E^A_tot + E^B_tot + E^C_tot − E^A+B+C_tot − 2E^bulk_tot (2)

where A, B, and C stand for vacancy, solute, and impurity, respectively. The special configuration for Sol–Vac–Imp complex is also illustrated in Fig. 1 since this configuration is considered as the most favorable one from the point of view of energy.²⁸ The verification for the most stable configuration is presented in Section 3.3 below. From a thermodynamic perspective, a negative binding energy indicates a repulsive interaction and a positive energy denotes an attractive interaction.

All computations of the defect energies in present work are obtained under the fixed periodicity vectors condition. It is well known that the results under fixed periodic boundary condition usually overestimate the defect energies due to the interaction of defects with its periodic images.^29,30 In order to determine this effect, the elastic correction described in ref. 29 is applied to remove the elastic interaction of the defect clusters with its periodic images. Therefore, our results determine the binding energies in both cases, considering the effect of elastic image interaction or not. Besides, the calculations of elastic correction in this study are performed at constant volume (no strain) condition which is recommended and more appropriate to obtain the converge results.

To determine the atomic size of each solute impurity, the solute volume V_sol is calculated using the following expression:³¹

V_sol = V_cont-sol − V_tot (3)

where V_cont-sol and V_tot are the volume of the supercell with and without a single solute atom, respectively.

3. Result and discussion

The preliminary knowledge about single impurity, vacancy, and solute atom is essential for the development of the following Sol–Imp and Sol–Vac–Imp interaction models. The formation energies of single vacancy, impurities (C, O and N), as well as solution energies of solute atoms in V bulk, had been investigated in our previous works.^16,17,32 The results showed that the formation energies of C, O and N atoms in V bulk are all negative, suggesting their dissolution processes are exothermic.^17,32 In addition, they prefer the octahedral interstices sites to the tetrahedral ones. The solution energies for solutes are strongly related to atomic size.^17,32 Specially, the solutes with larger atomic sizes (such as Sc, Y, Zr, La, and Hf) have higher positive solution energies, and thus are less energetically favorable. The vacancy formation energy in host V is calculated (2.704 eV),¹⁷ and is in agreement with other results.^33,34

3.1 The interaction between solutes and impurities

We start by studying the interaction between solute and single impurity atom. Fig. 2 shows the binding energies between the solutes and the impurities C or O at the 1nn to 4nn sites. In order to obtain the exact interactions, the results of Sol–Imp binding energies corrected or not by elastic model are all presented in the figure. For the case of Sol–C pairs as presented in Fig. 2(a), it can be seen that the Sol–C interactions corrected for large solutes (such as Sc, Ti, Y, Zr, La, and Hf) are different from others. The 1nn sites Sol–C binding present the largest repulsive interactions. As the distance increases, the repulsive interactions firstly decrease sharply at 2nn and 3nn site and then increase obviously, making a volcano-like variation tendency. Hence, as for the binding of large solutes and impurity C, there is an increasing repulsive interaction preventing the two species from approaching each other. On the contrary, for those 3d transition-metal elements (e.g. Cr, Mn, Fe, Co, and Ni), a V-like tendency of binding energies is found as the distance increases, with 2nn or 3nn sites as the least favorable ones. This difference between the solute–impurity binding energy of the 3d metals compared to the other metals maybe is relative to the bonding characteristic. We perform spin polarized calculation to study the magnetic effect and the result shows that the perturbations of the binding energy caused by the magnetic interactions are very small in V (less than 0.01 eV). In order to obtain a comparison, we present the charge density contour for the 1nn Fe–C and Ru–C pair as shown in the Fig. 3. It is obvious to see that the electron densities for C atom are confined in a small sphere and lead to outward expansion of the neighbor atoms. As for Fe–C pair, there is high charge density around Fe atom and demonstrate strong electronic interaction, which result in an attractive interaction between Fe and C in 1nn distance. On the contrary, the repulsive interaction exists in the 1nn Ru–C pair. To hold the overall tendency, as seen in Fig. 4, the 1nn Sol–C pair binding energies are nearly linearly correlated with the volume of solutes. This character indicates that the local strain may play a crucial role in Sol–C interaction. Locally tensile stress field is created when solutes are introduced to the crystal, and thus there is not enough space to accommodate 1nn interstitial impurity with large sized solutes. However, this locally tensile stress can be released when 3d transition-metal solutes are introduced into the host as their volumes are smaller than V (the atomic size of each solute can find in Fig. 4). Liu et al.²⁸ discussed the interactions of C atom with substitutional solutes in α-Fe. They concluded that the repulsive interactions of solute and C atom become obvious when the sizes of solutes increasing, which is in agreement with our conclusions.

Fig. 2 Calculated solute–impurity binding energies for the 3d, 4d, and 5d elements with C atom (a) or O atom (b) in the 1nn to 4nn octahedral site. Solid symbols refer to DFT uncorrected results and open symbols to the results corrected by elastic model.

Fig. 3 Charge density contour plots for 1nn Fe–C (a) and Ru–C (b) pair.

Fig. 4 Calculated 1nn Sol–C binding energy as a function of volume of solutes.

As shown in Fig. 2(b), it can be seen that the binding of Sol–O pairs exhibit a similar tendency to the case of Sol–C ones. Most of the Sol–O interactions are repulsive except for a few configurations involving large sized solutes (Sc, Ti, Y, Zr, La, and Hf). It is noteworthy that the repulsive interactions for 1nn Sol–O pairs are more obvious than those for the Sol–C ones. For instance, without corrected by elastic model, the binding energy for Au–O is −2.17 eV, but the energy is only −1.59 eV for Au–C. For large sized solutes (Sc, Y, Zr, La, and Hf), however, the Sol–O binding energies exhibit a much more interesting behavior. It needs to be emphasized that the attractive interactions for Sol–O configurations are stronger than for Sol–C ones, and thus O impurity prefers to stay at 2nn sites. Based on above analysis, we can infer that the large sized solutes are not directly binding with adjacent impurities due to the repulsive interaction between each other in 1nn distance. However, according to the results of experiment, the solutes Ti and Y are liable to combine with adjacent interstitial impurities (C, O, and N) to forming the precipitates.^3,7 In fact, Y was used as the scavenger for O and N impurities.³⁵

In order to elucidate the effect of elastic correction on the binding energies, the elastic correction energies as a function of volume of solute are summarized in Fig. 5. The correction energy is evaluated by removing the strain created by its periodic images.²⁹ Evidently, the correction energy for 1nn Sol–Imp binding exhibits a distinct positive dependency on the volume of solute. Especially, as for large size solutes, such as Y solute binding with C (or O) in 1nn distance, the elastic correction energy is as high as 0.42 eV (0.37 eV). In addition, based on Fig. 2, we can find that the discrepancy between corrected energy and uncorrected one is more obvious in 1nn distance than other nn configurations, indicating that the elastic correction energy achieve the maximum in 1nn distance for Sol–Imp binding. Therefore, we reach the conclusion that elastic correction energy cannot be neglected in studying the Sol–Imp interaction.

Fig. 5 Calculated elastic correction energy as a function of volume of solutes for binding of solute with C (a) or O (b) in the 1nn distance.

3.2 The interaction between vacancy and impurities

The binding energies (corrected by elastic model) of Vac–Imp pairs with different distances are calculated and summarized in Table 1. It can be seen that the binding energies are all positive at 1nn distance, suggesting that there is an attractive interaction for the vacancy binding with C or O, and thus the formation of stable Vac–Imp pairs is energetically favorable. Similar conclusions have been found for the binding of Vac–Imp pairs in Fe³⁶ and W.³⁷ Remarkably, the attractive interaction for Vac–O pair is strong in the 1nn distance than that of other neighbor distances, suggesting that vacancy can serve as a strong O trapper in the V crystal. Meanwhile, it is also noteworthy that the elastic correction energies are weak (as low as 0.01 eV) when the Vac–Imp pairs are formed at 1nn distance. Even for the 2nn Vac–C configuration, with the highest correction energy, its correction energy is still not high than 0.12 eV. These prominent features can be explained well by the vacancy effect, where the vacancies can efficiently relax the local strain stemmed from the dissolution of the impurity or solute atoms into bulk. This can explain why the effect of elastic correction on binding energy is so slight when there is a vacancy in the vicinity of defects.

Table 1 The binding energies for the vacancy with C atom or O atom in the 1nn to 3nn octahedral site are listed. Energies are expressed in eV. The values in parentheses represent the elastic correction energies of the corresponding defect structures corrected by elastic model

	1nn	2nn	3nn
Vac–C	0.03 (0.01)	−0.53 (0.11)	0.44 (0.05)
Vac–O	0.56 (0.01)	0.07 (0.09)	0.26 (0.06)

---

3.3 The interaction among solute, vacancy and impurity

In solid system, the vacancy is the simplest and the most common structural defect in metals, affecting the atomic transport of substitutional elements and the evolution of microstructures. Since most of the Sol–Imp configurations present repulsive interactions and there is an attractive interaction for the Vac–Imp pair, we attempt to introduce the vacancy to find the stable defect clusters, such as Sol–Vac–Imp complex. The binding energies for several possible configurations are presented in the Fig. 6 to find the most stable configuration of a Sol–Vac–Imp complex. As illustrated in Fig. 6, the complex configuration in Fig. 6(d) is the most stable one because it comes into being the maximum binding energies denoting an attractive interaction. Meanwhile, this configuration provides a good environment that both the solute and impurity are most strongly bound to the vacancy and the Sol–Imp is less repulsive.²⁸ Fig. 7 shows the binding energies for special Sol–Vac–Imp configurations with different solute atom. As illustrated in Fig. 7(a) and (b), it is evident that the distributions of binding energies are small in the middle but large at both ends. In contrast to the existed repulsive interaction of the Sol–Imp binding, the positive binding energies suggest that the Sol–Vac–Imp complex is energetically stable. Especially, the binding energies for large atomic sizes of solutes (such as Sc, Y, Zr, La, and Hf) exhibit larger positive values, indicating that the Sol–Vac–Imp complex is stabilized by the vacancy introduced since the corresponding 1nn site Sol–Imp binding presents strong repulsive interactions. In order to further elucidate the effect of elastic correction, we also present the uncorrected binding energies in the Fig. 7 to make a comparison. It can be seen that the difference between them is small. As mentioned above, we have the similar conclusions that the effect of elastic correction on the binding energy is slight when the vacancy is in existence around the defects.

Fig. 6 Configurations of the Sol–Vac–Imp complex and their binding energies for solute Y, Ru, and Ag binding with impurity (C or O) atom in the 1nn (a) to 5nn (e) octahedral site around the vacancy.

Fig. 7 The Sol–Vac–Imp binding energies for the 3d, 4d, and 5d elements with C atom (a) or O atom (b). Solid symbols refer to DFT uncorrected results and open symbols to the results corrected by elastic model.

According to the results of experiment, the solutes Ti and Y are liable to combine with adjacent interstitial impurities (C, O, and N) to forming the precipitates.^3,7 In fact, Y was used as the scavenger for O and N impurities.³⁵ Generally, the C, O and N interstitial impurities exhibit repulsive interaction with the surrounding lattice atoms in bcc V, and lead to outward expansion of the neighbor host atoms.¹⁵ Similar, when large sized solute (Sc, Y, Zr, La, and Hf) is introduced to the crystal, the local stress field is created. Hence, it is too difficult for solute atoms to integrate directly with impurities (C, O, or N) due to the strong repulsive interactions in 1nn site Sol–Imp binding. Meanwhile, solute atoms need to combine with other adjacent solute or vacancy so as to relax the local distortion. Then, the vacancy is an effective tool to relieve local strain, and also is considered as an important factor in trapping impurities^38,39 and stabilizing the nanoclusters.^40,41 In particular, solute Y usually relaxes to halfway between their original lattice site and the 1nn vacancy, forming a stable solute-centered divacancy.¹⁷ In conclusion, we can infer that the forming process of precipitates/nanoclusters in V alloys as follow: firstly, solute Y and Ti bond with the adjacent vacancy; and then Sol–Vac pair serving as a strong impurity trapper form the Sol–Vac–Imp complex defined the precipitates/nanoclusters in preliminary stage. In order to demonstrate this inference, we study the effects of the Ti–Vac pair on the C and O migration in V. As shown in Fig. 8, the migration barriers of C (O) atom decreases from 1.23 eV (1.18 eV) to 1.05 eV (0.69 eV) when impurity migrate from octahedral interstice site to the 2nn one of vacancy. Next, closing to the Ti–Vac pair, there are two different mechanisms relative to the trap process for C and O impurities. As for C impurity, due to the strongly repulsive interaction in 2nn Vac–C (−0.53 eV), the barrier is low to 0.2 eV so that C atom can spontaneously migrate to the tetrahedral interstice site around Ti–Vac pair and no need for energy outside but only the thermal vibration. On the contrary, the barrier for O is 0.73 eV and there is no spontaneous behavior. In addition, both C and O atoms, once the impurities are trapped by the Ti–Vac pair, it will be more difficult for the impurities to escape from this trap due to their larger barriers (1.17 eV and 1.27 eV) in the reverse direction. This fact reveals that the Sol–Vac pair can serve as a strong impurity trapper in V. This process is also consistent with experiment results that precipitates in V–4Cr–4Ti alloy are Ti-rich and most likely to be Ti–(O, N, C).⁷ Therefore, we can also infer that the vacancy is one of the main reasons for the enhanced formation of precipitates/nanoclusters in V–4Cr–4Ti.

Fig. 8 C (a) and O (b) diffusion energy curve with or without the presence of adjacent Ti–Vac pair in V. Corresponding migration pathways are presented in the figure. Both initial and final positions of the diffusion paths for C and O atoms are chosen as the octahedral interstitial sites, while each path passes through a tetrahedral interstitial site.

4. Summary

To uncover possible formation mechanisms of precipitate in V alloys, first-principles calculations have been performed to investigate the Sol–Imp interactions with or without the presence of vacancy. The binding energies for Vac–Imp are all positive at 1nn distance, suggesting that it is energetically favorable to form stable Vac–Imp pairs. For large sized solutes (Sc, Ti, Y, Zr, La, and Hf), the Sol–Imp binding energies present strong repulsive interactions when no vacancy is in existence in the vicinity of defects. A comprehensive study on the binding energy including numerous configurations led us to conclude that the elastic correction needs to be considered to relax the elastic interaction of the defect cluster with its periodic images. On the other hand, binding energies of Sol–Vac–Imp complexes show strong attractive interactions in the presence of vacancy. As for the formation of nanoscale precipitates during the preliminary stage, large sized solutes prefer to bond with the adjacent vacancy to relieve local strain; and then Sol–Vac pair serves as a strong impurity trapper to form Sol–Vac–Imp complex. Both C and O atoms, once the impurities are trapped by the Ti–Vac pair, it will be more difficult for the impurities to escape from this trap due to their larger barriers in the reverse direction. Based on the binding preference, we can infer that the vacancy is one of the main reasons for enhancing the formation of precipitates in V alloys.

Acknowledgements

This work is financially supported by the National Nature Science Foundation of China (51501063, 51301066, and 51371080), and the Chinese National Fusion Project for ITER (No. 2013GB114001).

References

1. B. A. Loomis and D. L. Smith, J. Nucl. Mater., 1992, 191–194, 84–91.
2. H. M. Chung, B. A. Loomis and D. L. Smith, J. Nucl. Mater., 1996, 239, 139–156.
3. T. Muroga, in Comprehensive Nuclear Materials, ed. R. J. M. Konings, Elsevier, Oxford, 2012, pp. 391–406,  DOI:10.1016/B978-0-08-056033-5.00094-X.
4. N. J. Heo, T. Nagasaka, T. Muroga and H. Matsui, J. Nucl. Mater., 2002, 307–311, 620–624.
5. J. M. Chen, T. Muroga, T. Nagasaka, Y. Xu, C. Li, S. Y. Qiu and Y. Chen, J. Nucl. Mater., 2004, 334, 159–165.
6. R. J. Kurtz, K. Abe, V. M. Chernov, V. A. Kazakov, G. E. Lucas, H. Matsui, T. Muroga, G. R. Odette, D. L. Smith and S. J. Zinkle, J. Nucl. Mater., 2000, 283–287, 70–78.
7. T. Muroga, T. Nagasaka, K. Abe, V. M. Chernov, H. Matsui, D. L. Smith, Z. Y. Xu and S. J. Zinkle, J. Nucl. Mater., 2002, 307–311, 547–554.
8. B. Zhu, S. Yang, M. Zhang, J. Ding, Y. Long and F. Wan, Mater. Charact., 2016, 111, 60–66.
9. T. Miyazawa, T. Nagasaka, Y. Hishinuma, T. Muroga, Y. Li, Y. Satoh, S. Kim and H. Abe, Plasma Fusion Res., 2013, 8, 1405166.
10. J. R. DiStefano and J. H. DeVan, J. Nucl. Mater., 1997, 249, 150–158.
11. M. Koyama, K. Fukumoto and H. Matsui, J. Nucl. Mater., 2004, 329–333, 442–446.
12. P. F. Zheng, J. M. Chen, T. Nagasaka, T. Muroga, J. J. Zhao, Z. Y. Xu, C. H. Li, H. Y. Fu, H. Chen and X. R. Duan, J. Nucl. Mater., 2014, 455, 669–675.
13. P. Zhang, J. Zhao, Y. Qin and B. Wen, J. Nucl. Mater., 2011, 419, 1–8.
14. P. Zhang, J. Zhao, Y. Qin and B. Wen, Nucl. Instrum. Methods Phys. Res., Sect. B, 2011, 269, 1735–1739.
15. R. Li, P. Zhang, X. Li, C. Zhang and J. Zhao, J. Nucl. Mater., 2013, 435, 71–76.
16. L. Gong, Q. Su, H. Deng, S. Xiao and W. Hu, Comput. Mater. Sci., 2014, 81, 191–198.
17. L. Deng, X. Zhang, J. Tang, H. Deng, S. Xiao and W. Hu, J. Alloys Compd., 2016, 660, 55–61.
18. G. Kresse and J. Furthmüller, Comput. Mater. Sci., 1996, 6, 15–50.
19. G. Kresse and J. Furthmüller, Phys. Rev. B: Condens. Matter Mater. Phys., 1996, 54, 11169–11186.
20. P. E. Blöchl, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 50, 17953–17979.
21. G. Kresse and D. Joubert, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 1758–1775.
22. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865–3868.
23. G. Henkelman and H. Jonsson, J. Chem. Phys., 2000, 113, 9978–9985.
24. X. Li, H. Zhang, S. Lu, W. Li, J. Zhao, B. Johansson and L. Vitos, Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 86, 014105.
25. D. Bolef, R. Smith and J. Miller, Phys. Rev. B: Condens. Matter Mater. Phys., 1971, 3, 4100.
26. X.-S. Kong, X. Wu, Y.-W. You, C. S. Liu, Q. F. Fang, J.-L. Chen, G. N. Luo and Z. Wang, Acta Mater., 2014, 66, 172–183.
27. X. Zhang, H. Deng, S. Xiao, Z. Zhang, J. Tang, L. Deng and W. Hu, J. Alloys Compd., 2014, 588, 163–169.
28. P. Liu, W. Xing, X. Cheng, D. Li, Y. Li and X.-Q. Chen, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 90, 024103.
29. C. Varvenne, F. Bruneval, M.-C. Marinica and E. Clouet, Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 88, 134102.
30. C. Varvenne, O. Mackain and E. Clouet, Acta Mater., 2014, 78, 65–77.
31. C. Wolverton, Acta Mater., 2007, 55, 5867–5872.
32. X. Zhang, J. Tang, L. Deng, H. Deng, S. Xiao and W. Hu, Scr. Mater., 2015, 100, 106–109.
33. T. Korhonen, M. Puska and R. Nieminen, Phys. Rev. B: Condens. Matter Mater. Phys., 1995, 51, 9526.
34. B. Medasani, M. Haranczyk, A. Canning and M. Asta, Comput. Mater. Sci., 2015, 101, 96–107.
35. P. F. Zheng, J. M. Chen, T. Nagasaka, T. Muroga, J. J. Zhao, Z. Y. Xu, C. H. Li, H. Y. Fu, H. Chen and X. R. Duan, J. Nucl. Mater., 2014, 455, 669–675.
36. T. Ohnuma, N. Soneda and M. Iwasawa, Acta Mater., 2009, 57, 5947–5955.
37. Y.-W. You, X.-S. Kong, X.-B. Wu, C. S. Liu, Q. F. Fang, J. L. Chen and G. N. Luo, RSC Adv., 2015, 5, 23261–23270.
38. D. Kato, H. Iwakiri and K. Morishita, J. Nucl. Mater., 2011, 417, 1115–1118.
39. N. Fernandez, Y. Ferro and D. Kato, Acta Mater., 2015, 94, 307–318.
40. A. Hirata, T. Fujita, Y. R. Wen, J. H. Schneibel, C. T. Liu and M. W. Chen, Nat. Mater., 2011, 10, 922–926.
41. C. L. Fu, M. Krčmar, G. S. Painter and X.-Q. Chen, Phys. Rev. Lett., 2007, 99, 225502.

---