Ab initio results for the solute cluster

Using the coherent bcc supercell model constructed for the Cu-rich clusters as described in Sect. 2, the interaction between Ni/Mn/Si and the vacancies in the Cu-rich clusters was investigated. The primary parameter was the vacancy formation energy in the different solute-atom environments. Table 1 lists the values calculated for the Cu and Fe matrices. The vacancy formation energy for α-Fe, calculated in this study, was 2.27 eV, which agreed well with that obtained from previous calculations [38-41]. The vacancy formation energy was significantly lower in the Cu cluster (0.73 eV), consistent with that obtained using a 128-atom bcc Cu supercell in a previous study [38] and with that at the center of a 15-atom bcc Cu cluster calculated by Sato et al. [42]. From a thermodynamic perspective, these data suggest that vacancies are preferentially located in Cu clusters rather than in the Fe matrix.

In this study, the solute cluster was first constructed by replacing two Cu atoms with one solute atom (Ni, Mn, or Si) and one vacancy. In this configuration, the distance between the solute atom and vacancy varied from the first nearest neighbor (1nn) to the fifth nearest neighbor (5nn), as shown in Fig. 1. The vacancy formation energies of solute clusters are shown in Fig. 2. Among all the solute clusters, the vacancy formation energy was significantly lower in the Cu-Si cluster than in the pure Cu cluster. The energy values were 0.60 eV and 0.58 eV when the Si vacancy distances were 1nn and 2nn, respectively. The energy values approached that in the pure Cu cluster (0.73 eV) when the distance was larger than 3nn. These results indicate that Si favors vacancy positioning in the solute cluster and that this effect is short-range. The 1nn vacancy formation energy in the Cu-Ni cluster is almost 0.07 eV higher than that in the pure Cu cluster. This suggests that Ni may slightly impede vacancy siting in the solute clusters. For Mn, the vacancy-formation energies indicate that Mn might weakly suppress vacancies in the solute cluster at 2nn. In conclusion, the single effects of both Ni and Mn on vacancy positioning in the cluster are weak. Figure 3 shows the binding energies between the solute atoms and vacancies in the solute clusters. These data are consistent with the calculated vacancy formation energies; an attractive interaction between solute atoms and vacancies corresponds to a lower vacancy formation energy in the solute cluster.

Fig. 1Full-screen View

(Color online) Configuration of the bcc supercell containing (a)–(e) one solute atom and one vacancy, and (f)–(h) two solute atoms and one vacancy. The matrix atoms are in blue, and the solute atom is in purple. Specifically, the configurations are: 1nn solute atom–vacancy (a), 2nn solute atom–vacancy (b), 3nn solute atom–vacancy (c), 4nn solute atom–vacancy (d), 5nn solute atom–vacancy (e), 2nn solute atom–solute atom and nearby vacancy (f), 3nn solute atom–solute atom and nearby vacancy (g), and 5nn solute atom–solute atom and nearby vacancy (h)

Fig. 2Full-screen View

(Color online) Vacancy formation energies in the solute clusters

Fig. 3Full-screen View

(Color online) Binding energies between vacancy and solute atom in the solute clusters

Considering the enriched contents of Ni, Mn, and Si in the solute clusters compared to the Fe matrix, the effects of increasing the solute atoms in the solute clusters were also investigated. The supercells were constructed by replacing the three Cu atoms with two solute atoms (Ni, Mn, or Si) and one vacancy. This configuration is illustrated in Fig. 1, where the two solute atoms are at 1nn of the vacancy, considering their short-range interaction, and the distances between the two solute atoms are 2nn, 3nn, and 5nn. Figure 4 shows the vacancy formation energies of the solute clusters. The vacancy formation energies in the Cu-2Si and Cu-2Mn clusters were lower than the 1nn values for the Cu-Si and Cu-Mn clusters, respectively. Additionally, the vacancy formation energy in the Cu-2Ni cluster was higher than that in the Cu-Ni cluster. Therefore, increasing the concentration of solute atoms enhanced the effects of Ni, Mn, and Si at 1nn of the vacancy on the positioning of vacancies within the solute clusters. This observation highlights the pronounced effect of Si addition, in contrast to the relatively minor impact of Ni and Mn additions.

Fig. 4Full-screen View

(Color online) Vacancy formation energies in the solute clusters containing 2 solute atoms

The introduction of solute atoms and vacancies into the simulated supercell disrupted the local atomic arrangement, leading to lattice distortion. The lattice distortion is calculated as follows:(3)where is the absolute value of the displacement of the matrix atom i and is calculated by comparing the supercell before and after relaxation. The lattice distortion caused by one solute atom and one vacancy in the solute clusters is shown in Fig. 5. D_matrix caused by Ni and vacancies, or by Mn and vacancies, is consistent with that caused by one vacancy in the pure Cu cluster, indicating that single Ni or Mn atoms minimize lattice distortion. Thus, if only the size effect is considered, then Si is expected to exhibit the highest vacancy formation energy. The presence of a vacancy in the lattice creates a traction zone, which is expected to constitute a preferential site for oversized atoms; the lowest binding energy should be between those of Si and the vacancy. Thus, analyzing the size effect is not sufficient for understanding the interactions between solute atoms and vacancies; in addition, the contribution of chemical interactions is non-negligible. Systematic analyses were conducted on surveillance specimens from Japanese PWRs [43, 44]. The mean atomic compositions of the clusters were 6Cu-10Ni-6Mn-7Si, 2Cu-11Ni-5Mn-9Si, and 0Cu-12Ni-6Mn-12Si for the 0.12 wt% Cu, 0.07 wt% Cu, and 0.04 wt% Cu steels, respectively. In the clusters, the Si concentration increased as the Cu concentration decreased, whereas the Ni and Mn concentrations remained unchanged, indicating a negative interaction between Si and Cu. This result is consistent with the findings of ab initio calculations because the presence of vacancies between Si and Cu may cause moderate repulsion between Si and Cu and thus reduce the total energy.

Fig. 5Full-screen View

(Color online) Lattice distortion caused by solute atoms and vacancy in the solute clusters

Ab initio results for the cluster–matrix interface

A previous atom probe study [22] performed a detailed cluster analysis by obtaining a composition profile across the clusters using the proxigram method. The predominant Cu content was observed inside the cluster, but the Ni, Mn, and Si contents were highest at the cluster–matrix interface. The interface was dominated by the Fe environment. In this study, to better understand the migration behavior of vacancies from the Fe matrix to the solute cluster across the interface, ab initio calculations of solute–vacancy interactions at the cluster–matrix interface were performed. The configuration of the interface was built using a method similar to that in a previous study on solute clusters (see Fig. 1). The vacancy formation energies at the interfaces containing Cu, Ni, Mn, or Si are shown in Fig. 6. The vacancy formation energies at the interface with one solute atom are lower than those in the Fe matrix when the vacancies are at 1nn and 2nn from the solute atom. When the distance between the solute atom and vacancy is 1nn, it is 0.21 eV, 0.06 eV, 0.13 eV, and 0.26 eV lower at the Fe-Cu, Fe-Ni, Fe-Mn, and Fe-Si interfaces, respectively than those in the Fe matrix. These results indicate that Cu, Ni, Mn, and Si favor vacancy migration into the solute cluster across the interface. The vacancy formation energies at the interface were close to those in the Fe matrix when the solute–vacancy distance became larger than 3nn, suggesting that these solute–vacancy interactions were short range, which agreed with a previous ab initio study [32]. The binding energies between solute atoms and vacancies at the interfaces are shown in Fig. 7, which corroborates the calculated vacancy formation energies. The results indicated that the Ni-vacancy binding energy was high only at 2nn (0.19 eV); this larger 2nn Ni-vacancy interaction was also observed in previous studies [32, 45, 46]. The binding energies between Cu, Ni, Mn, and Si and the vacancies were positive when the solute–vacancy distance was 5nn. Similar results were obtained by Messina et al. [32]. Olsson et al. [27] suggested that the larger 5nn binding energies were due to Friedel-like relaxation around the vacancies, which led to periodic oscillations in the lattice structure.

Fig. 6Full-screen View

(Color online) Vacancy formation energies at the cluster–matrix interface

Fig. 7Full-screen View

(Color online) Binding energies between vacancy and solute atom at the cluster–matrix interface

The effects of increasing the number of solute atoms at the cluster–matrix interface were investigated. The supercells were constructed by replacing the three Fe atoms with two solute atoms (Ni, Mn, or Si) and one vacancy. This configuration is shown in Fig. 1. Figure 8 presents the vacancy formation energies at the interfaces. The vacancy formation energies at the Fe-2Ni, Fe-2Mn, and Fe-2Si interfaces are lower than the 1nn results at the Fe-Ni, Fe-Mn, and Fe-Si interfaces. This indicates that increasing the number of solute atoms can enhance solute–vacancy interactions at the cluster–matrix interface.

Fig. 8Full-screen View

(Color online) Vacancy formation energies at the cluster–matrix interface containing 2 solute atoms

The lattice distortion caused by the solutes and vacancies at the cluster–matrix interface is shown in Fig. 9. In addition, the size effect alone does not suffice as a parameter to comprehend the interactions between solute atoms and vacancies, and the chemical interaction contributes significantly. The results related to Cu, Ni, Mn, and Si are in agreement with those of positron annihilation experiments [28, 47], which show an attractive interaction between the solute and vacancies. The interaction between Mn and the vacancies can be attributed to the magnetism of Mn, as detailed in Sect. 4.1.

Fig. 9Full-screen View

(Color online) Lattice distortion caused by solute atoms and vacancy at the cluster–matrix interface

Effect of magnetic state of Mn

The magnetic moments of Mn atoms in the Fe matrix are listed in Table 2. Mn can be in the ferromagnetic (FM) or antiferromagnetic (AFM) state. In the Fe matrix containing one Mn atom, the magnetic moment of Mn was negative, corresponding to an AFM state. This result is in agreement with that of a previous study [48], in which the calculated energy of an Fe matrix containing one Mn atom reveals that Mn in the AFM state corresponds to a lower substitutional energy; therefore, the AFM state is the ground state of Mn. Table 2 shows that after the addition of a second Mn atom, irrespective of the distance between the two Mn atoms (2nn, 3nn, or 5nn), their magnetic moments are negative (-2.50±0.30 μ_B). This result agrees well with the ab initio results reported by Schneider et al. [49], whose investigations show that two Mn atoms with a distance of 1nn or 2nn show antiferromagnetism. These findings are consistent with Radhakrishna et al. [50], who investigated the magnetism of Mn using diffuse neutron scattering in FeMn alloys with different Mn contents (3.15 at%, 5.89 at%, 8.83 at%). They found that the magnetic moments of Mn in the Fe matrix with varying Mn content were always 0.8±0.1 μ_B, indicating that the magnetic moments of Mn were independent of the number of Mn atoms. In addition, when one Ni or Si atom was close to the Mn atom, the Mn remained in AFM state, indicating that the solute atoms had little effect on the magnetism of the Mn atoms. Notably, when two Mn atoms are in close proximity (2nn), the energy of the supercell with AFM-AFM Mn atoms is only 0.05 eV lower than that with FM-AFM Mn atoms, which may result in the convergence of ab initio calculations to the FM-AFM state. However, after introducing one vacancy, the energy difference increased to 0.27 eV. A previous ab initio study [49] observed that the energy of a 128-atom Fe matrix containing one AFM Mn was lower than that with one FM Mn system by 0.05 eV and increased to 0.28 eV when one vacancy was introduced at the 1nn position of Mn. Considering the vacancy defects in RPV materials, the computational results suggest that Mn in the Fe matrix should be in AFM state. According to Fig. 7, the binding energy of 1nn Mn vacancy in the Fe matrix is 0.13 eV, indicating a short-range attraction between AFM Mn and vacancies. Chen et al. [51] calculated the binding energies between Mn and vacancy clusters V_x(x≤6) in an Fe matrix, and found that only Mn in AFM state attracted vacancies, which is consistent with the results of this study. This implies that only the AFM state of Mn at the cluster–matrix interface can facilitate vacancy attraction.

Table 3 lists the magnetic moments of the Mn atoms in the solute clusters. A comparison with the results shown in Table 2 demonstrates that after the Fe environment is replaced by the Cu environment, the magnetic states of the Mn atoms change from AFM to FM. This result indicates that the FM state of Mn in the solute clusters is the ground state, and the magnetic moments of Mn are influenced by the surrounding chemical environment. Even after adding another solute atom (Ni or Si), the magnetic moment of Mn remains positive, suggesting a minor influence of nearby solute atoms on the magnetic state of Mn. Additionally, in the solute clusters with two Mn atoms, regardless of the distance between the two Mn atoms (2nn, 3nn, or 5nn), the magnetic moments of Mn persist within the range of 3.55-3.66 μ_B, indicating that the magnetism of Mn is independent of the number of Mn atoms. The binding energies shown in Fig. 3 indicate that the 1nn Mn-vacancy interaction is weak in the solute clusters, in contrast to the attractive interaction between Mn and the vacancies in the Fe matrix. These results confirm that the magnetic state of Mn atoms can affect their interactions with vacancy defects. In solute clusters, Mn atoms are in the FM state, thus leading to weak interactions between Mn and the vacancies.

Synergistic effect of Ni and Mn

To investigate possible synergistic effects among Ni, Mn, and Si on vacancy aggregation, two different solute atoms and one vacancy were introduced in the supercell, and the structure is shown in Fig. 1. Table 4 summarizes the binding energies between different solute atoms and between solute atoms and vacancies in the Fe matrix. Evidently, the 1nn and 2nn binding energies between Ni and Mn atoms are -0.28 eV and -0.22 eV, respectively, indicating mutual repulsions. This result is consistent with those of previous ab initio calculations [46, 52] of interactions between Ni and Mn in a 128-atom supercell. This finding suggests that the NiMn structure is unstable in the Fe matrix. However, when one vacancy appears at 1nn of the solute atoms, the binding energy among Ni, Mn, and the vacancy (0.18 eV) indicates a strong attraction among them. This finding is consistent with the calculations of Whiting et al. [53]. Thus, vacancies can stabilize structures containing Ni and Mn. Furthermore, Bonny et al. [33] reported simulations, revealing that the tetrahedral structure of Ni₂Mn₂ is stable. Ab initio calculations by Yu et al. [24] indicated that clusters composed of Ni and Mn are thermodynamically stable, implying the presence of Ni and Mn atoms within the solute clusters. Table 4 shows that the 1nn binding energy of NiMn and the vacancy is 0.40 eV, which is significantly higher than the 1nn Ni-vacancy binding energy (0.06 eV) and the 1nn Mn-vacancy binding energy (0.13 eV). This result suggests that Ni and Mn have synergistic effects in attracting vacancies in the Fe matrix.

For Si, the binding energies of 2nn Ni-Si and 2nn Mn-Si were approximately 1.77 eV, and the Ni-Si-vacancy binding energy and Mn-Si–vacancy binding energies were less than 2.13 eV. Therefore, the interaction between solute atoms contributes the most to the attraction between solute atoms and vacancies. The 1nn NiSi-vacancy and MnSi-vacancy binding energies were 0.32 eV and 0.36 eV, respectively, which are close to the 1nn Si-vacancy binding energy (0.26 eV). Therefore, the attraction of NiSi or MnSi to nearby vacancies was mainly caused by the effect of the Si atom.

Table 5 summarizes the binding energies between the solute atoms and between the solute atoms and vacancies within the solute clusters containing two different solute atoms. The binding energy between Ni and Mn at 1nn distance was 0.09 eV, indicating attractive interaction. The interaction between 2nn Ni and Mn is weak (0.03 eV). However, when a vacancy was introduced at 1nn of solute atoms, the binding energy between Ni, Mn, and the vacancies reached 0.28 eV, suggesting an attraction between Ni, Mn, and the vacancies. This implies that introducing a vacancy significantly enhances the attractive interaction. This result agrees with the previous findings of Odette et al. [54]; they investigated the evolution of cluster nucleation using a theoretical model and found that Ni and Mn had a synergistic effect of mutual enhancement; that is, a higher Mn content promoted the enrichment of Ni in the solute clusters, whereas an increase in the number of Ni atoms could attract more Mn atoms. Chen et al. [22] performed atom probe tomography analyses of Fe-Cu-Ni-Mn alloys after thermal aging and neutron irradiation. Their results demonstrated that the number of Ni atoms and Mn atoms in solute clusters showed a 1:1 relation, which remained unchanged upon varying the Ni content (0.6-0.8 wt%), indicating a synergistic relationship between Ni and Mn. Table 5 also shows that the binding energy between NiMn and 1nn vacancies is 0.25 eV, which is significantly higher than those of 1nn Ni vacancies (-0.07 eV) and 1nn Mn vacancies (0.03 eV). These results indicate that the contribution of Ni or Mn alone in attracting vacancies is small; however, when Ni and Mn coexist, NiMn can facilitate vacancy aggregation. This result suggests that a synergistic effect between Ni and Mn facilitates vacancy aggregation in the solute clusters.

For Si, the Ni-Si-vacancy binding energy was 0.17 eV, and the Mn-Si-vacancy binding energy was 0.13 eV, indicating their strong attraction. As the 1nn Si vacancy binding energy is relatively high (0.14 eV), this attraction is attributed to the effect of Si. In addition, the binding energies of 1nn NiSi-vacancy (0.17 eV) and 1nn MnSi-vacancy (0.14 eV) were close to those of 1nn Si vacancy. This indicates that the effect of Si, rather than the synergistic effect of Ni-Si or Mn-Si, facilitates vacancy aggregation in solute clusters.

Implication on irradiation embrittlement

To develop a predictive model for radiation embrittlement, clarifying the physical mechanisms of the solute clusters and dislocation loops is imperative. In this study, we investigated the behavior of vacancies in three environments, i.e., Cu-rich clusters, Fe matrix, and a cluster matrix, using ab initio calculations. The basic idea of the modeling is that the Cu-rich cluster and cluster matrix have higher Ni, Mn, and Si contents than the Fe matrix. According to the vacancy formation energies listed in Table 1, Cu clusters can attract vacancies in the Fe matrix. Nagai et al. [28] observed Cu-V_n clusters (n≤6) produced by electron irradiation at room temperature in Fe-Cu alloys. Wang et al. [55] and Zhao et al. [56] reported that Cu and vacancies can form complexes in Fe-Cu alloys under proton irradiation. Bergner et al. [57] used the small-angle neutron scattering technique to show that clusters produced by neutron irradiation in Fe-Cu alloys contain approximately 84% Cu and 16% vacancies. Previous simulations [19, 20] revealed that Cu clusters in bcc Fe can attract vacancies and nearby interstitial defects. Hernández-Mayoral et al. [18] suggested that the formation of Cu-vacancy complexes induces the recombination of interstitial atoms and vacancies, leading to a decreased defect concentration, thereby inhibiting the growth of dislocation loops. For Cu-rich clusters containing Ni, Mn, and Si, the effect of Si as well as the synergistic effect of Ni and Mn promote the aggregation of vacancies in the solute clusters. In addition, Cu, Ni, Mn, and Si can promote the migration of vacancies across energy barriers from the cluster–matrix interface into the cluster. This observation suggests that vacancies can migrate from the Fe matrix to the solute clusters and thermodynamically aggregate. Therefore, compared with pure Cu clusters, Cu-rich clusters can further absorb vacancies and interstitial defects, thus reducing the concentration of point defects in the Fe matrix and suppressing the growth of dislocation loops while mitigating the irradiation embrittlement of the material.

In contrast, the irradiation embrittlement caused by Cu clusters is affected by the vacancy content in the clusters. Lambrecht et al. [58] conducted a PAS study on Fe-Cu alloys irradiated with 0.1 dpa neutrons, revealing that the Cu clusters formed in the early stage were the primary cause of irradiation hardening, which then became saturated. Further investigations [59] conducted tensile tests on Fe-Cu alloys after neutron irradiation and observed a decrease in the yield strength under neutron irradiation after the saturation of irradiation hardening. They then used PAS techniques and found that the vacancy content in the Cu clusters increased when irradiated with 0.2 dpa compared with 0.1 dpa, suggesting that an increase in the irradiation dose led to an increase in the vacancy content in the Cu clusters, thus resulting in a decrease in the hardening of the Cu clusters. This indicates that the obstacle strength of Cu clusters to moving dislocations decreases as the vacancy content in the clusters increases, thereby affecting irradiation embrittlement. In this study, it was found that solute elements (Ni, Mn, and Si) in solute clusters can facilitate the migration of vacancies across the cluster–matrix interface into the clusters, leading to an increase in the vacancy content in the solute clusters and a decrease in the obstacle strength of the clusters to moving dislocations. This feature consequently mitigates irradiation embrittlement of the material.

In addition, Soisson et al. [38] utilized ab initio calculations to obtain simulation parameters and subsequently performed Atomistic Kinetic Monte Carlo (AKMC) simulations on the migration of Cu clusters. They found that Cu clusters containing tens of atoms exhibited high mobility, mainly because of the capture of vacancies by the Cu clusters. Castin et al. [60] conducted AKMC simulations of the annealing process of Fe-Cu alloys and observed that Cu clusters with one vacancy had high mobility, and the recombination of mobile Cu clusters was identified as the primary mechanism of the growth and coarsening of Cu clusters. Jourdan et al. [61] employed MD simulations to study the precipitation of Cu in an Fe matrix and revealed that the introduction of the mobility of Cu clusters into the model accelerated the kinetics of precipitation. Thus, vacancy absorption by the Cu clusters enhances their mobility and accelerates the kinetics of Cu precipitation. Because Cu-rich clusters contain solute atoms (Ni, Mn, and Si) that can facilitate vacancy aggregation in the solute clusters, they can accelerate the kinetics of precipitation and growth of Cu-rich clusters, thereby affecting the irradiation embrittlement of high-Cu alloys.