First-Principles Study of Atomic Diffusion by Vacancy Defect of the L12-Al3M (M = Sc, Zr, Er, Y) Phase

Atomic diffusion by the vacancy defect of L12-Al3M (M = Sc, Zr, Er, Y) was investigated based on a first-principles calculation. The point defect formation energies were firstly evaluated. Then, the migration energy for different diffusion paths was obtained by the climbing-image nudged elastic band (CI-NEB) method. The results showed that Al atomic and M atomic diffusions through nearest-neighbor jump (NNJ) mediated by Al vacancy (VAl) were, respectively, the preferred diffusion paths in Al3M phases under both Al-rich and M-rich conditions. The other mechanisms, such as six-jump cycle (6JC) and next-nearest-neighbor jump (NNNJ), were energetically inhibited. The order of activation barriers for NNJ(Al-VAl) was Al3Zr < Al3Y < Al3Er < Al3Sc. The Al3Sc phase had high stability with a high self-diffusion activation barrier, while the Al3Zr and Al3Y phases were relatively unstable with a low self-diffusion activation energy. Moreover, the atomic-diffusion behavior between the core and shell layers of L12-Al3M was also further investigated. Zr atoms were prone to diffusion into the Al3Y core layer, resulting in no stable core-shelled Al3(Y,Zr), which well agreed with experimental observation.

Due to the high cost of Sc elements, low-cost elements that can form thermally stable Al 3 M phases were explored to replace Sc elements.A first-principles calculation showed that Er and Yb elements were candidate element to replace Sc elements [7,8].The Zr element can partially replace Sc element and was considered an ideal element to form the Al 3 Zr shell structure owing to the Al 3 Zr/Al interface with low interface energy and coherent strain energy [9].The typical core-shelled nanophases, such as Al 3 (Er,Zr) [10] and Al 3 (Yb,Zr) [11], were introduced in aluminum alloys to replace Al 3 (Sc,Zr) nanophases.The formation of these core-shelled Al 3 M phases was explained by the difference in diffusion rates between elements, that is, the rapid diffusion elements were enriched to form a core layer and the slow diffusion elements were segregated to form a shell layer [12].The core-double-shelled structure that was observed in Al-Sc-Er-Zr alloys contained an Er-enriched core surrounded by a Sc-enriched core and a Zr-enriched outer shell, obtaining a high coarsening resistant and high strength [5,13].The formation of the core-double-shelled structure well agreed with the prediction by the atomic diffusivity ordering of D Er > D Sc > D Zr, The Y element, as a Sc homologous element with similar physical and chemical properties, was a probable candidate element to replace the Sc element [14].Zhang et al. [15,16] showed that core-shelled Al 3 (Y,Zr) with a Y-rich core and Zr-rich shell can be formed during the early stage of aging in Al-Y-Zr alloys.The Al 3 Y phase acted as the heterogeneous core to accelerate the precipitation of Al 3 Zr, which well agreed with the atomic-diffusion control mechanism.However, atom probe tomography (APT) showed that Y and Zr atoms were randomly distributed in the L1 2 -Al 3 (Y,Zr), and hybrid structure, rather than coreshelled structure, was observed in the Al 3 (Y,Zr) phase after long-term aging.The strong binding energy between Y and Zr atoms was assumed to explain the formation of hybrid structure Al 3 (Y,Zr) [16].According to the authors' previous investigation, the core-shelled Al 3 Y/Al 3 Zr was thermodynamically unstable due to its high coherent strain energy of Al 3 Y/Al 3 Zr.A similar transition from a core-shelled structure to a hybrid structure during long-term aging was also observed in the Al-Yb-Sc alloy [17].Seidman et al. [17] suggested that the inter-diffusion of Yb and Sc resulted in a uniform distribution of elements throughout the precipitates.Thus, the mechanism of the L1 2 -Al 3 M phase with core-shelled structure or hybrid structure needed to be answered.
Atomic diffusion, especially mediated by vacancy, was very beneficial for understanding the microstructural stability of the L1 2 -Al 3 M phase [18].Although experimental methods were difficult to investigate atomic diffusion in intermetallic compounds [19,20], first-principles calculations can provide new insights into the microscopic mechanisms of atomic diffusion [18,20].First-principles calculations by Fan [21,22] showed that with the increase of the atomic number, the diffusion rate of rare-earth elements increased from Sc to Y, La, and then decreased to Lu. Shi et al. [23] investigated the atomic diffusion of pure and transition-element (TM)-doped L1 2 -Al 3 Sc based on first principles and found that under a strong Al-rich condition, the V Sc defect obtained low formation energy and the NNJ mechanism mediated by V Al was most favorable for Sc atomic diffusion.TM dopants increased diffusion activation energy for dominant Al 3 Sc diffusion.However, the atomic diffusion between the core layer and the shell layer for the core-shelled L1 2 -Al 3 M phase was far from fully understood.
The atomic-diffusion mechanism in the L1 2 -Al 3 M (M = Sc, Er, Y, Zr) phases was investigated based on the first-principles in the present work.The formation energy of point defects was calculated.Then, migration energy along possible diffusion was analyzed by the climbing-image nudged elastic band (CI-NEB) method [24].Thus, the diffusion activation energy was obtained.Furthermore, the atomic-diffusion behavior between the core and shell layers for the core-shelled L1 2 -Al 3 M phase was also further illustrated.The purpose of this study was to reveal the microscopic mechanism of atomic diffusion in the L1 2 -Al 3 M phases and core-shelled L1 2 -Al 3 M phase, providing a theoretical guidance for the development of high-performance aluminum alloys containing TM and RE elements.

Defect Formation Energy
In order to evaluate the difficulty of point defect formation, the formation energy of point defects in Al 3 M was calculated as [25] Here, E def total and E bulk total are the total energy of the defective supercell and the total energy of the defect-free supercell, respectively.n i represents the number of i atoms (i = Al or M) that increased (∆n i > 0) or decreased (∆n i < 0) when defects were formed, and µ i is the relative chemical potential of i atoms.
To maintain a stably balanced L1 2 phase, its chemical potential should meet the following requirement: Here, ∆H is the formation enthalpy of the unit chemical formula Al 3 M in the solid state; ∆µ Al and ∆µ M are the differences between the relative chemical potentials of Al and M atoms, respectively, and the chemical potentials of solid simple substance, which can be expressed as Here, µ bulk Al and µ bulk M are the chemical potentials of the metals Al and M, respectively, that is, the single-atomic energies in the elemental state.
In order to avoid the precipitation of the solid elements Al and M, the chemical potential of each atom in the defect phase should be less than that of the solid elements, that is, From the phase diagram of each Al-M system, Al 3 M was in equilibrium with the adjacent pure Al phase, as the non-stoichiometric ratio was Al-rich due to point defects.When the non-stoichiometric ratio was M-rich due to point defects, the Al 3 M phase was in equilibrium with the adjacent stoichiometric Al 2 M phase.In order to avoid the formation of the pure Al and Al 2 M secondary phase, the chemical potential should be limited by the following: Here, ∆H Al 2 M f is the formation enthalpy of the unit chemical formula Al 2 M in the solid state.
The formation energies of four kinds of defects in the L1 2 -Al 3 M phase were calculated, as shown in Figure 1.Under the Al-rich condition, the point defects of V Sc obtained low formation energy and were the main point defects for the Al 3 Sc phase, and the point defects of Al Er and V Er were the main point defects for the Al 3 Er phase.The Sc Al and Er Al defects were the main point defects for the Al 3 Sc phase and Al 3 Er phase owing to the lowest formation energy under Sc-rich and Er-rich conditions.The point defects of the Al 3 Sc phase under Al-rich and Sc-rich conditions were well consistent with Ref. [23].On the other hand, the change of the stoichiometric ratio had little effect on the formation energy of the point defects in the Al 3 Zr phase and Al 3 Y phase.The V Zr and Al Y defects obtained the lowest formation energy for the Al 3 Zr phase and Al 3 Y phase regardless of the Al-rich, Zr-rich, and Y-rich conditions.Furthermore, the defect energy of V Zr was a negative value, indicating that the Al 3 Zr phase was inclined to form stable V Zr vacancy defects under the Al-rich condition.The formation energy of the Y Al antisite was always the highest; thus, it was difficult to form Y Al antisite defects in the Al 3 Y phase.Shi et al. [23] suggested that the point defect formation energy was dependent on the electronic structure and the value of the electronic density of state (DOS) at the Fermi level.However, V Al was the primary point defect near the stoichiometry [23].Similar vacancy defects were reported in the Ni 3 Al phase [18], where vacancies defect on the Ni sublattice was the main point defect in the Ni 3 Al phase.

Al Atomic Migration
Figure 2a-c show the energy profiles for Al atomic diffusion mediated by VAl in the Al3M phase.The nearest-neighbor site Al around VAl can migrate to VAl through the symmetrical NNJ pathways (denoted as NNJ(Al-VAl)), and the energy profile was symmetrical due to the restoration of the local disordered structures (Figure 2a).The highest energies of the energy profile corresponding to the migration barrier were 0.913 eV, 0.914 eV, 0.647 eV, and 0.622 eV for Al3Sc, Al3Zr, Al3Er, and Al3Y, respectively, indicating that NNJ(Al-VAl) with the low migration barrier was the preferred diffusion path for Al3M phases owing to the direct jump to VAl.Furthermore, the migration barrier of Al3Sc was almost the same as that of Al3Zr, while Al3Er and Al3Y obtained low migration barriers.The different migration barrier for the NNJ path can be attributed to the different M atomic sizes.The Er and Y atoms had large atomic radii; thus, the Al3Er and Al3Y obtained high lattice gaps, where Al atoms can migrate through the large atomic gaps to VAl, obtaining a lower migration barrier.Al atoms that occupied the next-nearest-neighbor sites of V Al can migrate to V Al through two types of diffusion paths, denoted as NNNJ1 and NNNJ2, as shown in Figure 2b,c.As for the NNNJ1 path, the order of migration energy was Al 3 Zr > Al 3 Sc > Al 3 Er > Al 3 Y.Al atoms migrated through the quadrangle composed of the nearest-neighbor Al atoms, where the quadrangle gap became large with the increase in M atomic radius; correspondingly, the migration energy decreased with the increase in M atomic radius.However, the migration energies of the NNNJ2 path were almost the same and significantly increased owing to the high density of the quadrangle with two Al atoms and two M atoms.Compared with the NNJ migration, the Al atoms through the NNNJ1 and NNNJ2 paths needed to cross the quadrangle composed of four neighboring atoms, illustrating higher migration energy [23].Therefore, the tendency of Al migration by NNNJ was very low.Al atoms that occupied the next-nearest-neighbor sites of VAl can migrate to VAl through two types of diffusion paths, denoted as NNNJ1 and NNNJ2, as shown in Figure 2b,c.As for the NNNJ1 path, the order of migration energy was Al3Zr > Al3Sc > Al3Er > Al3Y.Al atoms migrated through the quadrangle composed of the nearest-neighbor Al atoms, where the quadrangle gap became large with the increase in M atomic radius; correspondingly, the migration energy decreased with the increase in M atomic radius.However, the migration energies of the NNNJ2 path were almost the same and significantly increased owing to the high density of the quadrangle with two Al atoms and two M atoms.Compared with the NNJ migration, the Al atoms through the NNNJ1 and NNNJ2 paths needed to cross the quadrangle composed of four neighboring atoms, illustrating higher migration energy [23].Therefore, the tendency of Al migration by NNNJ was very low.
The diffusion of Al atoms was also mediated by VM, including the NNJ and ASB migration paths.Figure 2d shows the migration-energy profile for the NNJ path of Al The diffusion of Al atoms was also mediated by V M , including the NNJ and ASB migration paths.Figure 2d shows the migration-energy profile for the NNJ path of Al atom mediated by V M (denoted as NNJ(Al-V M )), where the migration barriers mediated by V M were still higher than that by V Al .Meanwhile, the final state of migration was unstable due to the local disorder by the migration of Al atoms to V M .In this sense, NNJ(Al-V M ) was not the preferred migration path.As for the ASB migration (Figure 2e), Al M occupied the next-nearest-neighbor site of V M , and the nearest-neighbor Al atoms migrated to V M , forming V Al (step 1).Then, Al M migrated to the newly formed V Al (step 2), denoted as ASB(Al M -Al-V M ).The order of migration energy for AS and ASB was similar to that for NNNJ1 as Al 3 Zr > Al 3 Sc > Al 3 Er > Al 3 Y, which can be explained by the different M atomic radii.Although the disorder of the migration final state was restored to the migration initial state, the ASB migration of Al atoms mediated by V M was restricted due to the higher migration barrier of the NNJ migration by V Al .From the above discussion, the NNJ(Al-V Al ) path had the lowest migration energy and was the main migration pathway for Al atomic migration.

M Atomic Migration
The migration of M atoms mediated by V Al included the NNJ, AS, ASB, and 6JC paths.The migration of the nearest-neighbor M atoms jumped to V Al , denoted as NNJ(M-V Al ), as shown in Figure 3a.The migration barriers of NNJ(M-V Al ) were far higher than that of NNJ(Al-V Al ).Meanwhile, the final state of migration was unstable due to the local disorder with the formation of M Al and V M defects.However, the Y atoms needed to overcome the increasing energy barrier during the migration process, suggesting that this migration path of Y atoms was energetically prohibited.The AS migration path was another M diffusion path, where the MAl atoms directly migrated to VAl (denoted as AS(MAl-VAl)), as shown in Figure 3b.The migration barriers of the AS(MAl-VAl) path were very low compared with that of NNJ(M-VAl).Except for the Al3Sc phase, the final state energies of the Al3Zr, Al3Er, and Al3Y phases were negative, which was inconsistent with the fact that the local disorder restored to their initial state The AS migration path was another M diffusion path, where the M Al atoms directly migrated to V Al (denoted as AS(M Al -V Al )), as shown in Figure 3b.The migration barriers of the AS(M Al -V Al ) path were very low compared with that of NNJ(M-V Al ).Except for the Al 3 Sc phase, the final state energies of the Al 3 Zr, Al 3 Er, and Al 3 Y phases were negative, which was inconsistent with the fact that the local disorder restored to their initial state after M atomic migration.It suggested that the AS(M Al -V Al ) path did not exist for the Al 3 Zr, Al 3 Er, and Al 3 Y phases in terms of energy.Furthermore, the AS(Sc Al -V Al ) migration path was limited to some extent due to the difficulty of coexisting V Al and Sc Al in nearby locations.
For the ASB migration path, M Al atoms and V Al occupied the site of Al atoms (Figure 3c).The nearest-neighbor M firstly jumped to V Al , newly forming a V M and M Al atom; then, the M Al atom migrated to the new V M vacancy, resulting in a V Al (denoted as ASB(M Al -M-V Al )).Obviously, the local disorder of the final state was consistent with that of the initial state; therefore, the migration-energy curve of the M atoms was symmetric.These two migration steps corresponded to the two saddle-curve characteristics of M atomic-migration energy.The Zr atomic migration by the ASB path in the Al 3 Zr phase obtained the maximum migration energy with 1.754 eV, which was lower than that of NNJ(M-V Al ) with 2.059 eV.Meanwhile, the migration barriers of the Er, Y, and Sc atoms were nearly the same for the ASB path.However, the ASB pathway was limited due to the simultaneous presence of both M Al and V Al defects.
The 6JC path of M atomic migration mediated by V Al consisted of the straight 6JC path and the bent 6JC path, as shown in Figure 3d,e.The six steps in the straight 6JC and bent 6JC were described in detail in Ref. [23].The energy curves of M diffusion by straight 6JC and bent 6JC were symmetric due to the restoration of local disorder after the six-step migration process.The first step (M-V Al ) in the straight 6JC and bent 6JC paths was similar to that of NNJ(M-V Al ).In the first step, the migration energy of the Y atom increased during the migration process, while the migration energy of the Sc and Zr elements decreased after the migration to V Al .Thus, the Y atomic diffusion obtained far higher migration energy than the Er, Sc, and Zr atoms.The high migration barriers for the Y, Er, Sc, and Zr atoms indicated the straight 6JC and bent 6JC paths were energetically restricted.
The migration of M atoms mediated by V M is also shown in Figure 3f.The M atom at the next-nearest-neighbor site jumped to V M (denoted as NNNJ(M-V M )), and the energy profile of the M diffusion was symmetric owing to the restoration of local disorder.The Zr atom obtained the higher migration barrier than that of the Er, Sc, and Y atoms due to the dense structure of the Al 3 Zr phase.However, the Sc atomic diffusion had the lowest migration barrier, which did not agree with the effect of atomic size.Therefore, the migration barrier was related not only to the atomic radius of M but also to the electronic structure of the M atom [23].Furthermore, the migration barriers of M atoms for NNNJ(M-V M ) were much higher than that for NNJ(M-V Al ) and ASB(M Al -M-V Al ), suggesting that the NNNJ(M-V M ) migration path was not a preferred migration path.
From the above discussion, except the AS(Sc Al -V Al ) path being the preferred path for Sc atomic migration, the NNJ(M-V Al ) and ASB(M Al -M-V Al ) paths contributed to M atomic migration, while the straight 6JC, bent 6JC, and NNNJ(M-V M ) paths were energetically prohibited.

Diffusion Activation Barrier
In the process of vacancy-mediated atomic diffusion, the activation barrier had a decisive influence on atomic diffusion and can be expressed as [23]: Here, ∆E defect f is the defect formation energy of the initial state, and ∆E m is the migration barrier for different migration paths.
Table 1 shows that the calculated diffusion activation energies of the Al and M atoms in the Al 3 M phases.The calculation of the Al 3 Sc phase generally agreed with that of Ref. [23].
Under the Al-rich condition, the Al atomic-diffusion activation barriers of NNJ(Al-V M ) were lower than that of NNJ(Al-V Al ) in the Al 3 M phases, which was attributed to the low V M formation energies under the Al-rich condition.However, the diffusion path of NNJ(Al-V M ) was restricted due to the unstable final state (Figure 2d).Thus, NNJ(Al-V Al ) diffusion was the main diffusion path for Al atoms under both Al-rich and M-rich conditions.For M atomic diffusion, NNJ(M-V Al ) diffusion obtained the low activation barriers for the Sc, Zr, Er, and Y atoms under both Al-rich and M-rich conditions, thus becoming the energetically preferred diffusion path.The diffusion activation barriers of Y atoms were far higher than that of the Sc, Zr, and Er elements due to their large atomic radii.Although the diffusion barriers of AS(Sc Al -V Al ) and ASB(Er Al -Er-V Al ) were much lower under the M-rich condition, their contribution to Sc atomic and Er atomic diffusions was limited due to the difficulty of coexisting V Al and Sc Al , Er Al defects in nearby locations.
The activation barriers of Al atomic and M atomic diffusions mediated by V Al under the M-rich condition were generally lower than that under the Al-rich condition owing to the lower formation energy of V Al .The diffusion activation barriers of Al atoms in the Al 3 M phase were lower than that of M atoms.Moreover, ASB, NNNJ, straight 6JC, and bent 6JC mechanisms with high diffusion activation barriers were not factually executed for Al atomic and M atomic diffusions under both Al-rich and M-rich conditions.Therefore, the NNJ(Al-V Al ) diffusion under the M-rich condition was the most preferred diffusion mechanism in the Al 3 M phase, and the order of activation barriers was Al 3 Zr < Al 3 Y < Al 3 Er < Al 3 Sc.It should be noted that these calculations were Al-atomicself-diffusion activation barriers in the Al 3 M phase.It implied that the Al 3 Sc phase had high stability with a high self-diffusion activation energy, while the Al 3 Zr phase was relatively unstable with a low self-diffusion activation barrier, which well agreed with the fact that the Al 3 Zr phase transformed from the L1 2 structure to D0 23 structure at high temperatures [26].The addition of the Er, Y, Sc, and Zr elements in aluminum alloys typically formed L1 2 -Al 3 M phases with a core-shelled structure.Due to the low interface energy between the Al 3 Zr phase and aluminum matrix, the Al 3 Zr phase tended to form a shell layer, and the Al 3 N (N = Sc, Er, Y) phase was inclined to form a core layer, where the core-shell structure was denoted as Al 3 (N,Zr).Atomic diffusion between the Al 3 N core and Al 3 Zr shell was investigated in this section.The Zr atomic diffusion to the Al 3 N core and the N atomic diffusion to the Al 3 Zr shell were respectively calculated.The previous investigation showed that Zr atoms in the Al 3 Zr shell reciprocally substituted the site of N atoms in the Al 3 N core [27].
As described in Section 2.3, NNJ(M-V Al ) was the preferred diffusion path for Zr atoms in the Al 3 N core, where Zr atoms occupied the site of N atoms at the nearest-neighbor site of V Al .The formation energy of V Al was affected by Zr atomic substitution and can be expressed as Here, E def total (Zr N ) is the total energy of defective supercell with Zr substitution.µ Zr is the chemical potential of Zr atoms.
For the N atomic substitution for Zr atoms in the Al 3 Zr shell, the formation energies of V Al defect with N atomic substitution are expressed as Here, E def total (N Zr ) is the total energy of the defective supercell with N substitution.µ N is the chemical potential of N atoms.
Table 2 shows that the formation energy of the V Al defect with Zr substitution was within that of the pure Al 3 N phase between Al-rich and N-rich, and Zr substitution had little influence on the formation energy of the V Al defect due to the atomic radius of Al being close to that of Zr.As shown in Table 3, the Sc substitution for Zr atoms slightly increased the formation energy of the V Al defect, while the Er and Y substitutions significantly increased the formation energy of the V Al defect.Zr atoms were considered to migrate through the NNJ path mediated by VAl in the Al3N core, as shown in Figure 4a.The migration barriers of Zr diffusion in the Al3Sc, Al3Er, and Al3Y cores were 1.658 eV, 1.653 eV, and 1.570 eV, respectively, which were far lower than that in the Al3Zr phase with 2.059 eV.The order of migration barriers for Zr atoms in the Al3N core was Al3Sc > Al3Er >Al3Y.With the increase in the N atomic radius for the Al3N core, Zr atoms easily migrated due to the increase in the lattice gap.Thus, Zr atomic diffusion in the Al3Y core obtained the lowest migration energy.Figure 4b illustrates that the diffusion of Sc, Er, and Y atoms in the Al3Zr shell obtained migration barriers with 2.14 eV, 2.356 eV, and 2.53 eV, respectively, which were higher than that of Zr migration in the Al3Zr shell with 2.059 eV.The atomic-migration energy in the Al3Zr shell was sequentially Y > Er > Sc, which can be attributed to the high diffusion resistance for the large atomic radius.As the diffusion activation barrier included vacancy formation energy and atomicmigration energy, the diffusion activation energies of Zr atoms in the Al3Sc, Al3Er, and Al3Y cores were 2.852 eV, 2.84 eV, and 2.633 eV, respectively.Additionally, the diffusion activation barriers of the Sc, Er, and Y atoms in the Al3Zr shell were 3.024 eV, 3.492 eV, and 3.889 eV, respectively.Compared with the diffusions of the Sc, Er, and Y atoms into the Al3Zr shell, Zr atoms were more inclined to enter the Al3Sc, Al3Er, and Al3Y cores based on diffusion activation energy.Furthermore, Zr atoms preferred to diffuse into the Al3Y core, while the diffusion of Zr atoms into the Al3Er and Al3Sc cores required higher activation barriers.It revealed that Zr atoms in the Al3Zr shell were inclined to diffuse into the Al3Y core during the subsequent aging process, thus resulting in no typical core-shelled structure.However, Zr atoms were difficult to diffuse into Al3Sc and Al3Er cores due to their high diffusion activation barrier, thus maintaining a typical core-shelled structure.

2. 2 .
Figure 2a-c show the energy profiles for Al atomic diffusion mediated by V Al in the Al 3 M phase.The nearest-neighbor site Al around V Al can migrate to V Al through the symmetrical NNJ pathways (denoted as NNJ(Al-V Al )), and the energy profile was symmetrical due to the restoration of the local disordered structures (Figure 2a).The highest energies of the energy profile corresponding to the migration barrier were 0.913 eV, 0.914 eV, 0.647 eV, and 0.622 eV for Al 3 Sc, Al 3 Zr, Al 3 Er, and Al 3 Y, respectively, indicating that NNJ(Al-V Al ) with the low migration barrier was the preferred diffusion path for Al 3 M phases owing to the direct jump to V Al .Furthermore, the migration barrier of Al 3 Sc was almost the same as that of Al 3 Zr, while Al 3 Er and Al 3 Y obtained low migration barriers.The different migration barrier for the NNJ path can be attributed to the different M atomic sizes.The Er and Y atoms had large atomic radii; thus, the Al 3 Er and Al 3 Y obtained high lattice gaps, where Al atoms can migrate through the large atomic gaps to V Al , obtaining a lower migration barrier.Al atoms that occupied the next-nearest-neighbor sites of V Al can migrate to V Al through two types of diffusion paths, denoted as NNNJ1 and NNNJ2, as shown in Figure2b,c.As for the NNNJ1 path, the order of migration energy was Al 3 Zr > Al 3 Sc > Al 3 Er > Al 3 Y.Al atoms migrated through the quadrangle composed of the nearest-neighbor Al atoms, where the quadrangle gap became large with the increase in M atomic radius; correspondingly, the migration energy decreased with the increase in M atomic radius.However, the migration energies of the NNNJ2 path were almost the same and significantly increased owing to the high density of the quadrangle with two Al atoms and two M atoms.Compared with the NNJ migration, the Al atoms through the NNNJ1 and NNNJ2 paths needed to cross the

2. 4 .
Atomic Diffusion of Core-Shelled L1 2 -Al 3 (N,Zr) Zr atoms were considered to migrate through the NNJ path mediated by V Al in the Al 3 N core, as shown in Figure4a.The migration barriers of Zr diffusion in the Al 3 Sc, Al 3 Er, and Al 3 Y cores were 1.658 eV, 1.653 eV, and 1.570 eV, respectively, which were far lower than that in the Al 3 Zr phase with 2.059 eV.The order of migration barriers for Zr atoms in the Al 3 N core was Al 3 Sc > Al 3 Er >Al 3 Y.With the increase in the N atomic radius for the Al 3 N core, Zr atoms easily migrated due to the increase in the lattice gap.Thus, Zr atomic diffusion in the Al 3 Y core obtained the lowest migration energy.Figure4billustrates that the diffusion of Sc, Er, and Y atoms in the Al 3 Zr shell obtained migration barriers with 2.14 eV, 2.356 eV, and 2.53 eV, respectively, which were higher than that of Zr migration in the Al 3 Zr shell with 2.059 eV.The atomic-migration energy in the Al 3 Zr shell was sequentially Y > Er > Sc, which can be attributed to the high diffusion resistance for the large atomic radius.

Figure 4 .
Figure 4. Atomic-migration energy of core-shelled L12-Al3Zr(N): (a) the migration of Zr atom of Al3Zr shell layer to core Al3N phase and (b) the migration of N atom of Al3N core layer to shell Al3Zr phase.The blue arrow represented the path of atomic jump.

Figure 4 .
Figure 4. Atomic-migration energy of core-shelled L1 2 -Al 3 Zr(N): (a) the migration of Zr atom of Al 3 Zr shell layer to core Al 3 N phase and (b) the migration of N atom of Al 3 N core layer to shell Al 3 Zr phase.The blue arrow represented the path of atomic jump.As the diffusion activation barrier included vacancy formation energy and atomicmigration energy, the diffusion activation energies of Zr atoms in the Al 3 Sc, Al 3 Er, and Al 3 Y cores were 2.852 eV, 2.84 eV, and 2.633 eV, respectively.Additionally, the diffusion activation barriers of the Sc, Er, and Y atoms in the Al 3 Zr shell were 3.024 eV, 3.492 eV, and 3.889 eV, respectively.Compared with the diffusions of the Sc, Er, and Y atoms into the Al 3 Zr shell, Zr atoms were more inclined to enter the A l3 Sc, Al 3 Er, and Al 3 Y cores based on diffusion activation energy.Furthermore, Zr atoms preferred to diffuse into the Al 3 Y core, while the diffusion of Zr atoms into the Al 3 Er and Al 3 Sc cores required higher activation barriers.It revealed that Zr atoms in the Al 3 Zr shell were inclined to diffuse into the Al 3 Y core during the subsequent aging process, thus resulting in no typical core-shelled structure.However, Zr atoms were difficult to diffuse into Al 3 Sc and Al 3 Er cores due to their high diffusion activation barrier, thus maintaining a typical core-shelled structure.

Table 2 .
Formation energy of V Al defect in pure and Zr-substituted Al 3 N core.

Table 3 .
Formation energy of V Al defect in pure and N-substituted Al 3 Zr shell.

Table 3 .
Formation energy of VAl defect in pure and N-substituted Al3Zr shell.