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

Abstract

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 (V_Al) 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-V_Al) 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.

1. Introduction

Aluminum alloys containing transition metals (TM) and rare-earth (RE) elements are widely used in aerospace, high-speed trains, and automobiles [1,2,3]. The formation of L1₂-Al₃(RE,TM) nanophases with high thermal stability, such as the core-shelled Al₃(Sc,Zr) nanophase, can effectively inhibit the recrystallization process of aluminum alloys, achieving high strength toughness and corrosion resistance performance [2]. Based on the high thermostability and coarsening resistance of L1₂-Al₃(RE,TM) nanophases, Seidman et al. [4,5,6] developed an Al-RE-TM series of high-temperature aluminum alloys.

Due to the high cost of Sc elements, low-cost elements that can form thermally stable Al₃M 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 Al₃Zr shell structure owing to the Al₃Zr/Al interface with low interface energy and coherent strain energy [9]. The typical core-shelled nanophases, such as Al₃(Er,Zr) [10] and Al₃(Yb,Zr) [11], were introduced in aluminum alloys to replace Al₃(Sc,Zr) nanophases. The formation of these core-shelled Al₃M 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 Al₃(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 Al₃Y phase acted as the heterogeneous core to accelerate the precipitation of Al₃Zr, 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 L1₂-Al₃(Y,Zr), and hybrid structure, rather than core-shelled structure, was observed in the Al₃(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 Al₃(Y,Zr) [16]. According to the authors’ previous investigation, the core-shelled Al₃Y/Al₃Zr was thermodynamically unstable due to its high coherent strain energy of Al₃Y/Al₃Zr. 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 L1₂-Al₃M 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 L1₂-Al₃M 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 L1₂-Al₃Sc 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 Al₃Sc diffusion. However, the atomic diffusion between the core layer and the shell layer for the core-shelled L1₂-Al₃M phase was far from fully understood.

The atomic-diffusion mechanism in the L1₂-Al₃M (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 L1₂-Al₃M phase was also further illustrated. The purpose of this study was to reveal the microscopic mechanism of atomic diffusion in the L1₂-Al₃M phases and core-shelled L1₂-Al₃M phase, providing a theoretical guidance for the development of high-performance aluminum alloys containing TM and RE elements.

2. Results and Discussion

2.1. Defect Formation Energy

In order to evaluate the difficulty of point defect formation, the formation energy of point defects in Al₃M was calculated as [25]

$${\mathrm{E}}_{\mathrm{f}}^{\mathrm{d}\mathrm{e}\mathrm{f}}={\mathrm{E}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}^{\mathrm{d}\mathrm{e}\mathrm{f}}-{\mathrm{E}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}-\sum _{\mathrm{i}}\Delta {\mathrm{n}}_{\mathrm{i}}{\mathrm{\mu }}_{\mathrm{i}}$$

(1)

Here, ${\mathrm{E}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}^{\mathrm{d}\mathrm{e}\mathrm{f}}$ and ${\mathrm{E}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}$ 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 ($\Delta {\mathrm{n}}_{\mathrm{i}}>0$) or decreased ($\Delta {\mathrm{n}}_{\mathrm{i}}<0$) when defects were formed, and μ_i is the relative chemical potential of i atoms.

To maintain a stably balanced L1₂ phase, its chemical potential should meet the following requirement:

$$3\Delta {\mathrm{\mu }}_{\mathrm{A}\mathrm{l}}+\Delta {\mathrm{\mu }}_{\mathrm{M}}=\Delta {\mathrm{H}}_{\mathrm{f}}^{\mathrm{A}{\mathrm{l}}_3\mathrm{M}}$$

(2)

Here, $\Delta {\mathrm{H}}_{\mathrm{f}}^{\mathrm{A}{\mathrm{l}}_3\mathrm{M}}$ is the formation enthalpy of the unit chemical formula Al₃M in the solid state; $\Delta {\mathrm{\mu }}_{\mathrm{A}\mathrm{l}}$ and $\Delta {\mathrm{\mu }}_{\mathrm{M}}$ 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

$$\begin{array}{l}\Delta {\mathrm{\mu }}_{\mathrm{A}\mathrm{l}}={\mathrm{\mu }}_{\mathrm{A}\mathrm{l}}-{\mathrm{\mu }}_{\mathrm{A}\mathrm{l}}^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}\\ \Delta {\mathrm{\mu }}_{\mathrm{M}}={\mathrm{\mu }}_{\mathrm{M}}-{\mathrm{\mu }}_{\mathrm{M}}^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}\end{array}$$

(3)

Here, ${\mathrm{\mu }}_{\mathrm{A}\mathrm{l}}^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}$ and ${\mathrm{\mu }}_{\mathrm{M}}^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}$ 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,

$$\begin{array}{l}\Delta {\mathrm{\mu }}_{\mathrm{A}\mathrm{l}}\le 0\\ \Delta {\mathrm{\mu }}_{\mathrm{M}}\le 0\end{array}$$

(4)

From the phase diagram of each Al-M system, Al₃M 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 Al₃M phase was in equilibrium with the adjacent stoichiometric Al₂M phase. In order to avoid the formation of the pure Al and Al₂M secondary phase, the chemical potential should be limited by the following:

$$\begin{array}{l}\Delta {\mathrm{\mu }}_{\mathrm{A}\mathrm{l}}\le 0\\ 2\Delta {\mathrm{\mu }}_{\mathrm{A}\mathrm{l}}+\Delta {\mathrm{\mu }}_{\mathrm{M}}\le \Delta {\mathrm{H}}_{\mathrm{f}}^{\mathrm{A}{\mathrm{l}}_2\mathrm{M}}\end{array}$$

(5)

Here, $\Delta {\mathrm{H}}_{\mathrm{f}}^{\mathrm{A}{\mathrm{l}}_2\mathrm{M}}$ is the formation enthalpy of the unit chemical formula Al₂M in the solid state.

The formation energies of four kinds of defects in the L1₂-Al₃M 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 Al₃Sc phase, and the point defects of Al_Er and V_Er were the main point defects for the Al₃Er phase. The Sc_Al and Er_Al defects were the main point defects for the Al₃Sc phase and Al₃Er phase owing to the lowest formation energy under Sc-rich and Er-rich conditions. The point defects of the Al₃Sc 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 Al₃Zr phase and Al₃Y phase. The V_Zr and Al_Y defects obtained the lowest formation energy for the Al₃Zr phase and Al₃Y 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 Al₃Zr 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 Al₃Y 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 Ni₃Al phase [18], where vacancies defect on the Ni sublattice was the main point defect in the Ni₃Al phase.

2.2. Vacancy-Mediated Atomic Migration

2.2.1. Al Atomic Migration

Figure 2a–c show the energy profiles for Al atomic diffusion mediated by V_Al in the Al₃M 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 Al₃Sc, Al₃Zr, Al₃Er, and Al₃Y, respectively, indicating that NNJ(Al-V_Al) with the low migration barrier was the preferred diffusion path for Al₃M phases owing to the direct jump to V_Al. 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 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 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 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 Al₃Zr > Al₃Sc > Al₃Er > Al₃Y, 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.

2.2.2. 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 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 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 after M atomic migration. It suggested that the AS(M_Al-V_Al) path did not exist for the Al₃Zr, Al₃Er, and Al₃Y 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 Al₃Zr 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 Al₃Zr 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.

2.3. 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]:

$$\Delta {\mathrm{E}}_{\mathrm{a}}=\Delta {\mathrm{E}}_{\mathrm{f}}^{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}}+\Delta {\mathrm{E}}_{\mathrm{m}}$$

(6)

Here, $\Delta {\mathrm{E}}_{{}_{\mathrm{f}}}^{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}}$ is the defect formation energy of the initial state, and $\Delta {\mathrm{E}}_{\mathrm{m}}$ is the migration barrier for different migration paths.

Table 1. Diffusion activation barrier of L1₂-Al₃M.

System	Diffusion Atoms	Diffusion Mechanisms	Activation Barrier (eV)
			Al-Rich	Sc-Rich
Al3Sc	Al	NNJ(Al-VAl)	2.131–2.250 [23]	1.778–1.689 [23]
		NNJ(Al-VSc)	1.668–2.352 [23]	3.081–3.916 [23]
		NNNJ1(Al-VAl)	3.503–3.825 [23]	3.150–3.244 [23]
		NNNJ2(Al-VAl)	5.963–6.441 [23]	5.609–5.860 [23]
		ASB(AlSc-Al-VSc)	2.346–2.628 [23]	5.574–6.297 [23]
	Sc	NNJ(Sc-VAl)	2.900–3.124 [23]	2.547–2.543 [23]
		AS(ScAl-VAl)	3.956–4.144 [23]	1.178–1.148 [23]
		ASB(ScAl-Sc-VAl)	5.079–5.546 [23]	2.958–2.640 [23]
		NNNJ(Sc-VSc)	5.893–8.881 [23]	7.306–10.625 [23]
		S6JC(Sc----VAl)	5.489–4.759 [23]	5.135–6.504 [23]
		B6JC(Sc----VAl)	9.430–10.154 [23]	9.076–10.734 [23]
Al3Zr	Al	NNJ(Al-VAl)	1.776	1.572
		NNJ(Al-VZr)	1.175	1.788
		NNNJ1(Al-VAl)	3.376	3.172
		NNNJ2(Al-VAl)	6.183	5.830
		ASB(AlZr-Al-VZr)	1.736	3.166
	Zr	NNJ(Zr-VAl)	2.926	2.721
		AS(ZrAl-VAl)	-	-
		ASB(ZrAl-Zr-VAl)	4.666	3.645
		NNNJ(Zr-VZr)	6.029	6.642
		S6JC(Zr----VAl)	5.373	5.169
		B6JC(Zr----VAl)	8.755	8.550
Al3Er	Al	NNJ(Al-VAl)	1.856	1.722
		NNJ(Al-VEr)	1.748	2.149
		NNNJ1(Al-VAl)	1.969	1.888
		NNNJ2(Al-VAl)	6.187	6.053
		ASB(AlEr-Al-VEr)	2.849	5.932
	Er	NNJ(Er-VAl)	2.966	2.832
		AS(ErAl-VAl)	-	-
		ASB(ErAl-Er-VAl)	5.370	2.555
		NNNJ(Er-VZr)	6.451	6.853
		S6JC(Er----VAl)	5.350	5.216
		B6JC(Er----VAl)	9.220	9.086
Al3Y	Al	NNJ(Al-VAl)	1.730	1.649
		NNJ(Al-VY)	1.725	1.968
		NNNJ1(Al-VAl)	1.969	1.888
		NNNJ2(Al-VAl)	6.090	6.009
		ASB(AlY-Al-VY)	2.885	3.452
	Y	NNJ(Y-VAl)	4.507	4.426
		AS(YAl-VAl)	-	-
		ASB(YAl-Y-VAl)	5.347	4.942
		NNNJ(Y-VSc)	6.472	6.715
		S6JC(Y----VAl)	6.203	6.172
		B6JC(Y----VAl)	10.230	10.149

shows that the calculated diffusion activation energies of the Al and M atoms in the Al₃M phases. The calculation of the Al₃Sc 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 Al₃M 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 Al₃M 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 Al₃M phase, and the order of activation barriers was Al₃Zr < Al₃Y < Al₃Er < Al₃Sc. It should be noted that these calculations were Al-atomic-self-diffusion activation barriers in the Al₃M phase. It implied that the Al₃Sc phase had high stability with a high self-diffusion activation energy, while the Al₃Zr phase was relatively unstable with a low self-diffusion activation barrier, which well agreed with the fact that the Al₃Zr phase transformed from the L1₂ structure to D0₂₃ structure at high temperatures [26].

2.4. Atomic Diffusion of Core-Shelled L1₂-Al₃(N,Zr)

The addition of the Er, Y, Sc, and Zr elements in aluminum alloys typically formed L1₂-Al₃M phases with a core-shelled structure. Due to the low interface energy between the Al₃Zr phase and aluminum matrix, the Al₃Zr phase tended to form a shell layer, and the Al₃N (N = Sc, Er, Y) phase was inclined to form a core layer, where the core-shell structure was denoted as Al₃(N,Zr). Atomic diffusion between the Al₃N core and Al₃Zr shell was investigated in this section. The Zr atomic diffusion to the Al₃N core and the N atomic diffusion to the Al₃Zr shell were respectively calculated. The previous investigation showed that Zr atoms in the Al₃Zr shell reciprocally substituted the site of N atoms in the Al₃N core [27].

As described in Section 2.3, NNJ(M-V_Al) was the preferred diffusion path for Zr atoms in the Al₃N 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

$${\mathrm{E}}_{\mathrm{f}}^{\mathrm{d}\mathrm{e}\mathrm{f}}(\mathrm{Z}{\mathrm{r}}_{\mathrm{N}}^{})={\mathrm{E}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}^{\mathrm{d}\mathrm{e}\mathrm{f}}(\mathrm{Z}{\mathrm{r}}_{\mathrm{N}})-{\mathrm{E}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}-\sum _{\mathrm{i}}\Delta {\mathrm{n}}_{\mathrm{i}}{\mathrm{\mu }}_{\mathrm{i}}-{\mathrm{\mu }}_{\mathrm{Z}\mathrm{r}}$$

(7)

Here, ${\mathrm{E}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}^{\mathrm{d}\mathrm{e}\mathrm{f}}(\mathrm{Z}{\mathrm{r}}_{\mathrm{N}})$ 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 Al₃Zr shell, the formation energies of V_Al defect with N atomic substitution are expressed as

$${\mathrm{E}}_{\mathrm{f}}^{\mathrm{d}\mathrm{e}\mathrm{f}}({\mathrm{N}}_{\mathrm{Z}\mathrm{r}})={\mathrm{E}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}^{\mathrm{d}\mathrm{e}\mathrm{f}}({\mathrm{N}}_{\mathrm{Z}\mathrm{r}})-{\mathrm{E}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}^{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}-\sum _{\mathrm{i}}\Delta {\mathrm{n}}_{\mathrm{i}}{\mathrm{\mu }}_{\mathrm{i}}-{\mathrm{\mu }}_{\mathrm{N}}$$

(8)

Here, ${\mathrm{E}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}^{\mathrm{d}\mathrm{e}\mathrm{f}}({\mathrm{N}}_{\mathrm{Z}\mathrm{r}})$ is the total energy of the defective supercell with N substitution. μ_N is the chemical potential of N atoms.

Table 2. Formation energy of V_Al defect in pure and Zr-substituted Al₃N core.

	Formation Energy of VAl (eV)
	Al3Sc	Al3Er	Al3Y
Pure	0.863–1.217	1.074–1.208	1.026–1.107
Zr substitution	1.194	1.187	1.063

shows that the formation energy of the V_Al defect with Zr substitution was within that of the pure Al₃N 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. Formation energy of V_Al defect in pure and N-substituted Al₃Zr shell.

	Pure	Sc	Er	Y
VAl (eV)	0.661–0.865	0.884	1.136	1.359

, 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 V_Al 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 atomic-migration 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 A_l3Sc, 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.

3. Computational Methods

Based on density functional theory (DFT) [28], first-principles calculations were carried out by Vienna ab initio simulation package (VASP) software [29]. The projector augmented wave (PAW) with the Perdew–Burke–Ernzerh (PBE) method of generalized gradient approximation (GGA) was used to describe the exchange-correlation energy functional between electrons [30]. The electron configuration was described by the Al-3s²3p¹, Sc-3s²3p⁶4s¹3d², Zr-4s²4p⁶5s¹4d³, Er-6s²5p⁶5d¹, and Y-4s²4p⁶5s¹4d² valence states, respectively. The kinetic energy cutoff of the plane-wave basis and the size of the k-mesh for the Brillouin zone were tested for self-consistent convergence. The geometric structure was optimized by the Monkhorst-Pack k-point grids with linear k-mesh analytical values of less than 0.032π/Å. The total energy was calculated using the linear tetrahedron method with the Blöchl correction when the total energy converged to 10⁻⁴ eV/atom. The lattice constants (a₀) were predicted as 4.042 Å, 4.103 Å, 4.108 Å, and 4.232 Å for fcc-Al, L1₂-Al₃Sc, L1₂-Al₃Zr, and L1₂-Al₃Er, respectively, which were well consistent with Ref. [31].

There were two sublattices in the L1₂-Al₃M (M = Sc, Zr, Er, Y) unit cell, the Al sublattice located at the 3c position (0,0.5,0.5) and the M sublattice located at the 1a (position (0,0,0). Therefore, there were four types of primary point defects in Al₃M, including Al vacancy (V_Al), M vacancy (V_M), Al antisite (Al_M), and M antisite (M_Al). In order to reduce vacancy density and limit the interaction between defects, 108 atoms in a 3 × 3 × 3 supercell were used in this calculation.

The diffusion mechanism of the L1₂-Al₃M phase mediated by vacancy included from the nearest-neighbor to complex hopping sequences, such as nearest-neighbor jump (NNJ), next-nearest-neighbor jump (NNNJ), antistructural sublattice (AS), antistructural bridge (ASB), and 6-jump cycle (6JC) [23]. The CI-NEB method [24] was used to calculate the energy profile. A series of atomic positions were inserted between the initial and final states to construct the model, and then, each insertion point model was relaxed until the force threshold at the insertion point was 10⁻² eV/Å. By this method, the vacancy diffusion behavior of the L1₂-Al₃M phase was comparatively studied, and the atomic-diffusion interaction between the core layer and the shell layer for the core-shelled L1₂-Al₃M phase was also discussed.

4. Conclusions

Atomic-diffusion mechanisms in L1₂-Al₃M (M = Sc, Er, Y, Zr) phases were investigated based on a first-principles calculation. The main conclusions are summarized as follows:

(1)

NNJ(Al-V_Al) and NNJ(M-V_Al) diffusions were the energetically preferred diffusion paths under both Al-rich and M-rich conditions. The straight 6JC, bent 6JC, and NNNJ were significantly inhibited owing to their high activation barriers. Other diffusion paths, such as NNJ(Al-V_M), AS(Sc_Al-V_Al), and ASB(Er_Al-Er-V_Al), were limited due to the unstable final-state structure and the difficulty of coexisting V_Al and M_Al defects.

(2)

The order of activation barriers for NNJ(Al-V_Al) was Al₃Zr < Al₃Y < Al₃Er < Al₃Sc. The Al₃Sc phase had high stability with a high self-diffusion activation barrier, while the Al₃Zr phase was relatively unstable with a low self-diffusion activation energy.

(3)

Compared with the diffusion of the Sc, Er, and Y atoms in the Al₃Zr shell, Zr atoms were more inclined to diffuse into the Al₃Y, Al₃Er, and Al₃Sc cores, and the activation barriers were as follows: Al₃Y < Al₃Er < Al₃Sc. Thus, Zr atoms were prone to diffuse into the Al₃Y core, resulting in no core-shelled structure.