Structural Transition of Vacancy–Solute Complexes in Al–Mg–Si Alloys

Abstract

To theoretically examine the structural transition of vacancy–solute complexes in Al–Mg–Si alloys, we performed first-principles calculations for layered vacancy–solute complexes with additional Mg atoms. The central Mg atom in the additional Mg layer shifted to the Si layer with the increase in the number of Mg atoms to weaken the repulsive Mg–Mg interaction and to form Mg–Si bonds. When five Mg atoms were added to the layered vacancy–solute complex, the central Mg atom completely shifted to the Si layer, and a Mg vacancy was formed in the Mg layer, which indicated that the β″-eye is formed upon the addition of Mg atoms. We reproduced β″-eye formation from a solid solution with a vacancy using first-principles-based Monte Carlo simulations. Once the β″-eye was formed on the layered vacancy–solute complex, the process can be repeated by the formation of alternate Mg and Si layers along (010) β″. These results clearly indicate that the layered vacancy–solute complex plays an important role in β″-eye formation.

1. Introduction

The use of 6000 series Al–Mg–Si alloys for manufacturing automobile body panels has increased because such alloys exhibit hardening characteristics. The precipitation of the β″ phase during paint-bake treatment is mainly responsible for the hardening in Al–Mg–Si alloys [1,2,3]. However, the solute clusters formed during storage at room temperature, referred to as natural aging, suppresses the precipitation of the β″ phase [4,5,6]. In our recent study [7], we determined the stable structures of vacancy–solute complexes, V(Mg,Si)_n for n = 1–12, in Al–Mg–Si alloys using first-principles calculations and predicted the initial structures through multiple linear regression analysis. We observed that the formation energy of the vacancy–solute complex decreased with the increasing number of solute atoms, and the VMg₄Si₈ complex exhibited the lowest formation energy.

Regarding the β″ phase, theoretical studies have been carried out to investigate its structural properties [8,9,10,11,12,13,14], chemical bonding [8,9,14], and stability [10,11,13,14]. However, the structural transition to the β″ phase remains unclear. The formation of a layered MgSi phase in an Al matrix was reported using a multiscale approach [15]. However, the formation of the β″ phase, which requires a shift of Mg atoms to the Si layer, was not observed. Before we proceed to investigate the structural transition from the vacancy–solute complexes to the β″ phase, it is important to clarify the relationship between the vacancy–solute complexes and the β″-eye, which is the precursor of the β″-phase [2]. In this study, to investigate the structural transition from the vacancy–solute complexes to the β″-eye in Al–Mg–Si alloys, we performed first-principles calculations for the vacancy–solute complexes with additional Mg atoms in Al–Mg–Si alloys. To reproduce the structural transition of the vacancy–solute complexes from the solid solution in the Al–Mg–Si alloy, we performed first-principles-based Monte Carlo (MC) simulations.

2. Computational Methods

The atomic arrangement of the (001) plane of the Al matrix and the (010) plane of β″-Mg₅Si₆ [1,16] is shown in Figure 1a. The frame of the unit cell of β″-Mg₅Si₆ was fitted to the Al matrix. The number of atoms in the unit cell of β″-Mg₅Si₆ was the same as that in the adjacent area in the Al matrix. However, the Mg atom at the corner of the unit cell of β″-Mg₅Si₆ was located not on an face-centered cubic (FCC) lattice site but on an octahedral site of FCC as shown in the side view around the β″-eye in β″-Mg₅Si₆ in Figure 1b. Instead, an FCC lattice site in the middle of the a-axis of β″-Mg₅Si₆ is vacant. The octahedral site occupied by the Mg atom corresponds to the vacant site in the adjacent (010) plane. Therefore, considering its atomic arrangement, β″-Mg₅Si₆ can be considered as a structure in which an FCC lattice site occupied by a Mg atom is shifted to an octahedral site surrounded by four Si atoms in the adjacent plane. As a result of the shift of the Mg atom, the four Si atoms around the Mg atom are outwardly displaced, and the four Mg atoms around the vacant site are displaced inward from the ideal lattice sites of the Al matrix. The driving forces of the Mg atom shift possibly are the energy gain by the formation of Mg–Si bonds and the reduction of repulsive Mg–Mg interaction, as will be discussed later.

Figure 2 shows the structures of the most stable and second most stable VMg₄Si₈ complexes obtained in our previous study [7] and the β″-eye in β″-Mg₅Si₆; the second most stable VMg₄Si₈ complex exhibits a layered structure. The structural feature of the β″-eye is the presence of Mg atoms in the Si layer. The comparison of the structures of the layered VMg₄Si₈ complex, and the β″-eye indicates that the length of the Si–Si bond in the β″-eye is ≈1 Å longer than that in the layered VMg₄Si₈ complex, while the lengths of the Mg–Si and Mg–Mg bonds are identical. This is because the Si-Si bonds must be expanded to accommodate the Mg–Si bonds in the Si layer. To examine the structural transition of the layered VMg₄Si₈ complex due to the addition of Mg atoms, we performed the first-principles calculations for the layered VMg₄Si₈ + Mg_n complexes.

We employed the same size of the supercell that was used to calculate the formation energies of VMg₄Si₈ complexes [7]: a supercell composed of 108 lattice sites with an FCC structure (3 × 3 × 3 unit cell). The length of the supercell was determined using the calculated equilibrium lattice parameter of the aluminum matrix, 4.0396 Å, which is in good agreement with the experimental value of 4.0496 Å. We employed the Vienna ab initio simulation package (VASP) [17,18] along with the generalized gradient approximation of Perdew–Burke–Ernzerhof (PBE) [19]. Potentials based on the all-electron projector augmented wave (PAW) method were also used [20,21]. All calculations were performed with a 350 eV energy cutoff and a 4 × 4 × 4 k mesh in the Monkhorst–Pack scheme. The structural relaxation of the atoms was continued until the total energy change decreased below 0.001 eV. The supercell volume and shape parameters were fixed for consistency with the clustering behavior of the solute atoms: during solute cluster formation, both the solute atom concentration and matrix lattice parameter remained essentially unchanged.

The formation energy of the VMg_mSi_n complex, $E_f\left(V{\mathrm{Mg}}_m{\mathrm{Si}}_n\right)$, can be obtained as follows (with reference to an isolated vacancy and isolated solute atoms):

$$E_f\left(V{\mathrm{Mg}}_m{\mathrm{Si}}_n\right)=\frac{1}{m+n}\left[E\left(V,{\mathrm{Mg}}_m,{\mathrm{Si}}_n\right)+\left(m+n\right)E_0-E\left(V\right)-mE\left(\mathrm{Mg}\right)-nE\left(\mathrm{Si}\right)\right]$$

(1)

where $E\left(V,{\mathrm{Mg}}_m,{\mathrm{Si}}_n\right)$ is the total energy of the supercell with a VMg_mSi_n complex, $E_0$ is the total energy of the supercell of the perfect lattice, $E\left(V\right)$, $E\left(\mathrm{Mg}\right),$ and $E\left(\mathrm{Si}\right)$ are the total energies of the supercell with a vacancy, a Mg atom, and a Si atom, respectively.

The binding energy of the Mg atom added to the VMg₄Si₈ + Mg_n−1 complex is defined as follows:

$$E_b^{\mathrm{Mg}}=E\left(V,{\mathrm{Mg}}_{4+n},{\mathrm{Si}}_8\right)+E_0-E\left(V,{\mathrm{Mg}}_{4+n-1},{\mathrm{Si}}_8\right)-E\left(\mathrm{Mg}\right).$$

(2)

Negative (positive) values indicate energy gain (penalty).

To examine the reproduction of the vacancy–solute complexes from the solid solution in the Al–Mg–Si alloy, we performed the first-principles-based MC simulations. The temperature in the MC simulations was set at 200 K to avoid an excessive increase in the potential energy (shown in Appendix A, Figure A1). Metropolis–Hastings sampling was employed to determine the acceptance probability of the position swap [22]. The energy during the MC simulations was calculated using a k mesh reduced to a 2 × 2 × 2 k mesh.

3. Results and Discussion

Figure 3 shows the formation energies of VMg₄Si₈ + Mg_n (n = 1–5) complexes and the binding energy of a Mg atom to the VMg₄Si₈ + Mg_n−1 complex. For comparison, the calculated results for the most stable VMg₄Si₈ + Mg complex are shown in Figure 3, and the structures of the VMg₄Si₈ + Mg_n (n = 1–5) complexes are shown in Figure 4. The formation energy of VMg₄Si₈ + Mg₁ decreased as a result of the formation of four Mg–Si bonds. In Al–Mg–Si alloys, the Mg–Si bond possesses the lowest interaction energy around a vacancy and in the Al matrix [7]. In the VMg₄Si₈ + Mg₂ complex, both Mg–Si and Mg–Mg bonds were formed upon the addition of Mg atoms. The Mg–Mg interaction is repulsive in the Al matrix [7]. Thus, the formation energy of the VMg₄Si₈ + Mg_n complexes for n = 2–4 slightly increased owing to the positive interaction energy of the Mg–Mg bonds. However, the total formation energy of these complexes still decreased because the binding energy of the Mg atom to the VMg₄Si₈ + Mg_n−1 complex is negative. There is a steep drop both in the formation energy and the binding energy of the VMg₄Si₈ + Mg₅ complex. All possible sites in the most stable VMg₄Si₈ complex with additional Mg were examined; the calculated results of the most stable VMg₄Si₈ + Mg are shown in Figure 3. In the most stable VMg₄Si₈ + Mg complex, the additional Mg forms three Mg–Si bonds, whereas four Mg-Si bonds are formed in the layered VMg₄Si₈ + Mg complex. Therefore, the formation and binding energies of layered VMg₄Si₈ + Mg are lower than those of the most stable VMg₄Si₈ + Mg complex.

The central Mg atom in the top Mg layer of the VMg₄Si₈ + Mg_n complexes gradually shifted to the Si layer up to n = 4. In the layered VMg₄Si₈ + Mg₅ complex, the central Mg atom completely shifted to the Si layer, and a Mg vacancy was formed in the Mg layer, which indicates that the β″-eye was formed. The formation of the β″-eye was also observed using a larger supercell (shown in Appendix B, Figure A2). Figure 5 shows the average lengths of the Mg–Si, Si–Si, and Mg–Mg bonds in the central Mg and Si layers in the VMg₄Si₈ + Mg_n (n = 1–5) complexes. The average length of the Mg–Si bond remains almost constant at 2.80 Å, which is shorter than that in the Al matrix (2.86 Å). In the VMg₄Si₈ complex, the Si atoms were outwardly displaced. Therefore, the Si–Si bond is longer than the Mg–Si and the Mg–Mg bonds. The central Mg atom shifted to the Si layer with increasing Mg atoms to form Mg–Si bonds. In addition, the repulsive Mg–Mg interaction around the central Mg increased with the number of Mg atoms, which also contributed to the downward shift of the central Mg atom. The steep decrease in the formation and binding energies of the VMg₄Si₈ + Mg₅ complex arises from not only the formation of Mg–Si bonds but also the weakening of the repulsive Mg–Mg interaction. As a result of the shift of the central Mg atom, the average Si–Si bond length increased from 3.000 to 3.695 Å, which is 0.280 Å shorter than that in the β″-eye in β″-Mg₅Si₆. This is because the Al matrix suppresses the outward displacement of the Si atoms. The energy loss of the Si–Si bonds is not significant because the contribution of the Si–Si bond to the stability of the VMg₄Si₈ complexes is smaller than that of the Mg–Si, Al–Mg, and Al–Si bonds [7].

To evaluate the change in the stability of the β″-eye with increasing Mg atoms, we calculated the formation energy of the layered VMg₄Si₈ + Mg_n (n = 1–5) complexes with the β″-eye arrangement. The central Mg atom was fixed in the Si layer, which is equivalent to the structure of the β″-eye. The formation energies are shown in Figure 6 with those of the layered VMg₄Si₈ + Mg_n complexes. The formation energy of the β″-eye steeply decreases with increasing Mg atoms and turns out to be lower than that of the layered VMg₄Si₈ + Mg_n complexes for n = 5. The increase in the number of the Mg atoms, which possess the larger atomic radius than the Al atoms, induces the outward displacement of the Si atoms and the Al matrix. Therefore, the Si–Si bonds are expanded as shown in Figure 5, and the structure of the β″-eye becomes stable.

The structure in which the vacancy in VMg₄Si₈ is replaced by a Mg atom is equivalent to the L1₀ structure. However, in the L1₀ structure, the Mg atom does not shift to the adjacent Si layer because the Si–Si bonds are not expanded. Once the β″-eye is formed on the VMg₄Si₈ complex, the formation of the β″-eye can be repeated by the formation of alternate Mg and Si layers along [10] β″. These results clearly indicate that the layered VMg₄Si₈ complex plays an important role in the formation of the β″-eye. In the case wherein the first-principles calculation was performed for the VMg₄Si₈ + Mg₅ complex using an initial structure in which the Mg and Si atoms are located on the ideal FCC lattice sites, the Mg atom in the top Mg layer does not completely shift to the Si layer. This is because lattice relaxation is confined by symmetry restriction. Therefore, the atom-by-atom formation of the Mg layer is also important for the β″-eye formation on the VMg₄Si₈ + Mg₅ complex.

To reproduce the formation of the β″-eye from the solid solution in the Al–Mg–Si alloy, we performed the first-principles-based MC simulations using a supercell composed of 108 lattice sites with an FCC structure, including a vacancy and Mg and Si atoms. Figure 7 shows the change in the potential energy of Al₈₃Mg₁₂Si₁₂, in which the β″-eye was formed during the MC simulation. For comparison, we performed MC simulations for Al₈₄Mg₁₂Si₁₂, in which a vacancy was not included. The β″-eye was formed at 68 MC steps in Al₈₃Mg₁₂Si₁₂. Figure 8 shows the structures before and after the β″-eye formation. While the vacancy–solute complex that appeared in Al₈₃Mg₁₂Si₁₂ is not VMg₄Si₈ but VMg₄Si₇, the central Mg atom in the top Mg layer completely shifts to the Si layer. In contrast, the structure of Al₈₄Mg₁₂Si₁₂, in which a vacancy is not included, exhibits an L1₀-like layered structure, which agrees with the result of a multiscale approach [15]. If a Mg vacancy is formed in the L1₀ structure, the local structure around the Mg vacancy becomes similar to that of the layered VMg₄Si₈ complex. To examine the possibility of β″-eye formation around the Mg vacancy in the L1₀ structure, we investigated the change in the structure upon the addition of Mg atoms for the layered MgSi with a Mg vacancy. Figure 9 shows the structures in which up to five Mg atoms were located in the Si layer above the Mg vacancy. Although the central Mg atom shifted to the Si layer with increasing Mg atoms, the central Mg atom still existed between the Mg and Si layers. This is partly because the Si–Si bond was not well expanded to accommodate the Mg–Si bonds in the Si layer. The local structure around the Mg vacancies may be affected by the composition or size of the MgSi layers. Therefore, further research is necessary to clarify the possibility of the formation of the β″-eye on the MgSi layers. Note that the β″-eye was not formed on layered MgSi with a Mg vacancy in the present work.

The layered VMg₄Si₈ + Mg_n complex appeared in other compositions of the MC simulations, as shown in Figure 10. In Al₈₉Mg₆Si₁₂ and Al₈₉Mg₉Si₉, the layered VMg₄Si₈ + Mg₂ and VMg₄Si₈ + Mg₅ complexes were formed, respectively; in these complexes, the number of Mg atoms in the additional Mg layer was not sufficient to form the β″-eye. In the MC simulations performed in this study, the most stable VMg₄Si₈ complex was not observed, which suggests that the structural transition from the most stable VMg₄Si₈ complex is difficult to proceed. At lower temperatures, the amount of the layered VMg₄Si₈ complex decreases compared with that of the most stable VMg₄Si₈ complex, which is considered one of the reasons for the decrease in the peak hardness value obtained by the precipitation of the β″ phase by natural aging. As seen in the results of the MC simulations, more than 10 Mg atoms are required to reproduce the β″-eye formation. Therefore, to quantitatively evaluate the influence of Mg/Si ratios in the low concentration region on β″-eye formation efficiency, a larger supercell must be employed.

4. Conclusions

We investigated the stability and structure of vacancy–solute complexes VMg₄Si₈ with additional Mg atoms in Al–Mg–Si alloys using the first-principles calculations. The layered VMg₄Si₈ complex, which is the second most stable complex of VMg₄Si₈, becomes stable upon the addition of Mg atoms to the adjacent layers because of the formation of Mg–Si bonds. The central Mg atom in the additional Mg layer shifted to the Si layer with the increase in the number of Mg atoms to weaken the repulsive Mg–Mg interaction as well as to form Mg–Si bonds. When five Mg atoms were added to the layered VMg₄Si₈ complex, the central Mg atom completely shifted to the Si layer, and a Mg vacancy is formed in the Mg layer. Therefore, the layered VMg₄Si₈ + Mg₅ complex has the same structure as the β″-eye. These results indicate that the layered VMg₄Si₈ complex evolves into the β″-eye upon the addition of Mg atoms. In contrast, the layered MgSi with a Mg vacancy, which has a similar structure to the layered VMg₄Si₈ complex, does not evolve into the β″-eye. We confirmed the formation of the β″-eye from a solid solution with a vacancy using first-principles-based MC simulations. The β″ structure is composed of β″-eyes linked by Si–Si bonds. Therefore, to investigate the evolution from the β″-eye to the β″ phase, larger supercells are required in order to include more than one β″-eye, which will be examined in future work.