2. Computational Methods
The atomic arrangement of the (001) plane of the Al matrix and the (010) plane of β″-Mg
5Si
6 [
1,
16] is shown in
Figure 1a. The frame of the unit cell of β″-Mg
5Si
6 was fitted to the Al matrix. The number of atoms in the unit cell of β″-Mg
5Si
6 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″-Mg
5Si
6 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″-Mg
5Si
6 in
Figure 1b. Instead, an FCC lattice site in the middle of the
a-axis of β″-Mg
5Si
6 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″-Mg
5Si
6 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
VMg
4Si
8 complexes obtained in our previous study [
7] and the β″-eye in β″-Mg
5Si
6; the second most stable
VMg
4Si
8 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
VMg
4Si
8 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
VMg
4Si
8 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
VMg
4Si
8 complex due to the addition of Mg atoms, we performed the first-principles calculations for the layered
VMg
4Si
8 + Mg
n complexes.
We employed the same size of the supercell that was used to calculate the formation energies of
VMg
4Si
8 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,
, can be obtained as follows (with reference to an isolated vacancy and isolated solute atoms):
where
is the total energy of the supercell with a
VMg
mSi
n complex,
is the total energy of the supercell of the perfect lattice,
,
and
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
VMg
4Si
8 + Mg
n−1 complex is defined as follows:
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
VMg
4Si
8 + Mg
n (
n = 1–5) complexes and the binding energy of a Mg atom to the
VMg
4Si
8 + Mg
n−1 complex. For comparison, the calculated results for the most stable
VMg
4Si
8 + Mg complex are shown in
Figure 3, and the structures of the
VMg
4Si
8 + Mg
n (
n = 1–5) complexes are shown in
Figure 4. The formation energy of
VMg
4Si
8 + Mg
1 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
VMg
4Si
8 + Mg
2 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
VMg
4Si
8 + 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
VMg
4Si
8 + Mg
n−1 complex is negative. There is a steep drop both in the formation energy and the binding energy of the
VMg
4Si
8 + Mg
5 complex. All possible sites in the most stable
VMg
4Si
8 complex with additional Mg were examined; the calculated results of the most stable
VMg
4Si
8 + Mg are shown in
Figure 3. In the most stable
VMg
4Si
8 + Mg complex, the additional Mg forms three Mg–Si bonds, whereas four Mg-Si bonds are formed in the layered
VMg
4Si
8 + Mg complex. Therefore, the formation and binding energies of layered
VMg
4Si
8 + Mg are lower than those of the most stable
VMg
4Si
8 + Mg complex.
The central Mg atom in the top Mg layer of the
VMg
4Si
8 + Mg
n complexes gradually shifted to the Si layer up to
n = 4. In the layered
VMg
4Si
8 + Mg
5 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
VMg
4Si
8 + 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
VMg
4Si
8 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
VMg
4Si
8 + Mg
5 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″-Mg
5Si
6. 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
VMg
4Si
8 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
VMg
4Si
8 + 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
VMg
4Si
8 + 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
VMg
4Si
8 + 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
VMg
4Si
8 is replaced by a Mg atom is equivalent to the L1
0 structure. However, in the L1
0 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
VMg
4Si
8 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
VMg
4Si
8 complex plays an important role in the formation of the β″-eye. In the case wherein the first-principles calculation was performed for the
VMg
4Si
8 + Mg
5 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
VMg
4Si
8 + Mg
5 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 Al
83Mg
12Si
12, in which the β″-eye was formed during the MC simulation. For comparison, we performed MC simulations for Al
84Mg
12Si
12, in which a vacancy was not included. The β″-eye was formed at 68 MC steps in Al
83Mg
12Si
12.
Figure 8 shows the structures before and after the β″-eye formation. While the vacancy–solute complex that appeared in Al
83Mg
12Si
12 is not
VMg
4Si
8 but
VMg
4Si
7, the central Mg atom in the top Mg layer completely shifts to the Si layer. In contrast, the structure of Al
84Mg
12Si
12, in which a vacancy is not included, exhibits an L1
0-like layered structure, which agrees with the result of a multiscale approach [
15]. If a Mg vacancy is formed in the L1
0 structure, the local structure around the Mg vacancy becomes similar to that of the layered
VMg
4Si
8 complex. To examine the possibility of β″-eye formation around the Mg vacancy in the L1
0 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
VMg
4Si
8 + Mg
n complex appeared in other compositions of the MC simulations, as shown in
Figure 10. In Al
89Mg
6Si
12 and Al
89Mg
9Si
9, the layered
VMg
4Si
8 + Mg
2 and
VMg
4Si
8 + Mg
5 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
VMg
4Si
8 complex was not observed, which suggests that the structural transition from the most stable
VMg
4Si
8 complex is difficult to proceed. At lower temperatures, the amount of the layered
VMg
4Si
8 complex decreases compared with that of the most stable
VMg
4Si
8 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.