Prediction and characterization of an Mg-Al intermetallic compound with potentially improved ductility via orbital-free and Kohn-Sham density functional theory

Magnesium-aluminum (Mg-Al) alloys are lightweight and exhibit excellent mechanical properties, enabling a wide range of applications in automotive, aerospace, and electronic device industries [1]. However, the mechanical properties of Mg-Al alloys strongly depend on various precipitated Mg-Al intermetallic compounds. For example, the presence of brittle Mg₂Al₃ and Mg₁₇Al₁₂ compounds leads to welding cracks between Mg and Al interfaces that are susceptible to fracture [2]. As such, studying structure-property relationships of Mg-Al intermetallic compounds serves as a critical step in providing guidance for optimizing Mg-Al alloy mechanical properties.

Computational tools based on density functional theory (DFT) are reliable for characterizing the structures and mechanical properties of Mg-Al intermetallic compounds [3–7]. Here, we utilize two different DFT methods. The first is Kohn-Sham DFT (KSDFT) [8, 9], which uses both orbitals and electron density to evaluate properties. The second is orbital-free DFT (OFDFT) [10], which employs electron density as its only variable. KSDFT is more accurate than OFDFT because the kinetic energy of non-interacting electrons can be evaluated exactly via orbitals in the former, whereas an approximate kinetic energy density functional (KEDF) must be used in the latter. However, the number of metal atoms treatable with conventional KSDFT is dramatically smaller than with OFDFT, mainly due to the cubic scaling algorithm required to perform time-consuming orthonormalization of the KS orbitals. Instead, use of a KEDF enables OFDFT to scale quasi-linearly with system size. The tradeoff between efficiency and accuracy motivates use of both DFT methods when studying Mg-Al intermetallics.

Mg and Al can form a number of intermetallic compounds, as documented in well-known databases such as the American Society for Metals (ASM) Alloy Phase Diagram Database [11] and the Inorganic Crystal Structure Database (ICSD) [12]. These databases play important roles in the design and selection of Mg-Al alloys. The crystal structures archived in the databases are also useful as starting geometries for simulations. However, only a limited number of compounds are listed in the databases. Going beyond known Mg-Al intermetallic compounds to search for new ones with promising mechanical properties requires the predictive power of quantum mechanics calculations such as those employed here.

In this work, we combine the abovementioned strengths of OFDFT and KSDFT to explore other possible Mg-Al intermetallic compounds. We are specifically interested in Mg-rich Mg-Al alloys to maximize the light-weighting of metal alloys and furthermore are interested in precipitates that could form within the hexagonal-close-packed (hcp) Mg matrix. Those precipitates are likely to be subject to the hcp host lattice type constraint in order to avoid large interface energies. We first use the OFDFT formation energy as a screening parameter to find stable structures subject to the hcp host lattice constraint. We identify a new Mg-Al intermetallic compound Mg₃Al with the D0₁₉ hexagonal structure that has a small formation energy, slightly lower in energy than a hypothetical L1₂ face-centered cubic (fcc) structure previously used to fit an interatomic potential [7] that we also compare against. We then turn to KSDFT for more accurate characterization of Mg₃Al properties. Both D0₁₉ and L1₂ Mg₃Al are found to obey common criteria for metastability. We further predict that D0₁₉ Mg₃Al should exhibit superior ductility compared to hcp Mg and L1₂ Mg₃Al. The prediction and characterization of lightweight engineering alloys is thus facilitated via OFDFT for screening and KSDFT for the assessment of mechanical properties, as demonstrated here.

We adopted the cluster-expansion (CE) method as implemented in the Alloy Theoretic Automated Toolkit (ATAT) [13] in search of stable Mg-Al intermetallic structures. The energy of a system within the CE method can be written as [14]

$$\begin{eqnarray}&&E(\sigma )={J}_{0}+\displaystyle \sum _{i}{J}_{i}{\hat{S}}_{i}(\sigma )+\displaystyle \sum _{j\lt i}{J}_{ij}{\hat{S}}_{i}(\sigma ){\hat{S}}_{j}(\sigma )+\displaystyle \sum _{k\lt j\lt i}{J}_{ijk}{\hat{S}}_{i}(\sigma ){\hat{S}}_{j}(\sigma ){\hat{S}}_{k}(\sigma )+\mathrm{..}.\end{eqnarray} \tag{ 1 }$$

where J_i, J_ij, and J_ijk refer to the CE coefficients for the clusters consisting of one, two, and three atoms, respectively. ${\hat{S}}_{i}(\sigma )=\pm 1$ denotes that a site is occupied by either an Mg (+1) or Al (−1) atom. Together with the constant J₀, these CE coefficients were determined from OFDFT formation energies of 101 different Mg-Al compounds calculated with the PROFESS 3.0 package [15]. Specifically, 55 clusters up to quadruplets were used in the least square fitting process in order to minimize the difference between the CE and OFDFT energies. Originally proposed by Collony and Williams [16], this fitting procedure is called the Collony-Williams method or the structure inversion method [13]. We implemented a scripting interface between ATAT and PROFESS 3.0, whereby the OFDFT formation energies are automatically utilized by ATAT. All OFDFT calculations used periodic boundary conditions, the Wang-Teter (WT) KEDF [17], and bulk-derived local pseudopotentials (BLPSs) [18, 19] for Mg and Al [20] to describe the electron-ion interactions. We employed the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) functional [21] for electron exchange-correlation (XC) and a planewave basis kinetic energy cutoff of 1200 eV. OFDFT with the WT KEDF, these GGA BLPSs [20], and the PBE XC functional yields properties in reasonable agreement with experiment and KSDFT for hcp Mg, fcc Al, and four Mg-Al intermetallic compounds such as Mg₁₇Al₁₂ [22].

All KSDFT calculations were performed with the Vienna Ab-Initio Simulation Package (VASP, version 5.3.5) [23]. Projector-augmented-wave (PAW) [24, 25] potentials for Mg, Al, and Ni were used to account for the nuclei and frozen core electrons, with the 3 s states of Mg, 3s3p states of Al, and 3d4s states of Ni optimized self-consistently. We used a 500 eV planewave basis kinetic energy cutoff. Within the Monkhorst-Pack scheme [26], we used k-point grids of 12 × 12 × 12, 16 × 16 × 16, 16 × 16 × 16, 16 × 16 × 16, 8 × 8 × 8, 18 × 18 × 12, and 18 × 18 × 18 for eight-atom D0₁₉ Mg₃Al, four-atom D0₃ Mg₃Al, four-atom L1₂ Mg₃Al, four-atom L1₂ Ni₃Al, 29-atom Mg₁₇Al₁₂, two-atom hcp Mg, and one-atom fcc Al primitive cells, respectively. The elastic constants were calculated using a symmetry-general strain-stress method [27] as implemented in VASP. Increasing the k-point mesh density by up to 1.5 times converges the calculated elastic constants to within a range of 1 to 3 GPa (see supplemental material, table S1). Pugh's ratio (vide infra) is rather insensitive to the k-point density. The selected planewave kinetic energy cutoff and k-point meshes converge the total energy to within 1.0 meV/atom. The Brillouin-zone integration follows the Methfessel-Paxton method [28] with a smearing width of 0.2 eV. All lattice parameters and atomic coordinates were fully optimized until a force threshold of 0.01 eV/Å is reached.

We performed Monte Carlo (MC) simulations using the OFDFT-derived CE energy expression to calculate the order-disorder transition temperature for D0₁₉ and L1₂ Mg₃Al using the ATAT package. A cutoff radius of 15 Å was used to determine the size of the MC supercell. The simulation systems were cooled down from 800 K to 300 K with a step of 20 K. At each temperature, 10 000 MC steps were run before another 10 000 steps were used for averaging the total energy. The constant volume heat capacity was then calculated as the first derivative of the temperature-dependent energy.

We computed the phonon spectra of D0₁₉ and L1₂ Mg₃Al via PHONOPY [29], which post-processes the KSDFT force constants obtained from VASP. We used 3 × 3 × 3, 4 × 4 × 4, 4 × 4 × 4, 2 × 2 × 2, 6 × 6 × 6, and 6 × 6 × 6 supercells to calculate the force constants of D0₁₉ Mg₃Al, D0₃ Mg₃Al, L1₂ Mg₃Al, Mg₁₇Al₁₂, hcp Mg, and fcc Al, respectively. We calculated the temperature-dependent formation energies of both Mg₃Al structures based on the quasi-harmonic approximation [30], which requires volume-dependent phonon spectra data and total energies. 17 different Mg₃Al primitive cells with equilibrium lattice constants scaled by 0.98 to 1.02 with an incremental step of 0.0025 were used.

We employed KSDFT to calculate the generalized stacking fault energies (GSFEs) γ_GSFE of D0₁₉ and L1₂ Mg₃Al using surface slab models containing 16 atomic layers. A vacuum spacing of 15 Å was used to ensure that interactions between periodic images are negligible. To obtain a full γ_GSFE curve, we successively displace the atoms in the top eight layers of a relaxed surface slab and optimize only the out-of-plane coordinates of these atoms. Each surface slab has only one stacking fault and γ_GSFE therefore is determined as

$$\begin{eqnarray}&&{\gamma }_{{\rm{GSFE}}}=({E}_{{\rm{disp}}}-{E}_{0})/{A}_{0}\end{eqnarray} \tag{ 2 }$$

where E_disp is the energy of the slab with displaced atoms, E₀ is the energy of the reference slab before the atomic displacements, and A₀ is the cross-sectional area of the lateral unit cell.

We first search for potential stable Mg-Al intermetallic structures with convex hulls constructed on-the-fly by ATAT (figure 1(a)). We use the formation energy E_f of an Mg_xAl_1−x compound as a descriptor of structural stability. E_f is calculated as

$$\begin{eqnarray}&&{E}_{{\rm{f}}}={E}_{{{\rm{Mg}}}_{x}{{\rm{Al}}}_{1-x}}-x{E}_{{\rm{Mg}}}-(1-x){E}_{{\rm{Al}}}\end{eqnarray} \tag{ 3 }$$

where ${E}_{{{\rm{Mg}}}_{x}{{\rm{Al}}}_{1-x}},$ E_Mg, and E_Al refer to the total energy per formula unit of hcp Mg_xAl_1−x, hcp Mg, and hcp Al cells, respectively, with x denoting the molar composition of a Mg_xAl_1−x compound. ATAT conventionally uses the same (parent) crystal structure for the compounds with compositions of x = 0 and 1. We follow this convention by using the energy of structurally relaxed Al initially adopting the same structure as hcp Mg, with the latter considered to be the parent structure (equation (3)). The relaxed Al structure here is a local minimum that remains hcp but has lattice vectors optimized for Al. The OFDFT total energy of ground-state fcc Al is lower than that of hcp Al by 4.4 meV/atom.

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Figure 1(a) displays four sets of data points. The first data set denoted 'OFDFT calculations' contains the formation energies of 101 Mg-Al intermetallic compounds calculated according to equation (3). The structures of these compounds are automatically generated by ATAT based on an algorithm [31] that is capable of efficiently enumerating superlattices and atomic configurations conforming to the symmetry of the parent structure. The lattice vectors and atomic coordinates of the structures are completely relaxed to compute the formation energies within the constraint of the parent structure symmetry. The energies of these compounds computed with the CE method are assembled in the second data set with an equal number of (101) data points, denoted 'CE fits.' The excellent agreement between these two data sets is shown in figure 1(b), validating the accuracy of the fitted CE coefficients. ATAT then uses the CE coefficients to predict the formation energies of 5675 other Mg-Al intermetallic compounds with different structures and stoichiometries, as represented by the 'CE predictions' data set (figure 1(a)). These structures are also generated by the same abovementioned algorithm. The final results obtained (figure 1(a)) indicate that there are three intermediate 'ground state' structures located at the convex hull, corresponding to the chemical formulae: MgAl₂, Mg₃Al, and Mg₉Al.

Because we employ hcp Mg as the parent structure, the structure search is limited to hcp structures. As a result, the structural energies should only be considered useful for considering relative energies of precipitates constrained by the matrix to have an hcp structure. We therefore restrict our attention to the two compounds (Mg₃Al and Mg₉Al) with Mg concentrations on the Mg-rich side (x > ½) of the convex hull, which could maintain a hexagonal lattice. (One could use other structure search methods, such as the evolutionary search approach as implemented in, e.g., the USPEX package [32] to identify possible Mg-Al intermetallic compounds with different lattice types on the Al-rich side of the convex hull, if this was of interest.) In this work, however, we ended up studying only the properties of the Mg₃Al intermetallic compound for the following reasons. In addition, predicting that Mg₃Al has a lower formation energy than Mg₉Al, we find that disproportionation of Mg₉Al into D0₁₉ Mg₃Al and hcp Mg is thermodynamically favorable. In particular, we predict ${{\rm{Mg}}}_{9}{\rm{Al}}\to {{\rm{Mg}}}_{3}{\rm{Al}}+6{\rm{Mg}}$ to be exoergic by −8.0 meV/atom, suggesting that D0₁₉ Mg₃Al precipitates in hcp Mg could be at least metastable. We leave the study of mechanical properties of Mg₉Al to future work, as its structure is more complicated than either of the possible Mg₃Al structures, as can be seen in figure S2 in the supplemental material, available at stacks.iop.org/MSMS/25/075002/mmedia. More specifically, unlike Mg₃Al, Mg₉Al is not a simple superlattice structure of Mg but has a much lower symmetry. As a result, larger supercells with hundreds of atoms are required to calculate the generalized stacking fault energies to the same degree of accuracy done here (vide infra). We therefore focus first on the simpler D0₁₉ Mg₃Al and its potentially competing L1₂ phase.

We consider here the lowest energy hcp structure determined by our ATAT search, D0₁₉, as well as two hypothetical cubic structures, D0₃ and L1₂ [7], for Mg₃Al. Figure 2 displays top and side views of these Mg₃Al primitive unit cells. The D0₁₉ structure, according to the Strukturbericht notation [33], has six Mg and two Al atoms in its primitive cell. A typical feature of this superstructure is that the in-plane lattice constant a₀ is nearly twice that of hcp Mg (table 1). The D0₁₉ structure is adopted by many A₃B-type intermetallic compounds with Ni₃Sn as the prototype [33]. Several Mg-based intermetallic compounds such as Mg₃Cd also crystallize in this D0₁₉ structure [34]. The D0₃ and L1₂ structures consist of three Mg and one Al atoms in their primitive cells. These two postulated structures were used in [7] to generate a Mg-Al interatomic potential.

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Considerations of strain and electronegativity lend some physical insight into formation of the Mg₃Al phase. First, the atomic radii of Mg (1.50 Å) and Al (1.25 Å) [36] differ by more than 15%, a mismatch strain that disfavors solid solutions and favors intermetallic compounds instead [37]. Second, the electronegativities of the two species (Mg: 1.31; Al: 1.61 by the Pauling scale [38]) are somewhat different, which also drives formation of intermetallic compounds [37], via polar/metallic bonding. OFDFT predicts E_f values for D0₁₉, D0₃, and L1₂ Mg₃Al of −23.9, 16.2, and −31.4 meV/atom, respectively, upon replacing the energy of hcp Al in equation (3) with that of ground-state fcc Al. By contrast, KSDFT yields values for E_f of 3.9, 43.0, and 5.4 meV/atom for D0₁₉, D0₃, and L1₂ Mg₃Al, respectively. The accuracy limits of the WT KEDF used here for screening may cause discrepancies in the sign of E_f that can be remedied by using the more accurate Wang-Govind-Carter (WGC) KEDF [39–41], which yields E_f of the correct sign (2.8 and 0.9 meV/atom for D0₁₉ and L1₂ Mg₃Al, respectively). The WGC KEDF also improves the E_f for D0₃ Mg₃Al to 22.7 meV/atom. To test the sensitivity of E_f to the choice of XC functional, we also use KSDFT with the local density approximation (LDA) XC, which produces E_f of −4.2, 34.9, and −2.7 meV/atom, for D0₁₉, D0₃, and L1₂ Mg₃Al, respectively. This is consistent with the general over-binding and under-binding shortcomings of the LDA and PBE XC functionals [42], respectively. Both OFDFT and KSDFT formation energies, however, consistently show that both D0₁₉ and L1₂ Mg₃Al are on the verge of being stable compounds. By contrast, the D0₃ structure consistently exhibits relatively large positive formation energies. We thus rule out D0₃ as a competitive phase.

Because of the above discrepancies between the OFDFT and KSDFT formation energies, we hereafter use the more accurate KSDFT to study various properties of Mg₃Al. The KSDFT-PBE formation energies above suggest that the new D0₁₉ Mg₃Al structure is more stable than D0₃ and L1₂ Mg₃Al by 39.1 and 1.5 meV/atom, respectively. Because the L1₂ phase is so close in energy to the predicted ground-state D0₁₉ phase and because its fcc structure could enable it to form by means of a stacking fault within the hcp Mg host lattice, we continue to investigate its properties in what follows.

The formation energy defined in equation (3) contains no contributions from the zero-point energy (ZPE), which could alter the relative stability of D0₁₉ and L1₂ Mg₃Al, given their small E_f. We thus determine their ZPEs from KSDFT phonon spectra (figure 3). We then apply the quasi-harmonic approximation with volume-dependent phonon dispersions to compute the temperature-dependent free energies of D0₁₉ and L1₂ Mg₃Al, hcp Mg, and fcc Al. Figure 4(a) displays the variation of E_f with temperature. The ZPE contributions to the E_f of D0₁₉ and L1₂ Mg₃Al are −37.6 and −38.0 meV/atom, respectively. Such large ZPEs alter the sign of E_f from positive to negative. Increasing the temperature from 0 to 700 K further decreases E_f due to increasing entropy. E_f remains negative within a wide temperature range, confirming that decomposition of D0₁₉ or L1₂ Mg₃Al into fcc Al and hcp Mg is endoergic. The variation of the free energy difference per atom between D0₁₉ and L1₂ Mg₃Al with temperature (figure 4(b)) shows D0₁₉ Mg₃Al consistently exhibiting slightly free lower energies than L1₂ Mg₃Al, confirming D0₁₉ Mg₃Al's slightly greater stability.

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Mg₃Al is unlikely to decompose directly into fcc Al and hcp Mg. We therefore should examine other possible decomposition reactions before concluding that it is thermodynamically stable. The most likely reaction to consider is

$$\begin{eqnarray}&&12{{\rm{Mg}}}_{3}{\rm{Al}}\to {{\rm{Mg}}}_{17}{{\rm{Al}}}_{12}+19{\rm{Mg}}.\end{eqnarray} \tag{ 4 }$$

We select Mg₁₇Al₁₂ and Mg as possible products because the composition of Mg in Mg₃Al is between that in Mg₁₇Al₁₂ and hcp Mg. Mg₁₇Al₁₂ is additionally one of three intermetallic compounds (Mg₁₇Al₁₂, Mg₂₃Al₃₀, and Mg₂Al₃) that exist in the experimental phase diagram of Mg-Al alloys [43]. Figure S3 in the supplemental material shows the phonon spectrum of Mg₁₇Al₁₂. Using free energies calculated as discussed earlier, figure S4 in the supplemental material displays the temperature-dependent free energies for Mg₃Al decomposing to Mg and Mg₁₇Al₁₂. We find that the above reaction (equation (4)) at 298 K is exoergic by −17.9 and −18.7 meV/atom for D0₁₉ and L1₂ Mg₃Al, respectively, i.e., D0₁₉ or L1₂ Mg₃Al is susceptible to decomposing into Mg₁₇Al₁₂ and hcp Mg. This decomposition, however, requires an unknown reaction barrier, which we speculate could be large since the structure of Mg₁₇Al₁₂ (see figure S1 in the supplemental material) differs drastically from D0₁₉ and L1₂ Mg₃Al. However, currently we have no way to theoretically estimate the barrier to decomposition, so experiments will be needed to confirm or disprove our speculation.

It is of course possible that some other phase of different crystal morphology (non-hcp, non-cubic) not considered here might exist that has a lower formation free energy than that of D0₁₉ Mg₃Al. Again, we emphasize that our work focuses on Mg-rich Mg-Al alloys with the realistic physical constraints imposed by the Mg lattice. As such, it is very unlikely that adding a small amount of Al to hexagonal Mg at ambient conditions would transform the structure to another morphology incommensurate with hcp, due to the large strain it would engender. A non-equilibrium synthesis process such as the rapid quenching method [44] could be used to precipitate D0₁₉ or L1₂ Mg₃Al in hcp Mg, as was used to prepare another metastable intermetallic compound, MgAl₂ [45].

We further explore Mg₃Al's metastability by evaluating its dynamical and mechanical stability. Beyond enabling calculation of the ZPE, the calculated phonon spectra in figure 3 serve another important purpose: verifying a material's dynamical stability [46], which requires the presence of only real phonon modes in the first Brillouin zone. Indeed, D0₁₉ and L1₂ Mg₃Al's phonon modes are all real, confirming both phases' dynamical stability. Figure S5 in the supplemental material shows the phonon spectrum of D0₃ Mg₃Al. D0₃ Mg₃Al is expected to be dynamically unstable, consistent with its positive formation energy E_f of 43.0 meV/atom.

Mechanical stability of D0₁₉ and L1₂ Mg₃Al is assessed via calculation of their five (C₁₁, C₁₂, C₁₃, C₃₃, and C₄₄) and three (C₁₁, C₁₂, and C₄₄) independent elastic constants, respectively. The elastic constants for hcp D0₁₉ Mg₃Al must satisfy the following four constraints [47]:

$$\begin{eqnarray}&&{C}_{11}\gt | {C}_{12}| \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&2{C}_{13}^{2}\lt {C}_{33}({C}_{11}+{C}_{12})\end{eqnarray} \tag{ 6 }$$

and

$$\begin{eqnarray}&&{C}_{44}\gt 0.\end{eqnarray} \tag{ 7 }$$

For fcc L1₂ Mg₃Al, three criteria need to be met:

$$\begin{eqnarray}&&{C}_{11}-{C}_{12}\gt 0\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{C}_{11}+2{C}_{12}\gt 0\end{eqnarray} \tag{ 9 }$$

and

$$\begin{eqnarray}&&{C}_{44}\gt 0.\end{eqnarray} \tag{ 10 }$$

The calculated elastic constants of Mg₃Al in table 1 clearly satisfy the above four criteria, showing that both D0₁₉ and L1₂ Mg₃Al are mechanically stable.

To explore the thermal stability of D0₁₉ and L1₂ Mg₃Al, we also simulated their order-disorder phase transition via MC simulations using the OFDFT-derived CE energy expression. Figure 5(a) displays the temperature-dependent total energy and heat capacity of D0₁₉ Mg₃Al. The total energy rises monotonically with increasing temperature until an abrupt increase occurs at about 460 K. The discontinuous temperature dependence correspondingly appears in the temperature-dependent heat capacity shown in the inset, revealing that an order-disorder phase transition in D0₁₉ Mg₃Al should occur at ∼460 K. This temperature is sufficiently high to conclude once again that D0₁₉ Mg₃Al should be at least thermally metastable under ambient conditions. By contrast, the order-disorder transition temperature of L1₂ Mg₃Al is ∼320 K (figure 5(b)). This lower temperature is consistent with the calculated formation energies showing that D0₁₉ Mg₃Al is slightly more stable.

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Given the encouraging stability metrics above, we proceed to study the mechanical properties of D0₁₉ and L1₂ Mg₃Al and compare them to those of hcp Mg. We use Pugh's ratio B/G (B: bulk modulus; G: shear modulus) to first assess the expected average ductilities of D0₁₉ and L1₂ Mg₃Al and then compare them with that of hcp Mg. This ratio is commonly utilized as an indicator of a material's ductility [48]. According to Pugh, larger B/G implies better ductility. Table 1 lists the bulk and shear moduli converted from the calculated elastic constants according to the Voigt (B_V and G_V) and Reuss (B_R and G_R) approximations [35]. These two approximations give rise to upper and lower bounds on the moduli, respectively. Table 1 also reports the bulk and shear moduli based on the Hill approximation, which are the averaged values of the two bounds. The latter approximation is often called the Voigt-Reuss-Hill approximation [35]. For Mg and Mg₃Al, the differences between the Voigt and Reuss moduli are negligible. The predicted bulk moduli of D0₁₉ and L1₂ Mg₃Al are identical and considerably larger than Mg, whereas the two shear moduli deviate differently: D0₁₉ is smaller while L1₂ is larger than hcp Mg. As a result, the predicted Pugh's ratios B/G of D0₁₉ and L1₂ Mg₃Al are 2.65 and 1.96, larger than that of hcp Mg with its calculated B/G of 1.76. The theoretical B/G of hcp Mg is in reasonable agreement with the experimental value of 1.93, derived from the measured elastic constants [49], validating our modeling approach. Given the significantly larger B/G for D0₁₉ Mg₃Al, we can state with confidence that D0₁₉ Mg₃Al should be more ductile than hcp Mg, whereas the Pugh's ratio of L1₂ Mg₃Al implies that its ductility should be more similar to that of Mg.

Another indicator of ductility is provided by the GSFE, γ_GSF. We first consider the calculated γ_GSF curves of D0₁₉ Mg₃Al for two common slip systems on the basal plane, $(0001)\langle 10\bar{1}0\rangle$ and $(0001)\langle 11\overline{2}0\rangle$ (figures 6(a) and (b), respectively). Here, the energy scales on the two slip systems are quite different: the γ_GSF on the $(0001)\langle 10\bar{1}0\rangle$ system is much smaller than on the $(0001)\langle 11\overline{2}0\rangle$ system. It therefore is much more likely that D0₁₉ Mg₃Al will deform along the former slip system, similar to Mg: by slipping in this direction, atoms in different planes do not pass directly above each other and can thus minimize repulsions. Table 2 summarizes the four critical γ_GSF of D0₁₉ Mg₃Al and Mg on the $(0001)\langle 10\bar{1}0\rangle$ slip system, including the unstable stacking fault (γ_USF), stacking fault (γ_SF), unstable twinning (γ_UT), and twinning (γ_TE) energies. A stacking fault corresponds to the stacking sequence ABABABABCACACACA of the close-packed planes in a 16-layer slab model, whereas a twinning fault has the ABABABABCBABABAB stacking pattern. The unstable stacking fault γ_USF and unstable twinning γ_UT refer to the energy barriers to form the stacking and twinning faults, respectively. Table 2 shows that our calculated critical energies of Mg are consistent with previous KSDFT values [50, 51]. Importantly, the slightly reduced γ_USF and γ_UT for D0₁₉ Mg₃Al will ease formation of stacking and twinning faults, contributing to its ease of plastic deformation. The differences in stacking fault energies (SFEs) between D0₁₉ Mg₃Al and hcp Mg are small but should be reliable because our calculations are well converged numerically. However, the SFEs of hcp Mg are small in absolute terms, so it is better to compare the SFEs of hcp Mg and Mg₃Al in terms of percentage. For example, the reduction in the unstable SFE from hcp Mg to D0₁₉ Mg₃Al is about 10%. Pugh's ratio (table 1) is a stronger indicator, increasing from 1.76 for hcp Mg to 2.65 for D0₁₉ Mg₃Al, an increase of 51%. Combining the evaluations of both ductility metrics, D0₁₉ Mg₃Al should exhibit better ductility than Mg.

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Table 2.  Unstable stacking fault $({\gamma }_{{\rm{USF}}}),$ stacking fault $({\gamma }_{{\rm{SF}}}),$ unstable twinning $({\gamma }_{{\rm{UT}}}),$ and twinning $({\gamma }_{{\rm{TE}}})$ energies (mJ/m²) of hcp D0₁₉ Mg₃Al and hcp Mg on the $(0001)\langle 10\overline{1}0\rangle$ slip system and fcc L1₂ Mg₃Al on the $(111)\langle 11\overline{2}\rangle$ slip system. The results calculated in this work are in bold. Other KSDFT results from the literature are shown for comparison.

	${\gamma }_{{\rm{USF}}}$	${\gamma }_{{\rm{SF}}}$	${\gamma }_{{\rm{UT}}}$	${\gamma }_{{\rm{TE}}}$
D019 Mg3Al	84	29	107	27
L12 Mg3Al	91	19	123	90
hcp Mg	93	35	113	46
	92a	36a	111a	39a
	94b	28b	113b	39b

^aReference [50]. ^bReference [51].

For hexagonal D0₁₉ Mg₃Al and hcp Mg, we also computed the GSFE for the prismatic $(1\bar{1}00)\langle 11\bar{2}0\rangle$ slip system, which is the second lowest energy slip system. Figure 6(c) shows the corresponding GSFE curves, and the energy barriers are determined as 162 and 203 mJ m⁻² for D0₁₉ Mg₃Al and hcp Mg, respectively. We define an anisotropy ratio ʌ as the ratio between the energy barriers for the prismatic and basal slip systems. The ʌ values for D0₁₉ Mg₃Al and hcp Mg are predicted to be 1.93 and 2.18, respectively. A larger ʌ corresponds to a stronger anisotropy, which results in worse ductility. D0₁₉ Mg₃Al therefore also exhibits better ductility than hcp Mg by this metric, which is consistent with the assessment using Pugh's ratio.

The competitive stability of D0₁₉ and L1₂ Mg₃Al motivates investigation of the GSFE of L1₂ Mg₃Al on the favored fcc $(111)\langle 11\bar{2}\rangle$ slip system to assess the energy barrier to form hcp-stacked layers as in D0₁₉ Mg₃Al. We again use a 16-layer slab model with the initial stacking sequence of ABCABCABCABCABCA. As a result of the stacking fault, the stacking sequence becomes ABCABCABCBCABCAB, i.e., four hexagonally stacked layers (BCBC) occur in the fcc structure. Figure 6(d) shows the resulting GSFE curve of L1₂ Mg₃Al with the four critical energies given in table 2. We predict a barrier to form the stacking fault of 91 mJ m⁻², comparable to the barrier (84 mJ m⁻²) to form three fcc stacking layers (ABC) starting from the D0₁₉ structure. Both energy barriers are sufficiently small and consistent with the nearly degenerate energies of the two structures, suggesting that both structures could be present in such precipitates.

Beyond evaluating the GSFE, we also considered another important planar defect in ordered intermetallic phases that help determine ductility trends, namely, anti-phase boundaries (APBs) [52]. An APB would form in an ordered Mg₃Al precipitate by the passing of (partial) dislocations. The formation of an APB in an intermetallic compound introduces a compositional rearrangement, e.g., switched positions of Mg and Al atoms, which could be energetically costly and decrease ductility via setting up an additional effective barrier to dislocation motion. We therefore calculated the APB energies of L1₂ and D0₁₉ Mg₃Al with their $\{111\}$ and $\{0001\}$ planes shifted along the $\displaystyle \frac{1}{2}\langle 110\rangle$ and $\displaystyle \frac{1}{3}\langle \bar{1}2\bar{1}0\rangle$ vectors, respectively. The atomic coordinates of the surface slabs with and without an APB are given in the supplemental material. We found that the APB energies of L1₂ and D0₁₉ Mg₃Al are extremely small, 14 and 27 mJ m⁻², respectively. For comparison, we computed the APB energy of L1₂ Ni₃Al with our setup and found it to be an order of magnitude larger, 196 mJ m⁻², consistent with previous theoretical and experimental values (e.g., [53, 54]). The significantly smaller APB energies of L1₂ and D0₁₉ Mg₃Al imply that such precipitates will not inhibit dislocation motion beyond the barriers predicted by the GSFEs shown in figure 6, which already imply an improved ductility. This conclusion can be drawn because the APB energy is inversely related to the distance between two partial dislocations [52]; the expected large distance between partials in this case then suggests that the ductility of Mg with such Mg₃Al precipitates therefore will be determined by the passage of a single partial dislocation, whose ease of movement depends on the GSFEs already discussed.

To elucidate the electronic origin of the predicted enhanced ductility of D0₁₉ Mg₃Al over L1₂ Mg₃Al and hcp Mg, figure 7 compares the density of states (DOS) of a D0₁₉ Mg₃Al primitive cell, an fcc L1₂ Mg₃Al 2 × 1 × 1 supercell, and an hcp 2 × 2 × 1 Mg supercell (these supercells are used so as to have the same number of atoms in each material's unit cells). The number of states per simulation cell at the Fermi level is 3.98 for D0₁₉ Mg₃Al, 3.09 for L1₂ Mg₃Al, and 2.88 for hcp Mg. The larger the DOS at the Fermi level, the higher the metallicity [55], again indicating better ductility for D0₁₉ Mg₃Al over L1₂ Mg₃Al and Mg, consistent with its much higher Pugh's ratio and somewhat lower unstable SFEs.

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Using a CE method fit to energies from OFDFT calculations, we discovered a potentially useful new Mg-Al intermetallic compound, Mg₃Al, exhibiting a simple D0₁₉ hcp structure. This new intermetallic phase satisfies mechanical and dynamical stability criteria, indicating that it is at least metastable. D0₁₉ Mg₃Al should have better ductility than Mg and a hypothetical L1₂ Mg₃Al phase, according to its calculated Pugh's ratios and unstable SFEs. We attribute the improved ductility to enhanced metallicity, as evidenced by the increased (Al) DOS at the Fermi level. This trend is consistent with the slight polarization of electrons towards Al and away from Mg expected from their relative electronegativity. Bader charge analysis [56, 57] confirms the differentially increased electron density around each Al atom in D0₁₉ Mg₃Al (2.96 e/atom) versus in L1₂ Mg₃Al (2.82 e/atom).

The potential D0₁₉ precipitate structure has two engineering-related important implications. First, its superior Pugh's ratio and unstable SFE imply that D0₁₉ Mg₃Al precipitates in Mg-rich Mg-Al alloys should improve Mg-rich Mg-Al alloy ductility more than L1₂ precipitates would. Second, because the in-plane lattice constant (table 1) of D0₁₉ Mg₃Al is nearly twice of that of hcp Mg, only a small misfit strain δ (∼3%) would be induced between the host Mg matrix and the precipitate. A dislocation-based interface model can be used to estimate the interface energy E caused by this misfit strain [58]: E = Gbδ/2. Here, b is the Burgers vector of Mg, ∼2.5 Å [59]. The resulting E (78 mJ m⁻²) is sufficiently small to suggest formation of an almost coherent interface that should reduce the risk of fracture failure.

More generally, our work demonstrates the efficacy of coupling OFDFT to the CE method, while searching for new Mg-Al intermetallic compounds. Although the superior efficiency of OFDFT was not fully exploited here due to the simplicity of the binary Mg-Al system, requiring only 101 structures to obtain the CE coefficients, the structure search and characterization strategy demonstrated here is quite general and can be applied to design numerous other complicated lightweight alloys such as Mg-Li-Ca ternary alloys that are of great interest for engineering applications.

Other structure prediction methods [60], such as genetic algorithms, also could be coupled straightforwardly to OFDFT to predict other new complex alloys. The CE method implemented in ATAT has the disadvantage that the predicted structures are constrained to the parent structure symmetry. In contrast, genetic algorithms do not have such a constraint [61]. Developments in OFDFT and enhanced computational power have enabled efficient first-principles molecular dynamics simulations of phase transitions in samples containing more than 1000 metal atoms [62]. Peta-scale OFDFT computations recently demonstrated that the electronic ground state for more than one million metal atoms can be obtained in around one minute [63]. We therefore believe using OFDFT as the energy calculator for genetic-algorithm-based alloy optimization will soon be possible, even for large numbers of atoms.

We are grateful to the Office of Naval Research for funding (Grant No. N00014-15-1-2218). We acknowledge use of the TIGRESS high performance computer center at Princeton University. We thank Ms Nari Baughman and Dr Johannes M Dieterich for critical reading of this manuscript. We also thank Dr Qijun Hong for helping with the calculations of order-disorder transition temperatures.