First-principles study of the mechanical and thermodynamic properties of aluminium-doped magnesium alloys

Abstract

Magnesium–aluminum (Mg–Al) alloys are widely used in aerospace, automobile and medical equipment owing to their advantages of easy casting, high strength-to-mass ratio and good biocompatibility. The structural, mechanical, electronic and thermodynamic properties of Mg_xAl_y alloys (x + y = 16, x = 1, 2,…, 15) with varying Al-doping contents were studied using the first-principles method. In this work, the structures of Mg_xAl_y alloys were constructed by replacing Mg atoms in a supercell with Al atoms. The lattice parameters of the Al-doped Mg_xAl_y alloys decrease with an increasing Al content because of the smaller atomic size of Al than that of Mg. The calculated formation energies show that Mg₁₁Al₅, Mg₅Al₃ and Mg₉Al₇ have prominent structural stability. The analyses of the mechanical properties reveal that the doping of Al improves the ductility of Mg_xAl_y alloys. The elastic moduli increase with an increasing Al content, and Mg₉Al₇ has a notable ability to resist deformation, while Mg₁₁Al₅ and Mg₅Al₃ have better plasticity. The calculated results of their electronic properties reveal that Mg₁₁Al₅, Mg₅Al₃ and Mg₉Al₇ are good conductors without magnetism. Furthermore, CDD analyses show that the inner layer charges of Al atoms migrated to the outer layer, and the charges of Mg atoms accumulated significantly in the outer region of Al atoms. The Debye temperature of Mg₉Al₇ is higher than that of Mg₁₁Al₅ and Mg₅Al₃, indicating that it has better thermodynamic stability. Our findings would be helpful for the design of Mg–Al alloys with excellent mechanical and thermodynamic performances.

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1. Introduction

Magnesium (Mg) based alloys have the advantages of easy casting, high strength-to-mass ratio and good biocompatibility, which are widely used in fields such as the aircraft industry, automotive production and medical equipment.^1–5 Among all Mg-based alloys, magnesium–aluminum (Mg–Al) alloys hold an important and special position.^6–8 Firstly, the addition of Al could improve the flow ability, hot cracking resistance and mechanical properties of Mg alloys without significantly increasing their density.⁹ The density of Mg–Al alloys ranges from 1.74 to 2.7 g cm⁻³, with a maximum Young's modulus of 78.4 GPa, which has attracted widespread attention in the field of mechanical manufacturing.¹⁰ Secondly, the density and mechanical properties of Mg–Al alloys are close to that of human cortical bones, and they have good biocompatibility and biodegradability.^11,12 These advantages make them extremely suitable as repair materials for human bone injuries. Due to the special applications of Mg–Al alloys, their mechanical, thermodynamic and electronic properties have been extensively studied through experiments.^13–17 For example, Hsu et al.¹⁷ prepared several Mg–Al alloy electrodes with different Mg/Al mass ratios using powder metallurgy. They found that the Mg–Al alloy with 75 wt% has the highest efficiency in water electrolysis experiments, considering its anticorrosion property and material costs. Although many studies have been conducted on the properties of Mg–Al alloys, there are still significant challenges in exploring and preparing potential Mg–Al alloys with practical value because of the limitations in experimental equipment and technology. With the development of computer technology and scientific computing software, simulating the properties of materials at the atomic and electron level has become popular. Computational simulation is of great significance for the discovery of new materials and the improvement of material performance through specific designs. Doping other elements or clusters into materials is one of the efficient methods for discovering new high-performance materials, and many valuable works have been done in this regard.^18–23 For example, Abbasi et al.²⁰ studied the effects of elemental doping on the structural and electronic properties of stanene. Their results show that the influence of different doping elements on the electronic properties of the structure is significant, which provides a good idea for the design of electronic and optoelectronic devices in the future.

Many factors can affect the mechanical and thermodynamic properties of metal alloys, such as mass ratio, phase composition and grain boundaries.^24–27 Among these factors, the mass ratio of elements in alloys is fundamental and has a significant impact on their performance.^24,25 For Mg–Al alloys, some studies on mass ratios have also been reported.^28–30 Using the atomic substitution method, Zhou et al.²⁸ obtained the supercell of Mg₁₅Al and studied the mechanical and electronic properties. The calculated B/G ratio of pure Mg and Mg₁₅Al are 1.92 and 1.8, respectively, indicating that both structures are ductile. However, computational studies on discovering potential structures of Mg–Al alloys through doping procedures are still very rare. In this work, we aim to screen out potential structures with the lowest energy and stable state from massive initial Mg–Al supercells. Firstly, a large number of Mg–Al supercells with different Mg/Al ratios and substitution sites were constructed using the atomic substitution method. Then, potential stable structures were determined according to formation energies and the convex hull diagram. For the selected structures, we studied their structural features, electronic properties, and mechanical and thermodynamic properties based on density functional theory (DFT) schemes.³¹ Our work will provide guidance for designing new Mg–Al alloys and give a further understanding of their structural performance relationships.

2. Methods

The single crystal of Mg belongs to a hexagonal close-packed structure (space group P6₃/mmc). As shown in Fig. 1a, each unit cell of Mg contains 2 atoms. The lattice parameters are a = b = 3.2094 Å and c = 5.2105 Å, which were determined by Batchelder and Raeuchle.³² A 2 × 2 × 2 supercell with lattice parameters a = b = 6.4188 Å and c = 10.421 Å was constructed with a total of 16 Mg atoms, as shown in Fig. 1b. The site of each Mg atom can be doped by an Al atom, assuming that the lattice parameters have no change. To study the effect of the Mg/Al ratio on the mechanical and thermodynamic properties of Mg–Al alloys, structures of 15 proportions were constructed according to the formula Mg_xAl_y (x + y = 16, x = 1, 2,…, 15). For each Mg/Al ratio, the number of the potential structures is C^y₁₅, without considering the structural symmetry. Then, optimization calculations were performed on all structures to determine the structure with the lowest energy for each Mg/Al atomic ratio. To further evaluate and determine the stable structures, we calculated the formation energies and convex hull diagram of Mg_xAl_y structures with different Mg/Al ratios.

Fig. 1 The unit cell and supercell of Mg.

In this work, all of the calculations were performed using the Vienna ab initio simulation package (VASP).^33,34 The generalized gradient approximation (GGA) using the Perdew–Burke–Ernzerhof (PBE) function³⁵ was used to describe the exchange and correlation interactions, and the projector-augmented wave (PAW)^36,37 method was also used in this work. The kinetic-energy cutoff was set to 500 eV for the plane-wave basis set. All lattice parameters and atomic coordinates were fully relaxed until a force tolerance of 0.01 eV Å⁻¹ is reached. The Monkhorst–Pack k-point grids³⁸ used for all Mg_xAl_y structures are 8 × 8 × 12. The selected parameters ensure that the accuracy of the total energy is converged to within 1.0 meV per atom. The elastic constants were calculated by the stress–strain method.³⁹ Phonon calculations were carried out using the direct supercell method as implemented in the PHONOPY program⁴⁰ to determine the dynamical stability of Mg_xAl_y structures.

3. Results and discussion

3.1 Lattice parameters and formation energies

All optimized structures of Mg_xAl_y with the lowest energy were displayed by the Materials Studio software, as shown in Fig. 2. It is worth noting that the atom number of Mg_xAl_y structures discussed in this paper are all 16, and in order to unify writing standards, x and y are both the simplest ratios. For example, Mg₁₄Al₂ is abbreviated as Mg₇Al. The lattice constants, volume and density for Mg₁₆, fcc Al and Mg_xAl_y structures are listed in Table 1. When the number of doped Al atoms is less than 5, the structures with the lowest energy under each Mg/Al ratio show a bias of Al atom aggregation. The results reveal that the doping of Al atoms causes the change of Mg_xAl_y structures, and the values of a, c and c/a fluctuate and show a decreasing trend in general with increasing Al atoms. Due to the smaller atomic radii of Al compared with Mg, the volume of Mg_xAl_y decreases with increasing Al content. In addition, Al has a heavier atomic mass, which leads to an increase in density.

Fig. 2 The structures of Mg_xAl_y with the lowest energy.

Table 1 The lattice constants, volume and density of Mg₁₆, fcc Al and Mg_xAl_y alloys

Structures	a (Å)	c (Å)	c/a	V (Å^3)	ρ (g cm^−3)
Mg16	6.430	10.39	1.616	365.4	1.745
Mg15Al	6.349	10.28	1.619	358.7	1.792
Mg7Al	6.339	10.24	1.615	351.6	1.842
Mg13Al3	6.233	10.29	1.651	346.0	1.886
Mg3Al	6.194	10.03	1.619	333.6	1.971
Mg11Al5	6.164	10.02	1.626	327.7	2.022
Mg5Al3	6.172	9.936	1.610	321.9	2.074
Mg9Al7	6.102	9.772	1.601	314.4	2.140
MgAl	6.056	9.652	1.594	308.8	2.194
Mg7Al9	5.998	9.581	1.597	302.4	2.257
Mg3Al5	6.043	9.412	1.558	300.0	2.292
Mg5Al11	6.050	9.403	1.554	295.9	2.341
MgAl3	6.019	9.291	1.544	290.7	2.400
Mg3Al13	6.013	9.140	1.520	285.9	2.458
MgAl7	5.955	9.142	1.535	281.2	2.516
MgAl15	5.893	9.165	1.555	275.7	2.584
Al	4.044			66.17	2.712

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To preliminarily evaluate the stability of the structures, formation energies were calculated and a convex hull diagram was plotted. The formation energy of the Mg_xAl_y structure at 0 K is defined as follows,

E_f = (E(Mg_xAl_y) − xE(Mg) − yE(Al))/(x + y) (1)

where E(Mg_xAl_y) refers to the total energy of the Mg_xAl_y structure in the ground state, and E(Mg) and E(Al) represent the energy per atom of hcp Mg and fcc Al, respectively.⁴¹ Fig. 3 shows the calculated formation energies and convex hull of Mg_xAl_y structures with different Mg/Al ratios. As depicted in Fig. 3, when the number of Al atoms substituted in the Mg_xAl_y structures is greater than 11, the formation energies are positive, indicating that those structures are unstable. Due to the fact that structures with formation energies above the convex hull are meta-stable,⁴² Mg₁₁Al1₅, Mg₅Al₃ and Mg₉Al₇ were confirmed to be the most stable structures and adopted for further study.

Fig. 3 Calculated formation energy of Mg_xAl_y alloys as a function of Al-doping contents.

3.2 Elastic properties

In addition to the formation energy, another index of structural stability is mechanical stability, which can be studied by elastic constants. We calculated the elastic constants of Mg₁₆, fcc Al and Mg_xAl_y structures, as shown in Table 2. For Mg₁₆, the value of C₁₁ in this work is 2 GPa smaller than that of Wazzan et al.⁴⁴ measured through experiments. The values of C₁₂ and C₁₃ are also very close to the results obtained by Wazzan et al. with the largest discrepancy being 3.83%. However, the C₃₃ and C₄₄ values of Mg₁₆ are closer to the results obtained by Slutsky et al.⁴⁵ through experiments with the largest discrepancy of 3.16%. For fcc Al, the values of C₁₁ and C₁₂ in this work are very close to those obtained by Le Page et al.³⁹ by ab initio calculations with the largest discrepancy of 1.94%, while the value of C₄₄ is 3.1 GPa smaller than theirs. Considering the differences in experimental and computational methods, our calculated elastic constants C_ij have a good agreement with their results. Due to the addition of Al, the elastic constants of Mg_xAl_y alloys increase significantly, and the elastic constants of Mg₁₁Al₅, Mg₅Al₃ and Mg₉Al₇ increase with increasing Al content. Born's stability criteria⁴⁷ is also usually used to determine the mechanical stability of materials. According to Born, any applied strain should increase the energy of the stable ground-state solid. The eigenvalues of the stiffness tensor and its matrix are positive. Therefore, the elastic constants of crystals should satisfy certain conditions according to their symmetry. The formulas of mechanical stability criteria for hcp Mg₁₆, fcc Al and Mg_xAl_y structures are given in ESI.† From our calculated elastic constants, we confirm that Mg₁₆, fcc Al, Mg₁₁Al₅, Mg₅Al₃ and Mg₉Al₇ are mechanically stable.

Table 2 The elastic constants (GPa) of Mg₁₆, fcc Al and Mg_xAl_y alloys

Structures	C11	C12	C13	C33	C44
a Ref. 43.b Ref. 44.c Ref. 45.d Ref. 39.e Ref. 46.f Present work.
Mg^a	62.4	23.5	22.0	58.9	16.3
Mg^b	59.5	26.1	21.8	61.6	16.4
Mg^c	63.5	25.9	21.7	66.5	18.4
Mg16^f	57.5	27.1	21.0	68.6	18.1
Mg11Al5	70.7	37.5	30.5	79.4	12.9
Mg5Al3	73.7	38.5	31.6	80.7	14.5
Mg9Al7	76.9	44.6	31.4	88.3	21.6
Al^d	114.0	62.0			32.0
Al^e	120.0	61.0			28.0
Al^f	115.2	63.2			28.9

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Based on elastic constants, we also calculated the bulk modulus B, shear modulus G and Young's modulus E of Mg₁₆, fcc Al and Mg_xAl_y structures, as shown in Table 3. The elastic moduli and Poisson's ratio v were obtained using the Voigt–Reuss–Hill averaging scheme,^48–50

B = (1/2)(B_R + B_V) (2)

G = (1/2)(G_R + G_V) (3)

E = 9BG/(3B + G) (4)

v = (3B − 2G)/[2(3B + G)] (5)

where B_R and B_V represent the bulk moduli with the Reuss and Voigt averaging scheme, respectively. G_R and G_V represent the shear moduli with the Reuss and Voigt averaging scheme, respectively. The formulas for calculating B_R, B_V, G_R and G_V through C_ij are also given in ESI.† For Mg₁₆, the value of Young's modulus E is 45.6 GPa, which is 0.9 GPa smaller than the value calculated by Velikokhatnyi et al.⁴³ by first-principles calculations and 1.1 GPa larger than the result measured by Wazzan et al.⁴⁴ through experiments. The bulk modulus B of Mg₁₆ is 35.7 GPa, which is very close to the value calculated by Velikokhatnyi et al. However, the discrepancy is 2.52%, compared with the result from Wazzan et al.. In addition, the shear modulus G of Mg₁₆ is 17.7 GPa, which differs from the values of Velikokhatnyi et al. and Wazzan et al. by 0.4 and 0.5 GPa, respectively. In general, our results are very close to the values obtained by others through experiments and calculations, indicating that the method used in this work to calculate elastic properties is appropriate. For Mg_xAl_y, the elastic moduli increase with increasing Al content. It should be noted that E and G of Mg₁₁Al₅ and Mg₅Al₃ have similar values as Mg₁₆, while the bulk moduli B are larger. The doping of Al atoms changes the structure of the Mg supercell, and the results reveal that the effect of Al doping on the volume compression is greater than that of shear and tension. The elastic moduli of Mg₉Al₇ are significantly higher than those of Mg₁₁Al₅ and Mg₅Al₃, which is mainly because the Al atoms in Mg₉Al₇ supercell are more concentrated. B/G value can be used to characterize the ductility and brittleness of materials. According to the Pugh criterion,⁵¹ B/G values greater than 1.75 indicate ductility, while a value less than 1.75 characterizes brittleness. Our calculated B/G values show that Mg_xAl_y alloys have better ductility than pure Mg. In addition, the elastic moduli of Mg₁₁Al₅, Mg₅Al₃ and Mg₉Al₇ increase with increasing Al content, while the B/G values show an opposite trend. The results indicate that Mg₉Al₇ has the best ability to resist deformation, while Mg₁₁Al₅ and Mg₅Al₃ have better plasticity.

Table 3 The elastic moduli (GPa) of Mg₁₆, fcc Al and Mg_xAl_y alloys

Structures	E	B	G	v	B/G	A^U	Hv
a Ref. 43.b Ref. 44.c Present work.
Mg^a	46.5	35.4	18.1	0.28	1.96
Mg^b	44.5	36.6	17.2	0.29	2.06
Mg16^c	45.6	35.7	17.7	0.29	2.02	0.12	3.2
Mg11Al5	43.6	46.4	16.2	0.34	2.86	0.27	2.0
Mg5Al3	46.6	47.9	17.4	0.34	2.75	0.19	2.2
Mg9Al7	53.9	50.8	20.4	0.32	2.49	0.24	2.8
Al	74.6	80.5	27.7	0.35	2.91	0.01	2.9

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The elastic anisotropy of these structures was also studied according to a universal index of A^U = 5G_V/G_R + B_V/B_R − 6, which is related to the occurrence of micro-cracks in materials.⁵² The departure of A^U from zero (for isotropic crystal) defines the extent of crystal anisotropy and accounts for both the shear and the bulk contributions.⁵³ As shown in Table 3, the elastic anisotropy index shows that Mg₁₆ and Mg_xAl_y structures have considerable elastic anisotropy compared with fcc Al. The A^U value of Mg₁₁Al₅ is 0.27, indicating that the anisotropy is the most obvious. In addition, the hardness H_v of these three structures was evaluated according to the empirical correlation,⁵⁴

H_v = 0.92(G/B)^1.137G^0.708 (6)

where G and B represent the shear modulus and bulk modulus, respectively. Without considering the intercept term, the hardness formula obtained by Chen et al.⁵⁴ eliminates the possibility of unrealistic negative hardness. For Mg_xAl_y, the hardness of Mg₉Al₇ is larger than that of Mg₁₁Al₅ and Mg₅Al₃, as shown in Table 3. Hardness can be regarded as the comprehensive resistance of chemical bonds to indentation, indicating that the interaction between atoms was enhanced with increasing Al atoms.

3.3 Electronic properties

The electronic properties are of great significance for the development and application of new materials. In this work, we also employed the HSE06 hybrid functional⁵⁵ to calculate the electronic band structures of Mg₁₁Al₅ (a), Mg₅Al₃ (b) and Mg₉Al₇ (c) with the Fermi level shifted to zero, as shown in Fig. 4. The results reveal that the valence bands and conduction bands of these structures overlap at the Fermi level, resulting in a band gap value of 0. As is well known, conductivity occurs in one or more partially filled energy bands in metals, which possess both valence and conduction band properties. Therefore, our results also confirm that Mg–Al alloy is a good conductor. In addition, we also calculated the magnetization of the structures considering spin polarization. The calculated magnetization values of Mg₁₁Al₅, Mg₅Al₃ and Mg₉Al₇ are 0.0029, 0.0028 and 0.0027 A m⁻¹, respectively, which are much lower than the magnetization values of magnetic materials,^56–58 indicating that these structures are in a non-magnetic state.

Fig. 4 The electronic band structures of Mg₁₁Al₅ (a), Mg₅Al₃ (b) and Mg₉Al₇ (c) with the Fermi level shifted to zero.

To discuss the electronic structure of the system further, we also calculated the charge density difference (CDD) for Mg₁₁Al₅, Mg₅Al₃ and Mg₉Al₇. The CDD is one of the important ways to investigate electronic structure, which could help us understand the charge transfer induced by the doping of Al atoms. Usually, the charge density of the total system is obtained by subtracting the density of two discrete segments that make it,

Δρ = ρ_(Mg+Al) − ρ_(Mg) − ρ_(Al) (7)

where ρ_(Mg+Al) denotes the total charge density of the Mg_xAl_y structure. ρ_(Mg) and ρ_(Al) represent the charge densities of Mg and Al fragments taken from the Mg_xAl_y supercell, respectively. As shown in Fig. 5, the yellow and cyan regions represent the positive (electron accumulation) and negative (electron depletion) values, respectively. For Mg₁₁Al₅, Mg₅Al₃ and Mg₉Al₇, their CDD exhibits similar charge transfer behavior. The inner layer charges of Al atoms migrated to the outer layer, and the charges of Mg atoms also accumulated significantly in the outer region of Al atoms. Besides, the increase in the charge density at the middle of Al–Al domains reveals the formation of covalent bonds between these atoms. To observe the charge transfer of Al atoms at different sites in the Z-direction, two-dimensional cross-sectional color maps at different z-positions are also provided. As can be seen from Fig. 5, the charge transfer of Al atoms at different sites is similar; that is, they all migrate outward from the atomic center, with a maximum charge transfer value of −2.779 e Bohr⁻³.

Fig. 5 The charge density difference (CDD) of Mg₁₁Al₅ (a), Mg₅Al₃ (b) and Mg₉Al₇ (c) with the isosurface plot level of 0.002 obtained using VESTA software.⁵⁹ The yellow and cyan regions represent positive and negative values, respectively. Correspondingly, two-dimensional cross-sectional color maps at different z-positions are also provided.

3.4 Thermodynamic properties

The Debye temperature (θ_D) is closely related to the elastic constants, melting temperature and specific heat of materials, which can be used to characterize the strength of covalent bonds.⁶⁰ In this work, Debye temperature was calculated from the average sound velocity (v_m) as a function of longitudinal sound velocity (v_l) and shear sound velocity (v_t).^60,61

The calculated θ_D, v_l, v_t and v_m of Mg₁₁Al₅, Mg₅Al₃ and Mg₉Al₇ are listed in Table 4. The θ_D value increases with increasing Al content, and the value for Mg₉Al₇ increases by 7.19% compared with Mg₅Al₃. The results reveal that the covalent bond in Mg₉Al₇ is stronger than that of other phases. Hence, the mechanical and thermal stability of Mg₉Al₇ is the best among these three Mg_xAl_y structures, which agrees with the earlier calculated elastic properties. In addition, we further studied the phonon spectrum of the structures, which is another key criterion to characterize the stability of materials.⁶² The absence of virtual phonon modes means that the system is dynamically stable. As shown in Fig. 6, the phonon modes for Mg₁₁Al₅, Mg₅Al₃ and Mg₉Al₇ in the first Brillouin zone are all real, which is consistent with their negative formation energies and their elastic constants that satisfy Born's criteria. The corresponding phonon density of states (PDOS) was also given in order to visualize the distribution of phonon modes better. Many flat phonon bands were observed, giving rise to sharp peaks in the PDOS plots.

Table 4 Theoretically calculated thermal properties of Mg_xAl_y alloys, including longitudinal sound velocity (v_l), shear sound velocity (v_t), average sound velocity (v_m) and Debye temperature (θ_D)

Structures	vl (m s^−1)	vt (m s^−1)	vm (m s^−1)	θD (K)
Mg11Al5	5800.02	2831.13	3180.35	345.99
Mg5Al3	5854.61	2896.26	3251.07	355.80
Mg9Al7	6035.30	3086.31	3457.53	381.38

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Fig. 6 Phonon spectrum and PHDOS of Mg₁₁Al₅, Mg₅Al₃ and Mg₉Al₇.

Based on phonon properties, we studied the effect of temperature on heat capacity (C_v), entropy (S) and Gibbs free energy (G) of pure Mg, Al, Mg₁₁Al₅, Mg₅Al₃ and Mg₉Al₇. The thermodynamic properties were calculated by PHONOPY software under 0 GPa.⁴⁰ As shown in Fig. 7a, the heat capacity C_v increases with increasing temperature and tends to achieve a balance at higher temperatures. In the initial stage (<200 K), Mg and Al show the fastest and slowest increasing speed in C_v with temperature, respectively. The C_v curves of Mg₁₁Al₅, Mg₅Al₃ and Mg₉Al₇ are between that of Mg and Al and have similar responses to temperature. At about 200 K, the increasing speed of C_v starts to slow down with the change in temperature. After 500 K, C_v changes slightly with temperature and tends to achieve the maximum value of 25 J K⁻¹ mol⁻¹. The entropy S increases gradually with increasing temperature, as depicted in Fig. 7b. The entropy of Mg is greater than that of Al in the whole temperature range. The entropy–temperature curves of Mg₁₁Al₅, Mg₅Al₃ and Mg₉Al₇ follow the same pattern and have values between that of Mg and Al. From Fig. 7c, it is found that the Gibbs free energy G gradually decreases with increasing temperature. In the whole range of temperature, the values of G obey the following sequence: Al > Mg₉Al₇ > Mg₁₁Al₅ > Mg₅Al₃ > Mg.

Fig. 7 Thermodynamic properties of (a) heat capacity (C_V), (b) entropy (S) and (c) Gibbs free energy (G) for Mg₁₆, Al and Mg_xAl_y alloys.

4. Conclusion

In this work, the structures of Mg_xAl_y alloys (x + y = 16, x = 1, 2,…, 15) were constructed by atomic substitution method. Structural optimization calculations were performed to determine structures with the lowest energy at each Mg/Al ratio. Their formation energies, elastic moduli, electronic properties and thermodynamic properties were studied using DFT calculations. The lattice parameters of Mg_xAl_y alloys decrease with increasing Al content, which is mainly due to the smaller atomic size of the Al atom compared with that of Mg. According to the convex hull diagram, Mg₁₁Al₅, Mg₅Al₃ and Mg₉Al₇ have the best structural stability. The calculated bulk modulus B, shear modulus G, Young's modulus E and Poisson ratio v show that Mg₁₁Al₅, Mg₅Al₃ and Mg₉Al₇ have better ductility than pure Mg. Mg₉Al₇ has the largest values of elastic moduli, indicating that it has a better ability to resist deformation. The calculated results of their electronic properties reveal that Mg₁₁Al₅, Mg₅Al₃ and Mg₉Al₇ are good conductors without magnetism. Furthermore, the CDD analyses show that the inner layer charges of Al atoms migrated to the outer layer, and the charges of Mg atoms also accumulated significantly in the outer region of Al atoms. The Debye temperature of Mg₉Al₇ is significantly higher than that of Mg₁₁Al₅ and Mg₅Al₃, revealing better thermodynamic stability. In terms of mechanical and thermodynamic properties, Mg₉Al₇ is the most promising structure for application. Our computational results would be helpful for the design of Mg–Al alloys and give a further understanding of structure–property relationships.

Author contributions

Wenjie Zhu: data curation, investigation, and writing-original draft, methodology; Xingtao Ma: writing-review and editing, validation; Yarui Wang: writing-review and editing, validation; Chaoyong Wang: writing-review and editing; Wei Li: writing-review and editing. The descriptions are accurate and agreed upon by all authors.

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.

Conflicts of interest

There are no conflicts of interest to declare.

Acknowledgements

This work was supported by the Natural Science Foundation of Henan Province, the Foundation of Henan Educational Committee (Grant No. 232300420362).

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Footnote

† Electronic supplementary information (ESI) available: The mechanical stability criteria and formulas of elastic moduli are listed. Except for Mg₁₁Al₅, Mg₅Al₃ and Mg₉Al₇, the elastic constants, elastic moduli and thermal properties of Mg_xAl_y are also given in Tables S1, S2 and S3, respectively. See DOI: https://doi.org/10.1039/d4ra00470a

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