Atomic cluster structures, phase stability and physicochemical properties of binary Mg-X (X= Ag, Al, Ba, Ca, Gd, Sn, Y and Zn) alloys from ab-initio calculations

Intermetallics 95 (2018) 119–129

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Atomic cluster structures, phase stability and physicochemical properties of binary Mg-X (X= Ag, Al, Ba, Ca, Gd, Sn, Y and Zn) alloys from ab-initio calculations

T

Jinglian Dua,b, Ang Zhanga,b, Zhipeng Guoa,b,∗, Manhong Yanga,b, Mei Lic, Shoumei Xionga,b,∗∗ a

School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China Laboratory for Advanced Materials Processing Technology, Ministry of Education, Tsinghua University, Beijing 100084, China c Materials Research Department, Research and Innovation Center, Ford Motor Company, MD3182, P.O Box 2053, Dearborn, MI 48121, USA b

A R T I C L E I N F O

A B S T R A C T

Keywords: Mg-based alloys Atomic cluster structures Phase stability Physicochemical property Ab-initio calculations

Both structural and physicochemical properties of binary Mg-X (X = Ag, Al, Ba, Ca, Gd, Sn, Y, Zn) intermetallics were studied by performing ab-initio calculations. It was shown that except for Mg-Zn and Mg-Ba alloys, the mass density of the other Mg-X intermetallics changed linearly as the X-content. The local atomic structural features of Mg-X alloys could be well represented by the characteristic principal clusters, which denote the short-rangeorder structure of the Mg-X alloys. The coordination number (CN) of these atomic clusters changed in-between 8 and 16, and most were 12 and 14. The structural stability of Mg-Al, Mg-Ba, Mg-Ag, Mg-Ca, Mg-Sn, Mg-Y and MgGd intermetallics increased as the solute content, while that of Mg-Zn intermetallics decreased as the Zn-content. For each Mg-X alloy system, MgAl2, MgAg3, Mg17Ba2, Mg2Zn11, MgGd and MgY intermetallics had larger elastic moduli and higher hardness than the others. Besides, MgAg3 and MgZn2 exhibited better plasticity among these Mg-X intermetallics, as reflected by the Poisson ratio and Pugh ratio. All of these Mg-X intermetallics were both thermodynamically and mechanically stable phases, and exhibited conductive metallic features based on the band structures and density of states.

1. Introduction Magnesium alloys are preferable materials for various lightweight applications including automobile, areoplane and biomedical fields, because of their attractive performances such as high strength-toweight ratio, good machineability and environmentally friendliness etc. [1–6]. These excellent properties primarily originate from the intrinsic microstructures of magnesium alloys, among which the precipitated phases or the so-called intermetallics, together with the primary phase and the impurity segregation, have profound influences on the final performances of the alloy [7–14]. Because of the limited solubility of the additional elements (X) in Mg matrix (induced by chemical affinity differences), stable Mg-containing intermetallic particles or Mg-X precipitates will form during solidification, which significantly influence the microstructure patterns and subsequent mechanical properties of the magnesium alloys [15–19]. Extensive studies have been performed to investigate the intermetallics in different Mg-based alloy systems. Lunder et al. [20] investigated the role of Mg17Al12 phase on the corrosion behavior of the

∗

AZ91 alloy. Hu et al. [21] investigated the thermodynamic properties of the Mg-RE alloys using a modified embedded atom method. Min et al. [22] analyzed the valence electron structures of intermetallics containing calcium in Mg-Al-based alloys. Chuang et al. [23] reported ternary MgAlZn intermetallic alloys with high Vicker's hardness. Subsequently, the stability, elastic constants, and electronic properties of different intermetallics in binary and ternary magnesium alloy systems were studied systematically via first-principle calculations [24–31]. Besides, much more attention also focused on the hydrogen storage property of magnesium alloy, which is believed as one of the most promising candidates for environmental-protecting materials [32,33]. Despite much progress on the formation and performances of intermetallics in different Mg-based alloy systems, it is clear that understanding both atomic cluster structures and physicochemical properties is essential for the development of new Mg-based alloys. In this work, the characteristic atomic cluster structures representing short-rangeorder (SRO) features, as well as thermodynamic, mechanical and electronic properties of binary Mg-X (X = Ag, Al, Ba, Ca, Gd, Sn, Y and Zn) intermetallics were investigated by performing ab-initio calculations.

Corresponding author. School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China. Corresponding author. School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China. E-mail addresses: [email protected] (Z. Guo), [email protected] (S. Xiong).

∗∗

https://doi.org/10.1016/j.intermet.2018.02.005 Received 23 November 2017; Received in revised form 12 January 2018; Accepted 5 February 2018 0966-9795/ © 2018 Elsevier Ltd. All rights reserved.

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Table 1 Crystallographic information and mass density of the Mg-X (X = Ag, Al, Ba, Ca, Gd, Sn, Y, Zn) alloys, together with other reported values. Phase

Space group

Prototype

Pearson symbol

Unitcell lattice parameter (Å)

Mass density (kg/m3)

Reference

Mg

P63/mmc

Mg

hP2

MgAg

Pm3m

CsCl

cP2

MgAg3

Pm3m

AuCu3

cP3

Ag

Fm3m

Cu

cF4

Mg17Al12

I 43m

Mn

cI58

MgAl2

I 41/ amd

Ga2Hf

tI24

Mg23Al30

R3

Co5Cr2Mo3

hR53

Al

Fm3m

Cu

cF4

Mg17Ba2

R3m

Zn17Th2

hR19

Mg23Ba6

Fm3m

Mn23Th6

cF116

Mg2Ba

P63/mmc

MgZn2

hP12

Ba Mg2Ca

Im3m P63/mmc

W MgZn2

cI2 hP12

Ca

Fm3m

Cu

cF4

Mg2Gd

Fd3m

Cu2Mg

cF24

Mg3Gd

Fm3m

BiF3

cF16

MgGd

Pm3m

CsCl

cP2

Gd

P63/mmc

Mg

hP2

Mg2Sn

Fd3m

Cu2Mg

cF24

Sn

Fd3m

C

cF8

Mg24Y5

I 43m

Mn

cI58

Mg2Y

P63/mmc

MgZn2

hP12

MgY

Pm3m

CsCl

cP2

Y

P63/mmc

Mg

hP2

Mg2Zn11

Pm3

Mg2Zn11

cP39

MgZn2

P63/mmc

MgZn2

hP12

Zn

P63/mmc

Mg

hP2

a = 3.2094 a = 3.2065 a = 3.3306 a = 3.3302 a = 4.1609 a = 4.1090 a = 4.0857 a = 4.0863 a = 10.5296 a = 10.5438 a = 4.2004 a = 4.1320 a = 12.7663 a = 12.8254 a = 4.0495 a = 4.0500 a = 10.6179 a = 10.4970 a = 15.2086 a = 15.2130 a = 15.220 a = 6.6517 a = 6.660 a = 6.6650 a = 5.0190 a = 6.2458 a = 6.2250 a = 6.2390 a = 6.2340 a = 5.5820 a = 5.5884 a = 8.5571 a = 8.550 a = 7.3301 a = 7.310 a = 3.8043 a = 3.8245 a = 3.6315 a = 3.6330 a = 6.8204 a = 6.7620 a = 6.8250 a = 6.4912 a = 6.4892 a = 11.2622 a = 11.2780 a = 11.260 a = 6.0496 a = 6.0370 a = 3.7954 a = 3.810 a = 3.8030 a = 3.6451 a = 3.6475 a = 8.5240 a = 8.5525 a = 5.2150 a = 5.170 a = 2.6649 a = 2.6650

1736.67 1747.32 5940.04 5942.25 8019.61 8023.08 10505.2 10500.0 2096.46 2085.67 2357.61 2368.37 2219.97 2220.43 2698.85 2697.29 2251.37 2224.69 2611.31 2579.63 – 3044.96 3500.90 – 3607.39 1730.46 1738.68 1716.94 – 1530.62 1532.56 4364.44 4366.09 3881.70 3883.56 5474.79 5476.68 7915.26 7918.56 3502.42 3505.67 – 5764.72 5765.65 2389.69 2390.58 – 2934.03 2936.56 3438.53 3436.89 – 4477.84 4479.23 6175.65 6160.98 5155.64 5156.69 7136.86 7138.68

This work [35] This work [35] This work [35] This work [35] This work [35] This work [35] This work [35] This work [35] This work [43] This work [43] [30] This work [43] [30] This work This work [35] [43] [30] This work [35] This work [35] This work [35] This work [35] This work [35] This work [35] [30] This work [35] This work [35] [43] This work [35] This work [35] [30] This work [35] This work [35] This work [35] This work [35]

c = 5.2105 c = 5.2256

c = 24.9958 c = 26.6020 c = 21.7569 c = 21.7478

c = 15.5884 c = 15.5170

c = 10.5851 c = 10.5640 c = 10.5770 c = 10.0763 c = 10.1800 c = 10.0990 c = 10.0930

c = 5.7770 c = 5.7739

c = 9.8221 c = 9.7520

c = 5.7305 c = 5.7307

c = 8.4821 c = 8.50 c = 4.9468 c = 4.9470

alloys.

These additional elements, spanning much of the periodic table, are of great scientifical and technological significance for developing Mgbased alloys with favorable compositions and desired properties [15,16,24,28,34]. In particular, Al and Zn can improve both strength and ductility, Ca can enhance the creep resistance of Mg-Al alloys by replacing the detrimental Mg17Al12 phase with more stable Laves phases, Sn can improve the ductility in the bulk forming process, while the rare elements Y and Gd can improve both structure and property. The present investigations will provide great insight into understanding the atomic structures and physicochemical properties of Mg-based

2. Computational methodology The crystallographic information of binary Mg-X (X = Ag, Al, Ba, Ca, Gd, Sn, Y and Zn) intermetallics studied in this work were listed in Table 1 [35]. These Mg-X intermetallics included the MgX phase with simple structure, i.e. 2 atoms per primitive cell, and the Mg23X 6 phase with complicated structure, i.e. 116 atoms per primitive cell. All of the theoretical calculations were performed using the Vienne Ab initio 120

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showed the variation trend of the mass density (ρ) versus the soluteconcentration (i.e. mole-fraction of X) of these Mg-X intermetallics. Except for Mg-Ca alloys, the mass density of these Mg-X intermetallics increased as the X-concentration (at. %). This can be understood due to the fact that the mass density of Ag (10505.2 kg/m3), Gd (7915.26 kg/ m3), Zn (7136.86 kg/m3), Sn (5764.72 kg/m3), Y (4477.84 kg/m3), Ba (3607.39 kg/m3) and Al (2698.85 kg/m3) is larger than that of the matrix Mg (1736.67 kg/m3), whereas the mass density of Ca (1530.62 kg/m3) is smaller. Except for Mg-Zn and Mg-Ba alloys, the mass density of the other Mg-X alloys changed almost linearly as the Xcontent. In particular, the correlation between the X-content (c) and the mass dendity (ρ) was ρ = 8688.59c + 1663.05 for Mg-Ag alloys, ρ = 960.98c + 1713.26 for Mg-Al alloys, ρ = −219.43c + 1763.44 for MgCa alloys, for Mg-Gd alloys, ρ = 6010.71c + 2170.31 ρ = 3937.39c + 1917.98 for Mg-Sn alloys and ρ = 2693.83c + 1914.72 for Mg-Y alloys.

Simulation Package (VASP) based on density functional theory (DFT) [36,37]. The interaction between ions and valence electrons was modeled using the projector-augmented wave (PAW) [38]. The exchange and correlation interaction was described by the generalized gradient approximation (GGA) with the Perdew-Wang (PW91) parameterization [39]. The pseudopotentials employed in this work treated two valence electrons for magnesium (Mg 3s2), three for aluminum (Al 3s23p1), eleven for silver (Ag 4d105s1), four for tin (Sn 5s25p2), ten for barium (Ba 5s25p66s2), two for calcium (Ca 4s2), ten for gadolinium (Gd 4f86s2), three for yttrium (Y 4d15s2) and twelve for zinc (Zn 3d104s2). After convergence tests, a plane wave cutoff energy of 420 eV was used for Mg-Ag, Mg-Y, Mg-Gd and Mg-Zn alloys, while 450 eV for Mg-Al, MgBa, Mg-Ca and Mg-Sn alloys. Brillouin zone integrations were modeled by a Monkhorst-Pack k-points mesh [40], and the k-points separation in the Brillouin zone of reciprocal space was set as 0.01 Å−1 for each unitcell. All structures were fully relaxed with respect to volume as well as cell-internal and external coordinates. The total energy was converged to 5 × 10−7 eV/atom with respect to electronic, ionic and unitcell degrees of freedom. Besides, the calculation of dynamic matrix and its Fourier transform was performed using the density functional perturbation theory (DFPT) [41]. For the DFPT calculations, the 2 × 2 × 2 supercell was used for these Mg-X intermetallics. The interatomic force constants (IFCs) were acquired via the VASP and PHONOPY codes [36,42]. To ensure the computational accuracy, benchmark calculations were performed for the MgAg alloy phase. The lattice parameters after optimization (a = 3.3306 Å) for bulk MgAg phase agreed well with previously reported values (a = 3.3302 Å) [35], confirming the reliability of the computational scheme.

3.1.2. Atomic cluster structural characteristics Because of the structural heritability, the local atomic structures (i.e. the short range order-SRO) of crystalline derivatives are similar to that of the complex metallic alloys counterparts, such as quasicrystals, amorphous alloys and high entropy alloys [44,45]. Based on the “cluster-plus-glue-atom” model with cluster formula of [cluster](glue atom)x, the principal atomic clusters deduced from relevant intermetallics can be used to reflect the local SRO structural characteristics of the corresponding complex metallic alloys [46,47]. These characteristic principal clusters can further be used to design and optimize the composition of the complex metallic alloys based on relevant electronic rule and electrochemical potential equilibrium principle [48–50]. Accordingly, the characteristic principal clusters of these binary Mg-X intermetallics were determined from various primitive clusters centered by atoms with different occupation site in the unitcell based on the structural symmetry. In the “cluster-plus-glue-atom” model, the interatomic interaction inside the principal cluster is stronger than other parts, and the IFCs can reflect the magnitude of interatomic interaction. Accordingly, a central force field model method was developed to determine the cluster structure. Depending on this method, it is clear that for a given alloy phase, those atoms with the maximum IFCs act as the central atoms of the cluster, those atoms with the minimum IFCs serve as the glue atoms of the model, and those atoms with IFCs ranging between the maximum and the maximum values act as either the shell atoms of the cluster or the glue atoms of the “cluster-plus-glue-atom” model [51]. Accordingly, the principal atomic clusters representing the local atomic structural characteristics of an alloy phase can be determined with combination of the atomclose-packing principle [52]. The principal atomic clusters of MgAg, MgAg3, MgAl2, Mg17Al12, Mg2Ba, Mg17Ba2, Mg23Ba6, Mg2Ca, Mg2Gd, MgGd, Mg2Sn, Mg2Y, Mg24Y5, MgY, Mg2Zn11 and MgZn2 intermetallics were determined as the CN14 Mg7Ag8 cluster, CN12 Ag9Mg4 cluster, CN12 Mg3Al10 cluster, CN11 Al4Mg8 cluster, CN12 BaMg12 cluster, CN10 MgMg10 cluster, CN13 BaMg13 cluster, CN12 Mg7Ca6 cluster, CN12 Mg7Gd6 cluster, CN14 Gd7Mg8 cluster, CN8 SnMg8 cluster, CN8 Mg7Y2 cluster, CN9 YMg9 cluster, CN14 Y7Mg8, CN12 ZnZn12 and CN12 Zn7Mg6 cluster, as shown in Fig. 2. The horizontal axis denoted the type of atoms with different occupation site in the unitcell based on the symmetry, and the vertical axis denoted the corresponding IFCs for each type of atoms. Accordingly, their respective cluster formulas were denoted as [MgAg8Mg6](Mg), [Ag-Ag8Mg4](Ag)3, [Mg-Al10Mg2](Mg)2, [Al-Al3Mg8] (Al8Mg9), [Ba-Mg12](Ba5), [Mg-Mg10](Ba2Mg6), [Ba-Mg13](Ba5Mg10), [Mg-Mg6Ca6](Mg5), [Mg-Mg6Gd6](Mg5), [Gd-Mg8Gd6](Gd), [Sn-Mg8] (Sn3), [Mg-Y2Mg6](Y3/2), [Y-Mg9](Y4Mg15), [Y-Mg8Y6](Y), [Zn-Zn12] (Mg4Zn9) and [Zn-Zn6Mg6](Zn5). It is worth stressing that the first atom in the cluster represented the central atom. Taken the cluster formula: [Mg-Ag8Mg6](Mg) of MgAg phase as an example, the first part “Mg” denoted the central atom of the principal cluster, the second part

3. Results and discussion 3.1. Atomic structural information 3.1.1. Crystallographic structural parameters The crystal structure of the binary Mg-X intermetallics (including MgAg, MgAg3, MgAl2, Mg17Al12, Mg23Al30, Mg17Ba2, Mg23Ba6, Mg2Ba, Mg2Ca, Mg2Gd, Mg3Gd, MgGd, Mg2Sn, Mg24Y5, Mg2Y, Mg2Zn11 and MgZn2) was optimized firstly, together with that of the corresponding solvent Mg and solute X (X = Ag, Al, Ba, Ca, Gd, Sn, Y and Zn). Table 1 listed the lattice parameters and mass densities of these Mg-X alloys after optimization, along with those available values known from literature. The results indicated that the calculated values coincide well with those via experiment and/or theoretical studies [30,35,43]. Fig. 1

Fig. 1. The mass density versus the X-content (at. %) of the phases in binary Mg-X (X = Ag, Al, Ba, Ca, Gd, Sn, Y and Zn) alloy systems.

121

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Fig. 2. The principal atomic clusters of the intermetallic phases in binary Mg-X (X = Ag, Al, Ba, Ca, Gd, Sn, Y and Zn) alloy systems, determined from various basic primitive clusters via the central force field model [51]. The horizontal axis denoted the type of atoms in the unitcell of the intermetallics according to the symmetry, and the vertical axis denoted the corresponding IFCs of these atoms. Mg X Mg atoms and “b” X atoms in the equilibrium state, Esolid and Esolid were the total energy of one Mg atom and one X atom with its equilibrium lattice parameters, respectively. Given that the pressure effect on the condensed matter can be ignored and the calculations were performed at 0 K without entropy contribution, the formation energy was equal to the formation enthalpy [25]. A negative value of formation energy indicated that the formation process of the corresponding phase is an exothermic reaction [27]. Meanwhile, the more negative of the formation energy is, the stronger alloying ability and the more stable the phase is [31,54]. According to Eq. (1), the formation energy of these Mg-X intermetallics was calculated and the results were listed in Table 2, together with other values from literature. Fig. 3 showed the according stoichiometry-related formation energy of these Mg-X intermetallics. It is worth stressing that those stable phases at 0 K in each Mg-X systems were used to define the ground state convex hull in Fig. 3. The results indicated that MgAl2 and Mg2Gd intermetallics have formation energies above each of the corresponding convex hull in Fig. 3 by 2.073 kJ/mol atom and 1.389 kJ/mol atom, respectively, signifying that they were metastable phases at ambient environment and can further be decomposed into other stable phases. It has been generally accepted that those phases with more negative formation energy usually possess larger elastic moduli. For these Mg-X intermetallics, however, there was no direct one-to-one correspondence between the one with lower formation energy forming the ground-state convex hull

“Ag8Mg6” denoted the shell atoms (i.e. the surrounding atoms or coordination atoms of the principal cluster), and the third part “Mg” denoted the glue atoms of the “cluster-plus-glue-atom” model. The according results indicated that the coordination number (CN) of the characteristic principal clusters in these binary Mg-X intermetallics varied from 8 to 16, and most were 12 and 14 atomic cluster. Based on these binary Mg-X atomic cluster information, the local SRO structural characteristics of multicomponents Mg-based complex alloys can be understood, since the local atomic structures of complex metallic alloys inherit from their intermetallics counterparts [51,53]. 3.2. Phase stability and mechanical property 3.2.1. Thermodynamic stability To investigate the thermodynamic stability and alloying ability of these binary Mg-X alloys, the formation energy was calculated firstly based on the total energy of these intermetallics and the corresponding pure metals in their ground state. The formation energy of MgaXb phase was defined as the total energy difference between the MgaXb phase and the linear combination of the pure Mg and pure X at equilibrium state [43]: Mg Xb

Eforma

Mg Xb

= [Etotala Mg Xb

where Etotala

Mg X − (aEsolid + bEsolid ) ]/(a + b).

(1)

was the total energy of the MgaXb unitcell containing “a” 122

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Table 2 The formation energy and binding energy of the Mg-X (X = Ag, Al, Ba, Ca, Gd, Sn, Y, Zn) alloys, calculated in this work and known from the literature. Phase

Formation energy (kJ/mol atom)

Binding energy (eV/ atom)

Reference

Mg MgAg MgAg3 Ag Mg17Al12 MgAl2 Mg23Al30 Al

– −26.77 −18.28 – −3.24 −0.23 −2.92 – – −6.30 −6.76 −6.60 −7.55 −8.02 −7.50 −8.82 −9.00 −8.50 – −12.01 −12.85 −12.14 −12.30 – −7.54 −8.55 −8.91 – −19.24 −21.30 – −5.18 −8.24 −9.95 – −7.47 −13.95 –

1.49 2.35 2.55 2.65 2.39 2.89 2.71 3.58 3.56 1.60 – – 1.66 – – 1.72 – – 1.91 1.75 – – 1.75 1.90 2.57 2.33 3.08 4.48 2.43 2.24 3.71 2.05 2.56 3.07 4.45 1.25 1.38 1.11

This work This work This work This work This work This work This work This work [59] This work [43] [31] This work [43] [31] This work [43] [31] This work This work [43] [31] [30] This work This work This work This work This work This work [30] This work This work This work This work This work This work This work This work

Mg17Ba2

Mg23Ba6

Mg2Ba

Ba Mg2Ca

Ca Mg2Gd Mg3Gd MgGd Gd Mg2Sn Sn Mg24Y5 Mg2Y MgY Y Mg2Zn11 MgZn2 Zn

Fig. 4. Calculated binding energy of the phases in binary Mg-X (X = Ag, Al, Ba, Ca, Gd, Sn, Y and Zn) alloy systems.

separating the metallic crystals into the individual neutral free atoms at infinite separation, was another equilibrium thermodynamic parameter to measure the phase stability [26]. It was the total energy difference between the isolated atom and the crystal unitcell. Hence, the binding energy of MgaXb phase can be obtained via: Mg Xb

Ecoh a

Mg Xb

Mg X = (aEatom + bEatom − Ecrysta

)/(a + b).

(2)

Mg X where Eatom was the energy of one Mg atom, Eatom was the energy of one Mg X X atom, and Ecrysta b was the total energy of the MgaXb unitcell containing “a” Mg atoms and “b” X atoms in its ground state. Under this definition, a larger value of binding energy signifies a more stable phase and vice versa. Accordingly, the binding energy of these Mg-X intermetallics was obtained and the results were listed in Table 2. Fig. 4 showed the correlation between binding energy and X-content of these Mg-X intermetallics. Except for Mg-Zn alloys, the binding energy of the other Mg-X intermetallics increased as the X-content. A larger binding energy signified a higher stability of the crystal structure and a higher strength of metallic interactions between different elements [27]. Accordingly, it was deduced that the structural stability and metallic interaction of the Mg-Zn intermetallics decreased as the Zn-content, while that of the other Mg-X (X = Ag, Al, Ba, Ca, Gd, Sn and Y) intermetallics increased as the X-content.

3.2.2. Mechanical stability The mechanical stability of a phase can be predicted using the Born criterion, as formulated based on the corresponding single-crystal elastic constants (Cij) [55]. In this context, the elastic constants of these Mg-X intermetallics were calculated and listed in Table 3, together with those available values from the literature [43,56]. The single-crystal elastic constants were obtained by computing the total energy density as a function of suitable strains, and the total energies of the relevant phases have been calculated by imposing appropriate strains up to ± 1.25% at 0.25% interval. Based on the single-crystal elastic constants, the mechanical stability of these Mg-X intermetallics could be predicted accordingly. Among these Mg-X intermetallics, MgAl2 belongs to the tetragonal structure, MgAg, MgAg3, Mg17Al12, Mg23Ba6, Mg2Gd, MgGd, Mg3Gd, Mg2Sn, Mg24Y5, MgY and Mg2Zn11 belong to the cubic structure, while Mg2Ba, Mg17Ba2, Mg2Ca, Mg2Y and MgZn2 belong to the hexagonal structure. The mechanical stability criteria of the tetragonal structure [55] was given as:

Fig. 3. Calculated formation energy of the phases in binary Mg-X (X = Ag, Al, Ba, Ca, Gd, Sn, Y and Zn) alloy systems.

and that with the maxima bulk modulus. Furthermore, the negative formation energy implied that all of these Mg-X intermetallics were thermodynamically stable phases. Binding energy, which was defined as the required energy for 123

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Table 3 Calculated single-crystal elastic constants of the Mg-X (X = Ag, Al, Ba, Ca, Gd, Sn, Y, Zn) alloys, together with other available values from the literature. Phase

Single-crystal elastic constants (GPa)

Mg

C11 = 52.89 C11 = 58.1 C11 = 63.5 C11 = 87.36 C11 = 104.73 C11 = 123.11 C11 = 115.9 C11 = 107.03 C11 = 89.61 C11 = 94.59 C11 = 101.1 C11 = 39.60 C11 = 33.1 C11 = 39.8 C11 = 41.3 C11 = 40.0 C11 = 41.5 C11 = 58.37 C11 = 72.1 C11 = 11.18 C11 = 12.3 C11 = 60.06 C11 = 53.7 C11 = 59.5 C11 = 22.72 C11 = 22.1 C11 = 56.62 C11 = 55.5 C11 = 56.11 C11 = 75.79 C11 = 67.6 C11 = 72.97 C11 = 69.8 C11 = 56.18 C11 = 49.4 C11 = 74.44 C11 = 78.57 C11 = 73.9 C11 = 53.98 C11 = 51.8 C11 = 75.71 C11 = 78.0 C11 = 138.48 C11 = 97.05 C11 = 161.06 C11 = 159.5

MgAg MgAg3 Ag MgAl2 Mg17Al12 Al Mg2Ba

Mg23Ba6

Mg17Ba2 Ba Mg2Ca

Ca Mg2Gd MgGd Mg3Gd Gd Mg2Sn Sn Mg2Y Mg24Y5 MgY Y Mg2Zn11 MgZn2 Zn

C12 = 26.04 C12 = 27.6 C12 = 24.9 C12 = 54.82 C12 = 71.23 C12 = 80.85 C12 = 85.1 C12 = 36.07 C12 = 29.16 C12 = 64.66 C12 = 61.0 C12 = 13.78 C12 = 20.7 C12 = 14.5 C12 = 15.6 C12 = 15.2 C12 = 15.9 C12 = 18.29 C12 = 10.2 C12 = 7.98 C12 = 7.6 C12 = 17.71 C12 = 22.9 C12 = 17.8 C12 = 15.70 C12 = 15.3 C12 = 34.15 C12 = 34.86 C12 = 33.01 C12 = 23.25 C12 = 19.7 C12 = 25.09 C12 = 25.9 C12 = 27.10 C12 = 46.4 C12 = 28.74 C12 = 19.09 C12 = 22.1 C12 = 35.69 C12 = 35.8 C12 = 23.54 C12 = 24.8 C12 = 37.61 C12 = 66.43 C12 = 51.69 C12 = 56.0

C11 > 0, C33 > 0, C44 > 0, C66 > 0, C11 − C12 > 0, C11 > 0, C33 − 2C13 > 0, 2(C11 + C12) + C33 + 4C13 > 0.

Reference C13 = 23.91 C13 = 21.6 C13 = 20.0 C44 = 51.1 C44 = 52.25 C44 = 53.04 C44 = 42.1 C13 = 36.25 C44 = 20.58 C44 = 27.46 C44 = 25.4 C13 = 10.72 C13 = 9.5 C13 = 9.8 C44 = 16.8 C44 = 16.1 C44 = 17.2 C13 = 11.80 C13 = 11.6 C44 = 10.08 C44 = 10.5 C13 = 13.24 C13 = 10.1 C13 = 12.6 C44 = 14.86 C44 = 13.2 C44 = 25.5 C44 = 42.62 C44 = 41.42 C13 = 19.48 C13 = 24.9 C44 = 31.01 C44 = 31.1 C44 = 26.01 C44 = 15.4 C13 = 24.7 C44 = 16.15 C44 = 18.0 C44 = 39.36 C44 = 37.3 C13 = 21.66 C13 = 22.7 C44 = 36.68 C13 = 35.11 C13 = 66.04 C13 = 51.8

C33 = 97.66

C44 = 44.51

C33 = 45.94 C33 = 41.0 C33 = 46.2

C44 = 11.69 C44 = 11.0 C44 = 11.0

C33 = 61.86 C33 = 61.3

C44 = 18.07 C44 = 22.7

C33 = 65.53 C33 = 66.8 C33 = 66.0

C44 = 15.91 C44 = 14.6 C44 = 17.4

C33 = 83.57 C33 = 76.5

C44 = 21.94 C44 = 19.6

C33 = 79.33

C44 = 18.91

C33 = 85.59 C33 = 82.4

C44 = 24.73 C44 = 26.4

C33 = 135.4 C33 = 58.36 C33 = 57.0

C44 = 17.35 C44 = 26.32 C44 = 23.2

C66 = 33.86

This work [56] [29] This work This work This work [56] This work This work This work [56] This work [43] [30] This work [43] [30] This work [43] This work [56] This work [43] [30] This work [56] This work This work This work This work [56] This work [30] This work [56] This work This work [30] This work [30] This work [56] This work This work This work [56]

3.3. The physicochemical properties 3.3.1. The mechanical property To achieve better understanding on the mechanical properties of these Mg-X intermetallics, the elastic moduli including shear moduli (G), bulk moduli (K) and Young's moduli (E), together with Poisson's ratio (u) were evaluated by Voigt, Reuss and Hill (VRH) approximations based on the Cij values [57]. The formulas corresponding to G, K and E with respect to the VRH approximations were:

(4)

As shown in Table 3, the elastic constants of MgAg, MgAg3, Mg17Al12, Mg23Ba6, Mg2Gd, MgGd, Mg3Gd, Mg2Sn, Mg24Y5, MgY and Mg2Zn11 intermetallics satisfied the mechanical stability requirements according to Eq. (4), signifying that they were mechanically stable. The mechanical stability criteria of the hexagonal structure [55] was given as: 2 C44 > 0, C11 > C12 , (C11 + C12)C33 > 2C13 .

C44 = 17.36 C44 = 14.2 C44 = 19.3

Furthermore, it is worth stressing that although MgAl2 and Mg2Gd were predicted to be metastable phases at ambient environment, the calculated single-crystal elastic constants at 0 K also satisfied the mechanical stability criteria.

(3)

The results listed in Table 3 indicated that all the values of elastic constants for MgAl2 satisfied the mechanical stability restrictions according to Eq. (3), confirming that MgAl2 phase was mechanically stable. The mechanical stability criteria of the cubic structure [55] was given as:

C11 > 0, C44 > 0, C11 > C12 , C11 + 2C12 > 0.

C33 = 54.70 C33 = 64.7 C33 = 66

GVRH =

(5)

GV + GR K + KR 9∗KVRH , KVRH = V , EVRH = 2 2 1 + 3KVRH / GVRH

(6)

where GV, GR and KV,R were:

The elastic constants in Table 3 showed that these Mg-X intermetallics with the hexagonal structure meet the mechanical stability criteria of Eq. (5), indicating that Mg2Ba, Mg17Ba2, Mg2Ca, Mg2Y and MgZn2 intermetallics were mechanically stable phases. Accordingly, all of the Mg-X intermetallics studied here were mechanically stable.

GV =

5∗ (c11 − c12) ∗c44 c11 − c12 + 3∗c44 c + 2∗c12 , GR = , KV , R = 11 5 c44 + 3∗ (c11 − c12) 3 (7)

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Table 4 Polycrystalline shear moduli (G), bulk moduli (K), Young's moduli (E), Poisson ratio (u), G/K ratio and hardness (H) for the Mg-X (X = Ag, Al, Ba, Ca, Gd, Sn, Y, Zn) alloys. Phase

G (GPa)

K (GPa)

E (GPa)

u

G/K

H (GPa)

Reference

Mg

15.30 – 18.5 32.35 33.16 36.67 – 37.75 24.01 – 21.52 – 13.12 9.8 13.0 20.19 23.3 4.96 – 19.68 17.2 20.6 8.40 – 18.35 24.31 24.92 25.33 – 27.96 27.4 20.59 – 21.93 20.67 21.1 22.11 25.5 26.37 – 41.74 20.82 27.39 –

34.46 34.7 35.83 65.67 82.39 94.93 91.3 58.71 49.31 – 74.63 74.3 21.72 20.7 21.5 29.15 30.4 9.05 8.3 30.45 28.9 30.1 18.04 17.4 41.64 41.74 40.71 39.95 38.5 41.05 40.5 36.79 47.2 42.72 38.92 39.3 41.78 41.1 41.15 40.8 71.23 66.96 69.83 68.4

40.22 – 47.4 83.36 87.72 97.46 – 93.17 61.98 – 58.91 – 33.66 26.1 32.5 49.20 56.6 12.58 – 48.57 44.1 50.3 21.81 – 48.01 61.08 62.08 62.73 – 68.36 67.1 52.07 – 56.19 52.69 53.8 56.39 63.5 65.18 – 104.77 56.58 72.66 –

0.2766 – 0.28 0.2884 0.3226 0.3289 – 0.2352 0.2905 – 0.3685 – 0.3041 0.2901 – 0.2186

0.4440 – 0.5163 0.4926 0.4025 0.3863 – 0.6430 0.4870 – 0.2884 – 0.6040 0.49 0.6 0.6926

0.2682 – 0.2341 0.2457 – 0.2985 – 0.3079 0.2561 0.2458 0.2383 – 0.2225 – 0.2641 – 0.2808 0.2743 – 0.2751 – 0.2360 – 0.2549 0.3592 0.3266 –

0.5482 – 0.6464 0.61 0.68 0.4655 – 0.4407 0.5825 0.6120 0.6341 – 0.6811 0.6757 0.5597 – 0.5134 0.5312 0.5376 0.5292 0.6211 0.6407 – 0.5860 0.3109 0.3922 –

– – – 6.30 5.05 – – 9.42 5.22 4.35 ± 0.3 – – 4.72 – – 6.91 – – – 6.47 – – – – 3.97 6.48 6.97 – – 8.45 – – – 5.27 5.30 – 5.48 – – – 8.96 2.84 – –

This work [56] [29] This work This work This work [56] This work This work [2] This work [56] This work [43] [30] This work [43] This work [56] This work [43] [30] This work [56] This work This work This work This work [56] This work [30] This work [56] This work This work [30] This work [30] This work [56] This work This work This work [56]

MgAg MgAg3 Ag MgAl2 Mg17Al12 Al Mg2Ba

Mg17Ba2 Ba Mg2Ca

Ca Mg2Gd MgGd Mg3Gd Gd Mg2Sn Sn Mg2Y Mg24Y5 MgY Y Mg2Zn11 MgZn2 Zn

can be predicted from their elastic moduli, and those materials with larger elastic moduli usually have higher hardness [60,61]. Based on Chen's model [60], the correlation between elastic moduli (G and K) and hardness (H) can be written as:

Accordingly, the elastic moduli of these Mg-X intermetallics were obtained and listed in Table 4, together with those available values in literature. Fig. 5 (a) and (b) showed the variation of G and E with the Xcontent (at. %) of these Mg-X intermetallics, respectively. The results indicated that the intermetallics with the largest G and E values were MgAl2 (G = 37.75 GPa, E = 93.17 GPa), MgAg3 (G = 33.16 GPa, E = 87.72 GPa), Mg17Ba2 (G = 20.47 GPa, E = 50.18 GPa), Mg2Zn11 (G = 41.74 GPa, E = 104.77 GPa), MgGd (G = 24.31 GPa, E = 61.08 GPa), and MgY (G = 22.11 GPa, E = 56.39 GPa) for different Mg-X (X = Al, Ag, Ba, Zn, Gd and Y) alloys, respectively. It is well-known that Young's moduli E can be used as an effective parameter to measure the stiffness of materials, and a larger E value usually signifies a stiffer phase [58]. In this context, it was predicted that MgAl2, MgAg3, Mg17Ba2, Mg2Zn11, MgGd and MgY were the most stiff phases for each of the Mg-X (X = Al, Ag, Ba, Zn, Gd and Y) alloy systems. Although there was certain connection between the bulk moduli and the mass density of materials [59], the two physical parameters of the intermetallics in these Mg-X systems did no strictly follow the one-to-one corresponding correlation, as shown in Figs. 5(c) and 6(a). Bulk moduli K measures the deformation behavior of materials under hydropressure conditions [54]. Fig. 6(b) showed the correlation between bulk moduli and atomic volume of these Mg-X intermerallics, from which we can see that the bulk moduli decreased as the atomic volume. It has been generally accepted that the hardness of materials

H ≈ 1.887∗ (G3/ K 2)0.585.

(8)

Accordingly, the hardness of these Mg-X intermetallic phases was determined and shown in Table 4 and Fig. 5(d). The results indicated that MgAl2, MgAg3, Mg17Ba2, Mg2Zn11, MgGd and MgY were with the highest hardenss values among each Mg-X (X = Al, Ag, Ba, Zn, Gd and Y) alloy system. This confirmed that those intermetallics with the largest elastic moduli have the highest hardness among these Mg-X intermetallic phases, which agreed with previous predictions [54,59]. Taking G as the resistance to plastic deformation and K as the resistance to fracture, Pugh defined the G/K ratio as a criterion to evaluate the brittleness and toughness of phases [62]. The critical value of the G/K ratio was 0.57, i.e. phases with the G/K ratio larger than 0.57 were considered brittle, otherwise they were ductile. The effectiveness of this criterion has been confirmed for the intermetallics in different alloy systems [29,30,43]. Accordingly, the G/K ratio of these Mg-X intermetallics was obtained and shown in Table 4 and Fig. 7(a). The results indicated that MgAg, MgAg3, Mg17Al12, Mg2Gd, Mg2Y, Mg24Y5, MgY, and MgZn2 intermetallics have the G/K ratio lower than 0.57, signifying that they were ductile; while values of the G/K ratio for 125

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Fig. 5. Calculated elastic moduli including shear moduli (a), Young's moduli (b), bulk moduli (c), and hardness (d) versus the X-content (X = Ag, Al, Ba, Ca, Gd, Sn, Y and Zn) of the binary Mg-X intermetallic phases.

bulk properties in real materials. As another physical parameter to measure the ductility of materials [63], Poisson ratio (u) of these Mg-X intermetallics was calculated and listed in Table 4. Fig. 7(b) showed the correlation between G/K and u of these binary Mg-X intermetallics. The results indicated that these two ratios were exactly linear-dependent for the binary Mg-X intermetallics studied here, and the corresponding correlation can be formulated as:

MgAl2, Mg2Ba, Mg17Ba2, Mg2Ca, MgGd, Mg3Gd, Mg2Sn, and Mg2Zn11 intermetallics were larger than 0.57, indicating that they were brittle. It is worth stressing that the application scope of the brittleness and toughness in present discussion focused on the Mg-X (X = Al, Ag, Ba, Zn, Gd and Y) intermetallics. In practical cases, the brittleness and toughness of microscopic bulk properties are dominated by various factors like dislocation mobility. In this respect, future work is still required to investigate the brittleness and toughness of macroscopic

G / K = −2.733∗u + 1.284.

(9)

Fig. 6. Variation trend of the mass density (a), and the atomic volume (b) with the bulk moduli of the binary Mg-X (X = Ag, Al, Ba, Ca, Gd, Sn, Y and Zn) intermetallic phases.

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Fig. 7. Distribution of Pugh ratio (a), and correlation between Poisson ratio and Pugh ratio (b) of the binary Mg-X (X = Ag, Al, Ba, Ca, Gd, Sn, Y and Zn) intermetallic phases.

(deep valley near EF) resides between them. The structural stability of an alloy phase can be estimated from location of the Fermi surface in the DOS curve [65]. In general, the EF located at the valley in the bonding region implied that the phase has good stability, and the EF that lied on the valley in the antibonding states denoted a metastable phase [66]. As shown in Fig. 8, the EF lied to the left of the pseudogap (i.e. the bonding region) for most of the Mg-X intermetallics studied here, indicating they were the structurally stable phases. While the EF of MgAl2 lied to the right of the pseudogap (i.e. the antibonding region) in the DOS curve, indicating MgAl2 is a metastable phase and would decompose into other stable phases. These results coincided well with those predicted based on the thermodynamic analysis.

It has been generally accepted that materials with good plasticity were characterized by higher Poisson ratio. In this context, MgAg3 and MgZn2 phases possessed the best plasticity among these Mg-X intermerallics because of their relatively larger Poisson ratio of 0.3226 and 0.3529, respectively. Moreover, this result also revealed the correlation between G/K ratio and u for isotropic materials, as reported by Greaves [63].

3.3.2. The electronic property To investigate the electronic properties of these Mg-X intermetallics, the band structures and density of states (DOS) were obtained on the basis of their respective optimized structures, the calculated results of which were shown in Fig. 8. The red dash-dot line of zero-point energy implied the Fermi energy level (EF), which was defined as the highest energy level occupied by the valence electrons at 0 K. The band structures represented the energy of points along those high symmetrical directions of these Mg-X intermetallics. Fig. 8 showed that the valence band overlaps the conduction band at the Fermi surface for all of these Mg-X intermetallics, indicating they exhibit the conductive behavior. Meanwhile, the non-zero feature at the Fermi surface in the DOS signified that these Mg-X intermetallics presented the metallic conductivity. Furthermore, the bonding electron numbers per atom below the Fermi surface were 0.275, 0.273, 0.247, 0.129, 0.439, 0.178, 0.907, 0.532, 0.746, 3.385, 3.182, 5.826, 0.787, 1.150, 1.122, 0.175, 0.239 and 0.347 for MgAg, MgAg3, Mg17Al12, Mg23Al30, MgAl2, Mg23Ba6, Mg2Ba, Mg17Ba2, Mg2Ca, Mg2Gd, Mg3Gd, MgGd, Mg24Y5, Mg2Y, MgY, Mg2Sn, Mg2Zn11 and MgZn2, respectively. In general, the larger the number of bonding electrons indicate the stronger the interaction between charges and the better structural stability of the phase [64]. In this context, these results also reflected the structural stability of these Mg-X intermetallics, which agreed with that predicted by the binding energy in section 3.2.1. The interatomic bonding of materials will change from metallic to covalent characteristics as the increase of the distance from the nearest peak to zero energy symbolized bands in the DOS curve [65]. As shown in Fig. 8, these peaks were close to each other for these Mg-X intermetallics, which also signified the metallicity of these Mg-X alloys. The discrepancy in the stability of these Mg-X intermetallics may be attributed to the variation of bonding electron numbers at low-energy region below the Fermi level. The bonding states of Mg-Ag, Mg-Al, MgBa, Mg-Ca, Mg-Gd, Mg-Y, Mg-Sn and Mg-Zn intermetallics were occupied with the center of the band located at about 0.8 eV, 0.15 eV, 0.32 eV, 1.1 eV, 0.75 eV, 0.48 eV, 1.35 eV and 1.2 eV below the Fermi surface. The regions above the Fermi surface belong to the conduction band. Because of the strong hybridization, the entire DOS curve can be divided into the bonding and antibonding regions, and a pseudogap

4. Conclusion In summary, the atomic cluster structures, phase stability and physicochemical properties of binary Mg-X (X = Ag, Al, Ba, Ca, Gd, Sn, Y, Zn) intermetallics were investigated by performing ab-inito calculations. The optimized lattice parameters of these Mg-X intermetallics agreed well with those reported in literature. Except for Mg-Ba and MgZn intermetallics, the mass density of the other Mg-X intermetallics changed linearly as the X-content. The coordination number of the principal atomic clusters corresponding to these Mg-X intermetallics varied from 8 to 16, and most were 12 and 14. These binary Mg-X characteristic principal clusters can help to understand the local atomic structure of the multicomponents Mg-based complex alloys. The elastic moduli of these Mg-X intermetallics were predicted via VRH approximation based on the calculated elastic constants. For each Mg-X alloy system, MgAl2, MgAg3, Mg17Ba2, Mg2Zn11, MgGd and MgY phases exhibited larger elastic moduli and higher hardness than the others. The ductile and brittle behavior of these Mg-X intermetallics were investigated based on the Poisson ratio and Pugh criterion. Consequently, MgAg, MgAg3, Mg17Al12, Mg2Gd, Mg2Y, Mg24Y5, MgY, and MgZn2 intermetallics were ductile phases, while MgAl2, Mg2Ba, Mg17Ba2, Mg2Ca, MgGd, Mg3Gd, Mg2Sn and Mg2Zn11 intermetallics were brittle phases. The Pugh ratio and Poisson ratio of these Mg-X intermetallics followed a linear correlation of G/K = −2.7331u + 1.284. Furthermore, MgAg3 and MgZn2 had better plasticity than other intermetallics. All of these Mg-X intermetallics were conductors, and thermodynamically and mechanically stable. The present investigation provided important theoretical guidance for directing the development of new Mg-based alloys with favorable composition and desired properties. Acknowledgments This work was financially supported by the National Natural Science 127

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Fig. 8. Calculated band structures and density of states (DOS) for the binary Mg-X (X = Ag, Al, Ba, Ca, Gd, Y, Sn and Zn) intermetallics. The red dash-dot line of zero-point energy implies the Fermi energy level. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

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Foundation of China (51701104), the National Key Research and Development Program of China (2016YFB0301001), the Postdoctoral Science Foundation of China (2017M610884), the Tsinghua University Initiative Scientific Research Program (20151080370) and the UK Royal Society Newton International Fellowship Scheme. The authors acknowledge the National Laboratory for Information Science and Technology in Tsinghua University for access to supercomputing facilities.

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