Stability, Mechanical Properties and Anisotropic Elastic Properties of GaxMgy Compounds

The stability, mechanical properties and anisotropic properties of sound velocities of Ga2Mg5, GaMg2, GaMg, O-Ga2Mg, H-Ga2Mg and Ga5Mg2 are investigated systematically by the first-principles calculation. The cohesive energy and formation enthalpy are obtained and used to estimate the stability of the Ga-Mg binary compounds. GaMg compound is the most stable and has the lowest formation enthalpy as -0.162eV/atom of those GaxMgy compounds. The elastic constants of single crystal, hardness, bulk, shear, Young's modulus and Poisson's ratio of the polycrystalline crystal are obtained and used to estimate the mechanical properties. Ga5Mg2 and H-Ga2Mg have the lager bulk, shear and Young’s modulus and corresponding B/G is low. H-Ga2Mg is harder than the other compounds from the results of Poisson’s ratio. The anisotropic mechanical properties are discussed using the anisotropic index, two-dimensional planar projections on different planes of the bulk and Young's modulus. The Young's modulus of H-Ga2Mg shows the strongly anisotropy of mechanical properties and GaMg2 has the weakest anisotropy among all the compounds.


Introduction
Pure gallium is a soft metal with a low temperature melting point of 29.8 ºC, in recent years, gallium is mainly used in liquid metal alloys with a variety of applications, including chip cooling [1][2][3][4] , waste heat recovery 5,6 , electrical interconnects and contacts 7,8 , biomedical equipment 9 , kinetic energy harvesting 10 , thermal interface material 11,12 or printed electronics [13][14][15][16] .The eutectic gallium-indium binary alloy (EGaIn 75% gallium and 25% indium) and gallium-indium-tin ternary alloy (Galinstan 68.5% gallium, 21.5% indium, 10% tin) are the most common used non-toxic liquid metals today and also gallium rich Pd-Ga phases as supported liquid metal catalysts 17 .Due to the particular characteristics of gallium, the addition of other elements to the gallium alloy will open up new possibilities for design and applications, such as magnesium.Ga-Mg alloy for sacrificial anodes in seawater batteries 18,19 , hydrogen storage 20 and medical implants 21,22 .In Ga-Mg alloys, the effect of secondary phases (Mg 5 Ga 2 ) and impurities on the localized corrosion mechanism using AFM/ SKPFM is also studied 23 .However, the fundamental number of studies on Ga-Mg alloys is rather limited.There is some very early work regarding the phase diagram Ga-Mg summarized by Beck 24 , which was updated by Nayeb-Hashemi et al. 25 , Feng 26 and Meng 27 .Thermodynamics is the key component of Ga-Mg alloy, one of critically important thermodynamic data is the enthalpy of formation of the compounds, which can be obtained by first-principles calculations 28 .
To better control the design of a material with the desired properties, the thermodynamics and mechanical properties of gallium alloys are very necessary.In this work, the cohesive energy, formation enthalpy, mechanical properties and anisotropic elastic properties for all the Ga-Mg binary compounds based on Ga-Mg phase diagram are investigated by first principle calculations.The thermodynamic database and mechanical properties for the Ga-Mg system is helpful for the design of Gallium alloys.

Methods and Details
In this work, the whole calculations are carried out by first principle calculations which are based on density functional theory (DFT) as implemented in Cambridge sequential total energy package (CASTEP) code [29][30][31] .The crystal structures are optimized by a plane wave expansion method.By comparing ultrasoft and Norm-conserving pseudo potentials (NCPPs) , NCPPs are used to indicate the interactions between ionic core and valence electrons.The exchange correction energy is calculated by the generalized gradient-corrected (GGA) developed by Perdew, Burke and Ernzerhof (PBE) 32 .Monkhorst-Pack scheme is used for k-point sampling in the first irreducible Brillouin zone 33 .The 4s 2 4p 1 and 3s 2 are considered as valence electrons configurations for Ga and Mg, respectively.The Brillouin zone is sampled with the Monkhorst-Pack scheme 33 and the K point mesh is selected as 5×5×5 for all structures.The maximum energy cut off value of 450.0 eV is used for plane wave expansion in reciprocal space.The total energy changes during the optimization process are reduced to 1×10 -6 eV and the forces acting on distinct atom are converged to 0.05 eV/Å.In order to estimate the thermodynamic stability of Ga x Mg y compounds, the cohesive energy and formation enthalpy were calculated in this paper.The following expressions (Eq.( 1) and Eq.( 2) ) were estimated as following equations: (1) Where, E coh (Ga x Mg y ) and ΔH r (Ga x Mg y ) are the cohesive energy and formation enthalpy of Ga x Mg y per atom, respectively; E tot (Ga x Mg y ) is the total cell energy of Ga x Mg y phase; E iso refers to the total energy of an isolated Ga or Mg atom and E bin is the cohesive energy of crystal of Ga or Mg, respectively.The elastic constants of compounds in Ga x Mg y were calculated by stress-strain method, within namely Hooker's law.Several different strain modes were imposed on the crystal structure, and the Cauchy stress tensor for each strain mode was evaluated.Finally, the related elastic constants were identified as the coefficients in strain-stress relations as shown in Eq. (3) 34 : (3) where, C ij is the elastic constant, τ i and σ i are the shear stress and normal stress, respectively.The total number of independent elastic constants is determined by the symmetry of the crystal.In high symmetry system, the indispensable different strain patterns for the C ij calculations can be greatly reduced.

Stability
Fig. 1 illustrates the Ga-Mg equilibrium phase diagram 35 , in this phase diagram, the crystal structures of gallium magnesium including Ga 2 Mg 5 , GaMg 2 , GaMg, Ga 2 Mg, and Ga 5 Mg 2 are prepared from reference.For Ga 2 Mg, there are two polymorph structures, orthorhombic and hexagonal 36 , which are marked as O-Ga 2 Mg and H-Ga 2 Mg respectively.The other compounds contain three different types of lattice--tetragonal (Ga 5 Mg 2 and GaMg), orthorhombic (Ga 2 Mg 5 ) and hexagonal (GaMg 2 ) crystal classes.Fig. 2 shows the crystal structures of the Ga-Mg binary system.The calculated lattice parameters of optimized crystal structures and the described chemical stability of the intermetallic compounds of Ga-Mg binary system by cohesive energy and formation enthalpy are demonstrated in Table 1.Obviously, the crystal parameters of Ga-Mg compounds are in good agreement with other calculated values and experimental results.The calculated results are obtained at 0 K, but the experimental results are measured at room temperature.Moreover, when different exchange-correlation functions are used, the lattice parameters may be underestimated or overestimated.Therefore, the tiny deviation may result from the influence of thermodynamic effect on the crystal structure.The stability of the Ga-Mg binary compounds can be determined by cohesive energies and formation enthalpies.The results calculated by Eq. ( 1) and ( 2) are also tabulated in Table 1, the values of cohesive energy and formation enthalpy are negative.However, the chemical stability of Ga-Mg system compounds is determined by formation enthalpy.The lower the formation enthalpy, the more stable the compound.It can be seen that the formation enthalpy of GaMg (-0.162 eV/atom) is the lowest value, and indicating the most stable phase is GaMg in the Ga-Mg binary compounds.On the other hand, Ga 2 Mg 5 has the highest formation enthalpy as-0.120eV per atom, implying that it is less stable than other Ga x Mg y compounds.With the increase of Ga concentration in Ga x Mg y , the cohesive energy increasing, and the maximum and minimum value is -1.858 eV/atom and -3.701 eV/atom for α-Mg and α-Ga; respectively.Fig. 3 depicts the calculated and previously reported formation enthalpy of Ga x Mg y compounds.The calculated values in this work are in consistent with the available experimental data in Ref. 37,38 Because of different approximation method resulting in calculation accuracy, the values are a little difference with the values obtained by Hui Zhang et al 28 , and in Ref. 28,37,38 the enthalpy of formation is expressed as kJ/mol, we calculated the enthalpy of formation expresses as eV/atom.The stability of these compounds forms the following sequence: GaMg> O-Ga 2 Mg > H-Ga 2 Mg > Ga 5 Mg 2 > GaMg 2 > Ga 2 Mg 5 .In a word, GaMg is the most stable compound among Ga-Mg binary system.

E Ga Mg
x y   Cal.In Ref. 28 d Exp.In Ref. 40 e Exp.In Ref. 41 f Exp.In Ref. 42 g Exp.In Ref. 43 h Exp.In Ref. 44 i Exp.In Ref. 37 Materials Research

Mechanical properties
The response of crystals to external forces is determined by elastic constants, which is of great significance in practical applications.Therefore, it is necessary to study the elastic constants of Ga-Mg binary compounds for the mechanical properties.The elastic constants of Ga x Mg y compounds are determined by Eq. ( 3) based on the generalized Hook's law 45 , and the results are summarized in Table 2.According to Born-Huang's mechanical stability criterions, one condition is the strain energy must be positive for any homogeneous elastic deformation.The mechanical stability criterions can be expressed as 46 : Tetragonal system: (4) Orthorhombic system: (5) Hexagonal system: As shown in Table 2, the values of elastic constants satisfied the above criterions, which imply all the Ga-Mg binary compounds are elastically stable.H-Ga 2 Mg own the largest C 11 value as 136.9 GPa, which shows that H-Ga 2 Mg has high incompressibility under uniaxial stress along the crystallographic a axis (ε 11 The mechanical modulus such as bulk modulus (B), Young's modulus (E) and shear modulus (G) are evaluated by the elastic constants using Viogt -Reuss-Hill (VRH) approximation.The Viogt -Reuss-Hill (VRH) approximation is an average of Viogt and Reuss approximations, namely the upper and lower bounds to the elastic modulus, which provides the estimation for the mechanical properties of poly-crystalline materials from the known elastic constants for single crystal.Yong's modulus (E) and Poisson's ratio(σ) are estimated by following expressions [47][48][49][50] : Here, B V , B R and B VRH are the bulk modulus calculated by Voigt, Reuss and Voigt-Reuss-Hill approximation method, respectively.G V , G R and G VRH are the shear modulus calculated within Voigt, Reuss and Voigt-Reuss-Hill approximation method, respectively.In this paper, we also calculated the first order Lame constant (λ) and the second Lame constant (µ), namely compressibility and shear stiffness, using the following expressions 51 : (11) (12)   The related calculated values of Ga x Mg y binary compounds are showed in Table 3, and Fig. 4 illustrates the variations of elastic parameters of Ga x Mg y compounds.The greater B values of Ga 5 Mg 2 and H-Ga 2 Mg than other compounds; indicate that Ga 5 Mg 2 and H-Ga 2 Mg are the most difficult to be compressed under hydrostatic pressure in the Ga-Mg binary compounds, which is in consistent with the analysis of elastic constants.In addition, the bulk modulus of Ga is significantly larger than Mg.With the increase of Ga content, the bulk modulus of Ga-Mg alloy also increased, except O-Ga 2 Mg.Meanwhile, the G and E of Ga 5 Mg 2 and H-Ga 2 Mg are also larger than other compounds.The higher shear modulus is, the higher hardness of the compounds 52 .Because the intrinsic hardness is proportional to the shear modulus, the high hardness may correspond to Ga 5 Mg 2 and H-Ga 2 Mg.The ratio of B/G is used as an indicator for the ductility or brittleness of the compound; If the B/G values for the Ga x Mg y binary compounds are lower than the critical value as 1.75, the compounds are brittle.O-Ga 2 Mg has the largest B/G value as 2.71, while H-Ga 2 Mg has the lowest B/G value as 1.41 among the Ga-Mg binary compounds.The Vickers hardness (H v ) of Ga-Mg system is predicted by an empirical model which has better results for the anisotropic structures.The model is recently proposed by Chen et al. and expressed as follows 53 : (13)   Where k denotes the Pugh' s modulus ratio (k = G/B).The hardness of H-Ga 2 Mg is 9.05 GPa which is the largest in Ga-Mg system, while the value for O-Ga 2 Mg is 0.66 GPa as the smallest one among the Ga-Mg system.In Fig. 4, the B/G value is multiplied by the factor of 5 and Poisson's ratio is multiplied by the factor of 20 for a better illustration.The shear modulus decreases firstly and then increases when the atom percent of Ga exceed 30%.A sharp peak occurs at Ga 66.7 at% on the curve which presents the maximum shear modulus value for H-Ga 2 Mg, however, the minimum shear modulus value for O-Ga 2 Mg, simultaneously.With the Ga atom content increasing, the variation of shear modulus has the same tendency as the variation of Young's modulus and Vickers hardness.Furthermore, the hardness may be more sensitive to shear modulus than bulk modulus.As shown in Fig. 4, the trend of the B/G value curve is the same as the Poisson's ratio with increasing Ga atom content.However, the trend of Poisson's ratio and Vickers hardness are opposite.

Anisotropy of elastic properties
The anisotropy of mechanical properties is very important in the application of materials.The occurrence of microcracks in materials is often related to the elastic anisotropy.As a potential material, it is important to characterize the anisotropy of the mechanical properties of Ga-Mg compounds.In this work, six number of indices, including the universal anisotropy index (A U ), the percent anisotropy index (A B and Table 3.The bulk modulus (B, in GPa), shear modulus (G, in GPa), Yong's modulus (E, in GPa),poisson's ratio (σ, in GPa), the first order Lame constant (λ, in GPa), the second Lame constant (µ, in GPa) and the Vickers hardness (H v , in GPa) of Ga-Mg system.

Species
A G ) and the shear anisotropy factors ( A 1 , A 2 and A 3 ), are obtained by the following equations 54,55 : Where B V , B R , G V and G R are the bulk and shear modulus estimation within Voigt and Reuss methods, respectively.The values of unity for shear anisotropic factors indicate isotropic for a crystal, while the non-unity values imply anisotropy.The calculated results are shown in Table 4.The A B value for Ga 5 Mg 2 is zero and GaMg 2 has the largest value as 0.41% in Ga x Mg y compounds, which indicating that the anisotropy in bulk modulus of GaMg 2 is the strongest; But the index A B is not enough to identify the anisotropy, the index A U is considered as a better indicator than other indices, which can provide unique and consistent results for the mechanical anisotropic properties of Ga-Mg compounds.Obviously, the lowest A U value of GaMg 2 in Ga-Mg binary compounds, indicates the elastic modulus of GaMg 2 is not strongly dependent on the different orientations, and it is confirmed by the following A G value.In addition, besides α-Ga having the largest A G and A U as 12.6% and 1. Hexagonal crystal: Orthorhombic crystal: Where S ij are the elastic compliance constants, and l 1 , l 2 and l 3 are the directional cosines.For tetragonal crystal, the above equations ( 22) and ( 23) are also suitable by assuming S 11 =S 22 , S 44 =S 55 , and S 13 =S 23 .After substituting the relationships of the direction cosines in spherical coordinates Properties and Anisotropic Elastic Properties of Ga x Mg y Compounds with respect to θ and φ (l 1 =sinθ cosφ, l 2 =sinθ sinφ, l 3 =cosφ) into equations ( 22) and ( 23), we can obtain the projections of surface contour of bulk and Young's modulus shown in Fig. 5 and Fig. 6.From Fig. 5, the projections on the (001), (100) and ( 110 values.From the projections on the (001), ( 100) and ( 110) planes, we find that anisotropy of bulk modulus of GaMg 2 on (001) plane is stronger than that on (100) and (110) plane, but the result is in reverse for GaMg .On the other hand, it is evidence that the bulk modulus of these compounds show weak anisotropy because the planar projections of the gallium magnesium are all close to an ellipsoid.From Fig. 6, The Young's modulus on the (001), (100) and (110) planes show more anisotropic features than the bulk modulus due to the remarkable anisotropic geometry of the projections.Projections deviated from the regular ellipses on the (001), (100) and ( 110) planes indicate the strong anisotropy of Young's modulus for all the compounds.We may infer that the surface profiles of Young's modulus are anisotropic because their shapes deviate from the ideal sphere.In addition, the Young's modulus of H-Ga 2 Mg shows the strongly anisotropy of mechanical properties in Ga-Mg binary compounds.For GaMg, the anisotropy of Young's Table 4.The calculated universal anisotropic index (A U ), percent anisotropy (A B and A G ) and shear anisotropic factors (A 1 , A 2 and A 3 ) of Ga-Mg system.The most straightforward way to describe the elastic anisotropy is to plot the bulk and Young's modulus in two dimensions (2D) as a function of the crystallographic direction.The directional dependence of bulk and Young's modulus is given by 56,57  modulus on (100) plane is weaker than that on (001) plane thus the projection on (001) plane is strongly polarized, and the anisotropy of Young's modulus for H-Ga 2 Mg on ( 001) plane is weaker than that on (100) plane and the projection on (100) plane is strongly polarized, H-Ga 2 Mg shows the maximum Young's modulus along [010]and the value of Young's modulus in [100]direction are also larger than other compounds on (001) plane.In addition, GaMg 2 and H-Ga 2 Mg on the plane (100) are similar to those on the plane (110), which implies the analogous anisotropy of Young's modulus on these planes.Obviously, O-Ga 2 Mg shows the minimum Young's modulus along [001] direction.It can also be found that Ga 2 Mg 5 and O-Ga 2 Mg show the weakest anisotropy for Young's modulus on (001) and (010) planes, respectively.

Anisotropic sound velocity
The average sound velocity ν m is calculated by 58,59 : ν l and ν t are the longitudinal sound velocity and transverse sound velocity, respectively.
The following equations were used to calculate the bulk modulus (B) and shear modulus (G) previously obtained 60 .
(25) (26)   Table 5 shows the calculated acoustic velocities of Ga-Mg binary compounds.Ga 2 Mg 5 has the largest acoustic velocity among Ga-Mg binary compounds because it has the largest shear modulus and lowest density.
The acoustic velocity in a crystal is anisotropic which is determined by the symmetry of the crystal and propagation directions 54 .For example, the pure transverse and longitudinal modes can only be found for [100], [001] and [110] directions in a tetragonal crystal and the sound propagating modes in other directions are the quasi-transverse or quasi-longitudinal waves.In this work, we only consider the pure propagating modes for Ga x Mg y compounds and the acoustic velocities in the principal directions can be simply expressed as 61,62 : Tetragonal crystal: Orthorhombic crystal: Hexagonal crystal: Where ν t1 is the first transverse mode and ν t2 is the second transverse mode.The calculated results are presented in Table 6 and 7.The anisotropy of acoustic velocities also reveals the elastic anisotropy in these crystals.Some anisotropic, including sound velocity in different direction, can be expressed byC ij , that is,C ij in different direction represents different sound velocity.Thus, the more modulus of the direction, the higher speed of the sound.For example, the

Conclusions
In summary, the chemical stability, elastic properties, anisotropy of mechanical properties and anisotropic sound velocity of the Ga-Mg binary compounds have been investigated by first principles calculations.The cohesive energy and formation enthalpy of Ga x Mg y compounds show that the compounds are thermodynamically stable, GaMg is the most stable compound and has the lowest formation enthalpy with -0.1621eV/atom in Ga-Mg binary system, which is in good agreement with the experimental values.Ga 5 Mg 2 and H-Ga 2 Mg have the lager bulk, shear and Young's modulus as 60.3, 40.6 99.5 GPa and 60.1, 42.7, 103.6 GPa, respectively, and corresponding B/G is small.The results of Poisson's ratio varies from 0.21 for H-Ga 2 Mg to 0.34 for O-Ga 2 Mg, the lowest values of H-Ga 2 Mg imply that it is harder than other compounds.The Young's modulus of H-Ga 2 Mg shows the strongly anisotropy of mechanical properties and that of GaMg 2 the weakest anisotropy among all the compounds.Moreover, the hardness of Ga-Mg binary system is evaluated from 0.66 to 9.05 GPa.The results of anisotropic sound velocities showed C 11 , C 22 and C 33 determine the longitudinal sound velocities along [100], [010] and [001] directions, respectively, and C 44 , C 55 and C 66 correspond to the transverse modes.The results are helpful for the experiment design and application of Ga-Mg binary compounds in the future.

a
Department of Mechanical and Electrical Engineering, Hunan Biological and Electromechanical Polytechnics, Changsha, 400126, China b Faculty of Material Science and Engineering, Kunming University of Science and Technology, Kunming, 650093, China

Figure 3 .
Figure 3. Calculated enthalpies of formation plotted as a function of composition for the Ga-Mg system.
45 respectively, the largest A U value for O-Ga 2 Mg in binary compounds, suggest that O-Ga 2 Mg has the highest elastic anisotropy among the six Ga-Mg binary compounds.The anisotropy of the shear modulus is determined by A G , A 1 , A 2 and A 3 , and A 1 , A 2 and A 3 represent the anisotropy of the shear modulus in different crystal plane.H-Ga 2 Mg has the weakest anisotropy of the shear modulus in (100) plane and (010) plane, and A 1 and A 2 are 0.5, 0.5 respectively.The A 3 values of H-Ga 2 Mg GaMg 2 and GaMg are1.0,1.0 and 0.72 respectively.H-Ga 2 Mg and GaMg 2 has the same anisotropy of the shear modulus in (001) plane and GaMg has the lowest anisotropy of the shear modulus in (001) plane among all the Ga x Mg y compounds.

Figure 4 .
Figure 4.The variations of the elastic parameters of Ga-Mg compounds, note that B/G value is magnified by 5 times and σ value is magnified by 20 times to the initial value.

Table 1 .
The Lattice parameters ( a , b and c ), cohesive energy (E coh ) and formation enthalpy (ΔH r ) of Ga-Mg binary compounds.Species Space group Composition at.%Ga Lattice constants (Å) E coh ( eV/ atom) ΔH r (eV/atom) a b c ). O-Ga 2 Mg has the largest C 22 value as 102.8 GPa and Ga 5 Mg 2 has the largest C 33 value as 137.1 GPa, which shows that O-Ga 2 Mg and Ga 5 Mg 2 have high incompressibility under uniaxial stress along the crystallographic b axis (ε 22 ) and c axis (ε 33 ).C 44 , C 55 and C 66 represent the shearing modulus on (100), (010) and (001) crystal plane, respectively.Ga 5 Mg 2 and O-Ga 2 Mg show the largest C 44 40.1 GPa and smallest C 44 23.9 GPa, the shearing strength of O-Ga 2 Mg at (100) and (010) planes is weaker than (001) plane, while Ga 2 Mg 5 shows the largest shearing strength at (100) plane.

Table 2 .
The calculated elastic constants (C ij , in GPa) of Ga-Mg system.
) planes show details about the anisotropic properties of bulk modulus.It is obvious that the bulk modulus of H-Ga 2 Mg and O-Ga 2 Mg have a strong directional dependence.The bulk modulus of O-Ga 2 Mg in the [010] direction is larger than those in the [100] direction and [001] direction on (100) plane, which is in good agreement with the result of calculated elastic constants in which C 22 is much larger than C 11 and C 33 .Meanwhile, For GaMg 2 , the bulk modulus in the [001] direction is smaller than those in the [010] and [100] directions, because the value of C 33 is lower than C 11 and C 22 .GaMg 2 has the relatively strong anisotropy and the results are in good agreement with A B

Table 6 .
The anisotropic sound velocities of tetragonal Ga 5 Mg 2 and GaMg compounds.The unit of velocity (ν) is km/s.

Table 7 .
The anisotropic sound velocities of orthorhombic O-Ga 2 Mg, Ga 2 Mg 5 and hexagonal GaMg 2 , H-Ga 2 Mg compounds.The unit of velocity (ν) is km/s.