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Materials Research

Print version ISSN 1516-1439On-line version ISSN 1980-5373

Mat. Res. vol.22 no.2 São Carlos  2019  Epub Feb 18, 2019 


Stability, Mechanical Properties and Anisotropic Elastic Properties of GaxMgy Compounds

LinJing Liua

Liangchong Liana 

Jie Yub  * 

aDepartment of Mechanical and Electrical Engineering, Hunan Biological and Electromechanical Polytechnics, Changsha, 400126, China

bFaculty of Material Science and Engineering, Kunming University of Science and Technology, Kunming, 650093, China


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.

Keywords: Gallium alloy; First-principle calculation; Mechanical properties; Anisotropy

1. 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 cooling1-4, waste heat recovery 5,6, electrical interconnects and contacts7,8, biomedical equipment9, kinetic energy harvesting 10, thermal interface material11,12 or printed electronics13-16. 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 catalysts17. 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 batteries18,19, hydrogen storage20 and medical implants21,22. In Ga-Mg alloys, the effect of secondary phases (Mg5Ga2) and impurities on the localized corrosion mechanism using AFM/ SKPFM is also studied23. 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, Feng26 and Meng27. 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 calculations28.

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.

2. 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) code29-31. 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 zone33. The 4s24p1 and 3s2 are considered as valence electrons configurations for Ga and Mg, respectively. The Brillouin zone is sampled with the Monkhorst-Pack scheme33 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 GaxMgy 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:

EcohGaxMgy=EtotGaxMgyxEisoGayEisoMgx+y (1)

ΔHrGaxMgy=EtotGaxMgyxEbinGayEbinMgx+y (2)

Where, Ecoh(GaxMgy) and ΔHr (GaxMgy) are the cohesive energy and formation enthalpy of GaxMgy per atom, respectively; Etot (GaxMgy) is the total cell energy of GaxMgy phase; Eiso refers to the total energy of an isolated Ga or Mg atom and Ebin is the cohesive energy of crystal of Ga or Mg, respectively. The elastic constants of compounds in GaxMgy 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:

σ1σ2σ3τ4τ5τ6=C11C12C13C14C15C16C22C23C24C25C26C33C34C35C36C44C45C46C55C56C66ε1ε2ε3γ4γ5γ6 (3)

where, Cij is the elastic constant, τi and σi 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 Cij calculations can be greatly reduced.

3. Results and Discussion

3.1 Stability

Fig.1 illustrates the Ga-Mg equilibrium phase diagram35, in this phase diagram, the crystal structures of gallium magnesium including Ga2Mg5, GaMg2, GaMg, Ga2Mg, and Ga5Mg2 are prepared from reference. For Ga2Mg, there are two polymorph structures, orthorhombic and hexagonal36, which are marked as O-Ga2Mg and H-Ga2Mg respectively. The other compounds contain three different types of lattice--tetragonal (Ga5Mg2 and GaMg), orthorhombic (Ga2Mg5) and hexagonal (GaMg2) 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, Ga2Mg5 has the highest formation enthalpy as-0.120 eV per atom, implying that it is less stable than other GaxMgy compounds. With the increase of Ga concentration in GaxMgy, 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 GaxMgy compounds. The calculated values in this work are in consistent with the available experimental data in Ref.37,38Because of different approximation method resulting in calculation accuracy, the values are a little difference with the values obtained by Hui Zhang et al28, and in Ref.28,37,38the 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-Ga2Mg > H-Ga2Mg > Ga5Mg2> GaMg2> Ga2Mg5. In a word, GaMg is the most stable compound among Ga-Mg binary system.

Figure 1 The Ga-Mg equilibrium phase diagram. 

Figure 2 The crystal structures of the Ga-Mg binary system. (a) Ga2Mg5; (b) GaMg2; (c) GaMg; (d) Ga5Mg2; (e) H-Ga2Mg; (f) O-Ga2Mg5

Table 1 The Lattice parameters ( a , b and c ), cohesive energy (Ecoh) and formation enthalpy (ΔHr) of Ga-Mg binary compounds. 

Species Space group Composition at.% Ga Lattice constants (Å) Ecoh ( eV/atom) ΔHr(eV/atom)
a b c
α-Mg P63/mmc 0 3.209 3.209 5.211 -1.858 0
Ga2Mg5 Ibam 28.57 7.004b 13.663b 5.987b -2.504b -0.120b
7.017g 13.708g 6.020g -0.113i
7.036c 13.747c 6.028c -0.119c
GaMg2 P-62c 33.33 7.730b 7.730b 6.976b -2.605b -0.133b
7.794f 7.794f 6.893f -0.121i
7.805c 6.941c -0.130c
GaMg I41/aZ 50.00 10.523b 10.523b 5.539b -2.941b -0.162b
10.530e 5.530e -0.135i
10.690c 5.555c -0.142c
H-Ga2Mg P63/mmc 66.67 4.370b 4.370b 6.692b -3.229b -0.144b
4.343h 4.343h 6.982h
O-Ga2Mg Pbam 66.67 6.813b 16.259b 4.040b -3.234b -0.148b
6.802a 16.346a 4.111a -0.118i
6.868c 16.457c 4.139c -0.120c
Ga5Mg2 I4/mmm 71.43 8.568b 8.568b 7.125b -3.313b -0.140b
8.627d 7.111d -0.103i
8.725c 8.725c 7.178c -0.112c
α-Ga Cmca 100 3.706 4 4.8891 7.971 -3.701 0

aExp. In Ref.39

bThis work

cCal. In Ref.28

dExp. In Ref.40

eExp. In Ref.41

fExp. In Ref.42

gExp. In Ref.43

hExp. In Ref.44

iExp. In Ref.37

Figure 3 Calculated enthalpies of formation plotted as a function of composition for the Ga-Mg system. 

3.2 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 GaxMgy 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 as46:

Tetragonal system:

C11>0,C33>0,C44>0,C66>0,C11C12>0,C11+C33C13>0,2C11+C12+C33+4C13>0, (4)

Orthorhombic system:

C11+C12+C33+2C12+2C23>0,C11+C22>2C12,C22+C33>2C33,C11+C33>2C13,Cii>0i=16 (5)

Hexagonal system:

C11>0,C44C22>0,C11+C12C33>2C132 (6)

Table 2 The calculated elastic constants (Cij, in GPa) of Ga-Mg system. 

α-Mg Ga2Mg5 GaMg2 GaMg H-Ga2Mg O-Ga2Mg Ga5Mg2 α-Ga
C11 80.9 93.3 88.2 90.1 136.9 69.1 114.6 125.6
C22 - 80.7 - - - 102.8 - 93.8
C33 101.7 72.9 71.9 96.2 133.0 78.0 137.1 127.7
C44 47.2 37.9 25.6 36.5 29.5 23.9 40.1 12.2
C55 - 30.2 - - - 22.1 - 21.22
C66 - 23.9 - 21.7 - 29.3 34.6 16.8
C12 20.1 26.6 31.2 30.0 34.0 37.2 47.6 115.8
C13 1.1 27.2 31.1 35.7 16.8 56.6 20.4 26.1
C23 - 31.2 - - - 34.8 - 43.3
C16 4.3 0

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-Ga2Mg own the largest C 11 value as 136.9 GPa, which shows that H-Ga2Mg has high incompressibility under uniaxial stress along the crystallographic a axis (ε 11). O-Ga2Mg has the largest C 22 value as 102.8 GPa and Ga5Mg2 has the largest C 33 value as 137.1 GPa, which shows that O-Ga2Mg and Ga5Mg2 have high incompressibility under uniaxial stress along the crystallographic b axis (ε 22) and c axis (ε 33). C44, C55 and C66 represent the shearing modulus on (100), (010) and (001) crystal plane, respectively. Ga5Mg2 and O-Ga2Mg show the largest C 44 40.1 GPa and smallest C 44 23.9 GPa, the shearing strength of O-Ga2Mg at (100) and (010) planes is weaker than (001) plane, while Ga2Mg5 shows the largest shearing strength at (100) plane.

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 expressions47-50:

BVRH=12BV+BR (7)

GVRH=12GV+GR (8)



Here, BV , BR and BVRH are the bulk modulus calculated by Voigt, Reuss and Voigt-Reuss-Hill approximation method, respectively. GV , GR and GVRH 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 expressions51:

λ=σE1+σ12σ (11)

μ=E21+σ (12)

The related calculated values of GaxMgy binary compounds are showed in Table 3, and Fig.4 illustrates the variations of elastic parameters of GaxMgy compounds. The greater B values of Ga5Mg2 and H-Ga2Mg than other compounds; indicate that Ga5Mg2 and H-Ga2Mg 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-Ga2Mg. Meanwhile, the G and E of Ga5Mg2 and H-Ga2Mg are also larger than other compounds. The higher shear modulus is, the higher hardness of the compounds52. Because the intrinsic hardness is proportional to the shear modulus, the high hardness may correspond to Ga5Mg2 and H-Ga2Mg. 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 GaxMgy binary compounds are lower than the critical value as 1.75, the compounds are brittle. O-Ga2Mg has the largest B/G value as 2.71, while H-Ga2Mg has the lowest B/G value as 1.41 among the Ga-Mg binary compounds. The Vickers hardness (Hv) 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:

Hv=2k2G0.5853 (13)

Where k denotes the Pugh’ s modulus ratio (k = G/B). The hardness of H-Ga2Mg is 9.05 GPa which is the largest in Ga-Mg system, while the value for O-Ga2Mg 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-Ga2Mg, however, the minimum shear modulus value for O-Ga2Mg, 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.

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 (Hv, in GPa) of Ga-Mg system. 

Species α-Mg Ga2Mg5 GaMg2 GaMg H-Ga2Mg O-Ga2Mg Ga5Mg2 α-Ga
BV 34.2 46.3 48.3 53.3 60.2 56.3 60.3 59.5
BR 34.2 46.1 47.9 53.1 59.9 56.1 60.3 58.6
B 34.2 46.2 48.1 53.2 60.1 56.2 60.3 59.1
GV 41.0 29.2 26.3 30.6 44.7 23.1 41.5 26.9
GR 39.0 28.2 26.1 29.1 40.7 18.3 39.8 20.9
G 40.0 28.7 26.2 29.8 42.7 20.7 40.6 23.9
E 86.3 71.3 66.5 75.3 103.6 55.3 99.5 63.2
B/G 0.86 1.61 1.84 1.78 1.41 2.71 1.49 2.47
σ 0.08 0.24 0.27 0.26 0.21 0.34 0.23 0.32
λ 7.53 27.1 30.6 33.3 31.6 42.4 33.2 43.2
µ 40 28.7 26.2 29.8 42.7 20.7 40.6 23.9
Hv 17.8 5.16 3.64 4.39 9.05 0.66 7.99 1.44

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.  

3.3 Anisotropy of elastic properties

The anisotropy of mechanical properties is very important in the application of materials. The occurrence of micro-cracks 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 (AU), the percent anisotropy index (AB and AG) and the shear anisotropy factors ( A1 , A2 and A3), are obtained by the following equations54,55:

A1=4c44c11+c332c13for100plane (14)

A2=4c55c22+c332c23for100plane (15)

A3=4c66c11+c222c12for100plane (16)

AU=5GVGR+BVBR60 (17)



Where BV, BR, GV and GR 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 AB value for Ga5Mg2 is zero and GaMg2 has the largest value as 0.41% in GaxMgy compounds, which indicating that the anisotropy in bulk modulus of GaMg2 is the strongest; But the index AB is not enough to identify the anisotropy, the index AU 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 AU value of GaMg2 in Ga-Mg binary compounds, indicates the elastic modulus of GaMg2 is not strongly dependent on the different orientations, and it is confirmed by the following AG value. In addition, besides α-Ga having the largest AG and AU as 12.6% and 1.45 respectively, the largest AU value for O-Ga2Mg in binary compounds, suggest that O-Ga2Mg has the highest elastic anisotropy among the six Ga-Mg binary compounds. The anisotropy of the shear modulus is determined by AG, A1, A2 and A3, and A1, A2 and A3 represent the anisotropy of the shear modulus in different crystal plane. H-Ga2Mg has the weakest anisotropy of the shear modulus in (100) plane and (010) plane, and A1 and A2 are 0.5, 0.5 respectively. The A3 values of H-Ga2Mg GaMg2 and GaMg are1.0, 1.0 and 0.72 respectively. H-Ga2Mg and GaMg2 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 GaxMgy compounds.

Hexagonal crystal:

1B=S11+S12+S13S11+S12S13S33l32 (20)

1E=1l32S11+l34S33+l321l322S13+S44 (21)

Orthorhombic crystal:

1B=S11+S12+S13l12+S12+S22+S23l22+S13+S23+S33l32 (22)

1E=S11l14+S22l24+S33l34+2S12+S66l12l22+2S13+S55l12l32+2S23+S44l22l32 (23)

Where Sij are the elastic compliance constants, and l1 , l2 and l3 are the directional cosines. For tetragonal crystal, the above equations (22) and (23) are also suitable by assuming S11=S22, S44=S55, and S13=S23. After substituting the relationships of the direction cosines in spherical coordinates with respect to θ and φ (l1 =sinθ cosφ, l2=sinθ sinφ, l3 =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) planes show details about the anisotropic properties of bulk modulus. It is obvious that the bulk modulus of H-Ga2Mg and O-Ga2Mg have a strong directional dependence. The bulk modulus of O-Ga2Mg 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 GaMg2, 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. GaMg2 has the relatively strong anisotropy and the results are in good agreement with AB values. From the projections on the (001), (100) and (110) planes, we find that anisotropy of bulk modulus of GaMg2 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.

Table 4 The calculated universal anisotropic index (AU), percent anisotropy (AB and AG) and shear anisotropic factors (A1, A2 and A3) of Ga-Mg system.  

Species A1 A2 A3 AB AG AU
α-Mg 1.05 1.05 1 0. 06% 2.62% 0.2704
Ga2Mg5 1.35 1.33 0.79 0. 26% 1.82% 0.191251
GaMg2 1.04 1.04 1.00 0. 41% 0.38% 0.046665
GaMg 1.27 1.27 0.72 0. 19% 2.51% 0.261498
H- Ga2Mg 0.50 0.50 1.00 0. 25% 4.68% 0.496409
O-Ga2Mg 2.81 0.79 1.20 0. 18% 11.59% 1.31504
Ga5Mg2 0.76 0.76 1.03 0 2.09% 0.214562
α-Ga 0.24 0.63 0.40 0.76% 12.6% 1.450765

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 by56,57

Figure 5 (a)-(c) The (001), (100) and (110) planar projections of the bulk modulus of GaxMgy compounds, respectively.  

Figure 6 (a)-(c) The (001), (100) and (110) planar projections of the Young’s modulus of GaxMgy compounds, respectively.  

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-Ga2Mg shows the strongly anisotropy of mechanical properties in Ga-Mg binary compounds. For GaMg, the anisotropy of Young's 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-Ga2Mg on ( 001) plane is weaker than that on (100) plane and the projection on (100) plane is strongly polarized, H-Ga2Mg shows the maximum Young's modulus along [010] and the value of Young's modulus i n[100] direction are also larger than other compounds on (001) plane. In addition, GaMg2 and H-Ga2Mg 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-Ga2Mg shows the minimum Young’s modulus along the [001] direction. It can also be found that Ga2Mg5 and O-Ga2Mg show the weakest anisotropy for Young’s modulus on (001) and (010) planes, respectively.

3.4 Anisotropic sound velocity

The average sound velocity νm is calculated by58,59:

υm=132υl3+1υt313 (24)

νl and νt 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 obtained60.

υl=B+43Gρ (25)

υt=Gρ (26)

Table 5 shows the calculated acoustic velocities of Ga-Mg binary compounds. Ga2Mg5 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 directions54. 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 GaxMgy compounds and the acoustic velocities in the principal directions can be simply expressed as61,62:

Tetragonal crystal:

100=010100υl=C11ρ;001υt1=C44ρ;010υt2=C66ρ (27)

001001υl=C33ρ;100υt1=010υt2=C66ρ (28)

110110υl=C11+C12+2C662ρ;001υt1=C44ρ;110vt2=C11C122ρ (29)

Orthorhombic crystal:

100100υl=C11ρ;010υt1=C66ρ;001vt2=C552ρ (30)

010010υl=C22ρ;100υt1=C66ρ;001vt2=C44ρ (31)

001001υl=C33ρ;010υt1=C55ρ;010vt2=C44ρ (32)

Hexagonal crystal:

100100υl=C11C12ρ;010υt1=C11ρ;001vt2=C44ρ (33)

001001υl=C33ρ;100υt1=C44ρ;010vt2=C44ρ (34)

Where νt1 is the first transverse mode and νt2 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 ijin different direction represents different sound velocity. Thus, the more modulus of the direction, the higher speed of the sound. For example, the 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.

Table 5 The theoretical density (ρ, g/cm3), longitudinal sound velocity (νl, m/s), shear sound velocity (νt, m/s), average sound velocity (νm, m/s) of GaxMgy

Species ρ νl νt νm
α-Mg 1.74 7092.7 4794.6 5890.5
Ga2Mg5 3.03 5279.8 3077.6 3271.5
GaMg2 3.27 5039.1 2830.6 3689.7
GaMg 4.07 4778.5 2705.9 3519.7
H-Ga2Mg 4.92 4877.2 2946.0 3761.9
O-Ga2Mg 6.41 4152.0 2063.5 2766.2
Ga5Mg2 5.05 4760.3 2835.4 3636.0
α-Ga 6.41 3767.1 1930.9 2572.1

Table 6 The anisotropic sound velocities of tetragonal Ga5Mg2 and GaMg compounds. The unit of velocity (ν) is km/s. 

Direction [100] [001] [110]
[100]νl [001] νt1 [010] νt2 [001]νl [100] νt1 [010] νt2 [110]νl [001] νt1 [110] νt2
Ga5Mg2 4.764 2.818 2.618 5.210 2.618 2.618 4.788 2.818 2.576
GaMg 4.705 2.995 2.309 4.862 2.309 2.309 4.482 2.995 2.717

Table 7 The anisotropic sound velocities of orthorhombic O-Ga2Mg, Ga2Mg5 and hexagonal GaMg2, H-Ga2Mg compounds. The unit of velocity (ν) is km/s. 

Direction [100] [010] [001]
[100]νl [010] νt1 [001] νt2 [010]νl [100] νt1 [001] νt2 [001]νl [100] νt1 [010] νt2
O-Ga2Mg 3.770 2.455 2.132 4.596 2.455 2.216 4.005 2.132 2.216
Ga2Mg5 5.549 2.809 3.159 5.162 2.809 3.535 4.908 3.159 3.535
GaMg2 2.951 5.193 2.798 - - - - - - 4.689 2.798 2.798
H-Ga2Mg 3.233 5.275 2.449 - - - - - - 5.199 2.449 2.449

4. 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 GaxMgy 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. Ga5Mg2 and H-Ga2Mg 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-Ga2Mg to 0.34 for O-Ga2Mg, the lowest values of H-Ga2Mg imply that it is harder than other compounds. The Young's modulus of H-Ga2Mg shows the strongly anisotropy of mechanical properties and that of GaMg2 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 C11 , C22 and C33 determine the longitudinal sound velocities along [100], [010] and [001] directions, respectively, and C44, C55 and C66 correspond to the transverse modes. The results are helpful for the experiment design and application of Ga-Mg binary compounds in the future.

5. Acknowledgements

Project supported by Natural Science Foundation of Hunan province, China (2017JJ5040).

6. References

1 Ma KQ, Liu J. Heat-driven liquid metal cooling device for the thermal management of a computer chip. Journal of Physics D: Applied Physics. 2007;40(15):4722-4729. [ Links ]

2 Deng YG, Liu J. A liquid metal cooling system for the thermal management of high power LEDs. International Communications in Heat and Mass Transfer. 2010;37(7):788-791. [ Links ]

3 Deng Y, Liu J. Hybrid liquid metal-water cooling system for heat dissipation of high power density microdevices. Heat and Mass Transfer. 2010;46(11-12):1327-1334 [ Links ]

4 Vetrovec J, Litt AS, Copeland DA, Junghans J, Durkee R. Liquid metal heat sink for high-power laser diodes. In: Proceedings of SPIE - The International Society for Optical Engineering; 2013 Feb 2-7; San Francisco, CA, USA. 8605:8605E-1-7. [ Links ]

5 Dai D, Zhou YX, Liu J. Liquid metal based thermoelectric generation system for waste heat recovery. Renewable Energy. 2011;36(12):3530-3536. [ Links ]

6 Liu J, Li HY, inventors. A Thermal Energy Harvesting Device and its Fabrication Method. China Patent 201210241718.0. 2012. [ Links ]

7 Kim HJ, Son C, Ziaie B. A multiaxial stretchable interconnect using liquid-alloy-filled elastomeric microchannels. Applied Physics Letters. 2008;92(1):11904. [ Links ]

8 Cao A, Yuen P, Lin L. Microrelays With Bidirectional Electrothermal Electromagnetic Actuators and Liquid Metal Wetted Contacts. Journal of Microelectromechanical Systems. 2007;16(3):700-708. [ Links ]

9 Knoblauch M, Hibberd JM, Gray JC, van Bel AJ. A galinstan expansion femtosyringe for microinjection of eukaryotic organelles and prokaryotes. Nature Biotechnology. 1999;17(9):906-909. [ Links ]

10 Liu J, inventor. Piezoelectric Thin Film Electricity Generator and its Fabrication Method. China Patent 2012103225845. 2012. [ Links ]

11 Gao YX, Liu J. Gallium-based thermal interface material with high compliance and wettability. Applied Physics A. 2012;107(3):701-708. [ Links ]

12 Zhang Q, Liu J. Nano liquid metal as an emerging functional material in energy management, conversion and storage. Nano Energy. 2013;2(5):863-872. [ Links ]

13 Zhang Q, Zheng Y, Liu J. Direct writing of electronics based on alloy and metal (DREAM) ink: A newly emerging area and its impact on energy, environment and health sciences. Frontiers in Energy. 2012;6(4):311-340. [ Links ]

14 Gao Y, Li H, Liu J. Direct Writing of Flexible Electronics through Room Temperature Liquid Metal Ink. PLoS One. 2012;7(9):e45485. [ Links ]

15 Jeong SH, Hagman A, Hjort K, Jobs M, Sundqvist J, Wu Z. Liquid alloy printing of microfluidic stretchable electronics. Lab on a Chip. 2012;12(22):4657-4664. [ Links ]

16 Zheng Y, He Z, Gao Y, Liu J. Direct Desktop Printed-Circuits-on-Paper Flexible Electronics. Scientific Reports. 2013;3:1786-1792. [ Links ]

17 Taccardi N, Grabau M, Debuschewitz J, Distaso M, Brandl M, Hock R, et al. Gallium-rich Pd-Ga phases as supported liquid metal catalysts. Nature Chemistry. 2017;9:862-867. [ Links ]

18 Feng Y, Wang R, Peng C. Influence of Ga and In on microstructure and electrochemical properties of Mg anodes. Transactions of Nonferrous Metals Society of China. 2013;23(9):2650-2656. [ Links ]

19 Zhao J, Yu K, Hu Y, Li S, Tan X, Chen F, et al. Discharge behavior of Mg-4wt%Ga-2 wt%Hg alloy as anode for seawater activated battery. Electrochimica Acta. 2011;56(24):8224-8231. [ Links ]

20 Wu D, Ouyang L, Wu C, Wang H, Liu J, Sun L, et al. Phase transition and hydrogen storage properties of Mg-Ga alloy. Journal of Alloys and Compounds. 2015;642:180-184. [ Links ]

21 Xin Y, Hu T, Chu PK. Influence of test solutions on in vitro studies of biomedical magnesium alloys. Journal of Electrochemical Society. 2010;157(7):238-243. [ Links ]

22 Virtanen S. Biodegradable Mg and Mg alloys: Corrosion and biocompatibility. Materials Science and Engineering: B. 2011;176(20):1600-1608. [ Links ]

23 Mohedano M, Blawert C, Yasakau KA, Arrabal R, Matykina E, Mingo B, et al. Characterization and corrosion behavior of binary Mg-Ga alloys. Materials Characterization. 2017;128:85-89. [ Links ]

24 Predel B, Stein DW. Thermodynamic Investigation of the Gallium-Magnesium System. Journal of the Less Common Metals.1969;18(3):203-213. [ Links ]

25 Nayeb-Hashemi AA, Clark JB. The Ga-Mg (Gallium-Magnesium) system. Bulletin of Alloy Phase Diagrams. 1985;6(5):434-439. [ Links ]

26 Feng Y, Wang RC, Liu HS, Jin ZP. Thermodynamic reassessment of the magnesium-gallium system. Journal of Alloys and Compounds. 2009;486(1-2):581-585. [ Links ]

27 Meng FG, Wang J, Rong MH, Liu LB, Jin ZP. Thermodynamic assessment of Mg-Ga binary system. Transactions of Nonferrous Metals Society of China. 2010;20(3):450-457. [ Links ]

28 Zhang H, Shang S, Saal JE, Saengdeejing A, Wang Y, Chen LQ, et al. Enthalpies of formation of magnesium compounds from first-principles calculations. Intermetallics. 2009;17(11):878-885. [ Links ]

29 Hohenberg P, Kohn W. Inhomogeneous Electron Gas. Physical Review. 1964;136(3B):864-871. [ Links ]

30 Segall MD, Lindan PJD, Probert MJ, Pickard CJ, Hasnip PJ, Clark SJ, et al. First-principles simulation: ideas, illustrations and the CASTEP code. Journal of Physics: Condensed Matter. 2002;14(11):2717-2744. [ Links ]

31 Kresse G, Joubert D. From ultrasoft pseudopotentials to the projector augmented-wave method. Physical Review B. 1999;59(3):1758-1775. [ Links ]

32 Perdew JP, Burke K, Ernzerhof M. Generalized Gradient Approximation Made Simple. Physical Review Letters. 1996;77(18):3865-3868. [ Links ]

33 Monkhorst HJ, Pack JD. Special points for Brillouin-zone integrations. Physical Review B. 1976;13(12):5188-5192. [ Links ]

34 Feng J, Xiao B, Chen JC, Du Y, Yu J, Zhou R. Stability, thermal and mechanical properties of PtxAly compounds. Materials & Design. 2011;32(6):3231-3239. [ Links ]

35 Okamoto H. Ga-Mg (Gallium-Magnesium). Journal of Phase Equilibria and Diffusion. 2013;34(2):148. [ Links ]

36 Ellner M, Gödecke T, Duddek G, Predel B. Strukturelle und konstitutionelle Untersuchungen im galliumreichen Teil des Systems Magnesium-Gallium. Zeitschrift für anorganische und allgemeine Chemie. 1980;463(1):170-178. [ Links ]

37 Rossini FD. Selected values of chemical thermodynamic properties. Washington: United States National Bureau of Standards Publishing; 1965. [ Links ]

38 Nayeb-Hashemi AA, Clark JB. Phase diagrams of binary magnesium alloys. Materials Park: ASM International;1988. [ Links ]

39 Smith GS, Mucker KF, Johnson Q, Wood DH. The crystal structure of Ga2Mg. Acta Crystallographica. 1969;B25:549-553. [ Links ]

40 Smith GS, Mucker KF, Johnson Q, Wood DH. Crystal structure of Ga5Mg2. Acta Crystallographica. 1969;B25:554-557. [ Links ]

41 Schubert K, Gauzzi F, Frank K. Kristallstruktur Einiger Mg-B-3-phasen. Zeitschrift für Metallkunde. 1963;54(7):422-429. [ Links ]

42 Frank K, Schubert K. Kristallstruktur von Mg2Ga und Mg2Ti. Journal of the Less-Common Metals. 1970;20(3):215-221. [ Links ]

43 Schubert K, Frank K, Gohle R, Maldonado A, Meissner HG, Raman A, Rossteutscher W. Naturwissenschafte. 1963;50(2):41. [ Links ]

44 von Ellner M, Goedecke T, Duddek G, Predel B. Structure and Constitutional Studies in the Gallium-Rich Part of the Magnesium-Gallium System. Zeitschrift für Anorganische und Allgemeine Chemie. 1980;463:170-178. [ Links ]

45 Han J, Wang C, Liu X, Wang Y, Liu ZK, Jiang J. Atomic-Level Mechanisms of Nucleation of Pure Liquid Metals during Rapid Cooling. Chemphyschem. 2015;16(18):3916-3927. [ Links ]

46 Wu ZJ, Zhao EJ, Xiang HP, Hao XF, Liu XJ, Meng J. Crystal structures and elastic properties of superhard IrN2 and IrN3 from first principles. Physical Review B. 2007;76(5-1):054115. [ Links ]

47 Reuss A, Angew Z. Calculation of the flow limits of mixed crystals on the basis of the plasticity of monocrystals. Zeitschrift fur Angewandte Mathematik und Mechanik. 1929;9:49-58. [ Links ]

48 Hill R. The elastic behaviour of a crystalline aggregate. Proceedings of the Physical Society, Section A.1952;65(5):349-354. [ Links ]

49 Chong XY, Jiang YH, Zhou R, Feng J. Electronic structures mechanical and thermal properties of V-C binary compounds. RSC Advances. 2014;4(85):44959-44971. [ Links ]

50 Chong XY, Jiang YH, Zhou R, Feng J. The effects of ordered carbon vacancies on stability and thermo-mechanical properties of V8C7 compared with VC. Scientific Reports. 2016;6:34007-34016. [ Links ]

51 Gao F. Hardness estimation of complex oxide materials. Physical Review B. 2004;69(9):094113-094113. [ Links ]

52 Jiang X, Zhao J, Jiang X. Correlation between hardness and elastic moduli of the covalent crystals. Computational Materials Science. 2011;50(7):2287-2290. [ Links ]

53 Chen XQ, Niu H, Li D, Li Y. Modeling hardness of polycrystalline materials and bulk metallic glasses. Intermetallics. 2011;19(9):1275-1281. [ Links ]

54 Feng J, Xiao B, Zhou R, Pan W, Clarke DR. Anisotropic elastic and thermal properties of the double perovskite slab-rock salt layer Ln2SrAl2O7 (Ln= La, Nd, Sm, Eu, Gd or Dy) natural superlattice structure. Acta Materialia. 2012;60(8):3380-3392. [ Links ]

55 Xiao B, Feng J, Zhou CT, Jiang YH. Mechanical properties and chemical bonding characteristics of Cr7C3 type multicomponent carbides. Journal of Applied Physics. 2011;109(2):023507. [ Links ]

56 Panda KB, Chandran KSR. First principles determination of elastic constants and chemical bonding of titanium boride (TiB) on the basis of density functional theory. Acta Materialia. 2006;54(6):1641-1657. [ Links ]

57 Chong XY, Jiang YH, Zhou R, Feng J. Elastic properties and electronic structures of CrxBy as superhard compounds. Journal of Alloys and Compounds. 2014;610:684-694. [ Links ]

58 Kittel C, McEuen P. Introduction to Solid State Physics. New York: Wiley Publishing; 1986. [ Links ]

59 Chong XY, Jiang YH, Feng J. Mechanical properties, electronic structure and alkali-ion diffusion of Eldfellite-type AFe(SO4)2 (A=LiA=Li, Na, K) as potential cathode materials comparing with LiFePO4. Journal of Micromechanics and Molecular Physics. 2017;2(1):1750002. [ Links ]

60 Chong XY, Jiang YH, Zhou R, Feng J. First principles study the stability, mechanical and electronic properties of manganese carbides. Computational Materials Science. 2014;87:19-25. [ Links ]

61 Hearmon RFS. An Introduction to Applied Anisotropic Elasticity. London: Oxford University Press; 1961. [ Links ]

62 Chong XY, Jiang YH, Zhou R, Feng J. Stability, chemical bonding behavior, elastic properties and lattice thermal conductivity of molybdenum and tungsten borides under hydrostatic pressure. Ceramics International. 2015;42(2):2117-2132. [ Links ]

Received: September 19, 2018; Revised: November 19, 2018; Accepted: November 20, 2018


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