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 cooling^{1}^{-}^{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}^{-}^{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.
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) code^{29}^{-}^{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:
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}:
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.
3. Results and Discussion
3.1 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-0.120 eV 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.
Species | Space group | Composition at.% Ga | Lattice constants (Å) | E_{coh} ( eV/atom) | ΔH_{r}(eV/atom) | ||
---|---|---|---|---|---|---|---|
a | b | c | |||||
α-Mg | P63/mmc | 0 | 3.209 | 3.209 | 5.211 | -1.858 | 0 |
Ga_{2}Mg_{5} | 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 | ||||
GaMg_{2} | 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-Ga_{2}Mg | P63/mmc | 66.67 | 4.370b | 4.370b | 6.692b | -3.229b | -0.144b |
4.343h | 4.343h | 6.982h | |||||
O-Ga_{2}Mg | 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 | ||||
Ga_{5}Mg_{2} | 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 |
^{a}Exp. In Ref.^{39}
^{b}This work
^{c}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}
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 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:
Orthorhombic system:
Hexagonal system:
α-Mg | Ga_{2}Mg_{5} | GaMg_{2} | GaMg | H-Ga_{2}Mg | O-Ga_{2}Mg | Ga_{5}Mg_{2} | α-Ga | |
---|---|---|---|---|---|---|---|---|
C_{11} | 80.9 | 93.3 | 88.2 | 90.1 | 136.9 | 69.1 | 114.6 | 125.6 |
C_{22} | - | 80.7 | - | - | - | 102.8 | - | 93.8 |
C_{33} | 101.7 | 72.9 | 71.9 | 96.2 | 133.0 | 78.0 | 137.1 | 127.7 |
C_{44} | 47.2 | 37.9 | 25.6 | 36.5 | 29.5 | 23.9 | 40.1 | 12.2 |
C_{55} | - | 30.2 | - | - | - | 22.1 | - | 21.22 |
C_{66} | - | 23.9 | - | 21.7 | - | 29.3 | 34.6 | 16.8 |
C_{12} | 20.1 | 26.6 | 31.2 | 30.0 | 34.0 | 37.2 | 47.6 | 115.8 |
C_{13} | 1.1 | 27.2 | 31.1 | 35.7 | 16.8 | 56.6 | 20.4 | 26.1 |
C_{23} | - | 31.2 | - | - | - | 34.8 | - | 43.3 |
C_{16} | 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-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}). 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.
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}^{-}^{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}:
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}:
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.
Species | α-Mg | Ga_{2}Mg_{5} | GaMg_{2} | GaMg | H-Ga_{2}Mg | O-Ga_{2}Mg | Ga_{5}Mg_{2} | α-Ga |
---|---|---|---|---|---|---|---|---|
B_{V} | 34.2 | 46.3 | 48.3 | 53.3 | 60.2 | 56.3 | 60.3 | 59.5 |
B_{R} | 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 |
G_{V} | 41.0 | 29.2 | 26.3 | 30.6 | 44.7 | 23.1 | 41.5 | 26.9 |
G_{R} | 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 |
H_{v} | 17.8 | 5.16 | 3.64 | 4.39 | 9.05 | 0.66 | 7.99 | 1.44 |
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 (A_{U}), the percent anisotropy index (A_{B} and 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 1.45 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.
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 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) 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} 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.
Species | A_{1} | A_{2} | A_{3} | A_{B} | A_{G} | A^{U} |
---|---|---|---|---|---|---|
α-Mg | 1.05 | 1.05 | 1 | 0. 06% | 2.62% | 0.2704 |
Ga_{2}Mg_{5} | 1.35 | 1.33 | 0.79 | 0. 26% | 1.82% | 0.191251 |
GaMg_{2} | 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- Ga_{2}Mg | 0.50 | 0.50 | 1.00 | 0. 25% | 4.68% | 0.496409 |
O-Ga_{2}Mg | 2.81 | 0.79 | 1.20 | 0. 18% | 11.59% | 1.31504 |
Ga_{5}Mg_{2} | 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 by^{56}^{,}^{57}
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 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 i n[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 the [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.
3.4 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}.
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 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.
Species | ρ | ν_{l} | ν_{t} | ν_{m} |
---|---|---|---|---|
α-Mg | 1.74 | 7092.7 | 4794.6 | 5890.5 |
Ga_{2}Mg_{5} | 3.03 | 5279.8 | 3077.6 | 3271.5 |
GaMg_{2} | 3.27 | 5039.1 | 2830.6 | 3689.7 |
GaMg | 4.07 | 4778.5 | 2705.9 | 3519.7 |
H-Ga_{2}Mg | 4.92 | 4877.2 | 2946.0 | 3761.9 |
O-Ga_{2}Mg | 6.41 | 4152.0 | 2063.5 | 2766.2 |
Ga_{5}Mg_{2} | 5.05 | 4760.3 | 2835.4 | 3636.0 |
α-Ga | 6.41 | 3767.1 | 1930.9 | 2572.1 |
Direction | [100] | [001] | [110] | ||||||
---|---|---|---|---|---|---|---|---|---|
[100]ν_{l} | [001] ν_{t1} | [010] ν_{t2} | [001]ν_{l} | [100] ν_{t1} | [010] ν_{t2} | [110]ν_{l} | [001] ν_{t1} | [110] ν_{t2} | |
Ga_{5}Mg_{2} | 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 |
Direction | [100] | [010] | [001] | ||||||
---|---|---|---|---|---|---|---|---|---|
[100]ν_{l} | [010] ν_{t1} | [001] ν_{t2} | [010]ν_{l} | [100] ν_{t1} | [001] ν_{t2} | [001]ν_{l} | [100] ν_{t1} | [010] ν_{t2} | |
O-Ga_{2}Mg | 3.770 | 2.455 | 2.132 | 4.596 | 2.455 | 2.216 | 4.005 | 2.132 | 2.216 |
Ga_{2}Mg_{5} | 5.549 | 2.809 | 3.159 | 5.162 | 2.809 | 3.535 | 4.908 | 3.159 | 3.535 |
GaMg_{2} | 2.951 | 5.193 | 2.798 | - - | - - | - - | 4.689 | 2.798 | 2.798 |
H-Ga_{2}Mg | 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 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.