Stability, Mechanical Properties and Anisotropic Elastic Properties of Ga_xMg_y Compounds

Abstract

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 Ga_xMg_y 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 cooling^1-4, waste heat recovery ^5,6, electrical interconnects and contacts^7,8, biomedical equipment⁹, 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 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 batteries^18,19, hydrogen storage²⁰ and medical implants^21,22. 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) 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 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 Ga_xMg_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) ${\mathit{E}}_{\mathit{coh}}\left({\mathit{Ga}}_{\mathit{x}}{\mathit{Mg}}_{\mathit{y}}\right)=\frac{{\mathit{E}}_{\mathit{tot}}\left({\mathit{Ga}}_{\mathit{x}}{\mathit{Mg}}_{\mathit{y}}\right)-{\mathit{xE}}_{\mathit{iso}}\left(\mathit{Ga}\right)-{\mathit{yE}}_{\mathit{iso}}\left(\mathit{Mg}\right)}{\mathit{x}+\mathit{y}}$

(2) $\Delta {\mathit{H}}_{\mathit{r}}\left({\mathit{Ga}}_{\mathit{x}}{\mathit{Mg}}_{\mathit{y}}\right)=\frac{{\mathit{E}}_{\mathit{tot}}\left({\mathit{Ga}}_{\mathit{x}}{\mathit{Mg}}_{\mathit{y}}\right)-{\mathit{xE}}_{\mathit{bin}}\left(\mathit{Ga}\right)-{\mathit{yE}}_{\mathit{bin}}\left(\mathit{Mg}\right)}{\mathit{x}+\mathit{y}}$

Where, E_coh(Ga_xMg_y) and ΔH_r (Ga_xMg_y) are the cohesive energy and formation enthalpy of Ga_xMg_y per atom, respectively; E_tot (Ga_xMg_y) is the total cell energy of Ga_xMg_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_xMg_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) $\left(\begin{array}{c}{\mathit{\sigma }}_1\hfill \\ {\mathit{\sigma }}_2\hfill \\ {\mathit{\sigma }}_3\hfill \\ {\mathit{\tau }}_4\hfill \\ {\mathit{\tau }}_5\hfill \\ {\mathit{\tau }}_6\hfill \end{array}\right)=\left(\begin{array}{c}{\mathit{C}}_{11}{\mathit{C}}_{12}{\mathit{C}}_{13}{\mathit{C}}_{14}{\mathit{C}}_{15}{\mathit{C}}_{16}\hfill \\ {\mathit{C}}_{22}{\mathit{C}}_{23}{\mathit{C}}_{24}{\mathit{C}}_{25}{\mathit{C}}_{26}\hfill \\ {\mathit{C}}_{33}{\mathit{C}}_{34}{\mathit{C}}_{35}{\mathit{C}}_{36}\hfill \\ {\mathit{C}}_{44}{\mathit{C}}_{45}{\mathit{C}}_{46}\hfill \\ {\mathit{C}}_{55}{\mathit{C}}_{56}\hfill \\ {\mathit{C}}_{66}\hfill \end{array}\right)\left(\begin{array}{c}{\epsilon }_1\hfill \\ {\epsilon }_2\hfill \\ {\epsilon }_3\hfill \\ {\mathit{\gamma }}_4\hfill \\ {\mathit{\gamma }}_5\hfill \\ {\mathit{\gamma }}_6\hfill \end{array}\right)$

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 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 Ga_xMg_y compounds. With the increase of Ga concentration in Ga_xMg_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_xMg_y 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 (E_coh) and formation enthalpy (ΔH_r) 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

• ^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

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 Ga_xMg_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 as⁴⁶:

Tetragonal system:

(4) $\begin{array}{c}{\mathit{C}}_{11}>0,{\mathit{C}}_{33}>0,{\mathit{C}}_{44}>0,{\mathit{C}}_{66}>0,{\mathit{C}}_{11}-{\mathit{C}}_{12}>0,\hfill \\ {\mathit{C}}_{11}+{\mathit{C}}_{33}-{\mathit{C}}_{13}>0,2\left({\mathit{C}}_{11}+{\mathit{C}}_{12}\right)+{\mathit{C}}_{33}+4{\mathit{C}}_{13}>0,\hfill \end{array}$

Orthorhombic system:

(5) $\begin{array}{c}{\mathit{C}}_{11}+{\mathit{C}}_{12}+{\mathit{C}}_{33}+2{\mathit{C}}_{12}+2{\mathit{C}}_{23}>0,\hfill \\ {\mathit{C}}_{11}+{\mathit{C}}_{22}>2{\mathit{C}}_{12},{\mathit{C}}_{22}+{\mathit{C}}_{33}>2{\mathit{C}}_{33},\hfill \\ {\mathit{C}}_{11}+{\mathit{C}}_{33}>2{\mathit{C}}_{13},{\mathit{C}}_{\mathit{ii}}>0\left(\mathit{i}=1-6\right)\hfill \end{array}$

Hexagonal system:

(6) ${\mathit{C}}_{11}>0,{\mathit{C}}_{44}-{\mathit{C}}_{22}>0,\left({\mathit{C}}_{11}+{\mathit{C}}_{12}\right){\mathit{C}}_{33}>2{\mathit{C}}_{13}^2$

Table 2
The calculated elastic constants (C_ij, 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 expressions^47-50:

(7) ${\mathit{B}}_{\mathit{VRH}}=\frac{1}{2}\left({\mathit{B}}_{\mathit{V}}+{\mathit{B}}_{\mathit{R}}\right)$

(8) ${\mathit{G}}_{\mathit{VRH}}=\frac{1}{2}\left({\mathit{G}}_{\mathit{V}}+{\mathit{G}}_{\mathit{R}}\right)$

(9) $\mathit{E}=\frac{9{\mathit{B}}_{\mathit{VRH}}{\mathit{G}}_{\mathit{VRH}}}{3\mathit{B}{}_{\mathit{VRH}}+{\mathit{G}}_{\mathit{VRH}}}$

(10) $\mathit{\sigma }=\frac{3{\mathit{B}}_{\mathit{VRH}}-2{\mathit{G}}_{\mathit{VRH}}}{2\left(3{\mathit{B}}_{\mathit{VRH}}+{\mathit{G}}_{\mathit{VRH}}\right)}$

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 expressions⁵¹:

(11) $\mathit{\lambda }=\frac{\mathit{\sigma }\mathit{E}}{\left(1+\mathit{\sigma }\right)\left(1-2\mathit{\sigma }\right)}$

(12) $\mathit{\mu }=\frac{\mathit{E}}{2\left(1+\mathit{\sigma }\right)}$

The related calculated values of Ga_xMg_y binary compounds are showed in Table 3, and Fig.4 illustrates the variations of elastic parameters of Ga_xMg_y 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 Ga_xMg_y 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 (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) ${\mathit{H}}_{\mathit{v}}=2{\left({\mathit{k}}^2\mathit{G}\right)}^{0.585}-3$

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 (H_v, 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 (A_U), the percent anisotropy index (A_B and A_G) and the shear anisotropy factors ( A₁ , A₂ and A₃), are obtained by the following equations^54,55:

(14) ${\mathit{A}}_1=\frac{4{\mathit{c}}_{44}}{{\mathit{c}}_{11}+{\mathit{c}}_{33}-2{\mathit{c}}_{13}}\mathit{for}\left(100\right)\mathit{plane}$

(15) ${\mathit{A}}_2=\frac{4{\mathit{c}}_{55}}{{\mathit{c}}_{22}+{\mathit{c}}_{33}-2{\mathit{c}}_{23}}\mathit{for}\left(100\right)\mathit{plane}$

(16) ${\mathit{A}}_3=\frac{4{\mathit{c}}_{66}}{{\mathit{c}}_{11}+{\mathit{c}}_{22}-2{\mathit{c}}_{12}}\mathit{for}\left(100\right)\mathit{plane}$

(17) ${\mathit{A}}^{\mathit{U}}=5\frac{{\mathit{G}}_{\mathit{V}}}{{\mathit{G}}_{\mathit{R}}}+\frac{{\mathit{B}}_{\mathit{V}}}{{\mathit{B}}_{\mathit{R}}}-6\ge 0$

(18) ${\mathit{A}}_{\mathit{B}}=\frac{{\mathit{B}}_{\mathit{V}}-{\mathit{B}}_{\mathit{R}}}{{\mathit{B}}_{\mathit{V}}+{\mathit{B}}_{\mathit{R}}}$

(19) ${\mathit{A}}_{\mathit{G}}=\frac{{\mathit{G}}_{\mathit{V}}-{\mathit{G}}_{\mathit{R}}}{{\mathit{G}}_{\mathit{V}}+{\mathit{G}}_{\mathit{R}}}$

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 Ga₅Mg₂ is zero and GaMg₂ has the largest value as 0.41% in Ga_xMg_y compounds, which indicating that the anisotropy in bulk modulus of GaMg₂ 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 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 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-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 A_G, 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 Ga_xMg_y compounds.

Hexagonal crystal:

(20) $\frac{1}{\mathit{B}}=\left({\mathit{S}}_{11}+{\mathit{S}}_{12}+{\mathit{S}}_{13}\right)-\left({\mathit{S}}_{11}+{\mathit{S}}_{12}-{\mathit{S}}_{13}-{\mathit{S}}_{33}\right){\mathit{l}}_3^2$

(21) $\frac{1}{\mathit{E}}=\left(1-{\mathit{l}}_3^2\right){\mathit{S}}_{11}+{\mathit{l}}_3^4{\mathit{S}}_{33}+{\mathit{l}}_3^2\left(1-{\mathit{l}}_3^2\right)\left(2{\mathit{S}}_{13}+{\mathit{S}}_{44}\right)$

Orthorhombic crystal:

(22) $\begin{array}{c}\frac{1}{\mathit{B}}=\left({\mathit{S}}_{11}+{\mathit{S}}_{12}+{\mathit{S}}_{13}\right){\mathit{l}}_1^2+\left({\mathit{S}}_{12}+{\mathit{S}}_{22}+{\mathit{S}}_{23}\right){\mathit{l}}_2^2+\hfill \\ \left({\mathit{S}}_{13}+{\mathit{S}}_{23}+{\mathit{S}}_{33}\right){\mathit{l}}_3^2\hfill \end{array}$

(23) $\begin{array}{c}\frac{1}{\mathit{E}}={\mathit{S}}_{11}{\mathit{l}}_1^4+{\mathit{S}}_{22}{\mathit{l}}_2^4+{\mathit{S}}_{33}{\mathit{l}}_3^4+\left(2{\mathit{S}}_{12}+{\mathit{S}}_{66}\right){\mathit{l}}_1^2{\mathit{l}}_2^2+\hfill \\ \left(2{\mathit{S}}_{13}+{\mathit{S}}_{55}\right){\mathit{l}}_1^2{\mathit{l}}_3^2+\left(2{\mathit{S}}_{23}+{\mathit{S}}_{44}\right){\mathit{l}}_2^2{\mathit{l}}_3^2\hfill \end{array}$

Where S_ij 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 A_B 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 (A^U), percent anisotropy (A_B and A_G) 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 by^56,57

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

Figure 6
(a)-(c) The (001), (100) and (110) planar projections of the Young’s modulus of Ga_xMg_y 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 ν_m is calculated by^58,59:

(24) ${\mathit{\upsilon }}_{\mathit{m}}={\left[\frac{1}{3}\left(\frac{2}{{\mathit{\upsilon }}_{\mathit{l}}^3}+\frac{1}{{\mathit{\upsilon }}_{\mathit{t}}^3}\right)\right]}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

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 obtained⁶⁰.

(25) ${\mathit{\upsilon }}_{\mathit{l}}=\sqrt{\frac{\left(\mathit{B}+\left(\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)\mathit{G}\right)}{\mathit{\rho }}}$

(26) ${\mathit{\upsilon }}_{\mathit{t}}=\sqrt{\frac{\mathit{G}}{\mathit{\rho }}}$

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 Ga_xMg_y compounds and the acoustic velocities in the principal directions can be simply expressed as^61,62:

Tetragonal crystal:

(27) $\begin{array}{c}\left[100\right]=\left[010\right]\hfill \\ \left[100\right]{\mathit{\upsilon }}_{\mathit{l}}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{11}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.};\left[001\right]{\mathit{\upsilon }}_{\mathit{t}1}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{44}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.};\hfill \\ \left[010\right]{\mathit{\upsilon }}_{\mathit{t}2}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{66}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.}\hfill \end{array}$

(28) $\begin{array}{c}\left[001\right]\hfill \\ \left[001\right]{\mathit{\upsilon }}_{\mathit{l}}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{33}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.};\left[100\right]{\mathit{\upsilon }}_{\mathit{t}1}=\left[010\right]{\mathit{\upsilon }}_{\mathit{t}2}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{66}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.}\hfill \end{array}$

(29) $\begin{array}{c}\left[110\right]\hfill \\ \left[110\right]{\mathit{\upsilon }}_{\mathit{l}}=\sqrt{\raisebox{1ex}{$\left({\mathit{C}}_{11}+{\mathit{C}}_{12}+2{\mathit{C}}_{66}\right)$}\!\left/ \!\raisebox{-1ex}{$2\mathit{\rho }$}\right.};\hfill \\ \left[001\right]{\mathit{\upsilon }}_{\mathit{t}1}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{44}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.};\left[1\stackrel{─}{1}0\right]{\mathit{v}}_{\mathit{t}2}=\sqrt{\raisebox{1ex}{$\left({\mathit{C}}_{11}-{\mathit{C}}_{12}\right)$}\!\left/ \!\raisebox{-1ex}{$2\mathit{\rho }$}\right.}\hfill \end{array}$

Orthorhombic crystal:

(30) $\begin{array}{c}\left[100\right]\hfill \\ \left[100\right]{\mathit{\upsilon }}_{\mathit{l}}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{11}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.};\left[010\right]{\mathit{\upsilon }}_{\mathit{t}1}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{66}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.};\hfill \\ \left[001\right]{\mathit{v}}_{\mathit{t}2}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{55}$}\!\left/ \!\raisebox{-1ex}{$2\mathit{\rho }$}\right.}\hfill \end{array}$

(31) $\begin{array}{c}\left[010\right]\hfill \\ \left[010\right]{\mathit{\upsilon }}_{\mathit{l}}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{22}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.};\left[100\right]{\mathit{\upsilon }}_{\mathit{t}1}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{66}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.};\hfill \\ \left[001\right]{\mathit{v}}_{\mathit{t}2}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{44}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.}\hfill \end{array}$

(32) $\begin{array}{c}\left[001\right]\hfill \\ \left[001\right]{\mathit{\upsilon }}_{\mathit{l}}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{33}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.};\left[010\right]{\mathit{\upsilon }}_{\mathit{t}1}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{55}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.};\hfill \\ \left[010\right]{\mathit{v}}_{\mathit{t}2}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{44}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.}\hfill \end{array}$

Hexagonal crystal:

(33) $\begin{array}{c}\left[100\right]\hfill \\ \left[100\right]{\mathit{\upsilon }}_{\mathit{l}}=\sqrt{\raisebox{1ex}{$\left({\mathit{C}}_{11}-{\mathit{C}}_{12}\right)$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.};\left[010\right]{\mathit{\upsilon }}_{\mathit{t}1}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{11}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.};\left[001\right]{\mathit{v}}_{\mathit{t}2}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{44}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.}\hfill \end{array}$

(34) $\begin{array}{c}\left[001\right]\hfill \\ \left[001\right]{\mathit{\upsilon }}_{\mathit{l}}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{33}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.};\left[100\right]{\mathit{\upsilon }}_{\mathit{t}1}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{44}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.};\left[010\right]{\mathit{v}}_{\mathit{t}2}=\sqrt{\raisebox{1ex}{${\mathit{C}}_{44}$}\!\left/ \!\raisebox{-1ex}{$\mathit{\rho }$}\right.}\hfill \end{array}$

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 _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 (ν_l, m/s), shear sound velocity (ν_t, m/s), average sound velocity (ν_m, m/s) of Ga_xMg_y.

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 Ga_xMg_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. 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).

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Publication Dates

• Publication in this collection
  18 Feb 2019
• Date of issue
  2019

History

• Received
  19 Sept 2018
• Reviewed
  19 Nov 2018
• Accepted
  20 Nov 2018