Theoretical prediction of the fundamental properties for the ternary MgYZn and Mg0.9YZn1.06 alloys

Computational Materials Science 91 (2014) 315–319

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Theoretical prediction of the fundamental properties for the ternary MgYZn and Mg0.9YZn1.06 alloys Jianping Long ⇑,a, Lijun Yang a, Wen Huang b a b

College of Materials and Chemistry & Chemical Engineering, Chengdu University of Technology, Chengdu 610059, PR China College of Electronic Engineering, Chongqing University of Post and Telecommunications, Chongqing 400065, PR China

a r t i c l e

i n f o

Article history: Received 5 November 2013 Received in revised form 18 April 2014 Accepted 26 April 2014 Available online 2 June 2014 Keywords: Electronic properties Elastic properties Thermodynamic properties Minimum thermal conductivity

a b s t r a c t The fundamental properties of the ternary MgYZn and Mg0.9YZn1.06 (Mg–Y–Zn) alloys are investigated by using the first-principles density functional theory within the generalized gradient approximation (GGA). The calculated lattice constants are found in a good agreement with the available experimental data. The calculated elastic constants indicate that both of the Mg–Y–Zn alloys are mechanically stable. The shear modulus, Young’s modulus, Poisson’s ratio r, the ratio B/G and the universal anisotropy index are also calculated. According to these values, we have revealed that the Mg–Y–Zn alloys behave in a ductile manner and exhibit a strong anisotropy. Finally, the Debye temperature and minimum thermal conductivity are obtained. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction Magnesium–Zinc (Mg–Zn) alloys, which can be used as structural materials in the automobile, aerospace and electronics industries due to their low density, high specific resistance and good mechanical properties, have received increasing attention [1–4]. However, their industrial applications have been limited because conventional Mg–Zn alloys show poor corrosion resistance, poor ductility and low mechanical strength at high temperature. Thus both engineers and researchers have a strong interest to develop a new system that could improve the corrosion resistance, ductility and mechanical strength of Mg–Zn alloys. To improve these physical properties of Mg–Zn alloys, great endeavors have been made through adding rare-earth and/or transition-metal elements [5–9]. Reportedly, the addition of yttrium (Y) element to Mg–Zn alloys can improve their mechanical strength at room and high temperatures. The ternary Mg–Y–Zn alloys exhibit tensile yield strength, high mechanical performance, low eutectic temperature and high corrosion resistance at room temperature. The addition of Y element promotes the formation of intermetallic phase whose stability is retained up to high temperature. Depending on the Zn/Y ratios different ternary phases can be formed: LPSO phase (X-phase) Mg12YZn (hexagonal) [10,11]; I-phase Mg3YZn6 (icosahedral) [12,13]; Z-phase Mg28Y7Zn65 (hexagonal) [14,15]; W-phase Mg3Y2Zn3 (face center cubic, fcc) ⇑ Corresponding author. Tel.: +86 18502860022; fax: +86 28 84077930. E-mail address: [email protected] (J. Long). http://dx.doi.org/10.1016/j.commatsci.2014.04.057 0927-0256/Ó 2014 Elsevier B.V. All rights reserved.

[16,17]; H-phase MgYZn3 (hexagonal) [18,19]; Q (questionable)phase Mg38Y2Zn60 (decagonal) [20]. Many studies have been performed to investigate the physical and chemical properties of Mg–Y–Zn alloys through two major methods: the experimental method and theoretical method. The microstructure, creep behavior, electrical resistivity, magnetoresistance, phase equilibrium and transformations and specific heat of Mg–Y–Zn alloys were investigated with a wide range of experimental techniques, including high resolution transmission electron microscopy (HRTEM) [21], electron diffraction [22], environmental scanning electron microscope (ESEM) [23], X-ray diffraction [24] and atomic-resolution high-angle annular dark-field scanning transmission electron microscopy [25]. The phase stability, stacking fault energies, lattice vibration effect and thermodynamic model of Mg–Y–Zn alloys were investigated by first-principles density functional theory [26–29] and the CALPHAD [30] method. However, to the best of our knowledge, no theoretical and experimental investigations of the electronic structure, elastic properties, Debye temperature and thermal conductivity of the ternary MgYZn and Mg0.9YZn1.06 have been published. In this work, we calculated these physical properties of MgYZn and Mg0.9YZn1.06 by using the first-principles ultrasoft pseudopotential plane-wave method. 2. Calculation methods Our calculations were performed using the CASTEP code [31], which is based on the state-of-the-art of the density functional

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theory using the ultrasoft pseudopotential plane-wave (UPPW) method. The Perdew–Wang (PW91) [32]-GGA [33,34] was used for the exchange correlation potential. Using the UPPW method, 2p63s2 of Mg, 4d15s2 of Y and 3d104s2 of Zn were treated explicitly as valence electrons. In this study, the cutoff energy of plane-wave is 700 eV, which was large enough to obtain good convergence. In the Brillouin zone integrations, 8  8  8 Monkhorst–Pack k-points mesh were used. The structural parameters of MgYZn and Mg0.9YZn1.06 were calculated by using the Brodyden–Fletcher– Goldfarb–Shanno (BFGS) [35–38] method. 3. Results and discussions 3.1. Structure properties MgYZn and Mg0.9YZn1.06 alloys have a hexagonal structure with space groups P6-2m, as shown in Fig. 1. The calculated equilibrium lattice parameters and the atomic positions of MgYZn and Mg0.9YZn1.06 alloys are summarized in Table 1, together with the available experimental data for comparison. The lattice constants a = 7.57158 Å, c = 4.16593 Å of MgYZn and a = 7.61636 Å, c = 4.03937 Å of Mg0.9YZn1.06 respectively. They are in a good agreement with the experimental data, and the deviations from the experimental data are less than 2%. The reasons why there is a difference between theoretical values and experimental data are as follow: (1) the theoretical values have been got at 0 K while experimental data at 293 K; (2) It is well known that theoretical values obtained using the GGA method are consistently higher than the experimental data (see Table 1). 3.2. Electronic properties We have calculated the energy band structure of MgYZn and Mg0.9YZn1.06 alloys along high symmetry directions as shown in Fig. 2. From Fig. 2, it is observed that both of the alloys exhibit metallic characters as there are no band gap at the Fermi level, the valence and conduction band overlap significantly at Fermi level. The calculated total density of states (TDOS) and partial density of states (PDOS) of the ternary MgYZn and Mg0.9YZn1.06 alloys in the energy range between 4 eV and 4 eV are illustrated in Fig. 3. As it can be seen, the TDOS of MgYZn shows similar appearance as that of Mg0.9YZn1.06. For the MgYZn, the valence band (VB) are mainly composed of Mg-2p, Y-4d, Zn-3p states hybridized with small amount of Mg-3s, Y-5s, 4p and Zn-4s states. The conduction band (CB) are mainly composed of Mg-2p, Y-4d, Zn-3p states hybridized with small amount of Mg-3s, Y-4p and Zn-4s states. The calculated N(Ef) values of Mg-s, p, d, Y-s, p, d and Zn-s, p, d

Fig. 1. Crystal structure of MgYZn and Mg0.9YZn1.06 alloys.

states electrons at the Fermi level are shown in Table 2. As it can be seen, the N(Ef) values of MgYZn and Mg0.9YZn1.06 are significantly different. 3.3. Elastic properties Elastic constants of crystals provide a link between mechanical and dynamical behaviors. Also, they give important information concerning the elastic response of a crystal to an external pressure. To calculate the elastic constants, we have applied the nonvolume-conserving method. For the hexagonal crystals, its five independent elastic constants should satisfy the well-know Born stability criteria [40]

C 11 > 0;

C 33 > 0;

C 44 > 0;

ðC 11 þ C 12 ÞC 33 

2C 213

C 11  C 12 > 0; >0

ð1Þ

The computed elastic constants of MgYZn and Mg0.9YZn1.06 alloys are shown in Table 3. According to the above criteria, it is clear that the MgYZn and Mg0.9YZn1.06 alloys are mechanically stable at 0 K. From the Table 3, it can be seen that the elastic constant C 11 , which provides a measure of rigidity against unidirectional deformation along the a axis, is larger than elastic constant C33, which provides an estimation of the elastic response of material to a unidirectional pressure along the c axis, indicating that the MgYZn and Mg0.9YZn1.06 are more compressible along the c axis than along the a axis. For hexagonal structure, the computations of Voigt (GV) and Reuss shear modulus (GR) and Voigt (BV) and Reuss bulk modulus (BR) are [42]

1 1 ð2C 11 þ C 33  C 12  2C 13 Þ þ ð2C 44 þ C 66 Þ 15 5   2 1 BV ¼ C 11 þ C 12 þ 2C 13 þ C 33 9 2 GV ¼

BR ¼

GR ¼

ðC 11 þ C 12 ÞC 33  2C 213 C 11 þ C 12 þ 2C 33  4C 13 5C 2 C 44 C 66 2½3BV C 44 C 66 þ C 2 ðC 44 þ C 66 Þ

ð2Þ

ð3Þ

ð4Þ

; ð5Þ

C 2 ¼ ðC 11 þ C 22 ÞC 33  2C 213 The bulk modulus BH and shear modulus GH can be estimated by Voigt–Reuss–Hill approximation. The Young’s modulus E and Poisson’s ratio r can be computed by the following equations [43], respectively:

E¼

9BH GH 3BH þ GH

ð6Þ

r¼

3BH  2GH 2ð3BH þ GH Þ

ð7Þ

The ratio of bulk modulus to shear modulus of crystalline phases can predict the brittle and ductile behavior of materials. If B/G > 1.75 the material will behave in a ductile manner or else the material demonstrates brittleness [44]. The values of B, G, E, r and the ratio B/G are given in Table 4. The obtain B/G ratio is 2.08 for MgYZn and 1.95 for Mg0.9YZn1.06, according to those values MgYZn and Mg0.9YZn1.06 behave in a ductile manner. The Poisson’s ratio r provides more information about the characteristics of the bonding forces than any other elastic constants. The value of r is small for covalent materials (typically r = 0.1), and G = 1.1B. A typical value of r is 0.25 for ionic materials and

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J. Long et al. / Computational Materials Science 91 (2014) 315–319 Table 1 Calculated MgYZn and Mg0.9YZn1.06 alloys structure parameters and the atomic positions together with the experiment data. Compound MgYZn

Pearson symbol hP9

Cal.

2 ; S44 þ 2S11  2S12

Exp. [39]

rð2 1 1 0Þ ¼ rð01 1 0Þ ¼  Wyckoff positions

Lattice parameters (Å) Mg0.9YZn1.06 hP9

1  0Þ ¼ Gð01 1  0Þ ¼ Gð2 1

Wyckoff positions

Lattice parameters (Å)

Y(3f) (0.58813, 0, 0) Mg(3g) (0.24754, 0, 0.5) Zn1(2d) (0.3333, 0.6667, 0.5) Zn2 (1a) (0, 0, 0) a = 7.57158 c = 4.16593

Y(3f) (0.5866, 0, 0)

Y(3f) (0.5897, 0, 0) Mg/Zn1 (3g) (0.2479, 0, 0.5) Zn2 (2d) (0.3333, 0.6667, 0.5) Zn3 (1a) (0, 0, 0) a = 7.61636 c = 4.03937

Y(3f) (0.5865, 0, 0) Mg/Zn1 (3g) (0.2479, 0, 0.5) Zn2 (2d) (0.3333, 0.6667, 0.5) Zn3 (1a) (0, 0, 0) a = 7.58200 c = 4.11380

G = 0.6B; for metallic materials r is typically 0.33 and G = 0.4B [45]. The obtained r is 0.29 and 0.28 for MgYZn and Mg0.9YZn1.06 respectively. We can conclude that these alloys quite distinctly belong to the class of metallically bonding materials, which is also what would be expected from the metallic density of states. Elastic anisotropy is one of the most important parameters for estimating the mechanical properties of materials. The average Young’s modulus E, average shear modulus G and average 1  0Þ, ð0 1 1  0Þ and ð0 0 0 1Þ planes Poisson’s ratio r on the ð2 1 can be obtained using following relationships [46]:

Eð0 0 0 1Þ ¼

1 S33

rð0 0 0 1Þ ¼ 

S13 S33

1 S44

ð9Þ

ð10Þ

The computed Young’s modulus E, shear modulus G and Pois1  0Þ, ð0 1 1  0Þ and ð0 0 0 1Þ planes are son’s ratio r on the ð2 1 shown in Table 5. The results show that the anisotropy behavior of MgYZn and Mg0.9YZn1.06 are very significant due to the reason that the Young’s modulus E and shear modulus G are difference 1  0Þ, ð01 1  0Þ and the basal plane between the prismatic planes ð2 1 ð0 0 0 1Þ. For hexagonal crystal, compression anisotropy can be identified by Bc/Ba = (C11 + C12  2C13)/(C33  C13), where Bc and Ba represent bulk modulus along c-axis and a-axis direction, respectively. The calculated value of Bc =Ba is 0.91 for MgYZn and 1.10 for Mg0.9YZn1.06, it is concluded that these alloys exhibits anisotropy elasticity. Most recently, Ranganathan and Ostoja-Starzewski [47] introduced a concept of universal anisotropy index to measure the single crystal elastic anisotropy. The universal anisotropy index is given by:

Mg(3g) (0.2477, 0, 0.5) Zn1(2d) (0.3333, 0.6667, 0.5) Zn2 (1a) (0, 0, 0) a = 7.57900 c = 4.11590

AU ¼ 5

1  0Þ ¼ Eð0 1 1  0Þ ¼ 1 ; Eð2 1 S11

S12 þ S13 ; 2S11

Gð0 0 0 1Þ ¼

ð8Þ

GV BV þ 6 GR BR

ð11Þ

AU = 0 represents locally isotropic single crystals and AU > 0 denotes the extent of single crystal anisotropy. The calculated value of AU is 3.8 for MgYZn and 3.9 for Mg0.9YZn1.06, suggesting again these alloys have strong crystal anisotropy.

3.4. Thermodynamic properties The Debye temperature (hD) is not a strictly determined parameter, various estimates may be obtained through well established empirical or semi-empirical formulae. One of the semi-empirical formula can be used to estimate the values of Debye temperature

Table 2 The density of states N(Ef) (electron/eV) at the Fermi level of MgYZn and Mg0.9YZn1.06. Compounds

Mg(s)

Mg(p)

Mg(d)

Mg(tot)

Y(s)

Y(p)

Y(d)

Y(tot)

Zn(s)

Zn(p)

Zn(d)

Zn(tot)

tot

MgYZn Mg0.9YZn1.06

0.05 0.10

0.80 1.13

0 0.04

0.85 1.27

0.08 0.08

0.91 0.86

3.42 2.94

4.41 3.88

0.12 0.12

1.63 1.20

0.03 0.03

1.77 1.37

6.84 5.70

‘tot’: the total density of states.

Fig. 2. Calculated energy band structure of MgYZn and Mg0.9YZn1.06.

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J. Long et al. / Computational Materials Science 91 (2014) 315–319

Fig. 3. The total density of states of MgYZn and Mg0.9YZn1.06.

Table 3 The calculated elastic constants Cij (in GPa) of MgYZn and Mg0.9YZn1.06 alloys.

a b

Compounds

C11

C33

C44

C12

C13

MgYZn Mg0.9YZn1.06 Ref. [25]a Ref. [41]b

87.2 100.3 59.2 68.1

66.4 71.3 61.4 67.2

55.1 62.3 16.4 20.6

30.5 29.1 25.7 21.6

52.4 56.6 21.4 24.0

The elastic constants of Mg8Y2Zn. The elastic constants of Mg85Zn6Y9.

through elastic constants, averaged sound velocity (vm), longitudinal sound velocity (vl) and transverse sound velocity (vt) [49–53].

hD ¼

  1 h 3n NA q 3 vm k 4p M

ð12Þ

vm ¼

  13 1 2 1 þ 3 3 3 vt vl

ð13Þ

vl ¼ vt ¼

where h and k are Planck’s and Boltzmann’s constants; NA is Avogadro’s number; q is the density; M is the molecular weight, and n is the number of atoms in the unit cell. The calculated values of v m , v l , v t and hD of MgYZn and Mg0.9YZn1.06 at 0 K are given in Table 6. It is observed that the calculated Debye temperature of Mg0.9YZn1.06 larger than the MgYZn. Therefore, Mg0.9YZn1.06 is harder with a large wave velocity and has higher thermal conductivity than MgYZn. The Debye frequency v D is given by v D ¼ kB hD =h. On the basis of the Debye temperature hD , the Debye frequency of MgYZn and Mg0.9YZn1.06 is 6.13 THz and 6.55 THz, respectively. Thermal conductivity K is the property of a material that indicates its ability to conduct heat. However, in order to known if the material is a potential application for thermal barrier coating, its thermal conductivity needs to be investigated. Base on the Debye model, Clarke [54] and Liu et al. [55] suggested that the theoretical minimum thermal conductivity is given by:

K min ¼ kB v m

 1 B þ 43 G 2

ð14Þ

q  12 G

ð15Þ

q

M

!23 ð16Þ

q

where M is the average atomic weight which is the molecular weight M divided by the number of atoms in the unit cell. The calculated minimum thermal conductivity of MgYZn and Mg0.9YZn1.06 alloys are given in Table 6. It is observed that the calculated minimum thermal conductivity of Mg0.9YZn1.06 larger than the MgYZn.

Table 4 Calculated B (all in GPa), G (all in GPa), E (all in GPa), r and the ratio B/G of MgYZn and Mg0.9YZn1.06.

a b

Compounds

BV

BR

BH

GV

GR

GH

B/G

E

r

MgYZn Mg0.9YZn1.06 Refs.

56.8 61.8

56.8 61.8

56.8 61.8 38.0 [41]a 39.5

34.8 40.7

19.8 22.8

27.3 31.7

2.08 1.95

70.5 81.3 49.3 [48]b 102.3

0.29 0.28

The value of Mg85Zn6Y9. MgYZn0.5, MgY2Zn and MgY3Zn1.5 alloys at 673 K.

Table 5 1  0Þ, ð01 1  0Þ and ð0 0 0 1Þ planes, kc =ka and AU . Calculated the average Young’s modulus E, shear modulus G and Poisson’s ratio r on the ð2 1 Compounds

1  0Þ Eð2 1  0Þ Eð01 1

Eð0 0 0 1Þ

1  0Þ Gð2 1  0Þ Gð01 1

Gð0 0 0 1Þ

rð2 1 1 0Þ rð01 1 0Þ

rð0 0 0 1Þ

Bc/Ba

AU

MgYZn Mg0.9YZn1.06

43.35 50.99

19.80 21.89

37.44 45.31

55.10 62.26

0.37 0.37

0.44 0.44

0.91 1.10

3.8 3.9

J. Long et al. / Computational Materials Science 91 (2014) 315–319 Table 6 Calculated the density (q in g/cm3), transverse, longitudinal, average sound velocity (v t , v l , v m in m/s), the Debye temperatures (hD in K) and the minimum thermal conductivity (K min in Wm1 K1) of MgYZn and Mg0.9YZn1.06.

a b

Compounds

q

vt

vl

vm

hD

Kmin

MgYZn Mg0.9YZn1.06 Exp.

4.30145 4.44470

2517.1 2672.4

4653.3 4840.9

2809.1 2977.8

294.1 313.8 319 [24]a 325 [21]b

0.48 0.51

Zn58Mg36Y8. Zn50Mg42Y8.

4. Conclusions In present work, the elastic and thermodynamic properties of MgYZn and Mg0.9YZn1.06 have been studied by means of DFT within the GGA. The most relevant conclusions are summarized as follows: (1) The calculated lattice parameters of MgYZn and Mg0.9YZn1.06 alloys are in a good agreement with the experimental data, and deviated from measured data are less than 2%. (2) The mechanical properties like shear modulus and Young’s modulus are also calculated. From our results, we observe that these alloys are mechanically stable. (3) The Poisson’s ratio r and B/G ratio are calculated. According to these values, we have revealed that the MgYZn and Mg0.9YZn1.06 behave in a ductile manner. (4) The compression anisotropy Bc =Ba and the universal anisotropy index AU are obtained. Base on our calculated, we can conclude that MgYZn and Mg0.9YZn1.06 exhibits a strong anisotropy. (5) The Debye temperature and thermal conductivity are obtained. It is observed that the Debye temperature of Mg0.9YZn1.06 larger than the MgYZn. Therefore, Mg0.9YZn1.06 is harder and has higher thermal conductivity than MgYZn.

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