First-Principles Study on Stability and Mechanical Properties of Cr7C3

Rare Metal Materials and Engineering Volume 43, Issue 12, December 2014 Online English edition of the Chinese language journal Cite this article as: Rare Metal Materials and Engineering, 2014, 43(12): 2903-2907.

ARTICLE

First-Principles Study on Stability and Mechanical Properties of Cr7C3 Liu Yangzhen,

Jiang Yehua,

Zhou Rong

Kunming University of Science and Technology, Kunming 650093, China

Abstract: The lattice parameters, the stability, mechanical properties and the anisotropic sound velocity of hexagonal and orthorhombic Cr7C3 were investigated using first-principles calculations. The results of the cohesive energy and the formation enthalpy of these compounds indicate that they have thermodynamically stable structures. The elastic constants and the mechanical moduli of these compounds were estimated using the stress-strain method and Voigt-Reuss-Hill approximation, respectively. Moreover, the anisotropic properties of sound velocities and mechanical anisotropies of Hexa-Cr7C3 and ortho-Cr7C3 were explored. Key words: Cr7C3; first-principles; phase stability; mechanical property; Debye temperature

Transition metal carbides are of great technological importance in the cutting tool industry due to their excellent properties such as corrosion resistance, extreme stiffness, chemical stability and wear and oxidation resistances. Generally, such properties are related to the components, the crystal structure, the thermodynamic properties and the stability of these carbides. The properties of molybdenum and vanadium carbides have been investigated using experimental and theoretical methods[1-3]. Cr7C3 is one of the most important compounds and it has both orthorhombic and hexagonal crystalline structures [4, 5]. Several papers have reported the properties of Cr7C3 either by theoretical calculations or by empirical methods. For example, Rouault et al [5] found that Cr7C3 samples synthesized at high temperatures possess an orthorhombic structure, while those synthesized at lower temperatures have a hexagonal Ru7C3 structure. Jiang [6] investigated the structural, elastic and electronic properties of chromium carbides, they concluded that WC-type CrC exhibits the highest bulk and shear moduli and the lowest Poisson’s ratio, and is a potential low-compressibility and hard material among all chromium carbides. However, there are few reports on the properties of these forms and the relation between orthorhombic and hexagonal crystalline structures. In this paper, the lattice parameters, stability and mechanical properties of Ortho-Cr7C3 and

Hexa-Cr7C3 were calculated using first principles, and the relation between these structural forms was investigated. The crystalline structures of different Cr7C3 compounds are the following: Ortho-Cr7C3 crystal in the orthorhombic space group Pnma (No. 62) has four formula units (Z=4) per unit cell. Hexa-Cr7C3 crystal in the hexagonal space group P63mc (No. 186) has eight formula units (Z=8) per unit cell.

1

Calculation Method

Based on density functional theory (DFT), which implemented in CASTEP code [7], the first principles calculations were used to investigate the lattice parameters, stability and mechanical properties of Ortho-Cr7C3 and Hexa-Cr7C3. Ultrasoft pseudopotentials were used to represent the interaction between ionic cores and valence electrons. For Cr and C, the valence electrons were considered to be 3s23p63d54s1, 2s22p2, respectively. The exchange-correlation energy was calculated with GGA (generalized gradient approximation) of PBE (Perdew-Burke-Ernzerh) scheme [8]. A kinetic energy cut-off value of 450 eV was used for the plane wave expansions. The calculations in the first irreducible Brillouin zone were conducted by using 8×8×8 k point grid of Monkhorst-Pack scheme [9]. BFGS (Broyden-Fletcher-Goldfarb-Shannon)[10] optimized method was adopted to obtain the equilibrium crystal struc-

Received date: March 25, 2014 Foundation Item: National Natural Science Foundation of China (51171074, 51261013) Corresponding author: Jiang Yehua, Professor, Faculty of Materials Science and Engineering, Kunming University of Science and Technology, Kunming 650093, P. R. China, Tel: 0086-871-65180653, E-mail: [email protected] Copyright © 2014, Northwest Institute for Nonferrous Metal Research. Published by Elsevier BV. All rights reserved.

2903

Liu Yangzhen et al. / Rare Metal Materials and Engineering, 2014, 43(12): 2903-2907

elastic constants are illustrated in Table 2. There are five and nine independent elastic constants for Hexa-Cr7C3 and Ortho-Cr7C3, respectively. According to Born-Huang’s lattice dynamical theory, the mechanical stability criterions can be expressed as [11]: For hexagonal crystals: 2 C11>0, C44>0, C11-C12>0, (C11 + C12 )C33 − 2C13 > 0 (3) For orthorhombic crystals; C11+C22+C33+2C12+2C13+2C23>0 C23+C33-2C23>0, C11>0, C22>0, (4) C33>0, C44>0, C55>0, C66>0 From Table 2, it is obvious that the calculated elastic constants of Hexa-Cr7C3 and Ortho-Cr7C3 compounds satisfy above corresponding criterions, respectively, which implies that they are mechanically stable. Moreover, the calculated values are coincided with other theoretical and experimental values. Bulk modulus reflects the compressibility of the solid under hydrostatic pressure. The bulk modulus (B) and shear modulus (G) for crystal can be estimated within Voigt-Reuss-Hill methods [12] after evaluating the elastic constants; then the Young’s modulus and Poisson’s ratio can be calculated as follows [11]:

tures of carbides. The other convergence parameters were: (1) total energy changes were reduced to 1.0×10-6 eV; (2) Hellman-Feynman forces acting on distinct atoms were converged less than 0.5 eV/nm. In order to estimate the relative stability of two Cr7C3 compounds, the cohesive energy and formation enthalpy were investigated in this paper, the following expressions (Eq. (1) and (2)) were used for this calculations: Ecoh(Cr7C3)=Etot(Cr7C3)-7Eiso(Cr)-3Eiso(C) (1) ∆rH(Cr7C3)=Ecoh(Cr7C3)-7Ecoh(Cr)-3Ecoh(C) (2) Where Ecoh(Cr7C3) and ∆rH(Cr7C3) are the cohesive energy and formation enthalpy, respectively; Etot(Cr7C3) is the total energy of the Cr7C3 compounds; Ecoh(Cr) is the cohesive energy of Cr; Ecoh(C) is the cohesive energy of C; Eiso(Cr) is the total energy of a single Cr atom and Eiso(C) is the total energy of a single C atom.

2

Results and Discussion

2.1 Stability The lattice parameters, cohesive energy and formation enthalpy of Ortho-Cr7C3 and Hexa-Cr7C3 compounds are listed in Table 1. As can be seen from Table 1, each of the lattice parameters is overestimated by 3%. Our results match fairly closely the corresponding experimental ones at room temperature. If we take the other influences of the crystal structure into account, such as thermodynamic effects, our results could be in good agreement with the experimental findings. Therefore, the calculated parameters should be appropriate in this paper. The stability of compounds is described by cohesive energy and formation enthalpy. The calculated cohesive energy values of Ortho-Cr7C3 and Hexa-Cr7C3 are -9.88 eV/atom and -9.51 eV/atom, respectively. Therefore, the Ortho-Cr7C3 is more stable than Hexa-Cr7C3 from the point view of cohesive energy. However, the stability of compound is determined by formation enthalpy, so the negative value of formation enthalpy is indicated for the stable crystal structure. The lower the formation enthalpy, the more stable the compound. The values of formation enthalpy for Ortho-Cr7C3 and Hexa-Cr7C3 are -0.115 eV/atom and -0.087 eV/atom, respectively. Thus, the Ortho-Cr7C3 may be the most stable among them. 2.2 Mechanical properties The mechanical properties of the studied carbides, such as elastic moduli, elastic anisotropy and theoretical hardness, play an important role for the application as wear resistance materials. Therefore, the stress-strain approach, namely Hooker’s law, was employed to estimate the elastic constants. The calculated

E=

9 BVRH GVRH 3BVRH + GVRH

(5)

σ=

3BVRH − 2GVRH 2(3BVRH + GVRH )

(6)

Where BVRH, GVRH denote the bulk modulus and shear modulus estimated within Voigt-Reuss-Hill methods. The values of bulk moduli, shear moduli, Young’s moduli and Poisson’s ratio for Hexa-Cr7C3 and Ortho-Cr7C3 compounds are also presented in Table 2, which are in good agreement with other theoretical values. The bulk modulus of Hexa-Cr7C3 and Ortho-Cr7C3 are 301.2 and 309.4 GPa, respectively. All of them exhibit larger values than many other common carbides such as Fe3C (226.8[14]), TiC (249[15]), but smaller than diamond (436.8[16]) and WC (400.9[17]). Jiang and co-workers [18] proposed that the hardness of materials may be more sensitive to shear modulus than to bulk modulus. The larger the shear modulus (G) is, the higher the hardness of the compound. As indicated in Table 2, the shear modulus of Hexa-Cr7C3 is larger than that of Ortho-Cr7C3. therefore, it can be concluded that the hardness of Hexa-Cr7C3 may be higher than that of Ortho-Cr7C3.The Poisson’s ratio of Hexa-Cr7C3 and Ortho-Cr7C3 are 0.33 and 0.34, respectively, which are

Table 1 calculated lattice parameters, cohesive energy and formation enthalpy of Ortho-Cr7C3 and Hexa-Cr7C3 Species

a/nm

Hexa-Cr7C3

1.635 1.401

Ortho-Cr7C3

0.467 0.4526b

a

Reference [4]

b

Reference [5]

Cohesive energy/eV·atom-1

Formation enthalpy/eV·atom-1

1.635 1.401

a

0.368 0.4532

-9.51

-0.087

0.684 0.701b

1.184 1.2142b

-9.88

-0.115

b/nm a

c/nm a

2904

Liu Yangzhen et al. / Rare Metal Materials and Engineering, 2014, 43(12): 2903-2907

Table 2 Calculated values for elastic constants (Cij/GPa), bulk modulus (B/GPa), shear modulus (G/GPa), Young’s modulus (E/GPa), Poisson’s ratio (σ) and universal elastic anisotropy index of Hexa-Cr7C3 and Ortho-Cr7C3 Species Hexa-Cr7C3 Ortho-Cr7C3

C11

C12

C13

476.2

185.6

252.8

487a

203a

259a

C22 -

C23 -

142a

256.6

385.2

310.8

447.6

158.9

150.5

121.2

205.2b

391.9b

320.5b

438.8b

169.2b

175.6b

117.8b

G

B/G

E

σ

117.2

2.57

311.2

0.33

Ortho-Cr7C3

309.4

109.4

2.83

293.6

0.34

close to 0.3 and clearly indicate the strong metallic bonding in these compounds, and the observed deviations from 0.3 mainly because of covalent nature of Cr-C bonds in these compounds. B/G is generally used to illustrate the compound with ductility or brittleness. If the value lager than 1.75, the compound is ductile, otherwise, it is brittle. In our case, the two compounds are ductile, and Hexa-Cr7C3 may be relatively more brittle than Ortho-Cr7C3. Moreover, the brittle compound usually has high hardness. Therefore, it can be concluded that Hexa-Cr7C3 may have the higher hardness than Ortho-Cr7C3, which is in good agreement with above analysis. All single crystals in practice are anisotropic, so an appropriate parameter to characterize the extent of anisotropy is needed. However, the elastic anisotropy of a crystal can be characterized by many different ways; in this work, several anisotropic indexes, such as the universal anisotropic index (AU) [19] and the percent anisotropy (AB and AG) [20] are calculated. They following equations are used to evaluate them:

GV BV + −6 ≥ 0 GR BR BV − BR ⎧ ⎪ AB = B + B ⎪ V R ⎨ G G − R ⎪A = V ⎪⎩ G GV + GR AU = 5

-

210.3

B

Reference [13]

128a

C66 145.3

229.4b

301.2

Reference [6]

382a

C55

431.1

Species

b

C44 125.3

438.8b

Hexa-Cr7C3 a

C33 378.9

(7)

(8)

Where BV, BR, GV, and GR are the bulk modulus and shear modulus estimated within Voigt and Reuss methods, respectively. If all of the indexes in above equations are zero, it may be an isotropic structure. The large deviations from zero reveal highly mechanical anisotropic properties. The anisotropy index of Hexa-Cr7C3 and Ortho-Cr7C3 are shown in Table 3. The universal anisotropic index (AU) is a better indicator than other indicators, which provides unique and consistent results for the mechanical anisotropic properties of the compounds. The larger the value of AU is, the stronger the anisotropic of the compound. As can be seen from Table 3, the elastic anisotropic of Ortho-Cr7C3 is stronger than Hexa-Cr7C3.

Table 3 Calculated universal anisotropic index (AU) and percent anisotropy (AB and AG) of Hexa-Cr7C3 and Ortho-Cr7C3 AU 0.419 0.933

Species Hexa-Cr7C3 Ortho-Cr7C3

AB 0.001 0.013

AG 0.040 0.083

2.3 Debye temperature and anisotropic sound velocity The Debye temperature is a fundamental parameter of a material, linked to many physical properties such as specific heat, elastic constants and melting point[21]. The Debye temperature (ΘD) for Hexa-Cr7C3 and Ortho-Cr7C3 can be estimated from elastic constants; the expressions are given by Eq. (9)-(11) [22]: 13

ΘD =

h ⎡ 3n ⎛ N A ρ ⎞ ⎤ ⎜ ⎟ νm k ⎢⎣ 4π ⎝ M ⎠ ⎥⎦

⎡ 1 ⎛ 2 1 ⎞⎤ ν m = ⎢ ⎜ 3 + 3 ⎟⎥ ⎣ 3 ⎝ vl vt ⎠ ⎦ ⎧ 4 ⎞ ⎛ ⎪vl = ⎜ B + G ⎟ ρ 3 ⎠ ⎝ ⎨ ⎪ ⎩vt = G ρ

−

1 3

(9)

(10)

(11)

Where ΘD represents Debye temperature; h and k are Planck and Boltzmann constants, respectively; NA is the Avogadro’s number; n is the number of atoms in the molecule; M is the molecular weight; ρ is the density; B and G are the bulk modulus and shear modulus, respectively; vl is the longitudinal sound velocity and vt is the transverse sound velocity; vm is the average sound velocity. The calculated Debye temperature and sound velocities of Hexa-Cr7C3 and Ortho-Cr7C3 are shown in Table 4. Debye temperature reflects the strength of chemical bonding in the crystal structure and which is inverse to the molecular weight. As a result, the Debye temperature of Hexa-Cr7C3 is larger than that of Ortho-Cr7C3. Both vl and vt are correlated to bulk modulus and density, the compound with low density and high bulk modulus will have large sound velocities. Thus, the sound velocities of Hexa-Cr7C3 are larger than that of Ortho-Cr7C3. Since the thermal conductivity of the compound is proportional to the average sound velocity, thus one expects that

2905

Liu Yangzhen et al. / Rare Metal Materials and Engineering, 2014, 43(12): 2903-2907

Table 4 Sound velocities and Debye temperature of Hexa-Cr7C3 and Ortho-Cr7C3 ρ

vl

vt

vm

/g·cm-3

/m·s-1

/m·s-1

/m·s-1

Hexa-Cr7C3

6.22

8575.9

4340.8

5797.3

784.5

Ortho-Cr7C3

7.02

8053.1

3947.7

5305.9

747.6

Species

ΘD/K

Hexa-Cr7C3 will show a good thermal conductivity. The sound velocity in a solid is anisotropic which is determined by the symmetry of the crystal and propagation directions (Eq. (12) and (13)) [22]:

Cijkl ni nl − ρ v 2δ ik = 0

(12)

dw v(k ) = dk

(13)

[001] vl = [100] vt1 = [010] vt2 =

C44 ρ (15)

C33 ρ C44 ρ

Orthorhombic crystal class: [100]

[100] vl = [010] vt1 = [001] vt2 =

C11 ρ (16)

C66 ρ C55 ρ

[010]

For example, the pure transverse and longitudinal modes can only be found for [100], [010] and [001] directions in a orthorhombic crystal and the sound propagating modes in other directions are the quasi-transverse or quasi-longitudinal waves. In this study, the pure propagating modes of Hexa-Cr7C3 and Ortho-Cr7C3 are only considered: [100] and [001] directions for hexagonal crystal; [100], [010] and [001] directions for orthorhombic crystal. In different directions for crystals, the sound velocities can be written as: Hexagonal crystal class: [100]

[100] vl = ( C11 − C12 ) [010] vt1 = C11 ρ [001] vt 2 = C44 ρ

[100]

2ρ (14)

[010] vl = [100] vt1 = [001] vt2 =

C22 ρ C66 ρ

(17)

C44 ρ

[001]

[001] vl = [100] vt1 = [010] vt2 =

C33 ρ C55 ρ

(18)

C44 ρ

Where Cijkl is the elastic constants, ni and nl are the polarization and propagation directions, respectively, w is the phonon frequency and v is the sound velocity (vl: the longitudinal wave of sound velocity, vt1: the first transverse wave of sound velocity, vt2: the second transverse wave of sound velocity). The results are listed in Table 5.

Table 5 Anisotropic sound velocities of Hexa-Cr7C3 and Ortho-Cr7C3 compounds, and the unit of velocity (v) is km/s Direction Hexa-Cr7C3 Ortho-Cr7C3

3

[100]vl 4.833 7.836

[100] [010]vt1 8.749 4.155

[001]vt2 4.488 4.630

[010]vl 7.408

Conclusions

1) The cohesive energy and formation enthalpy of these compounds can be calculated using first principles and the results reveal that they are thermodynamically stable. Moreover, the Ortho-Cr7C3 is more stable than Hexa-Cr7C3. 2) The elastic constants and mechanical moduli can be obtained using stress-strain method and Voigt-Reuss-Hill approximation, respectively. The calculated elastic constants satisfy the Born-Huang’s stability criterions, which imply that these compounds are mechanically stable. 3) Several anisotropic indexes can be estimated to illustrate the mechanical anisotropy. The results show that the elastic anisotropic of Ortho-Cr7C3 is stronger than that of Hexa-Cr7C3. Moreover, the Debye temperatures of these compounds can be

[010] [100]vt1 4.155

[001]vt2 4.758

[001]vl 4.488 7.985

[001] [100]vt1 7.805 4.630

[010]vt2 4.488 4.758

also estimated, which indicate that the Debye temperature of Hexa-Cr7C3 is larger than that of Ortho-Cr7C3.

References 1 Liu Yangzhen, Jiang Yehua, Feng Jing et al. Physica B[J], 2013, 419: 45 2 Henriksson Koe, Sandberg N, Wallenius J. Applied Physics Letters[J], 2008, 93: 191912 3 Haines J, Léger J M, Chateau C et al. Journal of Physics: Condensed Matter[J], 2001, 13: 2447 4 Westgren A. Transactions Metallurgical Society AIME[J], 1961, 221: 479 5 Rouault M A, Herpin P, Frchart M R. Annales de Chimie France[J], 1970, 5: 461

2906

Liu Yangzhen et al. / Rare Metal Materials and Engineering, 2014, 43(12): 2903-2907

6 Jiang Chao. Applied Physics Letters[J], 2008, 92: 041909 7 Segall M D, Lindan P J D, Probert M J et al. Journal of Physics: Condensed Matter[J], 2002, 14: 2717 8 Perdew J P, Burke K, Ernzerhof M. Physical Review Letters[J], 1996, 77: 3865 9 Monkhorst H J, Pack J D. Physical Review B[J], 1976, 13: 5188 10 Pfrommer B G, Côté M, Louie S G et al. Journal of Computational Physics[J], 1997, 131: 233 11 Wu Zhijian, Zhao Erjun, Xiang Hongping et al. Physical Review B[J], 2007, 76: 054115 12 Zhou Wei, Liu Lijuan, Li Baoling et al. Computational Materials Science[J], 2009, 46: 921 13 Xiao B, Feng J, Zhou C T et al. Journal of Applied Physics[J], 2011, 109: 023507 14 Jiang J H, Kim I G, Bhadeshia H K D H. Computational Materials Science[J], 2009, 44: 1319

15 Yang Y, Lu H, Yu C et al. Journal of Alloys and Compounds[J], 2009, 485: 542 16 Liang Yongcheng, Guo Wanlin, Fang Zhong. Acta Physica Sinica[J], 2007, 56: 4847 17 Li Yefei, Gao Yimin, Xiao Bing et al. Journal of Alloys and Compounds[J], 2010, 502: 28 18 Jiang Xue, Zhao Jijun, Jiang Xin. Computational Materials Science[J], 2011, 50: 2287 19 Ranganathan S I, Starzewski M O. Physical Review Letters[J], 2008, 101: 055504 20 Ravindran P, Fast L, Korzhavyi P A et al. Journal of Applied Physics[J], 1998, 84: 4891 21 Wang Jiemin, Sun Jinfeng. Physica Status Solid (b)[J], 2010, 247: 921 22 Feng Jing, Xiao Bing, Chen Jing et al. Materials Design[J], 2011, 32: 3231

2907