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First-principles calculations of structural stability and mechanical properties of tungsten carbide under high pressure
Journal of Physics and Chemistry of Solids 75 (2014) 1234–1239
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Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
First-principles calculations of structural stability and mechanical properties of tungsten carbide under high pressure Xinting Li a, Xinyu Zhang a,n, Jiaqian Qin b,n, Suhong Zhang a, Jinliang Ning a, Ran Jing a, Mingzhen Ma a, Riping Liu a a b
State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, PR China Metallurgy and Materials Science Research Institute, Chulalongkorn University, Bangkok 10330, Thailand
art ic l e i nf o
a b s t r a c t
Article history: Received 2 February 2014 Received in revised form 6 May 2014 Accepted 3 June 2014 Available online 20 June 2014
The structural stability and mechanical properties of WC in WC-, MoC- and NaCl-type structures under high pressure are investigated systematically by first-principles calculations. The calculated equilibrium lattice constants at zero pressure agree well with available experimental and theoretical results. The formation enthalpy indicates that the most stable WC is in WC-type, then MoC-type finally NaCl-type. By the elastic stability criteria, it is predicted that the three structures are all mechanically stable. The elastic constants Cij, bulk modulus B, shear modulus G, Young's modulus E and Poisson's ratio ν of the three structures are studied in the pressure range from 0 to 100 GPa. Furthermore, by analyzing the B/G ratio, the brittle/ductile behavior under high pressure is assessed. Moreover, the elastic anisotropy of the three structures up to 100 GPa is also discussed in detail. & 2014 Elsevier Ltd. All rights reserved.
Keywords: C. High pressure D. Elastic properties D. Crystal structure
1. Introduction In recent decades transition metal carbides have drawn considerable attention owing to their outstanding mechanical and chemical properties such as high melting point, superior hardness and metallic conductivity [1–4]. Tungsten carbide, as one of such most promising engineering materials with the outstanding characteristics have been widely used in industrial machinery such as cutting tools and wearresistant coatings. Recent research demonstrates that tungsten carbide also plays an important role in multi-anvil high pressure systems as anvil materials and seats in diamond anvil cells [4,5]. WC exists in two polymorphic modifications: the low-temperature hexagonal (WC-type) and the high-temperature metastable cubic (NaCl-type) phase [6]. The NaCl-type structure can be synthesized by a rapid quenching process [7] or a plasma synthesis technique [8] at room temperature. Since W and Mo belong to the same group, we can suppose WC maybe also exists in MoC-type structure. In this work, our calculations indicate that WC might exist in MoC-type structure for its higher stability than the known NaCl-type structure. A number of studies have been performed to understand the properties of WC. Lee et al. [9] investigated the elastic constants of single crystal tungsten carbide under ambient conditions using
n
Corresponding authors. E-mail addresses: [email protected] (X. Zhang), [email protected] (J. Qin). http://dx.doi.org/10.1016/j.jpcs.2014.06.011 0022-3697/& 2014 Elsevier Ltd. All rights reserved.
high frequency ultrasonic pulse-echo measurements. Liu et al. [10] explored the structural and electronic properties of hexagonal WC by using the pseudopotential local-orbital method. Zhukov and Gubanov [11] confirmed the high bulk modulus (655 GPa) of hexagonal WC, which explains in part the superior properties of WC as a cutting material. Li et al. [12] calculated the chemical stability, elastic modulus, hardness and electronic structures of WC in WC- and NaCl-type structures by first-principles calculations. However, as a promising material for application in high pressure devices, the elastic properties of WC at high pressure have been little reported. Only Mishra et al. [13] presented the stability and elastic properties of WC in WC-type and NaCl-type phase under high pressure. Moreover, to our knowledge, no studies of the properties of WC in MoC-type structure have been reported. The present study is to systematically investigate the structure stability, mechanical properties and anisotropic behavior of WC with WC-, MoC- and NaCl-type structures under high pressure.
2. Computational methods All calculations in the present work were performed by firstprinciples calculation method based on the density functional theory (DFT), as implemented in the Vienna Ab initio Simulation Package (VASP) [14,15]. The projector-augmented wave (PAW) [16] method is employed to describe the electron-ion interactions. The exchange–correlation functional is described within the generalized
X. Li et al. / Journal of Physics and Chemistry of Solids 75 (2014) 1234–1239
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gradient approximation (GGA) of the Perdew–Burke–Ernzerhof (PBE) method [17]. The electron plane functions are expanded in plane-wave basis set with cutoff energy of 600.0 eV. Brillouin zone sampling was carried out using Monkhorst–Pack k-point meshes. The dense 13 13 12, 13 13 2 and 8 8 8 mesh parameters grids were utilized for WC-type, MoC-type and NaCl-type WC, respectively. The total energy was converged numerically to less than 1 meV/atom. The structures were optimized with the conjugategradient algorithm method. The elastic constants are defined by means of a Taylor expansion of the total energy, EðV; δÞ, for the system with respect to a small strain δ of the lattice primitive cell volume V. The energy of a strained system is expressed as follows [18]: ! EðV; δÞ ¼ EðV 0 ; 0Þ þV 0 ∑i τi ξi δi þ1=2∑C ij δi ξi δj ξj ;
ð1Þ
ij
where EðV 0 ; 0Þ is the energy of the unstrained system with equilibrium volume V 0 , τi is an element in the stress tensor, and ξi and ξj are the factors to take care of Voigt index.
3. Results and discussions
Fig. 2. The calculated formation enthalpies of three phases for WC under pressure.
Table 1 Calculated equilibrium lattice parameters (Å), equilibrium volume V0 (Å3/f.u.), formation enthalpy (eV/atom), bulk modulus B0 (GPa), its pressure derivative B0 and elastic constants (Cij) of WC compared with the previous experimental and theoretical results.
3.1. Structural properties NaCl-type
In order to investigate the thermodynamic stability of WC in the three considered structures, the total energy per formula unit as a function of cell volume are calculated. As shown in Fig. 1, the results reveal that the WC-type structure is the most stable phase and MoC-type is slightly more stable than NaCl-type WC. In order to check their stability further, we calculated the formation enthalpy at applied pressure (from 0 to 100 GPa) (shown in Fig. 2), which is expressed as ΔH ¼ H WC H W H C , where H WC , H W , and H C are the enthalpies of tungsten carbide, W metal, and C (the state is graphite under ambient pressure, and diamond under high pressure above 10 GPa), respectively. Several conclusions can be obtained from Fig. 1. First, at 0 GPa, the formation enthalpy is negative in WC-type, but positive in MoC- and NaCl-type, indicating WC-type phase is energetically favorable, while MoC- and NaCl-type structures are unstable under ambient pressure. Second, WC-type WC always has the minimum formation enthalpy in the whole applied pressure range implying its highest stability even at high pressure. Third, MoC-type WC possesses higher structural stability than NaCl-type for its lower formation enthalpy under high pressure. Fourth, with the increase of pressure, the formation enthalpy of each WC phase
Fig. 1. The calculated energies versus volumes of WC in NaCl-,WC- and MoC-type structures.
a0 c0 V0 Δr H B0 B0 C11 C12 C13 C33 C44
WC-type a
e
4.378(4.351 , 4.398 ) 20.985 0.319(0.341d) 375.2(374.4a,375.9d,) 4.38 755(696.5d) 191(215.5d)
100(122d)
MoC-type a
b
c
2.919(2.906 , 2.908 , 2.909 ) 2.844(2.825a, 2.841b, 2.829c) 20.990 0.139( 0.106d) 389.6(400.9a, 393d) 4.27 705(711.6d, 720f) 246(240.6d, 254f) 187(168d, 267f) 958(977.5d, 972f) 310(305.1d, 328f)
3.063 15.501 20.993 0.256 375.6 4.38 612 226 258 620 158
a
CASTEP calculation results in Ref. [19]. Experimental results measured by XRD in Ref. [20]. c Calculated by CASTEP with GGA scheme in Ref. [6]. d CASTEP method in Ref. [12]. e APW þLAPW method in Ref. [21]. f Experimental results measured by the high-frequency ultrasonic method in Ref. [9]. b
decreases, which illustrates that the stabilities of the three structures increase with increasing pressure. To study the ground-state properties, we calculated the equilibrium lattice constants and volumes of WC for the three structures. By fitting the energy-volume curves to the Birch–Murnaghan equation-of-state (EOS), we could obtain the bulk modulus and its pressure derivatives for the three considered structures. The structural fitting parameters and formation enthalpy are listed in Table 1, together with available experimental values and previous theoretical results [6,12,19–21] for comparison. As can be seen, our calculated fitting results at 0 GPa are in good agreement with the experiment values and theoretical data for WC- and NaCl-type WC, which confirms the reliability of our calculations. Also, the present formation enthalpy of WC-type and NaCl-type WC agree well with the data calculated with the plane wave pseudopotential method. For MoC-type WC, there is not available data for comparison, so our result could be a prediction for future studies. To compare the compressibility of the three considered phases of WC under pressure, the normalized volume V/V0 as a function of pressure is plotted in Fig. 3. The compression data for the wellknown superhard materials diamond and c-BN are also calculated for comparison. It is clearly found that all the three considered
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Fig. 3. The normalized volume of WC for three structures, c-BN and diamond as a function of pressure along with available experimental values [5]. The insert shows the enlarged view of normalized volume of WC-type structure and experimental data at the pressure range 0–30 GPa.
stability under hydrostatic pressure can be displayed as C 044 4 0, C 011 4 jC 012 j, ðC 011 þ C 012 ÞC 033 4 2C 02 13 . The mechanical stability criteria for cubic crystal under hydrostatic pressure are written as:C 011 4 0, C 044 4 0, C 011 4jC 012 j, ðC 011 þ 2C 012 Þ 4 0, where C 0ii ¼ C ii Pði ¼ 1; 3; 4Þ, C 012 ¼ C 12 þ P, C 013 ¼ C 13 þ P [25]. We verified that the three structures of WC all satisfy the mechanical stability criteria at the studied pressure, suggesting that they are mechanically stable up to 100 GPa. The elastic constants of the three structures as a function of pressure are presented in Fig. 5. It can be seen that, for all three phases, the elastic constants increase monotonically with increasing pressure. It is noteworthy that, for the hexagonal phases, C11 and C33 are much larger than the other constants, indicating they are very incompressible under uniaxial stress along x- or z- axis. From the elastic constants, the bulk modulus B, shear modulus G, Young's modulus E and Poisson's ratio ν can be estimated by the Voigt–Reuss–Hill approximation [26]. For a hexagonal structure, the Voigt bulk modulus (BV), Reuss bulk modulus (BR), Voigt shear modulus (GV) and Reuss shear modulus (GR) are expressed as the following formulae: 1 BV ¼ ½2ðC 11 þ C 12 Þ þ 4C 13 þ C 33 9
ð2Þ
BR ¼ C 2 =M
ð3Þ
GV ¼
1 ðM þ12C 44 þ 12C 66 Þ 30
ð4Þ
GR ¼
i h i 5h 2 C C 44 C 66 = 3BV C 44 C 66 þ C 2 ðC 44 þ C 66 Þ 2
ð5Þ
where
Fig. 4. The normalized lattice parameters versus pressure for WC in three structures.
structures are more incompressible than c-BN, but more compressible than diamond. We also noticed that WC-type structure has the highest compressibility among the three considered structures. The results indicate that the WC of the three structures are all ultra-incompressible material. Morever, the pressure dependence of normalized lattice parameters a/a0, c/c0 for the three structures is displayed in Fig. 4. It is clear that the equilibrium ratio a/a0 decreases quickly as pressure increases whereas c/c0 becomes moderate for the three considered structures, implying that the c-axis direction is stiffer than a-axis for both WC-type and MoCtype WC. 3.2. Mechanical properties Elastic constants play an important role in providing stability information and stiffness of materials. There are five independent elastic stiffness coefficients (C 11 ;C 33 ;C 44 ;C 12 ;C 13 ) for the hexagonal WC phase and three (C 11 ; C 44 ; C 12 ) for the cubic phase, which are computed with the strain–energy method [22–24]. The calculated results of these considered structures under high pressure are illustrated in Table 1 together with the available data [9,12]. It can be seen that our results are in good agreement with experimental data and other theoretical results for WC- and NaCl-type WC. The mechanical stability is a necessary condition for a crystal to exist. For a hexagonal crystal, the requirement of mechanical
M ¼ C 11 þ C 12 þ 2C 33 4C 13
ð6Þ
C 2 ¼ ðC 11 þ C 12 ÞC 33 2C 213
ð7Þ
C 66 ¼ ðC 11 C 12 Þ=2
ð8Þ
For a cubic structure, the elastic modulus values are defined as follows: BV ¼ BR ¼ ðC 11 þ 2C 12 Þ=3
ð9Þ
GV ¼ ðC 11 C 12 þ 3C 44 Þ=5
ð10Þ
GR ¼ 5ðC 11 C 12 ÞC 44 =½4C 44 þ 3ðC 11 C 12 Þ
ð11Þ
The polycrystalline modulus is the arithmetic average of the Voigt and Reuss moduli 1 BH ¼ ðBR þ BV Þ 2
ð12Þ
1 GH ¼ ðGR þ GV Þ 2
ð13Þ
For all considered structures, Young's modulus E and Poisson's ratio ν are obtained by the following expressions: E ¼ 9BG=ð3B þGÞ
ð14Þ
ν ¼ ð3B 2GÞ=½2ð3B þ GÞ
ð15Þ
Our calculated results are summarized in Table 2 along with other theoretical data [12]. It is obviously observed that the present results at zero pressure are consistent with previous data and the elastic moduli increase monotonically with increasing pressure. It is acknowledged that bulk modulus or shear modulus can be used to estimate the hardness of materials in an indirect way. Generally speaking, high hardness materials possess high bulk modulus and high shear modulus. It is distinct that WC-type WC has the largest bulk modulus (399 GPa) and shear modulus
X. Li et al. / Journal of Physics and Chemistry of Solids 75 (2014) 1234–1239
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Fig. 5. The elastic constants of WC in NaCl-,WC- and MoC-type structures as a function of pressure. Table 2 The calculated elastic modulus for WC under high pressure. P
NaCl-type
0 10 20 30 40 50 60 70 80 90 100 d
WC-type
MoC-type
B
G
E
ν
B
G
E
ν
B
G
E
ν
379 (376d) 423 465 507 548 588 626 665 704 742 779
154 (161d) 165 176 186 196 205 214 222 231 239 246
407 (443d) 439 469 498 525 551 576 601 624 647 669
0.321 (0.313d) 0.327 0.332 0.336 0.340 0.344 0.347 0.350 0.352 0.355 0.357
399 (393d) 442 484 525 566 605 644 682 720 757 794
282 (286d) 300 317 333 348 363 380 400 413 426 437
685 (691d) 734 780 824 867 909 953 1004 1041 1076 1108
0.214 (0.207d) 0.223 0.232 0.238 0.245 0.250 0.253 0.255 0.259 0.263 0.267
370
175
454
0.295
414 458 501 543 581 623 664 703 742 782
193 208 219 226 236 240 248 255 267 272
502 541 572 596 622 639 660 683 715 733
0.298 0.303 0.310 0.317 0.321 0.329 0.334 0.338 0.339 0.344
CASTEP method in Ref. [12].
(282 GPa) among the three types of WC, which suggests the ultraincompressible and hard characteristic of WC-type WC. It is worth noting that the bulk modulus of WC-type WC is higher than that of the synthesized ultra-incompressible materials such as ReB2 (354 GPa [27]), OsN2 (359 GPa [28]), IrN2 (327 GPa [29]) and PtN2 (272 GPa [30]), and the shear modulus is also higher than that of B6O (204 GPa) [31] and γ -B (227 GPa) [32], which are wellknown superhard materials. Young's modulus (E) and Poisson's ratio (ν) are the two important physical quantities describing the elastic behavior of materials. Young's modulus is defined as the ratio of the tensile stress to the corresponding tensile strain, and is used to provide a measure of the stiffness of solids. When the value of E is large, the material is stiff. From Table 2, we can see that WC-type WC possesses the largest stiffness among the three structures up to 100 GPa. Poisson's ratio ν reflects the stability of a crystal against shear deformation. Materials with bigger Poisson's ratio show better plasticity. It is evident that NaCl-type WC owns the best plasticity among the three competing structures in the pressure range from 0 to 100 GPa. It is known that the bulk modulus is the resistance to fracture, whereas the shear modulus is the resistance to plastic deformation [33], based on which the ratio of B/G was proposed by Pugh [34] to
Fig. 6. The ratio of B/G for WC-, MoC- and NaCl-type WC as a function of pressure.
predict the brittle or ductile behavior of materials. The critical value which separates ductile and brittle materials is about 1.75. The B/G ratio of three studied phases as a function of pressure is
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X. Li et al. / Journal of Physics and Chemistry of Solids 75 (2014) 1234–1239
shown in Fig. 6. It can be seen that, for all three structures, B/G increases monotonically with increasing pressure, indicating pressure can improve the ductility of these considered phases. For WC-type structure, the value reaches 1.75 at 80 GPa, implying WCtype WC transforms from brittle to ductile nature at 80 GPa. Nevertheless, for NaCl-type and MoC-type phases, the B/G ratio is always greater than 1.75, which indicates the two structures behave in a ductile manner in the pressure range from 0 to 100 GPa. 3.3. Elastic anisotropy The elastic anisotropy of crystals has an important implication in engineering science since it is highly correlated with the possibility of inducing microcracks in materials [35,36]. Therefore, it is important to calculate elastic anisotropy in order to improve its mechanical durability. Hence, we calculated the directional bulk modulus along the a axis (Ba) and c axis (Bc), which are defined as follows:
Table 4 The calculated anisotropy factors for three types of WC under pressure. P
0 10 20 30 40 50 60 70 80 90 100
NaCl-type
WC-type
MoC-type
A
ΔS1
ΔS2
ΔP
ΔS1
ΔS2
ΔP
ΔS1
ΔS2
0.517 0.488 0.464 0.445 0.429 0.415 0.403 0.392 0.383 0.375 0.367
2.822 3.121 3.402 3.676 3.936 4.192 4.443 4.688 4.928 5.167 5.397
0.354 0.320 0.294 0.272 0.254 0.239 0.225 0.213 0.203 0.194 0.185
1.359 1.346 1.335 1.325 1.317 1.310 1.294 1.280 1.276 1.274 1.272
1.039 1.014 0.993 0.975 0.960 0.947 0.940 0.933 0.922 0.912 0.902
1.351 1.388 1.420 1.449 1.472 1.491 1.471 1.453 1.473 1.495 1.522
1.013 0.999 0.989 0.978 0.976 0.957 0.961 0.961 0.956 0.952 0.957
1.137 0.896 0.777 0.687 0.611 0.573 0.530 0.497 0.477 0.459 0.437
0.816 0.983 1.080 1.152 1.269 1.305 1.406 1.486 1.526 1.567 1.655
calculated as follows:
ΔS 1 ¼
C 11 þ C 12 2C 13 C 33 C 13
ð21Þ
ΔS 2 ¼
2C 44 C 11 C 12
ð22Þ
dP Λ Ba ¼ a ¼ d a 2þα
ð16Þ
d P Ba ¼ Bc ¼ c dc α
ð17Þ
Λ ¼ 2ðC 11 þ C 12 Þ þ 4C 13 α þ C 33 α2
ð18Þ
ΔS 1 ¼
ð19Þ
The anisotropic factors for NaCl-type WC can be obtained from A ¼ ð2C 44 þ C 12 Þ=C 11 . If the material is completely isotropic, the value of A equals to 1, while any value larger or smaller than 1 indicates the degree of anisotropy. The larger deviation from one, means the higher anisotropy of materials. The calculated results for these considered phases are listed in Table 4. For the two hexagonal phases, it is easy to find that ΔP and ΔS1 decrease while ΔS2 increases with increasing pressure. The results indicate that WC in the two hexagonal structures is elastically anisotropic under high pressure. For NaCl-type WC, the value of A is smaller than one at zero pressure, and monotonically decreases with the increasing pressure, indicating the cubic phase possesses elastic anisotropy and the degree of anisotropy becomes gradually stronger under high pressure up to 100 GPa.
α¼
C 11 þ C 12 2C 13 C 33 C 13
The calculated results are presented in Table 3. The ratio Ba =Bc of WC- and MoC-type WC at 0 GPa are 0.748 and 0.889, respectively, which indicates WC-type WC is more anisotropic. It is clear that Ba and Bc of WC- and MoC-type WC increase with pressure increasing. Moreover, it is interesting to note that the directional bulk modulus along c axis is higher than that along a axis, implying that the compressibility along c axis is smaller than along a axis, which is consistent with the compression results and elastic constants calculation. The acoustic velocities can also be obtained from elastic constants by solving Christoffel equation [37]. Then, for the hexagonal phase the anisotropy of the compression wave (P) is obtained from the following equation:
ΔP ¼
C 33 : C 11
ð20Þ
The anisotropies of the wave polarized perpendicular to the basal plane (S1) and the polarized one in the basal plane (S2) are Table 3 The calculated directional bulk modulus Ba and Bc of WC-type and MoC-type WC under pressure. P
0 10 20 30 40 50 60 70 80 90 100
WC-type
MoC-type
Ba
Bc
Ba/Bc
Ba
Bc
Ba/Bc
1091 1212 1330 1445 1557 1667 1774 1881 1986 2089 2190
1458 1610 1756 1898 2039 2179 2317 2454 2589 2721 2849
0.748 0.753 0.757 0.761 0.763 0.765 0.766 0.766 0.767 0.768 0.769
1068 1187 1298 1401 1509 1638 1745 1850 1964 2073 2173
1200 1371 1550 1742 1916 2021 2175 2328 2450 2579 2727
0.889 0.866 0.837 0.804 0.788 0.811 0.802 0.795 0.802 0.804 0.797
For the cubic phase, ΔS1 and ΔS2 can be written as follows: 1
ΔS 2
¼
C 11 C 12 2C 44
ð23Þ
4. Conclusions The structural stability and mechanical properties of WC in three phases under high pressure are calculated by the plane-wave pseudopotentials within GGA in the frame of density functional theory. We have revealed that WC-type WC is energetically most stable among the considered phases, and MoC-type WC is slightly more stable than NaCl-type WC. The calculated equilibrium structural parameters and elastic modulus are in good agreement with previous experimental and theoretical results. The calculated elastic constants for three structures at all considered pressures satisfy the mechanical stability criteria, showing their mechanical stability. The elastic constants, bulk modulus, shear modulus, Young's modulus and Poisson's ratio of WC under pressure from 0 to 100 GPa are also calculated. The results indicate that WC-type WC is the most ultra-incompressible among the three considered structures. Furthermore, the ratio of B=G implies WC-type WC transforms from being brittle to ductile at 80 GPa, and NaCl-type and MoC-type WC keep a ductility manner up to 100 GPa. The elastic anisotropic factors for three types of WC suggest that they are all highly anisotropic.
X. Li et al. / Journal of Physics and Chemistry of Solids 75 (2014) 1234–1239
Acknowledgment This work was supported by the NBRPC (Grant 2013CB733000), NSFC (Grants 51171160/51171163). J.Q. would like to acknowledge support by Ratchadaphiseksomphot Endowment Fund of Chulalongkorn University (RES560530022-AM), and Key Laboratory of Metastable Materials Science and Technology, Yanshan University. References
[16] [17] [18] [19] [20] [21] [22] [23] [24]
[1] E.K. Storms, The Refractory Carbides, Academic Press, New York, 1967. [2] L. Toth, Transition Metal Carbides and Nitrides, Academic Press, New York, 1971. [3] P. Soni, G. Pagare, S.P. Sanyal, J. Phys. Chem. Solids 72 (2011) 810. [4] Y.W. Li, J.X. Yue, X. Liu, Y.M. Ma, T. Cui, G.T. Zou, Phys. Lett. A 374 (2010) 1880. [5] K.D. Litasov, A. Shatskiy, Y. Fei, A. Suzuki, E. Ohtani, K. Funakoshi, J. Appl. Phys. 108 (2010) 053513. [6] L. Liu, Y. Bi, J. Xu, X.R. Chen, Phys. B: Condens. Matter 413 (2013) 109. [7] R.H. Willens, E. Buehler, Appl. Phys. Lett. 7 (1965) 25. [8] P. Ronsheim, L. Toth, A. Mazza, E. Pfender, B. Mitrofanov, J. Mater. Sci. 16 (1981) 2665. [9] M. Lee, R.S. Gilmore, J. Mater. Sci. 17 (1982) 2657. [10] A.Y. Liu, R.M. Wentzcovitch, M.L. Cohen, Phys. Rev. B 38 (1988) 9483. [11] V. Zhukov, V. Gubanov, Solid State Commun. 56 (1985) 51. [12] Y.F. Li, Y.M. Gao, B. Xiao, T. Min, Z.J. Fan, S.Q. Ma, L.L. Xu, J. Alloys Compd. 502 (2010) 28. [13] V. Mishra, S. Chaturvedi, J. Phys. Chem. Solids 74 (2013) 509. [14] G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169. [15] G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758.
[25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
1239
P.E. Blöchl, Phys. Rev. B 50 (1994) 17953. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. L. Fast, J.M. Wills, B. Johansson, O. Eriksson, Phys. Rev. B 51 (1995) 17431. Y.F. Li, Y.M. Gao, B. Xiao, T. Min, Z.J. Fan, S.Q. Ma, D.W. Yi, Computational Mater. Sci. 50 (2011) 939. G.M. Amulele, M.H. Manghnani, S. Marriappan, X. Hong, F. Li, X. Qin, H.P. Liermann, J. Appl. Phys. 103 (2008) 113522. D.V. Suetin, I.R. Shein, A.L. Ivanovskii, J. Phys. Chem. Solids 70 (2009) 64. X.Y. Zhang, J.Q. Qin, X.W. Sun, Y.N. Xue, M.Z. Ma, R.P. Liu, Phys. Chem. Chem. Phys. 15 (2013) 20894. X.Y. Zhang, J.Q. Qin, J.L. Ning, X.W. Sun, X.T. Li, M.Z. Ma, R.P. Liu, J. Appl. Phys. 114 (2013) 183517. X.W. Sun, X.Y. Zhang, Y. Zhu, S.H. Zhang, J.Q. Qin, M.Z. Ma, R.P. Liu, J. Mater. Sci. 48 (2013) 7743. G.V. Sinko, N.A. Smirnov, J. Phys.: Condens. Matter 14 (2002) 6989. R. Hill, Proc. Phys. Soc. A 65 (1952) 349. Y.X. Wang, Appl. Phys. Lett. 91 (2007) 101904. Y.X. Wang, M. Arai, T. Sasaki, C.Z. Fan, Phys. Rev. B 75 (2007) 104110. H.P. Xiang, X.J. Liu, E.J. Zhao, J. Meng, Z.J. Wu, Appl. Phys. Lett. 91 (2007) 011903. R. Yu, Q. Zhan, X.F. Zhang, Appl. Phys. Lett. 88 (2006) 051913. Z.N. Ding, S.J. Zhou, Y.S. Zhao, Phys. Rev. B 70 (2004) 184117. J.Q. Qin, N. Nishiyama, H. Ohfuji, T. Shinmei, L. Lei, D.W. He, T. Irifune, Scr. Mater. 67 (2012) 257. P. Ravindran, L. Fast, P. Korzhavyi, B. Johansson, J. Wills, O. Eriksson, J. Appl. Phys. 84 (1998) 4891. S. Pugh, Philos. Mag. 45 (1954) 823. M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Clarendon Press, Oxford, 1954. A. Hao, X. Yang, L. Zhang, Y. Zhu, J. Phys. Chem. Solids 74 (2013) 1504. B.A. Auld, Acoustic Fields and Waves in Solids, Wiley, New York, 1973.
Contents lists available at ScienceDirect
Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
First-principles calculations of structural stability and mechanical properties of tungsten carbide under high pressure Xinting Li a, Xinyu Zhang a,n, Jiaqian Qin b,n, Suhong Zhang a, Jinliang Ning a, Ran Jing a, Mingzhen Ma a, Riping Liu a a b
State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, PR China Metallurgy and Materials Science Research Institute, Chulalongkorn University, Bangkok 10330, Thailand
art ic l e i nf o
a b s t r a c t
Article history: Received 2 February 2014 Received in revised form 6 May 2014 Accepted 3 June 2014 Available online 20 June 2014
The structural stability and mechanical properties of WC in WC-, MoC- and NaCl-type structures under high pressure are investigated systematically by first-principles calculations. The calculated equilibrium lattice constants at zero pressure agree well with available experimental and theoretical results. The formation enthalpy indicates that the most stable WC is in WC-type, then MoC-type finally NaCl-type. By the elastic stability criteria, it is predicted that the three structures are all mechanically stable. The elastic constants Cij, bulk modulus B, shear modulus G, Young's modulus E and Poisson's ratio ν of the three structures are studied in the pressure range from 0 to 100 GPa. Furthermore, by analyzing the B/G ratio, the brittle/ductile behavior under high pressure is assessed. Moreover, the elastic anisotropy of the three structures up to 100 GPa is also discussed in detail. & 2014 Elsevier Ltd. All rights reserved.
Keywords: C. High pressure D. Elastic properties D. Crystal structure
1. Introduction In recent decades transition metal carbides have drawn considerable attention owing to their outstanding mechanical and chemical properties such as high melting point, superior hardness and metallic conductivity [1–4]. Tungsten carbide, as one of such most promising engineering materials with the outstanding characteristics have been widely used in industrial machinery such as cutting tools and wearresistant coatings. Recent research demonstrates that tungsten carbide also plays an important role in multi-anvil high pressure systems as anvil materials and seats in diamond anvil cells [4,5]. WC exists in two polymorphic modifications: the low-temperature hexagonal (WC-type) and the high-temperature metastable cubic (NaCl-type) phase [6]. The NaCl-type structure can be synthesized by a rapid quenching process [7] or a plasma synthesis technique [8] at room temperature. Since W and Mo belong to the same group, we can suppose WC maybe also exists in MoC-type structure. In this work, our calculations indicate that WC might exist in MoC-type structure for its higher stability than the known NaCl-type structure. A number of studies have been performed to understand the properties of WC. Lee et al. [9] investigated the elastic constants of single crystal tungsten carbide under ambient conditions using
n
Corresponding authors. E-mail addresses: [email protected] (X. Zhang), [email protected] (J. Qin). http://dx.doi.org/10.1016/j.jpcs.2014.06.011 0022-3697/& 2014 Elsevier Ltd. All rights reserved.
high frequency ultrasonic pulse-echo measurements. Liu et al. [10] explored the structural and electronic properties of hexagonal WC by using the pseudopotential local-orbital method. Zhukov and Gubanov [11] confirmed the high bulk modulus (655 GPa) of hexagonal WC, which explains in part the superior properties of WC as a cutting material. Li et al. [12] calculated the chemical stability, elastic modulus, hardness and electronic structures of WC in WC- and NaCl-type structures by first-principles calculations. However, as a promising material for application in high pressure devices, the elastic properties of WC at high pressure have been little reported. Only Mishra et al. [13] presented the stability and elastic properties of WC in WC-type and NaCl-type phase under high pressure. Moreover, to our knowledge, no studies of the properties of WC in MoC-type structure have been reported. The present study is to systematically investigate the structure stability, mechanical properties and anisotropic behavior of WC with WC-, MoC- and NaCl-type structures under high pressure.
2. Computational methods All calculations in the present work were performed by firstprinciples calculation method based on the density functional theory (DFT), as implemented in the Vienna Ab initio Simulation Package (VASP) [14,15]. The projector-augmented wave (PAW) [16] method is employed to describe the electron-ion interactions. The exchange–correlation functional is described within the generalized
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gradient approximation (GGA) of the Perdew–Burke–Ernzerhof (PBE) method [17]. The electron plane functions are expanded in plane-wave basis set with cutoff energy of 600.0 eV. Brillouin zone sampling was carried out using Monkhorst–Pack k-point meshes. The dense 13 13 12, 13 13 2 and 8 8 8 mesh parameters grids were utilized for WC-type, MoC-type and NaCl-type WC, respectively. The total energy was converged numerically to less than 1 meV/atom. The structures were optimized with the conjugategradient algorithm method. The elastic constants are defined by means of a Taylor expansion of the total energy, EðV; δÞ, for the system with respect to a small strain δ of the lattice primitive cell volume V. The energy of a strained system is expressed as follows [18]: ! EðV; δÞ ¼ EðV 0 ; 0Þ þV 0 ∑i τi ξi δi þ1=2∑C ij δi ξi δj ξj ;
ð1Þ
ij
where EðV 0 ; 0Þ is the energy of the unstrained system with equilibrium volume V 0 , τi is an element in the stress tensor, and ξi and ξj are the factors to take care of Voigt index.
3. Results and discussions
Fig. 2. The calculated formation enthalpies of three phases for WC under pressure.
Table 1 Calculated equilibrium lattice parameters (Å), equilibrium volume V0 (Å3/f.u.), formation enthalpy (eV/atom), bulk modulus B0 (GPa), its pressure derivative B0 and elastic constants (Cij) of WC compared with the previous experimental and theoretical results.
3.1. Structural properties NaCl-type
In order to investigate the thermodynamic stability of WC in the three considered structures, the total energy per formula unit as a function of cell volume are calculated. As shown in Fig. 1, the results reveal that the WC-type structure is the most stable phase and MoC-type is slightly more stable than NaCl-type WC. In order to check their stability further, we calculated the formation enthalpy at applied pressure (from 0 to 100 GPa) (shown in Fig. 2), which is expressed as ΔH ¼ H WC H W H C , where H WC , H W , and H C are the enthalpies of tungsten carbide, W metal, and C (the state is graphite under ambient pressure, and diamond under high pressure above 10 GPa), respectively. Several conclusions can be obtained from Fig. 1. First, at 0 GPa, the formation enthalpy is negative in WC-type, but positive in MoC- and NaCl-type, indicating WC-type phase is energetically favorable, while MoC- and NaCl-type structures are unstable under ambient pressure. Second, WC-type WC always has the minimum formation enthalpy in the whole applied pressure range implying its highest stability even at high pressure. Third, MoC-type WC possesses higher structural stability than NaCl-type for its lower formation enthalpy under high pressure. Fourth, with the increase of pressure, the formation enthalpy of each WC phase
Fig. 1. The calculated energies versus volumes of WC in NaCl-,WC- and MoC-type structures.
a0 c0 V0 Δr H B0 B0 C11 C12 C13 C33 C44
WC-type a
e
4.378(4.351 , 4.398 ) 20.985 0.319(0.341d) 375.2(374.4a,375.9d,) 4.38 755(696.5d) 191(215.5d)
100(122d)
MoC-type a
b
c
2.919(2.906 , 2.908 , 2.909 ) 2.844(2.825a, 2.841b, 2.829c) 20.990 0.139( 0.106d) 389.6(400.9a, 393d) 4.27 705(711.6d, 720f) 246(240.6d, 254f) 187(168d, 267f) 958(977.5d, 972f) 310(305.1d, 328f)
3.063 15.501 20.993 0.256 375.6 4.38 612 226 258 620 158
a
CASTEP calculation results in Ref. [19]. Experimental results measured by XRD in Ref. [20]. c Calculated by CASTEP with GGA scheme in Ref. [6]. d CASTEP method in Ref. [12]. e APW þLAPW method in Ref. [21]. f Experimental results measured by the high-frequency ultrasonic method in Ref. [9]. b
decreases, which illustrates that the stabilities of the three structures increase with increasing pressure. To study the ground-state properties, we calculated the equilibrium lattice constants and volumes of WC for the three structures. By fitting the energy-volume curves to the Birch–Murnaghan equation-of-state (EOS), we could obtain the bulk modulus and its pressure derivatives for the three considered structures. The structural fitting parameters and formation enthalpy are listed in Table 1, together with available experimental values and previous theoretical results [6,12,19–21] for comparison. As can be seen, our calculated fitting results at 0 GPa are in good agreement with the experiment values and theoretical data for WC- and NaCl-type WC, which confirms the reliability of our calculations. Also, the present formation enthalpy of WC-type and NaCl-type WC agree well with the data calculated with the plane wave pseudopotential method. For MoC-type WC, there is not available data for comparison, so our result could be a prediction for future studies. To compare the compressibility of the three considered phases of WC under pressure, the normalized volume V/V0 as a function of pressure is plotted in Fig. 3. The compression data for the wellknown superhard materials diamond and c-BN are also calculated for comparison. It is clearly found that all the three considered
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Fig. 3. The normalized volume of WC for three structures, c-BN and diamond as a function of pressure along with available experimental values [5]. The insert shows the enlarged view of normalized volume of WC-type structure and experimental data at the pressure range 0–30 GPa.
stability under hydrostatic pressure can be displayed as C 044 4 0, C 011 4 jC 012 j, ðC 011 þ C 012 ÞC 033 4 2C 02 13 . The mechanical stability criteria for cubic crystal under hydrostatic pressure are written as:C 011 4 0, C 044 4 0, C 011 4jC 012 j, ðC 011 þ 2C 012 Þ 4 0, where C 0ii ¼ C ii Pði ¼ 1; 3; 4Þ, C 012 ¼ C 12 þ P, C 013 ¼ C 13 þ P [25]. We verified that the three structures of WC all satisfy the mechanical stability criteria at the studied pressure, suggesting that they are mechanically stable up to 100 GPa. The elastic constants of the three structures as a function of pressure are presented in Fig. 5. It can be seen that, for all three phases, the elastic constants increase monotonically with increasing pressure. It is noteworthy that, for the hexagonal phases, C11 and C33 are much larger than the other constants, indicating they are very incompressible under uniaxial stress along x- or z- axis. From the elastic constants, the bulk modulus B, shear modulus G, Young's modulus E and Poisson's ratio ν can be estimated by the Voigt–Reuss–Hill approximation [26]. For a hexagonal structure, the Voigt bulk modulus (BV), Reuss bulk modulus (BR), Voigt shear modulus (GV) and Reuss shear modulus (GR) are expressed as the following formulae: 1 BV ¼ ½2ðC 11 þ C 12 Þ þ 4C 13 þ C 33 9
ð2Þ
BR ¼ C 2 =M
ð3Þ
GV ¼
1 ðM þ12C 44 þ 12C 66 Þ 30
ð4Þ
GR ¼
i h i 5h 2 C C 44 C 66 = 3BV C 44 C 66 þ C 2 ðC 44 þ C 66 Þ 2
ð5Þ
where
Fig. 4. The normalized lattice parameters versus pressure for WC in three structures.
structures are more incompressible than c-BN, but more compressible than diamond. We also noticed that WC-type structure has the highest compressibility among the three considered structures. The results indicate that the WC of the three structures are all ultra-incompressible material. Morever, the pressure dependence of normalized lattice parameters a/a0, c/c0 for the three structures is displayed in Fig. 4. It is clear that the equilibrium ratio a/a0 decreases quickly as pressure increases whereas c/c0 becomes moderate for the three considered structures, implying that the c-axis direction is stiffer than a-axis for both WC-type and MoCtype WC. 3.2. Mechanical properties Elastic constants play an important role in providing stability information and stiffness of materials. There are five independent elastic stiffness coefficients (C 11 ;C 33 ;C 44 ;C 12 ;C 13 ) for the hexagonal WC phase and three (C 11 ; C 44 ; C 12 ) for the cubic phase, which are computed with the strain–energy method [22–24]. The calculated results of these considered structures under high pressure are illustrated in Table 1 together with the available data [9,12]. It can be seen that our results are in good agreement with experimental data and other theoretical results for WC- and NaCl-type WC. The mechanical stability is a necessary condition for a crystal to exist. For a hexagonal crystal, the requirement of mechanical
M ¼ C 11 þ C 12 þ 2C 33 4C 13
ð6Þ
C 2 ¼ ðC 11 þ C 12 ÞC 33 2C 213
ð7Þ
C 66 ¼ ðC 11 C 12 Þ=2
ð8Þ
For a cubic structure, the elastic modulus values are defined as follows: BV ¼ BR ¼ ðC 11 þ 2C 12 Þ=3
ð9Þ
GV ¼ ðC 11 C 12 þ 3C 44 Þ=5
ð10Þ
GR ¼ 5ðC 11 C 12 ÞC 44 =½4C 44 þ 3ðC 11 C 12 Þ
ð11Þ
The polycrystalline modulus is the arithmetic average of the Voigt and Reuss moduli 1 BH ¼ ðBR þ BV Þ 2
ð12Þ
1 GH ¼ ðGR þ GV Þ 2
ð13Þ
For all considered structures, Young's modulus E and Poisson's ratio ν are obtained by the following expressions: E ¼ 9BG=ð3B þGÞ
ð14Þ
ν ¼ ð3B 2GÞ=½2ð3B þ GÞ
ð15Þ
Our calculated results are summarized in Table 2 along with other theoretical data [12]. It is obviously observed that the present results at zero pressure are consistent with previous data and the elastic moduli increase monotonically with increasing pressure. It is acknowledged that bulk modulus or shear modulus can be used to estimate the hardness of materials in an indirect way. Generally speaking, high hardness materials possess high bulk modulus and high shear modulus. It is distinct that WC-type WC has the largest bulk modulus (399 GPa) and shear modulus
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Fig. 5. The elastic constants of WC in NaCl-,WC- and MoC-type structures as a function of pressure. Table 2 The calculated elastic modulus for WC under high pressure. P
NaCl-type
0 10 20 30 40 50 60 70 80 90 100 d
WC-type
MoC-type
B
G
E
ν
B
G
E
ν
B
G
E
ν
379 (376d) 423 465 507 548 588 626 665 704 742 779
154 (161d) 165 176 186 196 205 214 222 231 239 246
407 (443d) 439 469 498 525 551 576 601 624 647 669
0.321 (0.313d) 0.327 0.332 0.336 0.340 0.344 0.347 0.350 0.352 0.355 0.357
399 (393d) 442 484 525 566 605 644 682 720 757 794
282 (286d) 300 317 333 348 363 380 400 413 426 437
685 (691d) 734 780 824 867 909 953 1004 1041 1076 1108
0.214 (0.207d) 0.223 0.232 0.238 0.245 0.250 0.253 0.255 0.259 0.263 0.267
370
175
454
0.295
414 458 501 543 581 623 664 703 742 782
193 208 219 226 236 240 248 255 267 272
502 541 572 596 622 639 660 683 715 733
0.298 0.303 0.310 0.317 0.321 0.329 0.334 0.338 0.339 0.344
CASTEP method in Ref. [12].
(282 GPa) among the three types of WC, which suggests the ultraincompressible and hard characteristic of WC-type WC. It is worth noting that the bulk modulus of WC-type WC is higher than that of the synthesized ultra-incompressible materials such as ReB2 (354 GPa [27]), OsN2 (359 GPa [28]), IrN2 (327 GPa [29]) and PtN2 (272 GPa [30]), and the shear modulus is also higher than that of B6O (204 GPa) [31] and γ -B (227 GPa) [32], which are wellknown superhard materials. Young's modulus (E) and Poisson's ratio (ν) are the two important physical quantities describing the elastic behavior of materials. Young's modulus is defined as the ratio of the tensile stress to the corresponding tensile strain, and is used to provide a measure of the stiffness of solids. When the value of E is large, the material is stiff. From Table 2, we can see that WC-type WC possesses the largest stiffness among the three structures up to 100 GPa. Poisson's ratio ν reflects the stability of a crystal against shear deformation. Materials with bigger Poisson's ratio show better plasticity. It is evident that NaCl-type WC owns the best plasticity among the three competing structures in the pressure range from 0 to 100 GPa. It is known that the bulk modulus is the resistance to fracture, whereas the shear modulus is the resistance to plastic deformation [33], based on which the ratio of B/G was proposed by Pugh [34] to
Fig. 6. The ratio of B/G for WC-, MoC- and NaCl-type WC as a function of pressure.
predict the brittle or ductile behavior of materials. The critical value which separates ductile and brittle materials is about 1.75. The B/G ratio of three studied phases as a function of pressure is
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shown in Fig. 6. It can be seen that, for all three structures, B/G increases monotonically with increasing pressure, indicating pressure can improve the ductility of these considered phases. For WC-type structure, the value reaches 1.75 at 80 GPa, implying WCtype WC transforms from brittle to ductile nature at 80 GPa. Nevertheless, for NaCl-type and MoC-type phases, the B/G ratio is always greater than 1.75, which indicates the two structures behave in a ductile manner in the pressure range from 0 to 100 GPa. 3.3. Elastic anisotropy The elastic anisotropy of crystals has an important implication in engineering science since it is highly correlated with the possibility of inducing microcracks in materials [35,36]. Therefore, it is important to calculate elastic anisotropy in order to improve its mechanical durability. Hence, we calculated the directional bulk modulus along the a axis (Ba) and c axis (Bc), which are defined as follows:
Table 4 The calculated anisotropy factors for three types of WC under pressure. P
0 10 20 30 40 50 60 70 80 90 100
NaCl-type
WC-type
MoC-type
A
ΔS1
ΔS2
ΔP
ΔS1
ΔS2
ΔP
ΔS1
ΔS2
0.517 0.488 0.464 0.445 0.429 0.415 0.403 0.392 0.383 0.375 0.367
2.822 3.121 3.402 3.676 3.936 4.192 4.443 4.688 4.928 5.167 5.397
0.354 0.320 0.294 0.272 0.254 0.239 0.225 0.213 0.203 0.194 0.185
1.359 1.346 1.335 1.325 1.317 1.310 1.294 1.280 1.276 1.274 1.272
1.039 1.014 0.993 0.975 0.960 0.947 0.940 0.933 0.922 0.912 0.902
1.351 1.388 1.420 1.449 1.472 1.491 1.471 1.453 1.473 1.495 1.522
1.013 0.999 0.989 0.978 0.976 0.957 0.961 0.961 0.956 0.952 0.957
1.137 0.896 0.777 0.687 0.611 0.573 0.530 0.497 0.477 0.459 0.437
0.816 0.983 1.080 1.152 1.269 1.305 1.406 1.486 1.526 1.567 1.655
calculated as follows:
ΔS 1 ¼
C 11 þ C 12 2C 13 C 33 C 13
ð21Þ
ΔS 2 ¼
2C 44 C 11 C 12
ð22Þ
dP Λ Ba ¼ a ¼ d a 2þα
ð16Þ
d P Ba ¼ Bc ¼ c dc α
ð17Þ
Λ ¼ 2ðC 11 þ C 12 Þ þ 4C 13 α þ C 33 α2
ð18Þ
ΔS 1 ¼
ð19Þ
The anisotropic factors for NaCl-type WC can be obtained from A ¼ ð2C 44 þ C 12 Þ=C 11 . If the material is completely isotropic, the value of A equals to 1, while any value larger or smaller than 1 indicates the degree of anisotropy. The larger deviation from one, means the higher anisotropy of materials. The calculated results for these considered phases are listed in Table 4. For the two hexagonal phases, it is easy to find that ΔP and ΔS1 decrease while ΔS2 increases with increasing pressure. The results indicate that WC in the two hexagonal structures is elastically anisotropic under high pressure. For NaCl-type WC, the value of A is smaller than one at zero pressure, and monotonically decreases with the increasing pressure, indicating the cubic phase possesses elastic anisotropy and the degree of anisotropy becomes gradually stronger under high pressure up to 100 GPa.
α¼
C 11 þ C 12 2C 13 C 33 C 13
The calculated results are presented in Table 3. The ratio Ba =Bc of WC- and MoC-type WC at 0 GPa are 0.748 and 0.889, respectively, which indicates WC-type WC is more anisotropic. It is clear that Ba and Bc of WC- and MoC-type WC increase with pressure increasing. Moreover, it is interesting to note that the directional bulk modulus along c axis is higher than that along a axis, implying that the compressibility along c axis is smaller than along a axis, which is consistent with the compression results and elastic constants calculation. The acoustic velocities can also be obtained from elastic constants by solving Christoffel equation [37]. Then, for the hexagonal phase the anisotropy of the compression wave (P) is obtained from the following equation:
ΔP ¼
C 33 : C 11
ð20Þ
The anisotropies of the wave polarized perpendicular to the basal plane (S1) and the polarized one in the basal plane (S2) are Table 3 The calculated directional bulk modulus Ba and Bc of WC-type and MoC-type WC under pressure. P
0 10 20 30 40 50 60 70 80 90 100
WC-type
MoC-type
Ba
Bc
Ba/Bc
Ba
Bc
Ba/Bc
1091 1212 1330 1445 1557 1667 1774 1881 1986 2089 2190
1458 1610 1756 1898 2039 2179 2317 2454 2589 2721 2849
0.748 0.753 0.757 0.761 0.763 0.765 0.766 0.766 0.767 0.768 0.769
1068 1187 1298 1401 1509 1638 1745 1850 1964 2073 2173
1200 1371 1550 1742 1916 2021 2175 2328 2450 2579 2727
0.889 0.866 0.837 0.804 0.788 0.811 0.802 0.795 0.802 0.804 0.797
For the cubic phase, ΔS1 and ΔS2 can be written as follows: 1
ΔS 2
¼
C 11 C 12 2C 44
ð23Þ
4. Conclusions The structural stability and mechanical properties of WC in three phases under high pressure are calculated by the plane-wave pseudopotentials within GGA in the frame of density functional theory. We have revealed that WC-type WC is energetically most stable among the considered phases, and MoC-type WC is slightly more stable than NaCl-type WC. The calculated equilibrium structural parameters and elastic modulus are in good agreement with previous experimental and theoretical results. The calculated elastic constants for three structures at all considered pressures satisfy the mechanical stability criteria, showing their mechanical stability. The elastic constants, bulk modulus, shear modulus, Young's modulus and Poisson's ratio of WC under pressure from 0 to 100 GPa are also calculated. The results indicate that WC-type WC is the most ultra-incompressible among the three considered structures. Furthermore, the ratio of B=G implies WC-type WC transforms from being brittle to ductile at 80 GPa, and NaCl-type and MoC-type WC keep a ductility manner up to 100 GPa. The elastic anisotropic factors for three types of WC suggest that they are all highly anisotropic.
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Acknowledgment This work was supported by the NBRPC (Grant 2013CB733000), NSFC (Grants 51171160/51171163). J.Q. would like to acknowledge support by Ratchadaphiseksomphot Endowment Fund of Chulalongkorn University (RES560530022-AM), and Key Laboratory of Metastable Materials Science and Technology, Yanshan University. References
[16] [17] [18] [19] [20] [21] [22] [23] [24]
[1] E.K. Storms, The Refractory Carbides, Academic Press, New York, 1967. [2] L. Toth, Transition Metal Carbides and Nitrides, Academic Press, New York, 1971. [3] P. Soni, G. Pagare, S.P. Sanyal, J. Phys. Chem. Solids 72 (2011) 810. [4] Y.W. Li, J.X. Yue, X. Liu, Y.M. Ma, T. Cui, G.T. Zou, Phys. Lett. A 374 (2010) 1880. [5] K.D. Litasov, A. Shatskiy, Y. Fei, A. Suzuki, E. Ohtani, K. Funakoshi, J. Appl. Phys. 108 (2010) 053513. [6] L. Liu, Y. Bi, J. Xu, X.R. Chen, Phys. B: Condens. Matter 413 (2013) 109. [7] R.H. Willens, E. Buehler, Appl. Phys. Lett. 7 (1965) 25. [8] P. Ronsheim, L. Toth, A. Mazza, E. Pfender, B. Mitrofanov, J. Mater. Sci. 16 (1981) 2665. [9] M. Lee, R.S. Gilmore, J. Mater. Sci. 17 (1982) 2657. [10] A.Y. Liu, R.M. Wentzcovitch, M.L. Cohen, Phys. Rev. B 38 (1988) 9483. [11] V. Zhukov, V. Gubanov, Solid State Commun. 56 (1985) 51. [12] Y.F. Li, Y.M. Gao, B. Xiao, T. Min, Z.J. Fan, S.Q. Ma, L.L. Xu, J. Alloys Compd. 502 (2010) 28. [13] V. Mishra, S. Chaturvedi, J. Phys. Chem. Solids 74 (2013) 509. [14] G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169. [15] G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758.
[25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
1239
P.E. Blöchl, Phys. Rev. B 50 (1994) 17953. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. L. Fast, J.M. Wills, B. Johansson, O. Eriksson, Phys. Rev. B 51 (1995) 17431. Y.F. Li, Y.M. Gao, B. Xiao, T. Min, Z.J. Fan, S.Q. Ma, D.W. Yi, Computational Mater. Sci. 50 (2011) 939. G.M. Amulele, M.H. Manghnani, S. Marriappan, X. Hong, F. Li, X. Qin, H.P. Liermann, J. Appl. Phys. 103 (2008) 113522. D.V. Suetin, I.R. Shein, A.L. Ivanovskii, J. Phys. Chem. Solids 70 (2009) 64. X.Y. Zhang, J.Q. Qin, X.W. Sun, Y.N. Xue, M.Z. Ma, R.P. Liu, Phys. Chem. Chem. Phys. 15 (2013) 20894. X.Y. Zhang, J.Q. Qin, J.L. Ning, X.W. Sun, X.T. Li, M.Z. Ma, R.P. Liu, J. Appl. Phys. 114 (2013) 183517. X.W. Sun, X.Y. Zhang, Y. Zhu, S.H. Zhang, J.Q. Qin, M.Z. Ma, R.P. Liu, J. Mater. Sci. 48 (2013) 7743. G.V. Sinko, N.A. Smirnov, J. Phys.: Condens. Matter 14 (2002) 6989. R. Hill, Proc. Phys. Soc. A 65 (1952) 349. Y.X. Wang, Appl. Phys. Lett. 91 (2007) 101904. Y.X. Wang, M. Arai, T. Sasaki, C.Z. Fan, Phys. Rev. B 75 (2007) 104110. H.P. Xiang, X.J. Liu, E.J. Zhao, J. Meng, Z.J. Wu, Appl. Phys. Lett. 91 (2007) 011903. R. Yu, Q. Zhan, X.F. Zhang, Appl. Phys. Lett. 88 (2006) 051913. Z.N. Ding, S.J. Zhou, Y.S. Zhao, Phys. Rev. B 70 (2004) 184117. J.Q. Qin, N. Nishiyama, H. Ohfuji, T. Shinmei, L. Lei, D.W. He, T. Irifune, Scr. Mater. 67 (2012) 257. P. Ravindran, L. Fast, P. Korzhavyi, B. Johansson, J. Wills, O. Eriksson, J. Appl. Phys. 84 (1998) 4891. S. Pugh, Philos. Mag. 45 (1954) 823. M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Clarendon Press, Oxford, 1954. A. Hao, X. Yang, L. Zhang, Y. Zhu, J. Phys. Chem. Solids 74 (2013) 1504. B.A. Auld, Acoustic Fields and Waves in Solids, Wiley, New York, 1973.