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First-principles prediction of the hardness of fluorite TiO2
ARTICLE IN PRESS Physica B 404 (2009) 79–81
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
First-principles prediction of the hardness of fluorite TiO2 Wei Lu a,, Hai Wang a, Yongming Hu a,b, Haitao Huang a, Haoshuang Gu b a b
Department of Applied Physics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, PR China State Key Laboratory of Ferro and Piezoelectric Materials and Devices of Hubei Province, Faculty of Physics and Electronic Technology, Hubei University, Wuhan 430062, PR China
a r t i c l e in fo
abstract
Article history: Received 17 June 2008 Received in revised form 6 October 2008 Accepted 9 October 2008
We present first-principles calculations on the elastic constants, ideal tensile and shear strengths of cubic TiO2 with a fluorite structure (f-TiO2). The results show that f-TiO2 is mechanically stable at the ground-state structure. Both shear modulus and value of hardness predicted indicate that the hardness of f-TiO2 is comparable with TiN but is lower than TiB2. The ideal shear strength results suggest that the hardness of f-TiO2 is reduced because of the lower stress on the shear (111) /11¯ 0S slip system. & 2008 Elsevier B.V. All rights reserved.
Keywords: Ab-initio Hard material Mechanical properties TiO2
1. Introduction Synthesis and design of superhard materials are always of great scientific interest due to their variety of industrial applications [1]. With respect to synthesis of intrinsic superhard materials, nowadays, most work mainly focus on two aspects (1) compounds composed of light elements—boron, carbon, nitrogen and oxygen (B–C–N–O systems), such as BN, B6O and C3N4 and (2) introducing light elements into transition metals to possess higher bulk and shear modulus, such as ReB2, IrN2 [2–6]. Depending on that strategy, the results of various dioxides suggest that TiO2 could have a series of high-pressure phases with a hardness possibly approaching that of diamond [7,8]. TiO2 has a large number of crystalline polymorphs including rutile, anatase, brookite (the familiar natural titania phase) and many other highpressure forms [9]. To date, many high-pressure TiO2 phases have been successfully synthesized. The cubic TiO2 polymorph was proposed [10–13] and experimentally synthesized recently [9]. The experimental bulk modulus is 202 GPa of the cubic TiO2 phase with a fluorite structure (referred to as f-TiO2) as suggested by Mattesini et al. A recent theoretical investigation [14] shows that f-TiO2 is a highly incompressible solid with a large bulk modulus value (395 GPa) approaching that of ultrahard cotunnite TiO2 (431 GPa) [8]. This suggests that f-TiO2 may be a potential ultrahard material. However, the theoretical results within abinitio codes reported by Liang et al. [15] show that f-TiO2 could not be regarded as a potential candidate of superhard material. The Corresponding author.
E-mail address: [email protected] (W. Lu). 0921-4526/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2008.10.012
discrepancy in previous results inspires us to perform a detailed investigation on f-TiO2. In the present work, the mechanical stability of TiO2 with a fluorite structure was examined. The result shows that it is stable. Calculations of the hardness and intrinsic mechanical properties were performed to answer whether f-TiO2 is a potential ultrahard material.
2. Theoretical details The calculations presented in this work were performed within the density functional theory (DFT) using the full-potential linearized augmented plane wave method along with the local orbitals (lo) method with the package WIEN2K [16]. Both local density approximation (LDA) [17] the generalized gradient approximation (GGA) [18] were used. Relativistic effects were taken into account within the scalar-relativistic approximation. To obtain a satisfactory convergence, the basis function was expanded up to RMTKMax ¼ 8. The k-point sampling in the total Brillouin zone was conducted with a 10 10 10 mesh. The average values of LDA and GGA lattice constant and elastic constants were used because the LDA (GGA) usually underestimates (overestimates) the lattice constant and overestimates (underestimates) the elastic constants. Stress calculation was carried out using the normal-conserving pseudopotential with a plane wave basis set in the CASTEP code [19]. The GGA exchange correlation function, k point grid 9 9 9 in the Brillouin zone and energy cutoff of 500 eV were used for f-TiO2. The stress– strain dependence was calculated using the method described
ARTICLE IN PRESS 80
W. Lu et al. / Physica B 404 (2009) 79–81
previously [20,21] and was extensively used on strong solids recently [22–25]. The lattice vector was incrementally deformed in the direction of the applied strain. At each step, the atomic basis vectors orthogonal to the applied strain and the atomic site in the deformed unit cell were relaxed.
3. Results and discussion The lattice constants and bulk modulus that are determined by least squares fit of the E–V curves (total energy versus volume) calculated by the DFT to BirchMurnaghan’s equation of state are summarized in Table 1. It can be seen that the calculated lattice constants in the present work agree well with previous theoretical and experimental values. The bulk modulus with the LDA exchange correlation function in the present work agrees well with other theoretical results [11,15,26]. The GGA result of bulk modulus in this work is closer to the with experiment result [9] but obviously lower than the result reported by Swamy and Muddle [14]. Owing to the cubic structure of f-TiO2, there are only 3 independent elastic stiffness constants c11, c12 and c44. As a result, three types of strain, i.e., the volume change, the volumeconservative tetragonal change and rhombohedral distortion, were applied to the equilibrium structure to calculate these elastic constants. Table 2 lists the calculated elastic properties of f-TiO2. The mechanical stability of a crystal requires the strain energy to be positive, which for a cubic crystal implies [27] C 44 40; C 11 4jc12 j; and C 11 þ 2C 12 40 The calculated 3 independent elastic constants for f-TiO2 satisfy these stability criteria, thus suggesting that f-TiO2 is mechanically stable. As is known, the elastic constant C44 is one of the important parameters indirectly governing the intrinsic hardness. The present average elastic constant C44 is 58.6 GPa, which is consistent with the results (40–73 GPa) reported by Liang et al. [15]. Except for that, the polycrystalline shear modulus G is still considered another important parameter governing indentation hardness. The polycrystalline shear modulus of G ¼ 142.9 GPa, calculated from the single-crystal elastic constants based on the Table 1 Lattice constants and bulk moduli of f-TiO2. Method
a (A˚)
B (GPa)
Reference
LDA GGA PW-LDA GGA B3LYP BSTATE-LDA BSTATE-GGA LCAO-HF LCAO-LDA Experiment
4.744 4.841 4.860 4.831 4.822 4.729 4.822 4.794 4.748 4.870
293.6 249.3 282 395 390 324 272 331 308 202
This work This work [11] [14] [14] [15] [15] [26] [26] [9]
Voigt method [28], is obviously lower than the values (227–239 GPa) reported by Swamy and Muddle [14]. The polycrystalline shear modulus of f-TiO2 is comparable with that of TiN [29–31] but lower than that of TiB2 [31], which means that the hardness of f-TiO2 locates between TiN and TiB2. The estimat ion of theoretical hardness has been extensively developed and many models have been proposed [31–33]. Recently, a model based on first-principle calculations was proposed for calculating the hardness of covalent and ionic crystals [34]. Then, another approach was presented, that does avoid first-principle calculations, but gives a trustworthy value for many recently studied materials [35]. We calculate the hardness of f-TiO2 based on the semiempirical microscopic model (for detailed expression referring [35]). The radii of Ti and O are taken from Pearson’s handbook [36], that is, r(Ti) ¼ 1.46 and r(O) ¼ 0.89. In this calculation, we also use the same constants C ¼ 1450 and s ¼ 2.8 as the ones in Ref. [35]. The calculated hardness using the average lattice constants from GGA and LDA is Haver ¼ 17.1 GPa. The hardness of TiN and TiB2 calculated using the same method is HTiN ¼ 16.8 GPa and HTiB2 ¼ 31:6 GPa, respectively [35], which agrees with previous results [37]. The theoretical hardness with the result Hf -TiO2 ¼ 13 GPa was reported by Liang et al. [15] which is almost consistent with our present result. It can be seen that the hardness result of f-TiO2 is comparable with TiN but lower than TiB2. This agrees well with our prediction according to the polycrystalline shear modulus. In order to understand this phenomenon more clearly, ideal tensile and shear strengths of f-TiO2 were also calculated. To check the reliability of the present method, the ideal tensile and shear strengths of TiN are also computed. The ideal tensile stress along /1 0 0S, /11 0S and /111S are 30.2, 51.4 and 100.7 GPa, respectively, showing anisotropy of tensile strengths for TiN. A good agreement is achieved between the present result and that in the previous work [25]. The ideal shear stress on the easiest slip system (11 0) /11¯ 0S is 28.6 GPa, which also agrees well with previous results [22,25,38]. The lattice constant of f-TiO2 given by CASTEP is 4.81 A˚, which agrees well with previous results [11,14,15]. Fig. 1 shows the calculated strengths for f-TiO2 along /1 0 0S, /11 0S and /111S for tension and on (11 0) /11¯ 0S, (111) /11¯ 0S slip system for shear. The tensile strength shows anisotropy for f-TiO2. This phenomenon is similar to fcc-TiN, but the highest tensile strength for f-TiO2 is along /1 0 0S. The ideal shear stress on the (11 0) /11¯ 0S slip system of about 52.4 GPa for f-TiO2 is higher than that
Table 2 Calculated equilibrium lattice parameter, elastic stiffness constants cij (GPa), polycrystalline shear modulus G (GPa), Youngs modulus E (GPa), and Poissons ratio n of f-TiO2.
LDA GGA Ave.
a (A˚)
C11
C12
C44
G
E
n
4.744 4.841 4.793
670.3 589.98 630.1
103.5 79.6 91.6
68.6 48.5 58.6
154.5 131.2 142.9
394.1 334.9 364.5
0.275 0.276 0.276
Fig. 1. (Color online) Calculated tension and shear stress–strain relationships for f-TiO2.
ARTICLE IN PRESS W. Lu et al. / Physica B 404 (2009) 79–81
of TiN, but the ideal shear stress on (111) /11¯ 0S of about 15.7 GPa for f-TiO2 is lower than that of TiN [25]. As known, hardness measurements test the resistance of a material to permanent plastic deformation. The lower shear stress on (111) /11¯ 0S slip system for f-TiO2 will decrease the hardness of this material. So the hardness of f-TiO2 is only comparable with TiN but is lower than that of TiB2. To summarise, the elastic constants of cubic TiO2 with a fluorite structure suggest that f-TiO2 is mechanically stable. The shear modulus indicates that the hardness of f-TiO2 is comparable with some known hard materials, e.g. TiN, but is lower than TiB2. The theoretical hardness of f-TiO2 is Haver ¼ 17.1 GPa. The hardness prediction based on the semiempirical microscopic model is consistent with that of shear modulus. The hardness of f-TiO2 is reduced because of the lower shear stress on the (111) /11¯ 0S slip system.
Acknowledgement One of the authors (W. L.) would like to thank R.F. Zhang for valuable advice for stress–strain calculation and to Prof. He for providing the authority of using WIEN2k software to carry out this work. This work was supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region (Project: PolyU 5171/07E) and the Hong Kong Polytechnic University (Project no.G-YF71). References [1] R.B. Kaner, J.J. Gilman, S.H. Tolbert, Science 308 (2005) 1268. [2] D.M. Teter, R.J. Hemley, Science 53 (1996) 271. [3] H.-Y. Chung, M.B. Weinberger, J.B. Levine, A. Kavner, J.M. Yang, S.H. Tolbert, R.B. Kaner, Science 316 (2007) 436. [4] J. Haines, J.M. Leger, G. Bocquillon, Annu. Rev. Mater. Res. 31 (2001) 1. [5] V.V. Brazhkin, A.G. Lyapin, R.J. Hemley, Philos. Mag. A 82 (2002) 231.
81
[6] R. Yu, Q. Zhan, L.C. De Jonghe, Angew. Chem. Int. Ed. 46 (2007) 1136. [7] B. Silvi, A. Fahmi, C. Minot, M. Causa, Phys. Rev. B 47 (1993) 11717. [8] L.S. Dubrovinsky, N.A. Dubrovinskaia, V. Swamy, J. Muscat, N.M. Harrison, R. Ahuja, B. Holm, B. Johansson, Nature—London 410 (2001) 653. [9] M. Mattesini, J.S. de Almeida, L. Dubrovinsky, N. Dubrovinskaia, B. Johansson, R. Ahuja, Phys. Rev. B 70 (2004) 212101. [10] J. Haines, J.M. Le´ger, Phys. Rev. B 48 (1993) 13344. [11] J.K. Dewhurst, J.E. Lowther, Phys. Rev. B 54 (1996) R3673. [12] D.Y. Kim, J.S. de Almeida, L. Kocˇi, R. Ahuja, Appl. Phys. Lett. 90 (2007) 171903. [13] M. Mattesini, J.S. de Almeida, L. Dubrovinsky, N. Dubrovinskaia, B. Johansson, R. Ahuja, Phys. Rev. B 70 (2004) 115101. [14] V. Swamy, B.C. Muddle, Phys. Rev. Lett. 98 (2007) 035502. [15] Y.C. Liang, B. Zhang, J.Z. Zhao, Phys. Rev. B 77 (2008) 094126. [16] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Lutiz, WIEN2k, An augmented plane wave+local orbitals program for calculating crystal properties, in: K. Schwarz, T.U. Wien (Eds.), Austria, 2001, ISBN 3-9501031-1-2. [17] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. [18] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [19] G. Kresse, J. Furthmuller, Comput. Mater. Sci. 6 (1996) 15. [20] D. Roundy, C.R. Krenn, M.L. Cohen, J.W. Morris Jr., Phys. Rev. Lett. 82 (1999) 2713. [21] D. Roundy, C.R. Krenn, M.L. Cohen, J.W. Morris Jr., Philos. Mag. A 81 (2001) 1725. [22] S.H. Jhi, S.G. Louie, M.L. Cohen, J.W. Morris Jr., Phys. Rev. Lett. 87 (2001) 075503. [23] Y. Zhang, H. Sun, C.F. Chen, Phys. Rev. Lett. 93 (2004) 195504. [24] R.F. Zhang, S.H. Sheng, S. Veprek, Appl. Phys. Lett. 90 (2007) 191903. [25] R.F. Zhang, S.H. Sheng, S. Veprek, Appl. Phys. Lett. 91 (2007) 031906. [26] J. Muscat, V. Swamy, N.M. Harrison, Phys. Rev. B 65 (2002) 224112. [27] J.F. Nye, Physical Properties of Crystals, Oxford University Press, Oxford, 1985. [28] W. Voigt, Lehrbook Der Kristallphysik, seconded., Teubner, Leipsig, 1928. [29] I.N. Frantsevich, F.F. Voronov, S.A. Bokuta, in: I.N. Frantsevich (Ed.), Elastic Constants and Elastic Moduli of Metals and Insulators: Handbook, Naukova Dumka, Kiev, 1982, pp. 60–180. [30] D.G. Clerc, H.M. Ledbetter, J. Phys. Chem. Solids 59 (1998) 1071. [31] D.M. Teter, MRS Bull. 23 (1998) 22. [32] A.Y. Liu, M.L. Cohen, Science 245 (1989) 841. [33] F.M. Gao, J.L. He, E.D. Wu, S.M. Liu, D.L. Yu, D.C. Li, S.Y. Zhang, Y.J. Tian, Phys. Rev. Lett. 91 (2003) 015502. [34] A. Sˇimu˚nek, J. Vacka´rˇ, Phys. Rev. Lett. 96 (2006) 085501. [35] A. Sˇimu˚nek, Phys. Rev. B 75 (2007) 172108. [36] W.B. Pearson, The Crystal Chemistry and Physics of Metals and Alloys, Wiley, New York, 1972 (p. 151, Table 4-4). [37] V.V. Brazhkin, A.G. Lyapin, Philos. Mag. A 82 (2002) 231. [38] T. Liao, J.Y. Wang, Y.C. Zhou, Phys. Rev. B 73 (2006) 214109.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
First-principles prediction of the hardness of fluorite TiO2 Wei Lu a,, Hai Wang a, Yongming Hu a,b, Haitao Huang a, Haoshuang Gu b a b
Department of Applied Physics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, PR China State Key Laboratory of Ferro and Piezoelectric Materials and Devices of Hubei Province, Faculty of Physics and Electronic Technology, Hubei University, Wuhan 430062, PR China
a r t i c l e in fo
abstract
Article history: Received 17 June 2008 Received in revised form 6 October 2008 Accepted 9 October 2008
We present first-principles calculations on the elastic constants, ideal tensile and shear strengths of cubic TiO2 with a fluorite structure (f-TiO2). The results show that f-TiO2 is mechanically stable at the ground-state structure. Both shear modulus and value of hardness predicted indicate that the hardness of f-TiO2 is comparable with TiN but is lower than TiB2. The ideal shear strength results suggest that the hardness of f-TiO2 is reduced because of the lower stress on the shear (111) /11¯ 0S slip system. & 2008 Elsevier B.V. All rights reserved.
Keywords: Ab-initio Hard material Mechanical properties TiO2
1. Introduction Synthesis and design of superhard materials are always of great scientific interest due to their variety of industrial applications [1]. With respect to synthesis of intrinsic superhard materials, nowadays, most work mainly focus on two aspects (1) compounds composed of light elements—boron, carbon, nitrogen and oxygen (B–C–N–O systems), such as BN, B6O and C3N4 and (2) introducing light elements into transition metals to possess higher bulk and shear modulus, such as ReB2, IrN2 [2–6]. Depending on that strategy, the results of various dioxides suggest that TiO2 could have a series of high-pressure phases with a hardness possibly approaching that of diamond [7,8]. TiO2 has a large number of crystalline polymorphs including rutile, anatase, brookite (the familiar natural titania phase) and many other highpressure forms [9]. To date, many high-pressure TiO2 phases have been successfully synthesized. The cubic TiO2 polymorph was proposed [10–13] and experimentally synthesized recently [9]. The experimental bulk modulus is 202 GPa of the cubic TiO2 phase with a fluorite structure (referred to as f-TiO2) as suggested by Mattesini et al. A recent theoretical investigation [14] shows that f-TiO2 is a highly incompressible solid with a large bulk modulus value (395 GPa) approaching that of ultrahard cotunnite TiO2 (431 GPa) [8]. This suggests that f-TiO2 may be a potential ultrahard material. However, the theoretical results within abinitio codes reported by Liang et al. [15] show that f-TiO2 could not be regarded as a potential candidate of superhard material. The Corresponding author.
E-mail address: [email protected] (W. Lu). 0921-4526/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2008.10.012
discrepancy in previous results inspires us to perform a detailed investigation on f-TiO2. In the present work, the mechanical stability of TiO2 with a fluorite structure was examined. The result shows that it is stable. Calculations of the hardness and intrinsic mechanical properties were performed to answer whether f-TiO2 is a potential ultrahard material.
2. Theoretical details The calculations presented in this work were performed within the density functional theory (DFT) using the full-potential linearized augmented plane wave method along with the local orbitals (lo) method with the package WIEN2K [16]. Both local density approximation (LDA) [17] the generalized gradient approximation (GGA) [18] were used. Relativistic effects were taken into account within the scalar-relativistic approximation. To obtain a satisfactory convergence, the basis function was expanded up to RMTKMax ¼ 8. The k-point sampling in the total Brillouin zone was conducted with a 10 10 10 mesh. The average values of LDA and GGA lattice constant and elastic constants were used because the LDA (GGA) usually underestimates (overestimates) the lattice constant and overestimates (underestimates) the elastic constants. Stress calculation was carried out using the normal-conserving pseudopotential with a plane wave basis set in the CASTEP code [19]. The GGA exchange correlation function, k point grid 9 9 9 in the Brillouin zone and energy cutoff of 500 eV were used for f-TiO2. The stress– strain dependence was calculated using the method described
ARTICLE IN PRESS 80
W. Lu et al. / Physica B 404 (2009) 79–81
previously [20,21] and was extensively used on strong solids recently [22–25]. The lattice vector was incrementally deformed in the direction of the applied strain. At each step, the atomic basis vectors orthogonal to the applied strain and the atomic site in the deformed unit cell were relaxed.
3. Results and discussion The lattice constants and bulk modulus that are determined by least squares fit of the E–V curves (total energy versus volume) calculated by the DFT to BirchMurnaghan’s equation of state are summarized in Table 1. It can be seen that the calculated lattice constants in the present work agree well with previous theoretical and experimental values. The bulk modulus with the LDA exchange correlation function in the present work agrees well with other theoretical results [11,15,26]. The GGA result of bulk modulus in this work is closer to the with experiment result [9] but obviously lower than the result reported by Swamy and Muddle [14]. Owing to the cubic structure of f-TiO2, there are only 3 independent elastic stiffness constants c11, c12 and c44. As a result, three types of strain, i.e., the volume change, the volumeconservative tetragonal change and rhombohedral distortion, were applied to the equilibrium structure to calculate these elastic constants. Table 2 lists the calculated elastic properties of f-TiO2. The mechanical stability of a crystal requires the strain energy to be positive, which for a cubic crystal implies [27] C 44 40; C 11 4jc12 j; and C 11 þ 2C 12 40 The calculated 3 independent elastic constants for f-TiO2 satisfy these stability criteria, thus suggesting that f-TiO2 is mechanically stable. As is known, the elastic constant C44 is one of the important parameters indirectly governing the intrinsic hardness. The present average elastic constant C44 is 58.6 GPa, which is consistent with the results (40–73 GPa) reported by Liang et al. [15]. Except for that, the polycrystalline shear modulus G is still considered another important parameter governing indentation hardness. The polycrystalline shear modulus of G ¼ 142.9 GPa, calculated from the single-crystal elastic constants based on the Table 1 Lattice constants and bulk moduli of f-TiO2. Method
a (A˚)
B (GPa)
Reference
LDA GGA PW-LDA GGA B3LYP BSTATE-LDA BSTATE-GGA LCAO-HF LCAO-LDA Experiment
4.744 4.841 4.860 4.831 4.822 4.729 4.822 4.794 4.748 4.870
293.6 249.3 282 395 390 324 272 331 308 202
This work This work [11] [14] [14] [15] [15] [26] [26] [9]
Voigt method [28], is obviously lower than the values (227–239 GPa) reported by Swamy and Muddle [14]. The polycrystalline shear modulus of f-TiO2 is comparable with that of TiN [29–31] but lower than that of TiB2 [31], which means that the hardness of f-TiO2 locates between TiN and TiB2. The estimat ion of theoretical hardness has been extensively developed and many models have been proposed [31–33]. Recently, a model based on first-principle calculations was proposed for calculating the hardness of covalent and ionic crystals [34]. Then, another approach was presented, that does avoid first-principle calculations, but gives a trustworthy value for many recently studied materials [35]. We calculate the hardness of f-TiO2 based on the semiempirical microscopic model (for detailed expression referring [35]). The radii of Ti and O are taken from Pearson’s handbook [36], that is, r(Ti) ¼ 1.46 and r(O) ¼ 0.89. In this calculation, we also use the same constants C ¼ 1450 and s ¼ 2.8 as the ones in Ref. [35]. The calculated hardness using the average lattice constants from GGA and LDA is Haver ¼ 17.1 GPa. The hardness of TiN and TiB2 calculated using the same method is HTiN ¼ 16.8 GPa and HTiB2 ¼ 31:6 GPa, respectively [35], which agrees with previous results [37]. The theoretical hardness with the result Hf -TiO2 ¼ 13 GPa was reported by Liang et al. [15] which is almost consistent with our present result. It can be seen that the hardness result of f-TiO2 is comparable with TiN but lower than TiB2. This agrees well with our prediction according to the polycrystalline shear modulus. In order to understand this phenomenon more clearly, ideal tensile and shear strengths of f-TiO2 were also calculated. To check the reliability of the present method, the ideal tensile and shear strengths of TiN are also computed. The ideal tensile stress along /1 0 0S, /11 0S and /111S are 30.2, 51.4 and 100.7 GPa, respectively, showing anisotropy of tensile strengths for TiN. A good agreement is achieved between the present result and that in the previous work [25]. The ideal shear stress on the easiest slip system (11 0) /11¯ 0S is 28.6 GPa, which also agrees well with previous results [22,25,38]. The lattice constant of f-TiO2 given by CASTEP is 4.81 A˚, which agrees well with previous results [11,14,15]. Fig. 1 shows the calculated strengths for f-TiO2 along /1 0 0S, /11 0S and /111S for tension and on (11 0) /11¯ 0S, (111) /11¯ 0S slip system for shear. The tensile strength shows anisotropy for f-TiO2. This phenomenon is similar to fcc-TiN, but the highest tensile strength for f-TiO2 is along /1 0 0S. The ideal shear stress on the (11 0) /11¯ 0S slip system of about 52.4 GPa for f-TiO2 is higher than that
Table 2 Calculated equilibrium lattice parameter, elastic stiffness constants cij (GPa), polycrystalline shear modulus G (GPa), Youngs modulus E (GPa), and Poissons ratio n of f-TiO2.
LDA GGA Ave.
a (A˚)
C11
C12
C44
G
E
n
4.744 4.841 4.793
670.3 589.98 630.1
103.5 79.6 91.6
68.6 48.5 58.6
154.5 131.2 142.9
394.1 334.9 364.5
0.275 0.276 0.276
Fig. 1. (Color online) Calculated tension and shear stress–strain relationships for f-TiO2.
ARTICLE IN PRESS W. Lu et al. / Physica B 404 (2009) 79–81
of TiN, but the ideal shear stress on (111) /11¯ 0S of about 15.7 GPa for f-TiO2 is lower than that of TiN [25]. As known, hardness measurements test the resistance of a material to permanent plastic deformation. The lower shear stress on (111) /11¯ 0S slip system for f-TiO2 will decrease the hardness of this material. So the hardness of f-TiO2 is only comparable with TiN but is lower than that of TiB2. To summarise, the elastic constants of cubic TiO2 with a fluorite structure suggest that f-TiO2 is mechanically stable. The shear modulus indicates that the hardness of f-TiO2 is comparable with some known hard materials, e.g. TiN, but is lower than TiB2. The theoretical hardness of f-TiO2 is Haver ¼ 17.1 GPa. The hardness prediction based on the semiempirical microscopic model is consistent with that of shear modulus. The hardness of f-TiO2 is reduced because of the lower shear stress on the (111) /11¯ 0S slip system.
Acknowledgement One of the authors (W. L.) would like to thank R.F. Zhang for valuable advice for stress–strain calculation and to Prof. He for providing the authority of using WIEN2k software to carry out this work. This work was supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region (Project: PolyU 5171/07E) and the Hong Kong Polytechnic University (Project no.G-YF71). References [1] R.B. Kaner, J.J. Gilman, S.H. Tolbert, Science 308 (2005) 1268. [2] D.M. Teter, R.J. Hemley, Science 53 (1996) 271. [3] H.-Y. Chung, M.B. Weinberger, J.B. Levine, A. Kavner, J.M. Yang, S.H. Tolbert, R.B. Kaner, Science 316 (2007) 436. [4] J. Haines, J.M. Leger, G. Bocquillon, Annu. Rev. Mater. Res. 31 (2001) 1. [5] V.V. Brazhkin, A.G. Lyapin, R.J. Hemley, Philos. Mag. A 82 (2002) 231.
81
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