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First-principles study of the mechanical properties and phase stability of TiO2
Computational Materials Science 83 (2014) 114–119
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
First-principles study of the mechanical properties and phase stability of TiO2 Zhi-Gang Mei ⇑, Yi Wang, Shunli Shang, Zi-Kui Liu Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA
a r t i c l e
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Article history: Received 1 October 2013 Received in revised form 6 November 2013 Accepted 9 November 2013 Available online 28 November 2013 Keywords: TiO2 First-principles calculation Phase stability Phonon Mechanical properties.
a b s t r a c t We performed a density-functional theory study of the mechanical properties, phonon and phase stability of TiO2 in the structures of rutile, anatase, columbite, baddeleyite, OI, cotunnite, fluorite, and pyrite. Six exchange–correlation functionals were used to evaluate the structural and elastic properties of TiO2. The calculated bulk and shear moduli of TiO2 confirm that the cotunnite and fluorite phases are not as hard as traditional ultrahard materials, such as diamond. The predicted phonon spectra of the cubic phases of TiO2, i.e., the fluorite and pyrite phases, show that they are dynamically unstable at ambient conditions. However, the fluorite structure can be stabilized as a metastable phase at high pressures. The pressure-induced phase transitions of TiO2 are found to depend on the starting material. The predicted pressure-induced phase transition pressures and sequence are consistent with previous experimental and theoretical studies. From the calculated Gibbs energies, we investigated the pressure–temperature phase diagram of TiO2. The calculated phase equilibria are in good agreement with the available experimental results. The currently predicted phase diagram is expected to provide helpful guidance for the future synthesis of high-pressure phases in TiO2. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Titanium dioxide (TiO2) has been widely studied both experimentally and theoretically due to its scientific and technological importance [1–7]. In nature, there are several polymorphs existed for TiO2, such as rutile (P42/mnm), anatase (I41/amd), and brookite (Pbca) phases [3]. At elevated pressures, TiO2 experiences a series of structural phase transitions. Its high pressure polymorphs, i.e., columbite (TiO2 II, space group Pbcn) [3], baddeleyite (MI, P21/c) [4], orthorhombic I (OI, Pbca) [5], cotunnite (OII, Pnma) [8], and Fe2P-type (P62m) [7] were discovered at increased pressures. The cotunnite-structured TiO2 was reported to be the hardest oxide ever known [8]. Recently, cubic TiO2 was synthesized at high pressures and temperatures, which may serve as an important promising material for future generation solar cells [9]. Swamy and Muddle studied the crystal structures and compressibility of fluorite- and pyrite-structured TiO2 under varying pressures by firstprinciples calculations [6]. They suggested that fluorite TiO2 has the potential to be an ultrahard material, if it could be stabilized under ambient conditions. However, there are still a lot of debate on the hardness of the cotunnite and cubic phases of TiO2. Several density-functional theory (DFT) based first-principles studies show ⇑ Corresponding author. Current address: Argonne National Laboratory, Argonne, IL 60439, USA. Tel.: +1 630 252 2318. E-mail address: [email protected] (Z.-G. Mei). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.11.020
that the calculated bulk moduli of TiO2 phases are sensitive to the exchange–correlation functional used [5,6,10–12]. The structure and stability of the high-pressure phases of TiO2 are of particular interest in earth sciences, for these phases are an accessible analog of minerals in the earth’s mantle [7]. A number of experimental and theoretical studies have been done to understand the phase stabilities of TiO2 phases under pressure. High-pressure X-ray-diffraction and Raman spectroscopy studies have revealed that rutile and anatase transform to columbite and baddeleyite structures upon compression [4]. However, there has been considerable dispute over the phase equilibrium between rutile and columbite [13,14]. First-principles calculations have been widely used to investigate the phase stability of TiO2 [5,7,10–12], however, most of them are limited to zero temperature studies [15]. To clarify the ambiguity of the high-pressure phases of TiO2 as potential ultrahard materials, we performed a theoretical study of the mechanical properties, lattice dynamic and stability of various TiO2 phases by DFT. The study of the vibrational modes of TiO2 polymorphs is expected to determine the dynamical stabilities of the high-pressure phases at ambient pressure. Since it is still a challenging task to synthesize and characterize phases at high temperatures and high pressures, the equilibrium pressure–temperature phase diagram of TiO2 can provide useful information for the synthesis of the potential ultra-hard materials. The rest of the paper is organized as follows. The computational details of
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first-principles total energy and phonon spectra calculations are described in Section 2. The predictions of mechanical properties, phonon spectra, and pressure-induced phase transitions of TiO2 are discussed in Section 3. Finally, we give a summary of this work in Section 4. 2. Computational details The DFT calculations were performed by the projector augmented wave method [16] as implemented in the Vienna ab initio simulation package (VASP) [17,18]. The local density approximation (LDA) [19] and five different generalized gradient approximations (GGA) [20–24] were utilized to describe the exchange–correlation energy. The cell shapes and ionic positions of TiO2 phases were fully relaxed by the Gaussian smearing technique. Accurate total energy calculations were performed by means of the linear tetrahedron method with Blöchl’s correction [25]. In all cases the total energies are converged to 107 eV/cell with an energy cutoff of 500 eV. More details of the total energy calculations can be found in our previous work [26]. For accurate predictions of phonon frequencies of polar material, it is critical to include the long-range dipole–dipole interactions in order to study the LO-TO splitting presented [27]. The phonon frequencies of TiO2 polymorphs were studied by supercell approach as implemented in our Yphon code [28]. The Born effective charge tensor and electronic dielectric constant tensor were calculated using the linear-response method as implemented in VASP 5.2 [29]. To calculate force constants, we used an energy cutoff of 400 eV and 3 3 3 k-point mesh for the supercells of all TiO2 phases. Detailed setting of the supercell can be found elsewhere [26]. Ancillary calculations using denser k-point mesh and larger supercell, such as 5 5 5 k-point mesh and 3 3 3 supercell, were tested for the rutile phase. No significant difference was found. The calculated force constants within the supercell were used for the real-space component of the force constants. This approach makes full use of the accuracies of the force constants calculated in real space and the dipole–dipole interactions calculated in reciprocal space [28]. For completeness, we calculated the phonon dispersions and density of states (DOSs) of all TiO2 polymorphs using the same exchange–correlation functional, i.e., the best functional obtained from the test of the mechanical properties of TiO2. 3. Results and discussion 3.1. Mechanical properties We studied the elastic constants and related moduli of TiO2 polymorphs using six exchange–correlation (XC) functionals implemented VASP, i.e., LDA [19], PW91 [20], PBE [21], AM05 [22], rPBE [23], and PBEsol [24]. The modified PBE form, such as PBEsol, have been demonstrated to be more accurate in predicting the equilibrium properties of densely packed solids [24]. The bulk moduli calculated from elastic constants were also compared with those obtained from equation of state (EOS) fittings. The calculated equilibrium volume, lattice parameters, atomic position, and bulk moduli of rutile and anatase TiO2 are summarized in Table 1, together with experimental results [5,30–33]. Our calculation shows that the PBEsol functional provides the best description of the structural properties of rutile and anatase phases among all the exchange–correlation functionals studied. Since the elastic properties of a material are very sensitive to its equilibrium parameter, PBEsol should also provide the most accurate elastic properties of TiO2. An efficient stress–strain method [34] was used to calculate the single-crystal elastic constants of the rutile phase. As shown in Table
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2, PBEsol gives the best prediction of the elastic constants among all exchange–correlation functionals studied. Due to its accurate description of the bond length and lattice parameter, PBEsol functional is more adapted to the following study of the mechanical property, phonons, and phase stability of the high-pressure phases of TiO2. The intrinsic hardness of a material is correlated with its bulk modulus (B) and shear modulus (G). Of these two elastic moduli, B is easier to calculate by first-principles methods, i.e., from EOS fitting to the energy-volume relationship. Meanwhile, bulk modulus, shear modulus and Young’s modulus can be obtained from single-crystal elastic constants. Table 3 lists the calculated polycrystalline aggregate properties of TiO2 polymorphs, including bulk modulus BH, shear modulus GH, Young’s modulus EH, BH/GH ratio and Poisson’s ratio m according to the Voigt–Reuss–Hill approach [35]. The bulk moduli of TiO2 estimated from elastic constants are very close to those fitted from EOS, with difference less than 5%, except for the baddelyite phase. Fluorite- and cotunnite-structured TiO2 phases exhibit relative larger bulk modulus compared to other phases, however not large enough to be ultra-hard materials. Our results are consistent with the recent high-pressure experiments and first-principles calculations [11, 12,30,36,37]. The overestimated bulk modulus of the cotunnite phase by Dubrovinsky et al. [8] and the fluorite phase by Swamy and Muddle [6] may be attributed to the underestimated pressure derivative of the bulk modulus. The first pressure derivative of bulk modulus (B’) were predicted to be 1.35 for cotunnite [8] and 1.75 for fluorite [6], much smaller than the commonly accepted value, around 4. Nishio–Hamane et al. [37], found that very high B0 can be obtained if extremely small B0 (1–2) was assumed. By constraining B0 to 4.0, a different set of experiments did yield a B0 = 294 GPa for cotunnite, which is much close to our result. We believe that the reliable bulk modulus for the cotunnite and fluorite phases should lie within the range of 260–300 GPa. These values are significantly smaller than that of diamond (444 GPa), and thus we conclude that the hardness of the cotunnite and fluorite phases of are not comparable to traditional ultrahard materials, such as diamond. 3.2. Phonon properties In our previous work, we studied the phonon dispersion relations of all the high-pressure phases of TiO2, except for the cubic phases [26]. For the high-pressure polymorphs, it was discovered that there is no imaginary phonon frequency in the phonon dispersions, indicating that they are dynamically stable at ambient pressure. It suggests that the high-pressure polymorphs of TiO2 can be quenched to the ambient conditions as metastable phases. However, the phonon properties of the cubic phases of TiO2 are still not well studied. Here, we predicted the phonon dispersion relations of the fluorite and pyrite phases of TiO2 at their theoretical equilibrium volumes, as shown in Fig. 1. Imaginary phonon modes can be observed at X and W points in the Brillouin zone of the fluorite phase, while unstable modes appear in the whole Brillouin zone of the pyrite phase. These imaginary phonon modes indicate that both of the cubic phases are dynamically unstable at ambient pressure. We also studied the phonon density of state (DOS) of the fluorite and pyrite phases at several different pressures, as shown in Fig. 2. At zero pressure, the calculated phonon DOS of the fluorite phase shows a strong peak at negative frequencies. This peak in the negative region is mainly due to the vibrations of the Ti atoms, whereas the small and broad contribution of O atoms is spread over the whole range of the negative frequencies. Upon compression, the main peak in the negative frequency region decreases rapidly and eventually disappears completely around 55 GPa,
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Table 1 Calculated equilibrium properties of rutile and anatase TiO2 using different exchange–correlation functional, including volume (V0), lattice parameters (a, c/a), atomic position (u), and bulk moduli (B0). Phase
Method
V0 (Å[3]/TiO2)
a
c/a
u
B0 (GPa)
Rutile
Expt. [30,31] LDA PBE PW91 AM05 rPBE PBEsol
31.00–31.22 30.47 32.22 32.11 31.28 32.93 31.13
4.582–4.594 4.559 4.651 4.646 4.606 4.695 4.599
0.64–0.644 0.642 0.639 0.639 0.639 0.636 0.640
0.305 0.304 0.305 0.305 0.305 0.306 0.304
211–235 238 196 203 210 177 216
Anatase
Expt. [5,32,33] LDA PBE PW91 AM05 rPBE PBEsol
33.98–34.07 33.34 35.24 35.13 34.30 36.08 34.14
3.782–3.785 3.751 3.806 3.803 3.775 3.830 3.776
2.512–2.514 2.527 2.556 2.555 2.550 2.569 2.536
0.208 0.208 0.206 0.206 0.207 0.206 0.207
178–179 184 166 168 170 156 176
Table 2 Calculated elastic constants (Cij, in GPa) of rutile TiO2 with different exchange– correlation functional together with experimental results. Method
C11
C12
C13
C33
C44
C66
Source
Expt. (298 K) Expt. (300 K) Expt. (4 K) LDA PBE PW91 PBEsol
271 267 289 295 255 260 276
177 174 197 223 172 174 200
149 146 159 174 147 149 163
484 484 508 523 467 471 497
124 124 128 124 114 115 118
194 190 227 245 212 215 230
Fritz [46] Isaak [47] Manghnani [48] This work This work This work This work
Table 3 Calculated elastic properties of TiO2 phases, including polycrystalline aggregate properties of bulk modulus BH (GPa), shear modulus GH (GPa), Young’s modulus EH (GPa), BH/GH ratio and Poisson’s ratio m in Hill approach, and bulk modulus BEOS and pressure derivative of bulk modulus (B0 ) estimated from EOS fitting. Phase
BH
GH
EH
BH/GH
m
BEOS
B0
Rutile Anatase Brookite Columbite Baddeleyite OI Cotunnite Fluorite Pyrite
229 191 213 224 201 242 244 271 263
109 56 85 88 103 116 93 86 107
281 152 225 234 264 300 248 230 282
2.10 3.43 2.52 2.54 1.94 2.09 2.62 3.14 2.46
0.29 0.37 0.32 0.33 0.28 0.29 0.33 0.36 0.32
216 176 208 220 173 243 256 270 261
5.2 2.4 3.9 4.2 3.3 3.6 5.1 4.3 4.2
indicating that the fluorite phase can be stabilized by high-pressure. The phonon DOS of the pyrite phase also display a strong peak at negative frequencies, corresponding mostly to the vibrations of Ti atoms. As pressure increases, the Ti peak shifts towards
more negative region, and the peaks at negative frequencies remains at all calculated pressures. Special treatments such as chemical doping might be used to stabilize the fluorite phase to ambient pressure, similar to the approach used to stabilize the high temperature stable cubic ZrO2 to room temperature by doping yttria. We believe that the cubic phase of TiO2 synthesized by Mattesini et al. [9] is due to the contamination of the sample in the high-pressure experiment. Our calculations suggest that the fluorite structure can be assigned to the cubic phase observed in experiment. Another noticeable difference between the calculated phonon spectra of the fluorite and pyrite phases is that a band gap exists in the pyrite phase whereas it is absent in the fluorite phase. The phonon band gap in the pyrite does not change much even at very high pressures. The phonon gap in the pyrite phase can be ascribed to its different space group and coordination number of Ti atoms in comparison to the fluorite phase, which results in their unique phonon behaviors upon compression. 3.3. Pressure-induced phase transition at zero temperature The enthalpy change in the anatase to rutile transformation has been measured using different calorimetric techniques. However significant different results have been reported in the literature, from the exothermic values to the endothermic values [38]. The phase stability of TiO2 predicted by DFT is found to be sensitive to the treatment of XC. The relative stability between rutile and anatase, however, is not affected by the approximation used. Our calculations using different XC types show that the energy difference between rutile and anatase is always a positive value, ranging from 1.08 to 8.78 kJ/mol, which means that the anatase phase is more stable than the rutile phase at zero temperature. The current
Fig. 1. Calculated phonon dispersion relations of the cubic phases of TiO2, i.e., (a) fluorite and (b) pyrite, at the theoretical equilibrium volume.
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Fig. 2. Calculated phonon density of states of (a) the fluorite and (b) pyrite phases of TiO2 at different pressures.
Fig. 3. Calculated total energy for eight TiO2 polymorphs with respect to volume.
calculations are consistent with most of the recent theoretical studies [10,12]. The large differences for the experimental measurement might be ascribed to the large surface energy of anatase phase, which is not properly considered in the estimation of enthalpy change [39]. The zero temperature phase stability of TiO2 under high-pressure has been extensively studied using different theoretical methods [5,7,10–12,30,36]. We provide a similar DFT study here using the revised PBE form for XC, i.e., PBEsol. The obtained equilibrium properties from EOS fitting and transition pressure at zero temperature provide critical information for the following study of the phase stability of TiO2 at high temperatures. The total energies for TiO2 phases were calculated at a series of difference volumes, and then fitted by thirst-order Birch-Murnagham EOS fitting [40], illustrated in Fig. 3. In order to study the pressure-induced phase transitions, the calculated total energy of TiO2 was converted to enthalpy by H(P) = E(V) + PV, as shown in Fig. 4. The present calculation predicts that the anatase-to-columbite phase transformation occurs at 3.6 GPa, while the rutile and columbite phases are energetically very similar, with columbite having slightly lower energy at low pressure. Columbite is the most stable phase up to 9.9 GPa where the calculations predict a transformation to baddeleyite
Fig. 4. Calculated enthalpy of eight TiO2 polymorphs as a function of pressure, with the rutile phase as the reference state.
which in turn transforms to OI at 20.6 GPa, and to cotunnite at 39.3 GPa. Neither fluorite nor pyrite phase is a thermodynamically stable form in all the studied pressures up to 100 GPa, although the pyrite phase is more stable than the fluorite structure up to 24.2 GPa. Starting from anatase, the predicted pressure-induced phase transition sequence of TiO2 is anatase ? columbite ? baddeleyite ? OI ? cotunnite. In experiments, rutile phase, however, is more commanly used as the starting material. In that case, we predict a transition from rutile to baddeleyite at 6.9 GPa, to OI at 20.6 GPa, and to OII at 39.3 GPa, in close agreement with experiments [5,30,32,37,41,42]. Due to the small energy difference between rutile and columbite, it is worth studying the phase stability of TiO2 upon decompression. Our calculation shows that the baddeleyite phase transforms to columbite phase at 9.9 GPa with decreasing pressure, but never restore the rutile structure, in agreement with the experimental observation [30,37]. 3.4. Pressure–temperature phase diagram Using the calculated phonon DOS, the lattice vibrational free energies of TiO2 phases can be evaluated under the quasiharmonic
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Fig. 5. Calculated two-phase equilibrium line between rutile and anatase phases of TiO2. The blue circle is experimental data compiled by Chase [44].
Fig. 7. Calculated two-phase equilibrium line between columbite and baddeleyite phases. The blue dashed line is the two-phase equilibrium line measured by Tang et al. [42].
approximation, which were further used to study the lattice stability and phase transitions at finite temperatures. Using a similar approach as Ti [43], we predict the pressure–temperature (P–T) phase diagram of TiO2 by comparing the Gibbs energy of each phase calculated directly from phonon DOS. Fig. 5 shows the predicted phase equilibrium between rutile and anatase phases. The transition temperature between anatase and rutile at ambient pressure is predicted to be 1340 K, close to the experimental result of 1200 K compiled by Chase [44]. The transition temperature is determined by both the temperature dependence of the lattice vibrational energy and the zero temperature static energy. Since the thermodynamic properties of rutile and anatase phases are accurately predicted by our DFT calculations [26], the discrepancy of the transitional temperature should be from the uncertainty of lattice stability calculated at zero temperature, which is largely determined by the XC functional used. There has been considerable dispute over the phase equilibrium between rutile and columbite phases of TiO2. The high-pressure and high-temperature phase equilibrium between them has been studied by several authors [13,14,45]. Our predicted pressure– temperature two-phase equilibrium line (see Fig. 6) shows a positive slope, in agreement with results by Akaogi et al. [45] and Withers et al. [14], indicating that the two-phase equilibrium line with a negative slope measured by Olsen et al. [13] may be due to some technical errors. The currently predicted transition pressure between rutile and columbite at room temperature is very close to the measurement by Withers et al., while somewhat smaller
than the extrapolation from Akaogi et al.’s result if a linear twophase equilibrium line was assumed. This underestimation might be ascribed to the underestimated 0 K total energy of the rutile phase. Fig. 7 shows the calculated two-phase equilibrium line between the columbite and baddeleyite phases of TiO2, together with the experimental result measured by Tang et al. [42] The predicted transition pressure between columbite and baddeleyite at room temperature is 12.3 GPa, in an excellent agreement with experimental value of 11.8 GPa [42]. The two-phase equilibrium line between columbite and baddeleyite phases can be described by a linear relationship of T = 159.4 P-1637.6 (T in unit of K and P in unit of GPa), close to that measured by Tang et al., T = 188.7 P1919.5 [42]. By combining all the calculated two-phase equilibrium lines, we obtained the equilibrium pressure–temperature (P–T) diagram of TiO2 with six phases, as shown in Fig. 8. The fluorite and pyrite phases were not included, since thermodynamically they are not stable in the whole pressure and temperature range we studied. A general feature of the predicted P–T phase diagram of TiO2 is that all of the two-phase equilibrium lines show positive slopes, except that between the rutile and anatase phases. From thermodynamic point of view, we know the slope of the two-phase equilibrium in the P–T phase diagram is related to the changes of volume and entropy due to phase transition, by dT/dP = DV/DS. Since the volume changes are negative for the pressure-induced phase transitions,
Fig. 6. Calculated two-phase equilibrium line between rutile and columbite phases. The blue dashed line and read dash-dotted line are the two-phase equilibrium lines measured by Withers et al. [14] and Akaogi et al., [45] respectively.
Fig. 8. Calculated equilibrium pressure–temperature phase diagram of TiO2, together with experimental data [14,42,44,45].
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the entropy changes along the two-phase equilibrium line of rutile/ columbite, columbite/baddeleyite, baddeleyite/OI, and OI/cotunnite should be positive. However, the entropy change due to anatase–rutile transformation is negative. It should be pointed out that the current P–T diagram is predicted based on the thermodynamic equilibrium conditions and the kinetic information of phase transformations is not considered. Therefore, the high-pressure phases, such as OI and cotunnite, may need even higher temperatures than those suggested by the current calculations, to overcome the energy barriers encountered in phase transformations. 4. Summary We have studied the mechanical properties, dynamical and thermodynamic stabilities of TiO2 phases in order to evaluate the potential of the high-pressure phases of TiO2 as ultrahard materials. The elastic constants and moduli of the rutile and anatase phases were evaluated by six exchange–correlation functionals. The PBEsol functional shows the best description of the structural and elastic properties of rutile and anatase phases. Our predicted bulk and shear moduli of the nine phases of TiO2 confirm that the cotunnite and fluorite phases cannot be used as ultrahard materials. The calculated phonon spectra of the cubic phases of TiO2 show imaginary phonon modes in the Brillouin zones, indicating that they are dynamically unstable at ambient conditions. However, the fluorite phase might be stabilized as a metastable phase at high pressures. Treatment such as chemical doping may stabilize it to ambient pressure for future solar cell applications. The pressure-induced phase transitions of TiO2 are found to depend on the starting materials. The calculated total energy of the columbite phase is close to that of the rutile phase, which explains the observance of the columbite phase after releasing high pressure during experiment. The predicted phase transition pressures and sequence are consistent with previous experimental and theoretical studies. From the calculated Gibbs energies of the TiO2 phases, we predicted the two-phase equilibrium lines of the anatase/rutile, rutile/columbite, columbite/baddeleyite, and baddeleyite/cotunnite phases in the pressure–temperature phase diagram. The anatase phase of TiO2 is found to be more stable than the rutile phase at low temperatures and transforms to rutile phase at high temperatures. The calculated two-phase equilibrium line between rutile and columbite phases agrees with those measured by Akaogi et al. and Withers et al., while contradictory to the result by Olsen et al. Meanwhile, the two-phase equilibrium line between columbite and baddeleyite phases predicted in this work show an excellent agreement with experimental measurement. By combining all the two-phase equilibrium lines, we obtained the complete pressure–temperature phase diagram of TiO2 with total six phases. This phase diagram can be used to guide the synthesis of the high-pressure phases of TiO2. Acknowledgements This work is funded by the TIE project of the Center for Computational Materials Design (CCMD), a National Science Foundation (NSF) Industry/University Cooperative Research Center through Grant IIP-0737759 and the National Science Foundation (NSF) through Grant No. DMR-1006557. First-principles calculations were carried out on the LION clusters supported by the Materials
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Simulation Center and the Research Computing and Cyber infrastructure unit at the Pennsylvania State University. References [1] [2] [3] [4] [5]
[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]
[46] [47] [48]
A. Fujishima, K. Honda, Nature 238 (1972) 37. K.I. Hadjiivanov, D.G. Klissurski, Chem. Soc. Rev. 25 (1996) 61. J.C. Jamieson, B. Olinger, Science 161 (1968) 893. H. Sato, S. Endo, M. Sugiyama, T. Kikegawa, O. Shimomura, K. Kusaba, Science 251 (1991) 786. N.A. Dubrovinskaia, L.S. Dubrovinsky, R. Ahuja, V.B. Prokopenko, V. Dmitriev, H.P. Weber, J.M. Osorio-Guillen, B. Johansson, Phys. Rev. Lett. 87 (2001) 275501. V. Swamy, B.C. Muddle, Phys. Rev. Lett. 98 (2007) 035502. H. Dekura, T. Tsuchiya, Y. Kuwayama, J. Tsuchiya, Phys. Rev. Lett. 107 (2011) 045701. L.S. Dubrovinsky, N.A. Dubrovinskaia, V. Swamy, J. Muscat, N.M. Harrison, R. Ahuja, B. Holm, B. Johansson, Nature 410 (2001) 653. M. Mattesini, J.S. de Almeida, L. Dubrovinsky, N. Dubrovinskaia, B. Johansson, R. Ahuja, Phys. Rev. B 70 (2004) 212101. J. Muscat, V. Swamy, N.M. Harrison, Phys. Rev. B 65 (2002) 224112. X.A. Wu, E. Holbig, G. Steinle-Neumann, J. Phys.–Condens. Matter (2010) 295501. X.G. Ma, P. Liang, L. Miao, S.W. Bie, C.K. Zhang, L. Xu, J.J. Jiang, Phys. Status Solidi B-Basic Solid State Phys 246 (2009) 2132. J.S. Olsen, L. Gerward, J.Z. Jiang, J. Phys. Chem. Solids 60 (1999) 229. A.C. Withers, E.J. Essene, Y.X. Zhang, Cont. Mineral. Petrol. 145 (2003) 199. Z.-K. Liu, J. Phase Equilib. Diffus. 30 (2009) 517. P.E. Blöchl, Phys. Rev. B 50 (1994) 17953. G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169. G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758. J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. R. Armiento, A.E. Mattsson, Phys. Rev. B 72 (2005) 085108. Y. Zhang, W. Yang, Phys. Rev. Lett. 80 (1998) 890. J.P. Perdew, A. Ruzsinszky, G.I. Csonka, O.A. Vydrov, G.E. Scuseria, L.A. Constantin, X. Zhou, K. Burke, Phys. Rev. Lett. 100 (2008) 136406. P.E. Blöchl, O. Jepsen, O.K. Andersen, Phys. Rev. B 49 (1994) 16223. Z.G. Mei, Y. Wang, S.L. Shang, Z.K. Liu, Inorg. Chem. 50 (2011) 6996. Y. Wang, S. Shang, Z.-K. Liu, L.-Q. Chen, Phys. Rev. B 85 (2012) 224303. Y. Wang, J.J. Wang, W.Y. Wang, Z.G. Mei, S.L. Shang, L.Q. Chen, Z.K. Liu, J. Phys.– Condens. Matter 22 (2010) 202201. M. Gajdos, K. Hummer, G. Kresse, J. Furthmuller, F. Bechstedt, Phys. Rev. B 73 (2006) 045112. Y. Al-Khatatbeh, K.K.M. Lee, B. Kiefer, Phys. Rev. B 79 (2009) 134114. L.C. Ming, M.H. Manghnani, J. Geophys. Res. 84 (1979) 4777. T. Arlt, M. Bermejo, M.A. Blanco, L. Gerward, J.Z. Jiang, J.S. Olsen, J.M. Recio, Phys. Rev. B 61 (2000) 14414. V. Swamy, L.S. Dubrovinsky, J. Phys. Chem. Solids 62 (2001) 673. Z.G. Mei, S.L. Shang, Y. Wang, Z.K. Liu, J. Phys.–Condens. Matter 22 (2010) 295404. G. Simmons, H. Wang, Single Crystal Elastic Constants and Calculated Aggregate Properties, MIT press, Cambridge, MA, 1971. L. Koci, D.Y. Kim, J.S. de Almeida, M. Mattesini, E. Isaev, R. Ahuja, J. Phys.– Condens. Matter 20 (2008) 345218. D. Nishio-Hamane, A. Shimizu, R. Nakahira, K. Niwa, A. Sano-Furukawa, T. Okada, T. Yagi, T. Kikegawa, Phys. Chem. Miner. 37 (2010) 129. T. Mitsuhashi, O.J. Kleppa, J. Am. Ceram. Soc. 62 (1979) 356. A.A. Levchenko, G. Li, J. Boerio-Goates, B.F. Woodfield, A. Navrotsky, Chem. Mater. 18 (2006) 6324. Z.-G. Mei, S. Shang, Y. Wang, Z.-K. Liu, Phys. Rev. B 79 (2009) 134102. L. Gerward, J. Staun, J. Appl. Crystallogr. 30 (1997) 259. J. Tang, S. Endo, J. Am. Ceram. Soc. 76 (1993) 796. Z.G. Mei, S.L. Shang, Y. Wang, Z.K. Liu, Phys. Rev. B 80 (2009) 104116. M.W. Chase, NIST-JANAF Thermochemical Tables, American Institute of Physics, Washington, D. C., 1998. M. Akaogi, K. Kusaba, J.-I. Susaki, T. Yagi, M. Matsui, T. Kikegawa, H. Yusa, E. Ito, in: Y. Syono, M.H. Manghnani (Eds.), TERRAPUB American Geophyical Union, Washington, DC, 1992, p. 447. I.J. Fritz, J. Phys. Chem. Solids 35 (1974) 817. D.G. Isaak, J.D. Carnes, O.L. Anderson, H. Cynn, E. Hake, Phys. Chem. Miner. 26 (1998) 31. M.H. Manghnani, E.S. Fisher, W.S. Brower, J. Phys. Chem. Solids 33 (1972) 2149.
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
First-principles study of the mechanical properties and phase stability of TiO2 Zhi-Gang Mei ⇑, Yi Wang, Shunli Shang, Zi-Kui Liu Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA
a r t i c l e
i n f o
Article history: Received 1 October 2013 Received in revised form 6 November 2013 Accepted 9 November 2013 Available online 28 November 2013 Keywords: TiO2 First-principles calculation Phase stability Phonon Mechanical properties.
a b s t r a c t We performed a density-functional theory study of the mechanical properties, phonon and phase stability of TiO2 in the structures of rutile, anatase, columbite, baddeleyite, OI, cotunnite, fluorite, and pyrite. Six exchange–correlation functionals were used to evaluate the structural and elastic properties of TiO2. The calculated bulk and shear moduli of TiO2 confirm that the cotunnite and fluorite phases are not as hard as traditional ultrahard materials, such as diamond. The predicted phonon spectra of the cubic phases of TiO2, i.e., the fluorite and pyrite phases, show that they are dynamically unstable at ambient conditions. However, the fluorite structure can be stabilized as a metastable phase at high pressures. The pressure-induced phase transitions of TiO2 are found to depend on the starting material. The predicted pressure-induced phase transition pressures and sequence are consistent with previous experimental and theoretical studies. From the calculated Gibbs energies, we investigated the pressure–temperature phase diagram of TiO2. The calculated phase equilibria are in good agreement with the available experimental results. The currently predicted phase diagram is expected to provide helpful guidance for the future synthesis of high-pressure phases in TiO2. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Titanium dioxide (TiO2) has been widely studied both experimentally and theoretically due to its scientific and technological importance [1–7]. In nature, there are several polymorphs existed for TiO2, such as rutile (P42/mnm), anatase (I41/amd), and brookite (Pbca) phases [3]. At elevated pressures, TiO2 experiences a series of structural phase transitions. Its high pressure polymorphs, i.e., columbite (TiO2 II, space group Pbcn) [3], baddeleyite (MI, P21/c) [4], orthorhombic I (OI, Pbca) [5], cotunnite (OII, Pnma) [8], and Fe2P-type (P62m) [7] were discovered at increased pressures. The cotunnite-structured TiO2 was reported to be the hardest oxide ever known [8]. Recently, cubic TiO2 was synthesized at high pressures and temperatures, which may serve as an important promising material for future generation solar cells [9]. Swamy and Muddle studied the crystal structures and compressibility of fluorite- and pyrite-structured TiO2 under varying pressures by firstprinciples calculations [6]. They suggested that fluorite TiO2 has the potential to be an ultrahard material, if it could be stabilized under ambient conditions. However, there are still a lot of debate on the hardness of the cotunnite and cubic phases of TiO2. Several density-functional theory (DFT) based first-principles studies show ⇑ Corresponding author. Current address: Argonne National Laboratory, Argonne, IL 60439, USA. Tel.: +1 630 252 2318. E-mail address: [email protected] (Z.-G. Mei). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.11.020
that the calculated bulk moduli of TiO2 phases are sensitive to the exchange–correlation functional used [5,6,10–12]. The structure and stability of the high-pressure phases of TiO2 are of particular interest in earth sciences, for these phases are an accessible analog of minerals in the earth’s mantle [7]. A number of experimental and theoretical studies have been done to understand the phase stabilities of TiO2 phases under pressure. High-pressure X-ray-diffraction and Raman spectroscopy studies have revealed that rutile and anatase transform to columbite and baddeleyite structures upon compression [4]. However, there has been considerable dispute over the phase equilibrium between rutile and columbite [13,14]. First-principles calculations have been widely used to investigate the phase stability of TiO2 [5,7,10–12], however, most of them are limited to zero temperature studies [15]. To clarify the ambiguity of the high-pressure phases of TiO2 as potential ultrahard materials, we performed a theoretical study of the mechanical properties, lattice dynamic and stability of various TiO2 phases by DFT. The study of the vibrational modes of TiO2 polymorphs is expected to determine the dynamical stabilities of the high-pressure phases at ambient pressure. Since it is still a challenging task to synthesize and characterize phases at high temperatures and high pressures, the equilibrium pressure–temperature phase diagram of TiO2 can provide useful information for the synthesis of the potential ultra-hard materials. The rest of the paper is organized as follows. The computational details of
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first-principles total energy and phonon spectra calculations are described in Section 2. The predictions of mechanical properties, phonon spectra, and pressure-induced phase transitions of TiO2 are discussed in Section 3. Finally, we give a summary of this work in Section 4. 2. Computational details The DFT calculations were performed by the projector augmented wave method [16] as implemented in the Vienna ab initio simulation package (VASP) [17,18]. The local density approximation (LDA) [19] and five different generalized gradient approximations (GGA) [20–24] were utilized to describe the exchange–correlation energy. The cell shapes and ionic positions of TiO2 phases were fully relaxed by the Gaussian smearing technique. Accurate total energy calculations were performed by means of the linear tetrahedron method with Blöchl’s correction [25]. In all cases the total energies are converged to 107 eV/cell with an energy cutoff of 500 eV. More details of the total energy calculations can be found in our previous work [26]. For accurate predictions of phonon frequencies of polar material, it is critical to include the long-range dipole–dipole interactions in order to study the LO-TO splitting presented [27]. The phonon frequencies of TiO2 polymorphs were studied by supercell approach as implemented in our Yphon code [28]. The Born effective charge tensor and electronic dielectric constant tensor were calculated using the linear-response method as implemented in VASP 5.2 [29]. To calculate force constants, we used an energy cutoff of 400 eV and 3 3 3 k-point mesh for the supercells of all TiO2 phases. Detailed setting of the supercell can be found elsewhere [26]. Ancillary calculations using denser k-point mesh and larger supercell, such as 5 5 5 k-point mesh and 3 3 3 supercell, were tested for the rutile phase. No significant difference was found. The calculated force constants within the supercell were used for the real-space component of the force constants. This approach makes full use of the accuracies of the force constants calculated in real space and the dipole–dipole interactions calculated in reciprocal space [28]. For completeness, we calculated the phonon dispersions and density of states (DOSs) of all TiO2 polymorphs using the same exchange–correlation functional, i.e., the best functional obtained from the test of the mechanical properties of TiO2. 3. Results and discussion 3.1. Mechanical properties We studied the elastic constants and related moduli of TiO2 polymorphs using six exchange–correlation (XC) functionals implemented VASP, i.e., LDA [19], PW91 [20], PBE [21], AM05 [22], rPBE [23], and PBEsol [24]. The modified PBE form, such as PBEsol, have been demonstrated to be more accurate in predicting the equilibrium properties of densely packed solids [24]. The bulk moduli calculated from elastic constants were also compared with those obtained from equation of state (EOS) fittings. The calculated equilibrium volume, lattice parameters, atomic position, and bulk moduli of rutile and anatase TiO2 are summarized in Table 1, together with experimental results [5,30–33]. Our calculation shows that the PBEsol functional provides the best description of the structural properties of rutile and anatase phases among all the exchange–correlation functionals studied. Since the elastic properties of a material are very sensitive to its equilibrium parameter, PBEsol should also provide the most accurate elastic properties of TiO2. An efficient stress–strain method [34] was used to calculate the single-crystal elastic constants of the rutile phase. As shown in Table
115
2, PBEsol gives the best prediction of the elastic constants among all exchange–correlation functionals studied. Due to its accurate description of the bond length and lattice parameter, PBEsol functional is more adapted to the following study of the mechanical property, phonons, and phase stability of the high-pressure phases of TiO2. The intrinsic hardness of a material is correlated with its bulk modulus (B) and shear modulus (G). Of these two elastic moduli, B is easier to calculate by first-principles methods, i.e., from EOS fitting to the energy-volume relationship. Meanwhile, bulk modulus, shear modulus and Young’s modulus can be obtained from single-crystal elastic constants. Table 3 lists the calculated polycrystalline aggregate properties of TiO2 polymorphs, including bulk modulus BH, shear modulus GH, Young’s modulus EH, BH/GH ratio and Poisson’s ratio m according to the Voigt–Reuss–Hill approach [35]. The bulk moduli of TiO2 estimated from elastic constants are very close to those fitted from EOS, with difference less than 5%, except for the baddelyite phase. Fluorite- and cotunnite-structured TiO2 phases exhibit relative larger bulk modulus compared to other phases, however not large enough to be ultra-hard materials. Our results are consistent with the recent high-pressure experiments and first-principles calculations [11, 12,30,36,37]. The overestimated bulk modulus of the cotunnite phase by Dubrovinsky et al. [8] and the fluorite phase by Swamy and Muddle [6] may be attributed to the underestimated pressure derivative of the bulk modulus. The first pressure derivative of bulk modulus (B’) were predicted to be 1.35 for cotunnite [8] and 1.75 for fluorite [6], much smaller than the commonly accepted value, around 4. Nishio–Hamane et al. [37], found that very high B0 can be obtained if extremely small B0 (1–2) was assumed. By constraining B0 to 4.0, a different set of experiments did yield a B0 = 294 GPa for cotunnite, which is much close to our result. We believe that the reliable bulk modulus for the cotunnite and fluorite phases should lie within the range of 260–300 GPa. These values are significantly smaller than that of diamond (444 GPa), and thus we conclude that the hardness of the cotunnite and fluorite phases of are not comparable to traditional ultrahard materials, such as diamond. 3.2. Phonon properties In our previous work, we studied the phonon dispersion relations of all the high-pressure phases of TiO2, except for the cubic phases [26]. For the high-pressure polymorphs, it was discovered that there is no imaginary phonon frequency in the phonon dispersions, indicating that they are dynamically stable at ambient pressure. It suggests that the high-pressure polymorphs of TiO2 can be quenched to the ambient conditions as metastable phases. However, the phonon properties of the cubic phases of TiO2 are still not well studied. Here, we predicted the phonon dispersion relations of the fluorite and pyrite phases of TiO2 at their theoretical equilibrium volumes, as shown in Fig. 1. Imaginary phonon modes can be observed at X and W points in the Brillouin zone of the fluorite phase, while unstable modes appear in the whole Brillouin zone of the pyrite phase. These imaginary phonon modes indicate that both of the cubic phases are dynamically unstable at ambient pressure. We also studied the phonon density of state (DOS) of the fluorite and pyrite phases at several different pressures, as shown in Fig. 2. At zero pressure, the calculated phonon DOS of the fluorite phase shows a strong peak at negative frequencies. This peak in the negative region is mainly due to the vibrations of the Ti atoms, whereas the small and broad contribution of O atoms is spread over the whole range of the negative frequencies. Upon compression, the main peak in the negative frequency region decreases rapidly and eventually disappears completely around 55 GPa,
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Table 1 Calculated equilibrium properties of rutile and anatase TiO2 using different exchange–correlation functional, including volume (V0), lattice parameters (a, c/a), atomic position (u), and bulk moduli (B0). Phase
Method
V0 (Å[3]/TiO2)
a
c/a
u
B0 (GPa)
Rutile
Expt. [30,31] LDA PBE PW91 AM05 rPBE PBEsol
31.00–31.22 30.47 32.22 32.11 31.28 32.93 31.13
4.582–4.594 4.559 4.651 4.646 4.606 4.695 4.599
0.64–0.644 0.642 0.639 0.639 0.639 0.636 0.640
0.305 0.304 0.305 0.305 0.305 0.306 0.304
211–235 238 196 203 210 177 216
Anatase
Expt. [5,32,33] LDA PBE PW91 AM05 rPBE PBEsol
33.98–34.07 33.34 35.24 35.13 34.30 36.08 34.14
3.782–3.785 3.751 3.806 3.803 3.775 3.830 3.776
2.512–2.514 2.527 2.556 2.555 2.550 2.569 2.536
0.208 0.208 0.206 0.206 0.207 0.206 0.207
178–179 184 166 168 170 156 176
Table 2 Calculated elastic constants (Cij, in GPa) of rutile TiO2 with different exchange– correlation functional together with experimental results. Method
C11
C12
C13
C33
C44
C66
Source
Expt. (298 K) Expt. (300 K) Expt. (4 K) LDA PBE PW91 PBEsol
271 267 289 295 255 260 276
177 174 197 223 172 174 200
149 146 159 174 147 149 163
484 484 508 523 467 471 497
124 124 128 124 114 115 118
194 190 227 245 212 215 230
Fritz [46] Isaak [47] Manghnani [48] This work This work This work This work
Table 3 Calculated elastic properties of TiO2 phases, including polycrystalline aggregate properties of bulk modulus BH (GPa), shear modulus GH (GPa), Young’s modulus EH (GPa), BH/GH ratio and Poisson’s ratio m in Hill approach, and bulk modulus BEOS and pressure derivative of bulk modulus (B0 ) estimated from EOS fitting. Phase
BH
GH
EH
BH/GH
m
BEOS
B0
Rutile Anatase Brookite Columbite Baddeleyite OI Cotunnite Fluorite Pyrite
229 191 213 224 201 242 244 271 263
109 56 85 88 103 116 93 86 107
281 152 225 234 264 300 248 230 282
2.10 3.43 2.52 2.54 1.94 2.09 2.62 3.14 2.46
0.29 0.37 0.32 0.33 0.28 0.29 0.33 0.36 0.32
216 176 208 220 173 243 256 270 261
5.2 2.4 3.9 4.2 3.3 3.6 5.1 4.3 4.2
indicating that the fluorite phase can be stabilized by high-pressure. The phonon DOS of the pyrite phase also display a strong peak at negative frequencies, corresponding mostly to the vibrations of Ti atoms. As pressure increases, the Ti peak shifts towards
more negative region, and the peaks at negative frequencies remains at all calculated pressures. Special treatments such as chemical doping might be used to stabilize the fluorite phase to ambient pressure, similar to the approach used to stabilize the high temperature stable cubic ZrO2 to room temperature by doping yttria. We believe that the cubic phase of TiO2 synthesized by Mattesini et al. [9] is due to the contamination of the sample in the high-pressure experiment. Our calculations suggest that the fluorite structure can be assigned to the cubic phase observed in experiment. Another noticeable difference between the calculated phonon spectra of the fluorite and pyrite phases is that a band gap exists in the pyrite phase whereas it is absent in the fluorite phase. The phonon band gap in the pyrite does not change much even at very high pressures. The phonon gap in the pyrite phase can be ascribed to its different space group and coordination number of Ti atoms in comparison to the fluorite phase, which results in their unique phonon behaviors upon compression. 3.3. Pressure-induced phase transition at zero temperature The enthalpy change in the anatase to rutile transformation has been measured using different calorimetric techniques. However significant different results have been reported in the literature, from the exothermic values to the endothermic values [38]. The phase stability of TiO2 predicted by DFT is found to be sensitive to the treatment of XC. The relative stability between rutile and anatase, however, is not affected by the approximation used. Our calculations using different XC types show that the energy difference between rutile and anatase is always a positive value, ranging from 1.08 to 8.78 kJ/mol, which means that the anatase phase is more stable than the rutile phase at zero temperature. The current
Fig. 1. Calculated phonon dispersion relations of the cubic phases of TiO2, i.e., (a) fluorite and (b) pyrite, at the theoretical equilibrium volume.
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117
Fig. 2. Calculated phonon density of states of (a) the fluorite and (b) pyrite phases of TiO2 at different pressures.
Fig. 3. Calculated total energy for eight TiO2 polymorphs with respect to volume.
calculations are consistent with most of the recent theoretical studies [10,12]. The large differences for the experimental measurement might be ascribed to the large surface energy of anatase phase, which is not properly considered in the estimation of enthalpy change [39]. The zero temperature phase stability of TiO2 under high-pressure has been extensively studied using different theoretical methods [5,7,10–12,30,36]. We provide a similar DFT study here using the revised PBE form for XC, i.e., PBEsol. The obtained equilibrium properties from EOS fitting and transition pressure at zero temperature provide critical information for the following study of the phase stability of TiO2 at high temperatures. The total energies for TiO2 phases were calculated at a series of difference volumes, and then fitted by thirst-order Birch-Murnagham EOS fitting [40], illustrated in Fig. 3. In order to study the pressure-induced phase transitions, the calculated total energy of TiO2 was converted to enthalpy by H(P) = E(V) + PV, as shown in Fig. 4. The present calculation predicts that the anatase-to-columbite phase transformation occurs at 3.6 GPa, while the rutile and columbite phases are energetically very similar, with columbite having slightly lower energy at low pressure. Columbite is the most stable phase up to 9.9 GPa where the calculations predict a transformation to baddeleyite
Fig. 4. Calculated enthalpy of eight TiO2 polymorphs as a function of pressure, with the rutile phase as the reference state.
which in turn transforms to OI at 20.6 GPa, and to cotunnite at 39.3 GPa. Neither fluorite nor pyrite phase is a thermodynamically stable form in all the studied pressures up to 100 GPa, although the pyrite phase is more stable than the fluorite structure up to 24.2 GPa. Starting from anatase, the predicted pressure-induced phase transition sequence of TiO2 is anatase ? columbite ? baddeleyite ? OI ? cotunnite. In experiments, rutile phase, however, is more commanly used as the starting material. In that case, we predict a transition from rutile to baddeleyite at 6.9 GPa, to OI at 20.6 GPa, and to OII at 39.3 GPa, in close agreement with experiments [5,30,32,37,41,42]. Due to the small energy difference between rutile and columbite, it is worth studying the phase stability of TiO2 upon decompression. Our calculation shows that the baddeleyite phase transforms to columbite phase at 9.9 GPa with decreasing pressure, but never restore the rutile structure, in agreement with the experimental observation [30,37]. 3.4. Pressure–temperature phase diagram Using the calculated phonon DOS, the lattice vibrational free energies of TiO2 phases can be evaluated under the quasiharmonic
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Fig. 5. Calculated two-phase equilibrium line between rutile and anatase phases of TiO2. The blue circle is experimental data compiled by Chase [44].
Fig. 7. Calculated two-phase equilibrium line between columbite and baddeleyite phases. The blue dashed line is the two-phase equilibrium line measured by Tang et al. [42].
approximation, which were further used to study the lattice stability and phase transitions at finite temperatures. Using a similar approach as Ti [43], we predict the pressure–temperature (P–T) phase diagram of TiO2 by comparing the Gibbs energy of each phase calculated directly from phonon DOS. Fig. 5 shows the predicted phase equilibrium between rutile and anatase phases. The transition temperature between anatase and rutile at ambient pressure is predicted to be 1340 K, close to the experimental result of 1200 K compiled by Chase [44]. The transition temperature is determined by both the temperature dependence of the lattice vibrational energy and the zero temperature static energy. Since the thermodynamic properties of rutile and anatase phases are accurately predicted by our DFT calculations [26], the discrepancy of the transitional temperature should be from the uncertainty of lattice stability calculated at zero temperature, which is largely determined by the XC functional used. There has been considerable dispute over the phase equilibrium between rutile and columbite phases of TiO2. The high-pressure and high-temperature phase equilibrium between them has been studied by several authors [13,14,45]. Our predicted pressure– temperature two-phase equilibrium line (see Fig. 6) shows a positive slope, in agreement with results by Akaogi et al. [45] and Withers et al. [14], indicating that the two-phase equilibrium line with a negative slope measured by Olsen et al. [13] may be due to some technical errors. The currently predicted transition pressure between rutile and columbite at room temperature is very close to the measurement by Withers et al., while somewhat smaller
than the extrapolation from Akaogi et al.’s result if a linear twophase equilibrium line was assumed. This underestimation might be ascribed to the underestimated 0 K total energy of the rutile phase. Fig. 7 shows the calculated two-phase equilibrium line between the columbite and baddeleyite phases of TiO2, together with the experimental result measured by Tang et al. [42] The predicted transition pressure between columbite and baddeleyite at room temperature is 12.3 GPa, in an excellent agreement with experimental value of 11.8 GPa [42]. The two-phase equilibrium line between columbite and baddeleyite phases can be described by a linear relationship of T = 159.4 P-1637.6 (T in unit of K and P in unit of GPa), close to that measured by Tang et al., T = 188.7 P1919.5 [42]. By combining all the calculated two-phase equilibrium lines, we obtained the equilibrium pressure–temperature (P–T) diagram of TiO2 with six phases, as shown in Fig. 8. The fluorite and pyrite phases were not included, since thermodynamically they are not stable in the whole pressure and temperature range we studied. A general feature of the predicted P–T phase diagram of TiO2 is that all of the two-phase equilibrium lines show positive slopes, except that between the rutile and anatase phases. From thermodynamic point of view, we know the slope of the two-phase equilibrium in the P–T phase diagram is related to the changes of volume and entropy due to phase transition, by dT/dP = DV/DS. Since the volume changes are negative for the pressure-induced phase transitions,
Fig. 6. Calculated two-phase equilibrium line between rutile and columbite phases. The blue dashed line and read dash-dotted line are the two-phase equilibrium lines measured by Withers et al. [14] and Akaogi et al., [45] respectively.
Fig. 8. Calculated equilibrium pressure–temperature phase diagram of TiO2, together with experimental data [14,42,44,45].
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the entropy changes along the two-phase equilibrium line of rutile/ columbite, columbite/baddeleyite, baddeleyite/OI, and OI/cotunnite should be positive. However, the entropy change due to anatase–rutile transformation is negative. It should be pointed out that the current P–T diagram is predicted based on the thermodynamic equilibrium conditions and the kinetic information of phase transformations is not considered. Therefore, the high-pressure phases, such as OI and cotunnite, may need even higher temperatures than those suggested by the current calculations, to overcome the energy barriers encountered in phase transformations. 4. Summary We have studied the mechanical properties, dynamical and thermodynamic stabilities of TiO2 phases in order to evaluate the potential of the high-pressure phases of TiO2 as ultrahard materials. The elastic constants and moduli of the rutile and anatase phases were evaluated by six exchange–correlation functionals. The PBEsol functional shows the best description of the structural and elastic properties of rutile and anatase phases. Our predicted bulk and shear moduli of the nine phases of TiO2 confirm that the cotunnite and fluorite phases cannot be used as ultrahard materials. The calculated phonon spectra of the cubic phases of TiO2 show imaginary phonon modes in the Brillouin zones, indicating that they are dynamically unstable at ambient conditions. However, the fluorite phase might be stabilized as a metastable phase at high pressures. Treatment such as chemical doping may stabilize it to ambient pressure for future solar cell applications. The pressure-induced phase transitions of TiO2 are found to depend on the starting materials. The calculated total energy of the columbite phase is close to that of the rutile phase, which explains the observance of the columbite phase after releasing high pressure during experiment. The predicted phase transition pressures and sequence are consistent with previous experimental and theoretical studies. From the calculated Gibbs energies of the TiO2 phases, we predicted the two-phase equilibrium lines of the anatase/rutile, rutile/columbite, columbite/baddeleyite, and baddeleyite/cotunnite phases in the pressure–temperature phase diagram. The anatase phase of TiO2 is found to be more stable than the rutile phase at low temperatures and transforms to rutile phase at high temperatures. The calculated two-phase equilibrium line between rutile and columbite phases agrees with those measured by Akaogi et al. and Withers et al., while contradictory to the result by Olsen et al. Meanwhile, the two-phase equilibrium line between columbite and baddeleyite phases predicted in this work show an excellent agreement with experimental measurement. By combining all the two-phase equilibrium lines, we obtained the complete pressure–temperature phase diagram of TiO2 with total six phases. This phase diagram can be used to guide the synthesis of the high-pressure phases of TiO2. Acknowledgements This work is funded by the TIE project of the Center for Computational Materials Design (CCMD), a National Science Foundation (NSF) Industry/University Cooperative Research Center through Grant IIP-0737759 and the National Science Foundation (NSF) through Grant No. DMR-1006557. First-principles calculations were carried out on the LION clusters supported by the Materials
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Simulation Center and the Research Computing and Cyber infrastructure unit at the Pennsylvania State University. References [1] [2] [3] [4] [5]
[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]
[46] [47] [48]
A. Fujishima, K. Honda, Nature 238 (1972) 37. K.I. Hadjiivanov, D.G. Klissurski, Chem. Soc. Rev. 25 (1996) 61. J.C. Jamieson, B. Olinger, Science 161 (1968) 893. H. Sato, S. Endo, M. Sugiyama, T. Kikegawa, O. Shimomura, K. Kusaba, Science 251 (1991) 786. N.A. Dubrovinskaia, L.S. Dubrovinsky, R. Ahuja, V.B. Prokopenko, V. Dmitriev, H.P. Weber, J.M. Osorio-Guillen, B. Johansson, Phys. Rev. Lett. 87 (2001) 275501. V. Swamy, B.C. Muddle, Phys. Rev. Lett. 98 (2007) 035502. H. Dekura, T. Tsuchiya, Y. Kuwayama, J. Tsuchiya, Phys. Rev. Lett. 107 (2011) 045701. L.S. Dubrovinsky, N.A. Dubrovinskaia, V. Swamy, J. Muscat, N.M. Harrison, R. Ahuja, B. Holm, B. Johansson, Nature 410 (2001) 653. M. Mattesini, J.S. de Almeida, L. Dubrovinsky, N. Dubrovinskaia, B. Johansson, R. Ahuja, Phys. Rev. B 70 (2004) 212101. J. Muscat, V. Swamy, N.M. Harrison, Phys. Rev. B 65 (2002) 224112. X.A. Wu, E. Holbig, G. Steinle-Neumann, J. Phys.–Condens. Matter (2010) 295501. X.G. Ma, P. Liang, L. Miao, S.W. Bie, C.K. Zhang, L. Xu, J.J. Jiang, Phys. Status Solidi B-Basic Solid State Phys 246 (2009) 2132. J.S. Olsen, L. Gerward, J.Z. Jiang, J. Phys. Chem. Solids 60 (1999) 229. A.C. Withers, E.J. Essene, Y.X. Zhang, Cont. Mineral. Petrol. 145 (2003) 199. Z.-K. Liu, J. Phase Equilib. Diffus. 30 (2009) 517. P.E. Blöchl, Phys. Rev. B 50 (1994) 17953. G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169. G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758. J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. R. Armiento, A.E. Mattsson, Phys. Rev. B 72 (2005) 085108. Y. Zhang, W. Yang, Phys. Rev. Lett. 80 (1998) 890. J.P. Perdew, A. Ruzsinszky, G.I. Csonka, O.A. Vydrov, G.E. Scuseria, L.A. Constantin, X. Zhou, K. Burke, Phys. Rev. Lett. 100 (2008) 136406. P.E. Blöchl, O. Jepsen, O.K. Andersen, Phys. Rev. B 49 (1994) 16223. Z.G. Mei, Y. Wang, S.L. Shang, Z.K. Liu, Inorg. Chem. 50 (2011) 6996. Y. Wang, S. Shang, Z.-K. Liu, L.-Q. Chen, Phys. Rev. B 85 (2012) 224303. Y. Wang, J.J. Wang, W.Y. Wang, Z.G. Mei, S.L. Shang, L.Q. Chen, Z.K. Liu, J. Phys.– Condens. Matter 22 (2010) 202201. M. Gajdos, K. Hummer, G. Kresse, J. Furthmuller, F. Bechstedt, Phys. Rev. B 73 (2006) 045112. Y. Al-Khatatbeh, K.K.M. Lee, B. Kiefer, Phys. Rev. B 79 (2009) 134114. L.C. Ming, M.H. Manghnani, J. Geophys. Res. 84 (1979) 4777. T. Arlt, M. Bermejo, M.A. Blanco, L. Gerward, J.Z. Jiang, J.S. Olsen, J.M. Recio, Phys. Rev. B 61 (2000) 14414. V. Swamy, L.S. Dubrovinsky, J. Phys. Chem. Solids 62 (2001) 673. Z.G. Mei, S.L. Shang, Y. Wang, Z.K. Liu, J. Phys.–Condens. Matter 22 (2010) 295404. G. Simmons, H. Wang, Single Crystal Elastic Constants and Calculated Aggregate Properties, MIT press, Cambridge, MA, 1971. L. Koci, D.Y. Kim, J.S. de Almeida, M. Mattesini, E. Isaev, R. Ahuja, J. Phys.– Condens. Matter 20 (2008) 345218. D. Nishio-Hamane, A. Shimizu, R. Nakahira, K. Niwa, A. Sano-Furukawa, T. Okada, T. Yagi, T. Kikegawa, Phys. Chem. Miner. 37 (2010) 129. T. Mitsuhashi, O.J. Kleppa, J. Am. Ceram. Soc. 62 (1979) 356. A.A. Levchenko, G. Li, J. Boerio-Goates, B.F. Woodfield, A. Navrotsky, Chem. Mater. 18 (2006) 6324. Z.-G. Mei, S. Shang, Y. Wang, Z.-K. Liu, Phys. Rev. B 79 (2009) 134102. L. Gerward, J. Staun, J. Appl. Crystallogr. 30 (1997) 259. J. Tang, S. Endo, J. Am. Ceram. Soc. 76 (1993) 796. Z.G. Mei, S.L. Shang, Y. Wang, Z.K. Liu, Phys. Rev. B 80 (2009) 104116. M.W. Chase, NIST-JANAF Thermochemical Tables, American Institute of Physics, Washington, D. C., 1998. M. Akaogi, K. Kusaba, J.-I. Susaki, T. Yagi, M. Matsui, T. Kikegawa, H. Yusa, E. Ito, in: Y. Syono, M.H. Manghnani (Eds.), TERRAPUB American Geophyical Union, Washington, DC, 1992, p. 447. I.J. Fritz, J. Phys. Chem. Solids 35 (1974) 817. D.G. Isaak, J.D. Carnes, O.L. Anderson, H. Cynn, E. Hake, Phys. Chem. Miner. 26 (1998) 31. M.H. Manghnani, E.S. Fisher, W.S. Brower, J. Phys. Chem. Solids 33 (1972) 2149.