Ab initio Rutile-Cristobalite Transitions in Silicon Dioxide and Titanium Dioxide

Transcription

1 20 Ab initio Rutile-Cristobalite Transitions in Silicon Dioxide and Titanium Dioxide Moon, Timothy Y. ; Kroll, Peter Department of Chemistry and Biochemistry, The University of Texas at Arlington, Arlington, TX, USA Submitted August 22, 2009, revised October 20, 2009, accepted November 28, 2009, published online December 5, Abstract: Transitions between crystal structures are studied in order to discover polymorphs with novel applications. In this study, the use of nickel arsenide to sphalerite transition is proposed as a model for other transitions. This is accomplished by removing some of the interstitial atoms and performing the transition on the remaining atoms. The transition between the cristobalite and the rutile structures was derived using this method and applied to silicon dioxide and titanium dioxide. Ab initio calculations were performed on these transitions and data were collected on the thermodynamic and kinetic stability of the initial and end structures. Structural transformations are predicted at 5 GPa for α- cristobalite (SiO 2 ) and -0.4 GPa for rutile (TiO 2 ). Estimated upper boundaries of the activation energies of these transitions are 1.8 ev and 1.1 ev per formula unit SiO 2 and TiO 2, respectively. Westwood High School, Mellow Meadow Drive, Austin, TX correspondence to: Dr. Peter Kroll, pkroll@uta.edu, Department of Chemistry and Biochemistry, The University of Texas at Arlington, Box 19065, Arlington, TX

2 21 Introduction: Many crystal structures are based on a closest packing of spheres motif with additional interstitial atoms filling the spaces in between. The vast majority of these closest packings typically are found in two varieties: cubic closest packing (ccp) and hexagonal closest packing (hcp). Interstitial sites are typically coordinated by either four or six structural atoms, resulting in tetrahedral or octahedral coordination respectively. For instance, the rocksalt structure is a ccp with all of the octahedral interstitial sites filled and the wurtzite structure is an hcp with half of the tetrahedral interstitial sites filled. The transition between these forms has been investigated in several computational studies. 1,2 The same ccp-hcp motif can be found between nickel arsenide (NiAs) (hcp with all of the octahedral interstitial sites filled) and sphalerite (ccp with half of the tetrahedral interstitial sites filled). This is shown in Figure 1. However, this transition has not been thoroughly studied. Figure 1. Transition from a NiAs to a sphalerite structure. Yellow atoms are structural and green atoms are interstitials. The box depicts a unit cell. Notice that there are two interstitial sites in the sphalerite that are tetrahedral-coordinated, one up and one down. The octahedral-coordinated interstitial in the NiAs could have potentially moved into either one. It is possible to map this transition path onto structures related to NiAs or sphalerite by removing some of the interstitial atoms. For instance, the rutile structure (hcp, 50% of octahedra filled) can be derived from NiAs by removing half of the interstitial atoms. Similarly, the cristobalite structure (ccp, 25% of tetrahedra filled) can be generated from sphalerite. One system that is known to exhibit the rutile-cristobalite transformation is silicon dioxide. 3,4 Cristobalite is a mineral composed of SiO 2 that has been found in areas with volcanic rock. 5 Stishovite, a known highpressure polymorph of SiO 2, adopts the rutile structure. 6 The mineral rutile is a common polymorph of titanium dioxide. Although TiO 2 is not known to exhibit a cristobalite structure, a transition has been proposed based on its topology. 7 One way to trigger a rutilecristobalite structural transition is to apply pressure. At high pressure, compact structures, such as rutile, tend to be more thermodynamically favorable than open structures, such as cristobalite. Thus, given enough pressure, the transition will occur spontaneously. In this investigation, the pressure needed to trigger the transitions falls within the range of gigapascals. For reference, a gigapascal is approximately times standard atmospheric pressure. Although these conditions are extreme, they are not entirely unreasonable. Silicon dioxide within the Earth s upper mantle, for instance, experiences pressures up to 13.4 GPa. 8 Aside from their topological interest, silicon dioxide and titanium dioxide substances also have significant practical relevance. Silicon dioxide is prevalent in the Earth s crust and it is used to produce glass, concrete, and electronic components. Titanium dioxide is used as a white pigment and in self-cleaning glass. Therefore, if this research were continued and applied, it

3 22 could be highly beneficial. Our observations may, for one, explain phenomena witnessed in nature and industry. Furthermore, if the transition processes are well understood, the substances can be doped or otherwise manipulated to synthesize compounds more easily. In the end, it is hoped that our findings be used to find novel applications for these high-pressure polymorphs. Methods: The unit cells describing the rutile structure and the cristobalite structure do not contain the same number of atoms. Therefore, modified unit cells were used for the calculations. For the rutile structures, the lattice parameters a, b, and c were transformed to a, b, and c such that: a = a + b b = -a + b c = 2c For the cristobalite structures, the lattice parameters were modified such that: a = a + b b = -a + b c = c After these transformations were applied, both the rutile and the cristobalite contained sixteen structural and eight interstitial atoms. Calculations were conducted on the plane wave basis implemented by the Vienna ab initio simulation package (VASP). 9 Pseudopotentials for simulated atoms were produced with the projectoraugmented-wave (PAW) method implemented by Blöchl. 10,11 Functionals were based on generalized gradient approximation (GGA), specifically the Perdew-Wang 1991 GGA. 12 In order to accommodate the high electronegativity of the oxygen atoms in the compounds, the kinetic energy cutoff was maintained at 500 ev for all calculations. All simulations were at 0 K. The atomic positions and cell parameters of the ground state structures were optimized. In order to calculate the effect of pressure on the crystal structures, the scaling factors of the unit cells were modified. The atoms of these pressurized structures were then relaxed while the volume was kept constant. The transition pressure between structures was determined by finding the pressure at which both structures could exist at equilibrium, that is, when the difference in Gibbs energy was zero. Thus, the following relation was used: G = 0 = H T S However, because the calculations operated at absolute zero, the last term can be ignored. Thus, enthalpy was used as the determining thermodynamic factor of the transitions. Two linear transition paths were found between ideal NiAs and sphalerite structures due to the fact that any single interstitial atom within an octahedral interstitial of NiAs could potentially move to occupy one out of two tetrahedra within sphalerite. In order to apply these transitions to the rutilecristobalite transition, interstitial atoms were removed and the two NiAssphalerite transitions were applied to the remaining atoms. Structures along the linear transition paths were calculated while atomic positions and lattice parameters were kept constant. This single-point calculation method may overestimate the activation energy of the transitions.

4 23 Results and Discussion: SiO 2 The energy of a formula unit of α- cristobalite was calculated to be ev while that of stishovite (rutile-structured SiO 2 ) was found to be ev (see Figure 2). A high activation energy of 1.8 ev per formula unit indicates that this transition is rather slow. In nature, it is probable that this transformation occurs along a non-linear pathway, setting this activation energy as an upper bound. transition proposed by Huang, et al., involves a transition to an intermediate structure X-I around 20 GPa and to stishovite at 22 GPa. Thus, their results do not necessarily preclude the existence of a direct transition between α- cristobalite and stishovite around 5 GPa. It is of note that stishovite was formed when Tsuchida and Yagi applied a temperature above 1000 C to a sample to cristobalite at 15 GPa. This suggests that the high activation energy of the transition is the primary inhibiting factor of the transition at high-pressure. Figure 2. Energy of a formula unit of SiO 2 as it linearly transforms from α-cristobalite to stishovite. The calculated enthalpies of these two polymorphs of SiO 2 at several pressures are presented in Figure 3. The α-cristobalite exhibited a lower enthalpy, and thus was thermodynamically favored at lower pressures. However, at approximately 5.1 GPa, the enthalpies of both structures are equal, indicating that this is a transition pressure. Above this pressure, the stishovite is thermodynamically favored. This transition pressure is significantly lower than an experimentally derived pressure discovered by Tsuchida and Yagi 3 and corroborated by a computational study by Huang, et al. 4 The non-linear Figure 3. Enthalpy of a formula unit of SiO 2 over a range of pressures. TiO 2 The energy of a formula unit of rutile TiO 2 was calculated to be 26.6 ev while it was 26.4 ev for cristobalitestructured TiO 2 (see Figure 4). The activation energy of the transition, 1.1 ev per formula unit, is low enough for such a linear transition to be plausible. If the actual transition is nonlinear, it would have a lower activation energy. Thus, a rutile-cristobalite transition could happen if the thermodynamic balance shifts in favor of the cristobalite structure.

5 24 The rutile structure was calculated to have a lower enthalpy than the cristobalite structure at zero pressure. Because rutile is denser than cristobalite, it is unlikely that an increase in pressure will shift the thermodynamic balance. This is corroborated by the trend shown in Figure 5. The data suggests that the transition pressure is approximately -0.4 GPa. In other words, this transition may occur when sufficient tension is applied to rutile TiO 2. This amount of tension could probably be produced with current technology, albeit nonhydrostatically. This seems to be a promising area of research and should be investigated further. Conclusion: Figure 4. Energy of a formula unit of TiO 2 as it linearly transforms from rutile to a cristobalite structure. In this investigation, the transition between the cristobalite and the rutile structures was rationalized by removing atoms from the transition between the nickel arsenide and the sphalerite structures. Ab initio calculations predict a transformation in silicon dioxide from α-cristobalite to stishovite at approximately 5 GPa, although high activation energy may inhibit its progress. In addition, calculations predict a transformation in titanium dioxide from rutile to cristobalite at -0.4 GPa. Acknowledgements: I, Timothy Moon, would like to thank the Robert A. Welch Foundation for funding and coordinating my research experience. I also owe great gratitude to Dr. Peter Kroll for educating me in crystallography and computational chemistry as well as for the assistance he provided in supervising my project. References: Figure 5. Enthalpy of a formula unit of TiO 2 over a range of pressures. 1. Cai, J.; Chen, N. J. Phys.: Condens. Matter 2007, 19, Saib, S.; Bouarissa, N. Physica B 2007, 387, Tsuchida, Y; Yagi, T. Nature 1990, 347, Huang, L; Durandurdu, M.; Kieffer, J. Nat. Mater. 2006, 5, Howard, A.R. Am. Mineral. 1939, 24, Ross, N.L.; Shu, J.; Hazen, R.M. Am. Mineral. 1990, 75, O Keefe, M.; Hyde, B.G.; Acta Crystallogr. 1976, B32, Jeanloz, R. In Mantle Convection: Plate Tectonics and Global Dynamics; Peltier,

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