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The effect of cation-ordering on the elastic properties of majorite: An ab initio study
Earth and Planetary Science Letters 256 (2007) 28 – 35 www.elsevier.com/locate/epsl
The effect of cation-ordering on the elastic properties of majorite: An ab initio study Li Li a,b,⁎, Donald J. Weidner a,b , John Brodholt a , G. David Price a b
a Department of Earth Sciences, University College London, Gower Street, London WC1E6BT, UK Mineral Physics Institute, Department of Geosciences, University of New York at Stony Brook, Stony Brook, NY, 11790, USA
Received 18 July 2006; received in revised form 7 January 2007; accepted 7 January 2007 Available online 13 January 2007 Editor: R.D. van der Hilst
Abstract The effects of cation disorder and pressure on the structural and elastic properties of MgSiO3 majorite (Mj100) and MgSiO3– Mg3Al2Si3O12 solid solution (Py50Mj50) are modelled using the first principle simulations. Our results are consistent with the tetragonal phase as the stable structure for both compositions up to 25 GPa. Both pressure and disorder decrease the differences between c11 and c33, and between c44 and c66, indicating that the elastic properties move closer to cubic. The calculated bulk modulus and shear modulus are comparable with the reported experimental data. The bulk modulus of Mj100 varies little while the shear modulus decreases slightly with increasing cation disorder. The elastic properties of an ordered Py50Mj50 are nearly cubic in symmetry. Mg–Si disorder has much lower energy impact on Py50Mj50 than Mj100. In the Earth's mantle, variation of the Al content in the garnet will more significantly affect the seismic velocities than will the order/disorder of Si and Mg. © 2007 Elsevier B.V. All rights reserved. Keywords: majorite; cation disorder; elasticity; high pressure; seismic velocity
1. Introduction Majorite rich garnet is expected to be the second most abundant mineral in the Earth's transition zone [1]. The formation of majorite will significantly affect the density of subducted basaltic crust within a pyrolite mantle [2]. Thus, the properties of majorite and their dependence on chemical composition will have a significant impact on radial and lateral variations in seismic velocities and buoyancy in this region. The end-member, Mg–Si, majorite is among the few minerals that ⁎ Corresponding author. Mineral Physics Institute, Department of Geosciences, University of New York at Stony Brook, Stony Brook, NY, 11790, USA. E-mail address: [email protected] (L. Li). 0012-821X/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.epsl.2007.01.008
have ferroelastic phase transitions that are often accompanied by softening of elastic moduli or high acoustic attenuation [3,4]. This transition is marked by the ordering of magnesium and silicon cations on the octahedral sites accompanied by a cubic to tetragonal phase transformation. Sinogeikin et al. [5] suggest that there is a discontinuous decrease in the elastic moduli across this transition. Their experimental data, which span a range in the pyrope–majorite solid solution, were interpreted as having a small discontinuous decrease in the region where the tetragonal phase is stable. YeganehHaeri et al. [6], on the other hand, suggest a continuous decrease in elastic moduli as aluminium is replaced by an Mg–Si couple, with no discontinuity associated with the change in space group. This issue has an immediate implication for seismic velocities in the mantle which
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may have some dependence not only on the chemical composition of the garnet, but also on the degree of ordering of the Mg and Si on the octahedral site. Experimental studies are quite limited in clarifying this issue. The experimental data of [5] could also be explained, within the resolved errors, by a gradual compositional effect on the elastic moduli with no effect of the phase transition. Furthermore, X-ray studies are weak in defining the state of order between Mg and Si owing to the similarity of their structure factors. Here we present the results of ab initio calculations. They indicate that while the presence of the tetragonal phase is definitive evidence of the ordered structure, the tetragonal phase may be metrically cubic (all axes of the same length). Thus, the elastic properties may be determined by the ordered structure, but still appear to be cubic in terms of cell dimension thus rendering the experimental data even more ambiguous. However, we find that while disorder appears to lower the shear modulus, the effect of composition is by far the dominant variable affecting the seismic velocity in the pyrope–majorite solid solution. Majorite is unusual in that Mg and Si cations are significantly (i.e.N 10%) disordered. Laboratory studies have shown that majorite garnet can transform from cubic (Ia–3d) to tetragonal (I41/a) at elevated temperature and pressure [3]. The tetragonal MgSiO3 majorite garnet has Mg1(16f) and Mg2(8e) occupying the dodecahedral sites, Si1(4b), Si2(4a) and Si3(16f) occupying the tetrahedral sites, O1–O6(16f) occupies the corner of tetrahedral, and Mg(8c) and Si(8d) cations occupying the octahedral sites. Cation disordering occurs when Mg (8c) exchanges sites with Si(8d). We define here the 8c 8d compositional parameter x in Mgx8d Mg1−x Si1−x Six8c to represent the cations in the octahedral sites. Then complete ordering obtains for x = 0 (tetragonal structure) and disorder (cubic structure) occurs at x = 0.5. Angel et al. [7] report x = 0.2 for MgSiO3 majorite (0.2 Si on the Mg site). Even though the tetragonal structure has been reported for the synthesized majorite garnet [3,7–9], natural Mg–Fe rich majorite found in meteorite craters have been reported as being cubic (Ia–3d) [10,11] with Mg and Si randomly distributed. For garnet, order–disorder is associated with the tetragonal–cubic phase transition. In the Earth's mantle, majorite exists in solid solution with pyrope. An experimental study [12] has reported the tetragonal structure for the majorite–pyrope join when the majorite content is over 75 mol% and the cubic structure for the ones with more aluminium. The data are excellent in defining the cell dimensions, but still unable to define the state of order, and therefore unable to confirm the
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cubic space group. The presence of Al in the solid solution also affects the elasticity [5,6,13–19]. Typically, elastic properties decrease very modestly as majorite replaces pyrope in the solid solution. First principle simulation has successfully predicted structural and elastic properties of mantle minerals [20– 23]. This approach can define atom positions in a periodic cell and simulate the cation substitution and ordering, especially at high pressure. We specify the atom positions in a large cell. We use density functional theory to compute the enthalpy, structural, and elastic properties of majorite garnet with defined atom configurations at mantle pressure. To simulate disorder, we use a sample box with 160 atoms and perturb only a few of these atoms. The arrangement of atoms on a larger scale is assumed to be periodic with the sample box. Thus, we are restricted to define the properties of systems that have specific violations of the ordered space group. Howver, the calculations provide some insights that are not accessible experimentally. For example, X-ray diffraction cannot define the Si–O bond distances for the Si atoms that are on the Mg sites and yield only the average of the M–O bond distance is defined. The calculations do provide this information. 2. Computation method Computations were performed using the density functional theory (DFT) based VASP code with the projector augmented wave implementation [24,25] and a plane wave basis set with kinetic energy cut-off of 900 eV. The electron exchange and correlation is described within the generalized gradient approximation (GGA-PW91) [26,27]. The Γ-point is used for Brillouin zone sampling. We used a unit cell with I41/a symmetry containing 160 atoms (listed in Supplementary data Table S1 based on experimental data of [7]). These were used as the starting point for the calculations. The disordered atomic positions for different garnet models with x = 0, 1/8, 2/8, 4/8 are given in Table S2 of the Supplementary data. The four configurations for the x = 1/8 case represent all of the possible configurations for this degree of disorder with a cell containing 160 atoms. For higher values of disorder, the number of possible disordered arrangements grows rapidly. Not all possibilities were considered, rather the Mg and Si atoms were randomly chosen due to the high computation demand. A total of 10 configurations for majorite were considered. For each model, the lattice parameters, the cell shape, as well as ionic positions were allowed to relax. Calculations were performed between 0 and 30 GPa. Elastic moduli for the
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select models were calculated at high pressures (0, 20 and 30 GPa). To obtain accurate elastic moduli in the limit of zero strain, positive and negative strain of magnitude 1% were applied and corresponding stresses were deduced. Expanding the plane-wave cut-off energy to 1000 eV and the Monkhorst-Pack grids [28] by two kpoints in each direction varies the enthalpy by less than 1 meV/atom and the effect on the stress and elastic constants are insignificant. We estimate the calculational uncertainty of the elastic moduli to be less than 1 GPa. Similar calculations were carried out for a pyrope– majorite solid solution composition with 50 mol% of pyrope and 50 mol% of majorite. Two configurations (pm0 and pm2) were used (details of atom positions are given in the Supplementary data Table S3). In the octahedral site, pm0 has the four Mg atoms ordered; pm2 has two of the Mg atoms exchanged with two Si atoms. 3. Results and discussion It has been recognized by a number of studies that the GGA based calculation is precise in defining state properties as a function of V, but that it overestimates, by a constant value, the pressure for any volume [22,29]. Using the same approach from these studies, we have corrected for the volume offset by combining the volume (V)–bulk modulus (K) relationship of the model with the experimental V0 by fitting a Birch– Murnaghan relation to the V–K calculations for majorite. Since the variation in volume caused by cation disorder is small (∼0.1%) for the configuration mj1a, mj1c and mj4b, we fit the majorite models with the constraint that V0 = 1518.6 Å3 [7]. The parameters for
Fig. 1. Total energy (E) from DFT calculation of the unit cell garnet with GGA vs. the disorder parameter x. Diamond symbol represents the MgSiO3 majorite (Mj100); square symbol represents the MgSiO3– Mg3Al2Si3O12 solid solution (with composition of Py50Mj50). The energy increases monotonically with x for both compositions.
Fig. 2. Unit cell parameter a and c ratio as a function of disorder parameter x for garnet. For Mj100, solid diamond symbols are from this study, open square symbol represents data from [7]. Horizontal lines represent the data from [3] and [18], since their order parameters are not known. Solid triangle symbols represent calculated results for Py50Mj50.
the equation of state are then calculated. With the equation of state, we can calculate the pressures that correspond to the model volumes. We find that the GGA models consistently overestimate the pressure by 4.9 (2) GPa for each configuration regardless the pressure or the degree of disorder as well as for the pyrope–majorite compositions considered here. We indicate in the following text the corrected pressure Pc = P − 4.9 GPa for all compositions and configurations. It is common for garnets to have cubic symmetry (Ia– 3d). Majorite, with Si and Mg sharing the octahedral site copes with the different sizes of these cations by forming a tetragonal structure (I41/a), which has two crystallographically distinct octahedral sites Oc1 and Oc2 which can accommodate the different cation sizes. Temperature is an important agent for altering the degree of cation order. The cations can be “frozen” to its sites with a fast quench from high temperature [30,31]. Thus the disordered structure can be preserved at room temperature. In our calculation, we calculate the properties of such “frozen” disordered structures at 0 K. The total energies of the relaxed structures with the GGA are plotted in Fig. 1. The ordered structure has the lowest energy among all configurations plotted. The energies with the same disorder parameter represent the different sites of the Mg–Si pair. For MgSiO3 majorite (Mj100), the configuration with the higher degree of disorder has higher total energy. For MgSiO3–Mg3Al2Si3O12 solid solution (with composition of Py50Mj50), the variation of energy among the three degrees of cation ordering is smaller, indicating that the presence of Al lessens the energetic preference for cation ordering which is consistent with the observation that aluminium bearing garnets tend to be cubic at elevated temperatures [12].
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Fig. 3. Unit cell parameter ratio a/c vs. pressure for 4 configurations: ordered, mj1b, mj1d and mj4b.
We estimate the temperature that will accommodate specific disordered states by considering the configurational entropy [32]. For the pure majorite composition, this calculation implies that the degree of disorder will reach the value of 1/8 of the sites at about 2000 K. For the Py50Mj50 composition, the lowest energy system will be disordered (cubic) above 1500 K and ordered (tetragonal) below this temperature. While there is considerable uncertainty in these numbers since they are based on a small difference between two large numbers, they suggest that ordering may be significant at mantle conditions. The relaxed structure of all configurations is nearly tetragonal dimensionally (a = b bN c, α = β = γ = 90°) even though the space group of the disordered structures are necessarily triclinic. The unit cell parameter ratio a/c as a function of the disorder parameter x is plotted in Fig. 2. The a/c ratio from the previous experimental studies [3,7,18] is plotted for comparison with our results. The degree of disorder for the data of [3,7,18] are unknown however [7,18] samples are probably the same since they were grown at the same conditions. For most of the Mj100 calculations, the a/c ratio decreases with x, except configuration mj4b. The three data points for Py50Mj50 are slightly higher than but very close to 1. Furthermore, the effect of pressure on metrics of the cell is small as shown in Fig. 3. Table 1 indicates the effect of disorder on the octahedral bond lengths. The Si–O bond (1.815 Å) in
Fig. 4. Bond lengths for the ordered MgSiO3 majorite as a function of Pc. Plotted are the mean values of the calculation and experimental results [7].
Table 1 Bond length for octahedral sites in MgSiO3 majorite garnet X
0
1/8
2/8
Oc1(Mg)–O Oc2(Mg)–O Oc1(Si)–O Oc2(Si)–O
2.064
2.056 2.028 1.815 1.800
2.047 2.024 1.815 1.803
1.798
Fig. 5. Calculated single crystal elastic moduli cij as a function of degree of disorder x at Pc = −5 GPa (Fig. 5a) and 25 GPa (Fig. 5b).
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the Oc1 site for x = 1/8 is significantly longer than the Si–O bond in the ordered site (1.798 Å) but much shorter than the Mg–O bond in the Oc1 site (2.056 Å). The calculated mean Si–O bond length ranges from 1.796 Å to 1.815 Å which are close to the reported sixcoordinated Si–O distances range from 1.757 Å for stishovite [33] to 1.811 Å for silica-rich sodium pyroxene [34]. The Mg–O bond (2.028 Å) for x = 1/ 8 is significantly shorter than that in the ordered site (2.064 Å). The calculated mean Mg–O bond lengths fit into the range of mean Mg–O distance from 2.05 Å to 2.15 Å [35]. Angel et al. [7] refined the structure of MgSiO3 garnet from X-ray single crystal diffraction spectra.
Fig. 4 compares the X–O bond lengths reported by [7] with those calculated here for the completely ordered case. Our values are displayed as a function of pressure, Pc, while [7]'s results are for room pressure only. The excellent agreement between the calculated values for the tetrahedral sites and dodecahedral sites underscores the precision of the DFT calculations. The Mg–O octahedral bonds determined by [7] are slightly shorter that those determined here, while the Si–O octahedral bonds of [7] are longer. Angel et al. [7] pointed out that this is expected due to the disorder of Si onto the Mg site and Mg onto the Si site. Angel et al. [7] determined that the Si site is occupied by 0.8 Si atoms and 0.2 Mg atoms, but emphasized the uncertainty of this value owing to
Fig. 6. Calculated single crystal elastic moduli vs. pressure for 6 configurations (mj0, mj1b, mj2b, mj4a, mj4b and pm0).
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the similarity of the scattering factors of Si and Mg. Assuming that our calculations of the lengths for the Mg–O and Si–O bonds at 0 GPa pressure are accurate, and the lengths measured by [7] represent a simple weighted average of the bond lengths for the occupants of the sites, we can calculate the site occupancies. Based on the size of the Mg site, we conclude that the degree of order is 0.82, while the Si site size suggests 0.91. These values are in excellent agreement with the refinement of [7]. The single crystal elastic moduli as a function of x (given in Supplementary data Table S4) for 6 configurations (mj0, mj1b, mj2b, mj4a, mj4b and pm0) at Pc = − 5 GPa (Fig. 5a) and Pc = 25 GPa (Fig. 5b) are plotted. For both pressures, the shear moduli c66 varies significantly with cation disorder while c44 remains almost unchanged at − 5 GPa but increases at 25 GPa. At 25 GPa, c33 decreases with x while c11 varies little; At − 5 GPa, c11 decreases with x while c33 varies little. The different behaviour of c11 and c33 with pressure for the disordered majorite can also be seen in Fig. 6. The ordered majorite has the largest difference between c11 and c33, c44 and c66, c12 and c13. With increasing degree of cation disorder, the difference between these three pairs of cij decreases, eventually to zero, where the cubic symmetry dominates the elasticity. For the Py50Mj50 model, even though the Mg and Si are ordered, the elastic moduli are much closer to be cubic compared with ordered Mj100 model. For MgSiO3 majorite the bulk modulus (K) and its pressure derivative (K′) vary little with the cation disordering among the five configurations (mj0, mj1b, mj2b, mj4a, mj4b). This is summarized in the Supplementary data Table S5. Our calculated bulk moduli are consistent with various experimental groups [5,14, 15,18,36,37]. Our calculated shear modulus for ordered MgSiO3 majorite is slightly lower than the average of reported values [5,14,15,18,36] but within the scatter of the experimental data. The shear modulus of the disordered MgSiO3 majorite is in general slightly lower than for the ordered configuration with little variation among them. Of all of the configurations considered here, mj4b is most nearly metrically cubic for the relaxed structure (a = b = c). mj4b has the lowest shear modulus of all of the cases considered here, but is only about 3% lower than the average shear modulus. The pressure derivative of the shear modulus is quite constant among the various configurations, agreeing well with the experimental results of [15] while being considerably smaller than that of [36]. For the pyrope–majorite solid solution, our calculated results for Py50Mj50 are consistent with the
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experimental studies which have the same chemical composition [15,38] within the errors. There is a dependence of volume on the composition for pyrope– majorite solid solution [12]. In Fig. 7, we plot the ratio of volume to the Mj100 vs. the majorite mole fraction. Among the experimental studies [7,12,39], our results (Py50Mj50) follow the trend of this composition dependence of volume and fill in the middle composition. Two calculated points are plotted corresponding to pm0 and pm2 showing that the cation order–disorder does not have significant effect on the volume of the Py50Mj50 models. Comparing the pure majorite and the Py50Mj50 results, we generally conclude that the elastic moduli increase with increasing Al content. We conclude that the effect of composition change, while small, is stronger than the effect of ordering. This result is very important in modelling the seismic velocities from the mineralogy since the Al content is much easier to define than is the degree of Mg–Si order for all depths in the mantle. We conclude that we may have an error in the shear modulus less than about 2% for regions with pure majorite owing to ignorance of the state of order. However, in the mantle, the garnet composition will generally be closer to the 50:50 model, where the effect of ordering is very small. Elastic anisotropy of minerals contributes to the seismic velocity anisotropy. We have calculated the sound velocities of the majorite for each model from their single crystal elastic properties. Details of the results are found in the Supplementary data Table S6, which lists the maximum and minimum sound velocities and the anisotropy for each model at Pc = − 5 GPa and 25 GPa. In general, the ratio between the fastest
Fig. 7. Variation in cell volume (V/V0) as a function of composition for a series of materials along pyrope–majorite solid solution. V0 represents the volume for majorite (V0 = 1515.4 Å3 from [7]). 1:[39]; 2:[12]; 3: this study; 4:[7].
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4. Summary
Fig. 8. Acoustic velocities for Mj100 and Py50Mj50 at Pc = − 5 GPa and 30 GPa. In both cases, the Mg and Si atoms are in the ordered sites.
longitudinal velocity to slowest one (Ap) and the ratio between the fastest shear velocity to slowest one (As) is close to 1. At both pressures, the cation disorder decreases both Ap and As except model mj4a. The Py50Mj50 model has the lowest anisotropy among all the majorite models. Pyrope is the most isotropic. The velocity surface along various directions for Mj100 and Py50Mj50 are shown in Fig. 8 for Pc = − 5 GPa and 25 GPa. Fig. 8 contrast the effect of crystallographic direction on velocities with the effect of pressure on velocities for these two compositions. For longitudinal waves, the effect of pressure dominates for both compositions; for shear waves, the effect of anisotropy on velocities of Mj100 is comparable to the effect of 30 GPa pressure difference while Py50Mj50 is dominated by pressure since this composition is more isotropic.
Majorite garnet, the Al free end-member of mantle high pressure garnets, forms an ordered tetragonal phase. Density functional theory models at 0 K give insights into the properties of this material. The details of the crystal structure are well represented by the GGA based models. Volume and bulk modulus are independent of the degree of order, while the shear modulus decreases by less than 4% from the tetragonal, ordered structure to the disordered, cubic state. In contrast, the 50:50 pyrope: majorite solid solution shows little change in elastic moduli with ordering of the cations. However, the elastic moduli are Al dependent, with higher elastic moduli associated with more Al. We suggest that seismic modelling of a garnet bearing mantle need only be concerned with the Al content of the garnet. Variation of elastic moduli with ordering is determined to be a smaller effect than the compositional variations. Even though a previous study [12] detected the 50:50 composition to be cubic, it is difficult to determine the space group with X-ray diffraction when the distinction between the cubic and tetragonal structures for Py50Mj50 is small. Our models of Py50Mj50 predict that the tetragonal phase is the most stable at 0 K up to about 1500 K, yet the properties show very little variation in cell parameters or elastic properties with changes in cation order. Furthermore, the elastic moduli of the tetragonal structure appear to reflect a cubic symmetry and the c/a ratio is nearly unity. Generally, the symmetry of the elasticity tensor is nearly isotropic, suggesting that garnet, even with a preferred orientation, will not impose a seismically discernable anisotropy on the observed seismic signal. Acknowledgements This work is supported by NERC (Grant Nos. NER/ T/S/2001/00855; NER/O/S/2001/01227), and computer facilities are provided by NERC at University College London, and the High Performance Computing Facilities of the University of Manchester (CSAR) and the Daresbury Laboratory (HPCx). DJW acknowledges the Leverhulme Trust for support through the visiting Professor program. DJW and LL acknowledge NSF EAR-9909266, EAR0135551, EAR00135550. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j. epsl.2007.01.008.
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[20] B. Kiefer, L. Stixrude, R. Wentzcovitch, Normal and inverse ringwoodite at high pressures, Am. Mineral. 84 (1999) 288–293. [21] B. Kiefer, L. Stixrude, R. Wentzovitch, Calculated elastic constants and anisotropy of Mg2SiO4 spinel at high pressure, Geophys. Res. Lett. 24 (22) (1997) 2841–2844. [22] L. Li, J.P. Brodholt, S. Stackhouse, D.J. Weidner, M. Alfredsson, G.D. Price, Elasticity of (Mg, Fe)(Si, Al)O3–perovskite at high pressure, Earth Planet. Sci. Lett. 240 (2) (2005) 529–536. [23] L. Li, J.P. Brodholt, S. Stackhouse, D.J. Weidner, M. Alfredsson, G.D. Price, Electronic spin state of ferric iron in Al-bearing perovskite in the lower mantle, Geophys. Res. Lett. 32 (2005) L17307, doi:10.1029/2005GL023045. [24] G. Kresse, D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev., B 59 (1999) 1758–1775. [25] P.E. Blöchl, Projector augmented-wave method, Phys. Rev., B 50 (1994) 17953–17979. [26] J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Atoms, molecules, solids, and surfaces: applications of the generalized gradient approximation for exchange and correlation, Phys. Rev., B 46 (1992) 6671–6687. [27] J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Erratum: atoms, molecules, solids, and surfaces: applications of the generalized gradient approximation for exchange and correlation, Phys. Rev., B 48 (1993) 4978. [28] H.J. Monkhorst, J.D. Pack, Special points for Brillouin-zone integration, Phys. Rev., B 13 (1976) 5188. [29] A.R. Oganov, J.P. Brodholt, G.D. Price, The elastic constants of MgSiO3 perovskite at pressures and temperatures of the Earth's mantle, Nature (London) 411 (6840) (2001) 934–937. [30] S.A.T. Redfern, R.J. Harrison, H.S.C. O'Neil, D.R.R. Wood, Thermodynamics and kinetics of cation ordering in MgAl2O4 spinel up to 1600 °C from in situ neutron diffraction, Am. Mineral. 84 (1999) 299–310. [31] A. Pavese, G. Artioli, S. Hull, In situ powder neutron diffraction of cation partitioning vs. pressure in Mg0.94Al2.04O4 synthetic spinel, Am. Mineral. 84 (1999) 905–912. [32] A. Putnis, G.D. Price, Anonymous, Electron microscopy of shockproduced polymorphs of olivine and pyroxene in the Tenham chondrite Abstracts of papers; presented at the 42nd annual meeting, The Meteoritical Society, vol. 14 (4), 1979, p. 521. [33] R.J. Hill, M.D. Newton, G.V. Gibbs, A crystal chemical study of stishovite, J. Solid State Chem. 47 (1983) 185–200. [34] R.J. Angel, T. Gasparik, N.L. Ross, L.W. Finger, C.T. Prewitt, R.M. Hazen, A silica-rich sodium pyroxene phase with sixcoordinated silicon, Nature 335 (1988) 156–158. [35] J.R. Smyth, D.L. Bish, Crystal Structures and Cation Sites of the Rock-forming Minerals, Allen and Unwin, Boston, 1988. [36] G.D. Gwanmesia, G.L. Chen, R.C. Liebermann, Sound velocities in MgSiO3–garnet to 8 GPa, Geophys. Res. Lett. 25 (24) (1998) 4553–4556. [37] R.M. Hazen, R.T. Downs, P.G. Conrad, L.W. Finger, T. Gasparik, Comparative compressibilities of majorite-type garnets, Phys. Chem. Miner. 21 (5) (1994) 344–349. [38] J. Liu, G.L. Chen, G.D. Gwanmesia, R.C. Leibermann, Elastic wave velocities of pyrope–majorite garnets (Py62Mj38 and Py50Mj50) to 9 GPa, Phys. Earth Planet. Inter. 120 (1–2) (2000) 153–163. [39] M. Akaogi, S. Akimoto, Pyroxene–garnet solid-solution equilibria in the systems Mg4Si4O12–Mg3Al2Si3O12 and Fe4Si4O12– Fe3Al2Si3O12 at high pressures and temperatures, Phys. Earth Planet. Inter. 15 (1977) 90–106.
The effect of cation-ordering on the elastic properties of majorite: An ab initio study Li Li a,b,⁎, Donald J. Weidner a,b , John Brodholt a , G. David Price a b
a Department of Earth Sciences, University College London, Gower Street, London WC1E6BT, UK Mineral Physics Institute, Department of Geosciences, University of New York at Stony Brook, Stony Brook, NY, 11790, USA
Received 18 July 2006; received in revised form 7 January 2007; accepted 7 January 2007 Available online 13 January 2007 Editor: R.D. van der Hilst
Abstract The effects of cation disorder and pressure on the structural and elastic properties of MgSiO3 majorite (Mj100) and MgSiO3– Mg3Al2Si3O12 solid solution (Py50Mj50) are modelled using the first principle simulations. Our results are consistent with the tetragonal phase as the stable structure for both compositions up to 25 GPa. Both pressure and disorder decrease the differences between c11 and c33, and between c44 and c66, indicating that the elastic properties move closer to cubic. The calculated bulk modulus and shear modulus are comparable with the reported experimental data. The bulk modulus of Mj100 varies little while the shear modulus decreases slightly with increasing cation disorder. The elastic properties of an ordered Py50Mj50 are nearly cubic in symmetry. Mg–Si disorder has much lower energy impact on Py50Mj50 than Mj100. In the Earth's mantle, variation of the Al content in the garnet will more significantly affect the seismic velocities than will the order/disorder of Si and Mg. © 2007 Elsevier B.V. All rights reserved. Keywords: majorite; cation disorder; elasticity; high pressure; seismic velocity
1. Introduction Majorite rich garnet is expected to be the second most abundant mineral in the Earth's transition zone [1]. The formation of majorite will significantly affect the density of subducted basaltic crust within a pyrolite mantle [2]. Thus, the properties of majorite and their dependence on chemical composition will have a significant impact on radial and lateral variations in seismic velocities and buoyancy in this region. The end-member, Mg–Si, majorite is among the few minerals that ⁎ Corresponding author. Mineral Physics Institute, Department of Geosciences, University of New York at Stony Brook, Stony Brook, NY, 11790, USA. E-mail address: [email protected] (L. Li). 0012-821X/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.epsl.2007.01.008
have ferroelastic phase transitions that are often accompanied by softening of elastic moduli or high acoustic attenuation [3,4]. This transition is marked by the ordering of magnesium and silicon cations on the octahedral sites accompanied by a cubic to tetragonal phase transformation. Sinogeikin et al. [5] suggest that there is a discontinuous decrease in the elastic moduli across this transition. Their experimental data, which span a range in the pyrope–majorite solid solution, were interpreted as having a small discontinuous decrease in the region where the tetragonal phase is stable. YeganehHaeri et al. [6], on the other hand, suggest a continuous decrease in elastic moduli as aluminium is replaced by an Mg–Si couple, with no discontinuity associated with the change in space group. This issue has an immediate implication for seismic velocities in the mantle which
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may have some dependence not only on the chemical composition of the garnet, but also on the degree of ordering of the Mg and Si on the octahedral site. Experimental studies are quite limited in clarifying this issue. The experimental data of [5] could also be explained, within the resolved errors, by a gradual compositional effect on the elastic moduli with no effect of the phase transition. Furthermore, X-ray studies are weak in defining the state of order between Mg and Si owing to the similarity of their structure factors. Here we present the results of ab initio calculations. They indicate that while the presence of the tetragonal phase is definitive evidence of the ordered structure, the tetragonal phase may be metrically cubic (all axes of the same length). Thus, the elastic properties may be determined by the ordered structure, but still appear to be cubic in terms of cell dimension thus rendering the experimental data even more ambiguous. However, we find that while disorder appears to lower the shear modulus, the effect of composition is by far the dominant variable affecting the seismic velocity in the pyrope–majorite solid solution. Majorite is unusual in that Mg and Si cations are significantly (i.e.N 10%) disordered. Laboratory studies have shown that majorite garnet can transform from cubic (Ia–3d) to tetragonal (I41/a) at elevated temperature and pressure [3]. The tetragonal MgSiO3 majorite garnet has Mg1(16f) and Mg2(8e) occupying the dodecahedral sites, Si1(4b), Si2(4a) and Si3(16f) occupying the tetrahedral sites, O1–O6(16f) occupies the corner of tetrahedral, and Mg(8c) and Si(8d) cations occupying the octahedral sites. Cation disordering occurs when Mg (8c) exchanges sites with Si(8d). We define here the 8c 8d compositional parameter x in Mgx8d Mg1−x Si1−x Six8c to represent the cations in the octahedral sites. Then complete ordering obtains for x = 0 (tetragonal structure) and disorder (cubic structure) occurs at x = 0.5. Angel et al. [7] report x = 0.2 for MgSiO3 majorite (0.2 Si on the Mg site). Even though the tetragonal structure has been reported for the synthesized majorite garnet [3,7–9], natural Mg–Fe rich majorite found in meteorite craters have been reported as being cubic (Ia–3d) [10,11] with Mg and Si randomly distributed. For garnet, order–disorder is associated with the tetragonal–cubic phase transition. In the Earth's mantle, majorite exists in solid solution with pyrope. An experimental study [12] has reported the tetragonal structure for the majorite–pyrope join when the majorite content is over 75 mol% and the cubic structure for the ones with more aluminium. The data are excellent in defining the cell dimensions, but still unable to define the state of order, and therefore unable to confirm the
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cubic space group. The presence of Al in the solid solution also affects the elasticity [5,6,13–19]. Typically, elastic properties decrease very modestly as majorite replaces pyrope in the solid solution. First principle simulation has successfully predicted structural and elastic properties of mantle minerals [20– 23]. This approach can define atom positions in a periodic cell and simulate the cation substitution and ordering, especially at high pressure. We specify the atom positions in a large cell. We use density functional theory to compute the enthalpy, structural, and elastic properties of majorite garnet with defined atom configurations at mantle pressure. To simulate disorder, we use a sample box with 160 atoms and perturb only a few of these atoms. The arrangement of atoms on a larger scale is assumed to be periodic with the sample box. Thus, we are restricted to define the properties of systems that have specific violations of the ordered space group. Howver, the calculations provide some insights that are not accessible experimentally. For example, X-ray diffraction cannot define the Si–O bond distances for the Si atoms that are on the Mg sites and yield only the average of the M–O bond distance is defined. The calculations do provide this information. 2. Computation method Computations were performed using the density functional theory (DFT) based VASP code with the projector augmented wave implementation [24,25] and a plane wave basis set with kinetic energy cut-off of 900 eV. The electron exchange and correlation is described within the generalized gradient approximation (GGA-PW91) [26,27]. The Γ-point is used for Brillouin zone sampling. We used a unit cell with I41/a symmetry containing 160 atoms (listed in Supplementary data Table S1 based on experimental data of [7]). These were used as the starting point for the calculations. The disordered atomic positions for different garnet models with x = 0, 1/8, 2/8, 4/8 are given in Table S2 of the Supplementary data. The four configurations for the x = 1/8 case represent all of the possible configurations for this degree of disorder with a cell containing 160 atoms. For higher values of disorder, the number of possible disordered arrangements grows rapidly. Not all possibilities were considered, rather the Mg and Si atoms were randomly chosen due to the high computation demand. A total of 10 configurations for majorite were considered. For each model, the lattice parameters, the cell shape, as well as ionic positions were allowed to relax. Calculations were performed between 0 and 30 GPa. Elastic moduli for the
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select models were calculated at high pressures (0, 20 and 30 GPa). To obtain accurate elastic moduli in the limit of zero strain, positive and negative strain of magnitude 1% were applied and corresponding stresses were deduced. Expanding the plane-wave cut-off energy to 1000 eV and the Monkhorst-Pack grids [28] by two kpoints in each direction varies the enthalpy by less than 1 meV/atom and the effect on the stress and elastic constants are insignificant. We estimate the calculational uncertainty of the elastic moduli to be less than 1 GPa. Similar calculations were carried out for a pyrope– majorite solid solution composition with 50 mol% of pyrope and 50 mol% of majorite. Two configurations (pm0 and pm2) were used (details of atom positions are given in the Supplementary data Table S3). In the octahedral site, pm0 has the four Mg atoms ordered; pm2 has two of the Mg atoms exchanged with two Si atoms. 3. Results and discussion It has been recognized by a number of studies that the GGA based calculation is precise in defining state properties as a function of V, but that it overestimates, by a constant value, the pressure for any volume [22,29]. Using the same approach from these studies, we have corrected for the volume offset by combining the volume (V)–bulk modulus (K) relationship of the model with the experimental V0 by fitting a Birch– Murnaghan relation to the V–K calculations for majorite. Since the variation in volume caused by cation disorder is small (∼0.1%) for the configuration mj1a, mj1c and mj4b, we fit the majorite models with the constraint that V0 = 1518.6 Å3 [7]. The parameters for
Fig. 1. Total energy (E) from DFT calculation of the unit cell garnet with GGA vs. the disorder parameter x. Diamond symbol represents the MgSiO3 majorite (Mj100); square symbol represents the MgSiO3– Mg3Al2Si3O12 solid solution (with composition of Py50Mj50). The energy increases monotonically with x for both compositions.
Fig. 2. Unit cell parameter a and c ratio as a function of disorder parameter x for garnet. For Mj100, solid diamond symbols are from this study, open square symbol represents data from [7]. Horizontal lines represent the data from [3] and [18], since their order parameters are not known. Solid triangle symbols represent calculated results for Py50Mj50.
the equation of state are then calculated. With the equation of state, we can calculate the pressures that correspond to the model volumes. We find that the GGA models consistently overestimate the pressure by 4.9 (2) GPa for each configuration regardless the pressure or the degree of disorder as well as for the pyrope–majorite compositions considered here. We indicate in the following text the corrected pressure Pc = P − 4.9 GPa for all compositions and configurations. It is common for garnets to have cubic symmetry (Ia– 3d). Majorite, with Si and Mg sharing the octahedral site copes with the different sizes of these cations by forming a tetragonal structure (I41/a), which has two crystallographically distinct octahedral sites Oc1 and Oc2 which can accommodate the different cation sizes. Temperature is an important agent for altering the degree of cation order. The cations can be “frozen” to its sites with a fast quench from high temperature [30,31]. Thus the disordered structure can be preserved at room temperature. In our calculation, we calculate the properties of such “frozen” disordered structures at 0 K. The total energies of the relaxed structures with the GGA are plotted in Fig. 1. The ordered structure has the lowest energy among all configurations plotted. The energies with the same disorder parameter represent the different sites of the Mg–Si pair. For MgSiO3 majorite (Mj100), the configuration with the higher degree of disorder has higher total energy. For MgSiO3–Mg3Al2Si3O12 solid solution (with composition of Py50Mj50), the variation of energy among the three degrees of cation ordering is smaller, indicating that the presence of Al lessens the energetic preference for cation ordering which is consistent with the observation that aluminium bearing garnets tend to be cubic at elevated temperatures [12].
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Fig. 3. Unit cell parameter ratio a/c vs. pressure for 4 configurations: ordered, mj1b, mj1d and mj4b.
We estimate the temperature that will accommodate specific disordered states by considering the configurational entropy [32]. For the pure majorite composition, this calculation implies that the degree of disorder will reach the value of 1/8 of the sites at about 2000 K. For the Py50Mj50 composition, the lowest energy system will be disordered (cubic) above 1500 K and ordered (tetragonal) below this temperature. While there is considerable uncertainty in these numbers since they are based on a small difference between two large numbers, they suggest that ordering may be significant at mantle conditions. The relaxed structure of all configurations is nearly tetragonal dimensionally (a = b bN c, α = β = γ = 90°) even though the space group of the disordered structures are necessarily triclinic. The unit cell parameter ratio a/c as a function of the disorder parameter x is plotted in Fig. 2. The a/c ratio from the previous experimental studies [3,7,18] is plotted for comparison with our results. The degree of disorder for the data of [3,7,18] are unknown however [7,18] samples are probably the same since they were grown at the same conditions. For most of the Mj100 calculations, the a/c ratio decreases with x, except configuration mj4b. The three data points for Py50Mj50 are slightly higher than but very close to 1. Furthermore, the effect of pressure on metrics of the cell is small as shown in Fig. 3. Table 1 indicates the effect of disorder on the octahedral bond lengths. The Si–O bond (1.815 Å) in
Fig. 4. Bond lengths for the ordered MgSiO3 majorite as a function of Pc. Plotted are the mean values of the calculation and experimental results [7].
Table 1 Bond length for octahedral sites in MgSiO3 majorite garnet X
0
1/8
2/8
Oc1(Mg)–O Oc2(Mg)–O Oc1(Si)–O Oc2(Si)–O
2.064
2.056 2.028 1.815 1.800
2.047 2.024 1.815 1.803
1.798
Fig. 5. Calculated single crystal elastic moduli cij as a function of degree of disorder x at Pc = −5 GPa (Fig. 5a) and 25 GPa (Fig. 5b).
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the Oc1 site for x = 1/8 is significantly longer than the Si–O bond in the ordered site (1.798 Å) but much shorter than the Mg–O bond in the Oc1 site (2.056 Å). The calculated mean Si–O bond length ranges from 1.796 Å to 1.815 Å which are close to the reported sixcoordinated Si–O distances range from 1.757 Å for stishovite [33] to 1.811 Å for silica-rich sodium pyroxene [34]. The Mg–O bond (2.028 Å) for x = 1/ 8 is significantly shorter than that in the ordered site (2.064 Å). The calculated mean Mg–O bond lengths fit into the range of mean Mg–O distance from 2.05 Å to 2.15 Å [35]. Angel et al. [7] refined the structure of MgSiO3 garnet from X-ray single crystal diffraction spectra.
Fig. 4 compares the X–O bond lengths reported by [7] with those calculated here for the completely ordered case. Our values are displayed as a function of pressure, Pc, while [7]'s results are for room pressure only. The excellent agreement between the calculated values for the tetrahedral sites and dodecahedral sites underscores the precision of the DFT calculations. The Mg–O octahedral bonds determined by [7] are slightly shorter that those determined here, while the Si–O octahedral bonds of [7] are longer. Angel et al. [7] pointed out that this is expected due to the disorder of Si onto the Mg site and Mg onto the Si site. Angel et al. [7] determined that the Si site is occupied by 0.8 Si atoms and 0.2 Mg atoms, but emphasized the uncertainty of this value owing to
Fig. 6. Calculated single crystal elastic moduli vs. pressure for 6 configurations (mj0, mj1b, mj2b, mj4a, mj4b and pm0).
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the similarity of the scattering factors of Si and Mg. Assuming that our calculations of the lengths for the Mg–O and Si–O bonds at 0 GPa pressure are accurate, and the lengths measured by [7] represent a simple weighted average of the bond lengths for the occupants of the sites, we can calculate the site occupancies. Based on the size of the Mg site, we conclude that the degree of order is 0.82, while the Si site size suggests 0.91. These values are in excellent agreement with the refinement of [7]. The single crystal elastic moduli as a function of x (given in Supplementary data Table S4) for 6 configurations (mj0, mj1b, mj2b, mj4a, mj4b and pm0) at Pc = − 5 GPa (Fig. 5a) and Pc = 25 GPa (Fig. 5b) are plotted. For both pressures, the shear moduli c66 varies significantly with cation disorder while c44 remains almost unchanged at − 5 GPa but increases at 25 GPa. At 25 GPa, c33 decreases with x while c11 varies little; At − 5 GPa, c11 decreases with x while c33 varies little. The different behaviour of c11 and c33 with pressure for the disordered majorite can also be seen in Fig. 6. The ordered majorite has the largest difference between c11 and c33, c44 and c66, c12 and c13. With increasing degree of cation disorder, the difference between these three pairs of cij decreases, eventually to zero, where the cubic symmetry dominates the elasticity. For the Py50Mj50 model, even though the Mg and Si are ordered, the elastic moduli are much closer to be cubic compared with ordered Mj100 model. For MgSiO3 majorite the bulk modulus (K) and its pressure derivative (K′) vary little with the cation disordering among the five configurations (mj0, mj1b, mj2b, mj4a, mj4b). This is summarized in the Supplementary data Table S5. Our calculated bulk moduli are consistent with various experimental groups [5,14, 15,18,36,37]. Our calculated shear modulus for ordered MgSiO3 majorite is slightly lower than the average of reported values [5,14,15,18,36] but within the scatter of the experimental data. The shear modulus of the disordered MgSiO3 majorite is in general slightly lower than for the ordered configuration with little variation among them. Of all of the configurations considered here, mj4b is most nearly metrically cubic for the relaxed structure (a = b = c). mj4b has the lowest shear modulus of all of the cases considered here, but is only about 3% lower than the average shear modulus. The pressure derivative of the shear modulus is quite constant among the various configurations, agreeing well with the experimental results of [15] while being considerably smaller than that of [36]. For the pyrope–majorite solid solution, our calculated results for Py50Mj50 are consistent with the
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experimental studies which have the same chemical composition [15,38] within the errors. There is a dependence of volume on the composition for pyrope– majorite solid solution [12]. In Fig. 7, we plot the ratio of volume to the Mj100 vs. the majorite mole fraction. Among the experimental studies [7,12,39], our results (Py50Mj50) follow the trend of this composition dependence of volume and fill in the middle composition. Two calculated points are plotted corresponding to pm0 and pm2 showing that the cation order–disorder does not have significant effect on the volume of the Py50Mj50 models. Comparing the pure majorite and the Py50Mj50 results, we generally conclude that the elastic moduli increase with increasing Al content. We conclude that the effect of composition change, while small, is stronger than the effect of ordering. This result is very important in modelling the seismic velocities from the mineralogy since the Al content is much easier to define than is the degree of Mg–Si order for all depths in the mantle. We conclude that we may have an error in the shear modulus less than about 2% for regions with pure majorite owing to ignorance of the state of order. However, in the mantle, the garnet composition will generally be closer to the 50:50 model, where the effect of ordering is very small. Elastic anisotropy of minerals contributes to the seismic velocity anisotropy. We have calculated the sound velocities of the majorite for each model from their single crystal elastic properties. Details of the results are found in the Supplementary data Table S6, which lists the maximum and minimum sound velocities and the anisotropy for each model at Pc = − 5 GPa and 25 GPa. In general, the ratio between the fastest
Fig. 7. Variation in cell volume (V/V0) as a function of composition for a series of materials along pyrope–majorite solid solution. V0 represents the volume for majorite (V0 = 1515.4 Å3 from [7]). 1:[39]; 2:[12]; 3: this study; 4:[7].
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4. Summary
Fig. 8. Acoustic velocities for Mj100 and Py50Mj50 at Pc = − 5 GPa and 30 GPa. In both cases, the Mg and Si atoms are in the ordered sites.
longitudinal velocity to slowest one (Ap) and the ratio between the fastest shear velocity to slowest one (As) is close to 1. At both pressures, the cation disorder decreases both Ap and As except model mj4a. The Py50Mj50 model has the lowest anisotropy among all the majorite models. Pyrope is the most isotropic. The velocity surface along various directions for Mj100 and Py50Mj50 are shown in Fig. 8 for Pc = − 5 GPa and 25 GPa. Fig. 8 contrast the effect of crystallographic direction on velocities with the effect of pressure on velocities for these two compositions. For longitudinal waves, the effect of pressure dominates for both compositions; for shear waves, the effect of anisotropy on velocities of Mj100 is comparable to the effect of 30 GPa pressure difference while Py50Mj50 is dominated by pressure since this composition is more isotropic.
Majorite garnet, the Al free end-member of mantle high pressure garnets, forms an ordered tetragonal phase. Density functional theory models at 0 K give insights into the properties of this material. The details of the crystal structure are well represented by the GGA based models. Volume and bulk modulus are independent of the degree of order, while the shear modulus decreases by less than 4% from the tetragonal, ordered structure to the disordered, cubic state. In contrast, the 50:50 pyrope: majorite solid solution shows little change in elastic moduli with ordering of the cations. However, the elastic moduli are Al dependent, with higher elastic moduli associated with more Al. We suggest that seismic modelling of a garnet bearing mantle need only be concerned with the Al content of the garnet. Variation of elastic moduli with ordering is determined to be a smaller effect than the compositional variations. Even though a previous study [12] detected the 50:50 composition to be cubic, it is difficult to determine the space group with X-ray diffraction when the distinction between the cubic and tetragonal structures for Py50Mj50 is small. Our models of Py50Mj50 predict that the tetragonal phase is the most stable at 0 K up to about 1500 K, yet the properties show very little variation in cell parameters or elastic properties with changes in cation order. Furthermore, the elastic moduli of the tetragonal structure appear to reflect a cubic symmetry and the c/a ratio is nearly unity. Generally, the symmetry of the elasticity tensor is nearly isotropic, suggesting that garnet, even with a preferred orientation, will not impose a seismically discernable anisotropy on the observed seismic signal. Acknowledgements This work is supported by NERC (Grant Nos. NER/ T/S/2001/00855; NER/O/S/2001/01227), and computer facilities are provided by NERC at University College London, and the High Performance Computing Facilities of the University of Manchester (CSAR) and the Daresbury Laboratory (HPCx). DJW acknowledges the Leverhulme Trust for support through the visiting Professor program. DJW and LL acknowledge NSF EAR-9909266, EAR0135551, EAR00135550. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j. epsl.2007.01.008.
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