Earth and Planetary Science Letters

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1 Earth and Planetary Science Letters 271 (2008) Contents lists available at ScienceDirect Earth and Planetary Science Letters journal homepage: www. elsevier. com/ locate/ epsl Predicted glide systems and crystal preferred orientations of polycrystalline silicate Mg-Perovskite at high pressure: Implications for the seismic anisotropy in the lower mantle David Mainprice a,, Andréa Tommasi a, Denise Ferré b, Philippe Carrez b, Patrick Cordier b a Géosciences Montpellier, CNRS & Université Montpellier 2, Montpellier, France b Laboratoire de Structure et Propriétés de l'etat Solide, CNRS & Université des Sciences et Technologies de Lille, Villeneuve d'ascq, France A R T I C L E I N F O A B S T R A C T Article history: Received 7 November 2007 Received in revised form 26 March 2008 Accepted 27 March 2008 Available online 11 April 2008 Editor: R.D. van der Hilst Keywords: Perovskite ab initio dislocation creep glide systems crystal preferred orientation seismic anisotropy lower mantle D We use first first-principle methods and the Peierls Nabarro model to evaluate the resistance to glide, characterized by the Peierls stress, of glide systems for end-member MgSiO 3 Perovskite at mantle pressures. [010](100) is the easiest glide system in Mg-Perovskite at all pressures. Peierls stresses increase systematically with pressure for all systems except [001](010), indicating the importance of lattice friction at lower mantle pressures. The ratio of the maximum Peierls stress for each system relative to the [010](100) value defines their critical resolved shear stress (CRSS). These CRSS are used in a visco-plastic self-consistent homogenization model to predict the evolution of crystal preferred orientations (CPO) during deformation of polycrystalline Mg-Perovskite. In axial compression, [100] tends to align with the compression direction, in agreement with in situ observations in axial compression experiments. In simple shear, [010] concentrates near the shear direction and (100), although more dispersed, tends to align near the shear plane, consistent with the dominant activity of the easier [010](100) system. The calculated seismic anisotropy for a 100% Mg- Perovskite aggregate using the CPO in simple shear and the elastic constants of MgSiO 3 perovskite at lower mantle pressures and temperatures is weak (N3% for P-waves with and N2% for S-waves) and decreases with increasing temperature and pressure. P-waves show the fastest propagation parallel to the lineation and S- waves fast polarization is in the foliation at 38 GPa and normal to the lineation at 88 GPa. This weak anisotropy is consistent with global seismological observations of a nearly isotropic lower mantle. There are however two regions where strain-induced Mg-Perovskite CPO could contribute to anisotropy; a) low temperature regions in the uppermost lower mantle, where the predicted S-wave polarization anisotropy may attain 1.6% with a fast polarization parallel to the foliation, b) in high high-temperature domains in the D layer, where Mg-Perovskite may be the major stable phase, leading to polarization of fast S-waves normal to the lineation for propagation directions at high angle to the lineation and an apparent isotropy for all other propagation directions Elsevier B.V. All rights reserved. 1. Introduction Perovskite-structure MgSiO 3 is the most abundant mineral in the Earth, forming about 80% volume of the Earth's lower mantle (e.g., Hirose, 2002). In addition, Mg-Perovskite may also be locally stable in the D layer (2700 km 2890 km depth) just above the core core mantle boundary (CMB) (Hernlund et al., 2005; van der Hilst et al., 2007). The convective flow and the seismic properties in the deep mantle are thus strongly linked to the physical properties of MgSiO 3 Perovskite under very high pressure conditions. Global studies of seismic anisotropy over the last 10 yr have consistently suggested a roughly isotropic behaviour in the depth range between 670 km and Corresponding author. Tel.: ; fax: address: David.Mainprice@gm.univ-montp2.fr (D. Mainprice) km, that is over the entire 2030 km lower mantle section (e.g., Montagner and Kennett, 1996; Beghein et al., 2006; Panning and Romanowicz, 2006). These results have been interpreted as indicating dominant diffusion creep in the lower mantle (e.g. Karato et al., 1995). However, recent studies of olivine and wadsleyite deformation at high pressure have shown that almost isotropic seismic properties do not necessarily imply an absence of CPO and hence deformation by diffusion creep (Tommasi et al., 2004; Mainprice et al., 2005). In contrast to the lower mantle, the D layer is known to have significant seismic anisotropy at both the global and local scale (e.g., Panning and Romanowicz, 2006). The causes of this anisotropy (strain-induced CPO of posterovskite, oriented inclusions, layering) are still debated and a contribution of Mg-Perovskite cannot be ruled out. A good understanding of the deformation behaviour of minerals at the extreme pressure and temperature conditions of the Earth's deep X/$ see front matter 2008 Elsevier B.V. All rights reserved. doi: /j.epsl

2 136 D. Mainprice et al. / Earth and Planetary Science Letters 271 (2008) mantle is fundamental to evaluate the role of plastic deformation and of the resulting crystal preferred orientations in the development of seismic anisotropy. Deformation experiments on silicate Mg-Perovskite at lower mantle PT conditions remain thus a priority in mineral physics community, but to-date only preliminary data exist for silicate Mg-Perovskite (e.g., Chen et al., 2002; Merkel et al., 2003; Cordier et al., 2004; Wenk et al., 2004). Recent progress in high pressure and temperature techniques has allowed the deformation of orthorhombic (Pbnm) MgSiO 3 Perovskite structure in both internally heated multianvil apparatus (Cordier et al., 2004) and laser-heated diamond anvil cell (Wenk et al., 2004) at conditions of the shallow lower mantle. These studies highlighted the presence of dislocations and the development of CPO in the deformed samples. To obtain more information at experimentally accessible conditions, several studies have used structural oxide analogues of silicate Perovskite in deformation experiments to generate crystal preferred orientations in polycrystalline samples (e.g., CaIrO 3, Walte et al., 2007) in the expectation that dislocation mechanisms at the atomic scale will be the same as those in silicates. However the creep properties of perovskite-structured crystals do not seem to show a clear systematic trend (e.g., Poirier et al., 1989). In the present study, we use a new approach to investigate the deformation of orthorhombic (Pbnm) MgSiO 3 Perovskite. We use static (0 K) ab initio atomic scale modelling of dislocation glide systems in silicate Perovskite to predict the relative resistance to glide in the various systems at mantle pressures. These data are used as input for a polycrystalline visco-plastic self-consistent (VPSC) model to evaluate crystal preferred orientation (CPO) development during plastic deformation by dislocation glide. These calculations are performed for lower mantle pressures, ranging from 30 to 100 GPa, but 0 K temperature conditions, i.e., temperature conditions very different than those prevailing in the lower mantle. However, it has been shown that at lower mantle conditions, pressure plays a fundamental role on the elastic anisotropy of minerals, whereas temperature changes at lower mantle pressures bring only minor changes (see Karki et al., 2001; Mainprice, 2007). As both elasticity and dislocation glide at high pressure depend to first order on the atomic bonding in the compressed crystal structure, the present calculations should provide a good first order prediction of the plastic anisotropy of minerals at high pressure. Temperature dependent processes, such as diffusion-dependent dislocation climb or recrystallization, are not taken into account in the present models. Although these processes are probably important to deformation in the deep mantle, analysis of CPO formed during high-temperature deformation of olivine show that CPO development is essentially controlled by dislocation glide and that diffusion-dependent processes only change CPO intensity or allow a fast reorientation of the CPO in simple shear (Tommasi et al., 2000). 2. Computation of slip systems easiness and of crystal preferred orientation evolution 2.1. Evaluation of Peierls stresses through ab initio calculation of Generalised Stacking Faults The Peierls Nabarro (PN) model (Peierls, 1940; Nabarro, 1947) represents a useful and efficient approach to calculate the core properties of dislocations based on the assumption of a planar core (Schoeck, 2005). It has been shown to apply to a wide range of materials (Wang, 1996; Bulatov and Kaxiras, 1997; von Sydow et al., 1999; Lu et al., 2000, Lu, 2005) including, more recently, mineral structures (Miranda and Scandolo, 2005; Carrez et al., 2006; Carrez et al., 2007; Durinck et al., 2007). In the following, we give a brief description of the model (extended details can be found in Hirth and Lothe (1982) or more specifically in Carrez et al. (2006, 2007) and Ferré et al., 2007). With the Peierls Nabarro (PN) approach, the extend extent of a dislocation core can be calculated through the resolution of the well-know PN equation: Z K þl 1 dsðxvþ dxv¼ K Z þl qðxvþ 2p x xv dxv 2p x xv dxv¼ F ð SðxÞ Þ: ð1þ l l In this equation, the dislocation is treated as a continuous distribution of shear S(x) or as a continuous dislocation densities ρ(x) =ds/ dx in the glide plane of the defect (x corresponding to the coordinate along the displacement direction of the dislocation). As the misfit region around a dislocation core of inelastic displacement is restricted to vicinity of the glide plane, whereas linear elasticity applies far from it, the PN equation describes the fact that the dislocation density can be evaluated through the balance between elastic field and a restoring force of inelastic origin F in the vicinity of the glide plane. As a consequence, K corresponds to the energy coefficient of the dislocation of interest (K is function of the dislocation character and takes anisotropic elasticity into account) and F can be considered as the gradient of Generalised Stacking Fault (GSF) potential γ. Y FS ð Þ ¼ Y gradgðþ: S If the GSF potential γ is known, one can numerically solve the PN equation and thus determine the extension of the dislocation core. Combining the size of the dislocation with the GSF potential, we can evaluate the misfit energy W of the dislocation and define the Peierls stress σ p, i.e. the stress needed to move a straight dislocation from one Peierls valley to another (in the following equation, u is an arbitrary position of the dislocation of Burgers vector b in the Peierls valley of energy topography W). r p ¼ max 1 dwðuþ : ð3þ b du The goal of ab initio calculation is thus to accurately determine the GSF potential γ. First-principles calculations were performed using the ab initio total-energy calculation package VASP (Vienna Ab initiopackage) developed by Kresse and Hafner (1993, 1994). This code is based on the first-principles density functional theory and solves the effective one-electron Hamiltonian involving a functional of the electron density to describe the exchange-correlation potential. It gives access to the total energy of a periodic system with, as a single input, the atomic numbers of atoms. Computational efficiency is achieved using a plane wave basis set for the expansion of the single electron wave functions and fast numerical algorithms to perform self-consistent calculations. Within this scheme, we used the generalised gradient approximation (GGA) derived by Perdew and Wang (1992) and the all-electron projector augmented-wave method as implanted in the VASP code. Using this assumption, the outmost core radius for the Mg, Si and O atoms are 2, 1.9 and 1.52 a.u. respectively. Computation convergence is achieved in all simulations by using a single energy cut-off value of 600 ev for the plane wave expansion. In order to calculate the GSF energies, we used the supercell technique presented in Durinck et al. (2005a,b) or more specifically for the perovskite structure in Ferré et al. (2007). For a given glide system, the supercell is built on a Cartesian reference frame defined by the normal to the stacking fault plane and by the shear direction (i.e., the Burgers vector direction). All the supercells are built with one stacking fault and a vacuum buffer is introduced to isolate the stacking fault from periodic boundary condition images. The GSF are then calculated by imposing a given shear displacement to the atoms in the upper part of the supercell. The spacing between two sliding half crystals in the vicinity of the shear plane is allowed to relax in order to minimize the energy of the stacking fault. This is obtained by (i) keeping the supercell vectors fixed at the values obtained for a bulk system submitted to the pressure of interest, (ii) maintaining fixed atoms present on the two surfaces (to mimic the action of the surrounding bulk) and (iii) allowing relaxations of others atoms in the directions normal to the shear ð2þ

3 D. Mainprice et al. / Earth and Planetary Science Letters 271 (2008) Table 1 Peierls stresses (σ p ) and normalized critical resolved shear stresses (CRSS) for dislocation slip systems in orthorhombic Mg-Perovskite at pressures of 0, 30 and 100 GPa. σ p is the Peierls stress in GP Slip system Pressure 0 GPa Pressure 30 GPa Pressure 100 GPa Normalized CRSS [UVW](HKL) σ p (GPa) σ p σ p σ p σ p σ p P =0 GPa P=30 GPa P =100 GPa Screw Edge Screw Edge Screw Edge [100](010) [100](001) [010](100) [010](001) [001](100) [001](010) [001]{ 110} b 110N(001) b110n{ 110} σ p is the Peierls stress in GPa. The CRSS are calculated as the maximum Peierls stress of either the screw or edge dislocation orientation. The normalized CRSS is the CRSS of the slip system [UVW](HKL) divided by the CRSS of [010](100) at the same pressure. direction. Finally, for the GSF calculation, the first Brillouin zone is sampled using a Monkhorst Pack grid (Monkhorst and Pack, 1976) adapted for each supercell geometry in order to achieve the full energy convergence. In all calculations, the Monkhorst Pack grids were chosen to ensure an accuracy of convergence energy less than 0.1 mev Visco-plastic self-consistent (VPSC) model Crystal preferred orientations were predicted using an anisotropic visco-plastic self-consistent model originally developed by Lebensohn and Tomé (1993). The background of the model and its application to mantle minerals, such as olivine was described in detail by Tommasi et al. (2000) and a recent review of the method is given by Tomé and Lebensohn (2004), thus only a short overview is presented in the following section. The visco-plastic self-consistent approach allows both the microscopic stress and strain rate to differ from the corresponding macroscopic quantities. Strain compatibility and stress equilibrium are ensured at the aggregate scale. At the grain scale, deformation is accommodated by dislocation glide only; other mechanisms such as dynamic recrystallization and grain boundary sliding are not taken into account. The shear rate in a glide system s is related to the local deviatoric stress tensor s ij by a visco-plastic law: g s ¼ g 0 s s n s r s s 0 ¼ g 0 r s ij s ij s s 0! n s where γ 0 is a reference strain rate and n s, τ r s, τ 0 s and are respectively the stress exponent, the resolved shear stress, and the critical resolved ð4þ shear stress for the system s, whose orientation relative to the macroscopic stress axes is expressed by its Schmid tensor r s. The potentially active glide systems for Perovskite, their normalized critical resolved shear stresses are evaluated from ab initio modelling; they are presented in Table 1. We used a stress exponent of n=3 for all calculations. The problem lies in the calculation of a microscopic state (s, ɛ) for each grain, whose volume average determines the response of the polycrystal (Σ, D ). The 1-site approximation (Molinari et al., 1987) is used in the anisotropic VPSC formulation; the influence of individual neighbouring grains is not taken into account. Interactions between any individual grain and its surroundings are successively replaced by the interaction between an inclusion with similar crystal orientation and an infinite homogeneous equivalent medium (HEM), whose behaviour is the weighted average of the behaviour of all the other grains. This leads to: e ij D ij ¼ a M ijkl ðs kl P R kl Þ ð5þ where M is the interaction tensor and α is a constant used to parameterize the interaction between grains and the HEM. α=0 corresponds to the upper bound homogeneous strain (Taylor) model (see Lebensohn and Tomé, 1993), α=1 corresponds to the tangent model of Lebensohn and Tomé (1993), and α = infinity to the lower bound, stress equilibrium model (Chastel et al., 1993). The VPSC tangent model (α=1) has been applied here. In the present models, an aggregate of 1000 orientations, initially spherical in shape and randomly oriented, is deformed in simple shear Fig. 1. Left: Maximum Peierls stress for each slip system in Mg-Perovskite as a function of pressure. Note that the maximum Peierls stress increases with pressure, except for [001] (010) at 100 GPa pressure, indicating the lattice friction increases with pressure. Right: Single crystal plastic anisotropy indicated by the variation of normalized critical resolved shear stress (CRSS) for the various slip systems as a function of pressure. All CRSS are normalized relative to the easy glide [010](100) slip system at the studied pressure.

4 138 D. Mainprice et al. / Earth and Planetary Science Letters 271 (2008) and axial compression. The strain path is imposed by prescribing a constant macroscopic velocity gradient tensor L and a time increment, dt, set to achieve an equivalent strain of in each deformation step. The equivalent strain is defined as (Molinari et al., 1987): e eq ¼ R D eq ðþdt t where the Von Mises equivalent strain rate is: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D eq ¼ 2=3D ij D ij : ð7þ 3. Dislocations and glide systems of Mg-Perovskite at lower mantle pressures The results of the ab initio calculations at 30 GPa pressure were presented in detail elsewhere (Ferré et al., 2007). Here we will concentrate on the role of pressure on plastic anisotropy with additional calculations at 0 and 100 GPa pressure. The Peierls stresses for edge and screw dislocations for the nine glide systems studied in Mg-Perovskite at pressures of 0, 30 and 100 GPa are given in Table 1. The Peierls stress is a measure of the lattice resistance to glide. In our study the Peierls stress varies between 3 GPa for a [010](100) edge dislocation at P=0 GPa and 76.3 GPa for [100](001) screw dislocation at P=100 GPa. The [010](100) glide system is the easiest (lowest Peierls stress) at all pressures, but its Peierls stress increases from 3.0 GPa at 0 GPa to 23.0 GPa at 100 GPa. For [001](100), which is the most difficult system at P=100 GPa, the Peierls stress increases from 45.4 GPa to GPa as pressure increases from 0 to 100 GPa. The absolute values of Peierls stress for most systems increase between 7.7 and 2.5 times with increasing pressure to 100 GPa, the maximum Peierls stress for each system as a function of pressure is shown in Fig. 1. The increase of maximum Peierls stress with pressure indicates that lattice friction also increases with pressure. This is true for all ð6þ systems except [001](010), which Peierls stress decreases at 100 GPa pressure. The variation in the plastic anisotropy due to glide is more easily visualised using the normalized critical resolved shear stress (CRSS). Lattice friction being a major factor in plastic deformation of silicates, the CRSS is taken proportional to the Peierls stress of a dislocation glide system. In this study we have calculated the Peierls stress for two orientations on a dislocation loop, namely the edge and screw orientations, where the Burgers vector is respectively normal and parallel to the dislocation line. As the segment with the highest resistance will limit the dislocation motion we have taken the highest Peierls stress from the edge and screw orientations to be representative for the CRSS of the glide system. The CRSS of all glide systems have been normalized by the CRSS for the easy [010](100) system at each pressure. The histogram presented in Fig. 1 clearly shows that the range of normalized CRSS decreases with increasing pressure. It also shows that there are some relative changes in the glide resistance between individual systems. For example, the most difficult system at 0 and 30 GPa is [001](010), which at 100 GPa it is one of the three least resistant glide systems due to the dramatic drop in its Peierls stress with pressure. The decrease of the normalized CRSS with pressure indicated that the plastic mechanical anisotropy of the single crystal Mg-Perovskite decreases with increasing pressure. Note that according to our results (Fig. 1) the plastic anisotropy of single crystal Mg- Perovskite at ambient pressure (~0 GPa) and at lower mantle pressures are very different. The normalized CRSS presented in Table 1 have been used to build a VPSC model and to simulate the evolution of crystal preferred orientations in polycrystalline MgSiO 3. We modelled two endmember strain paths: simple shear, which is a common mode of deformation in the mantle, and axial compression to simulate the recent LHDAC results of Wenk et al. (2004). The different glide systems' activity in the three models (Fig. 2) reflects the change in normalized CRSS with pressure. The easy glide systems [010](100), Fig. 2. Polycrystalline plastic anisotropy indicated by the slip systems activity as a function of pressure in simple shear and axial compression.

5 D. Mainprice et al. / Earth and Planetary Science Letters 271 (2008) Fig. 3. Strength of crystal preferred orientation (CPO), measured by the J index (see text for details), as a function of finite strain and pressure in simple shear and axial compression. [010](001), and b 110N(001) and b110n{ 110} have the highest activities at all pressures (highlighted in Fig. 2). At 0 GPa glide systems activity in simple shear and axial compression are very similar. The most active system is not the easy glide [010](100), lowest Peierls stress system, but b 110N(001) because it has two glide directions [ 110] and [110] in the basal (001) plane. At higher pressures, 30 GPa and 100 GPa, the easy glide system [010](100) becomes the most active system and [010](001), b 110N(001) and b110n{ 110} have similar activities. In simple shear, b110n{ 110} is more active than b 110N (001), whereas in axial compression their activities are almost identical. The decrease in polycrystalline plastic anisotropy with pressure results in a decrease in the activity of the dominant glide system that is more marked for axial compression than for simple shear. 4. Crystal preferred orientation at lower mantle pressures The analysis of the variation of Peierls stresses and of CRSS as function of pressure (Fig. 1 and Table 1) shows that the latter has a major influence on the plastic anisotropy of Mg-Perovskite. This effect is not linear. The variation in CRSS is higher between 0 and 30 GPa than between 30 and 100 GPa. This leads to a smaller variation in slip systems' activity (Fig. 2). Another consequence of plastic anisotropy is the development of a strong crystal preferred orientation (CPO). A standard parameter for measuring the strength of CPO is the J index that represents the volume-averaged integral of the squared orientation densities, which is sensitive to peaks in the orientation distribution function (Bunge, 1982). The increase of the J index as a function of finite strain is indeed faster for simulations at 0 GPa than at 30 or 100 GPa (Fig. 3). Simple shear and axial compression CPO have similar values of the J index at the same equivalent strain and pressure. As expected, the slowest CPO evolution is observed at the highest pressure (100 GPa), but the difference with simulations at 30 GPa is small. However, the most remarkable feature is the similarity of predicted CPO for simple shear at pressures of 30 and 100 GPa (Fig. 4). The [010] direction aligned near the shear direction with maximum densities of 4.28 and 3.37 and the (100) poles are concentrated nearly normal to the shear plane with maximum densities of 2.78 and 2.51 at pressures of 30 and 100 GPa, respectively. This CPO is compatible with a high activity of the [010](100) system (Fig. 2) at lower mantle pressures. The [001] pole figures have lower maximum densities of 2.01 and 2.28 at pressures of 30 and 100 GPa, which reflect the lower activity of the [010](001) and b 110N(001) systems (Fig. 2). The stronger maximum density of [001] near the shear plane normal in the 100 GPa simulation is due to the increased activity of the [010](001) and b 110N(001) glide systems relative to [010](100) at this pressure. Yet, to first order we can consider that CPO of Mg-Perovskite at the top and bottom of the lower mantle will be similar. Mg-Perovskite CPO evolution in axial compression is characterized at a concentration of [100] axes parallel to the compression direction. This modelled CPO may be compared to those observed in situ by radial X-ray diffraction during laser-heated diamond anvil cell (DAC) experiments by Wenk et al. (2004). In these DAC experiments, San Carlos olivine has been axially compressed to 50 GPa in a sequence of increasing pressure and heating steps, in which olivine transformed to ringwoodite, and finally to perovskite at about 24 GPa. Analysis of the experimental Mg-Perovskite CPO show that it is similar in both distribution and intensity to the one predicted by our combined ab initio VPSC models for axial compression at 43 GPa at relatively low strain (Fig. 5). Given that only the finite strain was adjusted to fit the experimental data, the degree of similarity is very high. Thus, although Wenk et al. (2004) interpreted the strong alignment of Fig. 4. Crystal preferred orientation of Mg-Perovskite in simple shear using CRSS predicted by ab initio modelling at pressures of 30 and 100 GPa. Shear strain (γ) is 1.73 (Von Mises equivalent strain of 1.0). Maximum and minimum finite strain axes are marked X and Z, Horizontal black line is the XY flattening plane or foliation. Thick black arrows mark the dextral shear sense, SD is the shear direction and the inclined black line is the shear plane. Lower hemisphere equal area projection.

6 140 D. Mainprice et al. / Earth and Planetary Science Letters 271 (2008) Fig. 5. Comparison between Mg-Perovskite crystal preferred orientations (CPO) predicted by combined ab initio VPSC simulations for axial compression and measured during in situ diamond anvil cell (DAC) experiments. CPO is presented as inverse pole figures (IPFs) of the compression axis in crystallographic axes. [100] parallel to the compression axis as a phase transformation texture; our simulation suggests that dislocation glide is also a viable interpretation. 5. Seismic anisotropy in the lower mantle As Mg-Perovskite is volumetrically (ca 80%) the most important phase in the lower mantle, the seismic anisotropy of this layer will depend on the strain-induced CPO and on the single crystal elastic tensor of Mg-Perovskite, which depends on pressure and temperature. To explore the influence of temperature and pressure on the seismic anisotropy we will use the simple shear CPO calculated using the CRSS for 30 and 100 GPa (Fig. 4) and the elastic constants of Oganov et al. (2001) for temperatures of 1500 K and 3500 K and pressures of 38 GPa and 88 GPa. The CPO obtained for pressures of 30 GPa (100 GPa) is associated with the elastic constants at 38 GPa (88 GPa). We have calculated the seismic properties of Mg-Perovskite aggregates showing CPO formed at a large range of shear strains as well as two endmember geotherms, leading to either low (1500 K) or high temperatures (3500 K) in the lower mantle. This allows us to compare and contrast the effects of finite strain, temperature and pressure on the seismic anisotropy (Fig. 6). The P-waves propagation (V p ) anisotropy decreases strongly with increasing temperature; the anisotropy at 1500 K can be 2 times higher than at 3500 K at 88 GPa and 3.5 times higher at 38 GPa. V p anisotropy increases non-linearly with strain, except at low pressure and high temperature (3500 K and 38 GPa). The S S-wave polarization anisotropy is almost independent of temperature, being the most sensitive at 38 GPa where anisotropy decreases by 0.5% between 1500 K and 3500 K. At 88 GPa, there is no significant difference in S S-wave anisotropy for shear strains of 2 between 1500 K and 3500 K. The three dimensional distribution of seismic velocities and anisotropy at the top of the lower mantle estimated using the Mg- Perovskite CPO predicted for a shear strain (γ) of 1.73 at 30 GPa (Fig. 4) and single crystal elastic constants for a pressure 38 GPa and temperatures of 1500 K and 3500 K is shown in Fig. 7. P P-waves velocities (V p ) are fastest parallel to the direction of finite extension (X) and slowest parallel to the direction of finite compression (Z), i.e., normal to the foliation plane (XY) at both temperatures. The magnitude of the V p anisotropy decreases from 2.9% at 1500 K to only 0.6% at 3500 K. The S-wave anisotropy pattern shows an important variation with temperature. At 1500 K the maximum shear wave splitting (dv s ) is parallel to the intermediate finite strain direction (Y) and the fastest S-wave (V s1 ) is polarized parallel to the lineation (X direction). In contrast, at 3500 K the maximum shear wave splitting (dv s ) is nearly normal to the foliation, i.e., it lies in the direction of finite shortening (Z), and the fastest S-wave (V s1 ) is polarized normal to the lineation (X direction). The magnitude of the S-wave anisotropy is always low with a maximum of 1.6% and 1.1% at 1500 K and 3500 K, respectively. If the foliation is horizontal, vertically propagating SKS will show nearly no splitting at 1500 K, whereas at 3500 K the rather weak maximum (1.1%) is close to the direction of SKS wave propagation. At the bottom of the lower mantle (Fig. 8), the P-wave velocity distribution is the same as at the top (Fig. 7), with the fastest V p being parallel to X. The magnitude of the V p anisotropy is slightly higher Fig. 6. Maximum P-waves propagation (left) and S-waves polarization (right) anisotropy in Mg-Perovskite polycrystals deformed in simple shear as a function of strain using CRSS predicted at 30 and 100 GPa pressures. Single crystal elastic constants of Oganov et al. (2001) at pressures of 38 GPa and 88 GPa and temperatures of 1500 and 3500 K were used to simulate lower mantle PT conditions.

7 D. Mainprice et al. / Earth and Planetary Science Letters 271 (2008) Fig. 7. The predicted seismic anisotropy for the top of the lower mantle composed of 100% polycrystalline Mg-Perovskite with the CPO given in Fig. 5 at 30 GPa for horizontal flow (XY plane horizontal). The single crystal elastic constants given by Oganov et al. (2001) at P =38 GPa, T=1500 K and T=3500 K have been used. The cone between the black lines marks the SKS propagation directions in the lower mantle relative to a horizontal flow plane. X, Y and Z are the principle finite strain axes where XNYNZ. The black horizontal line is the XY plane of finite strain. Lower hemisphere equal area projection. Fig. 8. The predicted seismic anisotropy for the bottom of the lower mantle composed of 100% polycrystalline Mg-Perovskite with the CPO given in Fig. 5 at 100 GPa. The single crystal elastic constants given by Oganov et al. (2001) at P =88 GPa, T=1500 K and T= 3500 K have been used. The top two rows are for horizontal flow (XY horizontal). The bottom row is the high high-temperature case for vertical flow (XY vertical). The cone between the black lines marks the SKS propagation directions in the lower mantle relative to a horizontal flow plane. X, Y and Z are the principle finite strain axes where XNYNZ. The black horizontal line is the XY plane of finite strain. Lower hemisphere equal area projection.

8 142 D. Mainprice et al. / Earth and Planetary Science Letters 271 (2008) than at the top of the lower mantle: 3.1% and 1.8% at 1500 K and 3500 K, respectively. The S-wave anisotropy has a still lower magnitude at the bottom of the lower mantle: 0.9%, independent of temperature. In a hot lower mantle (3500 K), the propagation directions producing the maximum shear wave splitting and the orientation of the V s1 polarization are the same at the top and bottom of the lower mantle. If the flow planes are horizontal, the lower mantle may contribute about 1% of the anisotropy recorded by SKS waves. At 1500 K, a small (0.9%) contribution from the lower mantle to SKS anisotropy is also likely for one single propagation direction contained in the XZ plane. One should note however that the presence of other phases, for example 20% by volume of ferripericlase, is likely to reduce all the anisotropies reported here. 6. Discussion Although orthorhombic silicate Mg-Perovskite is the major constituent of the lower mantle and the most abundant mineral in the Earth, very little is known about its mechanical properties and deformation mechanisms at lower mantle temperature and pressure conditions. Global seismology constrains however the entire lower mantle to be the first order isotropic or very weakly anisotropic. Our understanding of the deformation of Perovskite is essentially conditioned by studies using analogue materials with the Perovskite structure that can be deformed at more accessible pressure and temperature conditions. For example, based on deformation experiments of a fine-grained oxide analogue of Perovskite at high temperature and ambient pressure that show superplastic diffusive creep and no CPO development, Karato et al. (1995) inferred that the lack of seismic anisotropy in the lower mantle was due to superplastic creep. The flow of the lower mantle by some sort of diffusive creep is still a widely held postulate in geodynamics (e.g. McNamara et al., 2001). Yet, although the use of analogues proved to be successful for the systematic study of elasticity, where it was first developed, its use to study plasticity has been questioned some time ago by one of their original proponents (Poirier et al., 1989) and the similarity of viscoplastic behaviour in the large series of perovskite perovskite-structure minerals is still to be established. Several attempts have been made to measure the CPO of deformed silicate Perovskite. In early axial compression experiments on silicate Perovskite in a diamond anvil cell at pressures of 35 GPa and ambient temperature by Meade et al. (1995), no measurable CPO was recorded, whereas olivine deformed under similar conditions developed CPO. Yet, the CPO was measured after decompression and removal of the samples from the diamond anvil cell. In a more sophisticated arrangement using a diamond anvil cell constructed for wide angle X-ray diffraction, Merkel et al. (2003) measured the CPO of silicate Perovskite in situ in axial compression at pressures up to 32 GPa and ambient temperature. They noted that the deviatoric axial component of stress (σ 1 σ 3 ) increased progressively from 0 to 10.9 GPa as the pressure was raised to 32.0 GPa, in agreement with previous mechanical studies at high pressure by Meade and Jeanloz (1990) and Chen et al. (2002). Merkel et al. (2003) X-ray measurements show no development of significant CPO in their sample. For comparison, our models show that at a pressure of 30 GPa the easy glide [010](100) has a maximum Peierls stress of 8.3 GPa (see Table 1) and its activation during axial compression will lead to rotation of pole to the (100) plane towards the compression axis. In contrast, the study by Wenk et al. (2004) using Perovskite synthesized in situ in the diamond anvil cell from transformed olivine did develop a CPO similar to the one predicted by our models (Fig. 5). In addition to these studies, Cordier et al. (2004) characterized dislocations in silicate Perovskite deformed in a multi-anvil press at lower mantle temperature and pressure conditions (1400 C and 25 GPa). As samples recovered from high pressure are too radiation sensitive for electron microscopy, they used the X-ray peak broadening technique to characterize the dislocations and identified two active slip systems: [010](001) and [100](001). These systems are respectively the third and the fourth easiest slip systems according to our Peierls stress calculations (Table 1). In summary, the majority of previous studies on silicate orthorhombic Perovskite report no or weak CPO, our calculations show that plastic anisotropy is reduced at lower mantle pressures and that the resulting CPO not as strong as the olivine one at comparable finite strains, but clearly measurable, as demonstrated in the most recent experiments of Wenk et al. (2004). Global seismology indicates that lower mantle is almost isotropic for the majority of its depth (Montagner and Kennett, 1996; Beghein et al., 2006; Panning and Romanowicz, 2006). Most observations are based on S-wave data, in agreement with our model predictions that show a maximum anisotropy for S-waves of only 1.6% for 1500 K and 1.1% for 3500 K. Yet both the global study of Montagner and Kennett (1996) and a regional study of Wookey et al. (2002) in the southwestern Pacific suggest that there may be some anisotropy in the uppermost lower mantle. The data of Wookey et al. (2002) were re-examined for the possibility of contamination by shear-coupled P- waves by Wookey and Kendall (2004). Although some of the shear waves splitting times were slightly reduced, the study confirmed the strong evidence of an anisotropy layer of about 100 km thickness in Fig. 9. Geodynamic model for the uppermost lower mantle in the Tonga Kermadec region based on the seismic anisotropy predicted using modelled Mg-Perovskite CPO and low temperature single crystal elastic constants and on seismic observations from Wookey et al. (2002). APM is the absolute plate motion. X, Y, and Z are the finite strain axes in Fig. 7. The circle is the pole figure of dv s anisotropy with the regions with high shear wave splitting in black.

9 D. Mainprice et al. / Earth and Planetary Science Letters 271 (2008) the uppermost lower mantle close to the Tonga Kermadec subduction zones (Fig. 9). Given that the S-wave anisotropy of Mg-Perovskite is stronger at lower temperatures (Fig. 7), the uppermost lower mantle surrounding fast subducting and hence very cold plates would be the most likely candidate for observable anisotropy. The Tonga Kermadec subduction zones have the fastest present-day subduction rates. Wookey et al. (2002) observed that the horizontally polarized S-waves were faster than the vertically polarized S-waves. To match these observations with our modelled CPO and the resulting seismic anisotropy at 1500 K (Fig. 7) would require horizontal flow and a propagation of S-waves normal to the flow direction. The S-waves were recorded from a source (very deep earthquakes) in the Tonga Kermadec subduction zones with seismometers in Australia for propagation directions, which are mainly East West (Fig. 9). The implied flow direction in the uppermost lower mantle region near the source would be approximately North South, which is coherent with present-day plate motion to the North of the eastern part of the Australian plate north of New Zealand. There are no seismic observations of anisotropy in the lowest lower mantle to our knowledge. However, if temperatures were high near the CMB Mg-Perovskite would be more stable than the Post- Perovskite. Anisotropy in the D layer has been observed by global seismology for some time (e.g. Montagner and Kennett, 1996). The global S-wave anisotropy in D, defined as the ratio between the squares of horizontally and vertically polarized S-waves velocities (ξ=v 2 SH/V 2 SV) is about 1.01 (e.g. Panning and Romanowicz, 2006). Indeed, the most commonly reported feature of D anisotropy is that the fastest S-waves are horizontally polarized. Can strain-induced CPO of Mg-Perovskite explain this observation? The S-wave anisotropy of polycrystalline Mg-Perovskite is very weak (maximum of 0.9% at 1500 K, Fig. 8). If the flow is horizontal, the anisotropy sampled by horizontally propagating S S-waves will be still lower (b0.5%). For high temperatures (3500 K), horizontal flow will produce a vertical polarization that is incompatible with seismic observations (middle row Fig. 8). However, vertical flow at high temperature (bottom row of Fig. 8) would match global observations for D layer with the fastest S- waves (V s1 ) polarized horizontally with an anisotropy of 0.9% and a symmetry that is very close to transverse isotropy (hexagonal) elastic symmetry (e.g. Kendall and Silver, 1996). The anisotropy in the horizontal plane varies with azimuth between ξ=1.01 and 1.02, entirely compatible with the values from global seismology. In addition, nearly vertically propagating SKS waves will not be contaminated by a hot lower mantle or D layer anisotropy because the anisotropy is less than 0.2% in the vertical direction. Hence, in hot regions of D where Mg-Perovskite is stable rather than Post- Perovskite, the observed transverse anisotropy could result from vertical flow of hot material (upwelling). Finally, pairs of discontinuities with anticorrelated topographies have been observed in some regions of D where the penetration of a cold slab into the D layer results in a relatively low temperatures at the core mantle boundary (Hernlund et al., 2005; van der Hilst et al., 2007). The velocity in the lens between the discontinuities is lower than the surrounding region. These observations have been interpreted as recording a Post-Perovskite-rich lens in a Perovskite matrix caused by the curved geotherm in D crossing the Clapeyron of the phase transition twice. In this case, a thin layer of Mg-Perovskite will occur just above the CMB due to the back-transformation from Post-Perovskite. However, our models show that if deformation near the CMB occurs essentially by horizontal shearing, horizontally propagating fast S-waves will have either a weak anisotropy with a vertical polarization or not be split (Fig. 9). This is contrary to most observations of seismic anisotropy in D that show a strong anisotropy for horizontally propagating S-waves with SH usually faster than SV (Panning and Romanowicz, 2006). Hence, we propose that the observed shear wave splitting observed in these domains must have other physical causes. 7. Conclusions We have undertaken first first-principles calculations of dislocation glide in silicate orthorhombic Mg-Perovskite and determined the Peierls stress and relative critical resolved shear stress (CRSS) of the possible slip systems at three pressures (0, 30 and 100 GPa). The plastic anisotropy of Mg-Perovskite varies strongly with pressure between 0 and 30 GPa, but at lower mantle pressures (between 30 GPa and 100 GPa) the variation is quite small. VPSC modelling of the deformation of a Mg-Perovskite aggregate using these slip systems and CRSS showed that the [010](100), b110n{ 110}, [010](001) and b 110N(001) systems are predominantly activated at lower mantle pressures in both simple shear and axial compression. Comparison of the combined ab initio/vpsc model predictions of CPO with those measured in situ by X-ray diffraction during axial compression in a diamond anvil cell shows a high degree of similarity with concentration of [100] axes parallel to the shortening direction. Predicted CPO in simple shear at lower mantle pressures of 30 and 100 GPa are similar; they show a strong alignment of the [010] axis in shear direction and of (100) close to the shear plane. The seismic anisotropy was calculated using the CPO predicted from the ab initio/vpsc model for a shear strain of 1.73 and single crystal elastic constants at lower mantle pressures and temperatures (P=38 and 88 GPa and T=1500 and 3500 K) predicted by ab initio molecular dynamics by Oganov et al. (2001). Seismic anisotropy is always weak, at low temperature (1500 K) results in higher P-wave anisotropy independent of the pressure. S-wave anisotropy is less sensitive to temperature. At the top of the lower mantle the anisotropy is relatively weak: 2.9% (0.6%) for P-waves and 1.6% (1.1%) for S S-waves at 1500 K (3500 K). The weak predicted seismic anisotropy implies that the isotropic behaviour observed in global seismology data may be explained without the need to involve mechanisms that do not produce CPO, such as diffusive or superplastic creep. As lower temperature results in higher anisotropy, regions of relatively low mantle temperatures, such as subduction zones, are the most likely to display measurable anisotropy in the lower mantle. Indeed, Wookey and Kendall (2004) measured significant S-wave delay times from a 100 km thick region in uppermost lower mantle in the Tonga subduction zone with the leading S-waves having a horizontal polarization. The analysis of these observations in the light of our model suggests that the travel path measured by Wookey et al. was the direction of maximum shear wave splitting and that the flow direction at the top of the lower mantle is N S, which is parallel to the present-day plate motion. Mg-Perovskite may also be an important phase in the D -layer in regions where the temperature is high. Both global and local observations suggest that D is mostly anisotropic with faster propagation of horizontally polarized S S-waves. The P-wave anisotropy predicted at the bottom of the lower mantle is very similar to the top, with the fastest P-wave parallel to the flow direction (X), but the anisotropy is slightly higher (3.1% at 1500 K and 1.8% at 3500 K). The S-wave anisotropy is very low ( 0.9%) for both 1500 K and 3500 K. For horizontal flow, only the lower temperature (1500 K) model reproduces the observed seismic anisotropy in D, but the anisotropy would be very weak (0.0 to 0.5% depending on the azimuth). An alternative hypothesis would be for vertical flow (upwelling) in the D layer, in regions where the temperature is high enough for Mg-Perovskite to be stable. Vertical flow would have the observed fastest shear velocity with horizontal polarization, with an anisotropy of 0.7 to 0.9% for horizontally propagating shear waves. 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