First principles investigation of the structural and elastic properties of hydrous wadsleyite under pressure

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114,, doi: /2008jb005841, 2009 First principles investigation of the structural and elastic properties of hydrous wadsleyite under pressure Jun Tsuchiya 1,2 and Taku Tsuchiya 1 Received 31 May 2008; revised 25 October 2008; accepted 1 December 2008; published 13 February [1] In order to clarify the effect of protonation of wadsleyite under high-pressure conditions, we determined defect structures of Mg SiO 4 H 0.25 (1Mg 2+ $ V 2 Mg +2H +, 1.65 wt % H 2 O), Mg 1.75 SiO 4 H 0.5 (2Mg 2+ $ 2V 2 Mg +4H,3.3wt%H 2 O) hydrous wadsleyite and their elastic properties by means of the density functional first principles method. Structural optimization calculations indicate that the most stable structures have monoclinic symmetry with magnesium M3 site vacancies. Protons are found to bond to the O1 site to align the OH dipoles along the edges of M3 vacancies. Calculated elastic constants, bulk and shear moduli, are found to decrease almost linearly with increasing water content but to increase linearly with increasing pressure. At 15 GPa and static 0 K condition, incorporation of 3.3 wt % H 2 O into wadsleyite, which corresponds to the maximum solubility of hydrous wadsleyite, reduces V P and V S by about 3.9 and 4.8%, respectively. This indicates that 1 wt % H 2 O hydration of wadsleyite corresponds to the temperature effects on bulk and shear moduli about 430 K (0 GPa) to 340 K (20 GPa) and 350 K (0 GPa) to 290 K (20 GPa), respectively. The transversely isotropic aggregates demonstrate largest positive polarization anisotropy V SH V SV when the c axis aligns vertically in both dry and wet cases. Citation: Tsuchiya, J., and T. Tsuchiya (2009), First principles investigation of the structural and elastic properties of hydrous wadsleyite under pressure, J. Geophys. Res., 114,, doi: /2008jb Introduction [2] The presence of volatile components like water in the deep Earth has been thought to significantly affect the mantle dynamics because the physical and chemical properties of materials such as atomic diffusivity, electrical conductivity, melting temperature, and elasticity are considerably changed [e.g., Karato, 1990; Hirth and Kohlstedt, 1996]. Smyth [1987] first pointed out that wadsleyite, which is considered to be a primary component of the upper part of mantle transition zone, can store a large amount of proton (H + ) in the crystal structure and could be a host of water in the deep Earth. Many high-pressure and temperature experiments have been conducted to date in order to clarify the water solubilities of the mantle constituent minerals [e.g., Bolfan-Casanova et al., 2000; Ohtani et al., 2004; Demouchy et al., 2005]. And those consistently found that at most a few wt % H 2 O can be incorporated into wadsleyite and also ringwoodite, a primary component of the deeper part of transition zone, at temperatures higher than 1200 C and pressures of about GPa [e.g., Ohtani et al., 2001]. [3] Knowledge of the effect of water on the mineral properties at the transition zone pressure and temperature conditions is therefore quite important to estimate the amount 1 Geodynamics Research Center, Ehime University, Matsuyama, Japan. 2 Now at Senior Research Fellow Center, Ehime University, Matsuyama, Japan. Copyright 2009 by the American Geophysical Union /09/2008JB005841$09.00 of water in the transition zone through interpreting seismic wave velocity observations. Although elastic moduli of hydrous wadsleyite and hydrous ringwoodite have recently been measured by Brillouin spectroscopy at ambient condition [Mao et al., 2008; Inoue et al., 1998], such information is still very limited at the transition zone pressure and temperature conditions. [4] Elastic and vibrational properties of Mg 2 SiO 4 dry wadsleyite have been reported using first principles calculation technique [Kiefer et al., 2001; Wu and Wentzcovitch, 2007]. First principles calculation is known as a powerful tool for investigating the physical properties of minerals in highpressure condition. However, high-pressure structural and physical properties of hydrous wadsleyite have not been calculated so far. [5] In order to investigate the physical properties of hydrous wadsleyite, we need to start with exploring the proton defect structures. After determining appropriate structure models which are reduced to the simplified ordered hydrogen structure, we study the equations of state, elasticity, seismic wave velocities and anisotropy of hydrous wadsleyite at high-pressure conditions using these models. Then, on the basis of the predicted physical properties, we further attempt to interpret seismological observations and discuss the amount of water in the mantle transition zone. [6] Smyth [1987] first pointed out that wadsleyite can be a host for hydrogen in the Earth. He calculated electrostatic potentials and Pauling bond strengths of wadsleyite and found that a special oxygen position O1 (Figure 1), which is surrounded only by Mg atoms, is electrostatically favorable 1of13

2 Figure 1. Crystal structure of wadsleyite. Yellow large, blue medium, and red small spheres are magnesium, silicon, and oxygen atoms, respectively. to a donor site. Then, he proposed a hypothetical structure of hydrous wadsleyite [Smyth, 1994]. The proposed defect structure was studied by quantum mechanical simulations [Winkler et al., 1995]. The stability of protonated O1 site in wadsleyite was confirmed by ab initio molecular dynamics [Haiber et al., 1997] and interatomic model potential calculations [Walker et al., 2006]. Since all of these computational studies were however performed with no consideration of pressure, the high-pressure energetic and mechanical properties were mostly unclear. Downs [1989] reported a bridging oxygen O2 site between corner-sharing SiO 4 tetrahedra as a candidate of donor site. Apart from these studies, the hydrogen docking sites of Mg 2 SiO 4 wadsleyite have been investigated on the basis of the topological analysis of the electron density distribution and were reported to potentially exist at all oxygen sites [Ross et al., 2003], not only around O1 and O2. In addition to these static properties, the OH stretching vibrations of hydrous wadsleyite measured by unpolarized and polarized FTIR spectroscopy indicated that the major OH stretching vibration can typically be interpreted by the protonation of O1 site [Kohn et al., 2002; Jacobsen et al., 2005]. However, it should be noted that there are other vibrational modes which cannot be explained by the protonation of O1 site. There are still several uncertainty to be determined in order to protonate O1 site, for instance, the orientation of the OH dipoles and the locations of associated cation vacancies. [7] Cation vacancy should be introduced in order to conserve the charge neutrality of the system when positively charged protons are incorporated in wadsleyite. Kudoh et al. [1996] reported that the defects in hydrous wadsleyite are mainly concentrated on magnesium M3 site. They concluded that the ratio of M site vacancy is M1:M2:M3 = 3 ± 3%:6 ± 4%:22 ± 1% in the 3.3 wt % H 2 O hydrous wadsleyite. It was reported that the crystal structure of wadsleyite slightly distorts from the orthorhombic to the monoclinic system by hydration (b [Smyth et al., 1997]). Cation vacancy ordering has been speculated as a possible cause of this monoclinic distortion. [8] The crystal structure and bulk modulus of hydrous wadsleyite have also been investigated [Yusa and Inoue, 1997; Holl et al., 2008]. These studies report that the primary structural change due to hydration is elongation of the b axis. In contrast, the cell length of a axis slightly decreases with hydration and the c length hardly changes [Yusa and Inoue, 1997; Holl et al., 2008]. Being consistent with these findings, Jacobsen et al. [2005] reported that the b/a axial ratio linearly increased with increasing the water content: b/a = C H2 O (in wt %) at least at ambient pressure. Also hydrous wadsleyite with 2.5 wt % H 2 O was reported to have zero-pressure bulk modulus of 155 ± 2 GPa [Yusa and Inoue, 1997]. This is 5 11% smaller than the bulk modulus of anhydrous composition, exhibiting a marked effect of water to elastically soften wadsleyite. Recent Brillouin scattering experiments at ambient pressure [Mao et al., 2008] demonstrated supportive results that the bulk and shear moduli of hydrous wadsleyite decreased linearly with increasing water content as B S0 = 170.9(9) 13.0(8) C H2 O, G 0 = 111.7(6) 7.8(4) C H2 O, where C H2 O means the H 2 O concentration in wt %. It is interesting how these effects of water change at high pressures. Are they enhanced or reduced? 2. Methods [9] Here we used the first principles calculation methods based on density functional theory. The generalized gradient approximation proposed by Perdew, Burke, and Ernzerhof (GGA-PBE) [Perdew et al., 1996] has been applied to represent the exchange and correlation potential. In this study, we do not use the local density approximation (LDA) because it has been shown to overestimate the strength of the hydrogen bond and therefore expected to be unsuitable for describing hydrogen bond systems [Hamann, 1997]. This deficiency is too large to be compensated by other factors such as temperature and quantum zero-point motion effects. [10] Calculation conditions are generally the same as in our previous studies on hydrous minerals [Tsuchiya et al., 2005, 2008]. The electronic wave function was expanded in plane waves using a kinetic energy cutoff of 80 Ry. Normconserving pseudopotentials [Troullier and Martins, 1991] have been employed to describe the ionic core potentials of Si, O, and H atoms, whereas Mg pseudopotential is generated by the method of U. von Barth and R. Car (unpublished material) [Karki et al., 2000]. These pseudopotentials have been extensively tested in our previous studies [e.g., Tsuchiya et al., 2004a, 2004b]. The irreducible Brillouin zone of a unit cell of hydrous wadsleyite are sampled on a Monkhorst-Pack mesh [Monkhorst and Pack, 1976]. Brillouin zone sampling is also carried out on Monkhorst- Pack mesh for a primitive cell of hydrous wadsleyite with space groups I2 and Pmmb. The effects of larger energy cutoffs and k point sampling on the calculated properties are found to be insignificant. All structural parameters are fully relaxed at a static (0 K) condition using damped variable cell shape molecular dynamics [Wentzcovitch, 1991] using the PWSCF code until residual forces became less than Ry per atomic unit. [11] Proton defects in hydrous wadsleyite should be introduced primarily by replacing Mg [Kudoh et al., 1996]. We therefore constructed the model crystals with compositions of Mg 15 Si 8 O 30 (OH) 2 (1Mg $ V Mg + 2H, 1.65 wt % H 2 O) and Mg 14 Si 8 O 28 (OH) 4 (2Mg $ 2V Mg + 4H, 3.3 wt % H 2 O) in 2of13

3 elastic constants of the most stable structures with above compositions using the stress-strain relations [Karki et al., 2001]. The magnitude of all applied strains was We confirmed that the linear relation was enough ensured for this strain range. 3. Results and Discussion [12] Dry wadsleyite structure has an orthorhombic symmetry with the space group Imma [Hazen et al., 2000]. There are 3 types of Mg sites (M1(4a), M2(4e) and M3(8g)) in wadsleyite. Among them, we consider only M2 and M3 vacancy models, since the M1 vacancy concentration is reported to be significantly low [Kudoh et al., 1996]. Figure 2. Crystal structure of hydrous wadsleyite (1.65 wt % H 2 O, 1V M2 model 1 and model 2). Smallest white spheres are hydrogen atoms. addition to the dry compound of Mg 16 Si 8 O 32 for comparison with protonated phases. The details of these model crystal structures are demonstrated in section 3. We have calculated 3.1. The 1.65 wt % H 2 O Model (Mg 15 Si 8 O 32 H 2 ) [13] Structural model of hydrous wadsleyite with 1.65 wt %H 2 O composition can be obtained by removing one Mg along with the incorporation of two protons into a wadsleyite unit cell (1Mg $ V Mg + 2H). For the model with M2 vacancy, we found two types of the stable protonation configurations (Table 1 and Figure 2). In both structures, protons stabilized near O1 owing to the electrostatic unbalance of O1 site in the wadsleyite structure [Smyth, 1987]. But model 1 has the lower energy than model 2 of about 0.4 ev cell 1. As shown in Figure 2, all the OH dipoles are found parallel to the c axis. Therefore, the orthorhombic symmetry is preserved by this type of hydration. [14] In the M3 vacancies models, we found three stable structures (Figure 3). Like the case of the M2 vacancy models, only the structures that O1 sites are protonated were stabilized. One has H + -V Mg associated vacancies and the other two have dissociated H + -V Mg. Among them, the structure with two OH dipoles oriented along the edges of V M3 has the lowest energy. This 1V M3 model 1 structure is found to be the most stable among the all structures we calculated including the M2 vacancy models (Table 1). We found that the energy difference between the most stable structure with the associated defect pairs (1V M3 model 1) and the metastable structures with the dissociated defect pairs (1V M3 model 2 and 3) is 0.9 ev cell 1. While the crystal structure with the M2 vacancy kept the orthorhombic symmetry, we found the cell of M3 vacancy model was distorted from orthorhombic to monoclinic by protonation (b = ) consistent with experimental reports [Smyth et al., 1997]. Table 1. Calculated Total Energy, Volume, and Cell Parameters of Dry and Hydrous Wadsleyite at 0 GPa (1.65 wt % H 2 O) a 1V M2 1V M3 Dry Imma Model 1 Model 2 Model 1 Model 2 Model 3 E (ev cell 1 ) - 15, , , , , V (Å 3 ) a (Å) b c b( ) B (GPa) (160.3) (149.4) - - B (4.4) (4.4) - - O-H1 (Å) OO O-H2 (Å) OO a Bulk modulus B 0 is obtained from Vinet (a third-order Birch-Murnahan) equation of state. 3of13

4 all of the stable structures have the O1 sites totally protonated. Therefore, this water content (3.3 wt % H 2 O) is very likely to be the maximum water solubility in wadsleyite. This is consistent with experimental findings [Inoue et al., 1995]. Among the M2 vacancy models of 3.3 wt % H 2 O composition, we could obtain the stable structures with the OH dipoles oriented parallel to the c axis(figure 4). Note that this 2V M2 model 1 structure is exactly identical to the structure proposed by Smyth [1994]. [16] In the wadsleyite unit cell, seven unequivalent ways are possible to choose two M3 vacancies. We have found that most of the stable configurations with M3 vacancies have the O1 protonation sites with the OH dipoles oriented along the edges of V M3 (Figure 5 and Table 2). This defect structure is the same as the structure of 1.65 wt % H 2 O composition with an M3 vacancy. Among these structure models, the lowest energy configuration is found to have the most homogeneous distribution of V M3 (Table 2, 2V M3 model 1). We excluded other three structures including adjacently located two M3 vacancies pairs from a model structure of hydrous wadsleyite, since we found those structures have energy more than 2.3 ev cell 1 higher than the structures with isolated M3 vacancy pairs. Figure 3. Crystal structure of hydrous wadsleyite (1.65 wt % H 2 O, 1V M3 model 1, model 2, and model 3) The 3.3 wt % H 2 O Model (Mg 14 Si 8 O 32 H 4 ) [15] The composition of 3.3 wt % H 2 O hydrous wadsleyite can be obtained by removing two Mg atoms in wadsleyite along with incorporating four hydrogen atoms. We found that Figure 4. Crystal structure of hydrous wadsleyite (3.3 wt % H 2 O, 2V M2 model 1 and model 2). 4of13

5 Figure 5. Crystal structure of hydrous wadsleyite (3.3 wt % H 2 O, 2V M3 model 1, model 2, model 3, and model 4). [17] Downs [1989] has reported that SiO 4 bridging oxygen O2 site can also be a host for hydrogen in wadsleyite. We also tested the protonation on this O2 site in both 1.65 and 3.3 wt % H 2 O compositions. We have performed a number of structural relaxation calculations of this model with changing the initial proton positions. However, protons always moved to near O1 during the relaxation and we could never obtain stable configuration of this protonation host. Protonations on the other oxygen sites O3 and O4 were also found to be unstable as well. [18] The lowest energy structure with 3.3wt % H 2 O composition (2V M3 model 1) can be assigned with the monoclinic symmetry (b 92.7 ) with the space group I2. This monoclinic distortion of hydrous wadsleyite has been also reported experimentally (e.g., b ) [Smyth et al., 1997; Kudoh and Inoue, 1999; Mao et al., 2008; Holl et al., 2008]. The distortion increases with increasing water content almost linearly (Tables 1 and 2), and therefore difference between calculated and experimental b may partially be caused by the difference of water contents in wadsleyite samples. Because of small energy differences among M3 vacancy models (Table 2, 2V M3 model 2, 3), disordering of the M3 vacancies seems likely. This may also cause the cancelling of the monoclinic distortion. [19] Averages of the calculated cell lengths of 2V M2 models are a = 5.795, b = , and c = Å, whereas those of 2V M3 models are a = 5.706, b = , and c = Å. 2V M3 vacancy models are found to give longer b than that of V M2 models (Table 2). Jacobsen et al. [2005] pointed out that b/a ratio increases with water the content of wadsleyite. Figure 6 shows the relationship between the b/a ratio and water content of wadsleyite calculated at 0 GPa. The b/a ratio of M3 vacancy model is found to increase with the water content, in excellent agreement with the experimental results. This relation might be used as calibration of water content in wadsleyite. On the other hand, M2 vacancy model does not show rapid increase of b/a ratio. This discrepancy between b/a ratio of M2 and M3 models clearly indicates that the major defects should exists on the M3 sites in wadsleyite. [20] Spectroscopic measurements have shown that there are roughly two types of the OH stretching vibration mode in hydrous wadsleyite [McMillan et al., 1991; Cynn and Hofmeister, 1994; Kohlstedt et al., 1996; Kohn et al., 2002; Jacobsen et al., 2005]. A major band exists at the frequency of 3350 cm 1 which decreases linearly with increasing pressure. A minor band observable at 3600 cm 1 remains nearly constant under high-pressure condition [Kleppe et al., 2001]. Our structural models indicate that there are two types of hydrogen bonds in M2 and M3 vacancy models. In the M2 vacancy models, the OH dipoles are oriented toward the O2 (Si 2 O 7 bridging oxygen) site. The distances between O1 and O2 are about 4.0 Å (Tables 1 and 2). These long OO lengths suggest that there are almost no hydrogen bonds formed in M2 vacancy models. In contrast, in the case of the M3 vacancy models, the hydrogen bond strength should be stronger than those of the M2 vacancy models, since the distances of OO pairs are approximately 3.0 Å (Tables 1 and 2). A correlation between the OO distance and OH vibrational frequency has been reported from empirical analyses of various hydrous minerals [Nakamoto et al., 1955]. When the OO distance is longer than 3.0 5of13

6 Table 2. Calculated Total Energy, Volume, and Cell Parameters of Hydrous Wadsleyite at 0 GPa (3.3 wt % H 2 O) a 2V M2 Model 1 Model 2 Model 1 Model 2 Model 3 Model 4 E (ev) 15, , , , , , V (Å 3 ) Space group Pmmb Pmmb I a(å) b c b( ) B (GPa) (137.7) B (4.5) O-H1 (Å) OO O-H2 (Å) OO O-H3 (Å) OO O-H4 (Å) OO a Bulk modulus B 0 is obtained from Vinet (a third-order Birch-Murnahan) equation of state. 2V M3 Å, the OH stretching frequency is less sensitive to pressure, while it is shorter than 3.0 Å, the OH stretching vibration frequency decreases with increasing pressure due to increasing of the hydrogen bond strength. Therefore, the major and minor peaks observed for hydrous wadsleyite can be reasonably interpreted as the OH stretching vibrations of the M3 and M2 vacancy models, respectively. Above discussion of spectroscopic properties of hydrous wadsleyite was based simply on the local environment around hydroxyls. Further calculations such as first principles lattice dynamics simulations [Tsuchiya et al., 2008] are needed for more details of vibrational properties. [21] As shown in above calculations, we have determined the most stable structural models of hydrous wadsleyite containing 1.65 wt % H 2 O(Mg 15 Si 8 O 30 (OH) 2 ) and 3.3 wt % H 2 O(Mg 14 Si 8 O 28 (OH) 4 ). In both compositions, two types of hydrogen defect structures, M2 and M3 vacancy models, can be identified. The structures with the M3 vacancy defects have lower energy than those with the M2 vacancies. This is consistent with the experiments of Kudoh et al. [1996] mentioned in introduction. We also found that the M3 vacancy defects induce a small distortion of b angle (b 92.7 ), which is also consistent with the experiments [Smyth et al., 1997]. In these vacancy models, protons bound to the O1 site to align the OH dipoles along the edges of Mg vacancies. However, many studies also pointed out the possibilities of the disordering of proton [Kohn et al., 2002; Ross et al., 2003; Jacobsen et al., 2005], for example IR measurements detected other minor OH vibrations suggesting the existence of other protonation site except for O1. Therefore, further studies concerning other protonation sites in hydrous wadsleyite should be conducted in order to investigate detailed protonation mechanisms in hydrous wadsleyite. Next, we report the high-pressure structural and elastic properties of hydrous wadsleyite using the crystal structures of the most stable models (1V M3 model 1 and 2V M3 model 1) Compression Behavior [22] Figure 7 shows variations of calculated cell lengths of dry, 1.65 and 3.3 wt % H 2 O wadsleyite at pressures from 0 to 40 GPa. The calculated cell lengths of the dry composition are slightly longer (about 0.02 Å) than the experimental values. This systematic overestimation is typically seen in the GGA calculation. The relative changes of the a, b, and c lengths by the incorporation of 3.3 wt % H 2 O are 0.03 Å, Å, and 0.04 Å at 0 GPa, respectively. These changes of cell lengths are consistent with the experimental study of hydrous wadsleyite [Yusa and Inoue, 1997; Holl et al., 2008]. For comparison, we also plot the cell lengths of the M2 vacancy model (2V M2 model 1, dashed lines in Figure 7) which has higher energy than 2V M3 model 1. The 2V M2 model 1 is found to have the a axis considerably longer than that of the dry phase, while the b length is rather comparable or shorter than that of the dry phase. This also suggests an irrelevance of this M2 vacancy model. [23] The equations of state of the dry, 1.65, 3.3 wt % H 2 O compositions are plotted in Figure 7. At 0 GPa, hydrous wadsleyite has slightly larger volumes than dry one primarily due to the presence of vacancy. This excess volumes of hydrous wadsleyite are rapidly compressed with increasing pressure, since the Mg vacancies are more compressible than MgO 6 octahedra. Least squares fittings of these data to a Vinet [Vinet et al., 1989] (a third-order Birch-Murnaghan) equation of state yield zero-pressure bulk modulus B 0 of Figure 6. The relationship between b/a ratio and water content of wadsleyite at 0 GPa. Dashed black line indicates b/a = C H2 O (in wt %) [Jacobsen et al., 2005]. 6of13

7 3.4. Elasticity [24] Using the most stable M3 vacancy model structures, we calculated elastic constants of hydrous wadsleyite. As mentioned before, these structural models with 1.65 and 3.3 wt % H 2 O compositions are slightly distorted to monoclinic symmetry. Therefore, there are = 13 independent elastic constants, though all the additional four elastic constants are very small (Table 3). The experimental elastic constants are now available at ambient conditions [Mao et al., 2008], we compare our results at 0 GPa with the experiment (Figure 8). All the elastic constants are found to decrease almost linearly with increasing the water content in wadsleyite. The experimental C 11,C 22,C 33 seem to decrease more than the present study with increasing the water content of wadsleyite. [25] We have also calculated the high-pressure elastic constants of dry, 1.65, and 3.3 wt % H 2 O hydrous wadsleyite as a function of pressure (Figure 9). Slight underestimations due to the nature of GGA are seen particularly in C 11,22,33. We found that these elastic constants increase almost linearly with increasing pressure at least up to 20 GPa. Figure 7. Pressure dependencies of cell lengths of dry and hydrous wadsleyite. Thin (red), medium-thick (green), and thick (blue) lines indicate calculated cell lengths of dry, 1.65, and 3.3 wt % H 2 O wadsleyite, respectively. Dashed line corresponds to that of M2 vacancy model of 3.3 wt % H 2 O composition. Open and full dots indicate experimental values of dry [Horiuchi and Sawamoto, 1981; Hazen et al., 2000] and hydrous wadsleyite containing 2.5 wt % H 2 O [Yusa and Inoue, 1997], respectively. dry, 1.65, and 3.3wt % H 2 O wadsleyite about (160.3) GPa, (149.4) GPa, and (137.7) GPa, respectively (Tables 1 and 2). These B 0 are found to decrease almost linearly with increasing the water content in wadsleyite. Since the difference of B 0 of dry wadsleyite between experiments [Hazen et al., 2000; Yusa and Inoue, 1997; Holl et al., 2008] and the present study are about 6%, the calculated B 0 of hydrous wadsleyite may also be underestimated as well by GGA. Figure 8. Dependencies of elastic constants on water contents of hydrous wadsleyite at 0 GPa. Experimental elastic constants are from Mao et al. [2008]. 7of13

8 Table 3. Calculated Elastic Constants of Dry and Hydrous Wadsleyite Water (wt % H 2 O) C 11 C 22 C 33 C 12 C 23 C 13 C 44 C 55 C 66 C 15 C 25 C 35 C 46 0 (0 GPa) (10 GPa) (20 GPa) (0 GPa) (10 GPa) (20 GPa) (0 GPa) (10 GPa) (20 GPa) Bulk and Shear Moduli and Velocities [26] Figure 10a and Table 4 show the aggregate bulk and shear moduli of dry and hydrous wadsleyite calculated from the Voigt-Ruess-Hill average. The reported zero-pressure bulk (B 0 ) and shear moduli (G 0 )ofdrymg 2 SiO 4 wadsleyite are and GPa, respectively [Li et al., 1996; Zha et al., 1997; Li et al., 1998; Isaak et al., 2007]. Our calculated moduli are 6 and 7% smaller than the respective experimental values due to the nature of GGA, and decreased with increasing the water content. From these results, relative effects of water on bulk and shear moduli are estimated as (@B/@C H2O )= 7.3 (0 GPa), 6.6 (10 GPa) and 5.8 (20 GPa) [GPa C 1 H2O ], and (@G/@C H2O )= 5.5 (0 GPa), 4.8 (10 GPa) and 4.5 (20 GPa) [GPa C 1 H2O ], where C H2 O indicates wt % H 2 O in wadsleyite. That effect of water on bulk modulus at 0 GPa ( 7.3 GPa wt % H 2 O 1 ) is significantly smaller than that of recent Brillouin scattering measurement ( 13.0 GPa wt % H 2 O 1 )[Mao et al., 2008]. This experiment has been conducted for hydrous wadsleyite containing up to 1.66 wt % H 2 O. Further Figure 9. Calculated elastic constants of wadsleyite under pressure. Thin (red), medium-thick (green), and thick (blue) lines correspond to dry, 1.65, and 3.3 wt % H 2 O wadsleyite, respectively. Experimental elastic constants of dry wadsleyite are indicated by black dots [Zha et al., 1997]. 8of13

9 Figure 10. (a) Calculated aggregate bulk and shear moduli (B and G), and (b) acoustic velocities (V P, V f and V S )of wadsleyite at 0 20 GPa, and 0 K conditions. Thin (red), medium-thick (green), and thick (blue) lines correspond to dry, 1.65, and 3.3 wt % H 2 O wadsleyite, respectively. Experimental elastic constants of dry wadsleyite are indicated by black dots [Zha et al., 1997]. calculations and experiments are needed for detailed comparison. [27] Figure 10b shows the longitudinal and shear wave velocities of dry pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and hydrous wadsleyite, p which ffiffiffiffiffiffiffiffi are determined by V P = ðb þ 4=3GÞ=r and V S = G=r. As well as the moduli, elastic wave velocities decrease almost linearly with increasing the amount of H 2 O, though the relative velocity contrasts between the dry and wet composition slightly decrease under high pressure. At 15 GPa and static 0 K condition, incorporation of 3.3 wt % H 2 O in wadsleyite reduces V P and V S by 3.9 and 4.8%, respectively Anisotropy [28] Information of the elastic anisotropy has been generally thought to be important for inferring the flow patterns in the mantle [e.g., Christensen, 1984]. It has been pointed out that water has the potential ability to alter the anisotropy of mantle minerals [Karato, 1995] because it changes the plastic deformation mechanisms. We determined the singlecrystal elastic wave velocities of hydrous wadsleyite by solving the Cristoffel s equation; detjc ijkl n j n l rv 2 d ik j =0. In this equation, n, r, V, and d ij are the propagation direction, density, wave velocity and Kronecker delta, respectively. Our single-crystal elastic velocities of dry wadsleyite by GGA are 0 6% smaller than those of previous first principles LDA calculation [Kiefer et al., 2001] and Brillouin spectroscopic measurement [Zha et al., 1997; Sawamoto et al., 1984] (Figure 11). Since C 33 is relatively smaller than C 11 and C 22 in both dry and hydrous wadsleyite (Figure 9), the fastest and slowest V P directions do not change with water content, and are along [100] and [010], and [001], respectively. The fastest V S direction of dry and wet wadsleyite is along [110], whereas there is no distinctively slow direction for V S. [29] Single-crystal azimuthal anisotropies for P (A P ) and S(A S ) waves, defined as A P =(V Pmax V Pmin )/V P 100 and A S =(V Smax V Smin ) 100, are plotted in Figure 12a. This single-crystal azimuthal anisotropy may give the upper limit on the realistic anisotropy of aggregates. Calculated azimuthal anisotropies for P and S waves are significant (more than 15%) at 0 GPa but rapidly decrease with increasing pressure. Incorporation of water in wadsleyite also reduces them especially for P wave at high pressures. However, they are found still relatively large at the pressures over 10 GPa particularly in V S. [30] Transverse anisotropy in V S is defined as A T S =(V SH V SV )/V S 100, where V SH (V SV ) is horizontally (vertically) polarized V S propagating horizontally. V S is the usual isotropic averages. Figure 12b shows the V S polarization anisotropy for the transversely isotropic aggregates with the a, b, and c vertical alignment. Figure 12b indicates that the transversely isotropic medium produces largest positive shear wave splitting in both dry and hydrous wadsleyites when the c direction aligns vertically. In contrast to the single-crystal anisotropy, these shear wave splittings increase with increasing pressure and also with the water content. 4. Geophysical Implications [31] In order to compare the calculated properties with the observed seismic velocity structures of mantle transition zone, we have estimated the high-temperature V P and V S along the adiabatic geotherm (1873 K at 660 km depth) by applying the experimental thermoelastic effects of Mg 2 SiO 4 wadsleyite (@B S /@T) P = [GPa K 1 ] and (@G/@T) P = [GPa K 1 ][Isaak et al., 2007]. [32] In Figure 13a, obtained high-temperature V P and V S of dry and hydrous wadsleyite (3.3 wt % H 2 O) are compared Table 4. Bulk and Shear Moduli, Acoustic Velocities of Dry and Hydrous Wadsleyite Water (wt % H 2 O) B (GPa) G (GPa) V P (km s 1 ) V S (km s 1 ) V f (km s 1 ) r (g cm 3 ) 0 (0 GPa) (10 GPa) (20 GPa) (0 GPa) (10 GPa) (20 GPa) (0 GPa) (10 GPa) (20 GPa) of13

10 Figure 11. Three single-crystal wave velocities (one P and two S waves) of dry (red) and 3.3 wt % H 2 O (thick blue) wadsleyite at 0, 10, and 20 GPa. Black full lines, black broken lines, gray broken lines, and open dots are calculated or scanned from Kiefer et al. [2001], Zha et al. [1997], Mao et al. [2008], and Sawamoto et al. [1984], respectively. with the seismological velocity models (PREM [Dziewonski and Anderson, 1981] and iasp91 [Kennett and Engdahl, 1991]). In Figure 13a, the elastic velocities of wadsleyite vary within the shaded areas depending on the water content. Calculated velocities of hydrous wadsleyite are primarily responsible to the seismic velocities in particular at the upper part of the transition zone. However, we of course need to consider the effects of coexistent mineral phases, the thermoelastic effects on dry and hydrous wadsleyite under highpressure conditions, and also the existence of other elements such as Fe for more quantitative discussion. [33] Since water decreases shear modulus G more than bulk modulus B, the calculated V P /V S ratio of hydrous wadsleyites containing 3.3 wt % H 2 O is about 1 2% higher than anhydrous one at the transition zone conditions (Figure 13b). Contrary to the calculated V P /V S ratios of dry and hydrous wadsleyite, those of the velocity models decrease with increasing pressure within the transition zone. Such negative slope of V P /V S ratio was not able to obtain even considering gradual decrease of water content from 3.3 to 0 wt % in wadsleyite with depth at the upper part of transition zone (gray arrow in Figure 13b). Therefore, Figure 12. (a) Single-crystal azimuthal anisotropy and (b) polarization anisotropy for transversely isotropic aggregate of dry and hydrous wadsleyite). Figure 13. (a) Calculated acoustic velocities (V P, and V S ) and (b) V P /V S ratio of dry and hydrous wadsleyite along adiabatic geotherm (1873 K at 660 km depth) using (@B S P = GPa K 1 and (@G/@T) P = GPa K 1 [Isaak et al., 2007]. Gray arrow indicates V P /V S ratio when water content of wadsleyite decreases from 3.3 to 0 wt % H 2 O with depth in the upper part of the transition zone. 10 of 13

11 Table 5. Calculated Perturbation Ratios Produced by Water Anomaly in Wadsleyite Pressure (GPa) R S/P R f/s R r/s other effects such as Fe, and other mineral phases might affect the V P /V S ratio in the transition zone. [34] The calculated perturbation ratios R S/P S V P, R f/s V f /@ln V S, and R r/s r/@ln V S produced by hydrous wadsleyites are shown in Table 5. These values seem to fall into the uncertainties of seismic observations and geodynamic modeling (e.g., R S/P 1.6, R f/s 0.3 in the transition zone [Masters et al., 2000; Karato and Karki, 2001]). [35] The lateral variation of seismic velocities has recently been observed in the mantle transition zone by means of for example the seismic tomography technique [e.g., Fukao et al., 2001, 2004; Zhao, 2001]. The slower velocity is usually interpreted by high temperature but it can also be explained by high water concentration. Comparing with the temperature effects reported by Isaak et al. [2007], effects of 1 wt % H 2 OonBand G are approximately comparable to the temperature increases of about 430 K (0 GPa) 340 K (20 GPa), and 350 K (0 GPa) 290 K (20 GPa), respectively. [36] Some studies attempted to separately determine these temperature and water anomalies in the upper mantle and transition zone in conjunction with other data such as depth of mantle discontinuities [Blum and Shen, 2004; Suetsugu et al., 2006] and seismic attenuation [Shito and Shibutan, 2003]. Using assumed temperature and water effects on the slowness of V P and the depth of 660-km discontinuity, temperature anomalies and water contents were estimated as low as 500 to 700 K and wt % H 2 O in the transition zone beneath Philippine Sea, and 300 to 4600 K and wt % H 2 O in the transition zone beneath western Japan [Suetsugu et al., 2006]. These water concentrations could be higher if using our velocity dependence on water content (@V P /@C H2 O 0.13 to 0.15 km s 1 wt % H 2 O 1 ) because it was significantly smaller than the parameter ( 0.2 km s 1 wt % H 2 O 1 ) which was determined on the basis of the experiments at ambient condition [Suetsugu et al., 2006]. [37] Polarization anisotropy V SH > V SV has been observed in the mantle transition zone beneath northern Australia and Fiji-Tonga [Tong et al., 1994; Chen and Brudzinski, 2003]. Fouch and Fischer [1996] have attributed the intermittent polarization anisotropy observable in the upper part of the transition zone beneath southern Kurils, Japan, Izu-Bonin, and Tonga to the lattice preferred orientation of wadsleyite developed under shear associated with the slab subduction. Our calculated polarization anisotropy indicates that the observed polarization anisotropy can be explained by the transversely anisotropic aggregate of both dry and hydrous wadsleyite with the vertical c principal axis. Since dry and hydrous wadsleyites show relatively similar azimuthal and polarization anisotropy, the difference of the rheological properties such as crystal-preferred orientation [e.g., Tommasi et al., 2004] between dry and hydrous wadsleyite may hold the key to whether hydrous wadsleyite is detectable by assessing seismic anisotropy. 5. Conclusions [38] We have determined the model crystal structures of hydrous wadsleyite on the basis of the density functional first principles calculation methods. We found several kinds of (meta)stable protonation sites with M2 and M3 vacancies, but the M3 vacancy model is more energetically favorable than the M2 vacancy model. We also conclusively confirmed that the electrostatically unstable O1 site is the major host of proton. The value 3.3 wt % H 2 O, which is thought to be the maximum solubility of water in wadsleyite, is achieved when all the O1 sites are protonated. The calculated structural changes due to the hydration are quite consistent with experimental results. The calculated bulk and shear moduli decrease almost linearly with increasing the water content while increase linearly with increasing pressure. At 15 GPa, the differences in V P and V S of dry and 3.3 wt % H 2 O phases were calculated to be 3.9 and 4.8%, respectively. Thus, concentration of 1wt % H 2 Oand Khightemperature anomaly both have similar contribution to the elastic moduli of wadsleyite. We also found that the transversely anisotropic aggregate of dry and hydrous wadsleyite with the vertical c principal axis shows polarization anisotropy V SH > V SV as observed beneath several places of circum Pacific. [39] Acknowledgments. 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