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Bridging the water solubility and ion diffusivity in the mantle silicates by a thermodynamic model

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Abstract

Water solubility in Mg-endmember silicates is suggested to correlate with cation diffusivity. This study proposes a thermodynamic model by introducing the concept of transient Frenkel defect for Mg ions to quantify the relationship between water solubility and Mg diffusivity in mantle silicates. The formation of the Mg transient Frenkel defect is regarded as the escape of Mg from Mg-O polyhedron, and the concentration of Mg transient vacancy is considered as its statistical probability in silicate minerals. The proposed thermodynamic model reveals that the formation energy for Mg transient vacancy (ΔGtr) is approximately ΔG/6 (ΔG is activation energy for Mg diffusion in silicates) for silicates with Mg-O6 octahedra, such as enstatite, forsterite, wadsleyite, and ringwoodite, and 5ΔG/12 for bridgmanite and post-perovskite. This observation implies a possible diffusion mode of Mg through transient vacancies. Our model connects water solubility and ion diffusivity in silicates.

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References

  • Ammann MW, Brodholt JP, Wookey J, Dobson DP (2010) First-principles constraints on diffusion in lower-mantle minerals and a weak D″ layer. Nature 465:462–465

    Google Scholar 

  • Bali E, Bolfan-Casanova N, Koga KT (2008) Pressure and temperature dependence of H solubility in forsterite: an implication to water activity in the earth interior. Earth Planet Sci Lett 268(3–4):354–363

    Google Scholar 

  • Berry AJ, Hermann J, O'Neill HSC (2004) The water site in mantle olivine. Geochim Cosmochim Acta 68:A36

    Google Scholar 

  • Berry AJ, Hermann J, O'Neill HSC, Foran GJ (2005) Fingerprinting the water site in mantle olivine. Geology 33:869–872

    Google Scholar 

  • Berry AJ, O'Neill HSC, Hermann J, Scott DR (2007a) The infrared signature of water associated with trivalent cations in olivine. Earth Planet Sci Lett 261:134–142

    Google Scholar 

  • Berry AJ, Walker AM, Hermann J, O'Neill HSC, Fordan GJ, Gale JD (2007b) Titanium substitution mechanisms in forsterite. Chem Geol 242:176–186

    Google Scholar 

  • Béjina F, Blanchard M, Wright K, Price GD (2009) A computer simulation study of the effect of pressure on mg diffusion in forsterite. Phys Earth Planet Inter 172:13–19

    Google Scholar 

  • Bolfan-Casanova N (2005) Water in the Earth’s mantle. Mineral Mag 69(3):229–257

    Google Scholar 

  • Brodholt J (1997) Ab initio calculations on point defects in forsterite (Mg2SiO4) and implications for diffusion and creep. Am Miner 82:1049–1053

    Google Scholar 

  • Caracas R, Panero WR (2017) Hydrogen mobility in transition zone silicates. Prog in Earth Planet Sci 4:9

    Google Scholar 

  • Chakraborty S, Farver JR, Yund RA, Rubie DC (1994) Mg tracer diffusion in synthetic forsterite and San Carlos olivine as a function of P, T and fO2. Phys Chem Miner 21:489–500

    Google Scholar 

  • Chakraborty S, Knoche R, Schulze H, Rubie DC, Dobson D, Ross NL, Angel RJ (1999) Enhancement of cation diffusion rates across the 410-kilometer discontinuity in Earth's mantle. Science 283:362–365

    Google Scholar 

  • Costa F, Chakraborty S (2008) The effect of water on Si and O diffusion rates in olivine and implications for transport properties and processes in the upper mantle. Phys Earth Planet Inter 166:11–29

    Google Scholar 

  • Demouchy S, Mackwell S (2003) Water diffusion in synthetic iron-free forsterite. Phys Chem Miner 30:486–494

    Google Scholar 

  • Demouchy S, Deloule E, Frost DJ, Keppler H (2005) Pressure and temperature dependence of the water solubility in Fe-free wadsleyite. Am Miner 90(7):1084–1091

    Google Scholar 

  • Demouchy S, Mackwell S (2006) Mechanisms of hydrogen incorporation and diffusion in iron-bearing olivine. Phys Chem Miner 33:347–355

    Google Scholar 

  • Demouchy S, Mackwell S, Kohlstedt DL (2007) Influence of hydrogen on Fe-Mg interdiffusion in (Mg,Fe)O and implications for Earth's lower mantle. Contrib Mineral Petrol 154:279–289

    Google Scholar 

  • Demouchy S, Tommasi A, Boffa-Ballaran T, Cordier P (2013) Low strength of Earth's uppermost mantle inferred from tri-axial deformation experiments on dry olivine crystals. Phys Earth Planet Inter 220:37–49

    Google Scholar 

  • Du Frane WL, Tyburczy JA (2012) Deuterium-hydrogen exchange in olivine: implications for point defects and electricity conductivity. Geochem Geophys Geosyst 13(3):Q03004

    Google Scholar 

  • Fair RB, Wortman JJ, Liu J (1984) Modeling rapid thermal diffusion of arsenic and boron in silicon. J Electrochem Soc 131(10):1387–2394

    Google Scholar 

  • Farber DL, Williams Q, Ryerson FJ (2000) Divalent cation diffusion in Mg2SiO4 spinel (ringwoodite), β phase (wadsleyite), and olivine: implications for the electrical conductivity of the mantle. J Geophys Res 105:513–529

    Google Scholar 

  • Faul UH, Cline CJ II, David EC, Berry AJ, Jackson I (2016) Titanium-hydroxyl defect-controlled rheology of the Earth's upper mantle. Earth Planet Sci Lett 452:227–237

    Google Scholar 

  • Fei H, Koizumi S, Sakamoto N, Hashiguchi M, Yurimoto H, Marquardt K, Miyajima N, Katsura T (2018) Mg lattice diffusion in iron-free olivine and implications to conductivity anomaly in the oceanic asthenosphere. Earth Planet Sci Lett 484:204–212

    Google Scholar 

  • Fei H, Wiedenbeck M, Yamazaki D, Katsura T (2013) Small effect of water on upper-mantle rheology based on silicon self-diffusion coefficients. Nature 498:213–216

    Google Scholar 

  • Fei H, Wiedenbeck M, Yamazaki D, Katsura T (2014) No effect of water on oxygen self-diffusion rate in forsterite. J Geophys Res 119:7598–7606

    Google Scholar 

  • Férot A, Bolfan-Casanova N (2012) Water storage capacity in olivine and pyroxene to 14 GPa: implications for the water content of the Earth's upper mantle and nature of seismic discontinuities. Earth Planet Sci Lett 349-350:218–230

    Google Scholar 

  • Griffin JM, Berry AJ, Frost DJ, Wimperis S, Ashbrook SE (2013) Water in the Earth's mantle: a solid-state NMR study of hydrous wadsleyite. Chem Sci 4:1523–1538

    Google Scholar 

  • Grüninger H, Armstrong K, Greim D, Boffa-Ballaran T, Frost DJ, Senker J (2017) Hidden oceans? Unraveling the structure of hydrous defects in the Earth's deep interior. J Am Chem Soc 139:10499–10505

    Google Scholar 

  • Grüninger H, Schmutzler A, Siegel R, Armstrong K, Frost DJ, Senker J (2018) Quantitative description of 1H SQ and DQ coherences for the hydroxyl disorder within hydrous ringwoodite. Phys Chem Chem Phys 20:15098

    Google Scholar 

  • Hae R, Ohtani E, Kubo T, Koyama T, Utada H (2006) Hydrogen diffusivity in wadsleyite and water distribution in the mantle transition zone. Earth Planet Sci Lett 243:141–148

    Google Scholar 

  • Hier-Majumder S, Anderson IM, Kohlstedt DL (2005) Influence of protons on Fe-mg interdiffusion in olivine. J Geophys Res 110:B02202

    Google Scholar 

  • Hirschmann MM, Aubaud C, Withers AC (2005) Storage capacity of H2O in nominally anhydrous minerals in the upper mantle. Earth Planet Sci Lett 236:167–181

    Google Scholar 

  • Hirth G, Kohlstedt DL (1996) Water in the oceanic upper mantle: implications for rheology, melt extraction and the evolution of the lithosphere. Earth Planet Sci Lett 144:93

    Google Scholar 

  • Holl CM, Smyth JR, Jacobsen SD, Frost DJ (2008) Effects of hydration on the structure and compressibility of wadsleyite, β-(Mg2SiO4). Am Miner 93:598–607

    Google Scholar 

  • Holzapfel C, Chakraborty S, Rubie DC, Frost DJ (2009) Fe-mg interdiffusion in wadsleyite: the role of pressure, temperature and composition and the magnitude of jump in diffusion rates at the 410 km discontinuity. Phys Earth Planet Inter 172:28–33

    Google Scholar 

  • Ingrin J, Blanchard M (2006) Diffusion of hydrogen in minerals. Rev Mineral Geochem 62:291–230

    Google Scholar 

  • Jacobsen SD, Demouchy S, Frost DJ, Ballaran TB, Kung J (2005) A systematic study of OH in hydrous wadsleyite from polarized FTIR spectroscopy and single-crystal X-ray diffraction: oxygen sites for hydrogen storage in Earth's interior. Am Miner 90:61–70

    Google Scholar 

  • Jaoul O, Béjina F, Élie F, Abel F (1995) Silicon self-diffusion in quartz. Phys Rev Lett 74(11):2038–2041

    Google Scholar 

  • Jollands MC, Padrón-Navarta JA, Hermann J, O'Neill HSC (2016) Hydrogen diffusion in Ti-doped forsterite and the preservation of metastable point defects. Am Miner 101:1571–1583

    Google Scholar 

  • Karki BB, Khanduja G (2007) A computational study of ionic vacancies and diffusion in MgSiO3 perovskite and post-perovskite. Earth Planet Sci Lett 260:201–211

    Google Scholar 

  • Karato S (2008) Deformation of earth materials: an introduction to the rheology of solid earth. Cambridge University Press, New York

    Google Scholar 

  • Karato S, Jung H (1998) Water, partial melting and the origin of the seismic low velocity and high attenuation zone in the upper mantle. Earth Planet Sci Lett 157(3–4):193–207

    Google Scholar 

  • Kawamoto T, Hervig RH, Holloway JR (1996) Experimental evidence for a hydrous transition zone in the early Earth's mantle. Earth Planet Sci Lett 142:587–592

    Google Scholar 

  • Keppler H, Bolfan-Casanova N (2006) Thermodynamics of water solubility and partitioning. Rev Mineral Geochem 62(1):193–230

    Google Scholar 

  • Kohlstedt DL, Keppler H, Rubie DC (1996) Solubility of water in the in the α, β and γ phases of (mg, Fe)2SiO4. Contrib Mineral Petrol 123:345–357

    Google Scholar 

  • Kohlstedt DL, Mackwell S (1998) Diffusion of hydrogen and intrinsic point defects in olivine. Z Phys Chem 207:147–162

    Google Scholar 

  • Kubo T, Shimojuku A, Ohtani E (2004) Mg–Fe interdiffusion rates in wadsleyite and the diffusivity jump at the 410-km discontinuity. Phys Chem Miner 31(7):456–464

    Google Scholar 

  • Kudo T, Ohtani E, Hae R, Shimojuku R (2006) Hydrogen diffusion in ringwoodite. Abstract of 2006 Annual Meeting of Japan Geoscience Union [in Japanese]. I143-P021, Chiba, May 14–18

  • Kudoh Y, Kuribayashi T, Mizobata H, Ohtani E (2000) Structure and cation disorder of hydrous ringwoodite, γ-Mg1.89Si0.98H0.30O4. Phys Chem Miner 27:474–479

    Google Scholar 

  • Lemaire C, Kohn SC, Brooker RA (2004) The effect of silica activity on the incorporation mechanisms of water in synthetic forsterite: a polarised infrared spectroscopic study. Contrib Mineral Petrol 147(1):48–57

    Google Scholar 

  • Litasov KD, Ohtani E, Kagi H, Jacobsen SD, Ghosh S (2007) Temperature dependence and mechanism of hydrogen incorporation in olivine at 12.5–14.0 GPa. Geophys Res Lett 34(16):130–144

    Google Scholar 

  • Litasov KD, Ohtani E, Langenhorst F, Yurimoto H, Kubo T, Kondo T (2003) Water solubility in mg-perovskites and water storage capacity in the lower mantle. Earth Planet Sci Lett 211:189–203

    Google Scholar 

  • Matveev S, O’neill HSC, Ballhaus C, Taylor WR, Green DH (2001) Effect of silica activity on OH−IR spectra of olivine: implications for low-α SiO2 mantle metasomatism. J Petrol 42(4):721–729

    Google Scholar 

  • Mei S, Kohlstedt DL (2000) Influence of water on plastic deformation of olivine aggregates 1. Diffusion creep regime. J Geophy Res 105(B9):21457–21469

    Google Scholar 

  • Mibe K, Kanzaki M, Kawamoto T, Matsukage KN, Fei Y, Ono S (2007) Second critical endpoint in the peridotite-H2O system. J Geophys Res 112:B3201

    Google Scholar 

  • Mierdel K, Keppler H (2004) The temperature dependence of water solubility in enstatite. Contrib Mineral Petrol 148(3):305–311

    Google Scholar 

  • Mierdel K, Keppler H, Smyth JR (2007) Water solubility in aluminous orthopyroxene and the origin of the asthenosphere. Science 315:364–368

    Google Scholar 

  • Mosenfelder JL, Deligne NI, Asimow PD, Rossman GR (2006) Hydrogen incorporation in olivine from 2-12 GPa. Am Miner 91:285–294

    Google Scholar 

  • Murakami M, Hirose K, Kawamura K, Sata N, Ohishi Y (2004) Post-perovskite phase transition in MgSiO3. Science 304:855–858

    Google Scholar 

  • Nomura R, Hirose K, Uesugi K, Ohishi Y, Tsuchiyama A, Miyake A, Ueno Y (2014) Low core-mantle boundary temperature inferred from the solidus of pyrolite. Science 343(6170):522–525

    Google Scholar 

  • Ohtani E, Mizobata H, Yurimoto H (2000) Stability of dense hydrous magnesium silicate phases in the systems Mg2SiO4-H2O and MgSiO3-H2O at pressures up to 27 GPa. Phys Chem Miner 27(8):533–544

    Google Scholar 

  • Padrón-Navarta JA, Hermann J, O'Neill HSC (2014) Site-specific hydrogen diffusion rates in forsterite. Earth Planet Sci Lett 392:100–112

    Google Scholar 

  • Padrón-Navarta JA, Hermann J (2017) A sub-solidus olivine water solubility equation for the Earth's upper mantle. J Geophys Res 122:9862–9880

    Google Scholar 

  • Panero WR (2010) First principles determination of the structure and elasticity of hydrous ringwoodite. J Geophy Res 115:B03203

    Google Scholar 

  • Paterson MS (1982) The determination of hydroxyl by infrared absorption in quartz, silicate glasses and similar materials. Bull Mineral 105:20–29

    Google Scholar 

  • Poirier JP (1985) Creep of crystals: high-temperature deformation processes in metals, ceramics and minerals. Cambridge University Press, New York

    Google Scholar 

  • Qin T, Wentzcovitch RM, Umemoto K, Hirschmann MM, Kohlstedt D (2018) Ab initio study of water speciation in forsterite: importance of the entropic effect. Am Miner 103:692–699

    Google Scholar 

  • Rauch M, Keppler H (2002) Water solubility in orthopyroxene. Contrib Mineral Petrol 143(5):525–536

    Google Scholar 

  • Schwandt CS, Cygan RT, Westrich HR (1998) Magnesium self-diffusion in orthoenstatite. Contrib Mineral Petrol 130:390–396

    Google Scholar 

  • Shimojuku A, Kubo T, Ohtani E, Nakamura T, Okazaki R (2010) Effects of hydrogen and iron on the silicon diffusivity of wadsleyite. Phys Earth Planet Inter 183:175–182

    Google Scholar 

  • Smyth JR (1987) β-Mg2SiO4: a potential host for water in the mantle. Am Mineral 72:1051–1055

    Google Scholar 

  • Smyth JR, Frost DJ, Nestola F, Holl CM, Bromiley G (2006) Olivine hydration in the deep upper mantle: effects of temperature and silica activity. Geophys Res Lett 33(15):311–324

    Google Scholar 

  • Smyth JR, Holl CM, Frost DJ (2003) Structural systematic of hydrous ringwoodite and water in Earth’s interior. Am Miner 88:1402–1407

    Google Scholar 

  • Stalder R, Skogby H (2003) Hydrogen diffusion in natural and synthetic orthopyroxene. Phys Chem Miner 30(1):12–19

    Google Scholar 

  • Stalder R, Ulmer P, Thompson A, Günther D (2001) High pressure fluids in the system MgO–SiO2–H2O under upper mantle conditions. Contrib Mineral Petrol 140(5):607–618

    Google Scholar 

  • Sun W, Yoshino T, Sakamoto N, Yurimoto H (2015) Hydrogen self-diffusivity in single crystal ringwoodite: implications for water content and distribution in the mantle transition zone. Geophys Res Lett 42:6582–6589

    Google Scholar 

  • Tollan PME, Smith R, O'Neill HSC, Hermann J (2017) The responses of the four main substitutioin mechanisms of H in olivine to H2O activity at 1050 °C and 3 GPa. Prog Earth Planet Sci 4:14

    Google Scholar 

  • Townsend JP, Tsuchiya J, Bina CR, Jacobsen SD (2016) Water partitioning between bridgmanite and post-perovskite in the lowermost mantle. Earth Planet Sci Lett 454:20–27

    Google Scholar 

  • Varotsos PA, Alexopoulos KD (1977) Calculation of the formation entropy of vacancies due to anharmonic effects. Phys Rev B 15:4111

    Google Scholar 

  • Varotsos PA, Alexopoulos KD (1980) Calculation of diffusion coefficients at any temperature and pressure from a single measurement. I Self diffusion Phys Rev B 22:3130

    Google Scholar 

  • Varotsos PA, Alexopoulos KD (1986) Thermodynamics of point defects and their relation with bulk properties. North-Holland, New York

  • Verma A, Karki BB (2009) Ab initio investigations of the native and protonic point defects in Mg2SiO4 polymorphs under high pressure. Earth Planet Sci Lett 285:140–149

    Google Scholar 

  • Walker AM, Hermann J, Berry AJ, O'Neill HSC (2007) Three water sites in upper mantle olivine and the role of titanium in the water weakening mechanism. J Geophys Res 112:B05211

    Google Scholar 

  • Walker AM, Woodley SM, Slater B, Wright K (2009) A computational study of magnesium point defects and diffusion in forsterite. Phys Earth Planet Inter 172:20–27

    Google Scholar 

  • Walker AM, Wright K, Slater B (2003) A computational study of oxygen diffusion in olivine. Phys Chem Miner 30:536–545

    Google Scholar 

  • Wang D, Mookherjee M, Xu Y, Karato S (2006) The effect of water on the electrical conductivity of olivine. Nature 443:977–980

    Google Scholar 

  • Wang Z, Hiraga T, Kohlstedt DL (2004) Effect of H+ on Fe-Mg interdiffusion in olivine, (Fe,Mg)2SiO4. Appl Phys Lett 85:209–211

    Google Scholar 

  • Watson GW, Wall A, Parker SC (2000) Atomistic simulation of the effect of temperature and pressure on point defect formation in MgSiO3 perovskite and the stability of CaSiO3. J Phys Condens Matter 12:8427–8438

    Google Scholar 

  • Withers AC, Hirschmann MM (2007) H2O storage capacity of MgSiO3 clinoenstatite at 8–13 GPa, 1,100–1,400 °C. Contrib Mineral Petrol 154(6):663–674

    Google Scholar 

  • Wright K (2006) Atomistic models of OH defects in nominally anhydrous minerals. Rev Miner Geochem 62:67–83

    Google Scholar 

  • Wright K, Price GD (1993) Computer simulations of defects and diffusion in perovskites. J Geophys Res 98:22245–22253

    Google Scholar 

  • Xu J, Yamazaki D, Katsura T, Wu X, Remmert P, Yurimoto H, Chakraborty S (2011) Silicon and magnesium diffusion in a single crystal of MgSiO3 perovskite. J Geophys Res 116:B12205

    Google Scholar 

  • Xu L, Mei S, Dixon N, Jin Z, Suzuki AM, Kohlstedt DL (2013) Effect of water on rheological properties of garnet at high temperatures and pressures. Earth Planet Sci Lett 379:158–165

    Google Scholar 

  • Xue X, Kanzaki M, Turner D, Loroch D (2017) Hydrogen incorporation mechanisms in forsterite: new insights from 1H and 29Si NMR spectroscopy and first-principles calculation. Am Miner 102:519–536

    Google Scholar 

  • Yang X (2015) OH solubility in olivine in the peridotite-COH system under reducing conditions and implications for water storage and hydrous melting in the reducing upper mantle. Earth Planet Sci Lett 432:199–209

    Google Scholar 

  • Yang X (2016) Effect of oxygen fugacity on OH dissolution in olivine under peridotite-saturated conditions: an experimental study at 1.5-7 GPa and 1100-1300 °C. Geochim Cosmochim Acta 173:319–336

    Google Scholar 

  • Yang X, Keppler H (2011) In-situ infrared spectra of OH in olivine to 1100 °C. Am Mineral 96:451–454

    Google Scholar 

  • Yang X, Keppler H, Li Y (2016) Molecular hydrogen in mantle minerals. Geochim Persp Lett 2:160–168

    Google Scholar 

  • Yang X, Liu D, Xia QK (2014) CO2-induced small water solubility in olivine and implications for properties of the shallow mantle. Earth Planet Sci Lett 403:37–47

    Google Scholar 

  • Yoshino T, Matsuzaki T, Yamashita S (2006) Hydrous olivine unable to account for conductivity anomaly at the top of the asthenosphere. Nature 443:973–976

    Google Scholar 

  • Zhang B, Shan S (2015) Thermodynamic calculations of Fe-mg interdiffusion in (mg, Fe)2SiO4 polymorphs and perovskite. J Appl Phys 117:054906

    Google Scholar 

  • Zhang B, Wu X, Xu J, Zhou R (2010) Application of the cBΩ model for the calculation of oxygen self-diffusion coefficients in minerals. J Appl Phys 108:053505

    Google Scholar 

  • Zhang B, Yoshino T, Wu X, Matsuzaki T, Shan S, Katsura T (2012) Electrical conductivity of enstatite as a function of water content: implications for the electrical structure in the upper mantle. Earth Planet Sci Lett 357-358:11–20

    Google Scholar 

  • Zhang B, Yoshino T, Zhao C (2019) The effect of water on Fe-mg interdiffusion rates in ringwoodite and implications for the electrical conductivity in the mantle transition zone. J Geophys Res 124:2510–2524

    Google Scholar 

  • Zhao YH, Ginsberg SB, Kohlstedt DL (2004) Solubility of hydrogen in olivine: dependence on temperature and iron content. Contrib Mineral Petrol 147:155

    Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 41304062, 41773056 and 41303048) and the Institute of Crustal Dynamics, China Earthquake Administration (Grant No. ZDJ2012-21). We thank editor Prof. Broekmans and two anonymous reviewers for their constructive comments that greatly improve the manuscript.

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  1. Key Laboratory of Crustal Dynamics, Institute of Crustal Dynamics, China Earthquake Administration, Beijing, 100085, China

    Junshan Xu

  2. Key Laboratory for High-Temperature and High-Pressure Study of the Earth’s Interior, Institute of Geochemistry, Chinese Academy of Sciences, Guiyang, 550081, Guizhou, China

    Baohua Zhang

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Xu, J., Zhang, B. Bridging the water solubility and ion diffusivity in the mantle silicates by a thermodynamic model. Miner Petrol 114, 1–13 (2020). https://doi.org/10.1007/s00710-019-00685-9

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