Constraints on Subduction Zone Fluid Compositions Based on Mineral Solubility and Dielectric Constant of H 2 O

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1 Running Title: Subduction Zone Fluid Constraints Constraints on Subduction Zone Fluid Compositions Based on Mineral Solubility and Dielectric Constant of H 2 O Jeffrey B. Creamer * Department of Earth Science University of California, Santa Barbara Frank J. Spera Department of Earth Science, University of California, Santa Barbara Institute for Crustal Studies, University of California, Santa Barbara * Corresponding author: jcreamer@umail.ucsb.edu 1

2 Abstract A method is developed to estimate the solute content of aqueous fluids in subduction zones combining the Born solvation model with laboratory solubility data including a model for the temperature-pressure variation of the relative permittivity (dielectric constant) of H 2 O. A molecular model for the dielectric constant of H 2 O for densities to 1400 kg/m 3 is calibrated using existing measurements, the dipole moment and water molecule diameter. Computed values are reported for the P-T range 0.5 to 4.5 GPa and 400 to 1400 o C. Solubility relations for SiO 2 -H 2 O and Mg 2 SiO 4 -CaMgSi 2 O 6 - H 2 O are mapped onto hot and cold subduction zone models. At depths relevant to magma genesis, the solubility of common phases at equivalent positions within hot and cold slabs can differ by <1 to >3 orders of magnitude. This impacts consideration of trace element redistribution via metasomatic fluid migration in the mantle wedge and the composition of subduction zone primary melts. 2

3 Introduction Aqueous fluids are ubiquitous in subduction zones and are critical for both magma generation and mantle metasomatism and hence the geochemical evolution of the Earth s upper mantle. Fluids transferred from the slab to the mantle wedge traverse strong lateral temperature and bulk composition gradients and are thus expected to carry a wide diversity and concentration of solutes (i.e. Kessel et al 2005). There are several approaches to constrain fluid solute content in subduction zones: analysis of fluid inclusions from exhumed slabs (i.e. Scambelluri and Philippot 2001); inverse modeling of arc basalts to constrain compositions of source components (including slab-derived fluids), and experimental solubility studies on simple (i.e. SiO 2 ; Manning 1994) or multicomponent (i.e. H 2 O-CO 2 -basalt; Holloway 1971) systems. Existing mineral solubility studies cover an incomplete patchwork of temperature, pressure and composition variations encountered in subduction zones. Thus, some basis for interpolation and extrapolation of mineral solubility is desired. The goal here is to assess the solute content of supercritical subduction zone fluids to conditions of pressures, P in range GPa and temperatures, T from o C relevant to hydrous phase dehydration and magma generation. The approach used in this study is the establishment of an empirical solubility scaling law based on identity of the solid phase, temperature T, and the relative permittivity or dielectric constant, ε(ρ,t), of water, the predominant species in subduction zone fluids Dielectric Constant Model 3

4 A fundamental parameter governing mineral solubility in a dipolar fluid is its dielectric constant. Fluids of high dielectric constant are good solvents because the dielectric fluid reduces the electrostatic forces holding the crystal together. At 20 o C and 1 bar the dielectric constant of water is 83 whereas at 500 C and 0.5 GPa, ε 19 (Heger et al 1980). Although increasing temperature decreases the solvent abilities of H 2 O, pressure has just the opposite effect. Using the results of this study, at 1000 C ε increases by a factor of three as pressure increases from 0.5 to 4.5 GPa. A useful functional form to scale mineral solubility based on the law of mass action and the Born solvation model is: [1] log m = A + 1 # T B+ C & % ( $ "( p,t) ' where m is the molality of the solute, T is temperature, ε is the dielectric constant and A, B, and C are empirically determined constants derived from mineral solubilities. Model equations for fluid dielectric constants are typically expressed as functions of fluid temperature and density rather than pressure. Since mineral solubility studies often report fluid pressure rather than density, conversion using an equation of state (EOS) for water is required. Pitzer and Sterner (1994) present an EOS for water valid to 10 GPa and 10,000 K. A calibrated model for the dielectric constant of water based on experimental data has previously been generated by Franck et al (1990) using existing experimental dielectric constant data (Heger et al 1980; Deul 1984). However, the expression utilized used by Franck et al (1990) reaches a numerical instability at water densities >1250 kg/m 3, restricting its geochemical application. As an alternative, a model for the dielectric constant of a generic fluid developed by Tani et al (1983) and modified by Valisko et al (2001) is not subject to numerical instabilities at high pressures 4

5 encountered in subduction zones, and is calibrated here for water following the procedure of Franck et al (see Fig. 1). Experimental data for the dielectric constant at subcritical conditions (<374 o C) are omitted from consideration. Molecular Dynamics simulations of the dielectric constant at high pressure and temperature (Wasserman et al 1995) are also omitted because they fail to conform to the behavior predicted by the Valisko et al 2001 model, even under conditions where they overlap with experimental data. This suggests that the rigid molecule potential used in the MD simulations of Wasserman et al is an inadequate representation of the orientational disorder of the water dipole in the pressure-temperature range relevant to subduction metasomatism. The molecular model used to predict the dielectric constant of water described by Valsiko et al (2001) approximates a generic dipolar fluid as a collection of spherically symmetric molecules with an imbedded dipole moment. The dielectric constant is related to the properties of the fluid according to + % 9I [2] "(p,t) = n 2 1+ f (n) y + g(n) y 2 + f (n)' & 16# $h(n) (. - * y 4, 2 0 ) / where y = 4πρµ 2 /9kT, and k is the Boltzmann constant, T is temperature (K), ρ = N/V is the number-density of water molecules, N is the number of water molecules in volume V, µ is the magnitude of the water dipole moment and I=(17π 2 /9)*( ρ* ρ* 2 )/( ρ* ρ* 2 ) where ρ* = ρσ 3 and σ is the molecular diameter. The refractive index functions f(n), g(n), and h(n) are defined f(n) = (n 2 +2)/(2n 2 +1), g(n) = (n 2 +2) 4 /(2n+1) 3 and h(n)= (2n 2-1) (n 2 +2) 4 /(2n 2 +1) 4 respectively and are functions of temperature and fluid density (Schiebener et al 1990). Fluid temperature is input from experimental data, and fluid mass-density is calculated from experimental temperature and pressure using the EOS from Pitzer and Sterner The 5

6 diameter of the water molecule (σ) and magnitude of the dipole moment (µ) are independent parameters, and were chosen to minimize residuals for experimental data for the dielectric constant of water. Here, σ=2.3 x m and µ=7.19 x Coloumbmeters were chosen to optimize experimental data. We note that the model equation developed by Tani et al 1983, which does not incorporate the fluid refractive index but is otherwise identical to the equation of Valsiko et al 2001, does not represent experimental data well. In distinction, by allowing for the density and temperature dependence of the index of refraction of water the fit is significantly improved and allows extrapolation. Interpolated and extrapolated values for the dielectric constant of water in the range 400 to 1400 o C and 0.5 to 4.5 GPa are reported in Table 1 and serve as provisional values until further laboratory measurements are made Solubility Scaling Five different mineral solubility datasets were evaluated to establish the utility of Eq. [1] (Fig. 2). These data are limited to solubility in pure H 2 O and one to three coexisting crystalline phases. Despite the significant range in experimental pressure (0.005 to 2.3 GPa), temperature (100 to 1000 o C), solute content (dilute to >12 wt % total solute), and a diversity of bulk compositions, a consistent relationship between log(m) and 1/T. [B+C/ε] exists for all observed conditions. The extrapolation of these scaling relationships should be limited to the stability field of the mineral species and by the second critical endpoint of the mineral-water systems (i.e. Keppler, 1996; Kessel et al. 2005). For example, the solubility of quartz is extended to regions of P-T space where coesite is the stable polymorph. We are not aware of experimental coesite solubility data. 6

7 Application to Subduction Zones The solubility scaling relationships for simple mineral-water systems can be applied to thermal models for subduction zones to gain insight into the range of solute concentrations both within an individual subduction zone and between different (i.e. hot and cold ) subduction zones. Because of extreme lateral gradients in temperature at fixed depth in and around a subducting slab, very strong mineral solubility gradients are anticipated. This expectation is indeed correct and, to our knowledge, has not been addressed previously. The thermal models analyzed here were developed by van Keken et al (2002) to include temperature and strain-rate dependent mantle viscosity, and are calibrated to approximate natural subduction zones based on observed or inferred parameters such as convergence rate, slab geometry, and forearc heatflow. The Cascadia and NE Japan subduction zone models were selected since they approximate global extremes for minimum and maximum penetration of isotherms into the mantle. Solubilities for the quartz and forsterite-diopside systems within the upper 10 km of subducting lithosphere and adjoining 10 km of mantle wedge are portrayed in Figure 3. Due to steep lateral thermal gradients, there are strong solubility gradients along hypothesized transport paths for slab-derived fluids migrating into the mantle wedge. As an extreme case, at a fixed depth of 100 km the solubility of rutile is 1000 times greater at the slab-mantle interface than at a position 3 km within the cold subducting lithosphere. Diopside-forsterite solubility is 10 to 100 times greater in hot than cold subduction zones at most equivalent positions within the slab for most depths < 100km. For example, the molality (moles of solute/kg of fluid) of diopside-forsterite within the 10 km lateral 7

8 section of slab at 80 km depth ranges from -4.3 and -2 log units for the cold slab and -2.2 to -1.8 for the hot slab Discussion Available mineral solubility data relevant to the deep lithosphere, the mantle wedge and adjacent descending slabs cover temperature-pressure-composition space in a patchwork manner. Methods of extrapolating available data based on theoretical or semitheoretical formulations are therefore useful in assessing in the solute content of subduction zone fluids. The realization that Born theory correlates the solubility of oxide and silicate geomaterials at diverse state conditions, and from dilute solutions to solutions with >12 wt% solute, lends credence to the conclusion that very large solubility gradients exist near the region of strong temperature contrast near the slab-wedge interface. There are no state conditions where the relationship between solute content of the fluid (log m) and the inverses of temperature (T) and dielectric constant (ε) is not linear and strongly correlated (fig. 2), so extrapolation using the solubility scaling relationships explored here (eq. [1]) is reasonable. The electrostatic approach to mineral solubility has the advantage of bypassing a traditional pitfall experienced using solubility models based on the free energy of solvation (i.e. Zhang and Frantz 2000) in that the model is not dependant upon the speciation of dissolved material (Walther and Schott 1988). With the eventual goal of treating real, complex fluids, a model in which solubility can be expressed independent of aqueous speciation will be particularly advantageous. The approach to mineral solubility outlined here could be applied to complex fluids. There has been a recent proliferation of 8

9 experimental mineral solubility studies with fluids in the system H-O-C-N-S-Cl-F. Mixing rules for the dielectric constant of complex fluids are well established (i.e. Looyenga 1965), but the major inhibition is the lack of knowledge of the dielectric constant of volatile species other than H 2 O. Further experimental measurement or Molecular Dynamics simulations using flexible molecule dipolar potentials for mixed fluids for determination of the relative permittivity at elevated temperatures and pressures should improve upon the reliability of the molecular model for dielectric constant for water presented here. While the estimated absolute value of the dielectric constant at a given P,T condition would be subject to change with the availability of further experimental data, the magnitude of the contrast in mineral solubility across the slab-wedge interface is so great that predicted solubility contrasts between hot and cold subduction zones are extremely unlikely to disappear. If the solubility variability for simple mineral-water systems for both inter- and intra-slab scenarios is indeed even grossly representative of real systems, there are important implications. For example, the volumetric proportion of slab-derived refractory material that ends up in parental arc basalts should vary strongly as a function of the slab temperature conditions Conclusion The method developed here to estimate mineral solubility as a function of temperature and pressure using the Born solvation model calibrated from experimental solubility data and a molecular model for the relative permittivity of dipolar H 2 O yields surprisingly well-correlated values for a wide range of experimental conditions (pressure, 9

10 temperature, and solute content). The H 2 O molecular model suggests that the dielectric constant of water varies between ~ 4 and 50, in the range 0.5 to 4.5 GPa and 400 to 1400 o C. Solubility relations suggest differences in the solute-content of slab fluids between hot and cold subduction zones varies by a factor of <10 to >100 within the slab at 100 km depth. The accuracy of this analogy to more realistic phase assemblages and mixed fluids in subduction environments is uncertain, but study of more complex fluids and mineral assemblages is a promising avenue to pursue in the future Acknowledgements The authors wish to thank Peter van Keken for his consultations with thermal modeling. Support from the NSF (NSF/ITR ) is gratefully acknowledged

11 Figure 1. The dielectric constant of water using the model of Valisko et al 2001 extrapolated as a function of fluid pressure along isotherms, superimposed on experimental data. Data from 400 o C are depicted as diamonds, data from 500 o C as squares and data from 550 o C as triangles. Higher temperature experimental measurements for ε do not currently exist. 11

12 o C GPa Table 1. Dielectric constant of water at selected P,T conditions based on the model developed as part of this study. 12

13 Figure 2. Calibration of solubility scaling laws. Experimentally determined solubilities are used to calculate constants in the model equation log(m) = A + 1/T [B+ C/ε]. Experimental Pressure (P) and Temperature (T) are labeled for each experimental set. a: Quartz (Manning 1994), b: Corundum (Walther 1997), c: Rutile (Audetát and Keppler 2005), d: Diopside-Forsterite (Macris and Manning 2005), and e: Paragonite-Quartz- Albite (Woodland and Walther 1987). Note that Diopside solubility is slightly incongruent since a minor amount of forsterite was precipitated. 13

218 219 220 221 222 223 224 225 226 227 Figure 3.

14 Figure 3. The top 10 km section of subducting slab and adjacent 10 km section of mantle wedge, between depths of 50 and 125 km, are considered in this figure (a). Within this section of the subduction zone, temperature fields modeled by van Keken et al 2002 for hot and cold subduction zones are displayed (b). The solubility, log (m), scaling relationships developed in this study for quartz (c) and diopside (d) are superimposed on these sections. Solubility data come from Manning 1994 and Macris and Manning 2005, respectively. 14

15 References Audetát, A. and Keppler, H. (2005). Solubility of rutile in subduction zone fluids, as determined by experiments in the hydrothermal diamond anvil cell. Earth and Planetary Sci. Letters, 232, Dandurand, J. and Schott, J. (1992). Prediction of Ion Association In Mixed-Crustal Fluids. J. Phys. Chem., 96, Deul, R. (1984) Dielektrizitatskonstante und Dichte von Wasser-Benzol-Mischungen bis 450 o C und 3000 bar, Thesis, Inst. Of Phys. Chem., Univ. Karlsruhe. Franck, E., Rosenzweig, S., and Christoforakos, M. (1990). Calculation of the Dielectric Constant of Water to 1000 o C and Very High Pressures. Ber. Bunsenges. Phys. Chem., 94, Heger, K., Uematsu, M., amd Franck, E.U. (1980) The Static Dielectric Constant of Water at High Pressures and Temperatures to 500 MPa and 550 o C. Ber. Bunsenges. Phys. Chem., 84, Holloway, J.R. (1971). Composition of Fluid Phase Solutes in a Basalt-H 2 O-CO 2 System. Geol. Soc. Am., 82, Kessel, R., Ulmer, P., Pettke, T., Schimdt, M.W., and Thompson, A.B., (2005). The water-basalt system at 4 to 6 GPa: Phase relations and second critical endpoint in a K-free ecolgite at 700 to 1400 o C. Earth and Planetary Science Letters, 237, Looyenga, H. (1965) Dielectric constants of heterogeneous mixtures. Physica, 31, 3, Manning, C.E. (1994). The solubility of quartz in H 2 O in the lower crust and upper mantle. Geochim et Cosmochim. Acta., 58, 22, Macris, C.A. and Manning, C.E. (2005) The Solubility of Diopside in Water at 10 to 15 kbar and 650 to 900 C, EOS Trans. AGU, 86(52), Fall Meet. Suppl., Abstract V31C Patey, G.N., Levesque, D., and Weis, J.J. (1979) Integral equation approximations for fluids of hard spheres with dipoles and quardupoles. Molecular Phys., 38, 5, Pitzer, K. S. and Sterner, S.M. (1994) Equations of state valid continuously from zero to extreme pressures for H 2 O and CO 2. J. Chem. Phys., 101, 4, Scambelluri, M. and Philippot, P. (2001). Deep fluids in subduction zones. Lithos, 55, 15

16 Schiebener, P., Straub, J., Levelt-Sengers, J.M.H., and Gallagher, J.S. (1990) Refractive index of water and steam as function of wavelength, temperature, and density. J. Phys. Chem. Ref. Data., 19, 3, Tani, A., Henderson, D., and Barker, J.A. (1983) Application of perturbation theory to the calculation of the dielectric constant of a dipolar hard sphere fluid. Molecular Phys., 48, 4, Valiskó, M., Boda, D., Liszi, J., and Szalai, I. (2001). Relative perimittivity of dipolar liquids and their mixtures. Comparison of theory and experiment. Phys. Chem. Chem. Phys., 3, Van Keken, P.E., Kiefer, B., and Peacock, S.M. (2002). High-resolution models of subduction zones: Implications for mineral dehydration reactions and the transport of water into the deep mantle. Geochem. Geophys. Geosys. 3, 10, /2001GC Walther, J.V. (1997) Experimental determination and interpretation of the solubility of corundum in H 2 O between 350 and 600 o C from 0.5 to 2.2 kbar. Geochim. Et Cosmochim. Acta, 61, 23, Walther, J. and Schott, J. (1988). The dielectric constant approach to speciation and ion paring at high temperature and pressure. Nature, 322, 14, Wasserman, E., Wood, B., and Brodholt, J. (1995). The static dielectric constant of water at pressures up to 20 kbar and temperatures to 1273K: Experiments, simulations, and empirical equations. Geochimica et Cosmochimica Acta, 59, 1, 1-6. Woodland, A.B. and Walther, J.V. (1987). Experimental determination of the solubility of the assemblage paragonite, albite, quartz in supercritical water. Geochim. Et Cosmochim. Acta, 51, Xie, Z. and Walther, J.V. (1992). Quartz solubilities in NaCl solutions with and without wollastonite at elevated temperatures and pressures. Geochim. Cosmochim. Acta., 57, Zhang, Y. and Frantz, J.D. (2000) Enstatite-forsterite-water equilibria at elevated temperatures and pressures. Am. Mineralogist, 85,

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GSA Data Repository 218145 Parolari et al., 218, A balancing act of crust creation and destruction along the western Mexican convergent margin: Geology, https://doi.org/1.113/g39972.1. 218145_Tables DR1-DR4.xls