https://www.perplex.ethz.ch/papers/connolly_epsl_05.pdf

Computation of phase equilibria by linear programming: A tool for geodynamic modeling and its application to subduction zone decarbonation

The models recognize that ZrSiO4, ZrTiO4,  and TiSiO4, but not ZrO2 or TiO2, are independently variable phase components in zircon. Accordingly, the equilibrium controlling the Zr content of rutile coexisting with zircon is ZrSiO4 = ZrO2 (in rutile) +  SiO2. The equilibrium controlling the Ti content of zircon is either ZrSiO4 + TiO2 = ZrTiO4 + SiO2 or TiO2 + SiO2 = TiSiO4, depending whether Ti substitutes for Si or Zr. The Zr content of rutile thus depends on the activity of SiO2
 $$(a_{\text{SiO}_{2}})$$
 as well as T, and the Ti content of zircon depends on $$a_{\text{SiO}_{2}}$$
  and 
 $$a_{\text{TiO}_{2}}$$
 as well as T. New and published experimental data confirm the predicted increase in the Zr content of rutile with decreasing $$a_{\text{SiO}_{2}},$$
 and unequivocally demonstrate that the Ti content of zircon increases with decreasing $$a_{\text{SiO}_{2}}$$
 . The substitution of Ti in zircon therefore is primarily for Si. Assuming a constant effect of P, unit $$a_{\text{ZrSiO}_{4}},$$
 and that $$a_{\text{ZrO}_{2}}$$
  and $$a_{\text{ZrTiO}_{4}}$$
 are proportional to ppm Zr in rutile and ppm Ti in zircon, [log(ppm Zr-in-rutile) + log
 $$a_{\text{SiO}_{2}}$$
 ] = A1 + B1/T(K) and  [log(ppm Ti-in-zircon) + log
 $$a_{\text{SiO}_{2}}$$
 − log
 $$a_{\text{TiO}_{2}}$$
 ] = A2 + B2/T, where the A and B are constants. The constants were derived from published and new data from experiments with $$a_{\text{SiO}_{2}}$$
 buffered by either quartz or zircon + zirconia, from experiments with $$a_{\text{SiO}_{2}}$$
 defined by the Zr content of rutile, and from well-characterized natural samples. Results are A1 = 7.420 ± 0.105;  B1 = −4,530 ± 111;  A2 = 5.711 ± 0.072;  B2 = −4,800 ± 86 with activity referenced to α-quartz and rutile at P and T of interest. The zircon thermometer may now be applied to rocks without quartz and/or rutile, and the rutile thermometer applied to rocks without quartz, provided that $$a_{\text{SiO}_{2}}$$
  and $$a_{\text{TiO}_{2}}$$
 are estimated. Maximum uncertainties introduced to zircon and rutile thermometry by unconstrained $$a_{\text{SiO}_{2}}$$
  and 
 $$a_{\text{TiO}_{2}}$$
 can be quantitatively assessed and are ≈60 to 70°C at 750°C. A preliminary assessment of the dependence of the two thermometers on P predicts that an uncertainty of ±1 GPa introduces an additional uncertainty at 750°C of ≈50°C for the Ti-in-zircon thermometer and of ≈70 to 80°C for the Zr-in-rutile thermometer.

New thermodynamic models and revised calibrations for the Ti-in-zircon and Zr-in-rutile thermometers

For petrological calculations, including geothermobarometry and the calculation of phase diagrams (for example, P–T petrogenetic grids and pseudosections), it is necessary to be able to express the activity–composition (a–x) relations of minerals, melt and fluid in multicomponent systems Although the symmetric formalism—a macroscopic regular model approach to a–x relations—is an easy-to-formulate, general way of doing this, the energetic relationships are a symmetric function of composition We allow asymmetric energetics to be accommodated via a simple extension to the symmetric formalism which turns it into a macroscopic van Laar formulation We term this the asymmetric formalism (ASF) In the symmetric formalism, the a–x relations are specified by an interaction energy for each of the constituent binaries amongst the independent set of end members used to represent the phase In the asymmetric formalism, there is additionally a "size parameter" for each of the end members in the independent set, with size parameter differences between end members accounting for asymmetry In the case of fluid mixtures, for example, H2O–CO2, the volumes of the end members as a function of pressure and temperature serve as the size parameters, providing an excellent fit to the a–x relations In the case of minerals and silicate liquid, the size parameters are empirical parameters to be determined along with the interaction energies as part of the calibration of the a–x relations In this way, we determine the a–x relations for feldspars in the systems KAlSi3O8–NaAlSi3O8 and KAlSi3O8–NaAlSi3O8–CaAl2Si2O8, for carbonates in the system CaCO3–MgCO3, for melt in the melting relationships involving forsterite, protoenstatite and cristobalite in the system Mg2SiO4–SiO2, as well as for fluids in the system H2O–CO2 In each case the a–x relations allow the corresponding, experimentally determined phase diagrams to be reproduced faithfully The asymmetric formalism provides a powerful and flexible way of handling a–x relations of complex phases in multicomponent systems for petrological calculations

Activity–composition relations for phases in petrological calculations: an asymmetric multicomponent formulation

[1] We present a new compilation of physical properties of minerals relevant to subduction zones and new phase diagrams for mid-ocean ridge basalt, lherzolite, depleted lherzolite, harzburgite, and serpentinite. We use these data to calculate H2O content, density and seismic wave speeds of subduction zone rocks. These calculations provide a new basis for evaluating the subduction factory, including (1) the presence of hydrous phases and the distribution of H2O within a subduction zone; (2) the densification of the subducting slab and resultant effects on measured gravity and slab shape; and (3) the variations in seismic wave speeds resulting from thermal and metamorphic processes at depth. In considering specific examples, we find that for ocean basins worldwide the lower oceanic crust is partially hydrated (<1.3 wt % H2O), and the uppermost mantle ranges from unhydrated to � 20% serpentinized (� 2.4 wt % H2O). Anhydrous eclogite cannot be distinguished from harzburgite on the basis of wave speeds, but its � 6% greater density may render it detectable through gravity measurements. Subducted hydrous crust in cold slabs can persist to several gigapascals at seismic velocities that are several percent slower than the surrounding mantle. Seismic velocities and VP/VS ratios indicate that mantle wedges locally reach 60–80% hydration. INDEX TERMS: 3040 Marine Geology and Geophysics: Plate tectonics (8150, 8155, 8157, 8158); 3660 Mineralogy and Petrology: Metamorphic petrology; 3919 Mineral Physics: Equations of state; 5199 Physical Properties of Rocks: General or miscellaneous; 8123 Tectonophysics: Dynamics, seismotectonics; KEYWORDS: subduction, seismic velocities, mineral physics, H2O

/pdf/subduction-factory-1-theoretical-mineralogy-densities-2416553cwa.pdf

Subduction factory 1. Theoretical mineralogy, densities, seismic wave speeds, and H 2 O contents

[1] Geodynamic models commonly assume equations of state as a function of pressure and temperature. This form is legitimate for homogenous materials, but it is impossible to formulate a general equation of state for a polyphase aggregate, e.g., a rock, as a function of pressure and temperature because these variables cannot distinguish all possible states of the aggregate. In consequence, the governing equations of a geodynamic model based on a pressure-temperature equation of state are singular at the conditions of low-order phase transformations. An equation of state as a function of specific entropy, specific volume, and chemical composition eliminates this difficulty and, additionally, leads to a robust formulation of the energy and mass conservation equations. In this formulation, energy and mass conservation furnish evolution equations for entropy and volume and the equation of state serves as an update rule for temperature and pressure. Although this formulation is straightforward, the computation of phase equilibria as a function of entropy and volume is challenging because the equations of state for individual phases are usually expressed as a function of temperature and pressure. This challenge can be met by an algorithm in which continuous equations of state are approximated by a series of discrete states: a representation that reduces the phase equilibrium problem to a linear optimization problem that is independent of the functional form used for the equations of state of individual phases. Because the efficiency of the optimization decays as an exponential function of the dimension of the function to be optimized, direct solution of the linearized optimization problem is impractical. Successive linear programming alleviates this difficulty. A pragmatic alternative to optimization as an explicit function of entropy and volume is to calculate phase relations over the range of pressure-temperature conditions of interest. Numerical interpolation can then be used to generate tables for any thermodynamic property as a function of any choice of independent variables. Regardless of the independent variables of the governing equations, a consistent definition of pressure, and the coupling of equilibrium kinetics to deformation, is only possible if the continuity equation accounts for dilational strain.

/pdf/the-geodynamic-equation-of-state-what-and-how-1b10vc4ms6.pdf

The geodynamic equation of state: What and how

The thermodynamic properties of 154 mineral end-members, 13 silicate liquid end-members and 22 aqueous fluid species are presented in a revised and updated data set. The use of a temperature-dependent thermal expansion and bulk modulus, and the use of high-pressure equations of state for solids and fluids, allows calculation of mineral-fluid equilibria to 100 kbar pressure or higher. A pressure-dependent Landau model for order-disorder permits extension of disordering transitions to high pressures, and, in particular, allows the alpha-beta quartz transition to be handled more satisfactorily. Several melt end- members have been included to enable calculation of simple phase equilibria and as a first stage in developing melt mixing models in NCKFMASH. The simple aqueous species density model has been extended to enable speciation calculations and mineral solubility determination involving minerals and aqueous species at high temperatures and pressures. The data set has also been improved by incorporation of many new phase equilibrium constraints, calorimetric studies and new measurements of molar volume, thermal expansion and compressibility. This has led to a significant improvement in the level of agreement with the available experimental phase equilibria, and to greater flexibility in calculation of complex mineral equilibria. It is also shown that there is very good agreement between the data set and the most recent available calorimetric data. kinetics which apply to determining directly the greatest majority of such equilibria in the laboratory, for forming solid solutions, and inclusion of aqueous and silicate melt species), and provides uncertainties especially at lower temperatures, as well as the diYculty of establishing reversals of reactions involving solid allowing the likely uncertainties on the results of thermodynamic calculations to be estimated. This is a solutions. The levels of precision and accuracy required of thermodynamic data in order to be able to forward- critical issue in that calculations using data sets should always involve uncertainty propagation to help evalu- model synthetic and natural mineral assemblages mean that the continuing upgrading and expansion of the ate the results. Because the experimental phase equilib- ria involve overlapping subsets of compositional space, data set by incorporation of new phase equilibrium constraints, calorimetry and new measurements of the derived thermodynamic data are highly correlated, and it is only the inclusion of the correlations which molar volume, thermal expansion and compressibility are more than justified. enables the reliable calculation of uncertainties on mineral reactions to be performed. Earlier work on mineral thermodynamic data sets for rock-forming minerals includes compilations of The thermodynamic data extraction involves using weighted least squares on the diVerent types of data

R. P Owell

Papers

An internally consistent thermodynamic data set for phases of petrological interest