An internally consistent thermodynamic dataset with uncertainties and correlations: 1. Methods and a worked example

Abstract This, the first two papers, sets out the philosophy and methods of determining an internally consistent thermodynamic dataset for minerals using the least squares method. The applicability of the least squares method is discussed, and it is applied to a small set of experimental equilibria in the system Na₂O–Al₂O₃–SiO₂–H₂O. The importance is stressed of defining not only the enthalpies of formation of minerals, but also the uncertainties and the correlations among them. The system which has been used as an illustration for this paper serves as a visual guide to the method, as it is small enough to represent graphically in two dimensions. In the paper which follows, we extend the method to a system of 60 equations (experimentally determined equilibria) involving 34 unknowns (enthalpies of formation of mineral end-members).     

An improved and extended internally consistent thermodynamic dataset for phases of petrological interest,involving a new equation of state for solids

The thermodynamic properties of 254 end‐members, including 210 mineral end‐members, 18 silicate liquid end‐members and 26 aqueous fluid species are presented in a revised and updated internally consistent thermodynamic data set. The PVT properties of the data set phases are now based on a modified Tait equation of state (EOS) for the solids and the Pitzer & Sterner (1995) equation for gaseous components. Thermal expansion and compressibility are linked within the modified Tait EOS (TEOS) by a thermal pressure formulation using an Einstein temperature to model the temperature dependence of both the thermal expansion and bulk modulus in a consistent way. The new EOS has led to improved fitting of the phase equilibrium experiments. Many new end‐members have been added, including several deep mantle phases and, for the first time, sulphur‐bearing minerals. Silicate liquid end‐members are in good agreement with both phase equilibrium experiments and measured heat of melting. The new dataset considerably enhances the capabilities for thermodynamic calculation on rocks, melts and aqueous fluids under crustal to deep mantle conditions. Implementations are already available in thermocalc to take advantage of the new data set and its methodologies, as illustrated by example calculations on sapphirine‐bearing equilibria, sulphur‐bearing equilibria and calculations to 300 kbar and 2000 °C to extend to lower mantle conditions.     

An enlarged and updated internally consistent thermodynamic dataset with uncertainties and correlations: the system K2O–Na2O–CaO–MgO–MnO–FeO–Fe2O3–Al2O3–TiO2–SiO2–C–H2–O2

We present, as a progress report, a revised and much enlarged version of the thermodynamic dataset given earlier (Holland & Powell, 1985). This new set includes data for 123 mineral and fluid end-members made consistent with over 200 P–T–X_CO2–f_O2 phase equilibrium experiments. Several improvements and advances have been made, in addition to the increased coverage of mineral phases: the data are now presented in three groups ranked according to reliability; a large number of iron-bearing phases has been included through experimental and, in some cases, natural Fe:Mg partitioning data; H₂O and CO₂ contents of cordierites are accounted for with the solution model of Kurepin (1985); simple Landau theory is used to model lambda anomalies in heat capacity and the Al/Si order–disorder behaviour in some silicates, and Tschermak-substituted end-members have been derived for iron and magnesium end-members of chlorite, talc, muscovite, biotite, pyroxene and amphibole. For the subset of data which overlap those of Berman (1988), it is encouraging to find both (1) very substantial agreement between the two sets of thermodynamic data and (2) that the two sets reproduce the phase equilibrium experimental brackets to a very similar degree of accuracy. The main differences in the two datasets involve size (123 as compared to 67 end-members), the methods used in data reduction (least squares as compared to linear programming), and the provision for estimation of uncertainties with this dataset. For calculations on mineral assemblages in rocks, we aim to maximize the information available from the dataset, by combining the equilibria from all the reactions which can be written between the end-members in the minerals. For phase diagram calculations, we calculate the compositions of complex solid solutions (together with P and T) involved in invariant, univariant and divariant assemblages. Moreover we strongly believe in attempting to assess the probable uncertainties in calculated equilibria and hence provide a framework for performing simple error propagation in all calculations in thermocalc, the computer program we offer for an effective use of the dataset and the calculation methods we advocate.     

Bayes estimation: A novel approach to derivation of internally consistent thermodynamic data for minerals,their uncertainties,and correlations. Part I: Theory

Computation of phase diagrams in mineral systems and quantitative geothermobarometry thrive on the availability and accuracy of internally consistent thermodynamic datasets for minerals. The prevailing two methodologies applied to derive them, mathematical programming (MAP) and least squares regression (REG), have their very specific advantages and deficiencies which are to some extent complementary. Bayes estimation (BE), the novel technique proposed here for obtaining internally consistent thermodynamic databases, can combine the advantages of both MAP and REG but avoid their drawbacks. It optimally uses the information on thermochemical, thermophysical, and volumetric properties of phases and experimental reaction reverals to refine the thermodynamic data and returns their uncertainties and correlations. Therefore, BE emerges as the method of choice. The theoretical background of BE, and its relation to MAP and REG, is explained. Although BE is conceptually simple, it can be computationally demanding. Fortunately, modern computer technology and new stochastic methods such as Gibbs sampling help surmount those difficulties. The basic ideas behind these methods are explored and recommendations for their use are made using the Al₂SiO₅ unary as an example. The potential of BE and its future perspective for application to multicomponent-multiphase systems appear very promising. For the convenience of readers not interested in the mathematical details of BE, an illustrative example is given in the Appendix to promote an intuitive understanding of what BE is all about.     

Bayes estimation: A novel approach to derivation of internally consistent thermodynamic data for minerals,their uncertainties,and correlations. Part II: Application

Internally consistent thermodynamic data, including their uncertainties and correlations, are reported for 22 phases of the quaternary system CaO-Al₂O₃-SiO₂-H₂O. These data have been derived by simultaneous evaluation of the appropriate phase properties (PP) and reaction properties (RP) by the novel technique of Bayes estimation (BE). The thermodynamic model used and the theory of BE was expounded in Part I of this paper. Part II is the follow-up study illustrating an application of BE. The input for BE comprised, among others, the a priori values for standard enthalpy of formation of the i-th phase, Δ_f H _i ⁰ , and its standard entropy, S _i ⁰ , in addition to the reaction reversal constraints for 33 equilibria involving the relevant phases. A total of 269 RP restrictions have been processed, of which 107 turned out to be non-redundant. The refined values for Δ_f H _i ⁰ and S _i ⁰ obtained by BE, including their 2σ-uncertainties, appear in Table 4; the Appendix reproduces the corresponding correlation matrix. These data permit generation of computed phase diagrams with 2σ-uncertainty envelopes based on conventional error propagation; Fig. 3 depicts such a phase diagram for the system CaO-Al₂O₃-SiO₂. It shows that the refined dataset is capable of yielding phase diagrams with uncertainty envelopes narrow enough to be geologically useful. The results in Table 4 demonstrate that the uncertainties of the prior values for Δ_f H _i ^Emphasis>0 , given in Table 1, have decreased by up to an order of magnitude, while those for S _i ⁰ improved by a factor of up to two. For comparison, Table 4 also lists the refined Δ_f H _i ⁰ and S _i ⁰ data obtained by mathematical programming (MAP), minimizing a quadratic objective function used earlier by Berman (1988). Examples of calculated phase diagrams are given to demonstrate the advantages of BE for deriving internally consistent thermodynamic data. Although P-T curves generated from both MAP and BE databases will pass through the reversal restrictions, BE datasets appear to be better suited for extrapolations beyond the P-T range explored experimentally and for predicting equilibria not constrained by reversals.