A new method to calculate end-member thermodynamic properties of minerals from their constituent polyhedra I: enthalpy, entropy and molar volume

The thermodynamic properties of silicate minerals can be described as a linear combination of the fractional properties of their constituent polyhedra. In contrast, given the thermodynamic properties of these polyhedra, the thermodynamic properties of minerals can be estimated, where only the crystallography of the mineral needs to be known. Such estimates are especially powerful for hypothetical mineral end‐members or for minerals where experimental determination of their thermodynamic properties is difficult. In this contribution the fractional enthalpy, entropy and molar volume for 35 polyhedra have been determined using weighted multiple linear regression analysis on a data set of published mineral thermodynamic properties. The large number of polyhedra determined, allows calculation of a much larger variety of phases than was previously possible and the larger set of minerals used provides more confident fractional properties. The OH‐bearing minerals have been described by partial and total hydroxide coordinated components, which gives better results than previous models and precludes the need of a S–V term to improve estimates of entropy. However, the fractional thermodynamic properties only give adequate results for silicate minerals and double oxides, and should therefore not be used to estimate the properties of other minerals. The thermodynamic properties of ‘new’ minerals are calculated from a linear stoichiometric combination of their constituent polyhedra, resulting in estimates generally with associated uncertainty of <5%. The quality of such data appears to be of sufficient accuracy for thermodynamic modelling as shown for meta‐bauxites from the Alps and the Aegean, where the effect of Zn on the P–T stability of staurolite can be both qualitatively and quantitatively reproduced.     

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 internally consistent thermodynamic data set for phases of petrological interest

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.     

A thermodynamic model for Ca–Na clinoamphiboles in Na2O–CaO–FeO–MgO–Al2O3–SiO2–H2O–O for petrological calculations

A calibration is presented for an activity–composition model for amphiboles in the system Na₂O–CaO–FeO–MgO–Al₂O₃–SiO₂–H₂O–O (NCFMASHO), formulated in terms of an independent set of six end‐members: tremolite, tschermakite, pargasite, glaucophane, ferroactinolite and ferritschermakite. The model uses mixing‐on‐sites for the ideal‐mixing activities, and for the activity coefficients, a macroscopic multicomponent van Laar model. This formulation involves 15 pairwise interaction energies and six asymmetry parameters. Calibration of the model is based on the geometrical constraints imposed by the size and shape of amphibole solvi inherent in a data set of 71 coexisting amphibole pairs from rocks, formed over 400–600 °C and 2–18 kbar. The model parameters are calibrated by combining these geometric constraints with qualitative consideration of parameter relationships, given that the data are insufficient to allow all the model parameters to be determined from a regression of the data. Use of coexisting amphiboles means that amphibole activity–composition relationships are calibrated independently of the thermodynamic properties of the end‐members. For practical applications, in geothermobarometry and the calculation of phase diagrams, the amphibole activity–composition relationships are placed in the context of the stability of other minerals by evaluating the properties of the end‐members in the independent set that are in internally consistent data sets. This has been performed using an extended natural data set for hornblende–garnet–plagioclase–quartz, giving the small adjustments necessary to the enthalpies of formation of tschermakite, pargasite and glaucophane for working with the Holland and Powell data set.     

Mixing properties of phengitic micas and revised garnet-phengite thermobarometers

Mixing properties for muscovite–celadonite–ferroceladonite solid solutions are derived from combining available experimental phase equilibrium data with tabulated thermodynamic data for mineral end‐members. When a partially ordered solution model is assumed, the enthalpy of mixing among the end‐members muscovite–celadonite–ferroceladonite is nearly ideal, although the Gibbs energies of muscovite–celadonite and muscovite–ferroceladonite solutions are asymmetric due to an asymmetry in the entropy of mixing. Thermodynamic consistency is achieved for data on phengite compositions inassemblages with (a) pyrope+kyanite+quartz/coesite (b) almandine+kyanite+quartz/coesite (c)talc+kyanite+quartz/coesite and (d) garnet–phengite pairs equilibrated both experimentally at high temperatures and natural pairs from low‐grade schists. The muscovite–paragonite solvus has been reanalysed using the asymmetric van Laar model, and the effects of the phengite substitution into muscovite have been quantitatively addressed in order to complete the simple thermodynamic mixing model for the solid solution among the mica end‐members. Results are applied to a natural pyrope–coesite–phengite–talc rock from the Western Alps, and to investigate the conditions under which biotite‐bearing mica schists transform to whiteschist‐like biotite‐absent assemblages for average pelite bulk compositions.     

Using estimated thermodynamic properties to model accessory phases: the case of tourmaline

Accessory phases and minor components in minerals are commonly ignored in thermodynamic modelling. Such an approach seems unwarranted, as accessory phases can represent a significant element reservoir and minor components can substantially change their host mineral's stability field. However, a lack of thermodynamic data prohibits assessment of these effects. In this contribution, the polyhedron method is used to estimate the thermodynamic properties of tourmaline, a common and widespread accessory phase, stable over a range of P–T–X conditions. The polyhedron method allows Δ H , S , V , C _P and V _m ( T , P ) properties to be estimated from a linear stoichiometric summation over the fractional properties of its polyhedron constituents. To allow for estimates of tourmaline, fractional thermodynamic properties for B^III and B^IV polyhedra were derived. Mixing contributions to molar volume were evaluated and symmetrical mixing parameters derived for Al-Mg, Al-Fe and Al-Li interaction on tourmaline's Y-site and T-site Al-Si interaction. Evaluation of the estimated properties using experimental and natural equilibria between tourmaline and melts, minerals and hydrothermal fluids, shows that reliable semi-quantitative results are obtained. The boron contents in fluids coexisting with tourmaline are calculated to within an order of magnitude of measured content, and where anchor-points are available, agreement improves to within a factor of 2. Including tourmaline in petrogenetic modelling of metamorphic rocks indicates that its presence leads to disappearance of staurolite and garnet, among others, and modifies the X _Mg of coexisting phases, in line with observations on natural rocks.     

Oxidized eclogites and garnet-blueschists from Oman: P–T path modelling in the NCFMASHO system

Eclogites and garnet‐blueschists exposed at the deepest structural levels of the Oman Mountains in north‐eastern Saih Hatat, Oman, indicate that the Arabian continental margin was subducted and subsequently exhumed. The peak metamorphic pressure has been a matter of debate for over a decade, with initial thermobarometric estimates, based on garnet–clinopyroxene–phengite barometry and the presence of radial cracks around quartz inclusions in garnet, yielding values in excess of 20 kbar; these estimates have been questioned by some researchers. The high‐pressure minerals (glaucophane, omphacite and epidote) contain significant amounts of ferric iron, previously postulated to displace the stability fields of the eclogite and blueschist assemblages to less extreme conditions. In the present study, we have calculated phase diagrams and pseudosections in the model system NCFMASHO, using the program thermocalc and the thermodynamic database of Holland and Powell, which incorporates data for Fe³⁺‐bearing end‐members. It is found that the phase compositions and modal abundances for typical bulk compositions are matched successfully at 520 ± 15 °C and 20 ± 1.6 kbar for the eclogites and 510–530 °C and 17–20 kbar for the garnet blueschists. These results support the original high‐pressure estimates for the eclogites, and indicate that crossitic amphibole and aegirine‐rich pyroxene do not necessarily reflect lower pressure conditions. The data set and activity models are applicable to other oxidized high (and ultra‐high) pressure mineral assemblages.     

The effect of Mn on mineral stability in metapelites revisited: new a–x relations for manganese‐bearing minerals

The a–x relations recently presented in White et al. ( 2014 , Journal of Metamorphic Geology, 32, 261–286) are extended to include MnO. This provides a set of internally consistent a–x relations for metapelitic rocks in the MnO–Na₂O–CaO–K₂O–FeO–MgO–Al₂O₃–SiO₂–H₂O–TiO₂–O₂ (MnNCKFMASHTO) system. The mixing parameters for the Mn‐bearing minerals were estimated using the micro‐? approach of Powell et al. ( 2014 , Journal of Metamorphic Geology, 32, 245–260). Then the Mn‐end‐member thermodynamic properties were calibrated using a database of co‐existing minerals involving literature data from rocks and from experiments on natural materials. Mn‐end‐members were calibrated for orthopyroxene, cordierite, staurolite, chloritoid, chlorite, biotite, ilmenite and hematite, assuming known properties for the garnet end‐member spessartine. The addition of MnO to phase diagram calculations results in a marked expansion of the stability of garnet‐bearing assemblages. At greenschist facies conditions garnet stability is extended down temperature. At amphibolite facies conditions, the garnet‐in boundary shifts to lower pressure. While the addition of MnO greatly influences the stability of garnet, it has relatively little effect on the stability of other common metapelitic minerals, with the resultant diagrams being topologically very similar to those calculated without MnO. Furthermore, the addition of MnO in the amounts measured in most metapelites has only a small effect on the mode of garnet, with calculated garnet modes remaining smaller than 1% in the P–T range outside its predicted Mn‐free P–T range.     

矿物热膨胀的晶体化学研究综述

热膨胀是矿物材料的重要属性之一,对它的研究在矿物学及材料科学中具有重要意义。前人关于矿物的热膨胀系数和晶体化学参数之间关系的研究表明,矿物的 结构、成分、化学键及原子价态、配位数、外层电子结构等对热膨胀系数都有不同程度的影响。晶体化学方法是研究矿物热膨胀系数的有效方法,但由 于矿物的热膨胀系数和晶体化学参数之间的定量关系尚未完全弄清,目前仅仅能对简单结构矿物的热膨胀系数进行计算,结构、成分较复杂的矿物的热膨胀系数计算尚缺乏好的晶体化学模型。为达到精确计算矿物热膨胀系数的目的,必须对矿物按照晶体化学特性进行分类。 在分类的基础上,把晶体化学方法和理论计算方法有效地结合起来。矿物的热膨胀性可按照 受热时结构有无旋转划分为键膨胀型和结构旋转型,过渡金属阳离子所形成的化合物根据d电子层电子云的分布进行分类。通过以上的分类,在计算不同类型矿物热膨胀系数时 ,可按矿物类型处理。对键膨胀型的矿物只需估计键的膨胀情况,而对于结构旋转型除考虑 键的膨胀性以外还要考虑配位多面体旋转的情况。     

Thermodynamic properties of San Carlos olivine at high temperature and high pressure

In this study, the thermal expansion and heat capacity of San Carlos olivine under high temperature and high pressure are reported. Combining accurate sound velocity data under different P–T conditions with density and heat capacity data at ambient pressure, the density, adiabatic bulk modulus, shear modulus, and most importantly, thermal expansion and heat capacity, of San Carlos are extracted to 14 GPa by a numerical procedure using classic thermodynamic relationships. These data are in agreement with published findings. To estimate the temperature gradient in the upper mantle, we also report the fitting equations of thermal expansion and heat capacity of San Carlos olivine as a function of both temperature and pressure to the P–T condition of the 410 km discontinuity, which provide the thermodynamic properties with increasing depth in the Earth’s interior.     

Prograde pressure–temperature paths in the pelitic schists of the Sambagawa metamorphic belt, SW Japan

Prograde P–T paths recorded by the chemistry of minerals of subduction‐related metamorphic rocks allow inference of tectonic processes at convergent margins. This paper elucidates the changing P–T conditions during garnet growth in pelitic schists of the Sambagawa metamorphic belt, which is a subduction related metamorphic belt in the south‐western part of Japan. Three types of chemical zoning patterns were observed in garnet: Ca‐rich normal zoning, Ca‐poor normal zoning and intrasectoral zoning. Petrological studies indicate that normally‐zoned garnet grains grew keeping surface chemical equilibrium with the matrix, in the stable mineral assemblage of garnet + muscovite + chlorite + plagioclase + paragonite + epidote + quartz ± biotite. Pressure and temperature histories were inversely calculated from the normally‐zoned garnet in this assemblage, applying the differential thermodynamic method (Gibbs' method) with the latest available thermodynamic data set for minerals. The deduced P–T paths indicate slight increase of temperature with increasing pressure throughout garnet growth, having an average dP/dT of 0.4–0.5 GPa/100 °C. Garnet started growing at around 470 °C and 0.6 GPa to achieve the thermal and baric peak condition near the rim (520 °C, 0.9 GPa). The high‐temperature condition at relatively low pressure (for subduction related metamorphism) suggests that heating occurred before or simultaneously with subduction.     

Garnet and spinel lherzolite assemblages in MgO–Al2O3–SiO2 and CaO–MgO–Al2O3–SiO2: thermodynamic models and an experimental conflict

The recent publication of an updated thermodynamic dataset for petrological calculations provides an opportunity to illustrate the relationship between experimental data and the dataset, in the context of a new set of activity–composition models for several key minerals. These models represent orthopyroxene, clinopyroxene and garnet in the system CaO–MgO–Al₂O₃–SiO₂ (CMAS), and are valid up to 50 kbar and at least 1800 °C; they are the first high‐temperature models for these phases to be developed for the Holland & Powell dataset. The models are calibrated with reference to phase‐relation data in the subsystems CaO–MgO–SiO₂ (CMS) and MgO–Al₂O₃–SiO₂ (MAS), and will themselves form the basis of models in larger systems, suitable for calculating phase equilibria in the crust and mantle. In the course of calibrating the models, it was necessary to consider the reaction orthopyroxene + clinopyroxene + spinel = garnet + forsterite in CMAS, representing a univariant transition between simple spinel and garnet lherzolite assemblages. The high‐temperature segment of this reaction has been much disputed. We offer a powerful thermodynamic argument relating this reaction to the equivalent reaction in MAS, that forces us to choose between good model fits to the data in MAS or to the more recent data in CMAS. We favour the fit to the MAS data, preserving conformity with a large body of experimental and thermodynamic data that are incorporated as constraints on the activity–composition modelling via the internally consistent thermodynamic dataset.     

Calculation of partial melting equilibria in the system Na2O–CaO–K2O–FeO–MgO–Al2O3–SiO2–H2O (NCKFMASH)

A thermodynamic model for haplogranitic melts in the system Na₂O–CaO–K₂O–Al₂O₃–SiO₂–H₂O (NCKASH) is extended by the addition of FeO and MgO, with the data for the additional end‐members of the liquid incorporated in the Holland & Powell (1998) internally consistent thermodynamic dataset. The resulting dataset, with the software thermocalc , is then used to calculate melting relationships for metapelitic rock compositions. The main forms for this are P–T and T–X pseudosections calculated for particular rock compositions and composition ranges. The relationships in these full‐system pseudosections are controlled by the low‐variance equilibria in subsystems of NCKFMASH. In particular, the solidus relationships are controlled by the solidus relationships in NKASH, and the ferromagnesian mineral relationships are controlled by those in KFMASH. However, calculations in NCKFMASH allow the relationships between the common metapelitic minerals and silicate melt to be determined. In particular, the production of silicate melt and melt loss from such rocks allow observations to be made about the processes involved in producing granulite facies rocks, particularly relating to open‐system behaviour of rocks under high‐grade conditions.     

流体的热力学前缘研究

总结了当前国内外关于流体的热力学前缘研究领域如下：（１）流体体系的ｐ－Ｖ－Ｔ－ｘ相关系研究，主要对象是Ｈ２Ｏ－ＣＯ２－盐类多组分体系高温高压下相图的实验和理论研究。（２）矿物在流体中的溶解度及溶解后在流体中溶解类型的形式和热力学性质——平衡常数（或Ｇｉｂｂｓ自由能）及各种偏摩尔性质的研究。（３）流体热力学模型化研究，已研制出大量的计算机软件，包括多种矿物、溶解类型的热力学数据库和模拟热液平衡、矿物溶解性质、反应路径和水—岩相互作用的实用程序。（４）超临界流体的相关系和化学反应等有许多特殊的性质，对认识地球内部的演化将有重要意义。（５）新技术新方法的发展，使分析单个矿物包裹体成分变成了现实。     

The ultra‐high pressure coronitic and pseudomorphous reactions in a metagranodiorite from the Brossasco‐Isasca Unit,Dora‐Maira Massif,western Italian Alps: a petrographic study and equilibrium thermodynamic modelling

Metagranodiorite samples from the Brossasco‐Isasca Unit, Dora‐Maira Massif, western Alps, show pseudomorphous and coronitic textures where igneous minerals were partially replaced by ultra‐high pressure (UHP) metamorphic assemblages. The original magmatic paragenesis consisted of quartz, plagioclase, K‐feldspar, biotite and minor phases. During UHP metamorphism, the plagioclase (site P) was replaced by zoisite, jadeite, quartz, K‐feldspar and kyanite, and coronitic reactions developed between biotite and adjacent minerals. At the original igneous biotite–quartz contact (site A), a single corona of poorly zoned garnet is developed, whereas at the biotite–K‐feldspar (site B) and biotite–plagioclase (site C) contacts, composite coronas are formed. Integration of results from petrographic observations, calculations of mineral stoichiometry and thermodynamic calculations of mineral stability has allowed the determination of the metamorphic reactions involved and the estimation of the metamorphic conditions, which reached as high as 24 kbar and 650 °C. Accurate microanalysis by energy‐dispersive spectroscopy (EDS) and statistical analysis of the data allowed us to identify, for the first time in a natural Na‐pyroxene of metagranitoid rocks, the end‐member Ca‐Eskola.     

Biogeochemical processes in a clay formation in situ experiment: Part E - Equilibrium controls on chemistry of pore water from the Opalinus Clay, Mont Terri Underground Research Laboratory, Switzerland

The chemistry of pore water (particularly pH and ionic strength) is an important property of clay rocks being considered as host rocks for long-term storage of radioactive waste. Pore waters in clay-rich rocks generally cannot be sampled directly. Instead, their chemistry must be found using laboratory-measured properties of core samples and geochemical modelling. Many such measurements have been made on samples from the Opalinus Clay from the Mont Terri Underground Research Laboratory (URL). Several boreholes in that URL yielded water samples against which pore water models have been calibrated. Following a first synthesis report published in 2003, this paper presents the evolution of the modelling approaches developed within Mont Terri URL scientific programs through the last decade (1997-2009). Models are compared to the composition of waters sampled during dedicated borehole experiments. Reanalysis of the models, parameters and database enabled the principal shortcomings of the previous modelling efforts to be overcome. The inability to model the K concentrations correctly with the measured cation exchange properties was found to be due to the use of an inappropriate selectivity coefficient for Na-K exchange; the inability to reproduce the measured carbonate chemistry and pH of the pore waters using mineral-water reactions alone was corrected by considering clay mineral equilibria. Re-examination of the measured Ca/Mg activity ratios and consideration of the mineralogical composition of the Opalinus Clay suggested that Ca/Mg cation exchange rather than dolomite saturation may control the ratio of these ions in solution. This re-examination also suggests that the Ca/Mg ratio decreases with increasing pore-water salinity. Several possible reasons for this are proposed. Moreover, it is demonstrated that feldspar equilibria must not be included in Opalinus Clay modelling because feldspars are present only in very small quantities in the formation and because Na/K ratios measured in pore water samples are inconsistent with feldspar saturation. The principal need to improve future modelling is additional or better data on rock properties, in particular: (i) a more detailed identification of phases in the Opalinus Clay that include redox-sensitive elements together with evaluation of their thermodynamic properties; (ii) an improved understanding of the distribution of celestite throughout the Opalinus Clay for Sr/SO₄ concentrations control; (iii) improvements in analytic and thermodynamic data for Ca-Mg rock cation exchange and mineral chemical properties and (iv) the measurement of composition and stability constants of clay minerals actually present in the formation.     

On thermobarometry

Thermobarometry, the estimation of the pressure–temperature ( P – T ) conditions of metamorphism, can be undertaken by using pseudosection calculations as well as by conventional methods. Conventional thermobarometry uses only the equilibrium thermodynamics of balanced reactions between end-members of minerals, combined with the observed compositions of minerals. In contrast, pseudosections involve a forward calculation of mineral equilibria for a given rock composition. When related to observed rock data such as mineral assemblages, mineral proportions and mineral compositions, pseudosections have the power to provide valuable additional thermobarometric information. This is because the rock composition provides added constraints on P – T , unavailable in conventional thermobarometry, such as when minerals in the mineral assemblage are no longer stable, or when additional minerals join the mineral assemblage. Considering both conventional and pseudosection thermobarometry, a minimum requirement is that they use the same thermodynamic data and activity–composition models for the minerals involved. A new thermocalc facility is introduced that allows pseudosection datafile coding to be used for conventional thermobarometry. Guidelines are given and pitfalls discussed relating to pseudosection modelling and conventional thermobarometry. We argue that, commonly, pseudosection modelling provides the most powerful thermobarometric tools.     

The glauconite–Fe‐illite–Fe‐smectite problem: a critical review

Iron silicate minerals are a significant component of sedimentary systems but their modes of formation remain controversial. Our analysis of published data identifies end‐member compositions and mixtures and allows us to recognize controls of formation of different mineral species. The compositional fields of glaucony, Fe‐illite, Fe–Al smectites are determined in the M⁺/4Si vs. Fe/Sum of octahedral cations (M⁺ = interlayer charge). Solid solutions could exist between these phases. The Fe–Al and Fe‐rich clay minerals form two distinct solid solutions. The earliest phases to be formed are Fe–Al smectites or berthierine depending on the sedimentation rate. Reductive microsystems appear in the vicinity of organic debris in unconsolidated sediments. The Fe is incorporated first in pyrite and then in silicates after oxidation. Potassium ions diffuse from the sea‐water–sediment interface. If not interrupted, the diffusion process is active until reaction completion is reached, i.e. formation of Fe‐illite or glauconite or a mineral assemblage (berthierine–nontronite) according to the available Al ion amounts in the microsystem. Mixed‐layer minerals are formed when the diffusion process is interrupted because of sedimentation, compaction or cementation. Despite the common belief of their value as palaeoenvironment indicators, these minerals can form in a variety of environments and over a period of millions of years during sediment burial.     

含铁闪锌矿热力学性质的推导

热容是矿物热力学性质的基础,通过热容测定,可以推导出矿物的其他热力学性质,本文在对３个含铁闪锌矿样品的热容进行实验测定的基础上,推导出它们的热力学性质,本文的研究结果为系统地研究闪锌矿及其成矿物理化学条件参考数据。