A new method to calculate end-member thermodynamic properties of minerals from their constituent polyhedra II: heat capacity, compressibility and thermal expansion

Thermodynamic calculations in petrology are generally performed at pressures and temperatures beyond the standard state conditions. Accurate prediction of mineral equilibria therefore requires knowledge of the heat capacity, thermal expansion and compressibility for the minerals involved. Unfortunately, such data are not always available. In this contribution we present a data set to estimate the heat capacity, thermal expansion and compressibility of mineral end‐members from their constituent polyhedra, based on the premise that the thermodynamic properties of minerals can be described by a linear combination of the fractional properties of their constituents. As such, only the crystallography of the phase of interest needs to be known. This approach is especially powerful for hypothetical mineral end‐members and for minerals, for which the experimental determination of their thermodynamic properties is difficult. The data set consists of the properties for 35 polyhedra in the system K–Na–Ca–Li–Be–Mg–Mn–Fe–Co–Ni–Zn–Al–Ti–Si–H, determined by multiple linear regression analysis on a data set of 111 published end‐member thermodynamic properties. The large number of polyhedra determined allows calculation of a much larger variety of phases than was previously possible, and the choice of constituents together with the large number of thermodynamic input data results in estimates with associated uncertainty of generally <5%. The quality of the data appears to be sufficiently accurate for thermodynamic modelling as demonstrated by modelling the stability of margarite in the CASH system and the position of the talc–staurolite–chloritoid–pyrope absent invariant point in the KMASH system. In both cases, our results overlap within error with published equivalents.     

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.     

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.     

Interactions between <Emphasis Type="Italic">Bacillus mucilaginosus</Emphasis> and silicate minerals (weathered adamellite and feldspar): Weathering rate,products, and reaction mechanisms

Bacillus mucilaginosus is a common soil bacterium,and usually used as a model bacterium in studying microbe-mineral interactions.Several reaction mechanisms of B.mucilaginosus weathering silicate minerals were proposed.However,the molecule mechanisms and detailed processes were still unclear.In this paper,bacterium-mineral interactions were studied in terms of variations in pH value over the experimental period,variations in mineral composition,weathering rates of silicate minerals and volatile metabolites in the culture medium,etc.,to further explore the bacterium-mineral interaction mechanisms.The results showed that B.mucilaginosus could enhance silicate mineral weathering obviously.The weathering rates were quite different for various kinds of silicate minerals,and the weathering rate of weathered adamellite could reach 150 mg/m2/d.Although B.mucilaginosus produced little acidic substance,pH in the microenvironment of bacterium-mineral complex might be far lower than that of the circumjacent environment;a large amount of acetic acid was found in the metabolites,and was likely to play an important role as a ligand.These results appear to suggest that acidolysis and ligand degradation are the main mechanisms of B.mucilaginosus dissolving silicate minerals,the formation of bacterium-mineral complexes is the necessary condition for the bacteria weathering silicate minerals,and extracelluar polysaccharides played important roles in bacterium-mineral interaction processes by forming bacterium-mineral complexes and maintaining the spe-cial physicochemical properties of microenvironment.     

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.     

Calc-silicate reactions and bedding-controlled isotopic exchange in the Notch Peak aureole, Utah: implications for differential fluid fluxes with metamorphic grade

Fluid compositions and bedding‐scale patterns of fluid flow during contact metamorphism of the Weeks Formation in the Notch Peak aureole, Utah, were determined from mineralogy and stable isotope compositions. The Weeks Formation contains calc‐silicate and nearly pure carbonate layers that are interbedded on centimetre to decimetre scales. The prograde metamorphic sequence is characterized by the appearance of phlogopite, diopside, and wollastonite. By accounting for the solution properties of Fe, it is shown that the tremolite stability field was very narrow and perhaps absent in the prograde sequence. Unshifted oxygen and carbon isotopic ratios in calcite and silicate minerals at all grades, except above the wollastonite isograd, show that there was little to no infiltration of disequilibrium fluids. The fluid composition is poorly constrained, but X(CO₂)_fluid must have been >0.1, as indicated by the absence of talc, and has probably increased with progress of decarbonation reactions. The occurrence of scapolite and oxidation of graphite in calc‐silicate beds of the upper diopside zone provide the first evidence for limited infiltration of external aqueous fluids. Significantly larger amounts of aqueous fluid infiltrated the wollastonite zone. The aqueous fluids are recorded by the presence of vesuvianite, large decreases in δ¹⁸O values of silicate minerals from c. 16‰ in the diopside zone to c. 10‰ in the wollastonite zone, and extensive oxidation of graphite. The carbonate beds interacted with the fluids only along margins where graphite was destroyed, calcite coarsened, and isotopic ratios shifted. The wollastonite isograd represents a boundary between a high aqueous fluid‐flux region on its higher‐grade side and a low fluid‐flux region on its lower‐grade side. Preferential flow of aqueous fluids within the wollastonite zone was promoted by permeability created by the wollastonite‐forming reaction and the natural tendency of fluids to flow upward and down‐temperature near the intrusion‐wall rock contact.     

A novel technique for characterizing thermal expansion in minerals

When a crystalline solid is subjected to a temperature increase, its constituent polyhedra may change in size and shape and rotate relative to one another. If the deformation can be approximated by a linear transformation of atomic coordinates, these changes can be quantitatively described in terms of second rank tensors. An iterative least-squares method is used to calculate strain and rotation tensors given the positions of the coordinating atoms of a polyhedron at two temperatures. The method is applied to polyhedral thermal expansions in silicate and oxide minerals.     

Dissolution and depletion of ferromagnesian minerals from Holocene tephra layers in an acid bog,New Zealand,and implications for tephra correlation

This study examines the depletion of ferromagnesian silicate minerals from a sequence of thin, distal, mainly rhyolitic tephra layers of Holocene age preserved in an acid peat bog (Kopouatai), North Island, New Zealand. The rate of such depletion has been fast, as indicated by the complete loss of biotite from one tephra layer (Kaharoa Tephra), in which it is normally dominant, in only ca. 770 yr. Chemical dissolution is advocated as the likely cause for the depletion, with amphiboles and other mineral grains commonly showing etch pits, microcaves, and other characteristic surface solution features. Theoretical thermodynamic and kinetic models show a marked increase in the rate of dissolution of all ferromagnesian minerals under conditions of low pH (< 4), but that where silica concentrations in solution are high the relative proportions of minerals remaining are unaffected. However, where concentrations of dissolved silica are low, as in most bog environments, the relative proportions of ferromagnesian minerals are affected as well as absolute amounts being decreased. Amphiboles are depleted relative to pyroxenes, consistent with kinetic studies. The results show that the identification and correlation of tephras on the basis of relative abundances of ferromagnesian minerals alone may be unreliable, and emphasise the need to use multiple criteria in such studies.     

Viscosity and configurational entropy of silicate melts

With the configurational entropy theory of relaxation processes of Adam and Gibbs (1965), one predicts that the viscosity depends on temperature according to log $\text{η = A}{}_{\mathrm{e}}\text{+}\text{B}{}_{\mathrm{e}}\text{TS}{}_{\mathrm{<mtext>conf</mtext>}}$, where S_conf is the configurational entropy of the liquid. Thermochemical calculations of S_conf performed for some mineral compositions show the importance of non-configurational contributions to the entropy differences between amorphous and crystalline phases. Except for the case of SiO₂, the available thermodynamic data indicate that the above equation for viscosity accounts quantitatively for the experimentally determined temperature dependence of the viscosity of silicate melts. The Adam and Gibbs theory also provides a simple rationale for the non linear variation of the logarithmic viscosity with composition in mixed alkali silicate liquids at low temperatures, the minimum of viscosity resulting from the contribution of the entropy of mixing to S_conf.     

A Model of Magmatic Crystallization

A computer model simulating fractional crystallization at oneatmosphere pressure incorporates nine broadly-defined minerals—magnetite,olivine, hypersthene, augite, quartz, plagioclase, orthoclase,leucite, and nepheline. The crystallization temperature of eachmineral is considered to be a smooth function of the compositionof the magmatic liquid. These mineral temperature equationsare obtained by multiple linear regression analysis of informationfrom published silicate systems and rock melting experiments.The nine equations are solved for any primary liquid, withinthe broad range of common magma types, to select the crystallizingmineral or minerals. Partition ratios from published experimentsand analyses of lavas and phenocrysts permit calculation ofthe composition of the crystallizing mineral assemblage. Subtractionof a small amount of that composition from the primary liquidyields a new liquid, which may be recycled to yield a sequenceof liquids during fractional crystallization. The crystallizationmodel handles assemblages of co-precipitating minerals, andcan trace progressive saturation in new minerals, substitutionof a new mineral for an old mineral, and cessation of crystallizationof a mineral. The sequences of minerals and liquids derivedfrom a broad set of primary liquids are geologically realistic,so the model is useful in predicting phenocrysts in volcanicrocks and events during crystallization of shallow intrusions.     

On the nature of the excess heat capacity of mixing

The excess vibrational entropy (ΔS _vib^ex) of several silicate solid solutions are found to be linearly correlated with the differences in end-member volumes (ΔV _i ) and end-member bulk moduli (Δκ _i ). If a substitution produces both, larger and elastically stiffer polyhedra, then the substituted ion will find itself in a strong enlarged structure. The frequency of its vibration is decreased because of the increase in bond lengths. Lowering of frequencies produces larger heat capacities, which give rise to positive excess vibrational entropies. If a substitution produces larger but elastically softer polyhedra, then increase and decrease of mean bond lengths may be similar in magnitude and their effect on the vibrational entropy tends to be compensated. The empirical relationship between ΔS _vib^ex, ΔV _i and Δκ _i , as described by ΔS _vib^ex = (ΔV _i  + mΔκ _i )f, was calibrated on six silicate solid solutions (analbite–sanidine, pyrope–grossular, forsterite–fayalite, analbite–anorthite, anorthite–sanidine, CaTs–diopside) yielding m = 0.0246 and f = 2.926. It allows the prediction of ΔS _vib^ex behaviour of a solid solution based on its volume and bulk moduli end-member data.     

基于假三元模型的透辉石-硬玉固溶体热力学性质计算

自然界中矿物多以固溶体形式存在,据其晶体化学特征计算热力学性质是开展矿物成因理论研究的基础。本文引入描述二元矿物固溶体热力学性质的假三元模型,计算得到了透辉石-硬玉固溶体系列的热力学性质。该模型通过构造一种高度有序的中间相,同时考虑长程和短程有序效应,基于热力学平衡态矿物固溶体自由能最低的规律,可以计算特定组分下矿物的平衡自由能、焓和熵等热力学参数。本文针对透辉石-硬玉固溶体体系,取绿辉石为其中间有序态,计算了其活度-成分关系和温度-组分相图等,发现绿辉石随温度升高的有序无序相变为一级相变,相变温度为1 148±25 K,与实验研究结果一致。本文获得的透辉石-绿辉石-硬玉体系的热力学参数可用于视剖面图方法研究MORB成分的岩石的榴辉岩相变质作用过程。     

First-principles calculations of equilibrium fractionation of O and Si isotopes in quartz,albite, anorthite,and zircon

In this study, we used first-principles calculations based on density functional theory to investigate silicon and oxygen isotope fractionation factors among the most abundant major silicate minerals in granites, i.e., quartz and plagioclase (including albite and anorthite), and an important accessory mineral zircon. Combined with previous results of minerals commonly occurring in the crust and upper mantle (orthoenstatite, clinoenstatite, garnet, and olivine), our study reveals that the Si isotope fractionations in minerals are strongly correlated with SiO₄ tetrahedron volume (or average Si–O bond length). The ³⁰Si enrichment order follows the sequence of quartz > albite > anorthite > olivine ≈ zircon > enstatite > diopside, and the ¹⁸O enrichment follows the order of quartz > albite > anorthite > enstatite > zircon > olivine. Our calculation predicts that measurable fractionation of Si isotopes can occur among crustal silicate minerals during high-temperature geochemical processes. This work also allows us to evaluate Si isotope fractionation between minerals and silicate melts with variable compositions. Trajectory for δ³⁰Si variation during fractional crystallization of silicate minerals was simulated with our calculated Si isotope fractionation factors between minerals and melts, suggesting the important roles of fractional crystallization to cause Si isotopic variations during magmatic differentiation. Our study also predicts that δ³⁰Si data of ferroan anorthosites of the Moon can be explained by crystallization and aggregation of anorthite during lunar magma ocean processes. Finally, O and Si isotope fractionation factors between zircon and melts were estimated based on our calculation, which can be used to quantitatively account for O and Si isotope composition of zircons crystallized during magma differentiation.     

Quantum mechanical potential surfaces and calculations on minerals and molecular clusters

Recently, ab-initio quantum mechanical potential surfaces calculated for silicate hydroxyacid molecules were used to extract covalent potentials for use in mineral physics calculations (Lasaga and Gibbs 1987). The calculations showed that these potentials are capable of generating the structure and physical properties of silicate minerals. In this paper we explore in more detail the suitability of various covalent potentials in mimicking the topography of the ab-initio potential surfaces. We also extend the use of such quantum-derived potentials in generating the structures of hydroxyacid dimers, trimers, and pentamers of silicate tetrahedra and in studying the structure and the dynamical properties of minerals and glasses.     

The effect of TiO2 and Fe2O3 on metapelitic assemblages at greenschist and amphibolite facies conditions: mineral equilibria calculations in the system K2O–FeO–MgO–Al2O3–SiO2–H2O–TiO2–Fe2O3

Mineral equilibria calculations in the system K₂O–FeO–MgO–Al₂O₃–SiO₂–H₂O–TiO₂–Fe₂O₃ (KFMASHTO) using thermocalc and its internally consistent thermodynamic dataset constrain the effect of TiO₂ and Fe₂O₃ on greenschist and amphibolite facies mineral equilibria in metapelites. The end‐member data and activity–composition relationships for biotite and chloritoid, calibrated with natural rock data, and activity–composition data for garnet, calibrated using experimental data, provide new constraints on the effects of TiO₂ and Fe₂O₃ on the stability of these minerals. Thermodynamic models for ilmenite–hematite and magnetite–ulvospinel solid solutions accounting for order–disorder in these phases allow the distribution of TiO₂ and Fe₂O₃ between oxide minerals and silicate minerals to be calculated. The calculations indicate that small to moderate amounts of TiO₂ and Fe₂O₃ in typical metapelitic bulk compositions have little effect on silicate mineral equilibria in metapelites at greenschist to amphibolite facies, compared with those calculated in KFMASH. The addition of large amounts of TiO₂ to typical pelitic bulk compositions has little effect on the stability of silicate assemblages; in contrast, rocks rich in Fe₂O₃ develop a markedly different metamorphic succession from that of common Barrovian sequences. In particular, Fe₂O₃‐rich metapelites show a marked reduction in the stability fields of staurolite and garnet to higher pressures, in comparison to those predicted by KFMASH grids.     

Interpreting prograde-growth histories of Al2SiO5 triple-point rocks using oxygen-isotope thermometry: an example from the Truchas Mountains, USA

Oxygen‐isotope compositions of kyanite, andalusite, prismatic sillimanite and fibrolite from the Proterozoic terrane in the Truchas Mountains, New Mexico differ from one another, suggesting that these minerals did not grow in equilibrium at the Al₂SiO₅ (AS) polymorph‐invariant point as previously suggested. Instead, oxygen‐isotope temperature estimates indicate that growth of kyanite, andalusite and prismatic sillimanite occurred at c. 575, 615 and 640 °C respectively. Temperature estimates reported in this paper are interpreted as those of growth for the different AS polymorphs, which are not necessarily the same as peak metamorphic temperatures for this terrane. Two distinct temperature estimates of c. 580 °C and c. 700 °C are calculated for most fibrolite samples, with two samples yielding clear evidence of quartz‐fibrolite oxygen‐isotope disequilibrium. These data indicate that locally, and potentially regionally, oxygen‐isotope disequilibrium between quartz and fibrolite may have resulted from rapid fibrolite nucleation. Pressures of mineral growth that were extrapolated from oxygen‐isotope thermometry results and calculated using petrological constraints suggest that kyanite and one generation of fibrolite grew during M1 at 5 kbar, and that andalusite, prismatic sillimanite and a second generation of fibrolite grew during M2 at 3.5 kbar. M1 and M2 therefore represent two distinct metamorphic events that occurred at different crustal levels. The ability of the AS polymorphs to retain δ¹⁸O values of crystallization make these minerals ideal to model prograde‐growth histories of mineral assemblages in metamorphic terranes and to understand more clearly the pressure–temperature histories of multiple metamorphic events.     

Recent developments and the future of low‐T calorimetric investigations in the Earth sciences: Consequences for thermodynamic calculations and databases

Low‐T calorimetry is an experimental science that measures the thermodynamic function heat capacity, C_p(T), from which the standard third‐law entropy (298.15 K), S°, is calculated. The recent technological development of relaxation calorimetry allows both new experimental strategies and types of C_p investigations to be made, which were previously not possible. The C_p measurements are fast and automated and can be made on mg‐sized mineralogical samples between 2 and 400 K. These advantages, when careful measurement procedures are used, permit better determinations of C_p(T) behaviour. The C_p of synthetic single‐crystal MgO was measured between 5 and 302 K, and S° calculated using relaxation calorimetry to further investigate the method's precision and accuracy. A number of synthetic and natural end‐member or nearly end‐member compositions of silicate garnet were investigated in the past via adiabatic calorimetry, an old and established technique, and more recently and extensively with the relaxation method. First C_p(T) and S° results, using relaxation calorimetry, have been obtained on spessartine (Mn₃Al₂Si₃O₁₂) and knorringite (Mg₃Cr₂Si₃O₁₂). Furthermore, reinvestigations on pyrope (Mg₃Al₂Si₃O₁₂), almandine (Fe₃Al₂Si₃O₁₂), grossular (Ca₃Al₂Si₃O₁₂) and andradite (Ca₃Al₂Si₃O₁₂), often on multiple samples, have resolved uncertainties and certain problems with published thermodynamic data. S° can be affected by various low‐T physical phenomena, such as cooperative magnetic phase transitions or Schottky anomalies at temperatures of <15 K, which were not described fully in some older adiabatic calorimetric studies. New C_p results show that small differences in the thermodynamic behaviour between some natural and synthetic silicates may exist as demonstrated by extensive work on grossular. Important and “new” research questions on the thermodynamic behaviour of minerals are coming to light and are being investigated. The C_p behaviour and S° values for six silicate garnet end‐members are analysed and the latter are compared to the “best fit or optimized” S° values given in various internally consistent thermodynamic databases. Conclusions are drawn on what types and directions of calorimetric study are required in order to obtain better thermodynamic property determinations of minerals, as well as achieving a better understanding of the underlying microscopic physical behaviour that determines the macroscopic C_p and S° functions.     

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.