ALBERT

Notes: Abstract The model for the thermodynamic properties of multicomponent pyroxenes (Part I) is calibrated for ortho- and clinopyroxenes in the quadrilateral subsystem defined by the end-member components Mg2Si2O6, CaMgSi2O6, CaFeSi2O6, and Fe2Si2O6. This calibration accounts for: (1) Fe-Mg partitioning relations between orthopyroxenes and augites, and between pigeonites and augites, (2) miscibility gap features along the constituent binary joins CaMgSi2O6-Mg2Si2O6 and CaFeSi2O6-Fe2Si2O6, (3) calorimetric data for CaMgSi2O6-Mg2Si2O6 pyroxenes, and (4) the P-T-X systematics of both the reaction pigeonite=orthopyroxene+augite, and miscibility gap featurs, over the temperature and pressure ranges 800–1500°C and 0–30 kbar. The calibration is achieved with the simplifying assumption that all regular-solution-type parameters are constants independent of temperature. It is predicated on the assumptions that: (1) the Ca-Mg substitution is more nonideal in Pbca pyroxenes than in C2/c pyroxenes, and (2) entropies of about 3 and 6.5 J/K-mol are associated with the change of Ca from 6- to 8-fold coordination in the M2 site in magnesian and iron C2/c pyroxenes, respectively. The model predicts that Fe2+-Mg2+ M1-M2 site preferences in C2/c pyroxenes are highly dependent on Ca and Mg contents, with Fe2+ more strongly preferring M2 sites both in Ca-rich C2/c pyroxenes with a given Fe/(Fe+Mg) ratio, and in magnesian C2/c pyroxenes with intermediate Ca/(Ca+Fe+Mg) ratios. The proposed model is internally consistent with our previous analyses of the solution properties of spinels, rhombohedral oxides, and Fe-Mg olivines and orthpyroxenes. Results of our calibration extend an existing database to include estimates for the thermodynamic properties of the C2/c and Pbca pyroxene end-members clinoenstatite, clinoferrosilite, hedenbergite, orthodiopside, and orthohedenbergite. Phase relations within the quadrilateral and its constitutent subsystems are calculated for temperatures and pressures over the range 800–1700°C and 0–50 kbar and compare favorably with experimental constraints.

Notes: Abstract A model is proposed for the thermodynamic properties of multicomponent pyroxenes in the composition space defined by the end-member component CaMgSi2O6 and the exchange components Fe(Mg)-1, TiAl2(MgSi2)-1, Fe3+(Al)-1, Fe3+Al(MgSi)-1, and Mg(Ca)-1. It is formulated for the simplifying assumptions that: (1) a molecular mixing type approximation describes changes in the molar configurational entropy associated with the coupled exchange substitutions TiAl2⇔MgSi2, Fe3+Al⇔MgSi, and Al2⇔MgSi (and their ferroan equivalents), and (2) Fe2+ and Mg2+, and Al3+ and Fe3+ display long-range non-convergent ordering between M2 and octahedral M1 sites, and octahedral M1 and tetrahedral sites, respectively. The molar vibrational Gibbs energy is described by a Taylor expansion of second degree in seven linearly independent composition and ordering variables, which is extended to third degree to account for asymmetry in the mixing of Ca and Mg, and Ca and Fe on the M2 site, and is further modified for the assumption that the standard state properties of Ca end-member components of clinopyroxenes are linearly dependent on the coordination number of Ca2+ on the M2 site. The model is shown to be consistent with miscibility gap feaures of pyroxenes in the system CaMgSi2O6−CaTiAl2O6−CaAl2SiO6. In subsequent papers, the model is calibrated for the simplifying assumptions that: (1) all regular-solution-type parameters are constants independent of temperature, (2) Pbca and C2/c end-members have identical heat capacities and coefficients of thermal expansion and compressibility, and (3) the heat capacities and coefficients of thermal expansion and compressibility are zero for all reciprocal reactions relating Pbca and pigeonite or high-calcium pyroxene C2/c endmember components.

Notes: Abstract A model is developed for the thermodynamic properties of Fe2+−Mg2+-aluminate-titanate-ferrite spinels of space group Fd3m. The model incorporates an expression for the configurational entropy of mixing which accounts for long-range order over tetrahedral and octahedral sites. Short-range order or departures from cubic symmetry are not considered. The non-configurational Gibbs energy is formulated as a second degree Taylor expansion in six linearly independent composition and ordering variables. The model parameters are calibrated to reproduce miscibility gap constraints, order-disorder phenomena in MgAl2O4 and MgFe2O4, and Fe2+−Mg2+ partitioning data between olivine and: (1) aluminate spinels; (2) ferrite spinels; (3) titanate spinels; (4) mixed aluminate-ferrite spinels. This calibration is achieved without invoking non-configurational excess entropies of mixing. The model predicts that the ordering state of FeAl2O4 is more normal than that of MgAl2O4. It also successfully accounts for heat of solution measurements and activity-composition relations in the constituent binaries. Phase equilibrium constraints require that the structure of Fe3O4 is more inverse than random at all temperatures and that Mg2+ has a strong tetrahedral site preference with respect to that of Fe2+. The analysis suggests that in the titanates short range order on octahedral sites may be significant at temperatures as high as 1300° C. Constraints developed from calibrating the thermodynamic properties of Fe2+−Mg2+-aluminatetitanate-ferrite spinel solid solutions permit extension of the database of Berman (1988) to include estimates of the end-member properties of hercynite (FeAl2O4), ulvöspinel (Fe2TiO4), MgFe2O4 and cubic Mg2TiO4. In constructing these estimates, provision is made for low-temperature magnetic entropy contributions and the energetic consequences of disordering the aluminates and the ferrites. These estimates are consistent with all of the available low-temperature adiabatic calorimetry, high-temperature heat content, and heat of solution measurements on the end-members. The analysis implies that there is a substantial heat capacity anomaly in the range 300°–900° C associated with disordering of the MgAl2O4 structure while that in FeAl2O4 becomes significant at temperatures above 700° C. The same heat capacity response in the ferrites indicates that the order/disorder transformation is coupled to the antiferromagnetic-paramagnetic transition in MgFe2O4 but takes place well above the ferrimagnetic-paramagnetic transition in magnetite. The proposed model is internally consistent with solution theory reported elsewhere for Fe2+−Mg2+ olivines and orthopyroxenes (Sack and Ghiorso 1989), rhombohedral oxides (Ghiorso 1990a) and the remaining end-member properties of Berman (1988).

Notes: Abstract A thermodynamic model for the Gibbs free energy of igneous pyroxenes with the general formula [Na, Ca, Fe2+, Mg]M2[Fe2+, Mg, Ti, Al, Fe3+]M1[Al, Fe3+, Si]TetSiO6 is calibrated from experimentally determined compositions of coexisting pyroxene and silicate melt. The model is based upon the general formulation, and relies upon the calibration of the “quadrilateral” subsystem, previously published by the present authors. The calibration database of pyroxene-liquid equilibria spans a broad spectrum of temperature, pressure and oxygen fugacity conditions, ranging from 1000°–1600°C, 0.001–30 kbar and iron-wüstite to air. Chemical potentials of endmember pyroxene components as well as exchange potentials between pyroxenes and coexisting liquids are defined utilizing the present authors' thermodynamic melt model. Model parameters are extracted from these relations by regression analysis. The resulting model and derivative endmember properties are internally consistent with an existing standard state thermodynamic database. The success of the model and its applicability to igneous petrogenesis are demonstrated by comparing calculated and experimentally determined liquidus compositions, temperatures and symmetry states for pyroxenes crystallizing from a variety of silicate melts, ranging in composition from tholeiites and angrites through rhyolites to potash ankaratrites.

Notes: Abstract A solution model is developed for rhombohedral oxide solid solutions having compositions within the ternary system ilmenite [(Fe 2+ s Ti 4+ 1−s ) A (Fe 2+ 1−s Ti 4+ s ) B O3]-geikielite [(Mg 2+ t Ti 4+ 1−t ) A (Mg 2+ 1−t Ti 4+ t ) B O3]-hematite [(Fe3+) A (Fe3+) B O3]. The model incorporates an expression for the configurational entropy of solution, which accounts for varying degrees of structural long-range order (0≤s, t≤1) and utilizes simple regular solution theory to characterize the excess Gibbs free energy of mixing within the five-dimensional composition-ordering space. The 13 model parameters are calibrated from available data on: (1) the degree of long-range order and the composition-temperature dependence of the $$R\bar 3c - R\bar 3$$ transition along the ilmenite-hematite binary join; (2) the compositions of coexisting olivine and rhombohedral oxide solid solutions close to the Mg−Fe2+ join; (3) the shape of the miscibility gap along the ilmenite-hematite join; (4) the compositions of coexisting spinel and rhombohedral oxide solid solutions along the Fe2+−Fe3+ join. In the course of calibration, estimates are obtained for the reference state enthalpy of formation of ulvöspinel and stoichiometric hematite (−1488.5 and −822.0 kJ/mol at 298 K and 1 bar, respectively). The model involves no excess entropies of mixing nor does it incorporate ternary interaction parameters. The formulation fits the available data and represents an internally consistent energetic model when used in conjuction with the standard state thermodynamic data set of Berman (1988) and the solution theory for orthopyroxenes, olivines and Fe−Mg titanomagnetite-aluminate-chromate spinels developed by Sack and Ghiorso (1989, 1990a, b). Calculated activity-composition relations for the end-members of the series, demonstrate the substantial degree of nonideality associated with interactions between the ordered and disordered structures and the dominant influence of the miscibility gap across much of the ternary system. The predicted shape of the miscibility gap, and the orientation of tie-lines relating the compositions of coexisting phases, display the effects of coupling between the excess enthalpy of solution and the degree of long-range order. One limb of the miscibility gap follows the composititiontemperature surface corresponding to the ternary $$R\bar 3 - R\bar 3c$$ second-order transition.

Notes: Abstract A new thermodynamic formulation of the Fe−Ti oxide geothermometer/oxygen barometer is developed. The method is based upon recently calibrated models for spinel solid solutions in the quinary system (Fe2+, Mg)(Al,Fe3+,Cr)2O4−(Fe2+, Mg)2TiO4 by Sack and Ghiorso, and rhombohedral oxides in the quaternary system (Fe2+,Mg,Mn)TiO3−Fe2O3 (this paper). The formulation is internally consistent with thermodynamic models for (Fe2+,Mg)-olivine and -orthopyroxene solid solutions and end-member thermodynamic properties tabulated by Berman. The constituent expressions account for compositional and temperature dependent cation ordering and reproduce miscibility gap features in all of the component binaries. The calibration does not account for the excess Gibbs energy resulting from compositional and temperature dependent magnetic ordering in either phase. This limits application of the method to assemblages that equilibrated at temperatures above 600° C. Practical implementation of the proposed geothermometer/oxygen barometer requires minimal use of projection algorthms in accommodating compositions of naturally occurring phases. The new formulation is applied to the estimation of temperature and oxygen fugacity in a wide variety of intermediate to silicic volcanic rocks. In combination with previous work on olivine and orthopyroxene thermodynamics, equilibration pressures are computed for a subset of these volcanics that contain the assemblage quartz, oxides and either ferromagnesian silicate. The calculated log10 f O 2-T relations are reflected in coexisting ferromagnesian mineral assemblages. Volcanics with the lowest relative oxygen fugacity (Δlog10 f O 2) are characterized by the assemblage olivine-quartz, those with slightly higher Δ log10 f O 2 s, by the assemblage orthopyroxene-quartz. The sequence proceeds with the necessary phases biotite-feldspar, then hornblende-quartz-clinopyroxene, and finally at the highest Δ log10 f O 2 s, sphene-quartz-clinopyroxene. Quantitative analysis of these trends, utilizing thermodynamic data for the constituent phases, establishes that, in most cases, the T-log10 f O 2value computed from the oxides is consistent with the compositions of coexisting silicate phases, indicating that phenocryst equilibrium was achieved prior to eruption. There is, however, considerable evidence of oxide-silicate disequilibrium in samples collected from more slowly cooled domes and obsidians. In addition, T-log10 f O 2trends from volcanic rocks that contain biotite and orthopyroxene are interpreted to imply a condition of Fe2+−Mg exchange disequilibrium between orthopyroxene and coexisting ferromagnesian silicates and melt. It is suspected that many biotite-feldspar-quartz-orthopyroxene bearing low temperature volcanic rocks inherit orthopyroxene xenocrysts which crystallized earlier in the cooling history of the magma body.