ALBERT

Notes: Abstract Lasaga's (1982) Master Equation for crystal growth is solved for multicomponent systems in situations which allow for coupled diffusion of melt species. The structure of the solution is explored in some detail for the case of a constant diffusion coefficient matrix. Incorporating these results, the growth of plagioclase is modeled in undercooled tholeiitic melts by approximating interface growth rates with (1) a reduced growth rate function and with (2) calculated solid-liquid solution properties obtained from the silicate liquid solution model of Ghiorso et al. (1983; appendix of Ghiorso 1985). For this purpose algorithms are provided for estimating the liquidus temperature or the chemical affinity of a multicomponent solid solution precipitating from a complex melt of specified bulk composition. Compositional trends in initial solids produced by successive degrees of undercooling are opposite to those predicted in the binary system NaAlSi3O8-CaAl2Si2O8. Calculations suggest that the solid phase and interface melt compositions rapidly approach a “steady state” for a given degree of undercooling. Consequently, the overall isothermal growth rate of plagioclase forming from tholeiitic melts appears to be entirely diffusion controlled. In magmatic systems the multicomponent growth equations allow for the formation of oscillatory zoned crystals as a consequence of the “couplingr” between interface reaction kinetics and melt diffusion. The magnitude of this effect is largely dependent upon the asymmetry of the diffusion coefficient matrix. Methods are described to facilitate the calibration of diffusion matrices from experimental data on multicomponent penetration curves. Experimental results (Lesher and Walker 1986) on steady state Soret concentration profiles resulting from thermal diffusion in MORB and andesitic liquids are analyzed using the theory of multicomponent linear irreversible thermodynamics. Under conditions where the entropy production is minimized, a linear relationship is derived between liquid chemical potentials and temperature. This relationship is utilized to evaluate the validity of the solution model of Ghiorso et al. (1983) in melts up to 300° C above their liquidus. The results indicate that configurational entropies are accurately modeled for MORB and andesite bulk compositions. The modeling fails in two four-component systems tested. Equations are derived which allow the calibration of multicomponent regular solution parameters from steady state Soret arrays. An algorithm is demonstrated which permits the calculation of steady state Soret concentration profiles, given an overall bulk melt composition and temperature gradient. This algorithm uses the liquid solution properties of Ghiorso et al. (1983) and constants obtained from the experimental measurements of Lesher and Walker (1986).

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 Estimates of the oxygen fugacity of the source regions of martian shergottites are obtained from coexisting Fe-Ti oxides and from Fe3+/Fe2+ ratios in pyroxenes. These estimates are one to four log10 units lower than previously reported values, and indicate that the shergottite source region has an oxidation state about two or three orders of magnitude below the quartz-fayalite-magnetite (QFM) oxygen reference buffer. An approximation of the major element composition of the shergottite source region is obtained utilizing the MELTS petrologic modeling software. A bulk composition of the shergottite source region is derived, which is consistent with the generation of sher-gottites upon 3% partial melting at 10 kbar. This composition is decidedly less enriched in Na2O and FeOTot than previous estimates. When compared to the bulk composition of the terrestrial mantle, the source region of the shergottites is lower in CaO, Al2O3, Na2O and TiO2, and higher in MgO.

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 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 Activity/composition relations are presented for high-structural state feldspars whose bulk compositions lie within the ternary system NaAlSi3O8 CaAl2Si2O8-KAlSi3O8. The expressions are parameterized from the data for coexisting feldspars of Seck (1971a) using an asymmetric regular solution approximation for the excess Gibbs free energy of mixing and an Al-avoidance model for the configurational entropy of solution. The solution properties of the plagioclase and alkali-feldspar binaries have been made to conform to the recent work of Thompson and Hovis (1979) and Newton et al. (1980). Using the proposed model the ternary feldspar solvus is extrapolated in temperature (up to 1,500° C) and pressure (up to 5kbars). A new two-feldspar geothermometer is presented which provides somewhat more reasonable estimates of crystallization temperatures than the equations and graphs of Stornier (1975), Powell and Powell (1977), Brown and Parsons (1981) and Haselton et al. (1983). In conjunction, some criteria are suggested for establishing the existence of “equilibrium” tie-lines between coexisting ternary feldspars in rhyolites and trachytes. Calculated values of the activity of KAlSi3O8 in plagioclase are examined in some detail. These compare favorably with independent estimates obtained from experimentally grown plagioclases precipitating at liquidus temperatures from igneous rocks of widely varying alkali contents.

Notes: Abstract A thermodynamic solution model is developed for minerals whose compositions lie in the two binary systems Mg2SiO4-Fe2SiO4 and Mg2Si2O6-Fe2Si2O6. The formulation makes explicit provision for nonconvergent ordering of Fe2+ and Mg2+ between M1 and M2 sites in orthopyroxenes and non-zero Gibbs energies of reciprocal ordering reactions in both olivine and orthopyroxene. The calibration is consistent with (1) constraints provided by available experimental and natural data on the Fe-Mg exchange reaction between olivine and orthopyroxene ± quartz, (2) site occupancy data on orthopyroxenes including both crystallographic refinements and Mössbauer spectroscopy, (3) enthalpy of solution data on olivines and orthopyroxenes and enthalpy of disordering data on orthopyroxene, (4) available data on the temperature and ordering dependence of the excess volume of orthopyroxene solid solutions, and (5) direct activity-composition determinations of orthopyroxene and olivine solid solutions at elevated temperatures. Our analysis suggests that the entropies of the exchange [Mg(M2)Fe(M1)⇔Fe(M2)Mg(M1)] and reciprocal ordering reactions [Mg(M2)Mg(M1)+ Fe(M2)Fe(M1)⇔Fe(M2)Mg(M1)+Mg(M2)Fe(M1)] cannot differ significantly (± 1 cal/K) from zero over the temperature range of calibration (400°–1300° C). Consideration of the mixing properties of olivine-orthopyroxene solid solutions places tight constraints on the standard state thermodynamic quantities describing Fe-Mg exchange reactions involving olivine, orthopyroxene, pyralspite garnets, aluminate spinels, ferrite spinels and biotite. These constraints are entirely consistent with the standard state properties for the phasesα-quartz,β-quartz, orthoenstatite, clinoenstatite, protoenstatite, fayalite, ferrosilite and forsterite which were deduced by Berman (1988) from an independent analysis of phase equilibria and calorimetric data. In conjunction with these standard state properties, the solution model presented in this paper provides a means of evaluating an internally consistent set of Gibbs energies of mineral solid solutions in the system Mg2SiO4-Fe2SiO4-SiO2 over the temperature range 0–1300° C and pressure interval 0.001–50 kbars. As a consequence of our analysis, we find that the excess Gibbs energies associated with mixing of Fe and Mg in (Fe, Mg)2SiO4 olivines, (Fe, Mg)3Al2Si3O12 garnets, (Fe, Mg)Al2O4 and (Fe, Mg)Fe2O4 spinels, and K(Mg, Fe)3AlSi3O10(OH)2 biotites may be satisfactory described, on a macroscopic basis, with symmetric regular solution type parameters having values of 4.86±0.12 (olivine), 3.85±0.09 (garnet), 1.96±0.13 (spinel), and 3.21±0.29 kcals/gfw (biotite). Applications of the proposed solution model demonstrate the sensitivity of petrologic modeling to activity-composition relations of olivine-orthopyroxene solutions. We explore the consequences of estimating the activity of silica in melts forming in the mantle and we develop a graphical geothermometer/geobarometer for metamorphic assemblages of olivine+orthopyroxene+quartz. Quantitative evaluation of these results suggests that accurate and realistic estimates of silica activity in melts derived from mantle source regions,P-T paths of metamorphism and other intensive variables of petrologic interest await further refinements involving the addition of “trace” elements (Al3+ and Fe3+) to the thermodynamic formulation for orthopyroxenes.

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 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.