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 We have developed models for the thermody-namic properties of nephelines, kalsilites, and leucites in the simple system NaAlSiO4−KAlSiO4−Ca0.5AlSiO4−SiO2−H2O that are consistent with all known constraints on subsolidus equilibria and thermodynamic properties, and have integrated them into the existing MELTS software package. The model for nepheline is formulated for the simplifying assumptions that (1) a molecular mixing-type approximation describes changes in the configurational entropy associated with the coupled exchange substitutions □Si?NaAl and □Ca? Na2 and that (2) Na+ and K+ display long–range non-convergent ordering between a large cation and the three small cation sites in the Na4Al4Si4O16 formula unit. Notable features of the model include the prediction that the mineral tetrakalsilite (“panunzite”, sensu stricto) results from anti-ordering of Na and K between the large cation and the three small cation sites in the nepheline structure at high temperatures, an average dT/dP slope of about 55°/kbar for the reaction over the temperature and pressure ranges 800–1050 °C and 500–5000 bars, roughly symmetric (i.e. quadratic) solution behavior of the K–Na substitution along joins between fully ordered components in nepheline, and large positive Gibbs energies for the nepheline reciprocal reactions and and for the leucite reciprocal reaction

Notes: Abstract The compositions of liquids coexisting with experimentally grown crystals of olivine, plagioclase, clinopyroxene, orthopyroxene, leucite, spinel, rhombohedral oxide, melilite and potassium feldspar are used to define, through mass action expressions of liquid/solid equilibrium, compositional derivatives of the Gibbs free energy of mixing of naturally occuring silicate liquids as a function of temperature, pressure and the fugacity of oxygen. The available experimental data describe these derivatives over a range of compositions which includes basic magmas. Therefore, for silicate liquids in this composition range, the topology of the Gibbs free energy of mixing can be approximated from experimental determinations of its derivatives. The majority of the existing thermodynamic data on the liquid phase is consistent with the application of regular solution theory to model the free energy of mixing. Strictly symmetric, temperature and pressure independent, regular solution interaction parameters are calibrated from this phase equilibrium data using regression techniques which have their basis in inverse theory. These techniques generate numerically stable interaction parameters which incorporate inter-variable correlation and account for experimental uncertainty. The regular solution model fits the available data on anhydrous silicate liquids to within the accuracy of the thermodynamic database +/−550 cals). Extensions to regular solution theory allow water solubility in more silica rich liquids to be modelled somewhat less accurately (+/−750 cals). The topology of the excess free energy of mixing surface is strongly asymmetric, possessing a single multicomponent saddle point which defines a spinodal locus. Given this prediction of a multicomponent spinode, a mathematical procedure based upon minimisation of the Gibbs free energy of mixing is developed for the calculation of the compositions of coexisting immiscible liquids. Predicted binodal compositions substantially agree with elemental liquid/liquid partitioning trends observed in lavas. Calculations suggest that an immiscible dome, in temperature-composition space, intersects the liquidus field of the magma type tholeiite. Immiscible phenomena are predicted at sub-liquidus temperatures for the bulk compositions of more basic or alkalic lavas, but are absent in more siliceous rock types for temperatures of the metastable liquid down to 900 K. The regular solution model is used in four petrological applications. The first concerns a prediction of the binary olivine-liquid phase diagram. The calculated geometry exhibits a minimum near Fa75, which, though not in accord with experimental results on the pseudobinary system, compares quite favorably with olivine-liquid phase equilibria interpreted from rhyolites, namely that the olivine phenocrysts of rhyolites are more iron rich than their coexisting liquids. The second petrological example concerns estimating the depth of the source regions of several basic lavas whose compositions cover a range from ugandite to basaltic andesite. The third application is a calculation of the saturation temperatures and compositions of plagioclase and olivine in four experimental basaltic liquids and a prediction of the liquidus temperatures and first phenocryst compositions of the Thingmuli lava series of Eastern Iceland. Lastly, enthalpies of fusion are computed for a variety of stoichiometric compounds of geologic interest. These demonstrate good agreement with calorimetrically measured quantities

Notes: Abstract The Big Jim complex is a concentrically zoned ultramafic to felsic plutonic complex which intruded the pelitic Chiwaukum schist. Most of the major plutonic rock types (from websterite through hornblendite, gabbronorite, hornblende gabbro and diorite, to granodiorite) enclose harzburgite and metaperidotite xenoliths similar to foliated metaperidotite lenses included in the Chiwaukum schist. The larger xenoliths preserve tectonite fabrics. All have Mg#'s (mole fraction MgO/(MgO+FeO*)) from 0.90 to 0.89, the same as those of Chiwaukum metaperidotites, and distinctly different from undeformed Big Jim dunite (Mg#'s 0.84 to 0.82) and websterite (0.82 to 0.78). Contact relations indicate widespread, stepwise replacement of harzburgite by pyroxenite, hornblendite, gabbro and diorite. Thermodynamic modelling using an expanded regular solution model for silicate liquids (Ghiorso 1985; Ghiorso and Carmichael 1985) predicts that reaction between olivine (Fo90) and a liquid with the composition of Big Jim diorite +1.5 wt% H2O, at 1,100° C and 3 kb, would produce websterite (Mg#'s 0.75 to 0.81) and dunite (0.79 to 0.82). This process is exothermic and results in a negative change in volume, since it increases total solid mass. Under conditions of decreasing temperature, modelled crystal fractionation with assimilation of olivine reproduces important features of the chemical variation observed in the Big Jim complex where crystal fractionation alone fails. The Big Jim complex has affinities with other ultramafic to felsic plutonic complexes such as the Bear Mountain complex (Snoke et al. 1981, 1982) and the Emigrant Gap complex (James 1971). The latter have wehrlite and clinopyroxenite, rather than websterite, but both have concentric zoning, with olivine-bearing rock types surrounded by successively more felsic pyroxenite, gabbro and diorite. In general, concentrically zoned complexes of this type may form where magma reacts with mantle-derived wall rock or ultramafic cumulates. Assimilation of peridotite in fractionating magma may be important in subduction-related magmatic arcs.

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 Numerical examples of the approach described in Part I of this series (Ghiorso, 1985) are presented in this paper. These examples include the calculation of the compositions and proportions of liquid and solid phases produced during (1) the equilibrium crystallization of a basaltic andesite at 1 bar, (2) the fractional crystallization of an olivine tholeiite at 1 bar and elevated pressures, (3) the fractional and equilibrium crystallization of an olivine boninite at 1 bar, and (4) the (a) isothermal and (b) isenthalpic assimilation of olivine (Fo90) into a liquid/solid assemblage of quartz dioritic composition at ∼1,125° C and 3 kbars. The numerical results on the crystallization of the basaltic andesite are verified by comparison with experimental data while those calculations performed using olivine tholeiitic and olivine boninitic compositions are favorably compared against whole rock and mineral analytical data and petrographic and field observations. In each of the examples presented, the heat effects associated with the modelled process are calculated (e.g. heat of crystallization, heat of assimilation), and free energies of crystallization are examined as a function of the degree of mineral supersaturation. The former quantities are on the order of 173 cal/grm for the cooling and fractional crystallization of an olivine tholeiite to a rhyolitic residuum (corresponding to a 400° C temperature interval). The latter represents an important petrological parameter, in that it quantifies the driving force for the rate of crystal growth and rate of nucleation in magmatic systems. Calculated free energies of crystallization are small (on the order of hundreds of calories per mole per 25° C of undercooling) which indicates that the kinetics of crystallization in magmatic systems are affinity controlled. Melt oxygen fugacity and the degree of oxygen metasomatism play a major role in controlling the fractionation trends produced from crystallizing basaltic liquids. Calculations suggest that in order to generate a silica rich residuum and the characteristic iron enrichment trend during the fractional crystallization of a tholeiitic basalt, the magma must crystallize esentially along $$f_{{\text{O}}_{\text{2}} } $$ buffer. This buffered state can be maintained by exchange of oxygen (via hydrogen diffusion) between the magma and the surrounding country rocks or by magmatic oxidation-reduction equilibria. Additional calculations indicate the possibility that oxygen exchange may be unnecessary if the magma contains sufficient sulfur to maintain the system along an S2/SO2 oxygen buffer during the initial stages of crystallization.

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 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.