Contrib Mineral Petrol (1996) 126: 1–24 C Springer-Verlag 1996

R.G. Berman ? L.Ya. Aranovich

Optimized standard state and solution properties of mineralsI. Model calibration for olivine, orthopyroxene, cordierite,garnet, and ilmenite in the system FeO-MgO-CaO-Al2O3-TiO2-SiO2

Received: 14 September 1994y Accepted: 20 March 1996

Abstract An internally consistent set of standard stateand mixing properties has been derived for olivine, or-thopyroxene, garnet, cordierite, and ilmenite in the sys-tem FeO-MgO-CaO-Al2O3-TiO2-SiO2-H2O from analy-sis of relevant phase equilibrium and thermophysicaldata. Solubility of Al2O3 in orthopyroxene is accountedfor in addition to Fe-Mg mixing. Added confidence inthe retrieved properties stems from the representationwithin reasonable uncertainties of data for seven linearlydependent Fe-Mg exchange equilibria, as well as nettransfer equilibria, among the above phases. Critical tosuccessful analysis was the extension of the mathemati-cal programming technique to include bulk compositionconstraints which force an observed assemblage of fixedcomposition to be stable at experimentally studied condi-tions. The final optimization reproduces the extremelytight constraints on endmember properties while invok-ing very simple macroscopic solution models that affordan excellent opportunity for extrapolation beyond thedata considered in this study. Compatibility among theexperimental data is improved markedly by incorpora-tion of recently published Cp data on pyrope and forster-ite. Electrochemical data defining the oxygen fugacityof Fe-Fa-Qz, Fa-Mt-Qz, and Mt-Hm allow excellentcompatibility of almandine thermochemical propertiesderived from phase equilibrium data obtained at bothreducing (Fe-Wst) and oxidizing (Hm-Mt) conditions.Analysis of the combined data involving endmembersand solid solutions removes many of the ambiguities inmixing property magnitudes that arise in analyses ofmore restricted sets of data. In addition, the consider-ation of the solid solution data allows further refinementof some endmember properties. Nonideal mixing

parameters, although correlated, are well defined by thecombination of experimental data, withGOl

ex.GIlmex .GGt

ex.GOpx

ex .GCdex , and 0.7,WOl

G ,4.1 kJyatom of iso-morphous Fe-Mg at 1000 K. Experiments defining theAl2O3 solubility of Opx in equilibrium with Gt andCd1Qz define negative Fe-Al interactions that have animportant effect on Fe-Mg partitioning in Opx. Applica-tions of this data set to high-grade metamorphic rocksare described in a companion paper, published as part IIof the present work.

Introduction

Considerable efforts have been made over the last decadeto derive internally consistent thermodynamic data ap-plicable to quantitative petrologic and thermobarometriccalculations (e.g. Berman 1988; Aranovich and Pod-lesskii 1989; Holland and Powell 1990; Perchuk 1991).Accumulation of a large number of high quality experi-mental data on petrologically significant mineral equi-libria, together with reasonably sophisticated mathemat-ical methods for their analysis, has led to the creation ofinternally consistent data sets of standard thermodynam-ic properties of the main rock-forming minerals (e.g.Berman 1988; Holland and Powell 1990). Application ofthese datasets for the purpose of deciphering pressuresand temperatures of geologic processes, although oftensuccessful, is not without shortcomings (e.g. Aranovichand Podlesskii 1989; Sack and Ghiorso 1989). First,these datasets have been derived largely without consid-eration of the growing body of high quality experimentaldata on solid solution bearing equilibria, thus omittingvery important information that may influence thederived standard state mineral properties. Secondly, anarbitrary combination of standard state thermodynamicproperties with solid solution models will not in generalgive results compatible with available experimental data,thus compromising the ability to discern reliably evenrelative differences in computedP-T conditions in anyparticular region. On the other hand, thermodynamic

R.G. Berman (✉)Geological Survey of Canada, 601 Booth Street,Ottawa, Ontario, Canada K1A 0E8

L.Ya. AranovichInstitute of Experimental Mineralogy, Chernogolovka, Russia

Editorial responsibility: K. Hodges

2

systematics based entirely on direct application of exper-imental data for specific equilibria in complex chemicalsystems (e.g. Aranovich 1991) suffer from the fact thatthe different equilibria may not be thermodynamicallyconsistent with one another or that the derived or impliedstandard state properties of minerals are often inconsis-tent with more directly constraining experiments in theboundary systems.

In order to meet the shortcomings of these two differ-ent approaches, endmember and solid solution propertiesmust be determined simultaneously from available exper-imental data. “Purists” may worry that the assumptionsand complexities inherent in treatment of solid solutionswill degrade the quality of endmember propertiesderived on their own. The reader should bear in mind,however, that the same set of constraints on endmemberproperties is retained for the combined analysis. With themathematical programming technique used to processall data in this study (see next section), the endmemberconstraints restrict the range of permissible solutionproperties that are compatible with all phase equilibriumdata, while the solid solution experiments also providefor refinement of the endmember properties, but onlyfrom within the range of the direct experimental con-straints on endmembers.

A general problem with treatment of phase equilibri-um data involving solid solutions is the nonuniqueness ofresulting solution properties (e.g. Hackler and Wood1989; von Seckendorff and O’Neill 1993). This resultsfrom the fact that phase equilibrium data yield con-straints on the differences in thermodynamic propertiesof reactants and products, and solution properties for onephase depend on assumed properties for other phasesinvolved in the analyzed equilibria. Calorimetric datacan be used to directly constrain mixing properties, butuncertainties associated with calorimetric measure-ments are large compared to the magnitude of the mixingenergetics (e.g. Newton et al. 1977; Chatillon-Colinetet al. 1983; Geiger et al. 1987). The most appropriateway to address this problem is not just to analyzesomecombinations of phase equilibria for which good experi-mental data have been obtained (e.g. Wood and Holloway1984; Wood 1987; Fei and Saxena 1986; Berman 1990),but to utilize all linear combinations of equilibria thathave been experimentally studied such that there are in-ternal checks on derived properties for all phases. Ef-forts that incorporate complete thermodynamic “cycles”like this should produce not only the “best” set of inter-nally consistent standard state and solution data, butshould also allow the best applications and extrapolationsof experimental results obtained in relatively simplechemical systems to natural systems of more general ge-ological interest. This paper reports the first results ofsuch a project, in which four complete “cycles” involvingseven Fe-Mg exchange equilibria as well as net transferreactions among olivine, orthopyroxene, cordierite, gar-net and ilmenite are analyzed in the system FeO-MgO-CaO-Al2O3-TiO2-SiO2-H2O (FMCATSH). The results ofthis study represent a key part of work in progress to

refine and extend the thermodynamic database presentedby Berman (1988). Applications of this data set to high-grade metamorphic rocks are described in a companionpaper (Aranovich and Berman 1996b).

Mathematical treatment

In keeping with the premise that a successful phase equi-librium experiment measureschangesin either mineralcompositions or proportions, we analyzed these data us-ing mathematical programming (MAP), a technique tosolve systems of inequalities that, in the present context,convey the sense of thechangein Gibbs free energy ofreaction or mole fraction product observed in each exper-iment. We consider this mathematical technique advanta-geous for treating phase equilibrium data for three pri-mary reasons. First, the technique provides the means toanalyze rigorously half-brackets, i.e. experiments inwhich a particular equilibrium was approached from on-ly one direction. In addition to points related to univari-ant equilibria discussed elsewhere (Berman et al. 1986;Berman 1988), analysis of half-brackets has paramountimportance in assessing phase equilibria involving solidsolutions because many portions of the experimentaldata base consist of half-brackets over significant com-positional ranges. Second, MAP allows the incorpora-tion of important experimental observations on solid so-lutions that we refer to as bulk composition constraints.These constraints are mathematical inequalities express-ing that assemblage A with specified mineral composi-tions (bulk composition) is more (or less) stable thanassemblage B with specified mineral compositions (seelater in this section). Third, MAP allows for explicit in-corporation of all experimental uncertainties, thusproviding a rigorous means to assess the compatibility ofdiverse types of data. Our own experience with regres-sion techniques (e.g. Aranovich et al. 1985) indicatesthat when all experimental data are not reproduced with-in their uncertainties, it is difficult to determine whetherit is the data weighting (implicit or otherwise) or a modelinadequacy that prevents appropriate representation ofall experimental observations. We have not attempted toestimate uncertainties in thermodynamic variables be-cause one of the most significant sources of uncertaintyis the systematic error related to which thermodynamicmodels are adopted and which experimental data out of aconflicting set are accepted. It is these systematic errors,which are not incorporated in statistical uncertainty esti-mates, that we attempt to define with the MAP tech-nique.

For an equilibrium that involves solid solutions, thebasic thermodynamic constraint can be expressed as thefollowing inequality:

DrGP,T5DrGOP,T1RTlnKX1RTlnKg_0 (1)

where KX and Kg are the products of the mole fractionsand activity coefficients, respectively, raised to the pow-

3

er of their stoichiometric reaction coefficients.DrGP,T isthe free energy of reaction atP andT. DrG

OP,T, the stan-

dard state (unit activity for pure minerals atP andT; unitfugacity atT for H2O and CO2) free energy change ofreaction atP andT, is expanded as

DrGOP,T5DrH

O1,2982TDr SO

1,2981&T298DrC

OPdT

2T &T298

DrCOP

TdT1&P

1DrVOdP (2)

The sign of inequality (Eq. 1) for each experimental runis defined by the direction in which the reaction proceed-ed during the run. This is determined most convenientlyby comparing the starting and final KX, although themost confidence is placed on experimental results inwhich the compositional changes of all phases of vari-able composition give a consistent sense of reaction di-rection.

A particularly vexing problem that arose early in thisstudy was the inability of the above formed constraintson net transfer equilibria to reproduce adequately thepositions of the divariant loops for Cd-Gt-Si-Qz and Ol-Opx-Qz (for mineral abbreviations see Table 1). Carlsonand Lindsley (1988) ascribed similar problems in fittingthe two-pyroxene pseudosolvus to the fact that the abovetype of constraints involve activity ratios for componentsin more than one phase, allowing the mathematical con-straints to be satisfied numerically while not reproduc-ing the experimental data for which the constraint iswritten. We solved this problem by writing “bulk compo-sition” constraints that, for a specific bulk composition,compare overall phase stabilities involving all compo-nents of each phase, rather than constraints like Eq. 1which relate chemical potentials of each component sep-

arately in compositional subsystems. For example,Bohlen and Boettcher (1981) performed an experiment at10008 C and 10 kbar in which they observed that Ol1Qzare stable at the bulk compositionXFe50.95, therebydefining the divariant field to be at more Mg-rich com-positions. We force Ol951Qz to be more stable thanOl961Opx901Qz by constraining the following massbalanced reaction

0.05 Ol9610.02 Opx9050.06 Ol9510.01 Qz (3a)

to haveDrGP,T,0. This yields

0.05 [0.96mOl96Fa 10.04mOl96

Fo ]10.02 [0.90mOpx90Fs

10.10mOpx90En ].0.06 [0.95mOl95

Fa 10.05mOl95Fo ]10.01mQz (3b)

wheremYni is the chemical potential of component i in

solid solution Y with fixed composition 100XFe5n. Inthis example, Ol96 is an olivine composition chosen asslightly more Fe-rich than Ol95 so that it represents thebulk composition when combined with Opx90, the or-thopyroxene composition in exchange equilibrium withOl96. Note that it is not sufficient to use the less con-straining free energy relations between Ol951Qz andOpx95 becauseGOpx951Qz. GOl961Opx901Qz. Bulk com-position constraints are also particularly useful in ac-counting for experimental observations on the disappear-ance or growth of a particular assemblage of fixed bulkcomposition. This type of constraint cannot be explicitlyincorporated with unconstrained least-square methods(e.g. Aranovich et al. 1985).

Final optimization of all derived thermodynamicproperties was achieved using a “least squares” objectivefunction (Berman et al. 1986; Engi 1987):

*

n1

i

(f12mi)2

s2i

1*

n2

j

(Gcalc,j2lnKx,j)2

s2lnKj

(4)

to minimize discrepancies between n1 derived (fi) anddirectly measured (mi) thermodynamic properties aswell as between n2 experimentally determined lnKX val-ues and predicted values (Gcalc). The latter are calculatedfrom rearrangement of Eq. 1 as:

Gcalc5(2DrGOP,T2RTlnKg)yRT5lnKX (5)

with KX and Kg written for the nominally determinedcompositions of coexisting phases for the given equi-librium. In the absence of quantitative estimates of KX

errors, the variance of lnKX was assumed to be equal to1% of lnKX.

Treatment of experimental data

Philosophy of selection

The primary objective of this study is to derive thermo-dynamic data that, by providing for interpolation be-tween and extrapolation beyond the available set of ex-perimental observations, can be used for reliable petro-logic calculations. In addition to being “internally con-

Table 1 Abbreviations, chemical formula of minerals and minals

Name Abbreviation Formula

Orthopyroxene Opx (Mg,Fe,Al) (Al,Si)O3Enstatite En MgSiO3Ferrosilite Fs FeSiO3Orthocorundum Ok Al2O3

Olivine Ol (Mg,Fe)2SiO4

Forsterite Fo Mg2SiO4

Fayalite Fa Fe2SiO4

Garnet Gt (Ca,Mg,Fe)3Al2Si3O12

Grossular Gr Ca3Al2Si3O12

Pyrope Py Mg3Al2Si3O12

Almandine Alm Fe3Al2Si3O12

Cordierite Cd (Mg,Fe)2Al4Si5O18

Mg-cordierite mCd Mg2Al4Si5O18

Fe-cordierite fCd Fe2Al4Si5O18

Ilmenite Il (Mg,Fe)TiO3

Ilmenite Ilm FeTiO3

Geikielite Gk MgTiO3

Anthophyllite Ant Mg7Si8O22(OH)2Spinel Sp MgAl2O4

Hercynite Hc FeAl2O4

Rutile Rt TiO2

Quartz Qz SiO2Sillimanite Si Al2SiO5

Corundum Co Al2O3

4

sistent”, these data should also reproduce diverse experi-mental observations within their respective uncertain-ties. Because any thermodynamic data are only as goodas the experimental data from which they are derived, anadditional objective of this study is to attempt to identifythe most compatible of the available experimental data aswell as outliers within the data.

In deriving thermodynamic data from relevant experi-mental data obtained with different methods and in dif-ferent laboratories, we use only the most reliable of thesedata in order to minimize the number of potential incon-sistencies among the different data sets. A critical step istherefore to establish criteria for assessing the reliabilityof these data so that there is a basis for trying to resolveany such inconsistencies when they arise. Besides ade-quate documentation of experimental apparati and theircalibrations, the most stringent criterion applied in thisstudy was that it be possible to discern exactly whatchanges were observed in each experiment, i.e. whatequilibria were responsible for the changes in mineralcompositions or proportions observed during an experi-ment, and in which direction these equilibria proceededduring each experiment. Translated into practical terms,this criterion excludes all data but those in which anassemblage of crystalline materials, capable of definingone or more equilibria, was observed to undergo changesin compositions or mineral proportions of a magnitudelarger than the uncertainties in deciphering suchchanges. Synthesis-type runs were not used because oftheir greater tendency towards formation of metastablerun products.

Constraints on end-member properties

Inspection of Eq. 1 indicates that parameters represent-ing analytical expressions for activity coefficients can bedetermined given experimental phase equilibrium data

defining KX and standard state thermodynamic datadefiningDrG

o. In the initial stages of this study, standardstate properties for all MAS phases were taken from theanalysis of Berman (1988) with the properties of FASTphases constrained by relevant univariant phase equi-libria. The final analysis presented below involved simul-taneous redetermination of MAS, MSH, and FAST stan-dard state properties (using data for equilibria (a) to (o)listed in Table 2 and MSH data described by Berman,1988) because the small (,1 kJymol) Gibbs energy ad-justments allowable by the relevant MAS experimentaldata permit much better representation of all experimen-tal constraints in the FMAST system. In addition, in-corporation of new Cp measurements for pyrope (Tequiet al. 1991) and forsterite (Gillet and Fiquet 1991) leadsto significantly improved representation of the overallexperimental phase equilibrium data set. Additionalchanges that have been incorporated into our analysis arecorrections to the entropy of anthophyllite (Hemingway1991), reassessment of the volume of anthophyllitebased on analysis of natural and synthetic amphiboles(Hirschmann et al. 1994), and recent determinationof the thermodynamic properties of spinel with equi-librium amount of disorder (Chamberlin et al. 1995). Incontrast to Berman (1988)’s use of fayalite, we usea-Feas the Fe anchor phase, employing careful EMF mea-surements (O’Neill 1987) to tie these properties to Fe-silicates.

All thermodynamic properties determined in thisstudy are consistent with those of CaO-K2O-Na2O-Al2O3-SiO2-H2O phases tabulated by Berman (1988).Phase equilibrium data published since completion ofthe Berman (1988) study that are relevant to this analysisbear on the stability of the aluminosilicate polymorphsand some Ca-silicates. The former data are shown in Fig.1, where it can be seen that the tabulated 1988 propertiesare compatible within experimental uncertainties withdirect observations by Bohlen et al. (1991) on the ky5sitransition (Fig. 1a) and by Harlov and Newton (1993) on

Table 2 Phase equilibriumdata used to constrain MASand FAST endmembers

' Equilibrium Authors

System: MgO-Al2O3-SiO2

(a)a Py1Fo54 En1Sp Danckwerth and Newton (1978); Perkins et al. (1981)(b) 6En12Si52Py12Qz Perkins (1983)(c) 3En1Co5Py Gasparik and Newton (1984)(d) mCd1Co52En13Si Newton (1972)(e) 5Fo1mCd510En12Sp Fawcett and Yoder (1966); Seifert (1974); Herzberg (1983)

System: FeO-Al2O3-SiO2-TiO2

(f) 2Fs5Fa1Qz Bohlen et al. (1980)(g) 2Alm14Si15Qz53fCd Mukhopadhyay andHoldaway(1994)(h) 3Hc13Si5Alm15Co Shulters and Bohlen (1989)(i) 3Hc15Qz5Alm12Si Bohlen et al. (1986)(j) Alm1Hm5Mt1Ky1Qz Harlov and Newton (1992)(k) 3Ilm1Si12Qz5Alm13Rt Bohlen et al. (1983b)(l) 2Ilm52Fe12Rt1O2 Shomate et al. (1946); Feenstra and Peters (1996)(m) 4Mt1O256Hm Myers and Eugster (1983); O’Neill (1988)(n) 3Fa1O252Mt13Qz O’Neill (1987)(o) 2Fe1Qz1O25Fa O’Neill (1987)

a Equilibria (a)–(e) involvealuminous Opx.

5

Our results of the MSH system are very similar tothose documented by Berman (1988), and the reader isreferred to the latter paper for detailed presentation ofthe MSH data. Significant differences between endmem-ber properties derived here and by Berman (1988) arediscussed in the “Results” section.

Constraints on solid-solution properties

With the MAP method, the Gibbs energy inequalities arewritten using explicit values for uncertainties inP, T, andall compositional parameters. Compositional uncertain-ties applied to each set of experimental data involving asolid solution phase were taken from authors’ evaluationwhen available or based on estimated minimum mi-croprobe errors ofXFe5Fey(Fe1Mg)5+0.005. Consid-ering the additional uncertainties related to composi-tional heterogeneity produced in all phase equilibriumstudies involving solid solutions, more reasonable mini-mum overall uncertainties areXFe5+0.01. In analyzingdata in which final compositions are heterogeneous, weassume that the most advanced compositions (thosewhich yield KX most different from that of the startingcompositions), not the average or most abundant compo-sitions, are closest to equilibrium.

Overlapping half-brackets after compositional adjust-ment suggests either underestimation of analytical un-certainties or real overstepping of an equilibrium (“path-looping” of Perkins and Newton 1980), whether the lat-ter is caused by the solution-reprecipitation process(Aranovich and Pattison 1995) or by different mecha-nisms and rates of the forward and backward reactions(Aranovich and Kosyakova 1987; Perkins and Newton1980). Any pair of runs that constitutes a reversal (twohalf-brackets) with overlapping compositions yields twomutually inconsistent limits that produce an infeasiblesolution to a MAP problem. To make them both compat-ible, three different approaches can be employed. Oneapproach, which assumes that the path-looping of thereaction is the only reason for the overlap, is to changethe sign of the reversal brackets arbitrarily, defining theirwidth as the amount of compositional overlap (Carlsonand Lindsley 1988). This approach cannot make use ofan experimental half-reversal (Carlson and Lindsley1988, p 244), a serious disadvantage because most of theexperiments concerning solid solutions are not reversedsensu stricto, but are more appropriately viewed as aseries of half-brackets.

A second approach is to adjust the uncertainty appliedto individual experimental points in order to eliminatethe inconsistencies. This approach is most in accord withthe Fe-Mg partitioning data used in this study. Becausethe Gibbs free energy of the equilibrated minerals is notmuch lower than that of the crystalline starting materials,and because of favorable reaction kinetics, equilibriumoverstepping does not appear to be a general phe-nomenon that affected a large number of experimentalruns. In contrast, minor relaxation of the errors assigned

Fig. 1a, b Comparison of recently published experimental datawith univariant curves in the system Al2O3–SiO2 computed withthermodynamic data of Berman (1988).a Data of Bohlen et al.(1991) for the equilibria Ky5Si and Ky5And. b Data of Harlovand Newton (1993) for the equilibria And5Co1Qz andKy5Co1Qz. Symbols show experimental data after adjustmentfor experimental uncertainties (direction of adjustment is awayfrom the equilibrium position).Opposite ends of connected linesshow nominal experimental half-brackets

metastable equilibria involving co1qz (Fig. 1b). The ex-periments of Zhu et al. (1994) on the equilibrium: cal-cite1qz5wollastonite1CO2 indicate increased stabilityof wollastonite with respect to calcite than given byBerman (1988). As calcite is not considered in this study,what is of direct concern here is Zhu et al.’s conclusion,based on the measured enthalpy of solution for anorthiteand analysis of other experimental brackets on the equi-librium grossular1quartz52wollastonite1anorthite,that their preferred enthalpies of formation for anorthite(299.5 kJymol), grossular (2329.55 kJymol), and wol-lastonite (21635.4 kJymol) are more in accord withthose given by Holland and Powell (1990;2101.2,2327.4, 21633.2) than by Berman (1988;296.5,2319.8,21631.5). Although this is true in terms of ab-solute differences, Zhu et al.’s preferred values are infact in excellent agreement with Berman (1988) valuesonce all are corrected for what amounts to a 3.5 kJymoldifference in CaO anchor value. Applying this change toZhu et al.’s preferred values yields296.0, 2319.05,21631.9 kJymol, differences for each phase that are lessthan 0.75 kJymol from Berman’s values.

6

to a small percentage (generally less than 5%) of thehalf-brackets is sufficient to obtain consistency of entiresets of experimental data. This approach places moreimportance on the scrutiny of all details of each experi-ment in order to discover possible sources of error, but ithas the distinct advantage to identifying possible outliersthat can lead to systematic errors in retrieved thermody-namic parameters.

A third approach is to treat the maximum composi-tional overlap observed in each particular experimentalstudy as two times the minimum compositional adjust-ment that is applied to all the experimental results of thatstudy. This practice, although to some degree arbitrary,is more in accord with either underestimated analyticaluncertainty or overstepping causing observed composi-tional overlaps, or both. It should be noted that this ad-justment permits consistency between two opposinghalf-brackets, but may not be of sufficient magnitude toallow consistency (feasible MAP solutions) among allexperiments of one study. This third approach was ap-plied to the experimental data bracketing the Al contentof Opx in equilibrium with Gt or Cd in the MAS andFMAS systems (Perkins et al. 1981; Kawasaki and Mat-sui 1983; Lee and Ganguly 1988) because sluggish reac-

tion kinetics make significant compositional overlappinga general feature of these experimental results.

Many experiments concerning solid solutions wereexcluded from our treatment because starting composi-tions were not reported. Experiments which producedinconsistent changes in mineral compositions (e.g. Fe-Mg exchange experiments involving two phases whichboth increased in Fe) were excluded from the final anal-ysis. Less weight was given to experiments in whichphases extraneous to the equilibrium under study ap-peared, because of the possibility that the entire assem-blage was not in equilibrium.

Experimental data for equilibria (A) to (U) listed inTable 3 were used in this study to determine solutionproperties. Specific points regarding these data are dis-cussed below:

Garnet-orthopyroxene

Perkins et al. (1981) obtained 46 pairs of reversals for the equi-librium (H): 3 En1Ok5Py in the MAS system. In most runs theyobserved an overlap in the final Opx composition, up to 0.7 wt%Al2O3. We found that+0.7 wt% was the minimum compositionaladjustment required to be applied to all data in order to ob-

Table 3 Solid solution equi-libria considered in this analy-sis

' System Reference

Fe-Mg exchange equilibria:

(A) 3Fs1Py53En1Alm Kawasaki and Matsui (1983); Harley (1984); Lee and Ganguly(1988); Eckert and Bohlen (1992)

(B) fCd12En5mCd12Fs Aranovich and Kosyakova (1987)(C) 3fCd12Py53mCd12Alm Aranovich and Podlesskii (1981, 1983); Perchuk and

Lavrent’eva (1983)(D) 3Fa12Py53Fo12Alm Hackler and Wood (1989)(E) 2Fs1Fo5Fa12En Medaris (1969); Fonarev (1981, 1987); Bohlen and Boettcher

(1981); Davidson and Lindsley (1989); Koch-Muller et al.(1992); von Seckendorff and O’Neill (1993)

(F) Fs1Gk5Ilm1En Hayob et al. (1993)(G) Fa12Ilm52Gk1Fo Andersen and Lindsley (1979); Bishop (1976, 1979, cit. after

Andersen et al. 1991); Andersen et al. (1991)

Net transfer equilibria involving Al content of Opx:

(H) 3En1Ok5Py Lane and Ganguly (1980); Perkins et al. (1981); Kawasaki andMatsui (1983); Harley (1984); Lee and Ganguly (1988)

(I) 3Fs1Ok5Alm Aranovich and Berman (1995, 1996a); Kawasaki and Matsui(1983); Harley (1984); Lee and Ganguly (1988)

(J) 2En12Ok13Qz5mCd Aranovich et al. (1983); Aranovich and Kosyakova (1987)(K) 2Fs12Ok13Qz5fCd Aranovich and Kosyakova (1987)(L) Sp1Fo5En1Ok Gasparik and Newton (1984)

Other net transfer equilibria:

(M) 2Py14Si16Qz53mCd Aranovich and Podlesskii (1981, 1983); Hensen (1977)(N) 2Alm14Si16Qz53fCd Aranovich and Podlesskii (1981, 1983); Hensen (1977)(O) Fa1Qz52Fs Fonarev and Korol’kov (1976); Fonarev (1987); Bohlen and

Boettcher (1981); Davidson and Lindsley (1989)(P) Fo1Qz52En Fonarev and Korol’kov (1976); Fonarev (1987); Bohlen and

Boettcher (1981); Davidson and Lindsley (1989)(Q) Fs1Rt5Ilm1Qz Hayob et al. (1993)(R) En1Rt5Gk1Qz Hayob et al. (1993)(S) Gros12Ky1Qz53An Wood (1988); Koziol and Newton (1989); Koziol (1990)(T) Gros12Alm53Fa13An Bohlen et al. (1983a)(U) 6Ilm13An13Qz5Gros Bohlen and Liotta (1986)

16Rt12Alm

7

tain internally consistent results for this set of data consideredalone.

We accepted six of the eight experiments reported by Kawasakiand Matsui (1983) for equilibrium (H) in the MAS system, elimi-nating those started from gel mixtures (run 571, in their Table 3)and compositionally uncharacterized synthesis products (run372). Only three of their seven runs in the FAS system, performedwith crystalline starting materials, have been used in this study toconstrain the position of equilibrium (I): 3 Fs1Ok5Alm. We ad-justed the pressure of Kawasaki and Matsui’s 50 kbar (nominal)runs to 46 kbar following the suggestions by Gasparik and Newton(1984) and Brey and Kohler (1990).

For the exchange equilibrium (A): 3 Fs1Py53 En1Alm, weaccepted 14 experiments of Kawasaki and Matsui (1983, theirTables 5 and 6) in the FMAS system, in which they employedpreanalyzed crystalline starting materials, ignoring all synthesis-type runs. For the same reason we have rejected most of the exper-imental points of Harley (1984) on equilibrium (A), acceptingthose five that were obtained on crystalline starting materialsplaced in graphite containers (Harley 1984; his Table 2). No runs inFe capsules were accpeted in this study because of the differentialrate of Fe change of Opx and Gt observed by Harley. In view of thescarcity of experiments defining Al2O3 solubility in FMAS Opx,we used Harley’s data for equilibrium (I) as part of the objectivefunction (4).

Lee and Ganguly (1988) used crystalline starting materialswith a PbO1PbF2 flux to promote the reaction rates, but did notreport details of any heterogeneity in compositions of run prod-ucts. All exchange equilibrium (A) data are mutually consistentusing the+0.01 compositional uncertainties estimated by the au-thors (Lee and Ganguly 1988; their Table 2). Nevertheless, we didnot utilize one half-bracket ('163) in which both Opx and Gtdecreased in Mg during the experiment.

Eckert and Bohlen (1992) used crystalline starting materialswithout any flux to facilitate reaction, except for one 9008 C brack-et with H2O added. The four brackets they report for the exchangeequilibrium (A) are internally consistent without any composition-al adjustments. Their reported Al2O3 contents of Opx, all ap-proached from undersaturation, require minimum adjustments of40% to be mutually compatible.

Recent data of Aranovich and Berman (1995, 1996a) offer thebest constraints on the position of equilibrium (I) in the FAS sys-tem. Aranovich and Berman (1995; 1996a) obtained nine reversalsof the Al2O3 content of FAS orthopyroxene in equilibrium withalmandine over theP-T range 12–20 kbar and 900–10008 C. Start-ing materials consisted of crystalline ferrosilite and Fs95Ok5 up to50mm in length, as well as the Fs95Ok5 oxide synthesis mix. Runproducts using the crystalline starting materials produced Al2O3-zoned Opx with most advanced compositions that converged with-in 0.002XOk without overlap. Run products of the oxide mixes alsoproduced heterogeneous Opx with a compositional range verysimilar to the reversal run brackets. The data presented by Ara-novich and Berman (1995; 1996a) also constrain the position ofthe FAS univariant equilibrium: Fe-Opx5Fa1Qz1Alm.

Cordierite1orthopyroxene1quartz

Aranovich et al. (1983) bracketed the Al2O3 content of Opx in thepresence of Cd in the MAS system, buffered by the equilibrium(J): 2En12Ok13Qz5mCd. In almost all of their reversal runs, anoverlap in the final Opx composition of 0.8–1.2 wt.% Al2O3 wasobserved.

Aranovich and Kosyakova (1987) determined Al2O3 contents ofFMAS Opx in equilibrium with Cd1Qz (equilibria (J) and (K)) aswell as Fe-Mg partitioning data (equilibrium (B)). No reversalswere reported for Al2O3 contents, however, because Al-free Opxwas used as the starting material in all experiments. For thepresent study we accepted 34 of their 36 runs, rejecting the tworuns which employed a starting mix of oxides.

Garnet1cordierite1sillimanite1quartz

The only experiments with this assemblage in FMAS (equilibria(M) and (N)) that utilized crystalline starting materials are thoseof Hensen (1977) and Aranovich and Podlesskii (1981, 1983). Wehave slightly reinterpreted the observations of the former author,selecting for this study the most advanced compositions of both Gtand Cd rather than the average ones referred to by Hensen. We alsoadjusted the final compositions for overlap (XMg50.002 in Gt andXMg50.02 in Cd) and corrected the nominal pressure of the runsby 210% as suggested by more recent piston-cylinder calibrations(e.g. Perkins et al. 1981).

From the work of Aranovich and Podlesskii (1981, 1983), wehave selected only those runs which produced appreciable changesin both garnet and cordierite composition. Runs in which bothphases moved in the same compositional direction (mostly fromMg-rich to Fe-rich compositions) have been used only to constrainthe exchange reaction (C). The maximum observed compositionaloverlaps are 0.015 for Gt and 0.01 for Cd. We also utilized observa-tions of these authors on the relative stability of the alternativephase assemblages in the form of bulk composition constraintsdiscussed above.

Olivine-garnet Fe-Mg exchange

Hackler and Wood (1989) presented reversed data at 10008 C,9.1 kbar, for equilibrium (D): 3Fa12Py52Alm13En, with garnetMgy(Mg1Fe)50.17–0.67. Synthetic garnets and olivines werehomogeneous to within 2.5 to 0.5 mol%, respectively. Reversalexperiments contained Gt:Ol ratios of 10: 1 by weight in order tominimize compositional changes in Gt. Their run products repre-sent homogeneous olivine and the most advanced of the slightrange of garnet compositions surrounding olivine. Final Gt com-positions shifted in the expected direction from nominal startingcompositions to compensate changes in Ol composition. In onlyone half-bracket was the change in composition greater than thestarting Gt heterogeneity range. These data are internally consis-tent using compositional uncertainties of 0.01.

Experimental data for this same equilibrium were also ob-tained by O’Neill and Wood (1979) over the temperature range1000–14008 C at 30 kbar. We did not analyze these data becausemost are synthesis runs, details of starting materials and run prod-ucts are not given, and they are presented graphically.

Olivine-orthopyroxene-(quartz)

The distribution of Fe-Mg between Opx and Ol (reaction E) andthe lower pressure stability limit of Opx relative to Ol1Qz (reac-tions O, P) have been extensively studied over the past few decades(Larimer 1968; Medaris 1969; Matsui and Nishizawa 1974; Fonar-ev and Korol’kov 1976; Fonarev 1981, 1987; Bohlen et al. 1980;Bohlen and Boetcher 1981; Davidson and Lindsley 1989; Koch-Muller et al. 1992; von Seckendorff and O’Neill 1993). Close ex-aminations reveals that not all of these experiments meet the strin-gent criteria imposed in this study. We rejected the data of Larimer(1968), Matsui and Nishizawa (1974), and Fonarev (1987) (exceptfor the 7508 C isotherm of the latter author) for equilibrium (E)because these authors either did not report starting mineral com-positions or performed unreversed synthesis experiments. We havealso not included the data of Koch-Muller et al. (1992) on reac-tions (O) and (P) at 950 and 10008 C because growth of Fs was notobserved at these temperatures. Because of some discrepanciesbetween X-ray composition equations used by different authors(Medaris 1969; Fonarev 1981, 1987; Koch-Muller et al. 1992), wehave attributed larger uncertainties (0.02 Mgy(Mg1Fe)) to thedata obtained by this method.

The most complete experimental study reviewed in the courseof this work is that of von Seckendorff and O’Neill (1993). Usingcrystalline starting materials and a BaO-B2O3 flux to promotereaction, they collected 46 half-brackets on reaction (E) ranging

8

Initially, activity coefficients for all phase compo-nents were computed with a third degree (asymmetric)Margules equation in the generalized form below(Berman and Brown 1984; Berman 1990):

nRTlngm5*WijkXiXjXk FQm

Xm

22G (6a)

Wijk5WGijk5WH

ijk2TWSijk1(P21)WV

ijk (6b)

In Eq. 6a, the summation is for all Margules parameters,n is the site multiplicity, and Qm is the number of i,j,ksubscripts that are equal to m. When combined with theWohl (1946, 1953) approximation for third degree equa-tions:

Cijk5(Wiij 1Wijj 1Wiik1Wikk1Wjjk1Wjkk)y22Wijk50 (6c)

Eq. 6a is compatible with a zero Wohl ternary interaction(Cijk) parameter (Berman 1990; Berman and Koziol1991). We prefer Eq. 6a–c to the mathematically equilva-lent expressions presented by Helffrich and Wood(1989), because of their ease of memorization, general-ization to any number of components, and conversion tocompact computer code.

Due to the general lack of direct constraints on excessentropy for all solid solutions, we considered a symmet-ric functional form (WS

iij 5WSijj ) to have the greatest reli-

ability for extrapolation outside theT-X range of existingexperimental data. In the final optimization, a symmetricMargules expression was also used for the excess en-thalpy of Ol, Opx, Cd, and Il. Additional specifics ofeach solid solution model used in this study are describedin the following sections.

Olivine

Fe-Mg mixing in olivine has been the subject of consid-erable experimental and theoretical investigation. Sackand Ghiorso (1989) found that available calorimetric andphase equilibrium data were most compatible with asymmetric regular solution parameter (WH) equal to10.17 kJy2 oxygen formula. More recent studies basedon various phase equilibria and calorimetry indicate val-ues in the range 3–5 kJy2 oxygen basis, with excess en-tropy less than 2 Jy2 oxygen basis (e.g. Hackler andWood 1989; Wiser and Wood 1991; von Seckendorff andO’Neill 1993; Kojitani and Akaogi 1994). In this studywe determined mixing properties of olivine simulta-neously with those of all other phases. Excess volumewas allowed to range between the extremes (0.011–0.045 Jybarymol) of various determinations (Schwab andKüstner 1977; Kawasaki and Matsui 1983). Minor order-ing of Mg-Fe21 between M1 and M2 sites (e.g. Ottonelloet al. 1990) has been ignored.

Ilmenite

Two different approaches have been applied recently formodelling solid solution properties in the system il-

from Fa03 to Fs98 and 900 to 11508 C. Olivine and orthopyroxenein run products were generally very homogeneous, with averagestandard deviations between 0.25 and 0.41 mol%. The width ofgrowth rims was 5–30mm, facilitating single phase microprobeanalysis. In 34 of 46 runs, run direction was unambiguously de-fined by sympathetic changes in both Ol and Opx compositionsthat were greater than analytical uncertainties. In the remaining 12runs, the observed changes in KD were also significantly largerthan uncertainties. Treated by themselves, these data are internallyconsistent with 0.002 Mgy(Mg1Fe) adjustment applied to all data.

Olivine-ilmenite Fe-Mg exchange

Experimental observations on compositions of coexisting olivineand ilmenite (exchange reaction G in Table 3) solid solutions equi-librated over a wide range of temperature, pressure and Mgy(Mg1Fe) have been reported by Andersen and Lindsley (1979),Andersen et al. (1991), and Bishop (1976, 1979; cited after An-dersen et al. 1991). Compositional uncertainties, although men-tioned (Andersen et al. 1991, p 436), have not been estimated bythe authors. The heterogeneity range in the representative runproducts shown by Andersen and Lindsley (1979, their Fig. 1)shows an overlap in most advanced compositions of about0.01 Mgy(Mg1Fe), which we applied as a minimum adjustment toall the data. Data by Bishop, as given in Table 8 of Andersen et al.(1991) require an additional 0.01 Mgy(Mg1Fe) adjustment to be-come internally consistent. To avoid problems related to the com-plex hematite solid solution in picroilmenite (e.g. Ghiorso 1990),we used for the present study only those experimental points whichshowed Hm contents less than 3 mol%, and recalculated them tothe Ilm-Gk binary. Any errors associated with neglecting Hmcomponent for this compositional range are believed to be signifi-cantly smaller than those related to the uncertainties in determin-ing equilibrium Mgy(Mg1Fe) in the coexisting minerals.

Orthopyroxene-ilmenite (+rutile,quartz)

Both exchange (equilibrium F) and net-transfer equilibria (equi-libria Q and R) operating in the SIRF (Silica-Ilmenite-Rutile-Fer-rosilite) system have been recently investigated by Hayob et al.(1993). The authors utilized crystalline mixtures of presynthesisedminerals as starting materials and buffered the run assemblagesnear iron-wüstite. Although synthetic ilmenites contain up to2 mol% hematite. product ilmenites contain less than 1 mol% ofHm. Product Opx was somewhat heterogeneous (Hayob et al.1993) with up to¥0.02 Mgy(Mg1Fe) compositional overlap inbracketing experiments. For our processing we accepted all thedata points reported by these authors, adjusted forP-T-XFe uncer-tainties (0.5 kbar, 58 C, 0.02XFe) quoted by Hayob et al. (1993).

Solid solution models

Any thermodynamic modelling employing experimentalphase equilibrium data involving solid solutions requiresanalytical formulation of Gex as a function of measuredparameters (T, P, Xi), and thus necessarily includes a cer-tain compromise between the complexity of the solidsolution models, the amount and quality of the data, aswell as the purposes of the modelling. In so far as ourmain goal is to produce reasonable phase diagrams aswell as geothermobarometric results, we tried as a firstapproximation the simplest solution models possible, in-creasing their complexity only when no feasible solutionto the whole data set could be reached without unjustifi-able enlargement of the experimental uncertainties.

9

menite-geikielite-hematite (Ghiorso 1990; Andersenet al. 1991). The departing point of the Andersen et al.(1991) treatment of Fe-Mg exchange equilibria il-menite-olivine and ilmenite-spinel corresponds to com-pletely ordered Ilm-Gk, while Ghiorso (1990) used aconvergent ordering model which assumes disorderingof Fe, Mg, and Ti (as well as Fe31 in the case of ternaryIlm-Gk-Hem solid solution) on structurally equivalentsites. This latter model was shown to be more successfulin modelling the ternary solution because of the explicitformulation of the entropy of mixing on the Ilm-Hem andGk-Hem joins (Ghiorso 1990). The advantages of thismodel are not as clear for the case of the Ilm-Gk binary.For example, it predicts considerable disorder in pureilmenite at a temperature above 10008 C (Ghiorso 1990,his Fig. 8) not supported by direct observations (e.g.Wechsler and Prewitt 1984). The predicted Gibbs freeenergy of mixing on the binary (Ghiorso 1990, his Fig. 9)exhibits very slight asymmetry and smooth temperaturedependence that should indicate relatively simple macro-scopic behavior of the solid solution. As far as our calcu-lations are restricted to essentially Fe31-free composi-tions, we have chosen for the Ilm-Gk solid solution asubregular solution model similar to that used by An-dersen et al. (1991), with the WV parameter fixed ac-cording to these authors at 0.0108 Jybar mole.

Cordierite

Cordierite has been successfully modelled as an idealone site solution with site multiplicity of 2 (Perchuk andLavrent’eva 1983; Aranovich and Podlesskii 1983; Ara-novich and Kosyakova 1987). Here we allowed for thepossibility of small deviations from ideality with a sym-metric (regular) Margules approximation. The increasedstability of cordierite resulting from introduction of H 2Omolecules into the structural channels has been account-ed for by applying McPhail et al.’s (1990) thermody-namic description of the reaction:

Mg2Al4Si5O1812H2O5MgAl2Si5O18? 2H2O

Following recent experimental observations (Boberskiand Schreyer 1990; Mukhopadhyay andHoldaway1994)we assumed no difference in the water solubility in Fe-and Mg-cordierite at similar P-T-conditions. Thus, thefollowing expressions are used for the activity of “drycordierite” end-members

amCd5(XCdMggmCd)

2(12XH2O)2

afCd5(XCdFegfCd)

2(12XH2O)2

whereXH2O is the number of moles of H2O per formulaunit of cordierite, divided by 2.

Garnet

Mixing properties of almandine-pyrope solutions havebeen the subject of considerable debate (e.g. Aranovich1983; Ganguly and Saxena 1984; Sack and Ghiorso1989; Berman 1990; Koziol and Bohlen 1992), with theweight of recent evidence favouring near-ideal behavior.All recent studies suggest larger excess enthalpies in Fe-rich compositions, the asymmetry in enthalpy mimick-ing that in the excess volumes. Although this study dealsprimarily with Fe-Mg exchange equilibria, phase equi-librium data on the Gr-Alm join place important con-straints on almandine standard state properties. For thisreason we evaluate mixing properties on this join whichis well constrained by both calorimetric (Geiger et al.1987) and phase equilibrium studies (equilibria (S)–(U)in Table 3; Bohlen et al. 1983; Bohlen and Liotta 1986;Koziol 1990). Our analysis of these data follows closelythat described by Berman (1990).

In order to be able to apply the results of this study tonatural Ca-Mg-Fe garnets, we present provisional mix-ing properties on the Gr-Py join based on two reversalsof the GASP equilibrium (Wood 1988) as well as en-thalpy of mixing data of Newton et al. (1977). FollowingBerman and Koziol’s (1991) analysis of GASP reversalswith Ca-Mg-Fe garnet (Koziol and Newton 1989), weassume the Wohl ternary interaction parameter (Cijk) isequal to zero (i.e. that Eq. 6c holds). While compatiblewith Koziol and Newton’s data, more rigorous testing ofthis assumption awaits further experimental constraintson Ca-Mg garnet as well as a broader analysis that in-cludes not only ternary GASP reversals, but also Fe-Mgexchange data involving ternary Ca-Mg-Fe garnet (e.g.Gt-Opx: O’Neill and Wood 1979; Gt-Cpx: Pattison andNewton 1989).

In this study we applied the subregular model to thePy-Alm, Gr-Alm, and Gr-Py joins. For the 12 oxygenformula, the activity of garnet components are given by

aPy5(XGtMggPy)

3

aAlm5(XGtFegAlm)3

aGr5(XGtCagGr)

3

with activity coefficients given by Eq. 6. Excess volumeparameters were taken from Berman’s (1990) evaluation,except for those on the Gr-Py binary which were takenfrom Ganguly et al.’s (1993) data.

Orthopyroxene

Orthopyroxene solid solution properties are at least ascontroversial as those for garnet (e.g. Sack 1980;Kawasaki and Matsui 1983; Aranovich and Kosyakova1987; Sack and Ghiorso 1989; Lee and Ganguly 1988;Chatillon-Colinet et al. 1983; Aranovich 1991; vonSeckendorff and O’Neill 1993). Both one-site and two-site models have been applied with success in these stud-ies, and one objective of this study was to compare re-sults of these different models.

10

1-Site model

With the one site model, orthopyroxene is treated as asimple mixture of the components MgSiO3, FeSiO3 andAl2O3 (orthocorundum) such that:

aen5(XOpxMg gEn)

aFs5(XOpxFe gFs)

aOk5(XOpxAl2O3

gOk)

with activity coefficients given by Eq. 6. This model hasbeen shown to be capable of reproducing the major phaserelationships involving Opx (Saxena 1981; Aranovichand Kosyakova 1987; Aranovich and Podlesskii 1989;Aranovich 1991; Lee and Ganguly 1988), and has theadvantage of not being dependent on the uncertainties inMg-Fe site occupancy constraints (e.g. Saxena 1983;Hawthorne 1983). The 1-site model is equivalent to a2-site model with complete Fe-Mg disorder (equal parti-tioning) between M1 and M2 sites.

Volumes for MAS and FAS Opx were constrained byavailable V-X data for orthopyroxene on the joins En-Ok(Stephenson et al. 1966; Chatterjee and Schreyer 1972;Danckwerth and Newton 1978; Doroshev et al. 1983)and Fs-Ok (Aranovich and Berman 1995). The dataavailable on the volume behavior of Fe-Mg Opx do notshow any significant deviation from linearity (e.g.Turnock et al. 1973; Chatillon-Colinet et al. 1983; Fon-arev 1987; Hayob et al. 1993).

Early on in this project it became evident that theone-site model taken in its simplest form (a regular solu-tion), although capable of reproducing the major phaserelationships involving Opx in the FMS and MAS sub-systems (see discussion under Opx subheadings below),was insufficient to reproduce available experimentalconstraints on Al2O3 solubility in Opx in the more com-plex system FMAS without unjustifiable enlargment ofthe experimental uncertainties. Because of the very lim-ited range of Opx Al2O3 contents covered by experimen-tal studies, we extended the above model by incorpora-tion of the Darken quadratic formalism (DQF).

The DQF model was proposed by Darken (1967a) andshown to work extremely well for analytical representa-tion of mixing properties of components in dilute binarysolid solutions even for the case when solute and solventare of significantly different nature (e.g. C dissolved inmetall alloys). This model was introduced to the petro-logical literature by Aranovich (1983, see also Appendixin Perchuk et al. 1985) and has been succesively appliedby Powell (1987) and Will and Powell (1992). The DQFwas derived for binary solutions (Darken 1967a), andDarken (1967b) and Aranovich (1991) expanded it forthe case of an arbitrary number of components dissolvedin one and the same solvent.

In binary systems, this model effectively treats twoterminal regions near each endmember as regular solu-tions involving a real and fictive endmember (Powell1987). Our application of the DQF model to the region ofdilute solubility of Al2O3 in Opx provides additional de-

grees of freedom in our analysis by utilizing endmemberproperties of “real” Al2O3 in MAS Opx and “imaginary”Al2O3 in FAS Opx. For ternary FMAS Opx, Darken’s(1967b) equations were modified to account for solutionof one component (Al2O3) in two different solvents (Enand Fs). For En and Fs, the corresponding expressionsare those given by Eq. 6 above. For Al2O3, activity coef-ficients given by Eq. (6;RTlngM argules

Al2O3) are modified as:

RT ln gAl2O35RT ln gM argules

Al2O31Fey(Fe1Mg)IFe–Al (7)

with IFe–Al expanded to IFe–Al5IH2T? IS1(P21) IV(Turkdogan and Darken 1968).

2-Site model

The two site model with non-convergent ordering of Mgand Fe atoms has been discussed in detail by Thompson(1969), Sack (1980) and Sack and Ghiorso (1989) forbinary Fe-Mg Opx and by Kawasaki and Matsui (1983)and Aranovich and Kosyakova (1987) for the ternary Fe-Mg-Al Opx. The notation of Sack and Ghiorso (1989) isused below in discussing results with this model.

Results

The primary variable with respect to thermodynamicmodels that was explored in this study was the differencebetween a one-site and two-site model for orthopyrox-ene. A somewhat surprising conclusion was that the1-site model with a symmetric, temperature-dependentexcess free energy reproduced the overall set of experi-mental observations more closely than the 2-site model.We found that 2-site models that were calibrated witheither the reversed Mössbauer data of Anovitz et al.(1988) or the in situ single crystal data of Yang andGhose (1994) could not reproduce all Fe-Mg exchangedata for Ol-Opx and Gt-Opx when these are adjusted formaximum compositional uncertainties of 0.01 Mgy(Mg1Fe). Nor were we able with these calibrations toobtain adequate representation of the Ol-Opx-Qz divari-ant field data. We found that use of different site occu-pancy data (e.g. Virgo and Hafner 1969; Saxena andGhose 1971; Besancon 1981; Domeneghetti and Steffen1992), and introduction of temperature-dependent ex-cess free energies on both sites did not substantiallychange these results. In contrast, the 1-site model withtemperature-dependent, symmetric free energy of mix-ing reproduces most of the Ol-Opx exchange data of vonSeckendorff and O’Neill (1993) with 0.005 Mgy(Mg1Fe) adjustments that are more in accord with theirquoted uncertainties, as well as the Ol-Opx-Qz data withexcellent fidelity (see below in Fe-Mg exchange sec-tion). On the basis of this modelling we conclude that thephase equilibrium data involving Fe-Mg Opx and theintracrystalline partitioning data are somewhat at oddswith one another.

11

The most likely reason for this conflict is that theavailable reversed Mössbauer data define site occupan-cies over a more limited temperature and compositionalrange compared to the phase equilibrium data, and ana-lytical interpolations and extrapolations from these dataappear to yield a poorer base level from which to makeempirical activity-composition corrections than theequal partitioning assumption. Gross inaccuracy ofMössbauer data (e.g. Seifert 1989) is precluded by re-

cent single crystal in situ determinations (Yang andGhose 1994) which offered general support for theMössbauer determinations of Anovitz et al. (1988). Yangand Ghose (1994) reported a polymorphic transition inOpx at about 9508 C which may further complicate ex-plicit account of cation occupancies over the 700–14008 C temperature range of the phase equilibrium.Comparison of the 1-site and 2-site results with respectto derived olivine solution properties (see below in

Table 4 Standard state ther-modynamic properties derivedin this study

Minal DfH1,298 S1,298 V1,298

(kJymol) (JyK ? mol) (Jybar? mol)

Almandine 25265.44 (1)a 341.51 (2)a 11.529 (2, 3)a

Anthophyllite 212074.47 (1) 537.40 (4) 26.330 (5)Cordierite 29161.48 (1) 416.23 (6) 23.311 (7)Fe-Cordierite 28430.55 (1) 482.18 (8) 23.706 (9)Fayalite 21477.17 (1) 151.73 (10, 11) 4.639 (3)Ferrosilite 21192.91 (1) 96.47 (13) 3.295 (14)Fosterite 22174.42 (1) 94.18 (15) 4.360 (16)Geikelite 21570.52 (1) 74.41 (17) 3.077 (7)Hematite 2826.74 (1) 87.36 (18) 3.027 (7)Hercynite 21945.36 (1) 123.13 (1) 4.080 (7)Ilmenite 21233.32 (1) 108.50 (19) 3.170 (7)Iron-a 0.00 (1) 27.45 (12) 0.709 (7)Iron-g 7.72 (1) 35.64 (1) 0.691 (7)Magnetite 21116.96 (1) 146.04 (20) 4.452 (7)Orthoenstatite 21546.04 (1) 66.18 (21) 3.133 (16)Orthocorundum 21634.95 (1) 33.93 (1) 3.123 (1)Pyrope 26284.74 (1) 268.80 (22) 11.311 (23)Rutile 2944.75 (1) 50.88 (24) 1.883 (7)Spinel 22302.16 (1) 80.37 (25) 3.978 (23)Talc 25898.96 (1) 261.21 (26) 13.610 (16)

a (1) Values derived in this study from analysis of phase equilibrium data; other values refined withphase equilibrium data are based on measurements or estimates of: (2) Anovitz et al. 1993; (3)Chatillon-Colinet et al. 1983; (4) Hemingway 1991; (5) Hirschmann et al. 1994; (6) Weller andKelley 1963; (7) Robie et al. 1967; (8) Holland 1989; (9) Mukhodpadhay andHoldaway1994; (10)Robie et al. 1982a; (11) Essene et al. 1980; (12) Engi and Feenstra (submitted for publication); (13)Bohlen et al. 1983; (14) Sueno et al. 1976; (15) Robie et al. 1982b; (16) Chernosky et al. 1985; (17)Shomate 1946; (18) Gronvold and Westrum 1959; (19) Anovitz et al. 1985; (20) Westrum and Gron-vold 1969; (21) Krupka et al. 1985; (22) Haselton and Westrum 1980; (23) Charlu et al. 1975; (24)Shomate 1947; (25) King 1955; (26) Stout and Robie 1963.

Table 5 Heat capacity coeffi-cients (JyK ? mol)a Minal k0 k1 (31022) k2 (31025) k3 (31027) Refb

Almandine 621.43 232.879 2150.810 221.187 (1)Anthophyllite 1233.79 271.340 2221.638 233.394 (2)Cordierite 954.39 279.623 223.173 237.021 (2)Fe Cordierite 983.48 284.037 218.703 28.568 (3)Fayalite 252.00 220.137 0.000 26.219 (2)Ferrosilite 174.20 213.930 24.544 23.771 (2)Forsterite 233.18 218.016 0.000 226.794 (4)Geikelite 146.20 24.160 239.998 40.233 (5)Hematitec 146.86 0.000 255.768 52.563 (2)Hercynite 251.77 220.444 213.483 13.150 (3)Ilmenite 150.00 24.416 233.237 34.815 (2)Iron-ad 51.87 23.794 225.430 50.680 (6)Iron-g 66.24 212.371 63.733 2106.040 (6)Magnetitec 207.93 0.000 272.433 66.436 (2)Orthoenstatite 166.58 212.006 222.706 27.915 (2)Orthocorundum 119.38 7.748 265.091 42.288 (3)Pyrope 590.90 228.270 2133.208 126.033 (7)Rutile 77.84 0.000 233.678 40.294 (2)Spinel 244.67 220.040 (8)Talc 664.11 251.872 221.472 232.737 (2)

a CP5k01k1T0.51k2T

21k3T3

(Berman and Brown 1985).b (1) Fit to data of Anovitzet al. (1993); (2) fit to datasummarized by Berman(1988); (3) estimated values;(4) fit to data of Gillet and Fi-quet (1991); (5) taken fromBerman and Brown (1985); (6)taken from analysis of Engiand Feenstra (submitted forpublication); (7) fit to data ofTequi et al. (1991); (8) takenfrom Chamberlin et al. (1995).c Lambda transition termsgiven by Berman (1988)d Lambda transition terms(Jymol) using equations 8–9 ofBerman (1988): l1520.22208;l250.00034823;T1bar

l51042 K;

k520.00057;DtH51000.

12

Table 6 Volume coefficientsaMinal v1 (3106) v2 (31012) Refb v3 (3106) v4 (31010) Refb

Almandine 20.570 0.434 (1) 18.599 74.711 (1)Anthophyllite 21.139 0.000 (1) 28.105 62.894 (1)Cordierite 21.158 0.000 (1) 3.003 18.017 (1)Fe Cordierite 21.700 0.000 (2) 4.265 0.000 (2)Fayalite 20.822 1.944 (1) 26.210 84.233 (1)Ferrosilite 20.911 0.303 (1) 31.406 80.400 (1)Forsterite 20.791 1.351 (1) 29.464 88.633 (1)Geikelite 20.529 0.000 (3) 23.314 88.328 (3)Hematite 20.479 0.304 (1) 38.310 1.650 (1)Hercynite 20.510 0.000 (2) 15.819 96.276 (2)Ilmenite 20.529 0.000 (1) 23.314 88.328 (1)Iron-a 20.602 0.000 (4) 45.071 14.104 (4)Iron-g 20.467 21.445 (4) 47.153 14.451 (4)Magnetite 20.582 1.751 (1) 30.291 138.470 (1)Orthoenstatite 20.749 0.447 (1) 24.656 74.670 (1)Orthocorundum 20.749 0.447 (5) 24.656 74.670 (5)Pyrope 20.576 0.442 (1) 22.519 37.044 (1)Rutile 20.454 0.585 (1) 25.716 15.409 (1)Spinel 20.489 0.000 (1) 21.691 50.528 (1)Talc 21.847 5.878 (1) 25.616 0.000 (1)

a VP,TyV1,2985v1(P-1)1v2(P-1)21v3(T-298)1v4(T-298)2 (units for v1-v4:K21; K22; b21; b22).b (1) Taken from, or fit to datasummarized by Berman(1988); (2) estimated values;(3) values equal to ilmenite;(4) taken from analysis ofEngi and Feenstra (submittedfor publication); (5) assumedequal to orthoenstatite.

Table 7 Mixing propertiesderived in this study Phase Parameter WH (Jymol) WS (JyK ? mol) WV (Jybar? mol)

Garnet 112 33470.00 18.79 0.17315Ca 25Mg 35Fe 122 68280.00 18.79 0.036

113 21951.40 9.43 0.170133 11581.50 9.43 0.090223 5064.50 4.11 0.010233 6249.10 4.11 0.060123 73298.30 32.33 0.281

Cordierite 12 21626.70 0.00 0.00015Mg 25Fe

Olivine 12 10366.20 4.00 0.01115Mg 25Fe

Ilmenite 12 1525.30 0.00 0.01015Mg 25Fe

Orthopyroxene 12 22600.40 21.34 0.00015Mg 25Fe 35Al 13 221878.40 0.00 20.386

23 232398.40 0.00 20.883

IFe-Al (Eq. 7) 26534.900 16.11110 0.175

Derived solution properties section) demonstrates thatthe two models yield remarkably similar macroscopicenergetics for the othopyroxene solid solution.

In view of the results summarized above our preferredthermodynamic properties given in Tables 4–7 were ob-tained with the 1-site Opx model. Details of the analysiswith this model and comparison of calculated and ob-served phase relationships are discussed below.

Equilibria constraining endmember thermodynamicproperties

Figures 2 and 3 summarize the major constraints im-posed by phase equilibrium data on Alm, Fa, Fs, fCd,and Hc endmember thermodynamic properties. The ex-perimental data for these equilibria are in good accordwith available thermophysical data and tightly constrain

the differences in standard state properties among thesephases. Apparent discrepancies between the calculatedslope and experimental brackets for the Fs5Fa1Qz equi-librium are within the small uncertainties (+300 bars)of the data. Our calculations support the CP data of al-mandine measured by Anovitz et al. (1993) over the dropcalorimetry results reported by Newton and Harlov(1993). All the phase equilibrium data involving al-mandine shown in Figs. 2 and 3 were obtained at reduc-ing conditions buffered at the Fe-Wüstite (Wst) assem-blage, except for the equilibrium:

Alm13Hm53Mt1Ky12Qz (j)

which Harlov and Newton (1992) found to occur approx-imately 4 kbar below the position computed with ther-modynamic data of Berman (1988). The position of equi-librium (j) depends on almandine properties, whichBerman (1988) derived from the Fe-Wst buffered data

13

Fig. 3 Comparison of available experimental data with univariantcurves in the system CaO-FeO-Al2O3-TiO2-SiO2 computed withthermodynamic data derived in this study. Symbols (as in Fig. 1)are for (from top to bottom): 2Fs5Fa1Qz (f illed diamonds), Gr12Alm16Rt56Ilm13An13Qz (open triangles), and Gr12Alm53An13Fa (open squares; filled triangles). Note that, at 3.9 kbar,Grafchikov and Fonarev (1986) produced growth of both productsand reactants in different experiments

Fig. 2 Comparison of available experimental data with univariantcurves in the system FeO-Al2O3-SiO2 computed with thermody-namic data derived in this study. Symbols as in Fig. 1

(Fig. 2), and on the difference between Hm and Mt,which Berman (1988) based on the data of Myers andEugster (1983). Using Robinson et al.’s (1982) evalua-tion of Hm and Mt properties, Anovitz et al. (1993) alsowere unable to reproduce the Harlov and Newton datawhile fitting the Fe-Wst buffered phase equilibrium data.

One possibility to explain the above discrepancy isthat almandine contained significantly more Fe31 in theHm-Mt buffered than the Fe-Wst buffered experiments.Woodland and O’Neill (1993) determined solution of ap-proximately 15% skiagite (Fe3Fe2Si3O12) component inalmandine at 30 kbar and 11008 C in the presence ofFe31-bearing slags. At the lower temperatures of Harlovand Newton’s experiments (650–9008 C), reducedskiagite solution would not account for the above dis-crepancy. In addition, Harlov and Newton’s (1992) mi-croprobe analyses of almandine show only minor Fe13

solubility. These observations suggest that the most like-ly source of discrepancy is in the calibration of theHm5Mt1O2 buffer, which is kinetically sluggish (e.g.O’Neill 1988) and which has yielded highly discordantresults in different studies (summarized by O’Neill1988). With Hm properties determined by the position ofHarlov and Newton’s experiments, the computed posi-tion of the Hm-Mt equilibrium is in perfect agreementwith O’Neill’s (1988) data between 1000 and 1150 K,offering strong support for O’Neill’s contention that his1000–1173 K data represent equilibrium values for theHM buffer. As discussed by O’Neill, deviations above1173 K are consistent with increased and well document-ed Fe2O3 solubility in Mt, while those below 1000 K can

be attributed to the extreme sluggishness of the Hm-Mtreaction.

Koziol’s (1990) reversed compositons of Gr-Alm gar-nets in equilibrium with anorthite-kyanite-quartz effec-tively define excess mixing properties on the Gr-Almbinary, although the magnitude of any excess entropy onthis join is not constrained by these data. When com-bined with equilibria that involve Gr33Alm67 garnet(Bohlen et al. 1983a; Bohlen and Liotta 1986;Grafchikov and Fonarev 1986; Perkins and Vielzeuf1992), more stringent bounds are placed on both al-mandine properties and excess entropy on the Gr-Almjoin (Anovitz and Essene 1987; Berman 1990; Perkinsand Vielzeuf 1992). If the brackets of Bohlen et al.(1983a) for equilibrium (T): Fa1An5Gr1Alm (Fig. 3)are accepted without adjustment beyond their reportedP-T uncertainties (+300 bars and 58 C), the minimumsymmetric excess entropy is about 5 Jmol21K21. Furtherrelaxation of the lowest temperature (7008 C) half-brack-et by as little as 200 bars permits zero excess entropy,although the slope of their brackets is most consistentwith positive excess entropy on this join.

Perkins and Vielzeuf (1992) report brackets for equi-librium (T) between 900 and 10508 C that supportBohlen et al.’s (1983a) reversals in this temperaturerange. Grafchikov and Fonarev (1986) also obtained rel-evant data in their study of the univariant equilbrium:Cpx1An5Gt1Fa1Qz which is degenerate to equilibri-

14

um (T) at the Gr33Alm67 composition. They pointed outthat equilibrium (T) must be tangential to their univari-ant curve at theP-T point of compositional degeneracywhich they estimate to be 7508 C. At higher and lowertemperatures the univariant curve must be at lower pres-sure than equilibrium (T). Figure 3 shows that their mostconstraining half-brackets, nominally at 4.1 kbar and7508 C, are mariginally compatible with Bohlen et al.’sdata considering uncertainties of both experimental datasets. Grafchikov and Fonarev’s preferred equilibrium lo-cation is at 3.9 kbar and 7508 C, based on growth of prod-ucts and reactants of equilibrium (T) in two differentexperiments at thisP-T. We favor the Grafchikov andFonarev bracket because the data were obtained usinghydrothermal pressure vessels, whereas Bohlen et al.’slowest temperature piston cylinder data may require aslight friction correction even for the low-strength NaClpressure assemblies they used. Direct experimental reso-lution of the conflicting 7508 C brackets would be ex-tremely useful. Although we allowed for a lower pressurethan Bohlen et al.’s 750 and 8008 C brackets, our opti-mized thermodynamic properties yield a excess entropy(9.43 JyKymol) in between estimates of Anovitz and Es-sene (1987) and Berman (1990).

Considerable controversy has existed about theP-Tlocation and slope of the equilibrium:

2Alm14Si15Qz53fCd (g)

in the FAS system (e.g. Currie 1971; Thompson 1976;Holdaway and Lee1977; Aranovich and Podlesskii 1981,1983). On the basis of their experimental data in theFMAS system, Aranovich and Podlesskii deduced a posi-tive P-T slope for equilibrium (g, Table 2) that has beenconfirmed by direct hydrothermal reversals obtained byMukhopadhyay andHoldaway(1994). Our calculationsare in good accord with both the new FAS data (Fig. 3)and the FMAS data discussed below.

Fe-Mg exchange and net transfer equilibria

Existing Gt-Opx Fe-Mg exchange data are in good over-all agreement within experimental uncertainties. Figures4a–b compare computed lnKD values with experimentaldata from four different studies. All experimental obser-vations are reproduced within 0.01 Mgy(Mg1Fe) uncer-tainties, with the exception of one experiment ('163;shaded square in Fig. 4a) of Lee and Ganguly (1988),which was omitted from the MAP analysis because bothGt and Opx increased in Mg' during this experiment.The combined analysis constrains the enthalpy change at258 C–1 bar for equilibrium (A) between 36.2 and 40.2,with an optimal value of 40.1 kJmol21. Larger valuesthat have been obtained elsewhere (e.g. Lee and Ganguly1988; Aranovich 1991), and which yield a greater tem-perature dependence for this exchange equilibrium, arenot permitted by the combination of constraints on end-member equilibria, entropies, and volumes. The enthalpychange of this exchange equilibrium estimated solely

Fig. 4 Variation of lnKD for equilibrium (A) with Fey(Fe1Mg)ratio in Opx. Curves were computed with thermodynamic dataderived in this study at the temperatures for which experimentaldata were collected (numbers and symbols at end of each curve)and a pressure of 25 kbar. Experimental observations have beennormalized to the same pressure. Symbols show nominal experi-mentally determined lnKD values. Opposite ends of connectedlines show lnKD values adjusted for 0.01 uncertainties inXFe ofboth Opx and Gt. Placement of lines above (below) symbols indi-cates equilibrium was approached from high (low) to low (high)KD. Computed curves are consistent with all data within theseuncertainties, except for run'163 (shaded square) of Lee andGanguly (1988) in which Gt and Opx show inconsistent composi-tional changes

from experimental data involving endmembers(33.61 kJmol21: Berman 1988) is outside of the rangedetermined above, a comparison that underscores the im-portance of deriving thermodynamic data for endmem-bers from the combination of data involving endmembersand solid solutions.

The exchange data for Gt-Cd (equilibrium C) are re-produced using 0.015 compositional adjustments for Gtand Cd, consistent with the amount of compositionaloverlap observed by Aranovich and Podlesski (1983).The computed two-phase loops at 700 and 7508 C arealso in good agreement with the data (Fig. 5). At 10 kbarand 10008 C, however, our calculated two-phase loop issomewhat more Fe-rich than Hensen’s (1977) bracket.This small discrepancy may be resolved by assuming asmaller friction correction than the 10% value commonlyassumed for talc pressure assemblies (e.g. Gasparik andNewton 1984).

All the exchange data for Opx-Cd (equilibrium B)were fitted within 0.01 uncertainties, with the exceptionof run '39y3 which produced a large compositionaloverlap with other runs. This datum was also not fittedby Aranovich and Kosyakova (1987) in their analysis.

The most constraining data set analyzed in this studyis that on the olivine-orthopyroxene exchange (equilibri-um E) combined with the olivine-orthopyroxene-quartznet transfer equilibria (O) and (P). Figure 6 demonstrates

15

Fig. 5 Comparison of computedP-XFe divariant loops for the as-semblage Cd1Gt1Si1Qz (solid curves7008 C, dashed7508 C)with experimental observations by Aranovich and Podlesskii(1983) (f illed squares7008 C,open squares7508 C). Symbols as inFig. 4 using pressure and compositional uncertainties of 200 barsand 0.015XFe

Fig. 6a–c Variation of lnKD for the equilibrium Fo12Fs52En1Fa (E) with Fey(Fe1Mg) ratio in Opx. Curves were computedusing thermodynamic data derived in this study at experimentaltemperature values andP516 kbar. Symbols as in Fig. 4 usingcompositional uncertainties of 0.005XFe for both Opx and Ol.Computed curves are consistent within+0.005XFe for all data

that all but one ('15, shaded symbol in Fig. 6b) of theexchange experiments of von Seckendorff and O’Neill(1993) are reproduced to within 0.005 Mgy(Mg1Fe), inexcellent agreement with their quoted uncertainties.These data also agree with the other Ol-Opx exchangedata accepted in this study within estimated uncertain-ties (see constraints on solid-solution properties sec-tion). Figure 7 shows that the tabulated thermodynamicproperties for Ol and Opx afford excellent representationof the divariant field determinations of Bohlen andBoettcher (1981), Fonarev (1987), and Davidson andLindsley (1989). In order to achieve this agreement, how-ever, it was necessary to adjust properties of Mg and Feendmembers given by Berman (1988) while maintainingconsistency with all phase equilibrium data constrainingthese endmember properties (Table 2), and to use thebulk composition constraints discussed above to fit eachhalf-bracket with the proper directional sense. Our over-all reproduction of the phase equilibrium data was alsoimproved by use of the new drop calorimetry measure-ments for forsterite (Richet and Fiquet et al. 1991) andpyrope (Richet et al. 1991). It should be noted as wellthat, in contrast with the value given by Berman (1988),the enthalpy difference of the reaction: 3 En1Co5Pyresulting from the present analysis falls within the 1s

uncertainties of calorimetric measurements of Eckertet al. (1992).

von Seckendorff and O’Neill (1993) have shown thattheir experimental data are most compatible with a

smaller entropy change for the Ol-Opx exchange equi-librium than calculated with either the database ofBerman (1988) or Holland and Powell (1990). The newheat capacity data summarized above markedly reducethis discrepancy (Fig. 8). The offset of von Seckendorffand O’Neill’s DG vsT curve to lower values than ours isdue to the strong correlation of standard state and mixingproperties combined with their preference for a morepositive heat of mixing in olivine than derived here.

Our results suggest that the disagreement of the Sackand Ghiorso (1989) calibration with experimental data atlow pressure on the Ol-Opx-Qz loop (Fig. 7b), as well aswith von Seckendorf and O’Neill’s (1993) exchangedata, results from their acceptance of (a) calorimetricdata (Chatillon-Colinet et al. 1983) and activity mea-surements (Sharma et al. 1987) to fix a much more posi-tive heat of mixing for orthopyroxene than obtained inthis study, and (b) endmember properties fixed at valuesgiven by Berman (1988).

Phase equilibrium data obtained by Hayob et al.(1993) on the Opx1Il1Rt1Qz assemblage place verytight constraints on the standard and mixing properties ofOpx and Il through operation of the Fe-Mg exchangeequilibrium (F) as well as the Fe and Mg net transfer

16

Fig. 8 Temperature dependence of the standard Gibbs free energyof exchange equilibrium (E). Curves were computed with thermo-dynamic data from Berman (1988), from von Seckendorff andO’Neill (1993), and from this study. Note similarity in slope be-tween the latter two sets of calculations, as well as their offsetwhich reflects differences in derived mixing propertiesFig. 7a, b Comparison of computedP-XFe divariant loops for the

assemblage Opx1Ol1Qz at 10008 C (a) and 8008 C (b) with avail-able experimental data (P-Tuncertainties not shown).Solid curveswere computed with thermodynamic data derived in this study.Dashed curveswere computed with thermodynamic data fromSack and Ghiorso (1989).Filled symbolsshow bracketing dataafter adjustment for 0.01XFe uncertainties in Opx and Ol.Oppo-site ends of connected linesshow starting compositions. Allf illedtriangleswithin divariant loops indicate starting and final compo-sitions within divariant loops. Additional experimental data fromBohlen and Boettcher (1981) indicate: Opx stable (X), Ol stable(1), Opx (open square) or Ol (open diamond) broke down to di-variant assemblage

are at the lower limit of Andersen et al.’s (1991) 12008 Cdata.

Equilibria controlling Al2O3 solubility in Opx

Sluggishness of the equilibration of Al2O3 in orthopy-roxene together with overstepping of equilibrium com-positions combine to yield significant uncertainties (10–50%) in equilibrium compositions. In the MAS system,computed Opx Al2O3 isopleths (Fig. 11) reproduce with-in their uncertainties the Al2O3 solubility data of Danck-werth and Newton (1978), Perkins et al. (1981), andKawasaki and Matsui (1983) for the Gt-Opx assemblage.In order to allow for slight curvature in these isopleths,we allowed for a small constant Cmax

P change of equili-brium (H). The above data are in excellent agreementwith reversed Al2O3 contents of Opx in equilibrium withCd1Qz in the MAS system at 750–8508 C and 1–5 kbar(Aranovich et al. 1983). Treated by themselves, the datafor equilibria (H) and (J) allow the En-Ok join to beideal, while the range of values permissible for WOpx

MgAlwhen all data are considered simultaneously is22.5 to228 kJmol21. Our preferred value resulting from finaloptimization is222 kJmol21. Use of this Al2O3 solubili-ty model permits good representation of MAS univariantequilibria involving orthopyroxene at both high pressurewith pyrope and low pressure with cordierite (Fig. 11).Not shown in Fig. 11 are brackets for equilibria (b), (c),and (d) that are also well represented.

equilibria (Table 3). As noted by Hayob et al., use of theBerman (1988) or Berman (1990) endmember propertiesdoes not allow perfect agreement with their data, but thesimultaneous determination of mixing properties andstandard state properties in this study leads to much im-proved results. Opx and Il compositions are all repro-duced within the uncertainties determined by Hayobet al. (Fig. 9). Apparent discrepancies in Fig. 10, particu-larly at 11008 C, are caused by ilmenite compositionsbeing calculated assuming no Fe31, while experimental-ly determined compositions are reported to have up to1 mol% hematite. The calculated lnKD for the Opx-Ilexchange equilibrium ranges from 15 to about 6 over thetemperature and compositional interval covered by theexperiments, in good agreement with Hayob et al.’s in-terpretations. The slightly larger KD values derived herestem from incorporation of phase equilibrium con-straints on the Ol-Il Fe-Mg exchange equilibrium (Fig.10). Derived KD values are in excellent agreement withAndersen and Lindsley’s (1979) data below 10008 C, but

17

Fig. 10 Variation of lnKD for the equilibrium 2Gk1Fa5Fo12Ilm(G) with Fey(Fe1Mg) ratio in Ol. Curves were computed usingthermodynamic data derived in this study at experimental temper-atures andP51 kbar (700–10008 C) and 1 bar (12008 C). Symbolsas in Fig. 4 using compositional uncertainties of 0.01XFe for bothIl and Ol

Fig. 9a–d Comparison of computedP-XFe divariant loops for theassemblage Opx1Rt1Il1Qz with experimental data of Hayobet al. (1993). Symbols as in Fig. 4 using reported uncertainties of0.02XFefor Opx (squares) and Il (triangles). Note that¥0.01 Fe31

in the experimental ilmenite improves agreement with the dis-played calculations which were performed on Fe31-free basis

Calculated isopleths of Al2O3 in the spinel peridotitefield (Fig. 11) are in excellent agreement with the mea-surements of Gasparik and Newton (1984). In contrast toBerman’s (1988) account of an unconstrained amount ofdisorder is spinel with use of a fixed increment to themeasured third law entropy, here we utilize the activitymeasurements and CP function of Chamberlin et al.(1995) to describe thermodynamic properties of spinelwith an equilibrium amount of disorder. Gibbs free ener-gy values for spinel retrieved in this study are in excep-tional agreement (,1 kJymol difference) with thosederived by Chamberlin et al. (1995).

Recent reversed experiments of Aranovich andBerman (1995, 1996a) on the solubility of Al2O3 in Opxin equilibrium with garnet in the FAS system between 12and 20 kbar are critical due to the lack of other reversalsfor high Fey(Fe1Mg) Opx. The positions of isoplethsfixed by these data (Fig. 12) are in excellent accord with

Fig. 11 Comparison of experimental data with univariant curves(bold) and Opx Al2O3 isopleths (light curveswith mol% Al2O3 inhexagons) in the system MgO-Al2O3-SiO2 computed with thermo-dynamic data derived in this study. Symbols as in Fig. 1.Dashedcurve is the position of the Fo1mCd5Sp1En equilibrium withanhydrous cordierite. Numbers insidesquares, triangles, andcir-clesgive experimental values of mol% Al2O3 in Opx

18

the lone other FAS reversal of Kawasaki and Matsui(1983) at 46 kbar. Although the combination of MASand FAS Al2O3 solubility reversals are also in good ac-cord with the three FMAS Gt-Opx reversals of Lee andGanguly (1988; Fig. 13a), they require significantly low-er Al2O3 contents than the measurements of Aranovichand Kosyakova (1987) at much lowerP-T for the Opx-Cd-Qz assemblage (Fig. 13b). Use of asymmetric andyortemperature-dependent mixing functions for Opx did notallow better agreement between the data for the two dif-ferent assemblages. Our predicted values are up to 50%lower than the data of Aranovich and Kosyakova. Nettransfer equilibria (H) and (I) control the solubility ofAl2O3 in Opx in equilibrium with garnet to considerablysmaller amounts (except for the most Mg-rich composi-tions), and with the opposite dependence on Mgy(Mg1Fe), compared to the Opx-Cd-Qz assemblage(equilibria J and K). Consideration of all data simulta-neously constrains the differenceWOpx

MgAl2WOpxFeAl in the

range from23.1 to 12.4 kJmol21. The optimized differ-ence (10.5 kJmol21), in agreement with thermodynamicanalyses on Opx and other silicates (Aranovich 1991;Mader and Berman 1992; Berman et al. 1995), displaysthe affinity of Fe for Al and is an important correction forexchange thermometers involving Opx.

Derived solution properties

The comparisons discussed above indicate that tabulatedthermodynamic data (Tables 4–7) provide excellent rep-resentation of the experimental data within their estimat-ed uncertainties. In this section, we discuss the derivedsolution properties of each mineral in the context of inde-pendent data sets that provide constraints on these prop-erties as well as the results of previous thermodynamicmodelling.

Olivine

Because solution properties for all phases derived fromphase equilibrium data are highly correlated with oneanother, we begin with a sensitivity analysis of olivinesolution properties in order to establish the range of mix-ing properties allowable by the combined set of experi-mental data. By far the most constraining data set is thatof von Seckendorff and O’Neill (1993). Consideration oftheir Fe-Mg exchange data together with experimentsinvolving endmembers fixes the range ofWOl

G (1000 K)between21.4 and 4.7 kJyatom. The shift in this range tosomewhat lower values than that computed by von Seck-endorff and O’Neill (2–8 kJyatom) is due to our consid-eration of the constraints on endmembers which, bydefining the standard free energy change of the exchangeequilibrium, limit the range of permissible mixing prop-erties. Addition of the Ol-Opx-Qz divariant field data

Fig. 12 Comparison of computed Opx Al2O3 isopleths with ex-perimentally determined values (Aranovich and Berman 1995;1996a) in the FAS system.RectanglesshowP-T, with uncertain-ties, of experiments withXok values approached from high andlow-Al Opx starting materials.Solid, half-filled,andopen rectan-gles represent experimental products with the assemblages Alm-Opx, Alm-Opx-Fa-Qz, and Alm-Fa-Qz, respectively

Fig. 13a, b Variation of Opx Al2O3 mole fraction versus Fey(Fe1Mg) of Opx for the assemblages Opx1Gt (a) andOpx1Cd1Qz (b). Curves are computed with thermodynamic dataderived in this study. Symbols show experimental data adjusted for20% uncertainties in Al2O3 contents.Opposite ends of connectedlines show starting Al2O3 contents. Only sets of experiments inwhich equilibrium was approached from under- and oversaturationare shown in (a). Numbers by curves and references in (a) give thepressure (kbar)ytemperature (8 C) at which the curves were com-puted and the data collected. All experiments approached equi-librium from undersaturation in (b)

19

narrows our calculated range ofWOlG (1000 K) only

slightly (21.1 to 4.6 kJyatom).In contrast to the above results with Ol and Opx, con-

sideration of the other data sets involving Ol providemuch less stringent bounds:214.8 to 19.8 kJyatom forthe Ol-Il and Gt-Ol exchange data considered simulta-neously. When these two data sets are combined, howev-er, with all other exchange datanot including Ol-Opx,the resultingWOl

G (1000 K) range (0.7 to 4.2 kJyatom) isin remarkable agreement with that permitted by the Ol-Opx data alone (21.4–4.7 kJyatom). Moreover, similarranges forWOl

G (1000 K) result with use of a 2-site Opxmodel (21.6 to 5.6 kJyatom). Even when all data setsinvolving Opx are excluded in entirety, essentially thesameWOl

G (1000 K) range (21.6 to 5.6 kJyatom) results,demonstrating that conclusions regarding olivine mixingproperties are not biased by our assumptions regardingthe Opx solid solution. Using the 1-site model, theWOl

G(1000 K) range allowed by all experimental data takentogether is 0.7 to 4.1 kJyatom. Our preferred free energyof mixing for Fe-Mg olivine solid solution at 1000 K(WOl

G 53.2 kJyatom) is in very good agreement with re-cent estimates (3.7+0.8 kJyatom, Wiser and Wood 1991;5.6+0.6 kJyatom, von Seckendorff and O’Neill 1993;1.5+0.3, Berman et al. 1995). Our somewhat lower val-ue than von Seckendorff and O’Neill is within their cal-culated 95% confidence limits for this value.

We found that a small excess entropy term(WOl

S 52.0 JyK-atom) significantly improved the overallrepresentation of phase equilibrium data. This is alsomore plausible from a molecular physics standpoint. Theso-called “first approximation” to the regular solutiontheory (e.g. Gokcen 1982; Aranovich 1991) suggeststhat excess entropy can be neglected only for the solu-tions with abs(WHyRT) ,, 1. Excess entropy also im-proves the correspondence of our derived excess en-thalpy (WH55.18 kJyatom) with calorimetric measure-ments of the enthalpy of mixing. Calibration of a sym-metric mixing model to the data of Wood and Kleppa(1981) results in WH57.08+1.8 kJyatom, but von Seck-endorff and O’Neill (1993) pointed out that exclusion ofone outlier in Wood and Kleppa’s data leads to a signifi-cantly smaller enthalpy of mixing (WH54.6 kJyatom).This latter value agrees with more recent data of Kojitaniand Akaogi (1994) that yield WH55.3+1.7 kJyatom.

Our Gex is significantly smaller than that of Sack andGhiorso (1989), who based their fitting on calorimetricconstraints combined with a limited set of phase equi-librium and natural partitioning data (see discussion invon Seckendorff and O’Neill 1993, p204). It is also notlarge enough to produce unmixing at low temperatureswhich has been proposed to explain the occurrence ofmm-scale lamellae in olivine grains of the Divnoe mete-orite (Petaev and Brearley 1994). Evans and Ghiorso(1995) used this explanation as support for the largernonideality of the Sack and Ghiorso (1989) olivine mod-el. The compositions of these lamellae range betweenFa23 and Fa29 with a difference of 1–1.5 mol% Fa be-tween adjacent lamellae. The formation of such Mg-rich

compositions by exsolution is inconsistent with the hightemperature phase equilibrium and calorimetric datasummarized above which both suggest symmetric excessmixing energetics. Similarly, Sack and Ghiorso’s (1989)olivine calibration predicts a symmetric solvus. Becauselamellae are not found in all olivine grains of the Divnoemeteorite (Petaev et al. 1994; Steele and Aranovich, un-published data) and because exsolution has not beenfound in olivine from more slowly cooled meteorites(Petaev et al. 1994), the origin of these lamellae remainsmysterious, and may relate more to the Divnoe meteor-ite’s particular deformation history than to an inherentinstability of olivine at low temperature.

Garnet

The mixing properties of garnet derived in this studyindicate small positive deviations from ideality acrossthe Fe-Mg join, in excellent agreement with other resultsbased on recent experimental studies (Lee and Ganguly1988; Hackler and Wood 1989; Koziol and Bohlen 1992)and with the model of Berman (1990). The preferredvalues derived here show the same sense of asymmetryfound in the latter two experimental studies, in the vol-ume and heat of mixing on this join (Geiger et al. 1987),as well as in more restricted thermodynamic assessmentsof Gt-Opx (Berman 1990) and Gt-Cpx (Berman et al.1995) exchange data. The allowance of small excess en-tropy on the Mg-Fe join improves representation of thehigh temperature Gt-Opx exchange data (see aboveResults section) and also results in predicted excessenthalpy values in accord (within uncertainties) withcalorimetric heat of mixing data of Geiger et al.(1987). These results contrast greatly with larger posi-tive deviations from ideality deduced by Ganguly andSaxena (1984) from analysis of natural assemblages andby Sack and Ghiorso (1989) from analysis of Gt-Opxexchange experiments combined with their Opx solutionmodel.

As discussed above, our present calculations suggestthat excess entropy on the Gr-Alm is midway betweenthe value of Berman (1988) and that proposed by Anovitzand Essene (1987) on the basis of Cressey et al.’s (1978)phase equilibrium data. The optimized value (9.43 JyK ? mol) also leads to reasonable agreement with the ex-cess enthalpy measurements of Geiger et al. (1987).

Cordierite

Previous work aimed at assessing cordierite stability re-lations assumed that Fe-Mg mixing is ideal (Perchuk andLavrent’eva 1983; Aranovich and Podlesskii 1983; Ara-novich and Kosyakova 1987). In this study we found thatadequate representation of the exchange data and the di-variant Gt-Cd loop, together with tight brackets on the Feendmember equilibrium (Mukhopadhyay andHoldaway1994) was most compatible with small negative devia-tions from ideality on this join (WCd

H 521.63 kJymol).

20

Orthopyroxene

Phase equilibrium data for the seven exchange couplesconsidered in this study are compatible with a symmet-ric, temperature dependent excess free energy of mixingin Fe-Mg orthopyroxene. The increasing value of WG

with increasing temperature agrees with the conclusionsof Hayob et al. (1993) based on their experimental obser-vations. They note that, although their preferred WH val-ue (3.6+4.9 kJymol) is positive, slightly negative valuesare also compatible within the uncertainties of theirphase equilibrium data. The same temperature depen-dence is also implied by available site occupancy data(Shi et al. 1992).

The small predicted negative excess free energies onthe Fs-En join differ markedly from the results of EMFmeasurements at 1000 K reported by Sharma et al.(1987), who obtained large positive Gex for the entireFe-Mg compositional range studied. Shortcomings ofthe technique used by these authors, however, have beenrecently discussed by von Seckendorff and O’Neill(1993). Although the differenceWOl

G 2WOpxG determined

in this study is almost identical to that determined by thelatter authors (¥4 kJyatom), our value forWOpx

G is morenegative than their preferred value due to the smallerWOl

Gdetermined in this study.

Our calibration yields negative enthalpies of mixingfor Opx that contrast with the results of solutioncalorimetry data at 7508 C (WH53966 Jymol; Chatillon-Colinet et al. 1983). It should be noted, however, that thecalorimetric measurements involve only three Fe-Mgsamples. Results are consistent with ideal mixing for twoof these samples within 1s uncertainties, and for allthree samples within 2s uncertainties. Chatillon-Col-inet et al.’s (1983) enthalpy of ordering experiments alsoindicate that if any of these samples retained a degree ofdisorder from their¥11208 C synthesis temperature,their heat of mixing would be more positive than that forsamples with an equilibrium ordering state. We considerthis possibility unlikely, however, in the light of the ki-netic data of Anovitz et al. (1988) which suggest that anequilibrium state of order is attained in less than the 2–3hours for samples which were equilibrated at 7508 C pri-or to the oxide-melt solution measurements. It shouldalso be recognized, that, while the phase equilibriumdata analyzed in this study tightly constrain the Gibbsfree energy of mixing of Opx, the data do not provide arobust separation of the excess enthalpy and entropy.

The negative deviations from ideality for both Hex andSex obtained in the present study are in accord with or-dered nature of Fe-Mg orthopyroxene, and agree wellwith the findings of Sack (1980) and Aranovich andKosyakova (1987), who modelled Opx as a two-site solu-tion with non-convergent ordering. The discrepancy be-tween the present results and those of Sack and Ghiorso(1989) is entirely related to the very large positive excessenthalpy of olivine accepted by these authors. Recentexperimental work (see above) does not support thisvalue.

Our calculations support the conclusions of variousexperimental studies (Harley 1984; Kawasaki and Mat-sui 1983; Aranovich and Kosyakova 1987) with respectto the important effect of aluminum on the mixing prop-erties of the orthopyroxene solid solution. Negative val-ues of the enthalpy of mixing on both enstatite – “or-thocorundum” and ferrosilite – “orthocorundum” joinsreflect a tendency for the solid solutions to form 1: 1 in-termediate compounds, although the fact that the Mg-Aljoin could be represented with ideal mixing indicatesthat this tendency is probably slight for the Mg system.The more negative value forWOpx

Fe-Al suggests that idealmixing models describing Al solubility in FMAS or-thopyroxene are in error. Given the relatively small stan-dard energy changes of the exchange reactions involvingthis mineral (for equilibrium (A),DrH

01,298540.1 kJy

mol), the difference betweenWOpxMg-Al and WOpx

Fe-Al is animportant factor in the distribution of Fe and Mg be-tween Opx and other minerals. For example, tempera-tures computed with equilibrium (A) are as much as¥1008 C higher when this correction is omitted for sam-ples with high Al2O3 Opx. In view of the simplified,regular solution model applied here to describe the mix-ing properties of aluminous Opx, both mixing and stan-dard properties of the “ortho-Al2O3” component shouldbe considered only as empirical fitting parameters. Nev-ertheless, our analysis found that the existing experimen-tal phase equilibrium data are represented as well withthis model as with models based on Tschermak’s compo-nents, and the present model has the computational ad-vantage of not requiring explicit specification of site oc-cupancies.

Ilmenite

Our retrieved mixing parameters for ilmenite-geikelitesolution involve symmetric excess entropy and enthalpy,in reasonable agreement with that determined by An-dersen et al. (1991). The somewhat larger absolute val-ues of these authors is related to their significantlyhigher estimates for the (asymmetric) entropy of mixing,and also to their acceptance of a largerWOl

G (7 kJyatomcompared to 3.2 kJyatom at 1000 K obtained in thisstudy). Direct comparison of the present results withGhiorso’s (1990) is prevented by the differences in themodels applied.

Conclusions

A particularly satisfying result of this study is that thederived thermodynamic properties given in Tables 4 and5 reproduce the Fe-Mg exchange data for all seven ex-change couples (Gt-Opx, Gt-Ol, Gt-Cd, Opx-Ol, Opx-Cd, Opx-Il, Ol-Il) within very reasonable estimates ofoverall uncertainties. This is particularly encouraging inas much as only four of these exchange couples are inde-pendent, with experimental data for the ohter three cou-ples providing a check, and added refinement of the

21

derived endmember and mixing properties. Moreover,the final optimization reproduces the extremely tightconstraints on endmember properties while invokingvery simple solution models that afford an excellent op-portunity for extrapolation beyond the data considered inthis study. The combination of experimental data on end-members and solid solutions define nonideal mixingparameters in the orderWOl

G .WIlmG .WGt

H .WOpxG .WCd

H ,with 0.7,WOl

G (1000 K),4.1 kJyatom of isomorphousFe-Mg. The results of experiments in MAS and FAS onthe Al2O3 solubility of Opx in equilibrium with Gt andwith Cd1Qz are in good agreement with the few rever-sals involving FMAS Gt but not the unreversed data forFMAS Cd. Further experimental work in the FMAS sys-tem is warranted to corroborate the applicability of thepresent analysis to predicting Al2O3 contents of FMASOpx at crustal conditions (see part II of this paper, Ara-novich and Berman 1996b).

We caution users of this newly derived data set, thatunlike the Berman (1988) data, which were derivedlargely without consideration of mixing properties, thepresent thermodynamic data for endmembers should on-ly be applied quantitatively to solid solutions in conjunc-tion with the presently derived mixing properties. To theextent that this study has correctly separated mixing andstandard energies, other calibrations of different mixingmodels with these standard state properties should be incloser accord with experimental observations than thedata given earlier by Berman (1988). Use of the presentstandard properties with different mixing propertieswould destroy the internal consistency of our calcula-tions, however, and can only be justified if the user per-forms calculations with different mixing properties thatdemonstrate compatibility with all relevant experimentalobservations. Applications of this data set to high-grademetamorphic rocks are described in a companion paper(Aranovich and Berman 1996b).

Acknowledgements We acknowledge and applaud the great num-ber of experimentalists from different labs around the world whoproduced the large set of high-quality phase equilibrium data uti-lized in this study. We are grateful to IGCP project'304 forproviding LYA with the opportunity to visit Canada, Terry Gordonfor introducing the authors to each other and helping to initiate thisproject, and the Geological Survey of Canada for financial assis-tance to LYA. LYA also thanks V.A. Zharikov for providing a 2-year leave from Chernogolovka in order to fulfill the project. Help-ful reviews of E.Froese and M. Hirschmann are much appreciated.This is Geological Survey of Canada contribution'41394.

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