Mixing properties of Ca-Mg-Fe-Mn garnets R. G. …American Mineralogist, Volume 75, pages 328-344, 1990 Mixing properties of Ca-Mg-Fe-Mn garnets R. G. BnnnaaN Geological Survey of

American Mineralogist, Volume 75, pages 328-344, 1990

Mixing properties of Ca-Mg-Fe-Mn garnets

R. G. BnnnaaNGeological Survey of Canada, 588 Booth Street, Ottawa, Ontario KIA 0E4, Canada

Ansrucr

A model for the mixing properties of quaternary Ca-Mg-Fe-Mn garnets is developed bymathematical programming analysis of reversed phase-equilibrium data as well as calo-rimetric and volumetric data. The positions of end-member equilibria are established byusing the thermodynamic data of Berman (1988) with minor revisions. Recent phase-equilibrium and calorimetric data strongly suggest excess entropy [5.08 J/(mol K)] onthe grossular-almandine join, but considerably smaller excess enthalpy than implied bythe data ofcressey et al. (1978).

Near-ideal mixing is deduced for the pyrope-almandine join from an analysis of threesets of orthopyroxene-garnet Fe-Mg exchange experiments. The derived mixing propertiesof Fe-Mg garnets are tested by calibrating annite properties from Ferry and Spear's (1978)experimental data on the garnet-biotite exchange equilibrium. These annite data lead toreasonable results for pressures computed from the equilibrium muscovite * almandine: annite * aluminosilicate + quartz, in contrast to results obtained with annite propertiesderived by using larger nonideal Fe-Mg mixing in garnet. Applications of this formulationof the garnet-biotite geothermometer yield most reasonable results when Zr"* : 0.

Excellent support for the derived mixing properties of garnet is demonstrated by (l)convergence ofdifferent equilibria involving various garnet components in a single P-Zregion for a two-pyroxene granulite, (2) calculating pressures with the equilibrium 3 an-orthite : grossular * 2 aluminosilicate * quartz that place the observed aluminosilicatepolymorph in natural assemblages within its computed stability field, and (3) computingpresswes as above and garnet-biotite temperatures that place rocks with more than onealuminosilicate phase in proximity to the appropriate aluminosilicate phase boundary.

INrnonucrroN

Garnet is one of the most important minerals used inpetrogenetic calculations because of its occurrence in awide range of bulk compositions over a wide range ofmetamorphic grades. Nevertheless, and in spite of con-siderable effort very little consensus has been reachedabout the validity of various models proposed for garnetsolution properties (compare, for example, Ganguly andSaxena, 1984, with Wood, 1987). This lack of consensusstems primarily from three sources. First, experimentalconstraints on solution properties come from direct mea-surements (calorimetry, xno) and phase-equilibrium datathat make the definition of a unique set of solution prop-erties highly sensitive to the weight given to each type ofdata. A related problem is the weight given to distribu-tion-coefficient (K") data of natural assemblages. Second,phase-equilibrium data (or natural Ko data) cannot beused to refine solution properties independently of stan-dard-state thermodynamic properties so that definitionof the former is sensitive to the choice of the latter. Third,considerable freedom exists in selecting model complex-ity and calibration of model parameters because of thenonnegligible uncertainties associated with all three typesof data.

0003404x/90/03044328$02.00 328

Several models for quaternary Ca-Mg-Fe-Mn garnetshave been proposed (Ganguly and Kennedy, 1974; Hodgesand Spear, 1982; Ganguly and Saxena, 1984), althoughGanguly and Saxena recommended caution in applica-tions of their model to garnets with Fe/Mg < 3. In spiteof extreme differences between the latter two models, bothare still commonly used by petrologists attempting to de-cipher P-7 conditions and P-T-t histories. Applicationof these and other models that apply to garnets of a lim-ited range of compositions invariably lead to diverse re-sults among which it is difficult for most petrologists toselect objectively. The purposes ofthis paper are to beginto resolve these differences by showing the range of so-lution properties that are consistent with available exper-imental data and to derive a revised garnet solution mod-el that can be applied with more confidence to a widerange of bulk compositions.

Phase-equilibrium experiments provide an importantsource of data from which to retrieve mixing properties,but the data derived are very sensitive to the choice ofstandard-state thermodynamic data used in the calcula-tions. Recently, Berman (1988) presented a set of stan-dard-state thermodynamic properties for 67 minerals inthe system NarO-KrO-CaO-MgO-FeO-FerOr-AlrOr-SiOr-TiOr-HrO-CO, that were optimized using the technique

BERMAN: MIXING PROPERTIES OF Ca-Mg-Fe-Mn GARNETS JZv

of mathematical programming (rrrnr). The internal con-sistency of these properties and their high degree of com-patibility with available experimental data (see Berman,1988, for supporting details) provide an excellent pointof departure to derive excess free energies of the garnetsolid solution. Direct measurements of mixing propertiesare utilized to distinguish the enthalpy, entropy, and vol-ume contributions to the free energies of mixing.

Eeuarroxs FoR MULTTcoMpToNENT soLUTroNs

This study utilizes the general excess equations (Eqs. 9and 22) derived by Berman and Brown (1984) for mul-ticomponent solutions. For a third-degree polynomial(asymmetric excess model), the activity coefficient of thernth component is written as follows:

p

nRTln ?. : > W,;.I(Q-X,X1X./X- - 2XIjX), (l)I

where each W refers to WG : WH - TWs * PWr, Q-isa term that counts the number of the i, 7, k subscriptsthat are equal to m (Table l; see also App. l), n is thenumber of sites on which mixing occurs, and p is equalto the number of excess parameters needed for a givensystem. This number and the identity of the parametersrepresent all possible permutations of the three subscriptsin a given component system. The major advantages ofthis formulation are ( I ) it can be memorized easily becausethe parameter subscripts serve as reminders of their coef-ficients, (2) the same equations apply to any number ofcomponents [or any degree of polynomial used to expressthe excess free energy if Eq. 22 of Berman and Brown(1984) is usedl, and (3) calculation ofactivity coefficientscanbe programmed with aminimum amount ofcomputercode while allowing for maximum generality.

For use in systems with more than two components, aspecific form of excess equation that was suggested byWohl (1946, 1953) has found widespread use (e.g., An-derson and Lindsley, l98l; Ganguly and Saxena, 1984;Fuhrman and Lindsley, 1988; Jackson, 1989). The Wohlequation can be recast in the form of the general excessequation (i.e., Eq. l) by substitution for the temary W,r,o:

W,.i.t : (W,,,' * Wij.j + Wi.i,k

* w,.u.u * wi.ik + wj.k)/2 - C,r.o. Q)

Equations I and 2 combine to give the same equationthat Anderson and Lindsley (1 98 I ) derived independent-ly by making the ternary excess parameter a function ofthird-degree constants in the initial polynomial used torepresent the excess free energy of solution. In the ab-sence of ternary data to calibrate the ternary Wohl Carterm, it can be seen that the Wohl equation approximatesthe ternary parameter given by Equation I as one-halfthe sum of the bounding binary parameters. This rela-tionship has been retained in this study as experimentaldata are available to calibrate ternary interactions onlyin the Ca-Mg-Fe system.

TABLE 1. O, terms in Equation 1 for garnet excess parametersgiven in Table 4

Parameter m:1 m : 2 m : 3 m = 4

MnrHopor,ocv

The technique of mathematical programming (Bermanet al., 1986) has been used to calibrate all relevant ther-modynamic properties. Accordingly, each half-bracket onthe position ofa phase equilibrium is cast as an inequalityconstraint on the Gibbs free energy of reaction, with thedirectional sense of the inequality constraint reflectingfrom which side the equilibrium has been approached(Fig. l). For the equilibrium

3 A n : G r + 2 K y * Q z , ( A )

(see App. 2 for abbreviations of mineral names) theseconstraints are written as

a,G s 0. (3)

Ifgarnet is the only phase that exhibits solid solution inthe experiment, this expression can be expanded (for asite multiplicity of three) to

A,Go + 3R?" ln 1o, * 3RT ln X* S 0. (4)

Upon rearrangement, it can be seen that each experimentoffers direct linear constraints on the set of Margules pa-rameters used to represent the activity-coefficient term ifthe standard-state properties and Xc, are known from ex-periment:

W'X' S - L,Go - 3RTln X*, (5)

where W' represents the set of Margules parameters andX' the sum of all mole-fraction coefficients resulting fromsubstitution of Equation 1. If any of the standard-stateproperties (H, S, V, Cr, a, 9) are unknown or in need ofrefinement, these terms remain as variables on the leftside of Relation 5.

The mathematical programming approach describedabove and discussed in more detail by Berman et al. (1986)offers several advantages over alternate methods oftreat-ing phase-equilibrium data. First, results using regressionanalysis are highly sensitive to the implicit or explicitweighting of the input data with no safeguard against vi-olation of specific phase-equilibrium brackets. With unp,the full range of solution properties consistent with a setof phase-equilibrium data can be explored without re-course to the iterative manual adjustment of input dataused by Lindsley et al. (1981, p. 166) or the trial and

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330 BERMAN: MIXING PROPERTIES OF Ca-Mg-Fe-Mn GARNETS

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Fig. l. Illustration ofthe procedure used to derive solutionproperties for garnet from constant temperature experiments onthe displacement of the equilibrium 3An : Gr + 2Ky + Qz.Crosses and arrowheads show the starting and final garnet com-positions, respectively. Uncertainties in pressure and final corn-positions are incorporated into the P-X* location ofconstraintsused in the analysis. Squares and diamonds represent inequalitybrackets with A,G < 0 and A,G > 0; respectively.

error calibration used by Saxena and Chatterjee (1986, p.828). Second, uncertainties in determining compositionsofexperimental products can be explicitly incorporated,thus allowing for realistic exploration of the constraintsprovided by any experimental data. Third, rrLlp makesuse of the constraints provided by half-brackets on theposition of an equilibrium without making the erroneousassumption that equilibrium has been reached. This lastpoint is very important, as most experimental studiesproduce half-brackets that cannot be combined into fullreversals.

Prior to the rraer analysis, excess-volume parameterswere fixed by analysis of measured volumes on each ofthe binary joins. Excess-entropy parameters were simi-larly fixed for the grossular-pyrope join, the only binaryfor which direct excess-entropy measurements have beenmade. Every attempt has been made to adhere to thephilosophy that the adopted solution model be only ascomplex as required by the experimental data and asso-ciated uncertainties.

Although inequality constraints representing 20 uncer-tainties of excess enthalpies could be analyzed simulta-neously with the phase-equilibrium data, this procedurehas not been followed here because of the repeated oc-currence of inconsistencies between phase-equilibriumdata and measured heats of mixing. Instead, phase-equi-librium data have been used to form the primary con-straints on enthalpies and entropies of mixing, whereas

900 1000 1100 1200 1500 1400

T e m p e r o t u r e ( " C )

Fig.2. Comparison of various experimental brackets for theequilibrium 3An : Gr + 2Ky + Qz with its position calculatedwith the thermodynamic data of Berman (1988). Open symbolsrepresent location of experiments after adjustment of nominalvalues (end of lines) for estimated P-I uncertainties. The dataofKoziol and Newton (1988a) and Wood (1988) have been ad-justed downward by 0.2 kbar for friction effects (see text). Theuncorrected data ofHensen et al. (l 975) and Cressey et al. (l 978)shown in this figure appear to require downward adjustment ofabout I kbar, presumably owing to friction effects.

heat-of-mixing data were used to optimize the final prop-erties with a nonlinear least-squares objective function(Berman et al., 1986). The lesser weight given to the latterdata can be rationalized by the common occurrence with-in individual sets of enthalpy measurements of outliersthat do not agree with 2o uncertainties with either smoothanalytical representations of the entire data set or withduplicate measurements (Figs. 3a, 6a, 8a). In contrast,taking account of compositional uncertainties in phase-equilibrium experiments generally suffices to correct forany overstepping ofinferred equilibrium boundaries in-dicated by nominal compositions, although inconsisten-cies among data sets can arise that need to be resolved.

Clr.nurroN oF BTNARY JorNs

Table 2 summarizes the phase-equilibrium data thathave been used to constrain thermodynamic properties.The thermodynamic activity of a mineral component iscalculated from the pressure difference between the po-sition of an equilibrium with end-member minerals andthe position of the same equilibrium with the solid so-lution. It is therefore important to establish and correctfor any differences among the positions of the end-mem-ber curves computed with the thermodynamic data usedin this study and the experimentally determined end-member curves. In this study preference is given to theposition of the thermodynamic end-member curves, asthe thermodynamic data have been refined using an ex-

tr'b

o Koziol ond Newton (' lgEEob ; H"."".

"t or. (1975)

a Cressey et ol (197E)o wood (19EE)

BERMAN: MIXING PROPERTIES OF Ca-Mg-Fe-Mn GARNETS 331

TABLE 2, Phase-equilibrium data used to derive garnet solution properties

Equilibrium Ref..Uncertainties used

in MAP analysisPressureassembly Pressure adlustments (see text)

(A) 3An: Gr + 2Ky + Qz

(B) An + 6l lm + 3Qz: Gr + 62Alm + 6Rt

(C) 3Fa + 3An : Gr + 2Alm 5

(D) 3En + Alm : 3Fs + Py 78

Y4 in. (1.905 cm)Naol

Va in. (1.905 cm)NaCl

Y2 in. (1.27 cm)talc

t/zin. (1.27 cmltalc

0.3 kbar, 5'C, 0.01 X@ 1 in. (2.54 cm)Nacl

0.3 kbar, 5'C, 0.01 Xo. 1 in. (2.54 cm)Naol

0.3 kbar, l(F C, 0.01 Xh, XM NaCl and CsCl10% kbar,20" C, 0.01 Xb, Xa, Tetrahedral anvil

I 1.0 kbar, 5' C, 0.01 X6, XM Talc and Pyrex

10 0.2 kbar, 5' C, 0.01 X^.", X", Cold seal

-0.2 kbar as suggested by Koziol andNewton (1988) for % in. (1.905+m) NaClcells

-0.2 kbar as suggested by Koziol andNewton (1988) tor Tq-in. (1 .905-cm) NaClcells; -0.3 kbar to account for An com-position

-1 kbar as suggested by comparison ofdata tor end-member curve (Fig. 2)

> -1 kbar as suggested by comparison otdata for end-memb€r curve (Fig. 2)

-0.5 kbar to ac@unt for final compositionsreported for An and llm

Only data with crystalline starting materialsused

Only data with crystalline starting materialsin C caosules used

1 0.3 kbar, 5'C, 0.01 Xc,

2 0.3 kbar,5 'C,0.01 Xu,

3 0.5 kbar, 5'C, 0.01 Xq

4 0.5 kbar, 5'C, 0.01 Xu,

(E) Ann + Py : Phl + Alm'References: (1) Koziot and Newton (1988a), (2) Wood (1988), (3) Hensen et al. (1975), (4) CressQl-et al. (1978), (5) Bohle_n et al. (1983a)' (6) Bohlen

and Liotta 0986),i7) Lee and Gangutyitseet, tiliHaaey 098ai, (9i xawasari and Matsui 0983), (10) Ferry and Spear (1978).

perimental database that extends far beyond the equilib-ria listed in Table 2 (see Berman, 1988, for details). Inaddition, this assumption leads to fulfillment of one ofthe goals of this work: to provide petrologists with aninternally consistent set of standard-state and solutionproperties of minerals that can be applied meaningfullyto natural assemblages.

Figure 2 shows the position of Equilibrium A comput-ed with the thermodynamic data of Berman (1988) incomparison with brackets determined in four studiesaimed at measuring grossular activity by the displacedequilibrium method. The computed curve agrees with theoverall distribution of Koziol and Newton's (1988a) data,which have been adjusted downward in pressure by 0.2kbar to account for their estimation of friction effects in3/+-in. (1.905-cm) NaCl pressure assemblies, although thecurve is displaced to the low-pressure uncertainty limit(0.54 kbar estimated by Koziol and Newton) of the 950and 1050 "C brackets. These small differences may beindicative ofslightly etreater friction effects than they es-timate or by upward displacement of the equilibriumcaused by incorporation of small amounts of Li (from theLirMoOo flux) into the feldspar structure. In either casethis small diference can be ignored in analyzing theirdisplaced phase equilibrium data, as the friction effectsshould be smaller at the lower pressures of the displacedequilibrium experiments, and contamination by Li shouldaffect both the displaced and the end-member data equal-ly.

At 1050 oC, Wood (1988) reversed Equilibrium A be-tween 23.0 and 23.5 kbar using a piston cylinder appa-ratus with a 3/q-in. (1.905-cm) NaCl pressure assembly.As these brackets are virtually identical to those foundby Koziol and Newton (1988a), Wood's data have also

been adjusted downward by the 0.2-kbar friction correc-tion recommended by Koziol and Newton for 3/-in. NaClpressure assemblies. Comparison of the 1300 and 1350oC brackets of Hensen et al. (1975) and Cressey et al.(1978) with the computed equilibrium curve (Fig. 1) in-dicates a l-kbar correction toward lower pressure. Thiscorrection represents a minimum value that should beapplied to their displaced equilibrium data collected atlower temperatures where friction effects would be ex-pected to increase.

Grossular-Almandine

Reversed experimental data relevant to analysis of theGr-Alm join have been obtained by Koziol and Newton(l9S8b) for Equilibrium A and by Bohlen and Liotta(1986) and Bohlen et al. (1983b), respectively, for equi-libria involving garnet of the composition Gr'rrAlmrrr:

A n + 6 I l m + 3 Q z : G r + 2 A l m + 6 R u ( B )

and

3 F a + 3 A n : G r + 2 A l m . ( C )

Retrieval of mixing properties from these data is criticallydependent on assumed standard-state thermodynamicdata. In order to reproduce all phase-equilibrium data asclosely as possible on this join, minor adjustments weremade to the standard state properties of almandine andilmenite given by Berman (1988) and derived withoutconsideration of mixing properties on this join. Theserevised standard state properties (Table 3) are consistentwith all other thermodynamic properties given by Ber-man (1988) and with the same phase equilibrium exper-iments used by Berman (1988) to derive these properties.

It is encouraging that these revisions lead to very close

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Fig. 3. Comparison of predicted and measured values for (a) heats of mixing and @) volumes of mixing on the grossular-almandine join. Error bars show 2o uncertainties that incorporate uncertainties in end-member values. Predicted curves are alsoshown in (a) for models ofGeiger et al. (1987; dashed curve) and Ganguly and Saxena (1984). Note in (b) that the suggestion ofnegative excess volume (Cressey et al., 1978) for almandine-rich compositions is not supported by more recent data.

15

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agreement of the enthalpy of the formation of almandine(- 5267 .216 kJlmol) with that (- 5261. .8 kJlmol) derivedfrom electrochemical measurements reported recently byWoodland and Wood (1989).

Although the data for Equilibria A and B are compat-

ible with near-ideal mixing with no excess entropy, thehighest temperature bracket for Equilibrium C requires aminimum symmetric entropy of about 9 J/(mol K). How-ever, this amount ofexcess entropy leads to excess heatsof mixing that are considerably larger than measured by

a Geiger e t o l . (1987)

Koziol (1988)CresseV et 6t. (1978)Geiger 'e t o l . (1987) '

BERMAN: MIXING PROPERTIES OF Ca-Mg'Fe-Mn GARNETS J J J

TABLE 3. Standard-state thermodynamic properties of minerals (at 1 bar and 298.15 K) modified from Berman (1988)

AH (kJ/mol) I [J(mol K)] V (Jlbat)

AlmandineAnnitellmenitePhlogopite

FeeAl2Si3O1,KFerAlSi30,o(OH),FeTiO,KMgrAlSbO,o(OH),

-5267.216'- 51 42.800t-1232.448'*-6210.391 $

340.007-420.000+1 08.6281334.346$

1 1 . s 1 1 t15.408+3.171

14.977t

Note..Heat-capacityandvolumefunctionsaregivenbyBerman(1988)forall mineralsexceptannite: Cp:727.21 .-_4775.O47-v - 13831 900 f-'z

+ 2119060OOO7+1b"in.l4mot K); estimation bisedoriexchangereaciiontophlogopite) V1",nlVx*,m,"n= 1 + (3.445 x 10+)(f - 298.15) - (1.697x 10{) (P - 1 ) (same as phlogopite).

. Properties revised on the basis of experimental data involving garnet solid solutions discussed in this paper.f Properties taken from Berman (1988).f Properties derived from garnetbiotit6 data of Ferry and Spear (1978) in conjunction with garnet properties given in Table 4 and the assumption of

ideal Fe-Mg mixing in biotite.$ Properties revised to account for data of Wood (1976) and Peterson and Newton (1988) for the equilibrium phlogopite + quartz : enstatite +

sanidine + HrO.

Geiger et al. (1987). A small adjustment (0.3 kbar) be-yond the uncertainty reported by Bohlen et al. (1983b)for this bracket permits a smaller excess entropy (5.08J/(mol K) and heats of mixing that are in good agreementwith the data of Geiger et al. (Fig. 3a). The amount ofthis adjustment is within the uncertainties of the ther-modynamic properties of minerals in Equilibrium C andmay indicate that minor revisions in some heat capacityand volume functions are necessary. The adopted solu-tion properties (Table 4) are consistent with both sets ofphase-equilibrium data for the AlmrrrGrrr. compositionGig. a) and with Koziol and Newton's (1988b) data (Fig.5). The data of Cressey et al. (1978) were not used be-cause they are clearly inconsistent with those of Kozioland Newton (Fig. 5) since similar garnet compositionswere produced about 3 kbar higher in the experiments ofCressey et al. The simplest interpretation is that this dis-crepancy results from much larger friction effects in theexperiments ofCressey et al. at 1000 "C than the l-kbarcorrection implied by comparison of their 1350-'C re-versal of the end-member equilibrium with the computedposition and with the other data sets (Fig. 2).

Excess volumes measured by Geiger et al. (1987) andKoziol (1988) are positive across the entire grossular-al-mandine join, contradicting the suggestion of negativeexcess volume for almandine-rich compositions (Cressey

TlaLE 4. Garnet solution properties (only nonzero values) to beused with Equations 1 and A1-44

Parameter Wa (J/mol) Ws U/(mol K)l WvQlbat)

et al., 1978; see Fig. 3b). The present two parameter rep-

resentation of the volume data is asymmetric towardgrossular-rich compositions, the same shape exhibited in

heats of mixing derived from the above analysis (Fig' 3a)'This parallelism can be viewed most simply as the en-

ergetic response to the observed volume mismatch.The model yields enthalpies of mixing that are less

strongly nonideal than the models of Ganguly and Saxena(1984) and Anovitz and Essene (1987). This differencecan be attributed to the previous workers' use of(a) phase

equilibrium data ofCressey et al. (1978)' which are sys-

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4000800 E50 900 950 1000 1050/ ^ ^ \Tempero ture ( "C)

Fig. 4. Comparison of experimental data for two equilibriainvolving Gr,rrAlm* garnets with positions computed withthermodynamic data from Tables 3 and 4. Syrnbols as in Fig. 2.The brackets for the high pressure equilibrium have been ad-justed down by 0.5 kbar, the displacement calculated (assuming

ideal mixing) for the reported compositions of anorthite andilmenite (Table 2). See text for discussion of the inconsistencywith the high temperature bracket for the Gr + 2Alm : 3An +3Fa equilibrium.

112 21 560122 69 200113 20320133 2620223 230233 3720123 58 825124 45424134 11 470234 1975

18.7918.795.085.08

0.100.10o.170.090.010.060.2650.1000.1300.035

23.8718.795.08

,Vote.' Component order: 1 : Gr, 2 : Py, 3 : Alm, 4 : Sp. Ternaryterms were calculated from binary parameters with Equation 2; ternaryparameters of Wohl equation, Cft, arc all equal to zero.

tr Bohlen ond Liotto (1986)

o Bohten et ot (1983b) n Ft$

Gr+2Ky+Qz

A lm -Gr(1000 'c)

tr Koziol ond Newton (1988b)

x Cressey et o l . (1978)

334

0 . 0 0 . 2 0 . 4 0 6 0 8 1 . 0

Mole F roc t i on Grossu lc r

Fig. 5. Comparison of experimental brackets for the dis-placement of the equilibrium 3An : Gr * 2Ky + ez in thealmandine-grossular system with its position calculated assum-ing ideal mixing (dashed curve) and with the thermodynamicdata given in Tables 3 and 4 (solid curve). Symbols with arrowsshow starting and final garnet compositions, respectively. Thefinal compositions and pressures have been adjusted for uncer-tainties (Table 2). Note the displacement to higher pressure ofthe results ofCressey et al. (1978) that suggests the need for alarge friction correction.

tematically higher in pressure than those of Koziol andNewton (see above), (b) Haselton and Newton's (1980)model for grossular-almandine excess volume that incor-porates negative departures from ideality for almandine-rich compositions, and (c) a larger excess-entropy param-eter for the grossular-almandine join [18.828 J/(mol K) inthe model of Ganguly and Saxena (1984) and in Anovitzand Essene's (1987) modell 34.2 J/(mol K) in Anovitzand Essene's model II]. In comparison to the analysis ofAnovitz and Essene (1987), the smaller excess entropyderived in this study largely reflects the effect ofadoptingstandard-state properties here that are in better agree-ment with the results of Bohlen et al. (1983a), whichshowed high-temperature reversals for the equilibrium

A l m * 3 R u : 3 I l m + S i + 2 e z

[compare Fig. I of Anovitz and Essene (1937) with Fig.44b of Berman (1988)1. Nevertheless, because of the ex-treme sensitivity of excess entropy to small perturbationsof both phase-equilibrium and thermodynamic data, ad-ditional experiments that span a wider range of temper-ature are required to fix this value more confidently.

Grossular-Pyrope

Excess volumes on the grossular-pyropejoin are poorlyconstrained by the data (Fig. 6b). Within the context of

BERMAN: MIXING PROPERTIES OF Ca-Mg-Fe-Mn GARNETS

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the amount of scatter within and the degree of corre-spondence among various data sets, it is difficult to justifya more complicated representation than the symmetricmodel adopted in this study. This model contrasts withthose that incorporate negative departures from ideality(e.g., Haselton and Newton, 1980; Ganguly and Saxena,1984) and near ideal mixing (Wood, 1988) near the py-rope end-member.

An excess entropy of 4.51 J/(mol K) was measured byHaselton and Westrum (1980) on a synthetic garnet ofcomposition Xo,: 0.4. Following Haselton and Newton(1980), and consistent with the above representation ofthe volume data, the simplest assumption is that the en-tropy is a symmetric function, giving W" : 18.79J/(mol K). In contrast, Wood (1988) argues that W"hasthe same dependence on composition as Wr, is thusstrongly asymmetric, and is nearly zero near the pyropeend-member.

Wood's reversals of garnet compositions Xo, : 0.128and 0.150 at 15 and 16 kbar agree with the half-bracketsof Hensen et al. (1975), but the combined data (Fig. 7)for garnet compositions with Xo. < 0.22 do not placevery stringent constraints on mixing properties across thisjoin. With the above approximations for excess entropyand volume, the enthalpy of mixing can be described asnearly symmetric to highly asymmetric toward pyrope-rich compositions. The latter alternative has been adopt-ed in this study because it produces heats of mixing com-patible with the measurements of Newton et al. (1977;Fig. 6a). It should be noted however that a symmetric fit(e.9., Wr, : 41.4 kJ/mol given by Hodges and Spear,1982) also represents these data reasonably well, but thisalternative would require a considerably smaller excessentropy than derived by Haselton and Westrum (1980)in order to reproduce the phase-equilibrium data. Furtherexperimental work involving more grossular-rich garnetand analysis of well-defined natural assemblages areneeded to reduce this remaining uncertainty.

Pyrope-Almandine

Recent volume measurements (Geiger et al., 1987) in-dicate near-ideal mixing for the pyrope-almandine bina-ry, consistent with the similarity in ionic radii of Fer* andMg'z*. The best representation of these data suggests smallpositive deviations from ideality with asymmetry to-wards the Fe end-member (Fig. 8b). The same sense ofasymmetry but with much larger magnitude is observedin the heat-of-mixing data of Geiger et al. (1987; Fig. 8a)and was deduced by Ganguly and Saxena (1984) from ananalysis of Mg-Fe fractionation among natural mineralpalrs.

A great deal of experimentally determined Mg-Fe ex-change data between garnet and other minerals can beused to derive garnet mixing properties on this join, butconclusions are highly dependent on the assumed prop-erties of the other minerals involved in the experiments.It seems clear that the best strategy is to process all datatogether so as to extract simultaneously a consistent set


0.2

335

14

T12I

1t\

b r\rn

t (Ji6-Y\-/ 4

r-

I\-o+-

XO

62IO

o.4 0.6 0.8

Mole Froction Grossulcr

0.04

o.o2

0.00

-o.o2

0.0 o.2 0.4 0.6 0.8 1.0

b Mole Frcction GrossulorFig. 6. Comparison of predicted and measured values for (a) heats of mixing and (b) volumes of mixing on the join pyrope-

grossular. Error bars show 2o uncertainties that incorporate uncertainties in end-member values. Note the large discrepancies amongvarious volume determinations (b) near both end-members.

-2

a

0.0

of solution properties for all minerals (Engi et al., 1984).This is the only procedure that can effectively reduce thepossibility of erroneous assumptions regarding the solu-tion properties of any one mineral misleading the inter-pretation of the properties of another mineral, in this case

garnet. Although the wisdom of this approach seems in-disputable, the magnitude of this task is considerable andbeyond the scope of this study. Instead the procedureadopted here is to derive garnet properties from consid-eration ofexperimental exchange data between orthopy-

Newton et o l . (1977)

t r Newton et o l . (1977)

-/

T o b l e 4

Deloney (19E1)Wood (1988)Newton' et rit. (1977)

a Hensen et o l . (1975)

o Wood (1988)

- -6\,1'r0"r/


-o 17-Y

q ) 1 6Lfa( r ' ) 1 5q)L

o_ 't4

0 . 0 0 1 0 . 2 0 . 3

M o l e F r o c t i o n G r o s s u l o r

Fig. 7. Comparison of experimental brackets for the dis-placement of the equilibrium 3An : Gr + 2Ky + Qz in thepyrope-grossular system with its position calculated assumingideal mixing (dashed curve) and with the thermodynamic datagiven in Tables 3 and 4 (solid curve). Symbols as in Fig. 5. Thedata of Wood have been corrected to lower pressure to accountfor the reduced activity ofanorthite in his experiments. The dataofHensen et al. (1975) have been corrected to account for thepresence of kyanite instead of sillimanite and adjusted down-ward by I kbar as suggested by comparison oftheir bracket withthe data of Koziol and Newton (1988a) for the end-membercurve (Fig. 2).

roxene and garnet, and then to form an independent testof the derived properties by applying them to naturalassemblages that do not contain orthopyroxene.

The assumption adopted in this study is that ortho-pyroxene can be well approximated at high temperatureas a two-site ideal solution. Many activity measurementsfor Mg-Fe orthopyroxenes suggest near-ideal mixing (e.g.,Kitayama and Katsura, 1968; Sakawa et al., 1978). Incontrast, recent data of Sharma et al. (1987) gathered atlower temperature (727 'C) indicate positive excess heatsof mixing, but these can be expected to more closely ap-proach ideality at higher temperatures. Many workershave also concluded that the two-site ideal mixing modelprovides the most reasonable thermobarometric resultsfor a wide range of natural assemblages (e.g., Wood andBanno, I 973; Newton, I 983; Perkins and Chipera, I 985).Such a model is admittedly oversimplified in that it ig-nores the effects of cation ordering for which there areconflicling descriptions (Davidson and Lindsley, 1989;Sack and Ghiorso, 1989). These effects are most impor-tant, however, at temperatures lower than those of themost constraining experiments used in the followinganalysis. An additional complication is introduced bypossible nonideal interactions related to Al solubility in

orthopyroxene. This should have a negligible effect in theanalysis because the most constraining experimental datadiscussed below involve pyroxenes with less than 2.75wto/o AlrOr.

Table 2 summarizes the three sets of experiments onthe equilibrium

3 En + Alm:3 Fs + Py. (D)

The data of Lee and Ganguly (1988) constitute the mostextensive set of data with Ko values reversed from bothdirections. Kawasaki and Matsui (1983) provide muchdala at I 100 and I 300 "C, but the present analysis utilizesonly those data in which the direction of change in K"values was reported and which were produced with crys-talline starting materials. The data of Harley (1984) aredifrcult to use with confidence because many of the ex-periments were conducted in Fe capsules and orthopy-roxene-garnet charges suffered differential iron addition.Comparison of the compositions of phases contained inFe anil C capsules along with textural criteria were usedby Harley to estimate corrections in Fe-Mg ratios of theFe capsule run products. Rather than base any conclu-sions on Harley's interpretations on the effects of Fe ad-dition and zoning profiles, I used only Harley's experi-ments with C capsules in this analysis.

Using the standard-state properties given by Berman(1988) with or without the almandine revision of thisstudy (see grossular-almandine section above), the com-bined data are nearly consistent with ideal mixing in or-thopyroxene and garnet (Figs. 9a, 9b). Only small dis-crepancies occur with the highest temperature data of allthree studies. In order to reproduce all valid experimentalhalf-brackets, it is necessary to allow for small positiveexcess heats of mixing in garnet that show the same senseof asymmetry as measured volumes and heats of mixing(Fig. 8) but which are much smaller in magnitude thanthose suggested by the calorimetric data. A closer repre-sentation of the excess enthalpies could be obtained byallowing for positive excess entropy, but the very smallobserved volumes of mixing are more consistent withsmaller excess heats retrieved here. In fact, the avetageof the derived mixing parameters is in reasonable accordwith a symmetrical excess enthalpy of 1.32 kJlmol pre-dicted with the empirical model of Davies and Navrotsky(1983) that is based on the correlation of volume mis-match with excess enthalpy. It should be noted that theexact values ofthe excess parameters derived here for thepyrope-almandine join are dependent on the computedCa-Fe mixing properties because the latter influence thederived thermodynamic properties of almandine.

Ganguly and Saxena (1984) derived Fe-Mg mixing pa-rameters based on a regression analysis of Ko values ofnatural assemblages; standard-state mineral propertiesused in their study were not reported. Their predictedmixing parameters (Fig. 8a) agree with the data of Geigeret al., but lead to large discrepancies with the orthopy-roxene-garnet dala at low X." (Fig. 9a). Because of thepossibility of introducing a systematic error in garnet


0.2 0.4 0.6 0.8

Mole Frcction Almondine

0.2 0.4 0.6 0.8

Mo le F rcctio n A lma ndine

J J I

14

10\-

ot_

-_Y

\-

IL-o_o-)

Xa)

XC)

I

2

-2

-6

l r \- t u

0.0a

o.o2

0.0 1

0.00

- t j . u l

0.0 1.0

Fig. 8. Comparison of predicted and measured heats (a) and volumes (b) of mixing on the join pyrope-almandine. Error barsshow 2o uncertainties that incorporate uncertainties in end-member values. In (a) heat of mixing curves are also shown for themodels ofGanguly and Saxena (1984) and Geiger et al. (1987; dashed curve).

properties as a result of the assumption of ideal mixing to test these new thermodynamic data against informa-in orthopyroxene, it is desirable to form an independent tion gleaned from natural assemblages.check on the Fe-Mg mixing properties derived above. The Fe-Mg exchange data of Ferry and Spear (1978)One effective method described below involves the use for the garnet-biotite equilibrium,of phase equilibrium data to extract thermodynamic datattral d"p".r-a on garnet Fe-Mg mixing properties and then

Ann * Py : Phl + Alm (E)

a Geiger et ol . ( 1987)

c+s(1989

Tob le 4

Py -A lnr

Py -A Inr

t r Geiger e t o l . (1987)

Py-Alm

Lee ond Gonguly (1988)

x

A

v

n

o


3.0

2 5

o_ox 2.oLON

av 1 5

Ln

- O o n-v ' '"

|.r)N

o)<

0.0 0.2 0 4 0.6 0.E

a Mo le F roc t i on A lmond ine

Fig. 9a. Comparison of ke and Ganguly's (1988) K" brack-ets for the equilibrium 3Fs + Py : 3En + Alm in the pyrope-almandine system with its position calculated with the ther-modynamic data given in Tables 3 and 4 (solid curves). Allexperimental data have been corrected to 25 kbar using the vol-ume data given in Tables 3 and 4 and by Berman (1988). Cal-culated curves and symbols marked on the right correspond tovarious experimental temperatures. Symbols show the extremesofexperimental K" data calculated \Mith 0.0 I uncertainties in themole fractions of analyzed garnet and pyroxene compositions.Connecting lines show experimental Ko values, unadjusted forcompositional uncertainties and point in the direction in whichequilibrium was approached during the course of each experi-ment. Note the large inconsistencies between the data and pre-dicted Ko's at 1050 and 1205 "C using the model of Ganguly andSaxena (1984; dashed curves) extrapolated to lower X* thantheir recommended limit (Xa- : 0.75).

have been used to derive the standard-state enthalpy offormation for annite with standard-state properties forthe other minerals in Equilibrium E given by Berman(1988) and in Table 3. Ideal Fe-Mg mixing in biotite isassumed following Mueller's (1972) analysis of naturalKo data and the experimental data of Wones and Eugster(1965). This assumption is also consistent with unit-cellrefinements of Hewitt and Wones (1975), which show nodeviation from a linear correlation between volume andcomposition on the annite-phlogopite join. Because theannite properties depend on gamet Fe-Mg mixing, twoalternative values have been extracted, one using theproperties derived in this study (Table 4, -5142.8 kI/mol) and the other (-5160.0 kJ/mol) using the Fe-Mggarnet mixing properties of Ganguly and Saxena (1984).The large difference between the values derived for anniteis a direct consequence ofthe differences in activity co-efficients calculated for pyrope and almandine with thetwo models noted above. Comparison of how closely thetwo sets ofannite-bearing equilibria correspond to other

o . o o . 2 0 4 0 . 6 0 . 8 1 0

b M o l e F r o c t i o n A l m o n d i n e

Fig. 9b. Comparison of Ko brackets for the equilibrium 3Fs+ Py : 3En + Alm in the pyrope-almandine system with itsposition calculated assuming ideal mixing (dashed curves) andwith the thermodynamic data given in Tables 3 and 4 (solidcurves). All experimental data have been adjusted to 25 kbarusing the volume data given in Tables 3 and 4 and by Berman(1988). Symbols as in Fig. 9a.

equilibria in applications to natural assemblages then formsan independent test of the proposed Fe-Mg mixing prop-erties.

For the purpose ofthis test, analytical data have beenused from muscovite-bearing pelites that straddle thekyanite-sillimanite isograd at Azure Lake, British Colum-bia (Pigage, 1982), and other muscovite-bearing assem-blages interpreted to have equilibrated near the Al-sili-cate triple point at Mount Moosilauke, New Hampshire(Hodges and Spear, 1982). Compositional homogeneityof minerals on a thin-section scale and systematic ele-ment partitioning between minerals suggest that equilib-rium was closely approached. In order to test the two setsof annite properties, comparison is made between pres-sures computed from Equilibrium A and the equilibrium

Ann * 2AlrSiOs + Qz:Ms * Alm. (F)

Activities of anorthite, muscovite, and biotite are com-puted from

ae.:1a.Xx0 + XA)2/4

aa1": 1*"X*X?a)Por,

X^: rot4171P" + Mg + Mn + Ti + 16rAl)

a*^: 7^.XtX]")?or,

X"": Fel(Fe + Mg + Mn + Ti + t6rAl)

with ^y*, YMs, and 7A"- computed with the models of

+ x Kowosoki ond Motsui (198J)

ovoa Hor ley ( 1984) tI

tr

A

X

Py -A lm


Fuhrman and Lindsley (1988), Chatterjee and Froese(1975), and Indares and Martingole (1985; model B), re-spectively. The latter model accounts for the nonidealeflects of Ti and t6lAl, discussed at greater length below.

Use of the preferred thermodynamic properties of an-nite along with those of garnet in Table 4 yields garnet-biotite temperatures between 555 and 6 l5 'C for the AzureLake pelites and between 485 and 535 "C for the MountMoosilauke pelites. Use of the second annite value alongwith Ganguly and Saxena (1984) garnet properties givessimilar ranges of temperatures, shifted about 50 .C higherowing largely to the effects of Mn. The general similarityof results with both models reflects the fact that both setsof calculations are internally consistent and have beencalibrated against the garnet-biotite experiments of Ferryand Spear (1978).

Figure l0 shows pressures computed with the preferredthermodynamic properties. For the Azure Lake pelites,Equilibrium F gives pressures that average I kbar higherthan Equilibrium A, although for three samples excellentagreement is achieved. For the Mount Moosilauke pe-lites, four samples give nearly coincident pressures, andtwo samples give pressures about 1.5 kbar higher withEquilibrium F. In marked contrast, use of the annite prop-erties derived with Ganguly and Saxena's (1984) garnetmodel yields pressures for sillimanite-bearing samples thataverage about 12 kbar higher with Equilibrium F thanwith Equilibrium A. For Azure Lake kyanite-bearing sam-ples, the discrepancies increase to about 22 kbar. Theseextreme differences in the results of the two sets of cal-culations are a direct consequence of the assumed Fe-Mggarnet mixing properties, which are transferred to anniteproperties through thermodynamic analysis of experi-mental data for the garnet-biotite equilibrium. The mag-nitude ofthese diferences reflects the low entropy ofEqui-librium F, particularly with kyanite, making it an extremelysensitive test of thermodynamic input data. The reason-able pressures computed with Equilibrium F offer strongsupport, completely independent of the assumed ortho-pyroxene properties, for the Fe-Mg mixing properties de-rived herein as compared to those of Ganguly and Saxena(1984). The same conclusion also applies to the highlyasymmetric model of Geiger et al. (1987) because of itssimilarity to the model of Ganguly and Saxena.

Although the above comparison supports the garnetproperties derived here, the small discrepancies apparentin Figure l0 between pressures computed with EquilibriaA and F wanant further attention. One complication, al-beit a second-order effect. is that the Indares and Martin-gole (1985) model does not specify the separate interac-tions of Ti and t6rAl with Fe and Mg in biotite, only theW*"r, - Wrnrand WaarptN - Ws4a6l differences. The re-sults shown in Figure l0 were computed assuming thatall Ti and rqAl interactions with Fe are zero. When in-teractions with Fe are assumed to be positive, pressurescomputed with Equilibrium F may be lowered by up toabout I kbar, improving results for most of the AzureLake samples but worsening them for most of the Mount

9000

8000

^ 7000a

6I oooo

q)

( /1 Jvvv

aq)

[ +ooo

3000

2000480 520 560 600 640 680

Tempero tu re ( "C )

Fig. I 0. Comparison of pressures derived from Equilibria Aand F for muscovite-bearing pelites from Azure Lake, BritishColumbia (Pigage, 1982), and Mount Moosilauke, New Hamp-shire (Hodges and Spear, 1982), using the thermodynamic dataof Tables 3 and 4 and from Berman (1988) along with otheractivity models discussed in the text. Temperatures are com-puted from the garnet-biotite equilibrium (Equilibrium E).Squares show the P-I intersections of Equilibria A and E. In-tersections between Equilibria F and E are shown by trianglesfor the Azure Lake samples and by diamonds for the MountMoosilauke samples.

Moosilauke samples. Until further work addresses theseparate interactions of Ti and t6lAl with Fe and Mg inbiotite, it will remain unclear whether the discrepanciesshown in Figure l0 are related to these effects, to calibra-tion of garnet mixing properties, or to subtle effects ofdisequilibrium in the natural samples.

Binary systems containing Mn

Following Ganguly and Kennedy (1974), ideal inter-actions are assumed on the almandine-spessartine binary,consistent with the very small difference in ionic radii ofFe2* and Mn2*. Recent displaced phase-equilibrium databy Koziol and Newton (1987) and Wood (1988) for Equi-librium A demonstrate that the grossular-spessartine joinis ideal within uncertainties (Fig. I l). Ganguly and Ken-nedy (1974) and Ganguly and Saxena (1984) derived alarge positive excess enthalpy of mixing on the pyrope-spessartine binary Q7656 J/mol) from their analyses ofMg-Fe fractionation between garnet and coexisting phas-es. On the other hand, Hodges and Spear (1982) conclud-ed that Zr*" must be close to zero in order to yieldgarnet-biotite temperatures close to the Al-silicate triplepoint for Mount Moosilauke pelites. The present for-

F;Mt. Moos i louke ond

Azure Loke Pe l i tes

C r - L 2 K r t + O z' . _ / | Y L

' , n Sp -Gr (1000"C)

o Koz io l ond Newton (1987)€

o wood (1988)


^ 1 6

o_o-Y "^v l o

a)L

i UaaoL

o_ 12

0.0 0.2 0.4 0.6 0.8 1.0

Mole F roc t i on Grossu lo r

Fig. I l. Comparison of experimental brackets for the equi-librium 3An : Gr + 2Ky + Qz in the spessartine-grossularsystem with its position calculated assuming ideal mixing. Sym-bols with arrows show starting and final garnet compositions,respectively.

mulation of the garnet-biotite geothermometer using theannite properties derived above also gives the most rea-sonable temperatures with ideal Mg-Mn mixing. For theseven Mount Moosilauke samples (0.09 < Xso < 0.171),calculated temperatures at the Al-silicate triple pointpressure (3.85 kbar) range from 485 to 535 "C, in agree-ment with the triple-point temperature of 505 "C. In con-trast, use of WMEM': 37656 J/mol produces temperaturesfrom 540 to 595 "C. Analysis of more Mn-rich pelites(0.08 < Xs" < 0.532) described by Raeside et al. (1988)with tIlMgM" : 0 also yields very reasonable garnet-rimbiotite temperatures for assemblages with chlorite-albite(390 "C), chlorite-plagioclase (425-455 "C), biotite (446-510'C), and sillimanite (485-610 .C). For the same as-semblages, use of W*"rro: 37656 J/mol yields 613 'C,

621-654 'C, 580-645 oC, and 660-790 'C. The latter setof temperatures is unrealistically high for the lower-graderocks and does not show the same consistent correlationwith increasing metamorphic grade.

CAl,rrnltroN oF TERNARy INTERACTIoNS

With the exception of the recent study by Koziol andNewton (1988b, 1988c) on Ca-Mg-Fe gamets, experi-mental data that can be used to constrain ternary garnetinteractions are almost entirely lacking. uer analysis ofthe preliminary experiments reported by Koziol andNewton (1988b) on displacement of Equilibrium A withCa-Mg-Fe garnet solutions suggests that these data arecompatible with a Wohl ternary CrB interaclion param-eter equal to zero, assuming the binary mixing properties

given in Table 4. Accordingly, and in marked contrast tothe model of Ganguly and Saxena (1984), the ternaryWohl parameters have been set to zero for this and theother three ternaries in the quaternary garnet system.

DrscussroN

One of the ultimate goals of theoretical petrology is toprovide thermodynamic data that allow one to calculatethe same unique pressure and temperature from any ofthe equilibria implied by a mineral assemblage that equil-ibrated at the same P and Z and for which accurate com-positional data are available (Powell and Holland, 1988;Berman and Brown, 1988). In order to achieve this goal,accurate solution models are needed, as well as accuratestandard-state properties. A great deal of attention hasbeen focused recently on providing refinements in bothtlpes of thermodynamic data in order to obtain compat-ibility of ditrerent geobarometers in various chemical sys-tems (e.g., Newton and Perkins, 1982; Anovitz and Es-sene, 1987; Moecher et a l . , 1988). Al though notindependent of standard-state thermodynamic data, mul-ti-equilibrium P-Z calculations provide one of the moststringent tests for solution models because, barring a for-tuitous cancellation oferrors, convergence in a single P-Zregion can only be achieved if all calculated componentactivity coefrcients are close to correct. A number ofgranulites have been studied, and Figure 1 2 shows typicalgraphical results of P-T calculations for a two-pyroxenegranulite studied by Coolen (1980). Using the activitymodels of Fuhrman and Lindsley (1988) for anorthite,Newton (1983) for clinopyroxene and orthopyroxene, andthis paper for garnet, together with standard-state data ofTable 3 and from Berman (1988), P-Z intersections ofmetastable and stable equilibria are produced within avery narrow P-7 window, spanning 20 .C and 0.4 kbar(small dotted region in Fig. l2). In contrast, use of theMoecher et al. (1988) or Ganguly and Saxena (l 984) mix-ing properties for garnet leads to much wider divergenceamong these intersections, ranging over 200 "C and 5.0kbar. This example highlights some of the major differ-ences that can be produced using diferent solution models,a point that is further emphasized by the fact that pres-sures computed with individual barometers shown inFigure 12 are up to 3 kbar higher using the garnet modelsof Moecher et al. (1988) and Ganguly and Saxena (1984).

A significant problem in testing solution models is thatconvergence in P-T values derived from various equilib-ria is only one of the conditions necessary to be satis-fied-some means of assessing the accuracy of the con-vergence zone must be devised. In addition, all mineralcompositions may not have quenched at the same pres-sure and temperature. Newton and Haselton (1981) andlater Ganguly and Saxena (1984) tested Al-silicate-bear-ing rocks by comparing pressures computed with Equi-librium A with the Al-silicate phase diagram. In orderthat results can be compared directly with these other twomodels, the same test is applied here. Although this testsuffers from the limitations discussed above. a minimum

Furuo Gronul i te MF-2E0


10000

a 8000a

m

o 6000LlaaQ)L

6_ a000

2000

341

- 12000ao-o

o 10000L

laaq)L

o_ 8000

400 500 600 700 800 900 1000

Tempero tu re ( "C )

Fig. 12. Calculated stable and metastable equilibria involv-ing garnet, plagioclase, orthopyroxene, clinopyroxene, and quartzfor granulite MF-280 of Coolen (1980). Plotted equilibria werecalculated urith standard-state thermodynamic data from Table3 and from Berman (1988) using the garnet mixing model ofMoecher et al. (1988). Dotted outlined areas show the range ofthe same intersections produced with the garnet mixing modelsof Ganguly and Saxena (1984) and of Table 4. Note the markedreduction in scatter between, and generally lower pressure, oftheP-T intersections calculated with the garnet mixing properties inTable 4.

number of standard-state properties are involved in thecalculation, and these are extremely well constrained byboth direct calorimetric measurements as well as phase-equilibrium data. The major sources of uncertainty inusing this test to assess garnet solution properties are theeffects of uncertainties in standard-state properties andplagioclase solution properties. Although Berman (1988)does not present uncertainties in standard-state proper-ties, pressures computed with Equilibrium A are viewedmore optimistically than concluded by Hodges andMcKenna (1987), because the process of thermodynamicanalysis results in significant refinement beyond the ex-perimental data concerned solely with this equilibrium.As pointed out by Hodges and McKenna (1987), the ef-fects of uncertainties in plagioclase solution propertiescannot be quantified because of the variety of differentassumptions that form the basis of alternate model cali-brations. An additional source of error is the nontrivialshift in the position of the Al-silicate phase boundariesfor any given rock caused by grain size and minor im-purities (e.g., Kerrick, 1987). Because the slope of thekyanite-sillimanite transition is almost parallel to that ofequilibrium A, errors in estimated equilibration temper-

500 600 700 800 900

Tempero ture ( 'C)

Fig. 13. Comparison of the calculated Al-silicate phase dia-gram with pressures computed from Equilibrium A using thegarnet mixing properties in Table 4 and standard-state thermo-dynamic data of Berman (1988). With the exception of circleddata, samples are the same as those used by Newton and Has-elton (1981), with their average temperature estimates. Circledfields are for rocks from Mount Moosilauke (Hodges and Spear,1982), Azure Lake @igage, 1982), and Wopmay orogen (trian-gles, St. Onge, 1984) that contain more than one Al-silicate poly-morph. For these samples, plotted P-T locations represent in-tersections ofEquilibrium A with the formulation ofthe garnet-biotite thermometer discussed in the text.

atures of the rock are of importance only in comparisonwith andalusite-sillimanite assemblages.

Use of the standard-state properties provided by Ber-man (1988) and the garnet solution properties in Table 4leads to good agreement between pressures computed withEquilibrium A and the calculated Al-silicate phase dia-gram (Fig. l3). As with the calibration of Newton andHaselton (1981), the only major discrepancy occurs for asillimanite-bearing granulite (calculated pressure : 10.0kbar) reported by Perkins (1979), but much of this dis-crepancy may be due to errors in the computed anorthiteactivity, as this sample contains the most anorthite-poorplagioclase (X^: 0.05) of all the samples plotted. Theremaining small inconsistencies are within the uncertain-ty of the pressure comparison, and it is satisrying to findthat such good agreement can be achieved without resortto the large ternary interaction parameters derived byGanguly and Saxena (1984) throueh optimization of thissame comparison.

A much more sensitive test than placing a given samplewithin one stability field can be applied to rocks thatcontain two or more Al-silicate polymorphs (Fig. l3). Al-though discrepancies are noted for individual MountMoosilauke samples, these are consistent with the ex-pected shifts in Al-silicate phase boundaries caused by

A

A

A

AA


the partitioning of Mn into andalusite relative to silli-manite and sillimanite relative to kyanite. Low Mn pe-lites from Azure Lake containing kyanite and sillimanitefall within 0.5 kbar of this transition. Excellent agreementis also found with andalusite, andalusite-sillimanite, andsillimanite-bearing pelites from Wopmay orogen (St.Onge, 1984), with the andalusite-sillimanite sample plot-ting within 0.2 kbar of this phase transition.

The above comparisons of computed pressures for Al-silicate assemblages and garnet-biotite temperatures forMn-rich samples suggest that the solution properties inTable 4 can be used reliably for geological applicationswith standard-state data from Table 3 and from Berman(1988). Nevertheless much additional work is required torefine these properties, especially excess entropies ofmix-ing. Carefully reversed phase-equilibrium experimentsthat extend over a wider range of temperature would bea worthwhile complement to direct measurements of ex-cess entropy. For the grossular-pyrope binary, experi-ments involving more Ca-rich compositions are required.Systematic volume measurements would also be helpfulin defining excess volumes from the large amount of scat-ter among the data already gathered. Until such experi-mental data become available, careful analysis of datafrom natural assemblages can be expected to provide ad-ditional constraints that can be incorporated into futurerefinements of the garnet solution model derived in thispaper.

AcxNowr,nocMENTs

I am very grateful to S. R. Bohlen, A. Koziol, R. C. Newton, and B. J.Wood for generously providing details of their experimental work priorto its publication. Thanks are also due to K. L. Currie, M. Engi, E. Froese,T. M. Gordon, A. M. Koziol, and J. L. Munoz for critical reviews and toD. McMullin for assisting with data analysis. Some financial support wasprovided through NSERC grant OGP0037234. This is Geological Surveyof Canada contribution 20089.

RrrnnnNcns crrEDAnderson, D.J., and Lindsley, D.H. (1981) A valid Margules formulatior

for an asymmetric ternary solution: Revision of the olivine-ilmenitothermometer, with applications. Geochimica et Cosmochimica Acta,45.847-853.

Anovitz, L.M., and Essene, E.J. (1987) Compatibility of geobarometersin the system CaO-FeO-Al,O,-SiO,-TiO, (CFAST): Implications forgarnet mixing models. Journal of Geology, 95, 633-645.

Berman, R.G. (1988) Internally-consistent thermodynamic data for stoi-chiometric minerals in the system NarO-KrO-CaO-MgO-FeO-FerOr-AlrO3-SiOr-TiOr-HrO-COr. Journal of Petrology, 29, 445-522.

Berman, R.G., and Brown, T.H. (1984) A thermodynamic model for mul-ticomponent melts, with application to the system CaO-AlrO!-SiOr.Geochimica et Cosmochimica Acta. 48. 661-678.

-(1988) A general method for thermobarometric calculations, witha revised garnet solution model and geologic applications. GeologicalSociety of America Abstracts with Programs, 20, A98.

Berman, R.G., Engi, M., Greenwood, H.J., and Brown, T.H. (1986) Der-ivation ofinternally consistent thermodynamic properties by the tech-nique of mathematical prograrnming, a review with application to thesystem MgO-SiOr-HrO. Journal of Petrology, 27, 1331-1364.

Bohlen, S.R., and Liotta, J.J. (l 986) A barometer for garnet amphibolitesand gamet granulites. Journal of Petrology, 27 , 1025-1034.

Bohlen, S.R., Wall, V.J., and Boettcher, A.L. (1983a) Experimental in-

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MANUSCRTPT REcETvED Apru 7, 1989MANUscRrpr AccEprED Nowrvrspn 22, 1989

AppBNorx I

The equations below can be used for the calculation of activ-ities in the grossular-pyrope-almandine-sp€ssartine system usingMargules parameters from Table 4.

Expansion ofEquation I yields (l : grossular, 2 : pyrope, 3: almandine, 4 : spessartine):

3RT ln 7*: W,,r(2X,X, - 2X?Xr) + Wrrr(X3 - 2XXZ)

+ wu3(2xrx3 - 2x?x3) + wr33(x? - 2xx3)

+ wil4(2xrx4 - 2x?x4) + w'oo(X? - 2xxz)

+ w2ne2x?x3) + w'33(-2xrx?) (Al)

+ w224e2x4x') + W244e2XzXZ)

+ w334(-2x?x4) + w3s(-2x3x?)

+ WB(X2X3 - 2xtx2x) + W124(X2X4 - 2XLX2)<4)

+ wr34(x3x4 - 2xtx3x) + wr34(-2xrx3x4)

3RT ln 7"": w,rr(X? - zxlxr) + w02(2xtx2 - 2xXZ)

+ wr3(-2x?x3) + wr33(-2x1x32)

+ wu4(-2x?x4) + wr44(-2xrx?)

+ W2B(2X2X3 - 2XZX) + Wr33(X3 - 2X,X3) (A2)

+ Wzz4(2X2X4 - zxl){) + W244(XZ - 2XrXZ)

+ W334(-2X?X4) + W344(-2X3X?)

+ wr23(xrx3 - 2xtx2x) + wr24(xrx4 - 2xrxrx4)

+ Wr34(-2XrX3Xo) + Wr34(X3X4 - 2X2X3){'4)


3RT ln 7^-: W,,z(-2X?Xr) + Wrrr(-2XrXZ)

+ Wr3(X? - 2X?X) + Wr33(2XrX3 - 2XtX?)

+ W1r4(-2X?Xo) + Wr44(-2XrX?)

+ Wrr3(X3 - 2)(7){) + W233(2X2X3 - 2Xr){?) (A3)

+ w224e2xlx) + w244e2x2x:4\

+ W334(2X3X4 - 2){3X) + W344(X? - 2:(3){?)

+ wr23(xrx2 - 2xtx2x3) + wr24(-2xrxrx4)

+ wr34(xrx4 - 2xrx3x4) * wrro(Xr& - 2x2x3x4)

3RT ln 7.0: W,,r(-2X?Xr) + W,rr(-2XrX;)

+ wr3(-2x?x3) + wr33(-2x'x3)

+ wr14(x? - 2x?x,) + wr44(2xrx4 - 2x'x3)

+ w2$e2xzx3) + wr33(-2xrx3) (A4)

+ w-o(Xr, - 2X3X) + W24(2X2){^4 - 2X}XZ)

+ W334(X3 - 2X?X) + W3g(2X3X4 - 2)(3X?)

+ wrB(-2xrx2x3) + wr24(xrx2 - 2xrx2x4)

+ wr34(xrx3 - 2xrx3x4) + w234(x2x3 - 2x2x3x4)

with ternary parameters given by Equation 2. Activities are com-puted by

ac.: (Xc,'?o)3

a", : (X."'7.r)3

am: (Xn- '1n-)3

aso : (Xsn'?so)3

AppnNnrx 2Abbreviations of mineral names:

AIm : almandineAn : anorthite

Ann : anniteEn : enstatiteFa : fayaliteFs : ferrosilite61 : gr.ossular

Ilm : ilmeniteKy : kyaniteMs : rnuscovitePhl : phlogopitePy : pyrope

Qz : quartzRt : rutileSi : sillimaniteSp : spessartine

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