American Mineralogist, Volume 78, pages 113-131, 1993 Octahedral excess mixing propertiesin biotite: A working model with applicationsto geobarometry and geothermometry Ar,nnnroE. PauNo Doucn Department of Geology, University of Georyia, Athens, Georgia 30602, U.S.A. A. DaNl JorrNsroN, Jlcx M. Rrcr Department of Geological Sciences, University of Oregon, Eugene, Oregon 97403, U.S.A. AssrRAcr Partial melting experiments on a natural metapelitic rock, reported by Patiflo Douce and Johnston (1991), found the assemblage biotite + garnet + aluminosilicate * quartz + ilmenite (or rutile) over the P-T range7- I 3 kbar, 825-97 5 'C. Compositions of coex- isting biotite and garnetwere usedto investigatethe nonideal solution behavior associated with mixing of Mg, Fe2*,Al, and Ti in the octahedral layer in biotite. Our results show that contents of t6lAl and t4rAI in biotite are partially decoupledand that solution of t6tAl can be effectively modeled by means of dioctahedral exchange components of the form AlrMg , (Mg dioctahedral) and AlrFe , (Fe dioctahedral). Because these exchange com- ponents are equivalent to differences between additive components with known thermo- dynamic properties (muscovite-phlogopiteand muscovite-annite, respectively)it is pos- sible to use net-transfer equilibria between these exchange components and garnet, aluminosilicate, and quartz to estimate the absolute values of the interaction parameters W**, and Wr.o,and of the differen cesW orr, - W*"r, ard W o,r, - Wo"rr. In order to account for the diferent octahedral site occupancies of phlogopite (or annite) and muscovite, we introduce an ideal mixing model for biotite that accounts for the nonequivalence of Ml and M2 octahedral sites in trioctahedral micas. Two sets of internally consistentinterac- tion parameters were then derived by linear programming analysis, incorporating different assumptions about Fe3* content in biotite in the experimental products. The values (in kJ/mol) for W*, WF"^b WAt, - W*"r, and Worr, - Wr,r, are 52.2ot 54.7,74.4 or 57,4, 47.8 or 65.1, and 60.I or 75.1, respectively, with uncertainties on the order of 10 kJ/mol for W-n, and WF"^t and 15 kJ/mol fot Worr, - W-"r, and Worr, - Wr.rr.Equilibria involving dioctahedral exchange components in biotite allow us to carry out thermoba- rometric estimates in metapelitic rocks that are independent of plagioclaseand of the (commonly low) grossularcontent of garnet. The biotite mixing model is applied in this way to generate internally consistentP- ?n estimatesfrom natural biotite * garnet * alu- minosilicate + qvafiz, biotite + cordierite + aluminosilicate * quartz, and biotite + garnet + cordierite + qvartz assemblages, which agreewith independent constraints on the values of theseintensive variables. Such agreement indicates that the chemical poten- tials of the exchange components AlrMg . and AlrFe , and of their linear combination MgFe , can be reliably estimated (at least within Al-rich bulk compositions) with the solution model that we propose. INrnooucrroN pressure, and compositional variables.This sameknowl- edgeis necessary ifwe want to understandbetter petro- Trioctahedral micas are amongthe most abundant maf- logic processes, such as crustal anatexisor the formation ic minerals in the continental crust, where they play ma- of granulites,which in many crustal rocks are controlled jor roles in both igneous and metamorphic petrologic toalargeextentbyterminalreactionsofthemicas. processes. The widespread distribution ofbiotite in pres- In spite ofthe obvious importance ofbiotite as a rock- sure-temperature-composition space makes it a poten- forming mineral, our knowledgeof its solution properties tially fruitful source of information about the values of is still rudimentary, in largepart because the composition thermodynamic intensive variables under which such space of biotite is so extensive.The complexities of bio- processes take place.Extraction ofsuch information from tite solid solution are reflected in the fact that an adequate mineral equilibria involving biotite, however, requires a description of natural trioctahedral micas requiresat least sound understandingof the thermodynamic behavior of five or six linearly independentcomponents (e.g., Dymek, this complex solid solution as a function of temperalure, 1983; Labotka, 1983;Guidotti, 1984; Hewitt and Abrecht, 0003-o04x/93 /oto24r I 3$02.00 I 13
Transcript
American Mineralogist, Volume 78, pages 113-131, 1993
Octahedral excess mixing properties in biotite: A working model withapplications to geobarometry and geothermometry
Ar,nnnro E. PauNo DoucnDepartment of Geology, University of Georyia, Athens, Georgia 30602, U.S.A.
A. DaNl JorrNsroN, Jlcx M. RrcrDepartment of Geological Sciences, University of Oregon, Eugene, Oregon 97403, U.S.A.
AssrRAcr
Partial melting experiments on a natural metapelitic rock, reported by Patiflo Douceand Johnston (1991), found the assemblage biotite + garnet + aluminosilicate * quartz+ ilmenite (or rutile) over the P-T range 7- I 3 kbar, 825-97 5 'C. Compositions of coex-isting biotite and garnet were used to investigate the nonideal solution behavior associatedwith mixing of Mg, Fe2*, Al, and Ti in the octahedral layer in biotite. Our results showthat contents of t6lAl and t4rAI in biotite are partially decoupled and that solution of t6tAl
can be effectively modeled by means of dioctahedral exchange components of the formAlrMg , (Mg dioctahedral) and AlrFe , (Fe dioctahedral). Because these exchange com-ponents are equivalent to differences between additive components with known thermo-dynamic properties (muscovite-phlogopite and muscovite-annite, respectively) it is pos-sible to use net-transfer equilibria between these exchange components and garnet,aluminosilicate, and quartz to estimate the absolute values of the interaction parametersW**, and Wr.o, and of the differen ces W orr, - W*"r, ard W o,r, - Wo"rr. In order to accountfor the diferent octahedral site occupancies of phlogopite (or annite) and muscovite, weintroduce an ideal mixing model for biotite that accounts for the nonequivalence of Mland M2 octahedral sites in trioctahedral micas. Two sets of internally consistent interac-tion parameters were then derived by linear programming analysis, incorporating differentassumptions about Fe3* content in biotite in the experimental products. The values (inkJ/mol) for W*, WF"^b WAt, - W*"r, and Worr, - Wr,r, are 52.2 ot 54.7, 74.4 or 57,4,47.8 or 65.1, and 60. I or 75.1, respectively, with uncertainties on the order of 10 kJ/molfor W-n, and WF"^t and 15 kJ/mol fot Worr, - W-"r, and Worr, - Wr.rr.Equilibriainvolving dioctahedral exchange components in biotite allow us to carry out thermoba-rometric estimates in metapelitic rocks that are independent of plagioclase and of the(commonly low) grossular content of garnet. The biotite mixing model is applied in thisway to generate internally consistent P- ?n estimates from natural biotite * garnet * alu-minosilicate + qvafiz, biotite + cordierite + aluminosilicate * quartz, and biotite +garnet + cordierite + qvartz assemblages, which agree with independent constraints onthe values of these intensive variables. Such agreement indicates that the chemical poten-tials of the exchange components AlrMg . and AlrFe , and of their linear combinationMgFe , can be reliably estimated (at least within Al-rich bulk compositions) with thesolution model that we propose.
INrnooucrroN pressure, and compositional variables. This same knowl-edge is necessary ifwe want to understand better petro-
Trioctahedral micas are among the most abundant maf- logic processes, such as crustal anatexis or the formationic minerals in the continental crust, where they play ma- of granulites, which in many crustal rocks are controlledjor roles in both igneous and metamorphic petrologic toalargeextentbyterminalreactionsofthemicas.processes. The widespread distribution ofbiotite in pres- In spite ofthe obvious importance ofbiotite as a rock-sure-temperature-composition space makes it a poten- forming mineral, our knowledge of its solution propertiestially fruitful source of information about the values of is still rudimentary, in large part because the compositionthermodynamic intensive variables under which such space of biotite is so extensive. The complexities of bio-processes take place. Extraction ofsuch information from tite solid solution are reflected in the fact that an adequatemineral equilibria involving biotite, however, requires a description of natural trioctahedral micas requires at leastsound understanding of the thermodynamic behavior of five or six linearly independent components (e.g., Dymek,this complex solid solution as a function of temperalure, 1983; Labotka, 1983; Guidotti, 1984; Hewitt and Abrecht,
0003-o04x/93 /oto24r I 3$02.00 I 13
t t 4 PATINO DOUCE ET AL.: MIXING PROPERTIES IN BIOTITE
1986). Compositions of natural trioctahedral micas showmixing of Mg, Fe2*, Fe3t, Al, Ti, and vacancies in octa-hedral sites, mixing of Si, Al, Fe3* and, possibly, Ti intetrahedral sites, mixing of K, Na, Ba, Ca, and vacanciesin interlayer sites, and mixing of OH-, O'-, F , and Cl-anions. Variable degrees ofcation ordering, especially be-tween divalent and tri- or tetravalent cations in the oc-tahedral layer (e.g., Bailey, 1984; Brigatti et al., 1991),must also be considered when modeling the ideal mixingbehavior of biotite. Even after such aspects of ideal mix-ing are accounted for, the excess energetic effects thatarise from nonideal interactions among the different ionsare likely to be important and must be determined andincorporated into solution models for trioctahedral micas.
In view of these difficulties, it is perhaps not surprisingthat utilization of mineral equilibria involving biotite ingeothermometry and geobarometry is commonly prob-lematic. A case in point is that of the geothermometerbased on the Fe-Mg exchange equilibrium between bio-tite and garnet, which was calibrated experimentally byFerry and Spear (1978) for biotite compositions along thephlogopite-annite join. It has become clear that temper-atures calculated with this geothermometer using naturaltrioctahedral micas whose compositions depart from thephlogopite-annite join are often inaccurate and inconsis-tent with temperatures obtained from other mineral equi-libria (e.g., Bohlen and Essene, 1980; Indares and Mar-tignole, 1985; Edwards and Essene, 1987; Chipera andPerkins, 1988; Essene, 1989). Although the solutionproperties of garnet are also incompletely characterized,these inconsistencies appear to be particularly severe inupper amphibolite and granulite facies rocks, which re-crystallized under pressure-temperature conditions atwhich biotite commonly incorporates important amountsof Ti, t0141, vacancies, halogens, and oxy-components (e.g.,Dymek, 1983; Guidotti, 1984).
A number of recent contributions have addressed theproblem of estimating excess mixing properties in biotiteand applying these mixing properties to improve the re-liability of the garnet-biotite geothennometer. Indares andMartignole (1985) modeled the mixing of 16rAl, torJi, t6rl![g,
and t6lFe in biotite with a regular solution model. Theyestimated the differences Wo.^ - WveN and Wr.r, - W-"r,from the compositions of natural garnet-biotite pairs, inwhich garnet cores were inferred to have equilibrated ata uniform temperature, during the peak of regional meta-morphism, with matrix biotite isolated from garnet crys-tals. Hoisch (1989, 1991) also applied a regular solutionmodel to the mixing of octahedral cations in biotite andestimated the values of the excess parameters W*^, Wr"or,ar'd W'o, from the compositions of natural metamorphicbiotites. The approach followed by Hoisch consisted of,in the first place, estimating equilibration temperaturesfrom garnet-biotite Fe-Mg exchange equilibrium (eitherassuming ideal behavior, as did Ferry and Spear, 1978,or incorporating the excess parameters of Indares andMartignole, I 9 8 5) and estimating equilibration pressuresfrom garnet + aluminosilicate + plagioclase + qvaftz
equilibrium. These pressure-temperature conditions werethen used to obtain biotite excess mixing parameters, uti-lizing the same biotite-garnet pairs from which pressuresand temperatures were calculated by performing multiplelinear regression on net-transfer equilibria involving thephlogopite, annite, eastonite, and siderophyllite compo-nents in biotite, the pyrope, almandine, and grossularcomponents in garnet, and anorthite component in pla-gioclase. The circular nature of this approach calls intoquestion the significance of the excess mixing parametersproposed by Hoisch (1991).
In another recent contribution, Williams and Gram-bling (1990) carried out a detailed statistical analysis ofthe effects of garnet and biotite compositions on the locusof the Fe-Mg exchange equilibrium between these twophases. They concluded that I6rAl does not make a sig-nificant contribution to the nonideality of Fe-Mg mixingin biotite, or, in other words, that the difference betweenFeAl and MgAl interaction parameters is relatively smallor negligible. In contrast, Williams and Grambling (1990)found that the effect of Ti on Fe-Mg mixing propertiesappears to be significant, but, because of multicolinearityin the data set used in their analysis, they could not ob-tain a reliable value for the contribution of Ti to the ex-cess free energy of Mg-Fe mixing in biotite.
In this contribution we present experimental results thathelp to elucidate the compositional changes that take placein biotite in bulk compositions saturated with quartz, alu-minosilicate, garatel, and titanium oxide phases, and weresolve some of the effects of these compositional changeson the excess Gibbs free energy of biotite. Our resultsconfirm that Ti enrichment in biotite coexisting with Ti-saturating phases is a strong function of temperature. Wealso show that t6tAl and torAl are at least partially decou-pled and that the content of t6tAl in biotite coexisting withgamet, aluminosilicate, and quartz can be reliably mod-eled by means of net-transfer equilibria between thesethree phases and dioctahedral components in biotite.Linear programming analysis of the loci of these equilib-ria in pressure-temperature-composition space providesinformation about Mg-Fe-Al-Ti mixing properties in thedioctahedral layer of biotite, making it possible to esti-mate the absolute values of the interaction parametersW*^, and Wo"o, and the differences Worr, - W-n, andWorr, - Wr"r,. We discuss applications of our results tothermometry and barometry of metapelitic rocks; in par-ticular, we show how the t6lAl content in biotite can beused as a reliable pressure indicator in assemblages thatbuffer the activity of alumina.
ExprnrvrnNTAl PRocEDURES
The results reported in this paper were derived from aseries of dehydration-melting experiments on a naturalmetapelitic rock, originally designed to study the pres-sure, temperature, and compositional dependence of meltfraction during crustal anatexis (Patifio Douce and John-ston, l99l). For the bulk composition studied, we found
PATINO DOUCE ET AL.: MIXING PROPERTIES IN BIOTITE
TaaLe 1, Experimental conditions and phase assemblages
Qtz + Als + Bio + Gar + llm MeltQtz + Als + Bio + Gar + llm MeltOtz + Als + Bio + Gar + llm MeltOtz + Als + Bio + Gar + llm MeltOtz + Als + Bio + Gar + llm MeltQtz + Als + Bio + Gar + llm + Rut MeltQtz + Als + Bio + Gar + llm MeltOtz + Als + Bio + Gar + llm MeltQtz + Als + Bio + Gar + llm MeltQtz + Als + Bio + Gar + llm + Rut MeltQtz + Als + Bio + Gar + Rut MeltQtz + Als + Bio + Gar + Rut MeltQtz + Als + Bio + Gar + Rut MeltQtz + Als + Bio + Gar + Rut MeltOtz + Als + Bio + Gar + Rut Melt
that biotite persisted over a wide pressure-temperaturerange during progressive dehydration melting and that itchanged very regularly in composition throughout thatrange. Because the pressure and temperature ofeach ex-periment are known and the grain size of the products islarge enough to allow accurate electron probe analyses ofall phases, it is possible to use these results to augmentour understanding ofthe solution properties ofbiotite.
All experiments were performed under vapor-absentconditions and with a single starting material, a naturalmetapelitic rock (HQ-36) from northern Idaho (U.S.A.).The mineral assemblage of HQ-36 (biotite * muscovite+ sillimanite + garnet + quartz + plagioclase * ilmen-ite) is described more fully in Patiflo Douce and Johnston(1991, their Table l), and it equilibrated at conditionsapproaching the second sillimanite isograd (Ruendal,1987). Experiments were conducted at7 and l0 kbar, atintervals of 25 "C above a minimum temperature of 825oC, and at 13 kbar at 900 and 950'C. Conditions anddurations of the experiments, as well as phase assem-blages of the experimental products, are given in Table1. Note that the muscovite-out temperature has been ex-ceeded in all experiments and that biotite is the only micapresent in the products.
All the experiments reported in this paper were con-ducted in a piston-cylinder apparatus with NaCl-basedcell assemblies 0.5 in. in diameter. The pressures reportedare nominal pressures (Heise gauge) and are believed tobe accurate to within 0.5 kbar (Bohlen, 1984). Experi-ments were pressurized at room temperature to 2 kbarbelow the target pressure and then heated to the targettemperature, and then the pressure was adjusted to itsnominal value. Because of thermal expansion, this finaladjustment corresponded in every case to a pressure re-lease (hot, piston out). Temperature was measured usingWroRer.-WnrRe, thermocouples relative to an electronicice point (0 "9 and controlled by a digital Eurotherm 808temperature controller. Temperature stability throughoutall experiments was better than + 5 'C. Samples wereground to less than l0 pm and contained in welded Aucapsules. Utilization of small sample volumes (10-15 mg)
resulted in the entire sample capsule being within 2-3mm of the thermocouple. Further details about the ex-perimental procedures can be found in Patiflo Douce andJohnston (1991).
Controlling /o, in high-pressure experiments relies onthe permeability of the capsule material to H, diffusionand on the presence ofa vapor phase in the experimentalcharge that frxes the /r,o. Because no vapor phase waspresent in our experimental charges, control of/o. in themwas not possible in practice. It has been found (e.g., Car-roll and Wyllie, 1990) that the graphite furnace in thetype of cell assembly used imposes redox conditions in-termediate between the Ni + NiO and QFM buffers. Theoxide mineral assemblage in the experimental products
[ilmenite (Hmr-o) t rutile, see tables] and the garnet com-positions (see below) are consistent with redox conditionsofthis order.
All successful experimental products were mounted inepoxy, sawed in hall and polished for electron beamanalysis. Quantitative analyses were performed with aCameca SX-50 microprobe utilizing a ZAF X-ray inten-sity reduction routine. Operating conditions were 15 kV,20 nA, for garnet and ilmenite and 15 kV, l0 nA, forbiotite, with a beam diameter of approximately I pm inevery case. Between 15 and 25 different crystals of eachphase (3-10 for the oxide phases) were analyzed in eachexperimental product. Mineral analyses are given in Ta-bles 2. 3. and 4.
Because none of the experiments were reversed, theattainment of equilibrium cannot be rigorously demon-strated. The long durations of the experiments (especiallyat the lower temperatures investigated, see Table 1), andthe presence of an HrO-bearing melt which acted as aflux, facilitated reaction between solid phases. Addition-ally, a number ofobservations support an acceptable ap-proach to equilibrium. Melt compositions and modes werefound to change very regularly and consistently through-out the full pressure-temperature range investigated (Pa-tiflo Douce and Johnston, 1991), and the same is true ofgarnet, biotite, and ilmenite compositions (Tables 2, 3,and 4, respectively). Melt and mineral compositions (ex-
lvote: Values are single (rim composition) analyses from the crystal with highest Mg/Fe in each experiment (see text). Mg/Fe (min) is the minimummeasured ratio for garnet rims in each experiment (see text) Typical uncertainties (counting statistics) are 1% (relative) for SiO. and FeO, 2ol" (relative)for AlrO3 and MgO, 8% (relative) for CaO and MnO. Stoichiometries calculated on the basis of eight cations and 12 O atoms; all residual negativecharge assigned to Fe3*. FerO3 wt% calculated from stoichiometry Activity coefiicients calculated with the model of Berman (1990), on a three-sitebasis.
cept garnet, discussed below) are also notably constantthroughout any given charge. Linear programming anal-ysis also shows that most of the experimental products(obtained over a temperature interval of more than 100oC and with durations ranging from 3-15 d) are internallyconsistent. Most importantly, application of the biotitesolution model derived from the experimental data tonatural assemblages yields P-I estimates that agree withindependent constraints for these intensive variables. Weargue that all these observations taken as a whole cannotbe easily dismissed as the product of coincidence in ex-periments that are far from equilibrium.
MrNnn.c,L coMPosITIoNS
Garnet
Garnet produced by the dehydration-melting reactionof biotite nucleated on garnet initially present in thecharges, but, because of slow cation diffusion rates, thecrystals were not homogenized. Two clearly distinct com-positional populations were therefore observed: corescorresponding to the initial, unreacted garnet and rimscorresponding to garnet formed during the experiment.Because garnet grain size is typically small (about l0-15pm) and the electron beam has a finite penetration (l-2pm), the possibility exists that rim analyses included somecore material from below the sample's surface. To min-imize this effect, we chose as representative of each ex-
periment the garnet rim analysis with the highest Mg/Feratio (i.e., with the Mg/Fe ratio most different from thatof the unreacted cores). These Mg/Fe ratios (molar pro-portions) are listed in Table 2, togelher with the lowestmeasured Mg/Fe ratio for rim analyses in each experi-ment (included to illustrate the variability observed). Thegarnet present in the starting material has molar Mg/Fe: 0.188 (Patiflo Douce and Johnston,l99l).
Garnet formulae were calculated from oxide percent-ages by assuming a total cation content of eight and atotal anionic charge of 24. All residual negative chargewas balanced with Fe3*, resulting in calculated Fe2*/(Fe3*+ Fe2*) ratios varying between 0.94 and 0.99 (Table 2).Similar Fe3+/Fe2+ ratios have been reported in electronprobe analyses of garnet from natural high-grade meta-morphic assemblages (e.g., Bohlen and Essene, 1980;Barbey et al., 1990; Sen Gupta et al., l99l). In order toascertain whether the calculated Fe3* values in the exper-imental products are reasonable, we used garnet compo-sitions to estimate fo, in the experiments from the fol-lowing net-transfer equilibrium (with standard-statethermodynamic properties from Berman, 1988, exceptfor andradite, which was taken from Holland and Powell,1990):
l2(Fe3+Al-,),". + 1OAlrSiO5 + 2SiO,
- 4(FerAlrSiro,r)r,. + 3O2 (l)
PATIfrIO DOUCE ET AL.: MIXING PROPERTIES IN BIOTITE rt7
and its solution in garnet is assumed to be ideal. Theresults ofthis exercise are shown in Figure 1, together -10with 6, values along the QFM butrer equilibrium. Mostexperiments at 7 and l0 kbar yield /". values about 2 logunits above the QFM buffer, which are consistent with _Izthe redox conditions measured by Canoll and Wyllie(1990) in NaCl-graphite cell assemblies. This result cre-ates confidence in the calculated Fe3* values. Results from 1iboth experiments at 13 kbar suggest considerably more ^--"oxidized conditions (up to 4 log units above QFM). It is ;not clear whether this result reflects a change in the be- .9havior of the cell assembly with pressure or whether it is -16
a consequence of ignoring a possibly significant excessvolume of mixing for the exchange component (Fe3*Al_r)*.
Biotite
Biotite analyses were also complicated by grain size (5-l0 pm), although in this case no compositional variationwas detected within individual biotite crystals. The mainissue was whether the biotite grain was too thin and someunderlying glass was included in the analysis. Because theglass compositions are very silica-rich (about 70 wto/o SiOr,Patiflo Douce and Johnston, 1991), this is detectable inthe silica content of the biotite analysis, and all biotiteanalyses yielding more than 39 wto/o SiO, were routinelydiscarded. All the remaining biotite analyses were aver-aged; these averages are listed in Table 3. The relativeerrors of these mean values (2 sd) are of essentially the
-20800 850 900 950 1000
T ("C)
Fig. l. The /", values in experiments (symbols) estimatedfrom garnet compositions (see text). For comparison, the curvesshow the loci of the QFM buffer at 7 , 10, and I 3 kbar (calculatedwith standard-state thermodynamic properties from Berman,1988). Redox conditions somewhat more oxidizing than QFMare consistent with independent measurements of /o, in NaCl-graphite cell assemblies (Carroll and Wyllie, 1990).
-18
r r 8
TABLE 3. Biotite compositions
P(kba4 7rfc) 825
PATIfrTO DOUCE ET AL.: MIXING PROPERTIES IN BIOTITE
1 0850
1 0825950
79259008758s0
sio,Tio,Alr03MgoCaOMnOFeO-NaroKrOr
TOTALFe'?r(Fe'?* + Fe3*)
SirnlAlr61AlTiFestMgFe2,NaKFOH
TorAL [4] + [6]cations
Si14lAl
16lAl
TiMgFe,tNaKF
OHTorAL [4] + [6]
cations
35.13.56
19.88.370.030.05
19.10.239.200.32
95.70.93
2.6561.344o.417o.2020.0680.9431.1220.0340.8880.o771.923b . / 5 b
Note: Wlo/" values are means of 5-15 analyses of different crystals. Typical relative uncertainties of the mean values (2 sd) are 4% (relative) for SiOr,K,O and FeO'; 6% (relative) for Al,O3 and MgO; 15% (relative) for TiO, and F; 25% (relative) for Na,O. FeO. : Total Fe as FeO. Fe'z+/(Fe'?+ + F93+)estimated from fH"o in melts and fo" from garnet compositions (see text). All stoichiometries calculated assuming total cation charge (except H) : +22.
same magnitude (4-6 relative o/0, see Table 3) for Sior,AlrOr, MgO, FeO, and KrO in all the experimental prod-ucts. Because the excitation volumes for these elementsare not all the same, similarity in the relative errors ofthe means is a good indication that the values shown inTable 3 represent essentially uncontaminated biotite.Larger relative errors for TiO2, F, and NarO are almostcertainly derived from the poorer counting statistics forthese elements, which are present in smaller concentrations.
Because all our analyses are electron beam analyses,the Fe3* and HrO contents of biotite were not deter-mined. Furthermore, vacancies are known to occur in thestructure of trioctahedral micas in high-grade metamor-phic rocks (e.g., Holdaway, 1980; Guidotti, 1984). Withthese limitations, recalculation of a biotite analysis intoa structural formula involves three unknown quantities,and a correct recalculation is impossible. Any recalcula-tion scheme chosen must therefore entail some assump-tions, which are discussed in the following paragraphs.
Hewitt and Wones (1984, p. 216-219), Partin (1984,in Hewitt and Wones, 1984), and Rebbert (1986) havediscussed the effect of/r, on the Fe3'/Fe2* ratio ofbiotite.They suggested that, in Al-rich biotite (with more thanabout l0 molo/o Tschermak's component), there are nomajor crystallographic constraints on the amount of Fe3*and that Fe3t content in such cases is primarily a functionof the fn, under which the mica equilibrated. Because allour experimental biotites contain more than l0 molo/oTschermak's component, we argue that the last conclu-sion is also applicable to them. The Fe3" content of biotitecan then be roughly estimated from the fo2 and .fn o inour experiments and the results ofthe authors cited above.In order to accomplish this, the /o, values calculated fromEquilibrium I were used. The /sro values were estimatedfrom HrO contents (calculated by mass balance) in theglasses coexisting with biotite (see Patiflo Douce andJohnston, l99l), by means of the HrO activity-compo-sition model for silicate melts of Burnham (1979). The
PATIfrIO DOUCE ET AL.: MIXING PROPERTIES IN BIOTITE l l 9
Notej Wty" values are means of at least three analyses of ditferent crystals. Typical relative uncertainties of the mean values (2 sd) are 0.5% (relative)for FeO and TiO, and 8% (relative) for MgO and MnO. Stoichiometries calculated on the basis of tvvo cations and three O atoms; all residual negativecharge assigned to Fe3r. Fe,O3 wt% calculated from stoichiometrv
r20 PATIfrO DOUCE ET AL.: MIXING PROPERTIES IN BIOTITE
12
3 r
& o s
0.6
3 r
o 0.6
N 0 4
E 0.3
!
o o z
0.1
9 0.5
N 0 4
II 0.3
aa o z
t a . . '. r a a
'+-
at a a
. . . . ' o 9
--{-
*ts-
a
with Fe3* in biotite
--{-
a
No Fe3* in biotite0 1 L
800 850 900 9s0Temp€rature ('C)
E50 900 950 1000
Temperature ('C)
Fig.2. Variations in t6tAl and excess t4rAl contents in exper-iments at 7 and 10 kbar (representative uncertainties shown bycrosses). Decoupling between I4rAl andt6rAl is apparent.
values of /"r, calculated from /or, .fu2o,3\d the Gibbs freeenergy of formation of HrO at the temperature of eachexperiment, were then used to estimate Fe2*/(Fe3* * Fe2*)in biotite after Hewitt and Wones (1984, p. 218) andRebbert (1986, p. 12-13). The resulting ratios, varyingbetween 0.92 and 1.0, are shown in Table 3.
In order to test the effect of ignoring Fe3* in modelingthe solution properties of biotite and also because theprocedure used to estimate Fe3* is subject to large uncer-tainties, two sets of structural formulae for biotite arepresented in Table 3. One set was calculated with the Fe3*values discussed above, and the other set assuming thatall Fe is Fe2*. The relative merits of these two assump-tions concerning Fe3* content in biotite are discussedfurther in connection with the different sets of mixingparameters arising from them. Both sets of formulae inTable 3 were calculated on the basis of a total cationiccharge in biotite (except H) of 22 (i.e., assuming that oxy-components are not present). Although the assumptionof no oxy-components may not be entirely justifled,given the high temperatures of the experiments, we be-lieve that this is safer than assuming an arbitrary amountof oxy-components.
An examination of Table 3 reveals that biotite presentin the experimental products departs from the idealphlogopite-annite join in four important respects. Theseare the presence of excess tolAl, the presence of t61Al andI6rTi, the substitution of F for OH, and the presence ofvacancies in interlayer sites. It is important to examinein some detail the behavior of Al because equilibria in-volving Al-bearing exchange components in biotite arecentral to our discussion of excess mixing parameters inbiotite. Figure 2 shows the variation in the contents ofr6rAl and excess lolAl (: t4lAl - l) as a function of tem-perature for the experiments at1 and l0 kbar. It is clearthat, at least within the assemblage garnet + aluminosil-icate + quartz + biotite, changes in the concentrationsof t6lAl and t4rAl in biotite are not coupled. This obser-vation implies that two linearly independent equilibria,involving at least two linearly independent componentsofbiotite, are required to account for the total Al content
04L.....--a0.15 0-2 0.25 0.3 035 0.4
0 . 4 '0.15 02 025 03 0.35 0.4
Ti per 12 ados Ti per 12 dioos
Fig. 3. Correlation between Fe2* and Ti contents in experi-mental biotite (representative uncertainties shown by crosses).The lines correspond to the equation Fe'?* : -2Ti + constant.The two points farther removed from the trend correspond tothe experimentsatg25 'C, 7 kbar and 975 "C, 10 kbar. The latterdata point was rejected by all linear-programming optimizationmodels.
of this phase. Previous workers (e.9., Robert, 19761'Dy'mek, 1983; Guidotti, 1984; Hewitt and Abrecht, 1986)have reached similar conclusions and have suggested thatthe two linearly independent Al-bearing exchange com-ponents that best describe biotite compositions are theTschermak's component, I6lAll4rAlR-rSi r, and the dioc-tahedral component, I6lAl2t6l!l?-3 (where R stands for Mgor Fe, depending on whether phlogopite or annite, re-spectively, is chosen as the additive component of bio-tite). The total Al content in biotite present in the exper-imental products can then be modeled by means of thefollowing three linearly independent net-transfer equilib-na:
(note that the garnet-biotite Fe-Mg exchange is a linearcombination of Eqs. 3 and 4).
Table 3 shows that the concentrations of Fe2* and Tiin biotite vary antithetically with temperature. The cor-relation between both compositional variables, shown inFigure 3, has a slope close to -2, which is not signifi-cantly affected by whether Fe3' is included in the biotiteformulae. Such correlation is consistent with a Ti-vacan-cy exchange component of the form t6lTil6rLlFe2-+2 (see alsoHoldaway et al., 1988). Although this particular exchangecomponent is not the only possible way of interpretingTi contents in biotite, Figure 3 does suggest that thesecontents can be modeled in pelitic compositions by equi-libria such as
depending on whether ilmenite or rutile, respectively, isthe Ti-saturating phase. We have found that the four ex-change components AlrMg ., AlrFe ,, AlrMg-,Si-,, andTiFe , are sufficient to model the mixing behavior of Al-Ti-Mg-Fe components in biotite. Of course, a differentset of four linearly independent exchange componentscould be equally satisfactory. Our particular choice ofcomponents is justified on the basis that it makes it pos-sible to estimate excess mixing properties in biotite in arelatively straightforward fashion.
Inn.ll, Acrryrry-coMposlTloN RELATToNSHTpS INALUMINOUS AND TITANIFEROUS BIOTITE
Mixing of octahedral cations in biotite is commonlymodeled as taking place on three indistinguishable sites.In order to evaluate whether this assumption is appro-priate, it is necessary to discuss cationic ordering in bio-tite. The three octahedral sites in trioctahedral micas arenot equivalent, the two M2 sites being smaller than thesingle Ml site (e.g., Bailey, 1984). This difference resultsin ordering ofcations with different sizes in the structureof biotite. As discussed by Bailey (1984), ordering be-tween Fe2* and Mg is seldom observed in biotite, butordering between these divalent cations and the smallerAl3* cation is common in aluminous biotite, with Al3tpreferentially entering the M2 sites. It thus appears rea-sonable, at least as a first approximation, to model theoctahedral layer in aluminous biotite by analogy with di-octahedral micas, in which t5tAl is restricted to the M2sites and vacancies required by charge balance to the Mlsite. Because Ti4* and Fe3* are closer in charge and sizeto Al3* than to the divalent cations, it also appears rea-sonable to assume that these two cations are partitionedinto the M2 sites. In the case of Ti4t, further support forthis hypothesis can be derived from the observation thatincorporation of this cation is accompanied by the for-mation of vacancies (cf. Fig. 3; also Holdaway et al., 1988;Abrecht and Hewitt, 1988) and that these vacancies shouldpreferentially form in the Ml site. It must be noted, how-ever. that recent structural refinements of Ti-rich and Al-poor biotite from alkaline igrreous rocks (Brigatti et al.,l99l) do not support our model for Ti site occupancy,but it is not clear whether such results are applicable tobiotite coexisting with Al-saturating phases. In any event,this discrepancy emphasizes the fact that caution mustbe exercised when applying the solution model proposedhere beyond the composition space of aluminous rneta-morphic and igneous rocks, and it underscores the needfor detailed structural studies of aluminous and titanif-erous biotite.
If ordering between the two kinds of octahedral sitestakes place in quaternary biotite solution (Mg-Fs'z+-toldl-Ti), then the assumption that mixing of these cations oc-curs on three indistinguishable sites is not correct and
should be abandoned. A better approach is to considerM2 and Ml as diferent crystallographic sites and to modelmixing on each site independently. Additionally, becauseof the observed decoupling between the behaviors of t6rAl
and t4lAl (cf. Fig. 2), mixing in the tetrahedral layer mustalso be considered independently and incorporated intothe ideal activity-composition expression for each com-ponent of the solution.
Each exchange component is equivalent to the differ-ence between two additive components. The two dioc-tahedral exchange components, which we will call Mg-dioctahedral (Me-dic) and Fe-dioctahedral (Fe-dic), aredefined by the differences (muscovite-phlogopite) and(muscovite-annite), respectively, whereas the Mg-Tscher-mak's exchange component (Mg-Tk) equals the differ-ence: (eastonite-phlogopite). The chemical potentials, idealactivities, and activity coefficients of these exchange com-ponents are defined in terms of the corresponding vari-ables for the additive components. Thus, for the Mg-di-octahedral exchange component:
These relationships show that, in order to calculate theideal activities of the exchange components chosen todescribe biotite composition, it is necessary to determineexpressions for the ideal activities ofphlogopite, annite,muscovite, and eastonite in a trioctahedral mica. A rig-orous application of the method of Kerrick and Darken(1975), with the cationic ordering assumptions discussedabove, leads to the following expressions for the contri-butions of tetrahedral, Ml, and M2 sites to the ideal ac-tivities of phlogopite, muscovite, and eastonite in biotitesolid solution (mole fractions are defined in Table 5).
Contributions from the tetrahedral site are calculatedby considering mixing of Si and Al on four indistinguish-able sites. The resulting tetrahedral contributions to theideal activities of muscovite, phlogopite, and eastonite
122
Teele 5. Mole fractions for activity-composition expressions
PATIfrIO DOUCE ET AL.:MIXING PROPERTIES IN BIOTITE
Tetrahedral sitesQ i
x , : " '" 4
r4tAlX,or: ___
4
Octahedral M1 site
Mox
",: M;J;; (2 6)
Fe.-Y . _ - ' s - 6 )
Mg + Ps'* '-
x " : 7 - >
Octahedral M2 sites
M o 2 - r 6 r A l - T i - F e 3 -v - -
M g + F e , 2
F e r * 2 _ r 6 r A l _ T i _ F e 3 *X'n":
G * r"*16lAl
X" o, : - l
Finally, mixing of Mg, Fe2*, Fe3*, Al, and Ti on twoindistinguishable M2 octahedral sites leads to
a';i;f2 : X?"*"
a]$;l$': : X3,o'
a'!"Yt : 4X"'MeX".At. ( 1 5 )
Because it is also assumed that no Fe2*-Mg ordering takesplace, the Mg/Fe2'ratio in Ml and M2 sites is the same.This premise is incorporated in the expressions for themole fractions of Mg and Fe2* shown in Table 5.
The ideal activity of each of the additive componentsphlogopite, muscovite, and eastonite is the product of thethree corresponding partial contributions listed above,together with the contributions from anions and interlay-er cations, resulting in
) \ 6a ron, : -#
x ?" usx* srx - ̂ |x i,X*x3*L I
) \ 6a 'A", : ; Xi .^ tXoX,, \ tXl ixkXLH
L I
a'A. : 64X,M2X,,.^tX,,MeXd^,X!,X*Xton,. (16)
An expression entirely analogous to that for phlogopitecan be written for annite by replacing all Mg mole frac-tions with the corresponding Fe2* mole fractions fromTable 5. Note that the phlogopite and annite activity-composition relationships collapse to the familiar activityexpressions if biotite composition is restricted to the puretrioctahedral annite-phlogopite join. For example, in thecase of phlogopi te, t6 lAl : T i : Fe3t :0; to l4 l : K: l ;Si : 3; OH : 2, and the activity expression becomes
x",-,: Ti
' ' 2
Notej Biotite stoichiometry on 12-anion basis; ) : total sum of octa-hedral plus tetrahedral cations.
(cid''') are
) \ 64l9ij: ' : dio;l' ' : -fi
X,,o,X1,
a:l'J" : l6xl'o,X?,' ( 1 3 )
This model for tetrahedral mixing is consistent with thesuggestion ofBailey (1984) that Si-Al ordering in the tet-rahedral layer of micas is not common, although recentspectroscopic studies (Circone et al., 1991) indicate thatlong-range Al-Si ordering may be the norm in Al-richbiotite. Tetrahedral contributions enter the activity-com-position expression of the Mg-Tk exchange component,but they cancel out in those ofthe dioctahedral exchangecomponents, which are the ones utilized in our determi-nation of excess mixing properties. Elimination of tetra-hedral contributions in the ideal activities ofthe diocta-hedral exchange components, however, is not strictlyjustified if Si-Al ordering difers in end-member musco-vite, phlogopite, and annite. In other words, the prelim-inary solution model developed here ignores possible en-ergetic effects arising from different degrees of Al-Siordering that could be coupled to the total Al content ofbiotite. Detailed experimental studies and structural re-finements are required to clarify this problem.
Following our assumptions about octahedral orderingin biotite, the contribution of Ml octahedral cations iscalculated by considering mixing of Mg, Fe2*, and vacan-cies on a single site:
aid.Mr : Al,oil"t : X",r"
4H#' : X"'
( l 7)
as expected.The ideal activities of the Mg-dioctahedral (a.r-o'") and
Fe-dioctahedral (ao"-.,.) exchange components are giv-en by
X?"^tXtr
,r*:(ffi)'
n l d. , u m u s
Qil , -ar: --- l l - :
q phl x?.MEx",Me( l 8)
( le)
in which all tetrahedral contributions cancel out. The ide-al activity of the Mg-Tschermak's component (a$,-r*)simplifies to
27 X".a1Xn'61
^ i d v 2 v, , 4 - r . A " A | A E
F Fe-d'c a:"" X 1.F,. X" ' r ,
a'"\.a' fr"-t*: i
:- phl 4X".*"X",
(20)
which preserves the contribution of tetrahedral Al-Si(14) mixing (compare Hoisch, l99l).
PATIfrIO DOUCE ET AL.: MIXING PROPERTIES IN BIOTITE t23
Nourln.Lr- MrxrNG BEHAVToR oF ALUMINous ANDTITANIFEROUS BIOTITE
Calculated values of the ideal activities of the diocta-hedral exchange components (a$r-o,. and c$_o,.) in biotitepresent in the experimental products are shown in Figure4 (for biotite stoichiometry calculated assuming that Fe3*is present, see Table 3). These ideal activities vary inregular and consistent ways with pressure and tempera-ture (with the possible exception of the experim enl" al" 97 5"C, l0 kbar), and this behavior is not affected if Fe3* inbiotite is neglected (not shown in Fig. 4). Such regulartrends support the choice of the Mg-dioctahedral and Fe-dioctahedral exchange components to model the t6tAlcontent in biotite. The importance of this result residesin the fact that Reactions 3 and 4 can also be written interms of additive components whose thermodynamicproperties are known, as follows:
KMgrAlSi,O,o(OHL + 2Al,SiO5 + SiO,
* KAlzAlSirO,o(OH). + Mg,Al,Si,O,, (21)
KFe.AlSi.O,0(OH), + 2AlrSiO5 + SiO,
* KAlzAlSirO,o(OH), + Fe,AlrSirO,r. (22)
The Gibbs free energy changes of Reactions 2l ard 22can be calculated at any given pressure and temperature,and from these free energy changes it is possible, by com-paring Reactions 2l and 22 with Reactions 3 and 4, re-spectively, to derive the chemical potentials of the Mg-dioctahedral and Fe-dioctahedral exchange componentsin biotite. Although other choices of exchange compo-nents could describe biotite composition as well as thedioctahedral components, no other choice leads to anequally reliable way of determining the actual values ofthe chemical potentials of the exchange components.
Assuming that quartz and aluminosilicate are purephases, the equilibrium constants for Reactions 3 and 4can be written as follows:
K t : a'#*-o*'Y -r-0. (Xii! )' ?o",
K t : Qld"-an^Y p"-0. (XFT )' "Y'-.
tt t a
l rr t a
f . ' oa o
aa
MgJi@bedral erchaqe mnPoneil
o o 1 L775 825 875
l d l r ( K )
001 -775 825 8.75 9
r d l r (K )
Fig. 4. Ideal activities of Mg-dioctahedral exchange compo-nent (Eq. I 8) and Fe-dioctahedral exchange component (Eq. 1 9),for biotite stoichiometries calculated with Fe3*. Representativeuncertainties (shown by crosses) estimated by propagating un-certainties in biotite compositions (see text). Note that, becauseof logarithmic and inverse scales, graphical size of error barschanges with coordinate values.
Values of ln Kf and In Kf, calculated from biotite andgarnet compositions in the experimental products, areshown in Table 6. These values include nonideality ingarnet calculated with the solution model of Berman(1990). The values of 70". and "y.r- calculated with thismodel (on a three-site basis, rather than the one-site basispresented by Berman) are shown in Table 2. Note thatthe activity coemcients for pyrope and almandine are closeto 1, reflecting the low Ca and Mn contents in the garnetsthat grew during the experiments. Also shown in Table 6are the values of ln K, and ln Ko, calculated from theequilibria among pure end-member additive components(Reactions 2l and 22, respectively) with standard-statethermodynamic properties from Berman (1988, 1990).The aluminosilicate polymorph used in calculating theseequilibrium constants was sillimanite at 7 and l0 kbarand kyanite at 13 kbar, and B-quartz was assumed to bethe stable SiO, polymorph in all the experiments.
Departure of the equilibrium positions from those pre-dicted by ideal behavior ofbiotite can be appreciated inFigure 5 (for biotite stoichiometry, assuming that Fe3* ispresent), as a function of pressure and of the inverse oftemperature. These diagrams show that solution of r6tAl
in magresium iron aluminum titanium biotite is far fromideal. The excess chemical potentials ofthe dioctahedralexchange components are in the range 30-45 kJ/mol (seeTable 6). For comparison, the standard-state molar Gibbsfree energy of the Mg-dioctahedral exchange component,defined as the difference between the standard-state mo-lar Gibbs free energies of muscovite and phlogopite, ison the order of 280 kJ/mol (at l0 kbar, 800 "C). Theexcess chemical potentials are clearly not negligible. Notefrom Figure 5, however, that Reactions 3 and 4 are al-most independent of temperature. This observation sug-gests that net-transfer reactions involving dioctahedralexchange components in biotite can be applied to geolog-ic barometry, provided that a model describing the chem-ical potentials of these components can be calibrated.
1
>& o-1
1
e 0 1
0 r l
a l '- + - a o ' t
_ a a a
Fe-didde&d qchege @Dponst
(23)
(24)
The excess chemical potentials of the Mg-dioctahedraland Fe-dioctahedral exchange components in biotite sol-id solution are then given by (respectively)
A working model for nonideal mixing ofMg-Fe-t6tAl-Ti in biotite
In order to determine values for cationic interpretationparameters from the observed excess chemical potentials,it is necessary to fit these excess chemical potentials to
some function of biotite composition. In the strict sense,because exchange components are being used and thesecomponents are defined as differences between additivecomponents, such a procedure will only yield differencesbetween interaction parameters. A number of simplifyingassumptions can be made, however, which can lead toreasonable estimates of the absolute values of some ofthese interaction parameters.
As a first approximation, we will assume that all non-ideality in biotite is restricted to the two M2 octahedralsites. Ignoring tetrahedral nonideality at this stage isjus-tified because tetrahedral contributions cancel out in theideal activity-composition expressions that we derivedfor the dioctahedral exchange vectors. Neglecting thenonideal contribution of the t6lMl site is a consequenceof our assumption about octahedral ordering in biotite,namely that only Fe2*, Mg, and vacancies mix in this site.Linear changes in unit-cell dimensions along the phlog-opite-annite join have been interpreted as implying thatFe2* and Mg mix almost ideally in binary biotite solidsolution (e.g., Wones and Eugster, 1965; Muellet, 1972:,Hewitt and Wones, 1975). Sack and Ghiorso (1989) havesuggested, however, that binary Fe-Mg mixing in biotiteis better described by a regular solution model with aninteraction parameter (W*.") on the order of 15-20 kJ/mol. Although this energetic contribution is not negligi-ble, it is small compared with those arising from Mg-Fe-Al-Ti mixing (see below). More importantly, however,
-{-
r l
. t t . l . la o o l o
3
2
1
0
- - ,
. 3 fI
" f- l
77575
-6
7 8.25 8.75 92510 1/ T (K)
8.25 8 75 9.25
ro ? r (K)
- l t r K 7 k b a r a h K : T k b u"lo K 10 kbar . ln K: l0 kbar-ln K 13 kb{ I lo K- 13 kber
Fig. 5. Measured equilibrium positions for net-transfer Re-actions 3 and 4 (Kf, EA. 27, and Kf, Eq. 28), compared withcalculated equilibrium positions for net-transfer Reactions 2land 22 among pure end-member additive components (K, andKo, curves, calculated with sillimanite at 7 and l0 kbar and ky-anite at 13 kbar). Values ofK* calculated for biotite stoichi-ometries including Fe3*. Representative uncertainties (shown bycrosses) estimated by propagating uncertainties in biotite andgamet compositions (see text).
'+.
a :: l::.
PATINO DOUCE ET AL.: MIXING PROPERTIES IN BIOTITE t25
the standard state thermodynamic properties of anniteemployed in this study (from Berman, 1990) rely heavilyon the assumption that Fe-Mg mixing in biotite in theexperiments of Ferry and Spear (1978) is ideal, so thatincluding nonideal Fe-Mg mixing terms in our model doesnot appear warranted.
The only remaining nonideality is that derived frommixing Mg, Fe2*, Al, Ti, and Fe3* in the two M2 sites.Because of the uncertainties involved in estimating Fe3*contents, this component will be considered in the cal-culation of mole fractions (cf. Table 5), but it will not beincluded in the mixing model. The excess free energy ofmixing can then be approximated by means of a quater-nary subregular model (e.g., Wohl, 1946; Thompson,1967; Helffrich and Wood, 1989), which can be simpli-fied considerably by assuming that the energetic contri-butions of ternary interactions are negligible comparedwith those of binary interactions. The latter assumptionis valid in many minerals, but it is not necessarily alwaystrue (see Helffrich and Wood, 1989). We believe, how-ever, that the density of our data set does not allow forthe inclusion of ternary parameters in the model and thatdoing so would amount to overfitting the data. Moreover,application of the model to rocks (beyond the P-Z-com-position space covered by the experiments) shows thatconsideration ofbinary interactions alone is sufficient togenerate reasonable estimates of the excess chemical po-tentials ofthe dioctahedral components. Ifwe neglect ter-nary interactions, the quaternary subregular model (afterHelffrich and Wood, 1989) results in the following ex-pressions for activity coefrcients calculated on a two-sitebasis (i.e., two M2 sites):
Note that mole fractions in these equations are mole frac-tions in the M2 octahedral sites (as defined in Table 5),and that, because the activity coefficients are defined ona two-site basis and all nonideality is assumed to residein these two sites, 1-,": ?er, 'ypnt: 'ywet and 7u"":7o".
Two additional simplifying assumptions can be intro-duced. First, we will assume that nonideal interactions inbiotite are symmetric, so that in every case W,, : W,,(where i and j stand for Mg, Fe, Al, or Ti). Second, inkeeping with the assumed ideality of Fe-Mg mixing, Zr*"
will be set equal to zero. The explicit formulations forthe excess chemical potentials of Mg-dioctahedral andFe-dioctahedral exchange components then simplifu to
p;ie.ai. : RZ ln 7rr-or. : RI(ln "yo, - ln 7rr)
: (X-" - Xo)W-nn + XF"WF.^I
+ xr,(w{T - w*"rr)
t il-or" : RI ln 7."-o," : RI(ln T4 - ln 7n")
: (Xr" - Xor)Wr"o, + XMEWM^\,
(32)
+ xri(wNr; - wo.rr). (33)
Examination of these equations shows that they containfour independent parameters: W**,, WF"A\, WNri - W*"r,and Worr, - Wr,rr.
Linear programming analysis of excess chemicalpotentials in biotite
Linear programming makes it possible to optimize thevalue of an objective function while maintaining consis-tency with inequality constraints provided by experimen-tal data (e.g., Gordon,1973). This technique was used tofit the excess chemical potentials shown in Table 6 to thesolution models described by Equations 32 and 33. In-teraction parameters derived by linear programming willin general not be unique solutions, but they will be inter-nally consistent and also consistent with the standard-state thermodynamic properties on which they are based.Because of these properties, the interaction parameterswill be useful in geological thermometry and barometry.Moreover, given estimates of their probable uncertain-ties, these interaction parameters will provide good start-ing points for further refinement.
In order to carry out a linear programming optimiza-tion, it is necessary to set up a system oflinear inequal-ities that constrain the model. This was accomplished bymaking use of the pressure and temperature uncertaintiesof each experiment. Experiments were spaced 25 "C apart(cf. Table l), and both mineral and melt compositionsand modes (see also Patiflo Douce and Johnston, 1991)change in smooth and regular ways between consecutiveexperiments. This behavior suggests that the temperatureuncertainty cannot be much larger than half the temper-ature spacing between consecutive experiments, so wehave adopted a value of +15 oC for this uncertainty(shown in Figs. 2, 4, 5). Uncertainty in pressure is esti-mated at + 500 bars. Pressure during the experiments waskept within 100 bars ofthe target pressure. Because fric-tion corrections in NaCl cell assemblies are small (e.g.,Bohlen, 1984), an uncertainty of +500 bars appears tobe quite conservative.
These uncertainties in temperature and pressure wereused to generate a system of linear inequalities in thefollowing manner. Let the Gibbs free energy change forthe Mg end-member additive component reaction (Re-action 2l) at the nominal P and f(PN,f*) of the experi-
r26 PATIfrTO DOUCE ET AL.: MIXING PROPERTIES IN BIOTITE
TABLE 7. Excess mixing parameters estimated by linear pro-grammrng
Model 1. Including Fe3* in biotitePreferred value (kJ/mol) 52.2 74.4 47.8 60.1Probable uncertainty
( kJ /mo l ) - 10 0 +10 .0 - 15 .0 +15 .0Number of ineoualities in data set: 50
Model 2. Assuming all Fe in biotite is Fel
change component in each experiment. A similar analysisleads to another two inequality constraints for the Fe-dioctahedral exchange component, except that the Gibbsfree energy changes and temperature and pressure inter-vals must in this case be chosen as follows:
a G R < A G R , < A G g < 0
for
PA: P* + AP, Tx: T* - AT
(42)
The set of four inequalities for each experiment wereused in conjunction with Equations 32 and 33 to definea feasible solution space for the four parameters: Z.o',WF"A, WNTi - W."r, and Worr, - Wr"r,.The values ofW*^, and Wr"o, wete constrained to be larger than zero,whereas the diferences Wo,r, - W*"rrand Worr, - Wo.r,were constrained to the interval -500 to 500 kJlmol. Inorder to choose the optimum solution within the feasiblesolution space, it is necessary to define an objective func-tion whose value can be either minimized or maximized.The study of Williams and Grambling (1990) of naturalAl-rich biotite concluded that the difference between themagnitudes of Mg-Al and Fe-Al excess free energies ofmixing is small or negligible. This conclusion, which isqualitatively consistent with the purported near-idealityof Fe-Mg mixing, suggests minimization of the followingobjective function, -F:
F: lW-"^, - Wo.orl. (44',)
W o t t - W o ' t -
wrno, wr.o, w*n , wr.r,
Preferred value (kJ/mol)Probable uncertainty
(kJ/mol) +3.0Number of inequalities in data set: 44
Note. Excess parameters for Mg-Fe,*-Al-Ti mixing on two M2 sites.Values are for mole of biotite calculated on a 12-anion basis. Probableuncertainties estimated by optimizing a set ot inequalities broadened bythe uncertainties in the ideal activities of dioctahedral comoonents in biotiteand almandine and pyrope components in garnet (see text). Internallyconsistent data sets for these broader inequalities contained 56 data pointsfor both models.
ment be AGft. If AP, AT are the pressure and temperatureuncertainties, then for this particular end-member reac-tion within the pressure-temperature range of the exper-iments the following relationships hold:
0 < a G R < a G $ < a c g (34)
forPo: P* + AP, fA : ZN + A?"
PB: PN - AP, fB : ZN - AZ (35)
54.7 57 4 65.1 75.1
+6.0 -6 .0 -15 .0 & : P * - a P , G : Z N + a Z . (43)
If the experimental product represents an equilibrium as- The values of the excess mixing parameters presented insemblage at some unknown pressure, P, and temperature, Table 7 were estimated by minimizing thJ value of Fz, within this interval, then within the feasible solution space defined by the inequal-
0 < AGR - LGorr: -Rf h K. < AGO (36) ity constraints (Eq. 4l and the equivalent relationship for
where AGp. is the Gibbs free energy chanse of Reacrion H*'l;t;"[1?h:n|Jffi*it"ffLi|l?Ti:T"?J;21 between pure end-member components at P and T. in alternative ways: minimizing the difference lW*"r, -Because wo.r,l, or the sum lw*"o, - wr.o,l r lw-"r, - wr"r,l.
ln K. : ln Kf + ltr 7*r-o," e:.) In every case we obtained mixing parameters that were
(see Eqs. 23 and27), Equarion 36 can be rearranged as f,'li,: :# 5lffit ::111il:1l#,J.l*li.,i,"ltil:follows: cause their values are not very sensitive to the particular
-AGg - RZ In Kf < RI ln "y.r-o,. choice of objective function'
= -AGl - R?"ln Kf (3g) Linear programmingwas performedby adding one da-tum (i.e., one pair of inequalities constraining either Re-
From this relationship, and noting (see Table 6) that action, 3 or 4) at a time, so as to group data points in
ln Kf < 0 (39) various.internally consistent sets. This routine was per-formed iteratively with the goal of finding the largest pos-
and that (see Eq. 35) sible internally consistent data set. Of the 15 experiments
TB = T .< T^ (40) Yl.9-lbr this study, only one (the experimenl at 975 "C,
l0 kbar, see Figs. 3-5) was always found to be inconsis-the following inequality is derived: tent with all the other experiments. This suggests that
-Gg - RI"lnKf = Rf lD^rrr_o," either the small amount of biotite (about 2o/o, Patiflo
= -aGR - RZo ln Kf. (41) ?:::-lj Johnston' l99l) present in this charge per-
r ' " sisted metastably or that some of the assumptions madeEquation 41 provides two inequality constraints for the in calculating its stoichiometry are no longer valid at theseexcess chemical potential of the Mg-dioctahedral ex- conditions. In particular, the anhydrous weight total for
this composition is very close to 1000/0, indicating thatbiotite in this experiment may be essentially anhydrous.
Two sets of interaction parameters were obtained(Models 1 and 2, Table 7), calculated with different as-sumptions about Fe3t in biotite. Model I was derivedusing Fe3* values in biotite from Table 3, whereas Model2 assumes no Fe3* in biotite. Discarding the always in-consistent data described in the preceding paragraph leavesa maximum possible total of 56 inequality constraints, ofwhich the largest internally consistent data sets contain50 and 44 inequalities for Models I and 2, respectively(see Table 7). These numbers indicate that the feasiblesolution spaces for both models are substantially over-determined, and therefore provide additional support infavor ofa reasonable approach to equilibrium during theexperiments. In both models the same garnet formulae(which include Fe3*, cf. Table 2) were used. Of course,assuming that biotite in equilibrium with Fe3*-bearinggarnet is itself free of Fe3* (i.e., Model 2) is not strictlyjustified, and this may be the reason why Model 2 isassociated with a smaller internally consistent data setthan Model l. In any event, comparison between bothsets of interaction parameters and comparison of P- Z es-timates for the same natural assemblages obtained withModels I and 2 (next section) will provide insight aboutthe effect that Fe3* content in biotite has on the calculatedpositions of biotite-bearing equilibria.
Uncertainties in the values of the interaction parame-ters were estimated as follows. The errors of the meanbiotite compositions (2 sd, cf. Table 3) were propagatedto obtain the uncertainties in the ideal activities of theMg-dioctahedral and Fe-dioctahedral exchange compo-nents (shown in Fig. 4). Uncertainties in the ideal activ-ities of garnet components were taken to be those arisingfrom counting statistics alone because garnet composi-tions are taken from single analyses. Uncertainties in theexcess chemical potentials of the dioctahedral exchangecomponents, arising from the uncertainties in the idealactivities of biotite and garnet, were characteristicallyfound to be about 1.5-2 kJ/mol. The uncertainties in theideal activities were also used to broaden the inequalityconstraints, and the new system of inequalities was op-timized by minimizing the objective function, F (Eq. aq.It was found that this led to internally consistent data setsthat included all data points in both models, except theexperiment at 975'C and l0 kbar. The interaction pa-rameters varied, relative to the values calculated initially,by 3-15 kJ/mol (see Table 7). Note that with this pro-cedure it is not possible to obtain an uncertainty intervalaround the preferred value. Rather, the results in Table7 suggest, for example, Ihat W* , is likely to be withinthe interval 42.2-52.2 kJlmol (for Model l) or 54.7-57.7kJ/mol(for Model 2).
Appr,rc.c,rroNs To THERMoMETRY ANDBAROMETRY OF NATURAL ASSEMBLAGES
The biotite solution model was tested by utilizing it toestimate internally consistent pressures and temperatures
t27
of crystallization of mineral assemblages reported in theliterature. The chosen assemblages are such that they al-low calculation of both pressure and temperature fromequilibria involving the dioctahedral exchange compo-nents in biotite (and their linear combination, FeMg ')'For example, the assemblage garnet + biotite * alumi-nosilicate * quartz permits calculation of pressure fromReaction 3 (see Fig. 5) and temperature from garnet-bi-
otite Fe-Mg exchange. From the assemblage cordierite +biotite * aluminosilicate * qvrtz, it is possible to cal-culate temperature from biotite-cordierite Fe-Mg ex-change and pressure from the net-transfer reaction
Finally, in the aluminosilicate-absent assemblage garnet+ cordierite + biotite + quartz, temperature can be es-timated from any of three Fe-Mg exchange equilibria andpressure from
Pressures determined from any of these equilibria relyon dioctahedral exchange components in biotite, and suchmethodology has not (to our knowledge) been rigorouslyapplied before. In this sense, the solution model that wepropose does not represent a refinement of models al-ready being used, but rather an altogether new way ofusing biotite in the estimation of intensive variables. Ifthe calculated pressures are consistent with independentpressure constraints, it would signify that the Mg-dioc-tahedral and Fe-dioctahedral exchange components canbe applied to thermobarometry of metamorphic and ig-neous rocks in which few other useful equilibria occur.Most importantly, because nonideality of the dioctahe-dral exchange components is quite large (cf. Fig. 5, Table6), the fact that reasonable pressure values are obtainedis a good indication that the solution model proposedhere yields acceptable approximations to the chemicalpotentials of these components.
Internally consistent pressures and temperatures weresolved simultaneously, by iterating between the corre-sponding exchange and net-transfer equilibria until con-secutive values of P and I were within specified conver-
PATIfrTO DOUCE ET AL.: MIXING PROPERTIES IN BIOTITE
r28 PATIfrIO DOUCE ET AL.: MIXING PROPERTIES IN BIOTITE
Fig. 6. Internally consistent P-1" conditions (symbols, esti-mated from Mg-dioctahedral + garnet + aluminosilicate + quartznet-transfer equilibrium and exchange of Fe and Mg betweengarnet and biotite) for sillimanite- and andalusite-bearing assem-blages from Hodges and Spear (1982) and sillimanite-bearingassemblages (+kyanite) from Pigage (1982). Estimated uncer-tainties (see text) shown by crosses. Rectangles show ranges inP-?'conditions estimated by Hodges and Spear (lower) and Pi-gage (upper).
gence intervals (50 bars and 5 "C). All standard-statethermodynamic properties and heat-capacity coefficientswere taken from Berman (1988, 1990). Compressibilitiesand coefrcients of thermal expansion were ignored be-cause their contributions to the Gibbs free energy changeof reaction are small compared with the uncertainties inthe biotite interaction parameters. A combined Newton-Raphson bisection root-finding algorithm (Press et al.,1989, p. 258) was used to solve the nonlinear equationsthat result from including the heat-capacity terms in theexpression for the Gibbs free energy change ofreaction.Mixing in garnet was modeled after Berman (1990),whereas cordierite was assumed to be anhydrous, and Fe-Mg mixing in it was considered ideal. Only assemblagescontaining sillimanite or andalusite were included in theanalysis of aluminosilicate-bearing equilibria. Assem-blages containing kyanite as the only aluminosilicatepolymorph are not suitable because Reactions 2l and 22with kyanite have relatively small volume changes andlarge entropy changes, so that their Clapeyron slopes aretoo steep to serve as practical barometers.
Figure 6 shows the results obtained from amphibolite-grade garnet + biotite + aluminosilicate + quartz assem-blages reported by Hodges and Spear (1982) and Pigage(1982). Although temperature estimates presented by theseauthors are not independent of biotite equilibria, thesestudies are particularly useful because P-7 conditions inboth cases are constrained by aluminosilicate phase re-lationships. Hodges and Spear (1982) estimated pressuresfrom the garnet + plagioclase * aluminosilicate * quartzequilibrium and proposed conditions (also suggested byfield evidence) in the neighborhood of the aluminosilicatetriple point (lower rectangle in Fig. 6). These conditionsare satisfactorily reproduced by Model 2 (which ignoresFe3* in biotite in the experimental products). The rocks
0400 500 600 700 800 900 1000
04m 500 600 7m E00 900 1000
Temperatue ('C) T€Epqatuc ('C)
i3lxlt t X:sl 8ffi]; S: t tli.iiffi!:ffi"*:f.'il'i"!1tr'}tii*t'f'o'-i**'
Fig. 7. Internally consistent P-7 conditions (symbols, esti-mated from Mg-dioctahedral + gamet + aluminosilicate +quartz, Mg-dioctahedral + cordierite + aluminosilicate + quartzand Mg-dioctahedral + garnet + cordierite + quartz net-transferequilibria and garnet-biotite or cordierite-biotite Fe-Mg ex-change equilibria) for assemblages from Grant and Frost (1990)and for sillimanite-bearing metapelitic assemblages from theManiwaki area (Indares and Martignole, 1984). Estimated un-certainties (see text) shown by crosses. Rectangles show rangesin P-T conditions estimated by Grant and Frost (lower) andIndares and Martignole (upper), in both cases constrained bybiotite-independent equi I ibria.
studied by Pigage (1982) traverse the sillimanite-kyanitephase boundary at a temperature (estimated on the basisof several equilibria in pelites and calc-silicates) of 605+ 50 "C (upper rectangle in Fig. 6). Model 2 correctlyplaces the P-Zconditions very close to the kyanite-silli-manite phase boundary within this temperature range. Inboth of these examples, Model I (which accounts for Fe3*in the experimental products) yields temperatures that are50 to 100'C too high (or pressures about I kbar too low)to be consistent with aluminosilicate phase relationships.
Examples of the application of the biotite solutionmodel to granulite-facies rocks are shown in Figure 7.Maximum metamorphic temperatures in the Maniwakiarea of the Grenville Province in Canada Ondares andMartignole, 1984), calculated with clinopyroxene-garnetFe-Mg exchange, range from 740 to 820 "C, whereas max-imum pressures are in the range 6.5-8.5 kbar, calculatedfrom pyroxene * plagioclase + garnet + qtJartz and pla-gioclase + garnet + sillimanite + q\artz equilibria. Ap-plication ofthe Ferry and Spear (1978) garnet-biotite geo-thermometer to these rocks, however, producestemperatures in the range 900-1065 oC, and this discrep-ancy led Indares and Martignole (1985) to develop anempirical calibration for garnet-biotite Fe-Mg exchangebased on samples from the Maniwaki area. Figure 7 showsinternally consistent pressures and temperatures calculat-ed from garnet + biotite * aluminosilicate * quartz as-semblages from the Maniwaki metapelites (data from In-dares and Martignole, 1984, and Martignole, personalcommunication). With the exception of two samples, bothModels I and2 satisfactorily reproduce the metamorphic
PATIfrTO DOUCE ET AL.:MIXING PROPERTIES IN BIOTITE
Moderl
/
./
t29
P-Z conditions estimated by Indares and Martignole(1984, upper rectangle in Fig. 7). This result is particu-larly important because it does not rely on any other bi-otite solution model to test the merits of the mixing mod-el proposed here.
The lower rectangle in Figure 7 shows the range in P-Tconditions for the contact aureole of the Laramie anor-thosite complex (Grant and Frost, 1990), also estimatedin part on the basis ofbiotite-absent equilibria. Pressurescalculated from different samples and with each of thethree biotite-bearing net-transfer equilibria (Equilibria 3,45, and 46) are remarkably constant (especially for Model2), whereas temperatures span a relatively wide interval.Such a pattern is consistent with a contact-metamorphicrecrystallization episode. Results from Model 2 are itreasonable agreement with the conditions estimated byGrant and Frost (1990), except for some samples thatyield temperatures that are lower than these authors' es-timates. This could be a consequence of continued op-eration (in some rocks) of both exchange and net-transferequilibria affecting biotite, during near-isobaric coolingof the contact aureole. Thus, pressure-temperature esti-mates obtained from biotite-bearing equilibria are notnecessarily inconsistent with the constraints provided byGrant and Frost (1990) but, rather, reflect different clos-ing temperatures (at constant pressure) for different rocks.
These four examples suggest that the biotite solutionmodel that we propose can satisfactorily predict thechemical potentials of the exchange components AlrMg-,and AlrFe-, (and hence of their linear combination,FeMg-,), at least within aluminous assemblages in whichbiotite composition is similar to that in the experimentalproducts (Table 3). Simultaneous analysis of exchangeand net-transfer reactions, which mediate equilibrium ofthese biotite exchange components with garnet, cordier-ite, aluminosilicate, and quartz, are capable of yieldingreasonable, internally consistent pressures and tempera-tures over a wide range of amphibolite- and granulite-grade conditions. In the examples investigated, these es-timates agree with independent P-7"constraints, althoughagreement appears to be better using Model 2, in whichFe3* in the experimental biotite is ignored. This does notnecessarily imply that Fe3* is not present in our experi-mental products. Rather, closer agreement with Model 2might be a consequence ofthe fact that Fe3* has also beenignored in the four studies discussed above. Note that inthe two cases in which results from Models I and 2 di-verge most (Fig. 6) temperatures calculated with ModelI are too high. If Fe3' were included in the biotite anal-yses ofHodges and Spear (1982) and Pigage (1982), thenestimated temperatures would decrease. Results fromModel I could then agree better with aluminosilicate phaserelationships than results from Model 2.
This hypothesis can be tested by using data from Wil-liams and Grambling (1990), who have determined (ei-ther directly or indirectly) Fe3* contents in biotite frommedium-grade metamorphic rocks from New Mexico.Peak metamorphic conditions in the Truchas Range ap-
mated from Mg-dioctahedral + gamet + aluminosilicate + quartznet-transfer equilibrium and exchange of Fe and Mg betweengarnet and biotite) for sillimanite- and andalusite-bearing assem-blages from the Truchas Range. Circles show results using Fer*estimates from Williams and Grambling (1990), squares showresults assuming that all Fe in biotite is Fe'?*. Estimated uncer-tainties (see text) shown by crosses. Rectangles show range inP-?"conditions estimated by Grambling et al. (1989).
pear to have been close to the aluminosilicate triple pointand are shown (after Grambling et al., 1989) by the rect-angles in Figure 8. Pressure-temperature conditions esti-mated with our model and with biotite compositions fromWilliams and Grambling (1990), which include Fe3*, arealso shown in Figure 8 (circles). In this case net-transferReaction 3 and garnet-biotite Fe-Mg exchange yield re-sults more consistent with aluminosilicate triple-pointconditions if Model I is used to account for nonidealityin biotite. Model 2, which ignores Fe3* in biotite in ourexperimental products, yields temperatures that are toolow (compare Fig. 6). We have also recalculated Williamsand Grambling's (1990) biotite compositions assumingthat all Fe is Fe2* and used these compositions to estimatoP-I conditions (Fig. 8, squares). Conditions calculatedfrom Model 2 are now close to the aluminosilicate triplepoint, whereas Model I appears to overestimate temper-ature (compare Fig. 6). These results suggest that esti-mated Fe3t contents in the experimental products areprobably reasonable and that Model I is a better repre-sentation of the excess mixing properties of biotite thanModel 2. However, unless Fe3+ is also determined in thebiotite to which the model is applied, Model 2 is likelyto generate better P-T estimates than Model l.
The effect ofthe likely uncertainties in the interactionparameters (cf. Table 7) on the pressure-temperature de-terminations was also investigated. In order to do so, wecalculated a new set of P-T values using interaction pa-rameters as removed from the values in Table 7 as theiruncertainties would allow. The new pressures were within500 and 350 bars (for Models I and 2, respectively) ofthe pressures calculated initially, whereas temperatureswere within 35 'C of the initial values for both models.For Model 2 new pressures and temperatures were con-sistently higher than those calculated initially, whereas no
130 PATIfrTO DOUCE ET AL.: MIXING PROPERTIES IN BIOTITE
such clear trend was observed for Model l. The uncer-tainties in pressure and temperature estimates derivedfrom dioctahedral exchange components are comparableto those associated with other mineral equilibria.
DrscussroN AND coNCLUsroNS
Although the results obtained from applying the biotitesolution model to natural assemblages are encouraging,it must be stressed that this is a working model, which isfar from providing a complete thermodynamic descrip-tion of biotite. Among its most obvious shortcomings arethe facts that the model was calibrated from a single bulkcomposition and over a restricted pressure-temperaturerange. Attempts were made to determine the temperatureand pressure dependencies ofthe interaction parameters(i.e., values for W" and W") from the experimental data.We concluded, however, that the relatively small numberof experiments, together with the restricted P-T rangecovered by them, did not permit a meaningful analysis.
Another problem that cannot be addressed in an en-tirely satisfactory manner arises from the unknown Fe3*content in the experimental products used to calibrate themodel. Table 7 shows that, whereas this uncertainty isnot likely to affect the calculated value of Wr*, in a sig-nificant way, estimates of 2."^,, Worr, - W*"-r, and Wor,- Wr"r, are quite sensitive to the assumed Fe3*-Fe2* dis-tribution in biotite. Sensitivity of the Fe-bearing param-eters is easy to understand. The estimates of Wo,r, - W*"r,are affected because W.*, and Wo"r, are implicitly cor-related when net-transfer Reactions 3 and 4 are solvedsimultaneously. Regardless of the Fe3* content of biotite,however, the large positive values obtained for the dif-ferences W^r, - W*"r, and Worr, - I,/."r, indicate thatthe excess free energy arising from Al-Ti mixing in biotiteis considerably greater than Mg-Ti and Fe-Ti excess freeenergies of mixing. This result is consistent with the Al-Ti avoidance observed in biotite from natural metapeliticassemblages containing Ti-saturating phases (cf. Guidot-r i , 1984 ) .
Yet another aspect of mixing in biotite that needs ad-ditional work is that ofcationic ordering and its effect onactivity-composition relationships. The purely macro-scopic model developed here relies on what appear to bereasonable assumptions about cationic ordering in micas.It would be preferable, however, to develop a mixingmodel that incorporates microscopic aspects of biotitemixing and in which activities are functions of both com-positional and ordering variables (cf. Ghiorso, 1990). Wesuggest that future experimental work should be designedso as to provide the required data base for such anapproach.
Our results demonstrate the feasibility of using net-transfer reactions involving Mg-dioctahedral and Fe-di-octahedral exchange components of biotite in barometryof assemblages that buffer the chemical potential of Al(e.g., containing at least a pair of the following phases:garnet, aluminosilicate, cordierite, staurolite, aluminatespinel, corundum). This should find important applica-
tions in medium- and high-grade metapelitic rocks, byproviding pressure estimates independent of commonlyused equilibria that involve anorthite and grossular. Theuse of dioctahedral components in biotite could be par-ticularly important in relatively low-pressure rocks, inwhich the grossular content of garnet is small (5 molo/o orless), and hence its activity coefficient very poorlyconstrained.
AcrNowr,nncMENTS
We thank M. Kohn, M.J. Holdaway, B. Mukhopadhyay, and R. Ber-man for reviewing an eailier version ofthis manuscript and offering valu-able and helpful comments, and we thank M. Kohn and D. Hewitt foroffering very helpful criticism of the iast version of the manuscript. Dis-cussions with Carolyn Rebbert helped A.E.P.D to better understand mix-ing of trioctahedral and dioctahedral micas. We also thank J Martignolefor making available unpublished mineral analyses that were necessary totest our model with the Maniwaki rocks and J. Grambling for providingdetailed information about mineral assemblages of the rocks from NewMexico The experiments used for this work were done while the seniorauthor was a graduate student at the University of Oregon supported byNSF gants EAR-8720150 and EAR-8915624 to Johnston. Most oftheprobe analyses were done at the University of Oregon, where the probefacility was funded by NSF grant EAR-8803960 and a matching grantfrom the Keck Foundation. Additional analyses were done at the Uni-versity of Georgia, where the probe facility was funded by NSF grantEAR-88 I 6748 and a matching grant from the University of Georgia Re-search Foundation.
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Qtz q\artzRut rutilea'd ideal activity of component iaiav contribution of site M to the ideal activity of com-
"Yi
lLi
pi
t4'AGR
ponent ,activity coemcient of component Ichemical potential of component istandard-state chemical potential of component iexcess chemical potential of component i
standard-state Gibbs free energy change of reactionat P-Z conditions labeled A
