Additional changes that have been incorporated into our analysis are corrections to the entropy of anthophyllite (Hemingway. 1991), reassessment of the volume ...
Contrib Mineral Petrol (1996) 126: 1–24
C Springer-Verlag 1996
R.G. Berman ? L.Ya. Aranovich
Optimized standard state and solution properties of minerals I. Model calibration for olivine, orthopyroxene, cordierite, garnet, and ilmenite in the system FeO-MgO-CaO-Al2O3-TiO2-SiO2
Received: 14 September 1994 y Accepted: 20 March 1996
Abstract An internally consistent set of standard state and mixing properties has been derived for olivine, orthopyroxene, garnet, cordierite, and ilmenite in the system FeO-MgO-CaO-Al2O3-TiO2-SiO2-H2O from analysis of relevant phase equilibrium and thermophysical data. Solubility of Al2O3 in orthopyroxene is accounted for in addition to Fe-Mg mixing. Added confidence in the retrieved properties stems from the representation within reasonable uncertainties of data for seven linearly dependent Fe-Mg exchange equilibria, as well as net transfer equilibria, among the above phases. Critical to successful analysis was the extension of the mathematical programming technique to include bulk composition constraints which force an observed assemblage of fixed composition to be stable at experimentally studied conditions. The final optimization reproduces the extremely tight constraints on endmember properties while invoking very simple macroscopic solution models that afford an excellent opportunity for extrapolation beyond the data considered in this study. Compatibility among the experimental data is improved markedly by incorporation of recently published Cp data on pyrope and forsterite. Electrochemical data defining the oxygen fugacity of Fe-Fa-Qz, Fa-Mt-Qz, and Mt-Hm allow excellent compatibility of almandine thermochemical properties derived from phase equilibrium data obtained at both reducing (Fe-Wst) and oxidizing (Hm-Mt) conditions. Analysis of the combined data involving endmembers and solid solutions removes many of the ambiguities in mixing property magnitudes that arise in analyses of more restricted sets of data. In addition, the consideration of the solid solution data allows further refinement of some endmember properties. Nonideal mixing R.G. Berman (✉) Geological Survey of Canada, 601 Booth Street, Ottawa, Ontario, Canada K1A 0E8 L.Ya. Aranovich Institute of Experimental Mineralogy, Chernogolovka, Russia Editorial responsibility: K. Hodges
parameters, although correlated, are well defined by the combination of experimental data, with GOl .GIlm .GGt ex ex ex Cd Ol .GOpx .G , and 0.7,W ,4.1 kJyatom of isoex ex G morphous Fe-Mg at 1000 K. Experiments defining the Al2O3 solubility of Opx in equilibrium with Gt and Cd1Qz define negative Fe-Al interactions that have an important effect on Fe-Mg partitioning in Opx. Applications of this data set to high-grade metamorphic rocks are described in a companion paper, published as part II of the present work.
Introduction Considerable efforts have been made over the last decade to derive internally consistent thermodynamic data applicable to quantitative petrologic and thermobarometric calculations (e.g. Berman 1988; Aranovich and Podlesskii 1989; Holland and Powell 1990; Perchuk 1991). Accumulation of a large number of high quality experimental data on petrologically significant mineral equilibria, together with reasonably sophisticated mathematical methods for their analysis, has led to the creation of internally consistent data sets of standard thermodynamic properties of the main rock-forming minerals (e.g. Berman 1988; Holland and Powell 1990). Application of these datasets for the purpose of deciphering pressures and temperatures of geologic processes, although often successful, is not without shortcomings (e.g. Aranovich and Podlesskii 1989; Sack and Ghiorso 1989). First, these datasets have been derived largely without consideration of the growing body of high quality experimental data on solid solution bearing equilibria, thus omitting very important information that may influence the derived standard state mineral properties. Secondly, an arbitrary combination of standard state thermodynamic properties with solid solution models will not in general give results compatible with available experimental data, thus compromising the ability to discern reliably even relative differences in computed P-T conditions in any particular region. On the other hand, thermodynamic
2
systematics based entirely on direct application of experimental data for specific equilibria in complex chemical systems (e.g. Aranovich 1991) suffer from the fact that the different equilibria may not be thermodynamically consistent with one another or that the derived or implied standard state properties of minerals are often inconsistent with more directly constraining experiments in the boundary systems. In order to meet the shortcomings of these two different approaches, endmember and solid solution properties must be determined simultaneously from available experimental data. “Purists” may worry that the assumptions and complexities inherent in treatment of solid solutions will degrade the quality of endmember properties derived on their own. The reader should bear in mind, however, that the same set of constraints on endmember properties is retained for the combined analysis. With the mathematical programming technique used to process all data in this study (see next section), the endmember constraints restrict the range of permissible solution properties that are compatible with all phase equilibrium data, while the solid solution experiments also provide for refinement of the endmember properties, but only from within the range of the direct experimental constraints on endmembers. A general problem with treatment of phase equilibrium data involving solid solutions is the nonuniqueness of resulting solution properties (e.g. Hackler and Wood 1989; von Seckendorff and O’Neill 1993). This results from the fact that phase equilibrium data yield constraints on the differences in thermodynamic properties of reactants and products, and solution properties for one phase depend on assumed properties for other phases involved in the analyzed equilibria. Calorimetric data can be used to directly constrain mixing properties, but uncertainties associated with calorimetric measurements are large compared to the magnitude of the mixing energetics (e.g. Newton et al. 1977; Chatillon-Colinet et al. 1983; Geiger et al. 1987). The most appropriate way to address this problem is not just to analyze some combinations of phase equilibria for which good experimental data have been obtained (e.g. Wood and Holloway 1984; Wood 1987; Fei and Saxena 1986; Berman 1990), but to utilize all linear combinations of equilibria that have been experimentally studied such that there are internal checks on derived properties for all phases. Efforts that incorporate complete thermodynamic “cycles” like this should produce not only the “best” set of internally consistent standard state and solution data, but should also allow the best applications and extrapolations of experimental results obtained in relatively simple chemical systems to natural systems of more general geological interest. This paper reports the first results of such a project, in which four complete “cycles” involving seven Fe-Mg exchange equilibria as well as net transfer reactions among olivine, orthopyroxene, cordierite, garnet and ilmenite are analyzed in the system FeO-MgOCaO-Al2O3-TiO2-SiO2-H2O (FMCATSH). The results of this study represent a key part of work in progress to
refine and extend the thermodynamic database presented by Berman (1988). Applications of this data set to highgrade metamorphic rocks are described in a companion paper (Aranovich and Berman 1996b).
Mathematical treatment In keeping with the premise that a successful phase equilibrium experiment measures changes in either mineral compositions or proportions, we analyzed these data using mathematical programming (MAP), a technique to solve systems of inequalities that, in the present context, convey the sense of the change in Gibbs free energy of reaction or mole fraction product observed in each experiment. We consider this mathematical technique advantageous for treating phase equilibrium data for three primary reasons. First, the technique provides the means to analyze rigorously half-brackets, i.e. experiments in which a particular equilibrium was approached from only one direction. In addition to points related to univariant equilibria discussed elsewhere (Berman et al. 1986; Berman 1988), analysis of half-brackets has paramount importance in assessing phase equilibria involving solid solutions because many portions of the experimental data base consist of half-brackets over significant compositional ranges. Second, MAP allows the incorporation of important experimental observations on solid solutions that we refer to as bulk composition constraints. These constraints are mathematical inequalities expressing that assemblage A with specified mineral compositions (bulk composition) is more (or less) stable than assemblage B with specified mineral compositions (see later in this section). Third, MAP allows for explicit incorporation of all experimental uncertainties, thus providing a rigorous means to assess the compatibility of diverse types of data. Our own experience with regression techniques (e.g. Aranovich et al. 1985) indicates that when all experimental data are not reproduced within their uncertainties, it is difficult to determine whether it is the data weighting (implicit or otherwise) or a model inadequacy that prevents appropriate representation of all experimental observations. We have not attempted to estimate uncertainties in thermodynamic variables because one of the most significant sources of uncertainty is the systematic error related to which thermodynamic models are adopted and which experimental data out of a conflicting set are accepted. It is these systematic errors, which are not incorporated in statistical uncertainty estimates, that we attempt to define with the MAP technique. For an equilibrium that involves solid solutions, the basic thermodynamic constraint can be expressed as the following inequality: DrGP,T5DrGO P,T1RT lnKX1RT lnKg_0
(1)
where KX and Kg are the products of the mole fractions and activity coefficients, respectively, raised to the pow-
3
er of their stoichiometric reaction coefficients. DrGP,T is the free energy of reaction at P and T. DrGO , the stanP,T dard state (unit activity for pure minerals at P and T; unit fugacity at T for H2O and CO2) free energy change of reaction at P and T, is expanded as O O T O DrGO P,T5DrH1,2982TDr S1,2981&298DrCP dT
2T &T298
DrCO P dT1&P1 DrV OdP T
(2)
The sign of inequality (Eq. 1) for each experimental run is defined by the direction in which the reaction proceeded during the run. This is determined most conveniently by comparing the starting and final KX, although the most confidence is placed on experimental results in which the compositional changes of all phases of variable composition give a consistent sense of reaction direction. A particularly vexing problem that arose early in this study was the inability of the above formed constraints on net transfer equilibria to reproduce adequately the positions of the divariant loops for Cd-Gt-Si-Qz and OlOpx-Qz (for mineral abbreviations see Table 1). Carlson and Lindsley (1988) ascribed similar problems in fitting the two-pyroxene pseudosolvus to the fact that the above type of constraints involve activity ratios for components in more than one phase, allowing the mathematical constraints to be satisfied numerically while not reproducing the experimental data for which the constraint is written. We solved this problem by writing “bulk composition” constraints that, for a specific bulk composition, compare overall phase stabilities involving all components of each phase, rather than constraints like Eq. 1 which relate chemical potentials of each component sep-
arately in compositional subsystems. For example, Bohlen and Boettcher (1981) performed an experiment at 10008 C and 10 kbar in which they observed that Ol1Qz are stable at the bulk composition XFe50.95, thereby defining the divariant field to be at more Mg-rich compositions. We force Ol951Qz to be more stable than Ol961Opx901Qz by constraining the following mass balanced reaction 0.05 Ol9610.02 Opx9050.06 Ol9510.01 Qz
to have DrGP,T,0. This yields Ol96 Opx90 96 0.05 [0.96 mOl Fa 10.04 mFo ]10.02 [0.90 mFs Opx90 Ol95 Ol95 10.10 mEn ].0.06 [0.95 mFa 10.05 mFo ]10.01 mQz
n1 i
Name
Abbreviation
Formula
Orthopyroxene Enstatite Ferrosilite Orthocorundum Olivine Forsterite Fayalite Garnet Grossular Pyrope Almandine Cordierite Mg-cordierite Fe-cordierite Ilmenite Ilmenite Geikielite Anthophyllite Spinel Hercynite Rutile Quartz Sillimanite Corundum
Opx En Fs Ok Ol Fo Fa Gt Gr Py Alm Cd mCd fCd Il Ilm Gk Ant Sp Hc Rt Qz Si Co
(Mg,Fe,Al) (Al,Si)O3 MgSiO3 FeSiO3 Al2O3 (Mg,Fe)2SiO4 Mg2SiO4 Fe2SiO4 (Ca,Mg,Fe)3Al2Si3O12 Ca3Al2Si3O12 Mg3Al2Si3O12 Fe3Al2Si3O12 (Mg,Fe)2Al4Si5O18 Mg2Al4Si5O18 Fe2Al4Si5O18 (Mg,Fe)TiO3 FeTiO3 MgTiO3 Mg7Si8O22(OH)2 MgAl2O4 FeAl2O4 TiO2 SiO2 Al2SiO5 Al2O3
(3b)
where mYn is the chemical potential of component i in i solid solution Y with fixed composition 100 XFe5n. In this example, Ol96 is an olivine composition chosen as slightly more Fe-rich than Ol95 so that it represents the bulk composition when combined with Opx90, the orthopyroxene composition in exchange equilibrium with Ol96. Note that it is not sufficient to use the less constraining free energy relations between Ol951Qz and Opx95 because GOpx951Qz . GOl961Opx901Qz. Bulk composition constraints are also particularly useful in accounting for experimental observations on the disappearance or growth of a particular assemblage of fixed bulk composition. This type of constraint cannot be explicitly incorporated with unconstrained least-square methods (e.g. Aranovich et al. 1985). Final optimization of all derived thermodynamic properties was achieved using a “least squares” objective function (Berman et al. 1986; Engi 1987): *
Table 1 Abbreviations, chemical formula of minerals and minals
(3a)
(f12mi)2 n2 (Gcalc,j2lnKx,j)2 1* s2i s2lnKj j
(4)
to minimize discrepancies between n1 derived (fi) and directly measured (mi) thermodynamic properties as well as between n2 experimentally determined lnKX values and predicted values (Gcalc). The latter are calculated from rearrangement of Eq. 1 as: Gcalc5(2DrGO P,T2RT lnKg)yRT5lnKX
(5)
with KX and Kg written for the nominally determined compositions of coexisting phases for the given equilibrium. In the absence of quantitative estimates of KX errors, the variance of lnKX was assumed to be equal to 1% of lnKX.
Treatment of experimental data Philosophy of selection The primary objective of this study is to derive thermodynamic data that, by providing for interpolation between and extrapolation beyond the available set of experimental observations, can be used for reliable petrologic calculations. In addition to being “internally con-
4
sistent”, these data should also reproduce diverse experimental observations within their respective uncertainties. Because any thermodynamic data are only as good as the experimental data from which they are derived, an additional objective of this study is to attempt to identify the most compatible of the available experimental data as well as outliers within the data. In deriving thermodynamic data from relevant experimental data obtained with different methods and in different laboratories, we use only the most reliable of these data in order to minimize the number of potential inconsistencies among the different data sets. A critical step is therefore to establish criteria for assessing the reliability of these data so that there is a basis for trying to resolve any such inconsistencies when they arise. Besides adequate documentation of experimental apparati and their calibrations, the most stringent criterion applied in this study was that it be possible to discern exactly what changes were observed in each experiment, i.e. what equilibria were responsible for the changes in mineral compositions or proportions observed during an experiment, and in which direction these equilibria proceeded during each experiment. Translated into practical terms, this criterion excludes all data but those in which an assemblage of crystalline materials, capable of defining one or more equilibria, was observed to undergo changes in compositions or mineral proportions of a magnitude larger than the uncertainties in deciphering such changes. Synthesis-type runs were not used because of their greater tendency towards formation of metastable run products.
Constraints on end-member properties Inspection of Eq. 1 indicates that parameters representing analytical expressions for activity coefficients can be determined given experimental phase equilibrium data Table 2 Phase equilibrium data used to constrain MAS and FAST endmembers
'
Equilibrium
defining KX and standard state thermodynamic data defining DrGo. In the initial stages of this study, standard state properties for all MAS phases were taken from the analysis of Berman (1988) with the properties of FAST phases constrained by relevant univariant phase equilibria. The final analysis presented below involved simultaneous redetermination of MAS, MSH, and FAST standard state properties (using data for equilibria (a) to (o) listed in Table 2 and MSH data described by Berman, 1988) because the small (,1 kJymol) Gibbs energy adjustments allowable by the relevant MAS experimental data permit much better representation of all experimental constraints in the FMAST system. In addition, incorporation of new Cp measurements for pyrope (Tequi et al. 1991) and forsterite (Gillet and Fiquet 1991) leads to significantly improved representation of the overall experimental phase equilibrium data set. Additional changes that have been incorporated into our analysis are corrections to the entropy of anthophyllite (Hemingway 1991), reassessment of the volume of anthophyllite based on analysis of natural and synthetic amphiboles (Hirschmann et al. 1994), and recent determination of the thermodynamic properties of spinel with equilibrium amount of disorder (Chamberlin et al. 1995). In contrast to Berman (1988)’s use of fayalite, we use a-Fe as the Fe anchor phase, employing careful EMF measurements (O’Neill 1987) to tie these properties to Fesilicates. All thermodynamic properties determined in this study are consistent with those of CaO-K2O-Na2OAl2O3-SiO2-H2O phases tabulated by Berman (1988). Phase equilibrium data published since completion of the Berman (1988) study that are relevant to this analysis bear on the stability of the aluminosilicate polymorphs and some Ca-silicates. The former data are shown in Fig. 1, where it can be seen that the tabulated 1988 properties are compatible within experimental uncertainties with direct observations by Bohlen et al. (1991) on the ky5si transition (Fig. 1a) and by Harlov and Newton (1993) on
Authors
System: MgO-Al2O3-SiO2 (a)a (b) (c) (d) (e)
Py1Fo54 En1Sp 6En12Si52Py12Qz 3En1Co5Py mCd1Co52En13Si 5Fo1mCd510En12Sp
Danckwerth and Newton (1978); Perkins et al. (1981) Perkins (1983) Gasparik and Newton (1984) Newton (1972) Fawcett and Yoder (1966); Seifert (1974); Herzberg (1983)
System: FeO-Al2O3-SiO2-TiO2
a
Equilibria (a)–(e) involve aluminous Opx.
(f) (g) (h) (i) (j) (k) (l) (m) (n) (o)
2Fs5Fa1Qz 2Alm14Si15Qz53fCd 3Hc13Si5Alm15Co 3Hc15Qz5Alm12Si Alm1Hm5Mt1Ky1Qz 3Ilm1Si12Qz5Alm13Rt 2Ilm52Fe12Rt1O2 4Mt1O256Hm 3Fa1O252Mt13Qz 2Fe1Qz1O25Fa
Bohlen et al. (1980) Mukhopadhyay and Holdaway (1994) Shulters and Bohlen (1989) Bohlen et al. (1986) Harlov and Newton (1992) Bohlen et al. (1983b) Shomate et al. (1946); Feenstra and Peters (1996) Myers and Eugster (1983); O’Neill (1988) O’Neill (1987) O’Neill (1987)
5
Our results of the MSH system are very similar to those documented by Berman (1988), and the reader is referred to the latter paper for detailed presentation of the MSH data. Significant differences between endmember properties derived here and by Berman (1988) are discussed in the “Results” section. Constraints on solid-solution properties
Fig. 1a, b Comparison of recently published experimental data with univariant curves in the system Al2O3–SiO2 computed with thermodynamic data of Berman (1988). a Data of Bohlen et al. (1991) for the equilibria Ky5Si and Ky5And. b Data of Harlov and Newton (1993) for the equilibria And5Co1Qz and Ky5Co1Qz. Symbols show experimental data after adjustment for experimental uncertainties (direction of adjustment is away from the equilibrium position). Opposite ends of connected lines show nominal experimental half-brackets
metastable equilibria involving co1qz (Fig. 1b). The experiments of Zhu et al. (1994) on the equilibrium: calcite1qz5wollastonite1CO2 indicate increased stability of wollastonite with respect to calcite than given by Berman (1988). As calcite is not considered in this study, what is of direct concern here is Zhu et al.’s conclusion, based on the measured enthalpy of solution for anorthite and analysis of other experimental brackets on the equilibrium grossular1quartz52wollastonite1anorthite, that their preferred enthalpies of formation for anorthite (299.5 kJymol), grossular (2329.55 kJymol), and wollastonite (21635.4 kJymol) are more in accord with those given by Holland and Powell (1990; 2101.2, 2327.4, 21633.2) than by Berman (1988; 296.5, 2319.8, 21631.5). Although this is true in terms of absolute differences, Zhu et al.’s preferred values are in fact in excellent agreement with Berman (1988) values once all are corrected for what amounts to a 3.5 kJymol difference in CaO anchor value. Applying this change to Zhu et al.’s preferred values yields 296.0, 2319.05, 21631.9 kJymol, differences for each phase that are less than 0.75 kJymol from Berman’s values.
With the MAP method, the Gibbs energy inequalities are written using explicit values for uncertainties in P, T, and all compositional parameters. Compositional uncertainties applied to each set of experimental data involving a solid solution phase were taken from authors’ evaluation when available or based on estimated minimum microprobe errors of XFe5Fey(Fe1Mg)5+0.005. Considering the additional uncertainties related to compositional heterogeneity produced in all phase equilibrium studies involving solid solutions, more reasonable minimum overall uncertainties are XFe5+0.01. In analyzing data in which final compositions are heterogeneous, we assume that the most advanced compositions (those which yield KX most different from that of the starting compositions), not the average or most abundant compositions, are closest to equilibrium. Overlapping half-brackets after compositional adjustment suggests either underestimation of analytical uncertainties or real overstepping of an equilibrium (“pathlooping” of Perkins and Newton 1980), whether the latter is caused by the solution-reprecipitation process (Aranovich and Pattison 1995) or by different mechanisms and rates of the forward and backward reactions (Aranovich and Kosyakova 1987; Perkins and Newton 1980). Any pair of runs that constitutes a reversal (two half-brackets) with overlapping compositions yields two mutually inconsistent limits that produce an infeasible solution to a MAP problem. To make them both compatible, three different approaches can be employed. One approach, which assumes that the path-looping of the reaction is the only reason for the overlap, is to change the sign of the reversal brackets arbitrarily, defining their width as the amount of compositional overlap (Carlson and Lindsley 1988). This approach cannot make use of an experimental half-reversal (Carlson and Lindsley 1988, p 244), a serious disadvantage because most of the experiments concerning solid solutions are not reversed sensu stricto, but are more appropriately viewed as a series of half-brackets. A second approach is to adjust the uncertainty applied to individual experimental points in order to eliminate the inconsistencies. This approach is most in accord with the Fe-Mg partitioning data used in this study. Because the Gibbs free energy of the equilibrated minerals is not much lower than that of the crystalline starting materials, and because of favorable reaction kinetics, equilibrium overstepping does not appear to be a general phenomenon that affected a large number of experimental runs. In contrast, minor relaxation of the errors assigned
6
to a small percentage (generally less than 5%) of the half-brackets is sufficient to obtain consistency of entire sets of experimental data. This approach places more importance on the scrutiny of all details of each experiment in order to discover possible sources of error, but it has the distinct advantage to identifying possible outliers that can lead to systematic errors in retrieved thermodynamic parameters. A third approach is to treat the maximum compositional overlap observed in each particular experimental study as two times the minimum compositional adjustment that is applied to all the experimental results of that study. This practice, although to some degree arbitrary, is more in accord with either underestimated analytical uncertainty or overstepping causing observed compositional overlaps, or both. It should be noted that this adjustment permits consistency between two opposing half-brackets, but may not be of sufficient magnitude to allow consistency (feasible MAP solutions) among all experiments of one study. This third approach was applied to the experimental data bracketing the Al content of Opx in equilibrium with Gt or Cd in the MAS and FMAS systems (Perkins et al. 1981; Kawasaki and Matsui 1983; Lee and Ganguly 1988) because sluggish reacTable 3 Solid solution equilibria considered in this analysis
'
System
tion kinetics make significant compositional overlapping a general feature of these experimental results. Many experiments concerning solid solutions were excluded from our treatment because starting compositions were not reported. Experiments which produced inconsistent changes in mineral compositions (e.g. FeMg exchange experiments involving two phases which both increased in Fe) were excluded from the final analysis. Less weight was given to experiments in which phases extraneous to the equilibrium under study appeared, because of the possibility that the entire assemblage was not in equilibrium. Experimental data for equilibria (A) to (U) listed in Table 3 were used in this study to determine solution properties. Specific points regarding these data are discussed below: Garnet-orthopyroxene Perkins et al. (1981) obtained 46 pairs of reversals for the equilibrium (H): 3 En1Ok5Py in the MAS system. In most runs they observed an overlap in the final Opx composition, up to 0.7 wt% Al2O3. We found that +0.7 wt% was the minimum compositional adjustment required to be applied to all data in order to ob-
Reference
Fe-Mg exchange equilibria: (A)
3Fs1Py53En1Alm
(B) (C)
fCd12En5mCd12Fs 3fCd12Py53mCd12Alm
(D) (E)
3Fa12Py53Fo12Alm 2Fs1Fo5Fa12En
(F) (G)
Fs1Gk5Ilm1En Fa12Ilm52Gk1Fo
Kawasaki and Matsui (1983); Harley (1984); Lee and Ganguly (1988); Eckert and Bohlen (1992) Aranovich and Kosyakova (1987) Aranovich and Podlesskii (1981, 1983); Perchuk and Lavrent’eva (1983) Hackler and Wood (1989) Medaris (1969); Fonarev (1981, 1987); Bohlen and Boettcher (1981); Davidson and Lindsley (1989); Koch-Muller et al. (1992); von Seckendorff and O’Neill (1993) Hayob et al. (1993) Andersen and Lindsley (1979); Bishop (1976, 1979, cit. after Andersen et al. 1991); Andersen et al. (1991)
Net transfer equilibria involving Al content of Opx: (H)
3En1Ok5Py
(I)
3Fs1Ok5Alm
(J) (K) (L)
2En12Ok13Qz5mCd 2Fs12Ok13Qz5fCd Sp1Fo5En1Ok
Lane and Ganguly (1980); Perkins et al. (1981); Kawasaki and Matsui (1983); Harley (1984); Lee and Ganguly (1988) Aranovich and Berman (1995, 1996a); Kawasaki and Matsui (1983); Harley (1984); Lee and Ganguly (1988) Aranovich et al. (1983); Aranovich and Kosyakova (1987) Aranovich and Kosyakova (1987) Gasparik and Newton (1984)
Other net transfer equilibria: (M) (N) (O)
2Py14Si16Qz53mCd 2Alm14Si16Qz53fCd Fa1Qz52Fs
(P)
Fo1Qz52En
(Q) (R) (S) (T) (U)
Fs1Rt5Ilm1Qz En1Rt5Gk1Qz Gros12Ky1Qz53An Gros12Alm53Fa13An 6Ilm13An13Qz5Gros 16Rt12Alm
Aranovich and Podlesskii (1981, 1983); Hensen (1977) Aranovich and Podlesskii (1981, 1983); Hensen (1977) Fonarev and Korol’kov (1976); Fonarev (1987); Bohlen and Boettcher (1981); Davidson and Lindsley (1989) Fonarev and Korol’kov (1976); Fonarev (1987); Bohlen and Boettcher (1981); Davidson and Lindsley (1989) Hayob et al. (1993) Hayob et al. (1993) Wood (1988); Koziol and Newton (1989); Koziol (1990) Bohlen et al. (1983a) Bohlen and Liotta (1986)
7 tain internally consistent results for this set of data considered alone. We accepted six of the eight experiments reported by Kawasaki and Matsui (1983) for equilibrium (H) in the MAS system, eliminating those started from gel mixtures (run 571, in their Table 3) and compositionally uncharacterized synthesis products (run 372). Only three of their seven runs in the FAS system, performed with crystalline starting materials, have been used in this study to constrain the position of equilibrium (I): 3 Fs 1Ok5Alm. We adjusted the pressure of Kawasaki and Matsui’s 50 kbar (nominal) runs to 46 kbar following the suggestions by Gasparik and Newton (1984) and Brey and Kohler (1990). For the exchange equilibrium (A): 3 Fs 1Py53 En1Alm, we accepted 14 experiments of Kawasaki and Matsui (1983, their Tables 5 and 6) in the FMAS system, in which they employed preanalyzed crystalline starting materials, ignoring all synthesistype runs. For the same reason we have rejected most of the experimental points of Harley (1984) on equilibrium (A), accepting those five that were obtained on crystalline starting materials placed in graphite containers (Harley 1984; his Table 2). No runs in Fe capsules were accpeted in this study because of the differential rate of Fe change of Opx and Gt observed by Harley. In view of the scarcity of experiments defining Al2O3 solubility in FMAS Opx, we used Harley’s data for equilibrium (I) as part of the objective function (4). Lee and Ganguly (1988) used crystalline starting materials with a PbO1PbF2 flux to promote the reaction rates, but did not report details of any heterogeneity in compositions of run products. All exchange equilibrium (A) data are mutually consistent using the +0.01 compositional uncertainties estimated by the authors (Lee and Ganguly 1988; their Table 2). Nevertheless, we did not utilize one half-bracket ('163) in which both Opx and Gt decreased in Mg during the experiment. Eckert and Bohlen (1992) used crystalline starting materials without any flux to facilitate reaction, except for one 9008 C bracket with H2O added. The four brackets they report for the exchange equilibrium (A) are internally consistent without any compositional adjustments. Their reported Al2O3 contents of Opx, all approached from undersaturation, require minimum adjustments of 40% to be mutually compatible. Recent data of Aranovich and Berman (1995, 1996a) offer the best constraints on the position of equilibrium (I) in the FAS system. Aranovich and Berman (1995; 1996a) obtained nine reversals of the Al2O3 content of FAS orthopyroxene in equilibrium with almandine over the P-T range 12–20 kbar and 900–10008 C. Starting materials consisted of crystalline ferrosilite and Fs95Ok5 up to 50 mm in length, as well as the Fs95Ok5 oxide synthesis mix. Run products using the crystalline starting materials produced Al2O3zoned Opx with most advanced compositions that converged within 0.002 XOk without overlap. Run products of the oxide mixes also produced heterogeneous Opx with a compositional range very similar to the reversal run brackets. The data presented by Aranovich and Berman (1995; 1996a) also constrain the position of the FAS univariant equilibrium: Fe-Opx5Fa1Qz1Alm. Cordierite1orthopyroxene1quartz Aranovich et al. (1983) bracketed the Al2O3 content of Opx in the presence of Cd in the MAS system, buffered by the equilibrium (J): 2En12Ok13Qz5mCd. In almost all of their reversal runs, an overlap in the final Opx composition of 0.8–1.2 wt.% Al2O3 was observed. Aranovich and Kosyakova (1987) determined Al2O3 contents of FMAS Opx in equilibrium with Cd1Qz (equilibria (J) and (K)) as well as Fe-Mg partitioning data (equilibrium (B)). No reversals were reported for Al2O3 contents, however, because Al-free Opx was used as the starting material in all experiments. For the present study we accepted 34 of their 36 runs, rejecting the two runs which employed a starting mix of oxides.
Garnet1cordierite1sillimanite1quartz The only experiments with this assemblage in FMAS (equilibria (M) and (N)) that utilized crystalline starting materials are those of Hensen (1977) and Aranovich and Podlesskii (1981, 1983). We have slightly reinterpreted the observations of the former author, selecting for this study the most advanced compositions of both Gt and Cd rather than the average ones referred to by Hensen. We also adjusted the final compositions for overlap (XMg50.002 in Gt and XMg50.02 in Cd) and corrected the nominal pressure of the runs by 210% as suggested by more recent piston-cylinder calibrations (e.g. Perkins et al. 1981). From the work of Aranovich and Podlesskii (1981, 1983), we have selected only those runs which produced appreciable changes in both garnet and cordierite composition. Runs in which both phases moved in the same compositional direction (mostly from Mg-rich to Fe-rich compositions) have been used only to constrain the exchange reaction (C). The maximum observed compositional overlaps are 0.015 for Gt and 0.01 for Cd. We also utilized observations of these authors on the relative stability of the alternative phase assemblages in the form of bulk composition constraints discussed above.
Olivine-garnet Fe-Mg exchange Hackler and Wood (1989) presented reversed data at 10008 C, 9.1 kbar, for equilibrium (D): 3Fa12Py52Alm13En, with garnet Mgy(Mg1Fe)50.17–0.67. Synthetic garnets and olivines were homogeneous to within 2.5 to 0.5 mol%, respectively. Reversal experiments contained Gt:Ol ratios of 10: 1 by weight in order to minimize compositional changes in Gt. Their run products represent homogeneous olivine and the most advanced of the slight range of garnet compositions surrounding olivine. Final Gt compositions shifted in the expected direction from nominal starting compositions to compensate changes in Ol composition. In only one half-bracket was the change in composition greater than the starting Gt heterogeneity range. These data are internally consistent using compositional uncertainties of 0.01. Experimental data for this same equilibrium were also obtained by O’Neill and Wood (1979) over the temperature range 1000–14008 C at 30 kbar. We did not analyze these data because most are synthesis runs, details of starting materials and run products are not given, and they are presented graphically.
Olivine-orthopyroxene-(quartz) The distribution of Fe-Mg between Opx and Ol (reaction E) and the lower pressure stability limit of Opx relative to Ol1Qz (reactions O, P) have been extensively studied over the past few decades (Larimer 1968; Medaris 1969; Matsui and Nishizawa 1974; Fonarev and Korol’kov 1976; Fonarev 1981, 1987; Bohlen et al. 1980; Bohlen and Boetcher 1981; Davidson and Lindsley 1989; KochMuller et al. 1992; von Seckendorff and O’Neill 1993). Close examinations reveals that not all of these experiments meet the stringent criteria imposed in this study. We rejected the data of Larimer (1968), Matsui and Nishizawa (1974), and Fonarev (1987) (except for the 7508 C isotherm of the latter author) for equilibrium (E) because these authors either did not report starting mineral compositions or performed unreversed synthesis experiments. We have also not included the data of Koch-Muller et al. (1992) on reactions (O) and (P) at 950 and 10008 C because growth of Fs was not observed at these temperatures. Because of some discrepancies between X-ray composition equations used by different authors (Medaris 1969; Fonarev 1981, 1987; Koch-Muller et al. 1992), we have attributed larger uncertainties (0.02 Mgy(Mg1Fe)) to the data obtained by this method. The most complete experimental study reviewed in the course of this work is that of von Seckendorff and O’Neill (1993). Using crystalline starting materials and a BaO-B2O3 flux to promote reaction, they collected 46 half-brackets on reaction (E) ranging
8 from Fa03 to Fs98 and 900 to 11508 C. Olivine and orthopyroxene in run products were generally very homogeneous, with average standard deviations between 0.25 and 0.41 mol%. The width of growth rims was 5–30 mm, facilitating single phase microprobe analysis. In 34 of 46 runs, run direction was unambiguously defined by sympathetic changes in both Ol and Opx compositions that were greater than analytical uncertainties. In the remaining 12 runs, the observed changes in KD were also significantly larger than uncertainties. Treated by themselves, these data are internally consistent with 0.002 Mgy(Mg1Fe) adjustment applied to all data. Olivine-ilmenite Fe-Mg exchange Experimental observations on compositions of coexisting olivine and ilmenite (exchange reaction G in Table 3) solid solutions equilibrated over a wide range of temperature, pressure and Mgy (Mg1Fe) have been reported by Andersen and Lindsley (1979), Andersen et al. (1991), and Bishop (1976, 1979; cited after Andersen et al. 1991). Compositional uncertainties, although mentioned (Andersen et al. 1991, p 436), have not been estimated by the authors. The heterogeneity range in the representative run products shown by Andersen and Lindsley (1979, their Fig. 1) shows an overlap in most advanced compositions of about 0.01 Mgy(Mg1Fe), which we applied as a minimum adjustment to all the data. Data by Bishop, as given in Table 8 of Andersen et al. (1991) require an additional 0.01 Mgy(Mg1Fe) adjustment to become internally consistent. To avoid problems related to the complex hematite solid solution in picroilmenite (e.g. Ghiorso 1990), we used for the present study only those experimental points which showed Hm contents less than 3 mol%, and recalculated them to the Ilm-Gk binary. Any errors associated with neglecting Hm component for this compositional range are believed to be significantly smaller than those related to the uncertainties in determining equilibrium Mgy(Mg1Fe) in the coexisting minerals. Orthopyroxene-ilmenite (+rutile,quartz) Both exchange (equilibrium F) and net-transfer equilibria (equilibria Q and R) operating in the SIRF (Silica-Ilmenite-Rutile-Ferrosilite) system have been recently investigated by Hayob et al. (1993). The authors utilized crystalline mixtures of presynthesised minerals as starting materials and buffered the run assemblages near iron-wüstite. Although synthetic ilmenites contain up to 2 mol% hematite. product ilmenites contain less than 1 mol% of Hm. Product Opx was somewhat heterogeneous (Hayob et al. 1993) with up to ¥0.02 Mgy(Mg1Fe) compositional overlap in bracketing experiments. For our processing we accepted all the data points reported by these authors, adjusted for P-T-XFe uncertainties (0.5 kbar, 58 C, 0.02 XFe) quoted by Hayob et al. (1993).
Solid solution models Any thermodynamic modelling employing experimental phase equilibrium data involving solid solutions requires analytical formulation of Gex as a function of measured parameters (T, P, Xi), and thus necessarily includes a certain compromise between the complexity of the solid solution models, the amount and quality of the data, as well as the purposes of the modelling. In so far as our main goal is to produce reasonable phase diagrams as well as geothermobarometric results, we tried as a first approximation the simplest solution models possible, increasing their complexity only when no feasible solution to the whole data set could be reached without unjustifiable enlargement of the experimental uncertainties.
Initially, activity coefficients for all phase components were computed with a third degree (asymmetric) Margules equation in the generalized form below (Berman and Brown 1984; Berman 1990): nRTlngm5*WijkXiXjXk
F G Qm 22 Xm
H S V Wijk5WG ijk5Wijk2TWijk1(P21)Wijk
(6a) (6b)
In Eq. 6a, the summation is for all Margules parameters, n is the site multiplicity, and Qm is the number of i,j,k subscripts that are equal to m. When combined with the Wohl (1946, 1953) approximation for third degree equations: Cijk5(Wiij1Wijj1Wiik1Wikk1Wjjk1Wjkk)y22Wijk50
(6c)
Eq. 6a is compatible with a zero Wohl ternary interaction (Cijk) parameter (Berman 1990; Berman and Koziol 1991). We prefer Eq. 6a–c to the mathematically equilvalent expressions presented by Helffrich and Wood (1989), because of their ease of memorization, generalization to any number of components, and conversion to compact computer code. Due to the general lack of direct constraints on excess entropy for all solid solutions, we considered a symmetric functional form (WSiij5WSijj ) to have the greatest reliability for extrapolation outside the T-X range of existing experimental data. In the final optimization, a symmetric Margules expression was also used for the excess enthalpy of Ol, Opx, Cd, and Il. Additional specifics of each solid solution model used in this study are described in the following sections. Olivine Fe-Mg mixing in olivine has been the subject of considerable experimental and theoretical investigation. Sack and Ghiorso (1989) found that available calorimetric and phase equilibrium data were most compatible with a symmetric regular solution parameter (WH) equal to 10.17 kJy2 oxygen formula. More recent studies based on various phase equilibria and calorimetry indicate values in the range 3–5 kJy2 oxygen basis, with excess entropy less than 2 Jy2 oxygen basis (e.g. Hackler and Wood 1989; Wiser and Wood 1991; von Seckendorff and O’Neill 1993; Kojitani and Akaogi 1994). In this study we determined mixing properties of olivine simultaneously with those of all other phases. Excess volume was allowed to range between the extremes (0.011– 0.045 Jybarymol) of various determinations (Schwab and Küstner 1977; Kawasaki and Matsui 1983). Minor ordering of Mg-Fe21 between M1 and M2 sites (e.g. Ottonello et al. 1990) has been ignored. Ilmenite Two different approaches have been applied recently for modelling solid solution properties in the system il-
9
menite-geikielite-hematite (Ghiorso 1990; Andersen et al. 1991). The departing point of the Andersen et al. (1991) treatment of Fe-Mg exchange equilibria ilmenite-olivine and ilmenite-spinel corresponds to completely ordered Ilm-Gk, while Ghiorso (1990) used a convergent ordering model which assumes disordering of Fe, Mg, and Ti (as well as Fe 31 in the case of ternary Ilm-Gk-Hem solid solution) on structurally equivalent sites. This latter model was shown to be more successful in modelling the ternary solution because of the explicit formulation of the entropy of mixing on the Ilm-Hem and Gk-Hem joins (Ghiorso 1990). The advantages of this model are not as clear for the case of the Ilm-Gk binary. For example, it predicts considerable disorder in pure ilmenite at a temperature above 1000 8 C (Ghiorso 1990, his Fig. 8) not supported by direct observations (e.g. Wechsler and Prewitt 1984). The predicted Gibbs free energy of mixing on the binary (Ghiorso 1990, his Fig. 9) exhibits very slight asymmetry and smooth temperature dependence that should indicate relatively simple macroscopic behavior of the solid solution. As far as our calculations are restricted to essentially Fe 31-free compositions, we have chosen for the Ilm-Gk solid solution a subregular solution model similar to that used by Andersen et al. (1991), with the W V parameter fixed according to these authors at 0.0108 J ybar mole. Cordierite Cordierite has been successfully modelled as an ideal one site solution with site multiplicity of 2 (Perchuk and Lavrent’eva 1983; Aranovich and Podlesskii 1983; Aranovich and Kosyakova 1987). Here we allowed for the possibility of small deviations from ideality with a symmetric (regular) Margules approximation. The increased stability of cordierite resulting from introduction of H 2O molecules into the structural channels has been accounted for by applying McPhail et al.’s (1990) thermodynamic description of the reaction: Mg2Al4Si5O1812H2O5MgAl2Si5O18 ? 2H2O
Following recent experimental observations (Boberski and Schreyer 1990; Mukhopadhyay and Holdaway 1994) we assumed no difference in the water solubility in Feand Mg-cordierite at similar P-T-conditions. Thus, the following expressions are used for the activity of “dry cordierite” end-members 2 2 amCd5(XCd Mg gmCd) (12XH2O) 2 2 afCd5(XCd Fe gfCd) (12XH2O)
where XH2O is the number of moles of H2O per formula unit of cordierite, divided by 2.
Garnet Mixing properties of almandine-pyrope solutions have been the subject of considerable debate (e.g. Aranovich 1983; Ganguly and Saxena 1984; Sack and Ghiorso 1989; Berman 1990; Koziol and Bohlen 1992), with the weight of recent evidence favouring near-ideal behavior. All recent studies suggest larger excess enthalpies in Ferich compositions, the asymmetry in enthalpy mimicking that in the excess volumes. Although this study deals primarily with Fe-Mg exchange equilibria, phase equilibrium data on the Gr-Alm join place important constraints on almandine standard state properties. For this reason we evaluate mixing properties on this join which is well constrained by both calorimetric (Geiger et al. 1987) and phase equilibrium studies (equilibria (S)–(U) in Table 3; Bohlen et al. 1983; Bohlen and Liotta 1986; Koziol 1990). Our analysis of these data follows closely that described by Berman (1990). In order to be able to apply the results of this study to natural Ca-Mg-Fe garnets, we present provisional mixing properties on the Gr-Py join based on two reversals of the GASP equilibrium (Wood 1988) as well as enthalpy of mixing data of Newton et al. (1977). Following Berman and Koziol’s (1991) analysis of GASP reversals with Ca-Mg-Fe garnet (Koziol and Newton 1989), we assume the Wohl ternary interaction parameter (Cijk) is equal to zero (i.e. that Eq. 6c holds). While compatible with Koziol and Newton’s data, more rigorous testing of this assumption awaits further experimental constraints on Ca-Mg garnet as well as a broader analysis that includes not only ternary GASP reversals, but also Fe-Mg exchange data involving ternary Ca-Mg-Fe garnet (e.g. Gt-Opx: O’Neill and Wood 1979; Gt-Cpx: Pattison and Newton 1989). In this study we applied the subregular model to the Py-Alm, Gr-Alm, and Gr-Py joins. For the 12 oxygen formula, the activity of garnet components are given by 3 aPy5(XGt Mg gPy) 3 aAlm5(XGt Fe gAlm) 3 aGr5(XGt Ca gGr)
with activity coefficients given by Eq. 6. Excess volume parameters were taken from Berman’s (1990) evaluation, except for those on the Gr-Py binary which were taken from Ganguly et al.’s (1993) data. Orthopyroxene Orthopyroxene solid solution properties are at least as controversial as those for garnet (e.g. Sack 1980; Kawasaki and Matsui 1983; Aranovich and Kosyakova 1987; Sack and Ghiorso 1989; Lee and Ganguly 1988; Chatillon-Colinet et al. 1983; Aranovich 1991; von Seckendorff and O’Neill 1993). Both one-site and twosite models have been applied with success in these studies, and one objective of this study was to compare results of these different models.
10
1-Site model With the one site model, orthopyroxene is treated as a simple mixture of the components MgSiO3, FeSiO3 and Al2O3 (orthocorundum) such that: aen5(XOpx Mg gEn) aFs5(XOpx gFs) Fe aOk5(XOpx Al2O3 gOk)
with activity coefficients given by Eq. 6. This model has been shown to be capable of reproducing the major phase relationships involving Opx (Saxena 1981; Aranovich and Kosyakova 1987; Aranovich and Podlesskii 1989; Aranovich 1991; Lee and Ganguly 1988), and has the advantage of not being dependent on the uncertainties in Mg-Fe site occupancy constraints (e.g. Saxena 1983; Hawthorne 1983). The 1-site model is equivalent to a 2-site model with complete Fe-Mg disorder (equal partitioning) between M1 and M2 sites. Volumes for MAS and FAS Opx were constrained by available V-X data for orthopyroxene on the joins En-Ok (Stephenson et al. 1966; Chatterjee and Schreyer 1972; Danckwerth and Newton 1978; Doroshev et al. 1983) and Fs-Ok (Aranovich and Berman 1995). The data available on the volume behavior of Fe-Mg Opx do not show any significant deviation from linearity (e.g. Turnock et al. 1973; Chatillon-Colinet et al. 1983; Fonarev 1987; Hayob et al. 1993). Early on in this project it became evident that the one-site model taken in its simplest form (a regular solution), although capable of reproducing the major phase relationships involving Opx in the FMS and MAS subsystems (see discussion under Opx subheadings below), was insufficient to reproduce available experimental constraints on Al2O3 solubility in Opx in the more complex system FMAS without unjustifiable enlargment of the experimental uncertainties. Because of the very limited range of Opx Al2O3 contents covered by experimental studies, we extended the above model by incorporation of the Darken quadratic formalism (DQF). The DQF model was proposed by Darken (1967a) and shown to work extremely well for analytical representation of mixing properties of components in dilute binary solid solutions even for the case when solute and solvent are of significantly different nature (e.g. C dissolved in metall alloys). This model was introduced to the petrological literature by Aranovich (1983, see also Appendix in Perchuk et al. 1985) and has been succesively applied by Powell (1987) and Will and Powell (1992). The DQF was derived for binary solutions (Darken 1967a), and Darken (1967b) and Aranovich (1991) expanded it for the case of an arbitrary number of components dissolved in one and the same solvent. In binary systems, this model effectively treats two terminal regions near each endmember as regular solutions involving a real and fictive endmember (Powell 1987). Our application of the DQF model to the region of dilute solubility of Al2O3 in Opx provides additional de-
grees of freedom in our analysis by utilizing endmember properties of “real” Al2O3 in MAS Opx and “imaginary” Al2O3 in FAS Opx. For ternary FMAS Opx, Darken’s (1967b) equations were modified to account for solution of one component (Al2O3) in two different solvents (En and Fs). For En and Fs, the corresponding expressions are those given by Eq. 6 above. For Al2O3, activity coefarg ules ficients given by Eq. (6; RTlng M ) are modified as: Al2O3 arg ules RT ln gAl2O35RT ln g M 1Fey(Fe1Mg)IFe–Al Al2O3
(7)
with IFe–Al expanded to IFe–Al5IH2T ? IS1(P21) IV (Turkdogan and Darken 1968).
2-Site model The two site model with non-convergent ordering of Mg and Fe atoms has been discussed in detail by Thompson (1969), Sack (1980) and Sack and Ghiorso (1989) for binary Fe-Mg Opx and by Kawasaki and Matsui (1983) and Aranovich and Kosyakova (1987) for the ternary FeMg-Al Opx. The notation of Sack and Ghiorso (1989) is used below in discussing results with this model.
Results The primary variable with respect to thermodynamic models that was explored in this study was the difference between a one-site and two-site model for orthopyroxene. A somewhat surprising conclusion was that the 1-site model with a symmetric, temperature-dependent excess free energy reproduced the overall set of experimental observations more closely than the 2-site model. We found that 2-site models that were calibrated with either the reversed Mössbauer data of Anovitz et al. (1988) or the in situ single crystal data of Yang and Ghose (1994) could not reproduce all Fe-Mg exchange data for Ol-Opx and Gt-Opx when these are adjusted for maximum compositional uncertainties of 0.01 Mgy (Mg1Fe). Nor were we able with these calibrations to obtain adequate representation of the Ol-Opx-Qz divariant field data. We found that use of different site occupancy data (e.g. Virgo and Hafner 1969; Saxena and Ghose 1971; Besancon 1981; Domeneghetti and Steffen 1992), and introduction of temperature-dependent excess free energies on both sites did not substantially change these results. In contrast, the 1-site model with temperature-dependent, symmetric free energy of mixing reproduces most of the Ol-Opx exchange data of von Seckendorff and O’Neill (1993) with 0.005 Mgy (Mg1Fe) adjustments that are more in accord with their quoted uncertainties, as well as the Ol-Opx-Qz data with excellent fidelity (see below in Fe-Mg exchange section). On the basis of this modelling we conclude that the phase equilibrium data involving Fe-Mg Opx and the intracrystalline partitioning data are somewhat at odds with one another.
11
The most likely reason for this conflict is that the available reversed Mössbauer data define site occupancies over a more limited temperature and compositional range compared to the phase equilibrium data, and analytical interpolations and extrapolations from these data appear to yield a poorer base level from which to make empirical activity-composition corrections than the equal partitioning assumption. Gross inaccuracy of Mössbauer data (e.g. Seifert 1989) is precluded by reTable 4 Standard state thermodynamic properties derived in this study
Minal
DfH1,298 (kJymol)
Almandine Anthophyllite Cordierite Fe-Cordierite Fayalite Ferrosilite Fosterite Geikelite Hematite Hercynite Ilmenite Iron-a Iron-g Magnetite Orthoenstatite Orthocorundum Pyrope Rutile Spinel Talc
25265.44 212074.47 29161.48 28430.55 21477.17 21192.91 22174.42 21570.52 2826.74 21945.36 21233.32 0.00 7.72 21116.96 21546.04 21634.95 26284.74 2944.75 22302.16 25898.96
cent single crystal in situ determinations (Yang and Ghose 1994) which offered general support for the Mössbauer determinations of Anovitz et al. (1988). Yang and Ghose (1994) reported a polymorphic transition in Opx at about 9508 C which may further complicate explicit account of cation occupancies over the 700– 14008 C temperature range of the phase equilibrium. Comparison of the 1-site and 2-site results with respect to derived olivine solution properties (see below in
S1,298 (JyK ? mol) (1)a (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1)
341.51 537.40 416.23 482.18 151.73 96.47 94.18 74.41 87.36 123.13 108.50 27.45 35.64 146.04 66.18 33.93 268.80 50.88 80.37 261.21
V1,298 (Jybar ? mol) (2)a (4) (6) (8) (10, 11) (13) (15) (17) (18) (1) (19) (12) (1) (20) (21) (1) (22) (24) (25) (26)
11.529 26.330 23.311 23.706 4.639 3.295 4.360 3.077 3.027 4.080 3.170 0.709 0.691 4.452 3.133 3.123 11.311 1.883 3.978 13.610
(2, 3)a (5) (7) (9) (3) (14) (16) (7) (7) (7) (7) (7) (7) (7) (16) (1) (23) (7) (23) (16)
a
(1) Values derived in this study from analysis of phase equilibrium data; other values refined with phase equilibrium data are based on measurements or estimates of: (2) Anovitz et al. 1993; (3) Chatillon-Colinet et al. 1983; (4) Hemingway 1991; (5) Hirschmann et al. 1994; (6) Weller and Kelley 1963; (7) Robie et al. 1967; (8) Holland 1989; (9) Mukhodpadhay and Holdaway 1994; (10) Robie et al. 1982a; (11) Essene et al. 1980; (12) Engi and Feenstra (submitted for publication); (13) Bohlen et al. 1983; (14) Sueno et al. 1976; (15) Robie et al. 1982b; (16) Chernosky et al. 1985; (17) Shomate 1946; (18) Gronvold and Westrum 1959; (19) Anovitz et al. 1985; (20) Westrum and Gronvold 1969; (21) Krupka et al. 1985; (22) Haselton and Westrum 1980; (23) Charlu et al. 1975; (24) Shomate 1947; (25) King 1955; (26) Stout and Robie 1963.
Table 5 Heat capacity coefficients (JyK ? mol)a
Minal
k0
k1 (31022)
k2 (31025)
k3 (31027)
Refb
CP5k01k1T0.51k2T21k3T3 (Berman and Brown 1985). b (1) Fit to data of Anovitz et al. (1993); (2) fit to data summarized by Berman (1988); (3) estimated values; (4) fit to data of Gillet and Fiquet (1991); (5) taken from Berman and Brown (1985); (6) taken from analysis of Engi and Feenstra (submitted for publication); (7) fit to data of Tequi et al. (1991); (8) taken from Chamberlin et al. (1995). c Lambda transition terms given by Berman (1988) d Lambda transition terms (Jymol) using equations 8–9 of Berman (1988): l1520.22208; 51042 K; l250.00034823; T1bar l k520.00057; DtH51000.
Almandine Anthophyllite Cordierite Fe Cordierite Fayalite Ferrosilite Forsterite Geikelite Hematitec Hercynite Ilmenite Iron-ad Iron-g Magnetitec Orthoenstatite Orthocorundum Pyrope Rutile Spinel Talc
621.43 1233.79 954.39 983.48 252.00 174.20 233.18 146.20 146.86 251.77 150.00 51.87 66.24 207.93 166.58 119.38 590.90 77.84 244.67 664.11
232.879 271.340 279.623 284.037 220.137 213.930 218.016 24.160 0.000 220.444 24.416 23.794 212.371 0.000 212.006 7.748 228.270 0.000 220.040 251.872
2150.810 2221.638 223.173 218.703 0.000 24.544 0.000 239.998 255.768 213.483 233.237 225.430 63.733 272.433 222.706 265.091 2133.208 233.678
221.187 233.394 237.021 28.568 26.219 23.771 226.794 40.233 52.563 13.150 34.815 50.680 2106.040 66.436 27.915 42.288 126.033 40.294
221.472
232.737
(1) (2) (2) (3) (2) (2) (4) (5) (2) (3) (2) (6) (6) (2) (2) (3) (7) (2) (8) (2)
a
12 Table 6 Volume coefficientsa
VP,TyV1,2985v1(P-1) 1v2(P-1)21v3(T-298) 1v4(T-298)2 (units for v1-v4: K21; K22; b21; b22). b (1) Taken from, or fit to data summarized by Berman (1988); (2) estimated values; (3) values equal to ilmenite; (4) taken from analysis of Engi and Feenstra (submitted for publication); (5) assumed equal to orthoenstatite. a
Table 7 Mixing properties derived in this study
Minal
v1 (3106)
v2 (31012)
Refb
v3 (3106)
v4 (31010)
Refb
Almandine Anthophyllite Cordierite Fe Cordierite Fayalite Ferrosilite Forsterite Geikelite Hematite Hercynite Ilmenite Iron-a Iron-g Magnetite Orthoenstatite Orthocorundum Pyrope Rutile Spinel Talc
20.570 21.139 21.158 21.700 20.822 20.911 20.791 20.529 20.479 20.510 20.529 20.602 20.467 20.582 20.749 20.749 20.576 20.454 20.489 21.847
0.434 0.000 0.000 0.000 1.944 0.303 1.351 0.000 0.304 0.000 0.000 0.000 21.445 1.751 0.447 0.447 0.442 0.585 0.000 5.878
(1) (1) (1) (2) (1) (1) (1) (3) (1) (2) (1) (4) (4) (1) (1) (5) (1) (1) (1) (1)
18.599 28.105 3.003 4.265 26.210 31.406 29.464 23.314 38.310 15.819 23.314 45.071 47.153 30.291 24.656 24.656 22.519 25.716 21.691 25.616
74.711 62.894 18.017 0.000 84.233 80.400 88.633 88.328 1.650 96.276 88.328 14.104 14.451 138.470 74.670 74.670 37.044 15.409 50.528 0.000
(1) (1) (1) (2) (1) (1) (1) (3) (1) (2) (1) (4) (4) (1) (1) (5) (1) (1) (1) (1)
WH (Jymol)
WS (JyK ? mol)
WV (Jybar ? mol)
Phase
Parameter
Garnet 15Ca 25Mg 35Fe
112 122 113 133 223 233 123
33470.00 68280.00 21951.40 11581.50 5064.50 6249.10 73298.30
18.79 18.79 9.43 9.43 4.11 4.11 32.33
0.173 0.036 0.170 0.090 0.010 0.060 0.281
Cordierite 15Mg 25Fe
12
21626.70
0.00
0.000
Olivine 15Mg 25Fe
12
10366.20
4.00
0.011
Ilmenite 15Mg 25Fe
12
1525.30
0.00
0.010
Orthopyroxene 15Mg 25Fe 35Al
12 13 23
22600.40 221878.40 232398.40
21.34 0.00 0.00
0.000 20.386 20.883
IFe-Al (Eq. 7)
Derived solution properties section) demonstrates that the two models yield remarkably similar macroscopic energetics for the othopyroxene solid solution. In view of the results summarized above our preferred thermodynamic properties given in Tables 4–7 were obtained with the 1-site Opx model. Details of the analysis with this model and comparison of calculated and observed phase relationships are discussed below. Equilibria constraining endmember thermodynamic properties Figures 2 and 3 summarize the major constraints imposed by phase equilibrium data on Alm, Fa, Fs, fCd, and Hc endmember thermodynamic properties. The experimental data for these equilibria are in good accord with available thermophysical data and tightly constrain
26534.900
16.11110
0.175
the differences in standard state properties among these phases. Apparent discrepancies between the calculated slope and experimental brackets for the Fs5Fa1Qz equilibrium are within the small uncertainties (+300 bars) of the data. Our calculations support the CP data of almandine measured by Anovitz et al. (1993) over the drop calorimetry results reported by Newton and Harlov (1993). All the phase equilibrium data involving almandine shown in Figs. 2 and 3 were obtained at reducing conditions buffered at the Fe-Wüstite (Wst) assemblage, except for the equilibrium: Alm13Hm53Mt1Ky12Qz
(j)
which Harlov and Newton (1992) found to occur approximately 4 kbar below the position computed with thermodynamic data of Berman (1988). The position of equilibrium (j) depends on almandine properties, which Berman (1988) derived from the Fe-Wst buffered data
13
Fig. 2 Comparison of available experimental data with univariant curves in the system FeO-Al2O3-SiO2 computed with thermodynamic data derived in this study. Symbols as in Fig. 1
(Fig. 2), and on the difference between Hm and Mt, which Berman (1988) based on the data of Myers and Eugster (1983). Using Robinson et al.’s (1982) evaluation of Hm and Mt properties, Anovitz et al. (1993) also were unable to reproduce the Harlov and Newton data while fitting the Fe-Wst buffered phase equilibrium data. One possibility to explain the above discrepancy is that almandine contained significantly more Fe31 in the Hm-Mt buffered than the Fe-Wst buffered experiments. Woodland and O’Neill (1993) determined solution of approximately 15% skiagite (Fe3Fe2Si3O12) component in almandine at 30 kbar and 11008 C in the presence of Fe31-bearing slags. At the lower temperatures of Harlov and Newton’s experiments (650–9008 C), reduced skiagite solution would not account for the above discrepancy. In addition, Harlov and Newton’s (1992) microprobe analyses of almandine show only minor Fe13 solubility. These observations suggest that the most likely source of discrepancy is in the calibration of the Hm5Mt1O2 buffer, which is kinetically sluggish (e.g. O’Neill 1988) and which has yielded highly discordant results in different studies (summarized by O’Neill 1988). With Hm properties determined by the position of Harlov and Newton’s experiments, the computed position of the Hm-Mt equilibrium is in perfect agreement with O’Neill’s (1988) data between 1000 and 1150 K, offering strong support for O’Neill’s contention that his 1000–1173 K data represent equilibrium values for the HM buffer. As discussed by O’Neill, deviations above 1173 K are consistent with increased and well documented Fe2O3 solubility in Mt, while those below 1000 K can
Fig. 3 Comparison of available experimental data with univariant curves in the system CaO-FeO-Al2O3-TiO2-SiO2 computed with thermodynamic data derived in this study. Symbols (as in Fig. 1) are for (from top to bottom): 2Fs5Fa1Qz (filled diamonds), Gr12 Alm16Rt56Ilm13An13Qz (open triangles), and Gr12Alm 53An13Fa (open squares; filled triangles). Note that, at 3.9 kbar, Grafchikov and Fonarev (1986) produced growth of both products and reactants in different experiments
be attributed to the extreme sluggishness of the Hm-Mt reaction. Koziol’s (1990) reversed compositons of Gr-Alm garnets in equilibrium with anorthite-kyanite-quartz effectively define excess mixing properties on the Gr-Alm binary, although the magnitude of any excess entropy on this join is not constrained by these data. When combined with equilibria that involve Gr33Alm67 garnet (Bohlen et al. 1983a; Bohlen and Liotta 1986; Grafchikov and Fonarev 1986; Perkins and Vielzeuf 1992), more stringent bounds are placed on both almandine properties and excess entropy on the Gr-Alm join (Anovitz and Essene 1987; Berman 1990; Perkins and Vielzeuf 1992). If the brackets of Bohlen et al. (1983a) for equilibrium (T): Fa1An5Gr1Alm (Fig. 3) are accepted without adjustment beyond their reported P-T uncertainties (+300 bars and 58 C), the minimum symmetric excess entropy is about 5 Jmol21K21. Further relaxation of the lowest temperature (7008 C) half-bracket by as little as 200 bars permits zero excess entropy, although the slope of their brackets is most consistent with positive excess entropy on this join. Perkins and Vielzeuf (1992) report brackets for equilibrium (T) between 900 and 10508 C that support Bohlen et al.’s (1983a) reversals in this temperature range. Grafchikov and Fonarev (1986) also obtained relevant data in their study of the univariant equilbrium: Cpx1An5Gt1Fa1Qz which is degenerate to equilibri-
14
um (T) at the Gr33Alm67 composition. They pointed out that equilibrium (T) must be tangential to their univariant curve at the P-T point of compositional degeneracy which they estimate to be 7508 C. At higher and lower temperatures the univariant curve must be at lower pressure than equilibrium (T). Figure 3 shows that their most constraining half-brackets, nominally at 4.1 kbar and 7508 C, are mariginally compatible with Bohlen et al.’s data considering uncertainties of both experimental data sets. Grafchikov and Fonarev’s preferred equilibrium location is at 3.9 kbar and 7508 C, based on growth of products and reactants of equilibrium (T) in two different experiments at this P-T. We favor the Grafchikov and Fonarev bracket because the data were obtained using hydrothermal pressure vessels, whereas Bohlen et al.’s lowest temperature piston cylinder data may require a slight friction correction even for the low-strength NaCl pressure assemblies they used. Direct experimental resolution of the conflicting 7508 C brackets would be extremely useful. Although we allowed for a lower pressure than Bohlen et al.’s 750 and 8008 C brackets, our optimized thermodynamic properties yield a excess entropy (9.43 JyKymol) in between estimates of Anovitz and Essene (1987) and Berman (1990). Considerable controversy has existed about the P-T location and slope of the equilibrium: 2Alm14Si15Qz53fCd
(g)
in the FAS system (e.g. Currie 1971; Thompson 1976; Holdaway and Lee 1977; Aranovich and Podlesskii 1981, 1983). On the basis of their experimental data in the FMAS system, Aranovich and Podlesskii deduced a positive P-T slope for equilibrium (g, Table 2) that has been confirmed by direct hydrothermal reversals obtained by Mukhopadhyay and Holdaway (1994). Our calculations are in good accord with both the new FAS data (Fig. 3) and the FMAS data discussed below. Fe-Mg exchange and net transfer equilibria Existing Gt-Opx Fe-Mg exchange data are in good overall agreement within experimental uncertainties. Figures 4a–b compare computed lnKD values with experimental data from four different studies. All experimental observations are reproduced within 0.01 Mgy(Mg1Fe) uncertainties, with the exception of one experiment ('163; shaded square in Fig. 4a) of Lee and Ganguly (1988), which was omitted from the MAP analysis because both Gt and Opx increased in Mg' during this experiment. The combined analysis constrains the enthalpy change at 258 C–1 bar for equilibrium (A) between 36.2 and 40.2, with an optimal value of 40.1 kJmol21. Larger values that have been obtained elsewhere (e.g. Lee and Ganguly 1988; Aranovich 1991), and which yield a greater temperature dependence for this exchange equilibrium, are not permitted by the combination of constraints on endmember equilibria, entropies, and volumes. The enthalpy change of this exchange equilibrium estimated solely
Fig. 4 Variation of lnKD for equilibrium (A) with Fey(Fe1Mg) ratio in Opx. Curves were computed with thermodynamic data derived in this study at the temperatures for which experimental data were collected (numbers and symbols at end of each curve) and a pressure of 25 kbar. Experimental observations have been normalized to the same pressure. Symbols show nominal experimentally determined lnKD values. Opposite ends of connected lines show lnKD values adjusted for 0.01 uncertainties in XFe of both Opx and Gt. Placement of lines above (below) symbols indicates equilibrium was approached from high (low) to low (high) KD. Computed curves are consistent with all data within these uncertainties, except for run '163 (shaded square) of Lee and Ganguly (1988) in which Gt and Opx show inconsistent compositional changes
from experimental data involving endmembers (33.61 kJmol21: Berman 1988) is outside of the range determined above, a comparison that underscores the importance of deriving thermodynamic data for endmembers from the combination of data involving endmembers and solid solutions. The exchange data for Gt-Cd (equilibrium C) are reproduced using 0.015 compositional adjustments for Gt and Cd, consistent with the amount of compositional overlap observed by Aranovich and Podlesski (1983). The computed two-phase loops at 700 and 7508 C are also in good agreement with the data (Fig. 5). At 10 kbar and 10008 C, however, our calculated two-phase loop is somewhat more Fe-rich than Hensen’s (1977) bracket. This small discrepancy may be resolved by assuming a smaller friction correction than the 10% value commonly assumed for talc pressure assemblies (e.g. Gasparik and Newton 1984). All the exchange data for Opx-Cd (equilibrium B) were fitted within 0.01 uncertainties, with the exception of run '39y3 which produced a large compositional overlap with other runs. This datum was also not fitted by Aranovich and Kosyakova (1987) in their analysis. The most constraining data set analyzed in this study is that on the olivine-orthopyroxene exchange (equilibrium E) combined with the olivine-orthopyroxene-quartz net transfer equilibria (O) and (P). Figure 6 demonstrates
15
Fig. 5 Comparison of computed P-XFe divariant loops for the assemblage Cd1Gt1Si1Qz (solid curves 7008 C, dashed 7508 C) with experimental observations by Aranovich and Podlesskii (1983) (filled squares 7008 C, open squares 7508 C). Symbols as in Fig. 4 using pressure and compositional uncertainties of 200 bars and 0.015 XFe
that all but one ('15, shaded symbol in Fig. 6b) of the exchange experiments of von Seckendorff and O’Neill (1993) are reproduced to within 0.005 Mgy(Mg1Fe), in excellent agreement with their quoted uncertainties. These data also agree with the other Ol-Opx exchange data accepted in this study within estimated uncertainties (see constraints on solid-solution properties section). Figure 7 shows that the tabulated thermodynamic properties for Ol and Opx afford excellent representation of the divariant field determinations of Bohlen and Boettcher (1981), Fonarev (1987), and Davidson and Lindsley (1989). In order to achieve this agreement, however, it was necessary to adjust properties of Mg and Fe endmembers given by Berman (1988) while maintaining consistency with all phase equilibrium data constraining these endmember properties (Table 2), and to use the bulk composition constraints discussed above to fit each half-bracket with the proper directional sense. Our overall reproduction of the phase equilibrium data was also improved by use of the new drop calorimetry measurements for forsterite (Richet and Fiquet et al. 1991) and pyrope (Richet et al. 1991). It should be noted as well that, in contrast with the value given by Berman (1988), the enthalpy difference of the reaction: 3 En1Co5Py resulting from the present analysis falls within the 1 s uncertainties of calorimetric measurements of Eckert et al. (1992). von Seckendorff and O’Neill (1993) have shown that their experimental data are most compatible with a
Fig. 6a–c Variation of lnKD for the equilibrium Fo12Fs52En 1Fa (E) with Fey(Fe1Mg) ratio in Opx. Curves were computed using thermodynamic data derived in this study at experimental temperature values and P516 kbar. Symbols as in Fig. 4 using compositional uncertainties of 0.005 XFe for both Opx and Ol. Computed curves are consistent within +0.005 XFe for all data
smaller entropy change for the Ol-Opx exchange equilibrium than calculated with either the database of Berman (1988) or Holland and Powell (1990). The new heat capacity data summarized above markedly reduce this discrepancy (Fig. 8). The offset of von Seckendorff and O’Neill’s DG vs T curve to lower values than ours is due to the strong correlation of standard state and mixing properties combined with their preference for a more positive heat of mixing in olivine than derived here. Our results suggest that the disagreement of the Sack and Ghiorso (1989) calibration with experimental data at low pressure on the Ol-Opx-Qz loop (Fig. 7b), as well as with von Seckendorf and O’Neill’s (1993) exchange data, results from their acceptance of (a) calorimetric data (Chatillon-Colinet et al. 1983) and activity measurements (Sharma et al. 1987) to fix a much more positive heat of mixing for orthopyroxene than obtained in this study, and (b) endmember properties fixed at values given by Berman (1988). Phase equilibrium data obtained by Hayob et al. (1993) on the Opx1Il1Rt1Qz assemblage place very tight constraints on the standard and mixing properties of Opx and Il through operation of the Fe-Mg exchange equilibrium (F) as well as the Fe and Mg net transfer
16
Fig. 7a, b Comparison of computed P-XFe divariant loops for the assemblage Opx1Ol1Qz at 10008 C (a) and 8008 C (b) with available experimental data (P-T uncertainties not shown). Solid curves were computed with thermodynamic data derived in this study. Dashed curves were computed with thermodynamic data from Sack and Ghiorso (1989). Filled symbols show bracketing data after adjustment for 0.01 XFe uncertainties in Opx and Ol. Opposite ends of connected lines show starting compositions. All filled triangles within divariant loops indicate starting and final compositions within divariant loops. Additional experimental data from Bohlen and Boettcher (1981) indicate: Opx stable (X), Ol stable (1), Opx (open square) or Ol (open diamond) broke down to divariant assemblage
equilibria (Table 3). As noted by Hayob et al., use of the Berman (1988) or Berman (1990) endmember properties does not allow perfect agreement with their data, but the simultaneous determination of mixing properties and standard state properties in this study leads to much improved results. Opx and Il compositions are all reproduced within the uncertainties determined by Hayob et al. (Fig. 9). Apparent discrepancies in Fig. 10, particularly at 11008 C, are caused by ilmenite compositions being calculated assuming no Fe31, while experimentally determined compositions are reported to have up to 1 mol% hematite. The calculated lnKD for the Opx-Il exchange equilibrium ranges from 15 to about 6 over the temperature and compositional interval covered by the experiments, in good agreement with Hayob et al.’s interpretations. The slightly larger KD values derived here stem from incorporation of phase equilibrium constraints on the Ol-Il Fe-Mg exchange equilibrium (Fig. 10). Derived KD values are in excellent agreement with Andersen and Lindsley’s (1979) data below 10008 C, but
Fig. 8 Temperature dependence of the standard Gibbs free energy of exchange equilibrium (E). Curves were computed with thermodynamic data from Berman (1988), from von Seckendorff and O’Neill (1993), and from this study. Note similarity in slope between the latter two sets of calculations, as well as their offset which reflects differences in derived mixing properties
are at the lower limit of Andersen et al.’s (1991) 12008 C data. Equilibria controlling Al2O3 solubility in Opx Sluggishness of the equilibration of Al2O3 in orthopyroxene together with overstepping of equilibrium compositions combine to yield significant uncertainties (10– 50%) in equilibrium compositions. In the MAS system, computed Opx Al2O3 isopleths (Fig. 11) reproduce within their uncertainties the Al2O3 solubility data of Danckwerth and Newton (1978), Perkins et al. (1981), and Kawasaki and Matsui (1983) for the Gt-Opx assemblage. In order to allow for slight curvature in these isopleths, we allowed for a small constant Cmax change of equiliP brium (H). The above data are in excellent agreement with reversed Al2O3 contents of Opx in equilibrium with Cd1Qz in the MAS system at 750–8508 C and 1–5 kbar (Aranovich et al. 1983). Treated by themselves, the data for equilibria (H) and (J) allow the En-Ok join to be ideal, while the range of values permissible for WOpx MgAl when all data are considered simultaneously is 22.5 to 228 kJmol21. Our preferred value resulting from final optimization is 222 kJmol21. Use of this Al2O3 solubility model permits good representation of MAS univariant equilibria involving orthopyroxene at both high pressure with pyrope and low pressure with cordierite (Fig. 11). Not shown in Fig. 11 are brackets for equilibria (b), (c), and (d) that are also well represented.
17
Fig. 10 Variation of lnKD for the equilibrium 2Gk1Fa5Fo12Ilm (G) with Fey(Fe1Mg) ratio in Ol. Curves were computed using thermodynamic data derived in this study at experimental temperatures and P51 kbar (700–10008 C) and 1 bar (12008 C). Symbols as in Fig. 4 using compositional uncertainties of 0.01 XFe for both Il and Ol
Fig. 9a–d Comparison of computed P-XFe divariant loops for the assemblage Opx1Rt1Il1Qz with experimental data of Hayob et al. (1993). Symbols as in Fig. 4 using reported uncertainties of 0.02 XFe for Opx (squares) and Il (triangles). Note that ¥0.01 Fe31 in the experimental ilmenite improves agreement with the displayed calculations which were performed on Fe31-free basis
Calculated isopleths of Al2O3 in the spinel peridotite field (Fig. 11) are in excellent agreement with the measurements of Gasparik and Newton (1984). In contrast to Berman’s (1988) account of an unconstrained amount of disorder is spinel with use of a fixed increment to the measured third law entropy, here we utilize the activity measurements and CP function of Chamberlin et al. (1995) to describe thermodynamic properties of spinel with an equilibrium amount of disorder. Gibbs free energy values for spinel retrieved in this study are in exceptional agreement (,1 kJymol difference) with those derived by Chamberlin et al. (1995). Recent reversed experiments of Aranovich and Berman (1995, 1996a) on the solubility of Al2O3 in Opx in equilibrium with garnet in the FAS system between 12 and 20 kbar are critical due to the lack of other reversals for high Fey(Fe1Mg) Opx. The positions of isopleths fixed by these data (Fig. 12) are in excellent accord with
Fig. 11 Comparison of experimental data with univariant curves (bold) and Opx Al2O3 isopleths (light curves with mol% Al2O3 in hexagons) in the system MgO-Al2O3-SiO2 computed with thermodynamic data derived in this study. Symbols as in Fig. 1. Dashed curve is the position of the Fo1mCd5Sp1En equilibrium with anhydrous cordierite. Numbers inside squares, triangles, and circles give experimental values of mol% Al2O3 in Opx
18
Fig. 12 Comparison of computed Opx Al2O3 isopleths with experimentally determined values (Aranovich and Berman 1995; 1996a) in the FAS system. Rectangles show P-T, with uncertainties, of experiments with Xok values approached from high and low-Al Opx starting materials. Solid, half-filled, and open rectangles represent experimental products with the assemblages AlmOpx, Alm-Opx-Fa-Qz, and Alm-Fa-Qz, respectively
the lone other FAS reversal of Kawasaki and Matsui (1983) at 46 kbar. Although the combination of MAS and FAS Al2O3 solubility reversals are also in good accord with the three FMAS Gt-Opx reversals of Lee and Ganguly (1988; Fig. 13a), they require significantly lower Al2O3 contents than the measurements of Aranovich and Kosyakova (1987) at much lower P-T for the OpxCd-Qz assemblage (Fig. 13b). Use of asymmetric andyor temperature-dependent mixing functions for Opx did not allow better agreement between the data for the two different assemblages. Our predicted values are up to 50% lower than the data of Aranovich and Kosyakova. Net transfer equilibria (H) and (I) control the solubility of Al2O3 in Opx in equilibrium with garnet to considerably smaller amounts (except for the most Mg-rich compositions), and with the opposite dependence on Mgy (Mg1Fe), compared to the Opx-Cd-Qz assemblage (equilibria J and K). Consideration of all data simultaneously constrains the difference WOpx 2WOpx in the MgAl FeAl 21 range from 23.1 to 12.4 kJmol . The optimized difference (10.5 kJmol21), in agreement with thermodynamic analyses on Opx and other silicates (Aranovich 1991; Mader and Berman 1992; Berman et al. 1995), displays the affinity of Fe for Al and is an important correction for exchange thermometers involving Opx.
Derived solution properties The comparisons discussed above indicate that tabulated thermodynamic data (Tables 4–7) provide excellent representation of the experimental data within their estimated uncertainties. In this section, we discuss the derived solution properties of each mineral in the context of independent data sets that provide constraints on these properties as well as the results of previous thermodynamic modelling.
Olivine
Fig. 13a, b Variation of Opx Al2O3 mole fraction versus Fey (Fe1Mg) of Opx for the assemblages Opx1Gt (a) and Opx1Cd1Qz (b). Curves are computed with thermodynamic data derived in this study. Symbols show experimental data adjusted for 20% uncertainties in Al2O3 contents. Opposite ends of connected lines show starting Al2O3 contents. Only sets of experiments in which equilibrium was approached from under- and oversaturation are shown in (a). Numbers by curves and references in (a) give the pressure (kbar)ytemperature (8 C) at which the curves were computed and the data collected. All experiments approached equilibrium from undersaturation in (b)
Because solution properties for all phases derived from phase equilibrium data are highly correlated with one another, we begin with a sensitivity analysis of olivine solution properties in order to establish the range of mixing properties allowable by the combined set of experimental data. By far the most constraining data set is that of von Seckendorff and O’Neill (1993). Consideration of their Fe-Mg exchange data together with experiments involving endmembers fixes the range of WOl (1000 K) G between 21.4 and 4.7 kJyatom. The shift in this range to somewhat lower values than that computed by von Seckendorff and O’Neill (2–8 kJyatom) is due to our consideration of the constraints on endmembers which, by defining the standard free energy change of the exchange equilibrium, limit the range of permissible mixing properties. Addition of the Ol-Opx-Qz divariant field data
19 Ol G
narrows our calculated range of W (1000 K) only slightly (21.1 to 4.6 kJyatom). In contrast to the above results with Ol and Opx, consideration of the other data sets involving Ol provide much less stringent bounds: 214.8 to 19.8 kJyatom for the Ol-Il and Gt-Ol exchange data considered simultaneously. When these two data sets are combined, however, with all other exchange data not including Ol-Opx, the resulting WOl (1000 K) range (0.7 to 4.2 kJyatom) is G in remarkable agreement with that permitted by the OlOpx data alone (21.4–4.7 kJyatom). Moreover, similar ranges for WOl (1000 K) result with use of a 2-site Opx G model (21.6 to 5.6 kJyatom). Even when all data sets involving Opx are excluded in entirety, essentially the same WOl (1000 K) range (21.6 to 5.6 kJyatom) results, G demonstrating that conclusions regarding olivine mixing properties are not biased by our assumptions regarding the Opx solid solution. Using the 1-site model, the WOl G (1000 K) range allowed by all experimental data taken together is 0.7 to 4.1 kJyatom. Our preferred free energy of mixing for Fe-Mg olivine solid solution at 1000 K (WOl 53.2 kJyatom) is in very good agreement with reG cent estimates (3.7+0.8 kJyatom, Wiser and Wood 1991; 5.6+0.6 kJyatom, von Seckendorff and O’Neill 1993; 1.5+0.3, Berman et al. 1995). Our somewhat lower value than von Seckendorff and O’Neill is within their calculated 95% confidence limits for this value. We found that a small excess entropy term (WOl 52.0 JyK-atom) significantly improved the overall S representation of phase equilibrium data. This is also more plausible from a molecular physics standpoint. The so-called “first approximation” to the regular solution theory (e.g. Gokcen 1982; Aranovich 1991) suggests that excess entropy can be neglected only for the solutions with abs(WHyRT) ,, 1. Excess entropy also improves the correspondence of our derived excess enthalpy (WH55.18 kJyatom) with calorimetric measurements of the enthalpy of mixing. Calibration of a symmetric mixing model to the data of Wood and Kleppa (1981) results in WH57.08+1.8 kJyatom, but von Seckendorff and O’Neill (1993) pointed out that exclusion of one outlier in Wood and Kleppa’s data leads to a significantly smaller enthalpy of mixing (WH54.6 kJyatom). This latter value agrees with more recent data of Kojitani and Akaogi (1994) that yield WH55.3+1.7 kJyatom. Our Gex is significantly smaller than that of Sack and Ghiorso (1989), who based their fitting on calorimetric constraints combined with a limited set of phase equilibrium and natural partitioning data (see discussion in von Seckendorff and O’Neill 1993, p204). It is also not large enough to produce unmixing at low temperatures which has been proposed to explain the occurrence of mm-scale lamellae in olivine grains of the Divnoe meteorite (Petaev and Brearley 1994). Evans and Ghiorso (1995) used this explanation as support for the larger nonideality of the Sack and Ghiorso (1989) olivine model. The compositions of these lamellae range between Fa23 and Fa29 with a difference of 1–1.5 mol% Fa between adjacent lamellae. The formation of such Mg-rich
compositions by exsolution is inconsistent with the high temperature phase equilibrium and calorimetric data summarized above which both suggest symmetric excess mixing energetics. Similarly, Sack and Ghiorso’s (1989) olivine calibration predicts a symmetric solvus. Because lamellae are not found in all olivine grains of the Divnoe meteorite (Petaev et al. 1994; Steele and Aranovich, unpublished data) and because exsolution has not been found in olivine from more slowly cooled meteorites (Petaev et al. 1994), the origin of these lamellae remains mysterious, and may relate more to the Divnoe meteorite’s particular deformation history than to an inherent instability of olivine at low temperature. Garnet The mixing properties of garnet derived in this study indicate small positive deviations from ideality across the Fe-Mg join, in excellent agreement with other results based on recent experimental studies (Lee and Ganguly 1988; Hackler and Wood 1989; Koziol and Bohlen 1992) and with the model of Berman (1990). The preferred values derived here show the same sense of asymmetry found in the latter two experimental studies, in the volume and heat of mixing on this join (Geiger et al. 1987), as well as in more restricted thermodynamic assessments of Gt-Opx (Berman 1990) and Gt-Cpx (Berman et al. 1995) exchange data. The allowance of small excess entropy on the Mg-Fe join improves representation of the high temperature Gt-Opx exchange data (see above Results section) and also results in predicted excess enthalpy values in accord (within uncertainties) with calorimetric heat of mixing data of Geiger et al. (1987). These results contrast greatly with larger positive deviations from ideality deduced by Ganguly and Saxena (1984) from analysis of natural assemblages and by Sack and Ghiorso (1989) from analysis of Gt-Opx exchange experiments combined with their Opx solution model. As discussed above, our present calculations suggest that excess entropy on the Gr-Alm is midway between the value of Berman (1988) and that proposed by Anovitz and Essene (1987) on the basis of Cressey et al.’s (1978) phase equilibrium data. The optimized value (9.43 Jy K ? mol) also leads to reasonable agreement with the excess enthalpy measurements of Geiger et al. (1987). Cordierite Previous work aimed at assessing cordierite stability relations assumed that Fe-Mg mixing is ideal (Perchuk and Lavrent’eva 1983; Aranovich and Podlesskii 1983; Aranovich and Kosyakova 1987). In this study we found that adequate representation of the exchange data and the divariant Gt-Cd loop, together with tight brackets on the Fe endmember equilibrium (Mukhopadhyay and Holdaway 1994) was most compatible with small negative deviations from ideality on this join (WCd 521.63 kJymol). H
20
Orthopyroxene Phase equilibrium data for the seven exchange couples considered in this study are compatible with a symmetric, temperature dependent excess free energy of mixing in Fe-Mg orthopyroxene. The increasing value of WG with increasing temperature agrees with the conclusions of Hayob et al. (1993) based on their experimental observations. They note that, although their preferred WH value (3.6+4.9 kJymol) is positive, slightly negative values are also compatible within the uncertainties of their phase equilibrium data. The same temperature dependence is also implied by available site occupancy data (Shi et al. 1992). The small predicted negative excess free energies on the Fs-En join differ markedly from the results of EMF measurements at 1000 K reported by Sharma et al. (1987), who obtained large positive Gex for the entire Fe-Mg compositional range studied. Shortcomings of the technique used by these authors, however, have been recently discussed by von Seckendorff and O’Neill (1993). Although the difference WOl 2WOpx determined G G in this study is almost identical to that determined by the latter authors (¥4 kJyatom), our value for WOpx is more G negative than their preferred value due to the smaller WOl G determined in this study. Our calibration yields negative enthalpies of mixing for Opx that contrast with the results of solution calorimetry data at 7508 C (WH53966 Jymol; ChatillonColinet et al. 1983). It should be noted, however, that the calorimetric measurements involve only three Fe-Mg samples. Results are consistent with ideal mixing for two of these samples within 1 s uncertainties, and for all three samples within 2 s uncertainties. Chatillon-Colinet et al.’s (1983) enthalpy of ordering experiments also indicate that if any of these samples retained a degree of disorder from their ¥11208 C synthesis temperature, their heat of mixing would be more positive than that for samples with an equilibrium ordering state. We consider this possibility unlikely, however, in the light of the kinetic data of Anovitz et al. (1988) which suggest that an equilibrium state of order is attained in less than the 2–3 hours for samples which were equilibrated at 7508 C prior to the oxide-melt solution measurements. It should also be recognized, that, while the phase equilibrium data analyzed in this study tightly constrain the Gibbs free energy of mixing of Opx, the data do not provide a robust separation of the excess enthalpy and entropy. The negative deviations from ideality for both Hex and Sex obtained in the present study are in accord with ordered nature of Fe-Mg orthopyroxene, and agree well with the findings of Sack (1980) and Aranovich and Kosyakova (1987), who modelled Opx as a two-site solution with non-convergent ordering. The discrepancy between the present results and those of Sack and Ghiorso (1989) is entirely related to the very large positive excess enthalpy of olivine accepted by these authors. Recent experimental work (see above) does not support this value.
Our calculations support the conclusions of various experimental studies (Harley 1984; Kawasaki and Matsui 1983; Aranovich and Kosyakova 1987) with respect to the important effect of aluminum on the mixing properties of the orthopyroxene solid solution. Negative values of the enthalpy of mixing on both enstatite – “orthocorundum” and ferrosilite – “orthocorundum” joins reflect a tendency for the solid solutions to form 1: 1 intermediate compounds, although the fact that the Mg-Al join could be represented with ideal mixing indicates that this tendency is probably slight for the Mg system. The more negative value for WOpx suggests that ideal Fe-Al mixing models describing Al solubility in FMAS orthopyroxene are in error. Given the relatively small standard energy changes of the exchange reactions involving this mineral (for equilibrium (A), DrH01,298540.1 kJy mol), the difference between WOpx and WOpx is an Mg-Al Fe-Al important factor in the distribution of Fe and Mg between Opx and other minerals. For example, temperatures computed with equilibrium (A) are as much as ¥1008 C higher when this correction is omitted for samples with high Al2O3 Opx. In view of the simplified, regular solution model applied here to describe the mixing properties of aluminous Opx, both mixing and standard properties of the “ortho-Al2O3” component should be considered only as empirical fitting parameters. Nevertheless, our analysis found that the existing experimental phase equilibrium data are represented as well with this model as with models based on Tschermak’s components, and the present model has the computational advantage of not requiring explicit specification of site occupancies. Ilmenite Our retrieved mixing parameters for ilmenite-geikelite solution involve symmetric excess entropy and enthalpy, in reasonable agreement with that determined by Andersen et al. (1991). The somewhat larger absolute values of these authors is related to their significantly higher estimates for the (asymmetric) entropy of mixing, and also to their acceptance of a larger WOl (7 kJyatom G compared to 3.2 kJyatom at 1000 K obtained in this study). Direct comparison of the present results with Ghiorso’s (1990) is prevented by the differences in the models applied.
Conclusions A particularly satisfying result of this study is that the derived thermodynamic properties given in Tables 4 and 5 reproduce the Fe-Mg exchange data for all seven exchange couples (Gt-Opx, Gt-Ol, Gt-Cd, Opx-Ol, OpxCd, Opx-Il, Ol-Il) within very reasonable estimates of overall uncertainties. This is particularly encouraging in as much as only four of these exchange couples are independent, with experimental data for the ohter three couples providing a check, and added refinement of the
21
derived endmember and mixing properties. Moreover, the final optimization reproduces the extremely tight constraints on endmember properties while invoking very simple solution models that afford an excellent opportunity for extrapolation beyond the data considered in this study. The combination of experimental data on endmembers and solid solutions define nonideal mixing parameters in the order WOl .WIlm .WGt .WOpx .WCd , G G H G H Ol with 0.7,WG (1000 K),4.1 kJyatom of isomorphous Fe-Mg. The results of experiments in MAS and FAS on the Al2O3 solubility of Opx in equilibrium with Gt and with Cd1Qz are in good agreement with the few reversals involving FMAS Gt but not the unreversed data for FMAS Cd. Further experimental work in the FMAS system is warranted to corroborate the applicability of the present analysis to predicting Al2O3 contents of FMAS Opx at crustal conditions (see part II of this paper, Aranovich and Berman 1996b). We caution users of this newly derived data set, that unlike the Berman (1988) data, which were derived largely without consideration of mixing properties, the present thermodynamic data for endmembers should only be applied quantitatively to solid solutions in conjunction with the presently derived mixing properties. To the extent that this study has correctly separated mixing and standard energies, other calibrations of different mixing models with these standard state properties should be in closer accord with experimental observations than the data given earlier by Berman (1988). Use of the present standard properties with different mixing properties would destroy the internal consistency of our calculations, however, and can only be justified if the user performs calculations with different mixing properties that demonstrate compatibility with all relevant experimental observations. Applications of this data set to high-grade metamorphic rocks are described in a companion paper (Aranovich and Berman 1996b). Acknowledgements We acknowledge and applaud the great number of experimentalists from different labs around the world who produced the large set of high-quality phase equilibrium data utilized in this study. We are grateful to IGCP project '304 for providing LYA with the opportunity to visit Canada, Terry Gordon for introducing the authors to each other and helping to initiate this project, and the Geological Survey of Canada for financial assistance to LYA. LYA also thanks V.A. Zharikov for providing a 2year leave from Chernogolovka in order to fulfill the project. Helpful reviews of E. Froese and M. Hirschmann are much appreciated. This is Geological Survey of Canada contribution '41394.
References Andersen DJ, Lindsley DH (1979) The olivine-ilmenite thermometer. Proc 10th Lunar Planet Sci Conf, Geochim Cosmochim Acta suppl. 1: 493–507 Andersen DJ, Bishop FC, Lindsley DH (1991) Internally consistent solution models for Fe-Mg-Mn-Ti oxides: Fe-Mg-Ti oxides and olivine. Am Mineral 76: 427–444 Anovitz LM, Essene EJ (1987) Compatibility of geobarometers in the system CaO-FeO-Al2O3-SiO2-TiO2 (CFAST): implications for garnet mixing models. J Geol 95: 633–645 Anovitz LM, Treiman AH, Essene EJ, Hemingway BS, Westrum EFJ, Wall VJ, Burriel R, Bohlen SR (1985) The heat capacity
of ilmenite and phase equilibrium in the system Fe-Ti-O. Geochim Cosmochim Acta 49: 2027–2040 Anovitz LM, Essene EJ, Dunham WR (1988) Order-disorder experiments on orthopyroxenes: implications for orthopyroxene speedometer. Am Mineral 73: 1060–1073 Anovitz LM, Essene EJ, Metz GW, Bohlen SR, Westrum Jr EF, Hemingway BS (1993) Heat capacity and phase equilibria of almandine, Fe3Al2Si3O12. Geochim Cosmochim Acta 57: 4191–4204 Aranovich LY (1983) Biotite – garnet equilibria in metapelites. I. Thermodynamics of solid solutions and minal reactions. In: Zharikov VV, Fed’kin V (eds) Ocherki fiziko chimicheskoi petrologii. Nauka Press, Moscow, pp 14–33 Aranovich LY (1991) Mineral equilibria of multicomponent solid solutions. Nauka Press, Moscow Aranovich LY, Berman RG (1995) Al2O3 solubility in orthopyroxene in equilibrium with almandine in the FeO-Al2O3-SiO2 system. Current Res Geol Surv Can Pap 1995-E:271–278 Aranovich LY, Berman RG (1996a) A new orthopyroxene-garnet thermometer, based on reversed Al2O3 solubilities in orthopyroxene in the FeO-Al2O3-SiO2 system. Am Mineral (in press) Aranovich LY, Berman RG (1996b) Optimized standard state and solution properties of minerals: II. Comparisons, predictions, and applications. Contrib Mineral Petrol 126: 25–37 Aranovich LY, Kosyakova NA (1987) The cordierite5orthopyroxene1quartz equilibrium: laboratory data on and thermodynamics of ternary Fe-Mg-Al orthopyroxene solid solutions. Geochem Int 24: 111–131 Aranovich LY, Pattison DRM (1995) Reassessment of the garnetclinopyroxene Fe-Mg exchange thermometer: I. Reanalysis of the Pattison and Newton (1989) run products. Contrib Mineral Petrol 119: 16–29 Aranovich LY, Podlesskii KK (1981) Experimental study of the garnet1sillimanite1quartz5cordierite equilibrium. Dokl Akad Nauk SSSR 281: 1440–1443 Aranovich LY, Podlesskii KK (1983) The cordierite-garnet-sillimanite-quartz equilibrium: experiments and calculations. In: Saxena SK (ed) Kinetics and equilibrium in mineral reactions. Springer, Berlin Heidelberg New York, pp 173–198 Aranovich LY, Podlesskii KK (1989) Geothermobarometry of highgrade metapelites: simultaneously operating reactions. In: Cliff RA, Yardley BWD, Daly JS (eds) Evolution of metamorphic belts. Blackwell Scientific Publications, Oxford, pp 45–62 Aranovich LY, Kosyakova NA, Van KV (1983) Experimental study of the equilibrium cordierite1orthopyroxene1quartz in the system MgO-Al2O3-SiO2 (in Russian). Dokl Akad Nauk SSSR 269: 1174–1177 Aranovich LY, Belonozhko AB, Kosyakova NA, Kopylov PA (1985) The technique for computation of thermodynamic parameters of solid solutions from data on phase equilibria using non-linear least square method. Geochem Int 7: 1052–1054 Berman RG (1988) Internally-consistent thermodynamic data for minerals in the system Na2O-K2O-CaO-MgO-FeO-Fe2O3Al2O3-SiO2-TiO2-H2O-CO2. J Petrol 29: 445–522 Berman RG (1990) Mixing properties of Ca-Mg-Fe-Mn garnets. Am Mineral 75: 328–344 Berman RG, Brown TH (1984) A thermodynamic model for multicomponent melts, with application to the system CaO-Al2O3SiO2. Geochim Cosmochim Acta 45: 661–678 Berman RG, Brown TH (1985) The heat capacity of minerals in the system Na2O-K2O-CaO-MgO-FeO-Fe2O3-Al2O3-SiO2TiO2-H2O-CO2: representation, estimation, and high temperature extrapolation. Contrib Mineral Petrol 85: 168–183 Berman RG, Koziol AM (1991) Ternary excess properties of grossular-pyrope-almandine garnet and their influence in geothermobarometry. Am Mineral 76: 1223–1231 Berman RG, Engi M, Greenwood HJ, Brown TH (1986) Derivation of internally-consistent thermodynamic data by the technique of mathematical programming, a review with application to the system MgO-SiO2-H2O. J Petrol 27: 1331–1364 Berman RG, Aranovich LY, Pattison DRM (1995) Reanalysis of the garnet – clinopyroxene Fe-Mg exchange thermometer. II. Thermodynamic analysis. Contrib Mineral Petrol 119: 30–42
22 Besancon JR (1981) Rate of cation disordering in orthopyroxenes. Am Mineral 66: 965–973 Bishop FC (1976) Partitioning of Fe21 and Mg between ilmenite and some ferromagnesium silicates. PhD Thesis, University of Chicago, Chicago, Illinois, 137 p Bishop FC (1979) Mg-Fe21 partitioning between olivine and ilmenite. Trans Amer Geophys Union 60: 420 Boberski C, Schreyer W (1990) Synthesis and water contents of Fe21-bearing cordierites. Eur J Mineral 2: 565–584 Bohlen SR, Boettcher AL (1981) Experimental investigations and geological applications of orthopyroxene geobarometry. Am Mineral 66: 951–964 Bohlen SR, Liotta JJ (1986) A barometer for garnet amphibolites and garnet granulites. J Petrol 27: 1331–1364 Bohlen SR, Essene EJ, Boettcher AL (1980) Reinvestigation and application of olivine-orthopyroxene-quartz barometry. Earth Planet Sci Lett 47: 1–10 Bohlen SR, Metz GW, Essene EJ, Anovitz LM, Westrum EFJ, Wall VJ (1983) Thermodynamics and phase equilibrium of ferrosilite as a potential oxygen barometer in mantle rocks. EOS Trans Am Geophys Union 64: 350 Bohlen SR, Wall VJ, Boettcher AL (1983a) Experimental investigation and application of garnet granulite equilibria. Contrib Mineral Petrol 83: 52–61 Bohlen SR, Wall VJ, Boettcher AL (1983b) Experimental investigations and geological applications of equilibria in the system FeO-TiO2-Al2O3-SiO2-H2O. Am Mineral 68: 1049–1058 Bohlen SR, Dollase WA, Wall VJ (1986) Calibration and applications of spinel equilibria in the system FeO-Al2O3-SiO2. J Petrol 27: 1143–1156 Bohlen SR, Montana A, Kerrick DM (1991) Precise determinations of the equilibria kyanite5sillimanite and kyanite5andalusite and a revised triple point for Al2SiO5. Am Mineral 76: 677–680 Brey GP, Kohler T (1990) Geothermobarometry in four-phase lherzolites II. New thermobarometers, and practical assessment of existing thermobarometers. J Petrol 31: 1353–1378 Carlson WD, Lindsley SH (1988) Thermochemistry of pyroxenes in the join Mg2Si2O6-CaMgSi2O6. Am Mineral 73: 242–252 Chamberlin L, Beckett JR, Stolper E (1995) Palladium oxide equilibration and the thermodynamic properties of MgAl2O4 spinel. Am Mineral 80: 285–296 Charlu TV, Newton RC, Kleppa OJ (1975) Enthalpies of formation at 970 K of compounds in the system MgO-Al2O3-SiO2 from high temperature solution calorimetry. Geochim Cosmochim Acta 39: 1487–1497 Chatillon-Colinet C, Newton RC, Perkins D III, Kleppa OJ (1983) Thermochemistry of (Fe21, Mg)SiO3 orthopyroxene. Geochim Cosmochim Acta 47: 1597–1603 Chatterjee ND, Schreyer W (1972) The reaction enstatite1sillimanite5sapphirine1quartz in the system MgO-Al2O3-SiO2. Contrib Mineral Petrol 36: 49–62 Chernosky JVJ, Day HW, Caruso LJ (1985) Equilibria in the system MgO-SiO2-H2O: I Experimental determination of the stability of Mg-anthophyllite. Am Mineral 70: 223–236 Chou I-M (1978) Calibration of oxygen buffers at elevated P and T using the hydrogen fugacity sensor. Am Mineral 63: 690– 703 Cressey G, Schmid R, Wood BJ (1978) Thermodynamic properties of almandine-grossular garnet solid solutions. Contrib Mineral Petrol 67: 397–404 Currie KL (1971) The reaction: 3 cordierite52 garnet14 sillimanite15 quartz as a geological thermometer in the Opinicon Lake region, Ontario. Contrib Mineral Petrol 33: 215–226 Danckwerth A, Newton RC (1978) Experimental determination of the spinel peridotite to garnet peridotite reaction in the system MgO-Al2O3-SiO2 in the range 900–11008 C and Al2O3 isopleths of enstatite in the spinel field. Contrib Mineral Petrol 66: 189–201 Darken LS (1967a) Thermodynamics of binary metallic solutions. Trans Metal Soc AIME 239: 80–89 Darken LS (1967b) Thermodynamics of ternary metallic solutions. Trans Metall Soc AIME 239: 90–96
Davidson PM, Lindsley DH (1989) Thermodynamic analysis of pyroxene-olivine-quartz equilibria in the system CaO-MgOFeO-SiO2. Am Mineral 74: 18–30 Domeneghetti MC, Steffen G (1992) M1, M2 site populations and distortion parameters in synthetic Mg-Fe orthopyroxenes from Mössbauer spectra and X-ray structure refinements. Phys Chem Mineral 19: 298–306 Doroshev AM, Turkin AI, Kuznetscov GN, Malinovskii IY (1983) Synthesis and unit cell parameters of the orthopyroxene in the system MgO-Al2O3-Cr2O3-SiO2. Zap Vseso Mineral Obschestva 112: 363–367 Eckert JO, Bohlen SR (1992) Reversed experimental determinations of the Mg-Fe21 exchange equilibrium in Fe-rich garnetorthopyroxene pairs. EOS Trans Am Geophys Union 73: 608 Eckert JO, Newton RC, Kleppa OJ (1992) Enthalpy change of the reaction 3y2 enstatite1corundum5pyrope. Geochim Cosmochim Acta 56: 3971–3978 Engi M (1987) Towards a thermodynamic data base for geochemistry: consistency and optimal representation of the stability relations in mineral systems. Habilitation, University of Bern Essene EJ, Wall VJ, Westrum EFJ (1980) Thermodynamic properties and phase equilibria for fayalite. Geol Soc Am Abstr Prog 58: 422 Evans BW, Ghiorso MS (1995) Thermodynamics and petrology of cummingtonite. Am Mineral 80: 649–663 Fawcett JJ, Yoder HSJ (1966) Phase relationships of the chlorites in the system MgO-Al2O3-SiO2-H2O. Am Mineral 51: 353– 380 Feenstra A, Peters T (1996) Experimental determination of activities in FeTiO3-MnTiO3 ilmenite solid solution by redox reversals. Contrib Mineral Petrol 126: 121–132 Fei Y, Saxena SK (1986) A thermochemical data base for phase equilibria in the system Fe-Mg-Si-O at high pressure and temperature. Phys Chem Mineral 13: 311–324 Fonarev VI (1981) Thermodynamic functions of olivine and orthopyroxene solid solutions. Geochem Int 18: 142–157 Fonarev VI (1987) Mineral equilibria in Precambrian iron formations. Nauka Press, Moscow Fonarev VI, Korol’kov GJ (1976) Olivine-orthopyroxene-quartz geobarometer (experimental data). Dokl Akad Nauk SSSR 238: 687–690 Ganguly J, Saxena SK (1984) Mixing properties of aluminosilicate garnets: constraints from natural and experimental data, and applications to geothermo-barometry. Am Mineral 69: 88– 97 Ganguly J, Chen W, O’Neill HSC (1993) Syntheses, volume, and structural changes of garnets in the pyrope-grossular join: implications for stability and mixing properties. Am Mineral 78: 583–593 Gasparik T, Newton RC (1984) The reversed alumina contents of orthopyroxene in equilibrium with spinel and forsterite in the system MgO-Al2O3-SiO2. Contrib Mineral Petrol 85: 186–196 Geiger CA, Newton RC, Kleppa OJ (1987) Enthalpy of mixing of synthetic almandine-grossular and almandine-pyrope garnets from high-temperature solution calorimetry. Geochim Cosmochim Acta 51: 1755–1763 Ghiorso MS (1990) Thermodynamic properties of hematite-ilmenite-geikielite solid solutions. Contrib Mineral Petrol 104: 645–667 Gillet P, Fiquet G (1991) High-temperature thermodynamic properties of forsterite. J Geophys Res 96: 11805–11816 Gokcen NA (1982) Multicomponent regular solutions. Scri Metall 16: 723–727 Grafchikov AA, Fonarev VI (1986) Experimental evidence on coexisting garnet, clinopyroxene and quartz in the FeO-CaOAl2O3-SiO2-H2O system. Geochem Int 63: 158–169 Gronvold F, Westrum EFJ (1959) a-ferric oxide: low temperature heat capacity and thermodynamic functions. J Am Chem Soc 81: 1780–1783 Hackler RT, Wood BJ (1989) Experimental determination of Fe and Mg exchange between garnet and olivine and estimation of Fe-Mg mixing properties in garnet. Am Mineral 74: 994– 999
23 Harley SL (1984) The solubility of alumina in orthopyroxene coexisting with garnet in FeO-MgO-Al2O3-SiO2 and CaO-FeOMgO-Al2O3-SiO2. J Petrol 25: 665–696 Harlov DE, Newton RC (1992) Experimental determination of the reaction 2magnetite12kyanite14quartz52almandine1O2 at high pressure on the magnetite-hematite buffer. Am Mineral 77: 558–564 Harlov DE, Newton RC (1993) Reversal of the metastable kyanite1corundum1quartz and andalusite1corundum1quartz equilibria and the enthalpy of formation of kyanite and andalusite. Am Mineral 78: 594–606 Haselton HTJ, Westrum EJF (1980) Low-temperature heat capacities of synthetic pyrope, grossular, and pyrope60grossular40. Geochim Cosmochim Acta 44: 701–709 Hawthorne FC (1983) Quantitative characterization of site-occupancies in minerals. Am Mineral 68: 287–306 Hayob JL, Bohlen SR, Essene EJ (1993) Experimental investigation and application of the equilibrium rutile1orthopyroxenee5quartz1ilmenite. Contrib Mineral Petrol 115: 18– 35 Helffrich G, Wood BJ (1989) Subregular model for multicomponent solutions. Am Mineral 74: 1016–1022 Hemingway BS (1991) Thermodynamic properties of anthophyllite and talc – corrections and discussion of calorimetric data. Am Mineral 76: 1589–1596 Hensen BJ (1977) Cordierite-garnet bearing assemblages as geothermometers and barometers in granulite facies terranes. Tectonophysics 43: 73–88 Herzberg CT (1983) The reaction forsterite1cordierite5aluminous orthopyroxene1spinel in the system MgO-Al2O3-SiO2. Contrib Mineral Petrol 84: 84–90 Hirschmann M, Evans BW, Yang H (1994) Composition and temperature dependence of octahedral-site ordering in cummingtonite-grunerite as determined by X-ray diffraction. Am Mineral 79: 862–877 Holdaway MJ, Lee SM (1977) Fe-Mg cordierite stability in high-grade pelitic rocks based on experimental, theoretical, and natural observation. Contrib Mineral Petrol 63: 175– 198 Holland TJB (1989) Dependence of entropy on volume for silicate and oxide minerals: a review and a predictive model. Am Mineral 74: 5–13 Holland TJB, Powell R (1990) An internally consistent thermodynamic dataset with uncertainties and correlations: the system K2O-Na2O-CaO-MgO-MnO-FeO-Fe2O3-Al2O3-TiO2-SiO2-CH2-O2. J Metamorphic Geol 8: 89–124 Kawasaki T, Matsui Y (1983) Thermodynamic analyses of equilibria involving olivine, orthopyroxene and garnet. Geochim Cosmochim Acta 47: 1661–1679 King EG (1955) Heat capacities at low temperatures and entropies at 298.16 K of crystalline calcium and magnesium aluminates. J Phys Chem 59: 218–219 Koch-Muller M, Cemic L, Langer K (1992) Experimental and thermodynamic study of Fe-Mg exchange between olivine and orthopyroxene in the system MgO-FeO-SiO2. Eur J Mineral 4: 115–135 Kojitani H, Akaogi M (1994) Calorimetric study of olivine solid solutions in the system Mg2SiO4-Fe2SiO4. Phys Chem Mineral 20: 536–540 Koziol AM (1990) Activity-composition relationship of binary Ca-Fe and Ca-Mn garnets determined by reversed, displaced phase equilibrium experiments. Am Mineral 75: 319–327 Koziol AM, Bohlen SR (1992) Solution properties of almandinepyrope garnet as determined by phase equilibrium experiments. Am Mineral 77: 765–773 Koziol AM, Newton RC (1989) Grossular activity-composition relationships in ternary garnets determined by reversed, displaced phase equilibrium experiments. Contrib Mineral Petrol 103: 423–433 Krupka KM, Robie RA, Hemingway BS, Kerrick DM, Ito J (1985) Low-temperature heat capacities and derived thermodynamic properties of anthophyllite, diopside, enstatite, bronzite, and wollastonite. Am Mineral 70: 249–260
Lane DL, Ganguly J (1980) Al2O3 solubility in orthopyroxene in the system MgO-Al2O3-SiO2: a reevaluation, and mantle geotherm. J Geophys Res 85: 6963–6972 Larimer JW (1968) Experimental studies on the system Fe-MgOSiO2-O2 and their bearing on the petrology of chondritic meteorites. Geochim Cosmochim Acta 32: 1187–1207 Lee HY, Ganguly J (1988) Equilibrium compositions of coexisting garnet and orthopyroxene: experimental determinations in the system FeO-MgO-Al2O3-SiO2, and applications. J Petrol 29: 93–113 Mader UK, Berman RG (1992) Amphibole thermobarometry: a thermodynamic approach. Current Res Geol Surv Can Pap 921E:393–400 Matsui Y, Nishizawa O (1974) Iron(II)-magnesium exchange equilibrium between olivine and calcium-free pyroxene over a temperature range 8008 C to 13008 C. Bull Soc Fr Mineral Cristallogr 97: 122–130 McPhail DC, Berman RG, Greenwood HJ (1990) Experimental and theoretical constraints on aluminum substitution in magnesian chlorite, and a thermodynamic model for H2O in magnesian cordierite. Can Mineral 28: 859–874 Medaris LG Jr (1969) Partitioning of Fe and Mg between coexisting synthetic olivine and orthopyroxene. Am J Sci 267: 945–968 Mukhopadhyay B, Holdaway MJ (1994) Cordierite-garnet-sillimanite-quartz equilibrium: I. New experimental calibration in the system FeO-Al2O3-SiO2-H2O and certain P-T-XH2O relations. Contrib Mineral Petrol 116: 462–472 Myers J, Eugster HP (1983) The system Fe-Si-O: oxygen buffer calibrations to 1,500 K. Contrib Mineral Petrol 82: 75–90 Newton RC (1972) An experimental determination of the highpressure stability limits of magnesian cordierite under wet and dry conditions. J Geophys Res 80: 398–420 Newton RC, Harlov DE (1993) Standard thermodynamic properties of almandine. Can Mineral 31: 391–399 Newton RC, Charlu TV, Kleppa OJ (1977) Thermochemistry of high pressure garnets and clinopyroxenes in the system CaOMgO-Al2O3-SiO2. Geochim Cosmochim Acta 41: 369–377 O’Neill HSC (1987) Quartz-fayalite-iron and quartz-fayalitemagnetite equilibria and the free energy of formation of fayalite (Fe2SiO4) and magnetite (Fe3O4). Am Mineral 72: 67– 75 O’Neill HSC (1988) Systems Fe-O and Cu-O: thermodynamic data for the equilibria Fe-“FeO”, Fe-Fe3O4, “FeO”-Fe3O4, Fe3O4Fe2O3, Cu-Cu2O, and Cu2O-CuO from emf measurements. Am Mineral 73: 470–486 O’Neill HSC, Wood BJ (1979) An experimental study of Fe-Mg partitioning between garnet and olivine and its calibration as a geothermometer. Contrib Mineral Petrol 70: 59–70 Ottonello G, Princivalle F, Della Giusta A (1990) Temperature, composition, and fO2 effects on interstite distribution of Mg and Fe21 in olivines: experimental evidence and theoretical interpretation. Phys Chem Mineral 17: 301–312 Pattison DRM, Newton RC (1989) Reversed experimental calibration of the garnet-clinopyroxene Fe-Mg exchange thermometer. Contrib Mineral Petrol 101: 87–103 Perchuk LL (1991) Derivation of a thermodynamically consistent set of geothermometers and geobarometers for metamorphic and magmatic rocks. In: Perchuk LL (ed) Progress in metamorphic and magmatic petrology. Cambridge University Press, Cambridge, pp 93–111 Perchuk LL, Lavrent’eva IV (1983) Experimental investigation of exchange equilibria in the system cordierite-garnet-biotite. In: Saxena SK (ed) Kinetics and equilibrium in mineral reactions. Springer, Berlin Heidelberg New York, pp 199–239 Perchuk LL, Aranovich LY, Podlesskii KK, Lavrent’eva IV, Gerasimov VY, Kitsul VI, Korsakov LP, Berdnikov NV (1985) Precambrian granulites of the Aldan shield, eastern Sibiria, the USSR. J Metamorphic Geol 3: 265–310 Perkins D (1983) The stability of Mg-rich garnet in the system CaO-MgO-Al2O3-SiO2 at 1000–13008 C and high pressure. Am Mineral 68: 355–364 Perkins D, Newton RC (1980) The compositions of coexisting pyroxenes and garnet in the system CaO-MgO-Al2O3-SiO2 at
24 900–1000 deg. C and high pressures. Contrib Mineral Petrol 75: 291–380 Perkins D, Vielzeuf D (1992) Reinvestigation of fayalite1anorthite5garnet. Contrib Mineral Petrol 111: 260–263 Perkins D, Holland TJB, Newton RC (1981) The Al2O3 contents of enstatite in equilibrium with garnet in the system MgO-Al2O3SiO2 at 15–40 kbar and 900–16008 C. Contrib Mineral Petrol 78: 99–109 Petaev MI, Brearley AJ (1994) Exsolution in ferromagnesian olivine of the Divnoe Meteorite. Science 266: 1545–1547 Petaev MI, Barsukova LD, Lipschutz ME, Wang M-S, Arishkin AA, Clayton RN, Mayeda TK (1994) The Divnoe meteorite: petrology, chemistry, oxygen isotopes and origin. Meteoritics 29: 182–199 Powell R (1987) Darken’s quadratic formalism and thermodynamics of minerals. Am Mineral 72: 1–11 Richet P, Fiquet G (1991) High-temperature heat capacity and premelting of minerals in the system. MgO-CaO-Al2O3-SiO2. J Geophys Res 96: 445–456 Robie RA, Bethke PM, Beardsley KM (1967) Selected x-ray crystallographic data, molar volumes, and densities of minerals and related substances. US Geol Surv Bull 1248 Robie RA, Finch CB, Hemingway BS (1982a) Heat capacity and entropy of fayalite (Fe2SiO4) between 5.1 and 383 K: comparison of calorimetric and equilibrium values for the QFM buffer reaction. Am Mineral 67: 463–469 Robie RA, Hemingway BS, Takei H (1982b) Heat capacities and entropies of Mg2SiO4, Mn2SiO4 and Co2SiO4 between 5 and 380 K. Am Mineral 67: 470–482 Robinson GRJ, Haas JLJ, Schafer CM, Haselton HTJ (1982) Thermodynamic and thermophysical properties of selected phases in the MgO-SiO2-H2O-CO2, CaO-Al2O3-SiO2-H2O-CO2, and Fe-FeO-Fe2O3-SiO2 chemical systems, with special emphasis on the properties of basalt. US Geol Surv Open File Report 83–79 Sack RO (1980) Some constraints on the thermodynamic mixing properties of Fe-Mg orthopyroxenes and olivine. Contrib Mineral Petrol 71: 2257–2269 Sack RO, Ghiorso MS (1989) Importance of considerations of mixing properties in establishing an internally consistent thermodynamic database: thermochemistry of minerals in the system Mg2SiO4-Fe2SiO4-SiO2. Contrib Mineral Petrol 102: 41– 68 Saxena SK (1981) The MgO-Al2O3-SiO2 system: free energy of pyrope and Al2O3-enstatite. Geochim Cosmochim Acta 45: 821–825 Saxena SK (1983) Problems of two-pyroxene geothermometry. Earth Planet Sci Lett 65: 382–388 Saxena SK, Ghose S (1971) Mg21-Fe21 order-disorder and the thermodynamics of the orthopyroxene crystalline solution. Am Mineral 56: 532–559 Schwab RG, Kustner D (1977) Prazisionsgitterkonstantenbestimmung zur Festlegung rontgenographischer Bestimmungskurven für synthetischen Olivin der Mischkristallreihe ForsteritFayalit. Neues Jahrb Mineral Monatsh 1977: 205–215 Seifert F (1974) Stability of sapphirine: a study of the aluminous part of the system MgO-Al2O3-SiO2-H2O. J Geophys Res 82: 173–204 Seifert S (1989) Recent advances in the mineralogical applications of the 57Fe Mosbauer effect. In: Salje EKH (ed) Physical properties and thermodynamic behaviour of minerals. Reidel, Dordrecht, pp 687–703 Sharma KC, Agrawal RD, Kapoor ML (1987) Determination of thermodynamic properties of (Fe,Mg)-pyroxenes at 1000 K by the emf method. Earth Planet Sci Lett 85: 302–310 Shi PF, Saxena SK, Sundman B (1992) Sublattice solid solution model and its application to orthopyroxene (Mg,Fe)2Si2O6. Phys Chem Mineral 18: 393–405 Shomate CH (1946) Heat capacities at low temperatures of the metatitanates of iron, calcium and magnesium. J Am Chem Soc 68: 964–966 Shomate CH (1947) Heat capacities at low temperatures of titanium dioxide (rutile and anatase). J Am Chem Soc 69: 218–219
Shomate CH, Naylor BF, Boericke FS (1946) Thermodynamic properties of ilmenite and selective reduction of iron in ilmenite. US Geol Surv Rep Invest 3864 Shulters JC, Bohlen SR (1989) The stability of hercynite and hercynite-gahnite spinels in corundum- or quartz-bearing assemblages. J Petrol 30: 1017–1031 Stephenson DA, Schar CB, Smith IV (1966) Unit cell volumes of synthetic orthoenstantite and low clinoenstantine. Mineral Mag35: 834–846 Stout JW, Robie RA (1963) Heat capacity from 11 to 300 K, entropy and heat of formation of dolomite. J Phys Chem 67: 2248–2252 Sueno S, Cameron M, Prewitt CT (1976) Orthoferrosilite: high temperature crystal chemistry. Am Mineral 61: 38–53 Tequi C, Robie RA, Hemingway BS, Neuville DR, Richet P (1991) Melting and thermodynamic properties of pyrope (Mg3Al2Si3O12). Geochim Cosmochim Acta 55: 1005–1010 Thompson AB (1976) Mineral reactions in pelitic rocks: 1. Prediction of P-T-X(Fe-Mg) phase relations. Am J Sci 276: 401–424 Thompson JB Jr (1969) Chemical reactions in crystals. Am Mineral 54: 341–375 Turkdogan ET, Darken LS (1968) Thermodynamics of binary metallic solutions. Part II. Trans Metal Soc AIME 242: 1997–2005 Turnock AC, Lindsley DH, Grover JE (1973) Synthesis and unit cell parameters of Ca-Mg-Fe pyroxenes. Am Mineral 58: 50–59 Virgo D, Hafner SS (1969) Fe21-Mg – order-disorder in heated orthopyroxenes. Mineral Soc Am Spec Pap 2: 67–81 von Seckendorff V, O’Neill HSC (1993) An experimental study of Fe-Mg partitioning between olivine and orthopyroxene at 1173, 1273 and 1473 K and 1.6 Gpa. Contrib Mineral Petrol 113: 196–207 Wechsler BA, Prewitt CT (1984) Crystal structure of ilmenite (FeTiO3) at high temperature and at high pressure. Am Mineral 69: 176–185 Weller WW, Kelley KK (1963) Low-temperature heat capacities and entropies at 298.15 K of a˚ kermanite, cordierite, gehlenite, and merwinite. US Geol Surv Rep Invest 6343 Westrum EFJ, Gronvold F (1969) Magnetite (Fe3O4). Heat capacity and thermodynamic properties from 5 to 350 K, low temperature transition. J Chem Thermodyn 1: 543–557 Will TM, Powell R (1992) Activity-composition relationships in multicomponent amphiboles: an application of Darken’s quadratic formalism. Am Mineral 77: 954–966 Wiser NM, Wood BJ (1991) Experimental determination of activities in Fe-Mg olivine at 1400 K. Contrib Mineral Petrol 108: 146–153 Wohl K (1946) Thermodynamic evaluation of binary and ternary liquid systems. Trans Am Inst Chem Eng 42: 215–249 Wohl K (1953) Thermodynamic evaluation of binary and ternary liquid systems. Chem Eng Prog 49: 218–219 Wood BJ (1987) Thermodynamics of multicomponent systems containing several solid solutions. In: Carmichael ISE, Eugster HP (eds) Thermodynamic modelling of geologic systems: minerals, fluids and melts. (Mineralogical Society of America Short Course). Mineralogical Society of America, Washington DC, pp 71–96 Wood BJ (1988) Activity measurements and excessvolume relationships for pyrope-grossular garnets. J Geol 296: 721–729 Wood BJ, Holloway JR (1984) A thermodynamic model for subsolidus equilibria in the system CaO-MgO-Al2O3-SiO2. Geochim Cosmochim Acta 48: 159–176 Wood BJ, Kleppa OJ (1981) Thermochemistry of forsterite-fayalite olivine solutions. Geochim Cosmochim Acta 45: 569–581 Woodland AB, O’Neill HSC (1993) Synthesis and stability of 31 Fe21 3 Fe2 Si3O12 garnet and phase relations with Fe3Al2Si3O12 31 2Fe21 Fe 3 2 Si3O12 solutions. Am Mineral 78: 1002–1015 Yang H, Ghose SH (1994) In-situ Fe-Mg order-disorder studies and thermodynamic properties of orthopyroxenes, (Mg,Fe)2Si2O6. Am Mineral 79: 633–643 Zhu H, Newton RC, Kleppa OJ (1994) Enthalpy of formation of wollastonite (CaSiO3) and anorthite (CaAl2Si2O8) by experimental phase equilibrium measurements and high-temperature solution calorimetry. Am Mineral 79: 134–144
