Contrib Mineral Petrol (1995) 119:197-212 , 9 Springer-Verlag 1995

Mark S. Ghiorso 9Richard O. Sack Chemical mass transfer in magmatic processes IV. A revised and internally consistent thermodynamic model for the interpolation and extrapolation of liquid-solid equilibria in magmatic systems at elevated temperatures and pressures

Received: 1 February 1994 / Accepted: 26 August 1994

Abstract A revised regular solution-type thermodyn- systems open to oxygen are detcrmined by directly amic model for twelve-component silicate liquids in the specifying the/o2 or the T-P-fo~ (or equivalently H-P-f;~, system SiOa-TiO2 A1203-Fe203-Cr203-FeO MgO S-P-fo2, T-V-L) evolution path. Calculations are per- CaO-Na20-K20 P2Os-H20 is calibrated. The formed by constrained minimization of the appropriate model is referenced to previously published standard thermodynamic potential. Compositions and propor- state thermodynamic properties and is derived from tions of solids and liquids in the equilibrium assem- a set of internally consistent thermodynamic models for blage are computed. solid solutions of the igneous rock forming minerals, including: (Mg, Fe 2 +, Ca)-olivines, (Na, Mg, Fe 2 +, Ca) M2 (Mg, Fe 2+, Ti, Fe 3+, A1)M1 (Fe 3+, A1, Si)2T~a'O6- Introduction and motivation pyroxenes, (Na,Ca,K)-feldspars, (Mg, Fe 2 +) (Fe 3 +, A1, Cr)20r Fe2+)2 TiO4 spinels and (Fe2+,Mg, In this paper, we present a substantial revision of the Mn 2 ~-)TiO3-Fe203 rhombohedral oxides. The calib- thermodynamic model of Ghiorso et al. (1983) and ration utilizes over 2,500 experimentally determined announce new computer software for the calculation of compositions of silicate liquids coexisting at known chemical mass transfer in magmatic systems (e.g. temperatures, pressures and oxygen fugacities with Ghiorso 1985; Ghiorso and Carmichael 1985; Ghiorso apatite _+ feldspar +_ leucite _+ olivine + pyroxene and Kelemen 1987). Impetus to revise our earlier work + quartz __+ rhombohedral oxides _+ spinel _+ whit- began with a realization that the shortcomings of the lockite + water. The model is applicable to natural model of Ghiorso et al. (1983) stem from a lack of magmatic compositions (both hydrous and anhyd- internally consistent end member thermodynamic data rous), ranging from potash ankaratrites to rhyolites, and from inadequate models of activity-composition over the temperature (T) range 900~ ~ C and pres- relations of the igneous rock-forming minerals. In pre- sures (P) up to 4 GPa. The model is implemented as paring the revision, wc have addressed these difficulties a software package (MELTS) which may be used to by adopting the thermodynamic database of Berman simulate igneous processes such as (1) equilibrium or (1988), extending it to include internally consistent fractional crystallization, (2) isothermal, isenthalpic or thermodynamic models for the relevant igneous solid isochoric assimilation, and (3) degassing of volatiles. solutions (Ghiorso 1990a; Ghiorso and Sack 1991; Phase equilibria are predicted using the MELTS pack- Sack and Ghiorso 1989, 1991a, b, 1994a, b, c). In addi- age by specifying bulk composition of the system and tion to these improvements, the database of liquid either (1) T and P, (2) enthalpy (H) and P, (3) entropy solid experiments upon which the liquid mixing prop- (S) and P, or (4) T and volume (V). Phase relations in erties are estimated has been greatly expanded in cover- age of temperature, pressure and liquid composition, and new algorithms (Ghiorso and Carmichael 1987) M. S. Ghiorso (I~) implementing "phase-absent" constraints have been Department of Geological Sciences, AJ-20, utilized in calibration. The resulting model is a robust University of Washington, Seattle, WA 98195, USA description of the Gibbs free energy of mixing of natu- R. O. Sack ral silicate liquids, applicable to magmatic composi- Department of Earth and Atmospheric Sciences, tions (both hydrous and anhydrous), ranging from Purdue University, West Lafayette, IN 47907, USA potash ankaratrites to rhyolites, over the temperature Editorial responsibility: T. L. Grove range 900~ ~ C and pressures up to 4 GPa. 198

Before discussing specific data sources and algo- erties. This led to a systematic discrepancy in modeled rithms utilized in the construction of our model, it is Fe2+(Mg)_l exchange potentials between liquid and appropriate to motivate these issues with a summary of coexisting ferromagnesian silicates. Another symptom the thermodynamic basis for calibration. We are ulti- of the lack of an internally consistent database is poor mately interested in generating models which quantify extrapolation of certain mineral-liquid stability rela- the relative stability of solid and liquid phases in ig- tions at very high or very low temperatures and at neous systems. The intent is to construct multicompo- elevated pressures. nent phase diagrams which account for all the major In this paper the deficiencies of the model of elements in the system and which are rooted in a ther- Ghiorso et al. (1983) are systematically addressed. We lnodynamic formulation. The latter is essential, as our summarize first the basic thermodynamic relations and knowledge of solid-liquid equilibria in igneous systems the revised and enlarged liquid-solid experimental requires both extrapolation and interpolation to en- database, describe the selection of standard state end compass the full range of compositions, temperatures member thermodynamic properties and solid solution and pressures relevant to igneous petrogenesis. Al- models, and then develop a revised and internally con- though the Gibbs free energy of the system is the sistent calibration for the liquid. critical descriptor of phase stability, its direct calib- ration is unfeasible. However, experimental determina- tion of phase compositions, equilibrated at known tem- Basic thermodynamic relations and calibration method peratures and pressures, serve to identify "points" in composition space of mutual tangency to thc Gibbs We seek a calibration of the molar Gibbs free energy (G) of natural surfaces of coexisting solid and liquid. This condition silicate liquids, as defined by the expression (Ghiorso et al. 1983): arises as a manifestation of the equality of chemical potentials in a system of heterogeneous phases which = Z X i~io + RTZ x, lnX~ + 21 V Z W,.~XiXj coexist in a state of equilibrium. If thermodynamic i--1 i--1 i~lj--1 properties of the coexisting solids are taken as known, then in principle, an internally consistent description of + RTEXwln Xw + (1 Xw)ln(l - Xw)], (1) the Gibbs free energy of the liquid may be derived from where R is the gas constant, T is the temperature in absolute, X i i8 the experimental liquid-solid database. In practice, the mole fraction of the i ~h thermodynamic component in the liquid three sorts of complications arise: (1) solution models (n is the total number of these components), Xw is the mole fraction for solid phases may not account for all the compo- of water,/~o is the standard state 1 chemical potential of the i th com- ponent and W~,~ are temperature, pressure-independent "regular- nents present in the liquid (e.g. Na or K in olivine) and solution" type interaction parameters. The last describe non-ideal consequently, will leave undefined certain critical solution behavior in the liquid. Wc assume Wi, j = Wj,~ and that the slopes of the tangent surface of the Gibbs energy of the Wi,i are zero. The validity of Eq. (1) in accounting for volumes and liquid in just those directions of composition space, (2) heat capacities of mixing for magmatic composition melts as well as the experimental database may be so narrowly focused solid-liquid phase relations and experiments on water saturation in multicomponent silicate liquids has been argued thoroughly in the as to span a relatively small range of liquid composi- literature (e.g. Ghiorso and Carmichael 1980; Nicholls 1980; tions, thereby providing insufficient constraints to Ghiorso et al. 1983; Ghiorso 1987; Kress and Carmichae11988, 1989, extrapolate the liquid Gibbs surface to other composi- 1991; Langc and Carmichael 1987, 1989, 1990; Lange and Navrotsky tions of petrologic interest, and (3) the adopted solution 1992). These arguments will not be repeated here. Our objective in models for the solid phases may be based upon mu- calibrating Eq (l) is to find optimal values of the W~.j. We achieve this objective by constraining compositional derivatives of G with tually inconsistent thermodynamic properties, which experimental data on the compositions of silicate liquids that are introduces conflicting constraints on the derived prop- saturated or under saturated with one or more solid phases erties of the liquid. _+ water. The derivative of (3 with respect to Xk (composition) is Ghiorso et al. (1983) encountered all three of these related via Darken's equation 2 to the chemical potential, Itk. From complications in calibrating their model for the Gibbs Eq. 1, the chemical potential of the k ~h component is given by: free energy of solution of natural silicate liquids. They addressed the first two problems by utilizing numerical gk=g~ fXiWi, k techniques for extraction of model parameters which i-1 have their basis in generalized inverse theory (Lawson and Hanson 1974; Ghiorso 1983). These techniques -- " Wi,jXjXj + RTIn(1 X~) (2a) provide numerically stable values of model parameters, i Ij but they do not resolve the issue of the limited scope of the solid-solution models employed. Consequently, 1 Unit activity of thc pure substance at any temperature and chemical potentials of certain liquid components such pressure as TiO2, Fe203, MnO and P2Os are very poorly de- 2 For Eq. (1), Darken's relation is given by pLk = G + fined by their calibration. Perhaps more importantly, " (Si,k _ Xi) ~--sG , where 81,j is the Kronecker delta (Ghiorso Ghiorso et al. (1983) did not address the issue of inter- i=l 7', P, Xj~j nal inconsistency of end member thermodynamic prop- 1990b) 199

and for water by: condition that the tangent hyperplane which emanates from the liquid composition and extends throughout component space, may ncver intersect the Gibbs surfaces of phases in which the liquid is g~ = g,~ + 2RTlnXw + XiWi.~ ~ ~ WiaXiXj 9 (2b) undersaturated. This is equivalent to the condition that A~, in Eq. (5) i=l i= j=l bc always positive. Rearranging Eq. (5), we obtain:

Constraints on the liquid chemical potentials (and indirectly, the n model parameters, W~,j) are achieved by writing a statement of the - AC_;~-- RT ~ Vp,k[lnXk + ln(1 -- Xw)] law of mass action for each expcrimental datum that provides either k=l (1) the composition of some phase that coexists with a liquid of known composition under imposed values of T and pressure (P) or A,~ - R TIn a~ (2) the knowledge that a particular phase is under saturated under these conditions. In general, thc reaction of phase q~ with the melt is described by the set of p-relations: Ik l ( Vp'kk'ixi ;, xx0]wi ' (7) n q~p-- ~ VmkCk, (3) where the "unknowns" are again grouped on the right-hand-side of k=l the expression, and this time include the a,p. Note that the a~ s proxy for the composition of the solid phase which idcntifies the ~linimal where (pp rcfcrs to the pth end member component of thc phase of energctic separation of its Gibbs surface and the tangential hyper- interest (e.g. Mg2SiO4 in olivine), ck refers to the Ibrmula for the plane of the liquid. These a~s (or equivalently, the mole fractions of k th component in the liquid, and vp. k to the stoichiometric reaction end member solid components) are unknowns because this composi- coefficicnt for the k ~ liquid component. From Eq. (3), the law of tion has not been experimentally determined. The general method of mass action yields calibration which utilizes phasc absent constraints requires non- linear least squares analysis subject to lincar equality constraints and linear inequality bounds. Aacpp = A% = AS~ q- L Vp, klnak -- RTlna,. (4) An approach to model calibration that takes advantage of in- k=l formation on both saturated and under saturated phases provides a way of exploiting the database of liquid-solid-vapor experiments where AG,, is the Gibbs Free energy change of reaction 3, which is to the fullest extent. Prior attempts at model calibration (Ghiorso equal to the~ ncgative of the chemical affinity (A~,) of the reaction and et al. 1983) have not recognized the significance of the under- is zero at phase saturation. AG ~ is the corresponding quantity in the saturated condition. Acknowlcdging these constraints proves crucial standard state, a k is thc activ~:ty of the k th liquid component and to the calibration of a robust modcl for the liquid. a,~, is the activity of the pth end member in the phase (equal to one for a pure phase). Eq. 4 may be expanded and rearranged with the aid of Eq. (2) to yield an expression for calibration of the model para- meters (Wi,;): Enlarged and revised liquid.selid-vapordatabase i1 - AG~ - RT ~ Vp, kElnXk + ln(1 -- X~.)] Experimental data utilized in the calibration of the k=l liquid model parameters are summarized in Tables = A,, R T In ao, 1 and 2 3. The database consists of compositions of coexisting glass (silicate liquid) and solid phases n rl 1 n n n (_+ water) equilibrated under known temperature, k~l i=1 k=l i=lj=l (5) pressure and oxygen fugacity conditions. Table 1 sum- marizes data from the literature. T, P and log Eq. (5) is linear in A~, and the Wia but is nonlinear in G,; For Jo~ ranges of the experiments are provided. For each calibration purposes, the most straightforward implementation of eq literature source, the number (N) of experiments in- (5) is to consider the first case mentioncd above, where the composi- tions of both the liquid and associated solid or vapor phase have volving a particular solid are indicated as well as the been experimentally determined at some known 7' and P. Then, Eq. standard deviation of the residuals (k J) of these experi- (5) may be rearranged, placing "known" quantities on the left-hand- ments resulting from the model calibration (see calib- side (1.h.s.) of the equation and the "unknowns" (W~,j) on the right- ration section below). hand-side, to yield: In constructing the literature database we have searched for experimental data that meet the following

-- AGO -- RT L Vp, k[lnXk + ln(1 -- Xw)] + RTlnae, minimal criteria: (1) experiments must be on silicate k=l melts with compositions approximating those observed in nature, not simple system liquids, (2) complete corn- ~ i=lL j~llk~l( vp'k~k,ixi - 2Vp'kXiXj)1 1 Wi'J (6) positions of coexisting solid and liquid phases must be determined, (3) the temperature and pressure of the where A,, is taken to be zero as the assemblage is in equilibrium. experiment must be known, (4) there must be documen- Multiplc statements of Eq. (6) represent a problem in linear least tation that supports the assumption that coexisting squares analysis for the parameters Wi, j. The 1.h.s. of Eq. (6) becomes the dependent variable in this analysis. Now let us consider the second case, and suppose that an experimentally investigated liquid is known to be under saturated with a particular phase under 3 These tables are available fiom the authors and may bc obtained specified 7', P conditions. This information may also be used to via anonymous FTP from internet node Jbndue.geology.washing- constrain the liquid model parameters. In effect, the shape of the Wn.edu. A complete data filc, including liquid and solid composi- liquid Gibbs surface (and hence the W~,i) must be consistent with the tions, is also available from this server 200 solid and liquid phases are in chemical equilibrium, and argue that complications are simply swept away by (5) the oxygen fugacity of the experiment must be fortuitous cancellation in magmatic composition melts, known. it remains unarguable that the best thermodynamic We have not included experimental data collected model for interpolation and extrapolation of experi- on synthetic systems (systems where any one of the mental data is the simplest one that adequately re- oxides SiO2, A1203, Fe203, FeO, MgO, CaO, Na20, produces the data on the system of interest. From a K20 is absent). This exclusion pertains, for example to pragmatic perspective, there are simply not enough liquid-solid data on the Fe-absent "basalt" tetrahedron, data available on all of the compositional subsystems of and to data from two and three component systems. interest to igneous petrologists to justify formulation of a The reasons for this decision stem from the fact that we comprehensive model. Such a model could never be are interested in generating a simple thermodynamic calibrated. Accordingly, we focus upon melts with mag- model which works for magmatic composition liquids. It matic compositions and warn the reader against using is not our objective to calibrate a formulation which our model on silicate liquids not represented in nature. accommodates the very interesting but otherwise pe- In Table 2 we present previously unpublished re- trologically irrelevant bounding compositional subsys- sults of one atmosphere crystallization experiments. tems. The reason that this is an issue is because workers These experiments were designed to expand the range who have focused their efforts on thermodynamic of liquid compositions covered by the literature modeling of simple melt systems (sec Berman and database. Starting materials were finely ground rock Brown 1987, for a synthesis and review) have demon- powders of olivine andesites, tholeiitic and alkalic ba- strated that quite complicated models are required to salts, basanites, minettes, tephrites, potash ankaratrites, achieve the same level of data reproducibility as simplcr melilitites and leucitites. Samples were ground to pow- formulations (e.g. Eq. 1) provide in magmatic liquids. It der, pressed into pellets using polyvinyl alcohol as appears to be unequivocal, that out of the chemical a binding agent, and welded to loops constructed of complexity of magmatic liquids arises a remarkable 0.13 mm Pt wire alloycd with quantities of Fe appro- degree of energetic simplicity. Although skeptics might priate for minimizing Fe-loss from the charge. These

Fig. 1 Summary of 1 10- experimental conditions for liquid-solid assemblages in the 0~ calibration database. 1,423 lO0 anhydrous "anhydrous" experiments and hydrous 04 -4i 170 "hydrous" (water saturated) O ' experiments are plotted. 90 O1 -8i o 80- -12 70- A -161 (n ~,,,,i .... I .... I'',,I .... 60- .Q 800 1200 1600 2000 T (~ 50- I:L I anhydrous 40- 300 nhydrous

30- II =,., ~ ,..~ OO el 2ool 20- 100 l . 10J . -I1,-~. IJ. ~ ~-~'- 0 ..-,- h., .., _ 800 1200 1600 2000 -5 -2.5 0 2.5 5 7,5 T (~ ANNO 201 were suspended in a vertical quenching furnace, drop- of the crust and upper mantle. In Fig. 1 we plot inten- quenched into water and subsequently analyzed, fol- sive variables that characterize the experiments of the lowing the methods outlincd by Sack et al. (1987) and calibration database. Note in particular, that the range Gee and Sack (1988). Oxygen fugacity was controlled of oxygen fugacities covered by these experiments is by CO/CO2 gas mixing or by opening the furnace to thirtcen orders of magnitude. In Fig. 2, liquid composi- the atmosphere. tions from the calibration database are displayed on Le This database of liquid-solid-vapor equilibria con- Maitre (1984) style classification diagrams. stitutes a ten-fold increase, in turns of the number of Although greatly expanded in coverage of natural regression equations (e.g. Eq. 6), over that used in the compositions, the calibration database still suffers from previous calibration (Ghiorso et al. 1983). More impor- inadequacies. The vast majority of data have been tantly, the current database spans the range of melt collected upon "basalt"-like liquid compositions compositions represented by silicate magmatic liquids and havc been acquired at low pressure ( < 0.3 GPa). and includes T, P and logfo ~ conditions representative The latter is in part a consequence of the fact that

Fig. 2a, b Summaryof liquid 16 compositions for liquid-solid assemblages in the calibration Tephriphonolite 9 9~ o Phn,i,oO O e / / a database. The TAS classification 14. schemeis after Le Maitre (1984). /+ .oh+ 1,423 "anhydrous" experiments 9 9149%'' 9 o9 .~" ~ ""V Trachyte / and 170 "hydrous" (water ~o 12, saturated) experimentsare Ph~176 r'te ~? ~;ii ~o,3~rachya ndesite / plotted, a Anhydrous 10. experiments, b Water-saturated 0 Trachybasalt.....~, ~ ~ "2,~N~. . ~ 9 9 9J Alkali Rhyolite expcrimcnts. .. ~__~..~ ..'.. \ ./.~ ,hyom 8. 9 9 oJ I 9 Foiditee " 9""#~::: -=~.~ :+'/ -I- 9 .~ ' + t,t 0 6.

Tephrite 9 /~~2~.-"]+" 9I +. -It '~'1~. ~- t" f.*l~*$ " _ . \ Z 4. Ba san ire = ~" .,,'~.[ =~,l~.I-).o. Da cite

2- 9.= ) Pic,.o,.,aa~. , . .. _''ZI'. _ .__ __ +,s,,,t!~Basaltic \ "~. ~ "~.. "~-- Andeeite " ~~,,."T I Andeeite '~ 0 9 9 ' ' i I" +-'-'-i +-~'---i '~ ,-,-i , , , 'nl~ I " ' I ' " " I .... I .... I " 3O 35 40 45 50 55 60 65 70 75 SiO2 wt %

16 Tephriphonolite~ ~ Phonolite/ b 14 /Alkali Trachyte 12 /" ...... ::~

10 O Trachybasalt /. X 4" .~"~ ~9149 ~~:''JA,~,,Bh,o,,,~ !, ~, 04 / \ /,.., %'S-\ ,,,o,,,~ 8 § Fo,0,,.s ( \ X 9 "." _~/:" 9 9 O 6 04 , ,-?:\ t~ Z 4 A ~"':'" I" oa~ \ 2 Ba~lt I i~ I B~,,ic \

0 "+~ .... ,1 ,++'i ,\ 3O 35 40 45 50 55 60 65 70 75 SiO 2 wt % 202

a significant fraction of published high-pressure experi- tent values of model parameters, these activity/com- mental results on natural compositions are unusable position relations must be internally consistent with the for the purpose of liquid calibration. This is due either adopted standard state propertics. For the majority of to a poorly constrained or unconstrained knowledge of igneous solid solution series, this is not a trivial require- oxygen fugacity (hence liquid ferric-ferrous ratio) or to ment, and we have expended considerable effort to an experimental geometry or duration which precluded achieve this goal. The problem is that the values as- attainment of equilibrium between liquid and solid sumed for mixing parameters in a given thermodyn- during the run. Perhaps the greatest difficulty with the amic description of a mineral solution series, generally current calibration database is the absence of quantit- depend on the choice of thermodynamic data for the ative data (i.e. determinations of compositions of min- end member components of that series and often erals coexisting with liquids) for water under saturated (through constraining heterogeneous phase relations) conditions and the paucity of usable data for melts upon models for other solid solutions. containing dissolved CO2. The adopted activity/composition models are inter- nally consistent, in the sense of the previous paragraph, with the end member database of Berman (1988). They include formulations for feldspar [CaAI2Si208- End member component thermodynamicproperties (Na,K)A1Si3Os; Elkins and Grove 1990], olivine [Ca(Mg,Fe)SiO4-(Mg,Fe)2SiO4; Hirschmann 1991], Calibration of the melt interaction parameters, utilizing orthopyroxene [(Mg,Fe)2Si206; Sack and Ghiorso either Eq. (6) or (7), requires evaluation of standard 1989], pyroxene [(Mg,Fe 2+)2Si206-Ca(Mg,Fe 2+) state Gibbs energy changes for reactions between end Si206-CaTi /2 (Mg,Fe)~/2(A1, Fe3+)SiO6-Ca(A1, member solid and liquid components. It is crucial to Fe 3 +)(A1,Fe 3 +)SiO6-Na(A1,Fe 3 +)SizO(,; Sack and the success of the calibration that these end member Ghiorso 1994a,b,c), rhombohedral oxides [-(Mg,Mn, thermodynamic properties be internally consistent, Fe2+)TiO3-(Fe3+)zO3; Ghiorso 1990a, Ghiorso and otherwise the solution interaction parameters (Wi,j) Sack 1991], and spinel [(Mg,Fe2+)(AI,Cr,Fe3+)204- calibrated from them will be forced to accommodate (Mg, Fe2+)2TiO4; Sack and Ghiorso 1991a, b]. relative errors in the standard state properties. We address this, concern by adopting solid end member properties consistent with the database of Berman (1988). In the appendix we summarize data on the end Calibration procedure and results member properties needed for calibration (Table A1), and discuss our internally consistent extensions to Ber- Model parameters are obtained by least squares ana- man (1988) for those end member compositions not lysis of a set of equations of the form of Eqs. (6) and (7) considered in his study. In the appendix we also discuss constructed from the experimental database (Tables the extension of Berman (1988) to afford calculation of 1 and 2), the adopted end member thermodynamic Gibbs energies of end member liquid components. This properties (Table A1) and the assumed activity-com- extension is dictated by two criteria: (1) the liquid position relations for mineral solid solutions. The components must be chosen so as to span the composi- chemical reactions relevant to this analysis are pro- tion space of natural silicate liquids with positive mole vided in Table 3. The indicated stoichiometric coeffi- fractions, otherwise a more elaborate entropy of mixing cients give the appropriate values of Vp, k necessary for model than that embodied in Eq. (1) would need to be the evaluation of Eqs. (6) and (7). In constructing these formulated, and (2) the liquid components should be equations, we have calculated the ferric-ferrous ratio of chosen to have stoichiometries identical to solid phases the liquid according to the algorithm of Kress and whose reference properties are tabulated by Berman Carmichael (1991). (1988) and for which the enthalpy/entropy of fusion has Regression equations generated from phase present been experimentally determined. These two criteria or equilibrium coexistence data (e.g. Eq. 6) are linear in motivate our choice of liquid components, and these the model parameters, Wi,j. Equations constructed are indicated along with their thermodynamic proper- from phase absent constraints (e.g. Eq (7)) are linear in ties in Table A1. Specific details are discussed in the A,p and W~.j but non-linear in the X~p. Each of the Appendix. imposed phase absent constraints generates an un- known reaction affinity, and a set of unknown mole fractions of end member solid components. In addition, Thermodynamic models for solid solutions each imposed phase absent constraint generates a bound condition that its affinity be positive and that Evaluation of Eqs. (6) and (7) for the purposes of model thc sum of the component mole fractions be unity. calibration, assumes the availability of activity/com- For a calibration database of respectable size, the position relations for the solid phases identified in the imposition of phase absent constraints can rapidly experimental database. In order to extract self consis- produce a regression problem with enormous numbers 203

Table 3 Reactions used in calibrating the model Phase/Component Reaction: solid = liquid Olivine Fayalite Fe2SiO4 = Fe2SiO,~ Forsterite MgzSiO4 = MgzSiO4 Pyroxene Diopside CaMgSi206 = ~SiO2 + ~MgzSiO~ -I- CaSiO3 Enstatite MgzSi206 = Mg2SiO4 + SiO2 Hcdenbergite CaFeSi206 = 89 + ~F%SiO, + CaSiO3 Feldspar Albite NaAISi3Os = ~SiO2 + 89 + 89 Anorthite CaAI2Si208 = SiO2 + A1203 + CaSiO3 Sanidinc KAISi308 = 2SiOz + KA1SiO4 Quartz SiO2 = SiO2 Tridymite Leucite KAISi206 = SiO2 + KA1SiO4 Corundum AI203 = A1203 Spinel Chromitc FeCr204 + ~Mg2SiO4 = MgCr204 + aFe2SiO4 Hercynite FeAI204 + 89 = A1203 + 89 Magnetite Fe304 + ~SiO2 = Fe203 + 89 Spinel MgAI204 + 89 = A1203 + 89 Ulv~spinel FezTiO~ + SiO2 = TiO2 + Fe2SiO4 Rhombohedral Oxide Geikielite MgTiO3 + ~SiO2 = TiO2 + 89 Hematite Fe203 = Fea()3 Ilmenite FeTiO3 + 89 = TiOz + 89 Whitlockite Ca3(PO4)2 = Ca3(PO4)2 Apatitc Ca,(PO4)3(OH) + 89 = 89 + ~Ca~(PO4)2 + ~H20 Water HzO (liquid or gas) = H20 (dissolved in melt)

of equations and unknowns. For example, if 1,000 dent" variables of the regrcssion problem, and third liquids in the calibration database were known to be choosing the minimal subset of these new variables that under saturated with feldspar (NaA1Si30~-KA1Si3Os predict the dependent variable at an acceptable level of CaA12Si2Os), this would generate 3,000 statements of precision. This is the method of singular value analysis Eq. (7). There would be 4,066 parameters in the regres- (SVA) which we have used previously (Ghiorso et al. sion problcm: 66 Wij, 3000 feldspar mole fractions, and 1983; Ghiorso 1983) on linear regression problems. 1,000 reaction affinities. The least-squares solution Here, we apply the singular value method to the non- would be subject to 1,000 equality constraints specify- linear algorithm by solving a series of related singular ing that the mole fraction sum for each "absent" feld- value decompositions (SVD) on the subset of linear spar be unity, and 1,000 inequality bounds imposing regression parameters during each step of the non-lin- positivity on each of the reaction affinities. Formidable ear minimization. The algorithm is numerically stable, non-linear regression problems of this scale may be requires a minimal amount of computer time and stor- solved using the software package of Murtagh and age and can accommodate a regression problem like Saunders (1987). Their algorithm takes advantage of the one described here with greater than 50,000 equa- the fact that most of the regression parameters (4,000 of tions and 40,000 parameters. the 4,066 in our example of feldspar) only appear in In analyzing the calibration database we adopt the a very small number of regression equations and are following procedure. An initial solution to the Wi,j is coupled only through the derived values of the Wi,j. obtained by evaluating only the phase-present (Eq. 6) Such methods are known as tcchniques in sparse non- constraints. This generates a linear sub problem in only linear optimization. 66 parameters and approximately 4,600 regression We have extended the algorithm of Murtagh and equations. The number of phase-present regression Saunders (1987) to account for the high degree of cor- equations is restrictcd by including only solid-solution relation between thc Wi4 model parameters. Recogni- end members whose mole fraction exceeds 0.05 in the tion of this correlation is essential if values of extracted reported phase. Thus, an experimental determination interaction parameters are expected to be stable to of coexisting olivine and liquid will result in a regres- extrapolation and interpolation of the calibration sion equation for the Mg2SiO4 (forsterite) component if database. The method of cxtension involves first identi- the mole fraction of Mg in the olivine exceeds 0.05, fying the level of correlation between the "indepcndent" Fe2SiO4 (fayalite) if Xw > 0.05, and CaMgSiO4 if variables, second constructing uncorrelated linear com- X~ > 0.05 and Xc~.,z > 0.05. This restriction assures binations of the Wi.j which become the new "indepen- that compositional uncertainties at low concentrations 204 do not overwhelm the model calibration. We solve the to be expected, and removal of this correlation moti- linear rcgression sub problem using SVA. This proced- vates our choice for reporting residuals in Fig. 3 and ure identifies and resolves correlations between the Table 1. The number of regression equations of each independent variables. The solution to this linear prob- type of phase-present constraint is also indicated in the lem is used as an initial guess to the larger non-linear Table. Note that in Fig. 3, the brackets are four regression problem which incorporates the phase-ab- standard deviations wide and are centered about the sent constraints. We analyze the larger problem using average error for that phase/data source. The overall the modified algorithm of Murtagh and Saunders standard deviation of residuals is 2.72 kJ in the depend- (1987). The solution of the non-linear problem serves to ent variable (1.h.s. of Eq. 6) on 4,666 statements of identify active phase absent-constraints, i.e. a constraint phase-present constraints. If we remove the within whose derived affinity is zero, indicating apparent satu- experiment correlation (as above), this value drops to ration. The active phase-absent constraints are then 1.3 kJ on 2,540 experiments. This translates roughly individually examined and relaxed within an "uncer- to an ability to back-predict the liquidus temperature tainty window" estimated from residuals on phase- of a given experiment to within 10 ~ or equivalently, present equations involving the same phase. For to calculate the composition of the liquidus phase to example, if the standard deviation of residuals asso- within 3 mol%. ciated with fitting leucite-present data is 1 k J, then an The important generalizations to be reached from uncertainty window of ___ 2 kJ ( _+ 2 cy) is applied to the the analysis of residuals of the model calibration are leucite absent equations. The logic here is that at the 2cy that (l) the average residual and standard deviation of level, the leucite saturation surface cannot be deter- residuals for each phase are on the whole comparable mined to better than 2 kJ by the model and conse- to that of the overall regression, and that (2) the with- quently, enforcement of phase absent criteria must in-phase variation of residuals shows no obvious cor- recognize this statistical uncertainty. The calibration relation to experimental study. The first generalization procedure is deemed to converge when there are fewer demonstrates the absence of inter-phase systematic bias than 2.2% (one-tailed 2~ probability) active phase ab- and is violated only for the silica minerals (Fig. 3). We sent constraints outside of the uncertainty window. The suspect that there may be equilibration problems in the entire calibration procedure is software driven and is high-pressure experiments involving quartz and that available as part of the MELTS package, described the offset in tridymite residuals may be mostly at- below. tributed to a systematic bias in modeling the activity of Optimal values of the model parameters are pro- SiO2 in the extremely iron rich liquids from which this vided in Table 4. Residuals on the phase-present equa- phase precipitated. The latter would point to a defi- tions are categorized by phase and data source in Fig. 3 ciency in model formulation. Alternatively, these sys- and are enumerated in Table 1. For simplicity, we have tematic offsets may offer evidence that the adopted grouped residuals for solid solution end members by averaging each experiment and computing a standard - deviation of the averages. This representation is ad- Fig. 3 Summary of residuals to the calibration database categorized equate in that individual end member residuals show by experimentalist and phase. The average overall residual for the parameter calibration is zero. Residuals for each phase/experi- no systematic bias with respect to the mean, i.e. the mentalist plotted in the figure should be compared to this average. average residual on each end member in a solid solu- These residuals are represented by a bracket centered about the tion is approximately that of the solid solution as averagc for the group of experiments in each category and are a whole. Average residuals for individual experiments delimited by _+ 2 standard deviations about this average. Numbers are not correlated, but end member residuals within of experiments in each category are summarized in Table 1. Resid- uals display no systematic relation with temperature, pressure or a given phase/experiment are highly correlated. This is liquid composition.

Table 4 Regular solution interaction parameters for the liquid phase (values in k J)

SiO2 TiO2 26267 A1203 - 39120 29450 Fe20 3 8110 84757 - 17089 MgCrzO 4 27886 - 72303 - 31770 216O6 F%SiO4 23661 5209 - 30509 - 179065 - 82972 MgzSiO 4 3421 -- 4178 - 32880 - 71519 46049 - 37257 CaSiO3 - 864 - 35373 - 57918 12077 30705 12971 - 31732 Na2SiO3 - 99039 - 15416 - 130785 - 149662 113646 90534 - 41877 13247 KA1SiO4 33922 -- 48095 - 25859 57556 75709 23649 22323 17111 6523 Ca3(PO~)2 61892 25939 52221 - 4214 5342 87410 - 23209 37070 15572 17101 H20 30967 81879 - 16098 31406 28874 35634 20375 -- 96938 10374 43451 SiO2 TiOz A120~ F%O3 MgCr204 Fe2SiO4 Mg2SiO4 CaSiO3 NazSiO3 KA1SiO~ Ca3(PO4)2 H20 205

Agee & Walker (1990) i ,, J3rndt (19..77~ - I- ---I uaker~ E~qCller(1987) - O I I S~. i Barn s 1 86 c i ...... i Ba.& ~ ~ro/,e11991~ - I ~, e i i I-- Bartelsetal. (1991) " i I ~'~ .... P X i --t -V ~,--I% "d I- n -i S e n i...- "1 p ~ ~ I e i I I a i i i i r i.- 4 i---i I i i I I I

i , I- i"~ -.t I- I-.-.- i , },- ,,~: i i I I i i-4 p-.._ I I I- I m I i i- I t-

i i m.i H I t ,i Mean ~1 !=i~r i I I m- m Meen 199~1 I--- i I [ Murck("Oampbell(1986& I I'-- I "~ ielsen et 1 , I I I Reader 1974) Boeder ~ 9eynolds (.i 991 .) ' -~, ~eck & Carmlchael('1984) i - i ~,ack & Ghlorso~1994c) i i IL ~eck et al. (1987) I, I ---'d Seilert et a1.(1988) toper 1 77 ! H ~tolp,sr 11~8~1 ,' ' l Takatlashi (1980) I--I aka ashl I 6 i I Taka~ashl ~ ~'~iro(1983) t----I Thompson 1974,1975) I To may et ~.1..(1987) ..-'t i-- d I 9reiman (luau) f-- Us~]er & Glazoer(1989) i- -t I---- L WalKer et al. (1973) H This Paper I ---t i, I I I I He z 1976 ii i K~le~en e~ aL (1990) Luhr {1990); HOLISh& Luhr (199~ P..- -t ~isson k& ~rove 1992a i--.-- ~lS8Oe ~ L=rove 1992h ~ a -8-4048 -8-4048 -8~048 -8 -4 0 4 8 kJ

Baker & Eggler (1987) ~ , Barnes (1986) -- O ~ r Biggar et al. (1971) I I W Delano (1980) P h ---4 e h Fram & Longhi (1999) X m U i Gee & Sack (1988) C t Grove & Beat"/(1980) H Grove & Juster (1989) I-- I i I Grove et al. (1982) H t O .... Juster et el, (1989) .-I I e C Longhi & Pan (1988) Longhi & Pan (1989) k Longhi et aL (1983) ...... I-- .-I i Mahood & Baker (1986) t Meen {1987) e Sack & Carmichael, I- Sack & Ghiorso (1994c) Sack et al, (1987) I.-,- -,~ I- -I F"i---- I Ussler & Glazner 0989) -I This Paper Helz (1976) I,-,- Kelemen et al. (1990) I

-8-4048 -8 -4 0 4 8 -8 -4 0 4 8 -8 -4 0 4 8 kJ

Fram & Longhi (1992) Grove & Undstey (1979) H C a Grove & Raudsepp ('~978~ -- t -- a W Johnson ('t 986) r H .,~ O P a Longhi & Pan (1988) __ i __ r a t Stolper (1977) d q u I t e Walker at al, (1973) U I rl Bumham&Jahns(1962) -- 'Y -- i r p- Hamilton et al. (1964) m a --d t r _ l---- Helz (1976) _ i u e Khitarov eta], (1959) t -- - t -- _m i -,i- .4 Luhr (1990) Z i.-.t Naney {1983) e Sisson & Grove (1992a) I-el i.----- Sisson & Grove (1992b) i------8-4048 -8-4048 -8-4048 -8 -4 0 4 8 kJ 206 temperature dependence to the heat capacity of the conditions. These include temperature, pressure, bulk SiO2 liquid component (see appendix) may be inappro- composition and methods or standards used for micro- priate for natural composition melts; the temperature probe analysis. In this regard it is worth emphasizing, spread of the tridymite saturation experiments is very that unlike our previous model calibrations (Ghiorso small and averages about 1000~ below the fusion et al. 1983), no experimental data were arbitrarily ex- temperature of cristobalite. It is even possible that the cluded from the fitting procedure on the basis of poor appreciable alkali contents of natural tridymites may quality of fit. be the origin of this energetic offset. There are too few From a statistical perspective, not all 66 model data on silica mineral-melt saturation to pursue a solu- parameters reported in Table 4 are of equal importance tion to this puzzle. From the second generalization we to a solution of the regression problem. A quantitative conclude that residuals for a given phase are not understanding of the significance of the various para- particularly sensitive to variations in experimental meters may be gleaned by examining a singular value decomposition (SVD) produced during the final stage Fig. 4 Visual representation of the Singular Value Decomposition of calibration. As described above, SVD reassembles (SVD) of the parameter calibration matrix constructed from the the model parameters into an equal number of linear calibration database. Each pie diagram dcnotes an independent combinations or eigenvectors. These eigenvectors are variable of the parameter calibration problem with the significance constructed so that each represents a truly independent (% variance) of that variable indicated as a numeric valuc displayed above or below each pie. The wedges of each pie indicate the relative or uncorrelated fraction of the variance of the indepen- contribution of each liquid rcgular solution interaction parameter dent variables. Moreover, the fraction of variance ac- (Wi.j in the text). All 66 W~,js contribute to each pie diagram, with counted by each of the eigenvectors is provided in the only the dominant contributions labcled. Minor contributions are analysis. In Fig. 4 we display a representation of the shaded. Mutually dominant W~,js within a given pie diagram indicate first twelve cigenvectors of the model calibration pres- a high degree of statistical correlation. The wedges are labeled using the following scheme: (1) Si, Ti, A1, Fe3 +, Cr, Fe2 +, Mg, Ca, Na, K, ented here. Each eigenvector is represented by a H20 refer to the liquid components SiO2, TiO2, Al2Oa, Fe203, pie-diagram. The numeric label attached to each pie FeCr~O4, Fe2SiO4, Mg2SiO,~, CaSiO3, Na2SiO3, KAISiO~ and corresponds to the percentage of the total variance H20 respectively, and (2) a W~.j is indicated by the symbol i-j, i,e. accounted for by that eigenvalue. The first twelve eigen- WS~O~,Aa~O~ is denoted Si-A1. There arc a total of 66 independent vectors displayed in Fig. 4 account for 95% of the total linear combinations of the Wi,.i, but only the first 12 are shown. These constitute over 95% of the total variance in the calibration variance of the independent variables. The pie wedges database. indicate the proportion of variance attributable to each

r

,,'" 4,

IK

~

Si.H20 ~ H20-Fe2

SI Mg 207

of the model parameters. Thus, the principal eigenvec- quently, revision of the model calibration presented in tor accounts for 28.25% of the total variance and is this paper is inevitable as new experimental data be- mainly composed of contributions from Wsio~,AI~O3, come available and as new internally consistent ther- Wsio~,MgaSiO,,, Wsio~, Fe~SiO4, and Wsio~,c~sio~. Fig. 4 re- modynamic formulations for solid phases arc built veals that the most dominant and consequently most upon the adopted standard state database (Table A1). statistically secure of the calibrated interaction para- Furthermore, such revisions may be accomplished by meters are the ones which constitute the eigenvectors of users of the MELTS package. Significant improve- highest variance. These involve mainly interactions ments in the present calibration await high-quality with the SiO2 component, which is not surprising, as experimental data on compositions of liquid and solid SiO2 has more variability than any other composi- phases equilibrated at elevated pressure under condi- tional variable. Additionally, Fig. 4 demonstrates the tions of known oxygen fugacity. The calibration extent of parameter correlation; parameters corres- database is still too highly skewed toward anhydrous ponding to dominant pie wedges in a given pie diagram low-pressure experiments and there is a real need for are highly correlated. These correlations are not always experiments in water-bearing systems, particularly in- immediately intuitive in that they cannot be easily volving water undersaturated melts. Improvements in interpreted in terms of simple solid-liquid reaction the crystallization simulations await construction of stoichiometry. Our final model calibration is construc- a thermodynamic model for high-Ca igneous ted from 56 eigenvectors of the possible 66 W~,j combi- amphiboles, the formulation of a model for ternary nations. This cutoff accounts for 99% of the allowable feldspars that incorporates temperature and composi- variance. The Wi,j parameters are reconstructed fi'om tionally induced symmetry transformations, and the these 56 eigenvectors following standard SVA methods development of internally consistent solution theory (Lawson and Hanson 1974). for the feldspathoid minerals. These extensions are un- der development. One extremely productive future direction of MELTS involves incorporation of trace element partitioning between liquid and solid. Hir- The MELTSsoftware package schmann and Ghiorso (1994) have begun to pursue this avenue of research by extending the calibration pro- The best way to demonstrate the consequences of the parameter posed here to model Ni, Co and Mn partitioning calibration discussed above is to utilize the derived coefficients to between liquid and olivine. Their approach is very forward model crystaMiquid cquilibria of well known bulk composi- tions. This has the advantage of displaying the consequences of the successful in systematizing and extrapolating trace ele- calibration in terms of calculated liquidi, phase compositions and ment distribution coefficients over a wide-range of tem- proportions much more familiar concepts than uncertainties in peratures and compositions. The key to the incorpora- end member chemical potentials. To facilitate calculations of this tion of trace elements into MELTS is the description of sort we have developed MELTS, a new software package for modeling chemical mass transfer in magmatic systems. MELTS is trace element energetics in the solid solution. Coupled based upon algorithms lbr energy minimization described by substitutions and cation ordering in the solid lead to Ghiorso et al. (1983), Ghiorso (1985), and Ghiorso and Kelemen highly complex liquid/solid partitioning behavior. (1987) and incorporates new procedures for determination of the High quality experimental data on liquid-solid distri- saturation state and stability of strongly nonidcal solid solutions bution coefficients, collected over a broad range of with respect to silicate melt (Ghiorso 1994). It supersedes and ex- pands the capabilities of the previously released modeling software, compositions and temperatures (e.g. Nielsen et al. SILMIN. Capabilities of the MELTS package include (1) revising 1994), are required but are not sufficient to achieve and/or updating the parameter calibration for the liquid (using the a generic model. Equal attention must be directed at methods outlined above) on the basis of new or revised calibration understanding the crystal chemical constraints of trace datasets, (2) modeling equilibrium or fractional crystallization of element substitution in the solid itself. Future research specified bulk compositions, (3) modeling crystallization paths under constrained T-fo~ total enthalpy, total entropy, total volume or any on the encrgetics of trace element substitution in pyro- combination of these constraints, and (4) modeling assimilation or xenes and spinets is prerequisite to extending MELTS magma mixing in evolving magmatic systems. The MELTS software to include a generic trace element partitioning model package may be obtained via anonymous FTP from intcrnet node for igneous systems. jbndue.geology.washinglon.edu. MELTS is intended to be run inter- actively on a "UNIX"-level workstation. It utilizes an Xll/Motif menu-driven graphical user interface and is written in ANSI C. Acknowledgements We are indebted to Marc Hirschmann and Victor Kress for their interest and contributions to this work and to Tim Grove who shared experimental results with us prior to publication. The manuscript benefitted from helpful reviews Future directions by Becky Lange and Tim Grove. Material support for this investi- gation was provided by the National Science Foundation The MELTS software package has been designed with through grants EAR84-51694 and EARgl-04714 (MSG) and the aim of automating the process of model recalibra- EAR89-04270 and EAR92-19083 (ROS). MELTS was developed over a three year period as a direct result of external research tion and facilitating the incorporation or revision of funding and a generous equipment grant fl-oln Digital Equipment new thermodynamic models for solid phases. Conse- Corporation. 208

The paucity of data on entropies of fusion of geologically Appendix relevant materials forces us to adopt a number of values which are little more than educated guesses and could have attached uncer- Summary of adopted thermodynamic data tainties on the order of 20% of the indicated value. In the context of the model proposed in this paper however, we think thcse estimates Thermodynamic data for end member solid components are taken are adequate l'or two reasons: (1) estimates of AS~ion in conjunction for the most part from Berman (1988). Internally consistent exten- with our adoptcd model for the entropy of mixing in the liquid, allow us to reproduce many of the measured entropies of fusion of sions to this database are sumanarized in Table A1. Apparent molar Gibbs li'ee energies of formation of solid end members are calculated more complex compounds, and (2) these estimates are highly corre- lated to the derived values of the exccss enthalpy of mixing of the from the tabulated quantities according Eqs 6, 13 and 20 of Berman liquid and are therefore, rendered internally consistent by model (1988). The reference tcmperature (Tr) is taken to be 298 K and the calibration to the adopted reference set of solid properties. Another reference pressure (PD 1 bar (0.0001 GPa). way o1" expressing this consistcncy is to note that we see no temper- The adopted thermodynamic properties of perovskite have not ature dependence to the rcsiduals of any solid-liquid equilibrium been evaluated for internal consistency with Berman (1988). The relations over a range of temperatures that span 800 ~ C. However, it reference enthalpy and entropy of Berman's (1988) anorthite is important to realize that the estimated values of ASfu~ion-~tabulated (CaAlzSizOs) have been adjusted, using the calorimetric data of in AI, and in fact many of thc calorimetrically determined values Carpenter (1988), in order to evaluate the properties of metastable reported in this table, may only be internally consistent with Berman C1 anorthite. This is necessary, as the compositional dependence of (1988) when used in conjunction with our proposed model. Caveat the Ii - > C1 phase transition in plagioclase is not accounted tbr in emptor. solution thcory for the feldspars modeled by Elkins and Grove Liquid heat capacities tabulated in Table A1 are obtained largely (1990) and adopted in this paper. Accepting Berman's (1988) anor- from the algorithms of Lange and coworkers (Lange and Car- thite in conjunction with Elkins and Grove's (1990) model without michael 1990; Langc and Navrotsky 1992) and are to bc understood making this correction, has the cffect of destabilizing the calcium as partial molar heat capacities of the end member component in component of the feldspar, particularly when XA, is greater than 0.8. "magmatic" composition liquids. Note that the adopted heat capa- The thermodynamic properties of water are calculated according to city of molten SiOz is modeled as temperature dependent below an algorithm devised by Berman (1988, personal communication.). 1480 K (the glass transition temperature) following Richer et al. This algorithm is based upon the Haar equation of state. (1982). We have incorporated this temperature dependence into our Apparent molar Gibbs energies of formation of liquid compo- model in order to avoid a logical inconsistency that is inhcrent to nents are calculated according to our earlier calibration (Ghiorso et al. 1983). Simple extrapolation of the liquid properties of molten silica down from the fusion point AGapv,-0T,P = AI~f~ T,,p, -~- f"~" ~0,p, sol dT -~- TfusionASfusion.0 using a constant Ce results in the liquid being more stable than the r solid (quartz, tridymite or cristobalite) at temperatures below about ;T C O'liq'qf~ T ( Tf...... --0, sol 1200 ~ C. One might argue from this observation that temperature +5 ...... ,, ~,--T ~o.,,+ [dr, C~d T dependent heat capacities should be adopted for all the "liquid" components below their respective glass transition temperatures. However, the glass transition temperaturc is unknown for most of T ~0, liq -o r Cy,:' dT~ + AG ~ the adopted components and in practice the correction between the + AS~o,, + ) ...... r ) ' CF lbr the mctastable "super cooled liquid" (the constant Cp) and that of the glass is inconsequential with respect to the total energy diflbrencc between solid and mctastable "liquid". In the case of SiO2, - ,7o [(0vo, I the illogical result manifests from the fact that the temperature integral of the heat capacity difference between glass and super- cooled liquid is of the same order as the enthalpy of fusion. Volumetric properties of the liquid are from Lange and Car- + (~)(T--1673)](P~ -- P,P + ~) michael (1990) and Kress and Carmichael (1991), except for values of (02V~ These have been estimated (V. C. Kress, personal com- munication) by visually comparing volumetric properties calculated (~2 g~ (63 Pr 102 Pr2P f) from +\~p2j 2 + 2 where the properties of the solid are determined as above, By this vo.., = vo,. + i (p_/'r) + \ 2 pm + method, thermodynamic properties of the liquid components are anchored to those of the indicated solid phases (Table A 1). With the exception of the unadjusted calorimetrically determined values for to those calculated from the Birch Murnaghan equation Na2SiO 3 (sodium metasilicate), the latter are internally consistent with Berman (1988). We have adopted entropies of fusion from o 3K F/'vr~ ,,\~ _ (vr e,']'~] a number of sources and in some cases have estimated values from correlation algorithms (see Stebbins et al. 1984 and footnotes to Table AI). Note in particular, our adopted value for the entropy of fusion of Ibrsteritc, 57.2 J/K. This value is based upon an estimate of x. { 1 3(4 - K , -o 1]} the enthalpy of fusion of forsterite published by Navrotsky et al. 4 L\vo.M (1989), which is in turn derived from data on the enthalpy of solution of molten CaAI2Si2Os Mg2SiO,~ liquids at 1500 ~ C. Our value dif- fers from that reported by Navrotsky et al. (1989; 53 • 9 J/K) in that Here K is defined for each liquid component as we have adjusted their calculations to reflect our adopted values of -0 the heat capacities of both forsterite and Mg2SiO ~ liquids. Within g i/,, p, the expected uncertainty of + 9 J/K, our adopted value is consistent K- with the lower bound (65.3 + 6 J/K) of the recently published measurement of Richet et al. (1993). Table A1 Additional thermodynamic properties of endmember solids and liquids. Volumetric properties of the liquid are from Lange and Carmichael (1990) and data lbr the solids are from Berman (1988) unless footnoted

SolidPhase Formula Al-I~)(h) S~ V~ ko k, xl0 -2 k~• S k3xl0 V v, xl0 ~ ve• ~2 v3xt06 v4xl0 ~~ Tt(K) zXHt(h) l~xl0-" lax10 ~

Clinoenstatite MgzSizO~ - 3081636 (~1 137.570 ("5 6.330 (") 333.16 -24.012 45.412 55.830 0.749 0.447 24.656 74.670 Hedenbergitc (") CaFeSi206 -2845389 171.431 6.789 307.89 - 15.973 69.925 93.522 -0.993 1.4835 31.371 83.672 Alumino- CaTi~,2Mg~,,2 --3275265 143.745 6.356 297.50 13.5596 --67.022 75.908 -0.870 2.171 22.250 52.863 Buffonitc~5 A1SiO~ Buffonite(u5 CaTi~,2Mg,,z --2836709 176.557 6.722 303.91 -14.t767 --43.565 35.252 0.870 2.171 22.250 52.863 FeSiO6 Esseneite ~u5 CaFeAISiOo 2860211 158.991 6.722 317.11 17.333 --51.097 54.222 --0.870 2.171 22.250 52.863 Anorthite CaA12Si2Os -4213249 ~r 207.223 ~~ 10.075 439.37 37.341 - 31.702 1.272 3.176 10.918 41.985 Sodium- Na~SiOa -- 1561427 {aS 113.847 (~5 234.77 (~5 -22.189 (~ 13.530 (~ rnetasilicate Nepheline (t~ NaA1SiO4 2087976 124.200 5.422 205.24 -7.599 108.383 208.182 - 2.221 600.0 31.552 0.2 467.15 241.835 --102.784 339.448 Kalsilite (f5 KAISiO~ 2111814 133.965 5.989 186.00 -- 131.067 213.893 - 1.567 0.0 23.487 140.257 800.15 1154 -7.0965 21.682 Lc~cite (r) KA1SizO0 - 3012026 210.704 9.263 271.14 -9.441 78.572 95.920 - 1.557 1.274 ~2.517 Perovskite (~5 CaTiO~ 1660630 (~5 93.640 ("5 3.363 (g> 150.49 (~) --6.213 (~) -43.010 l~) 1530 {u) 2301.2 (h) Chromite (~ FeCr~O4 - 1445490 142.676 4.401 {~) 236.874 16.796 - 16.765 Hercynite (1) FeAI20~ 1947681 115.362 4.075 235.190 - 14.370 -46.913 64.564 Ulvospinel~ FezTiO4 - 1488500 185.447 4.682 (g) 249.63 18.174 --5.453 Geikielite ~ MgTiO~ -1572560 74560 3.086 (g5 146.20 -4.160 - 39.998 40.233 0.584 1.230 27.248 29.968 Hematite Fe203 822000 ~> 87.400 (-i~ 3.027 146.86 - 55.768 52563 --0.479 0.304 38.310 1.650 955 1287 7.403 27.921 Pyrophanite (k) MnTiO~ -- 1350707 o5 104.935 (~) 3.170 150.00 4.416 -- 33.237 34.815 0.584 1.230 27.248 29.968 Whitlockite Ca~(PO,~)z 4097169 ~5 235.978 (~ 9.762 ~ 402.997 (v~ -- 28.084 (p) -32.623 <~) 1373(v) 14059 ~p) 2.5427 (~) 19.255 (v5 Apatite ~'> Ca~(PO4)3OH 6694689 398.740 16.403 758.81 --64.806 44.794

Liquid Reference Solid Tfu~i,,(i) AN~ o,(J,/K) C~ V~ at 1673 K (0V~ x104 (~xTr~ x l0S \OP~Tj|--I x 108 \~-jx 101~ Component Phase \ 8 7--~]v \ ~P ]~

SiO2 (amorphous) 1999 (q5 4.46 (q5 81.373 Iq5 2.690 0.0 - 1.89 1.3 3.6 ~r) TiO2 rutile 1870 r 35.824 (~5 109.2 (') 2.316 7.246 - 2.310 5.0 (o AIzO~ corundum 2320 ('~ 48.61 (~5 170.3 ~t5 3.711 2.62 -- 2.26 2.7 4.0 ~5 Fe203 hematite 1895 ~g5 60.41 (~) 240.9 (g~ 4.213 9.09 2.53 3.1 4.4 (r) MgCr204 picrochromite 2673 (') 73.22 (w) 335.1 ~) 5.358 (~) 1.171 ~x~ 2.26 t~5 1.8 (• 4.7 {~) FegSiO4 fayalite 1490 (~> 59.9 (-~) 240.2 5.420 5.84 -- 2.79 2.3 14.6 ('5 MgaSiO4 forsterite 2163 (~5 57.2 ("~> 271.0 4.980 5.24 - 1.35 - 1,3 4.U 5 CaSiO3 pseudowolla- stonitc 1817 (ub) 31.5 (bb) 172.4 4.347 2.92 - 1.55 1.6 3.9 (~) Na2SiO3 sodktm metasilicate 1361 (y5 38.34 (y> 180.2 5.568 7.41 - 4.29 5.3 8.4 (~) KA1SiO4 kalsilite 2023 ~~ 24.5 (aa) 217.0 6.838 7.27 - 6.40 - 4.6 12.1 ~) Ca3(PO4) 2 whitlockite 1943 ~) 35.690 m'5 574.7 ~1 H~O {ha5 (liquid)

(")Sack and Ghiorso (1994b); IbSSack and Ghiorso (1994c); ~C)H ref (11) from Berman (1988) adjusted to reflect I1 .-- > C1 transition (Carpenter, 1988); (a)Stull and Prophet (1971); I~SBerrnan and Brown (1985); (f)Berman (personal communiation), additional first order phase transition in nepheline: T~ = t180.15 K, AHt = 2393 J; (g)Robie et al. (1978); I~)Naylor and Cook (1946); (0Sack and Ghiorso (1991 b); (J~Ghiorso (1990a); (k~Berman (1988), analogy with FeTiO3; 1~SEstimated in conjunction with Ghiorso and Sack (1991); (m)Stephenson and Smith (1968); ~'>Zhu and Sverjensky (1991); (~ and Milner (1935);/V~Fitted to data tabulated by Kelley (1960); (45Richet et al. (1982), Att}= - 901554, go = 48.475, below 1480 K, C ~ = 127.3 - 10.777 x 10-3 T + 4.3127 x lOS/T z - 463.8/,/T; (r~Estimated, see text; (~?Samsonov (1982); (')Lange and Navrotsky1 (1992); (usestimated from Robie et al.'s (1978) data on of ASru~io.-0 of Fe304 and Fee; (')Levin et al. (1964); (W)estimatcd a posteriori by examination of temperature dependent calibration resiudals on FeCrzO4-rich spinels; (~)properties taken as idel~tical to those of MgFe204; (Y)Stebbins et al. (1984); <~)Bowen and Anderson (1914); ("")corrected value, based upon data in Navrotsky et al. (1989); (Ub)Adamkovi6ovfi et al. (1980); (~ (1980, p. 257); (da)estimated by StcbNns et al. (1984); (~ (1972); (mestimated as the average Agr~ of Mg3(PO4)z and Ba3(PO4)2 (Weast, 1972); (~"~)estimated from data on liquid Ces of Ca2P207 and CaPzO6 (Wcast, 1972); (hh)see text 210 and K' is taken to be 5 (Kress and Carmichael 1991). The Birch- Biggar GM, O'Hara MJ, Peckett A, Humphries DJ (1971) Lunar Murnaghan equation provides an excellent means of extrapolating lavas and the achondrites: Petrogenesis of protohypersthene the volumetric properties of liquids to quite elevated pressures. basalts in the maria lava lakes. Proc Lunar Planet Sci Conf However, it is non-linear in volume, which makes its use in thermo- 2:617 643 dynamic formulations cumbersome. The optimal values of (D2rr176 Bowen NL, Andersen O (1914) The binary system MgO-SiOz. Am ~P'~) reported in Table A1 allow our simplified polynomial repre- J Sci 37:487-500 sentation of liquid volumes to reproduce results calculated from the Burnham CW, Jahns RH (1962) A method liar determining the Birch-Murnaghan equation to pressures in excess of 4 GPa. Exten- solubility of water in silicate liquids. Am J Sci 260:721 745 sion of the higher order pressure derivative of volume to the model Carpenter M (1988) Thermochemistry of aluminum/siliconordering of Lange and Carmichael (1990) was necessitated in order to main- in feldspar minerals, In: Salje EKH (cd.) Physical properties and tain modcled liquid densities to bc less than those of associated solid thermodynamic behavior of minerals. Reidel, Dordrecht, pp phases at pressures up to 4 GPa. 265 323 Thermodynamic properties of the H20 component in molten Chen H-K, Delano JW, Lindsley DH (1982) Chemistry and phase silicate liquids are modeled following the method of Nicholls relations of VLT volcanic glasses from Apollo 14 and Apollo 17. (1980; Ghiorso et al. 1983). We adopt his value for the reference state Proc Lunar Planet Sci Conf 13 (J Geophys Res 87 Supple- entropy (152.63 J/tool-K) and optimize a value of the reference ment):A171-A181 enthalpy (-279.992 k J/tool) to satisfy best the experimental data on Delano JW (1977) Experimental melting relations of 63545, 76015, water solubility discussed above. We modify Nicholls' (1980) func- and 76055. Proc Lunar Planet Sci Conf 8:2097 2123 tion for the volume integral contribution to the molar Gibbs energy Delano JW (1980) Chemislry and liqnidus phase relations of Apollo of dissolved H20 by subtracting the Gibbs energy of supercritical 15 red glass: Implications for the deep lunar interior. Proc Lunar water tabulated by Robie et al. (1978) and adding the identical Planet Sci Conf 11:251 288 quantity from Berman (1988, see previous paragraphs). This insures Elkins LT, Grove TL (1990) Ternary feldspar experiments and the thermodynamic properties of the H20 component of Table A1 thermodynamic models. Am Mineral 75:54~559 are internally consistent with those of the supercritical fluid. Fram MS, Longhi J (1992) Phase equilibria of dikes associated with Proterozoic anorthosite complexes. Am Mineral 77:605 616 Fujii T, Bougalt H (1983) Melting relations of an abyssal tholciite and the origin of MORBs. Earth Planet Sci Lctt References 62:283-295 Gee LL, Sack RO (1988) Experimental petrology of melilite Adamkovib,)vS~ K, Kosa L, Proks I (1980) The heat of fusion of nephelinites. J Petrol 29:1233-1255 calcium silicate. Silikaty 24:193-201 Ghiorso MS (1983) LSEQIEQ: A FORTRAN IV subroutine pack- Agee CB, Walker D (1990) Aluminum partitioning between olivine age for the analysis of multiple linear regression problcms with and ultrabasic silicate liquid to 6 GPa. Contrib Mineral Petrol possibly deficient pseudorank and linear equality and inequality 105:243 254 constraints. Computers and Geosciences 9:391 416 Arndt NT (1977) Partitioning of nickel between olivine and ultra- Ghiorso MS (1985) Chemical mass transfer in magmatic processes I. basic and basic komatiite liquids. Canlegic Inst Washington Thermodynamic relations and numerical algorithms. Contrib Yearb 76:553 557 Mineral Petrol 90:107-120 Baker DR, Eggler DH (1987) Compositions of anhydrous and hy- Ghiorso MS (1987) Modeling magmatic systems: Thermodynamic drous melts coexisting with plagioclase, augite, and olivine or relations. In: Carmichael ISE, Eugster HP (eds) Thermodynamic low-Ca pyroxene fi'om 1 arm to 8 kbar: applications to the modeling of geological materials: minerals, fluids and melts. Aleutian volcanic center of Atka, Amer Mineral 72:12-28 (Reviews in Mineralogy vol 17). Minerological Society of Amer- Barnes SJ (1986) The distribution of chromium among ortho- ica, Washington. DC, pp 443 465 pyroxene, spinel and silicate liquid at atmospheric pressure. Ghiorso MS (1990a) Thermodynamic properties of hematite- Geochim Cosmochim Acta 50:1889 1909 ihnenite-geikielite solid solutions. 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Contrib Mineral Petrol system NazO-K20-CaO-MgO-FeO-Fe203-A1203-SiOz-TiO2- 90:121-141 H20-CO2: representation, estimation, and high temperature ex- Ghiorso MS, Carmichael ISE (1987) Modeling magmatic systems: trapolation. Contrib Mineral Petrol 89:168 183 petrologic applications. In: Carmichael ISE, Eugster HP (eds) Berman RG, Brown TH (1987) Development of models for multi- Thermodynamic modeling of geological materials: minerals, component melts: Analysis of synthetic systems, In: Carmichael fluids and melts. (Rcviews in Mineralogy vol. 17) Mineralogical ISE, Eugster HP, (eds). Thermodynamic modeling of geological Soceity of America, Washington DC, pp 467~499 materials: minerals, fluids and melts (Reviews in Mineralogy vol. Ghiorso MS, Kelemen PB (1987) Evaluating reaction stoichiometry 17). 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Ghiorso MS, Sack RO (1991) Fe-Ti oxide geothermometry: thermo- Kelemcn PB, Joyce DB, Webster JD, Holloway JR (1990) Reaction dynamic formulation and the estimation of intensive variables in between ultramafic rock and fractionating basaltic magma. I1. silicic magmas. Contrib Mineral Petrol 108:485-510 Experimental investigation of reaction between olivine tholeiite Ghiorso MS, Carmichacl ISE, Rivers ML, Sack RO (1983) The and harzburgite at 1150 1050~ and 5 kb. J Petrol 31:99 134 Gibbs free energy of mixing of natural silicate liquids; an Kelley KK (1960) Contributions to the data on theoretical meta- expanded regular solution approximation for the calculation of lurgy: part 13, High temperature heat content, heat capacity and magmatic intensive variables. Contrib Mineral Petrol entropy data for the elements and inorganic compounds. 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