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