# Activity–composition relations for phases in petrological calculations: an asymmetric multicomponent formulation

## Authors

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- Received:
- Accepted:

DOI: 10.1007/s00410-003-0464-z

- Cite this article as:
- Holland, T. & Powell, R. Contrib Mineral Petrol (2003) 145: 492. doi:10.1007/s00410-003-0464-z

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## Abstract

For petrological calculations, including geothermobarometry and the calculation of phase diagrams (for example,* P*–*T* petrogenetic grids and pseudosections), it is necessary to be able to express the activity–composition (*a*–*x*) relations of minerals, melt and fluid in multicomponent systems. Although the symmetric formalism—a macroscopic regular model approach to* a*–*x* relations—is an easy-to-formulate, general way of doing this, the energetic relationships are a symmetric function of composition. We allow asymmetric energetics to be accommodated via a simple extension to the symmetric formalism which turns it into a macroscopic van Laar formulation. We term this the asymmetric formalism (ASF). In the symmetric formalism, the* a*–*x* relations are specified by an interaction energy for each of the constituent binaries amongst the independent set of end members used to represent the phase. In the asymmetric formalism, there is additionally a "size parameter" for each of the end members in the independent set, with size parameter differences between end members accounting for asymmetry. In the case of fluid mixtures, for example, H_{2}O–CO_{2}, the volumes of the end members as a function of pressure and temperature serve as the size parameters, providing an excellent fit to the* a*–*x* relations. In the case of minerals and silicate liquid, the size parameters are empirical parameters to be determined along with the interaction energies as part of the calibration of the* a*–*x* relations. In this way, we determine the* a*–*x* relations for feldspars in the systems KAlSi_{3}O_{8}–NaAlSi_{3}O_{8} and KAlSi_{3}O_{8}–NaAlSi_{3}O_{8}–CaAl_{2}Si_{2}O_{8}, for carbonates in the system CaCO_{3}–MgCO_{3}, for melt in the melting relationships involving forsterite, protoenstatite and cristobalite in the system Mg_{2}SiO_{4}–SiO_{2}, as well as for fluids in the system H_{2}O–CO_{2}. In each case the* a*–*x* relations allow the corresponding, experimentally determined phase diagrams to be reproduced faithfully. The asymmetric formalism provides a powerful and flexible way of handling* a*–*x* relations of complex phases in multicomponent systems for petrological calculations.

## Introduction

In petrological mineral equilibria calculations, it is necessary to formulate the activity–composition (*a*–*x*) relations of multicomponent phases (minerals, fluids and melt). For example, calculation of the* PT* grid and pseudosections in the system Na_{2}O–CaO–K_{2}O–FeO–MgO–Al_{2}O_{3}–SiO_{2}–H_{2}O in White et al. (2001) involves phases which are solutions amongst three end members (alkali feldspars, plagioclase, garnet), four end members (orthopyroxene, biotite), and eight end members (silicate melt). In this context, the regular solution model and its expression as the symmetric formalism (SF, e.g. Powell and Holland 1993) has proved to be a powerful tool in representing the thermodynamics of phases. The main advantages of the SF are (1) that* a*–*x* relations of multicomponent phases may be derived from a knowledge of the constituent binary joins without recourse to ternary contributions, and (2) that the models are macroscopic, and so avoid the problems associated with formulating the many pair-wise microscopic (atomistic) interactions in complex multisite phases.

The principal disadvantage of the SF is that it is a symmetric model, and therefore cannot deal accurately with asymmetric mixtures, as reflected in such features as solvi which are skewed towards one end of a binary system. For many practical purposes, regions of a real asymmetric solid solution can be treated as a fictive symmetric solution; however, particularly in phase diagram calculations, a proper asymmetric solution model has become a necessity. An older model from the literature which has fallen into disuse, the van Laar model, is resuscitated and reformulated in this paper into a convenient form for multicomponent asymmetric solutions. It has both the advantages mentioned above for the symmetric formalism as well as bringing a considerable degree of flexibility to dealing with real asymmetric solutions. Although the van Laar model has been used in the geological literature before (e.g. Powell 1974, 1978; Saxena and Fei 1988; Shi and Saxena 1992; Aranovich and Newton 1999), it has not found favour as a general asymmetric model for solid solutions. Our reformulation of the van Laar model, coined the asymmetric formalism (ASF), allows it to be used in a straightforward and powerful way. This reformulation is outlined, then applied to the representation of the thermodynamics of K–Na–Ca feldspars, CaCO_{3}–MgCO_{3} carbonates, H_{2}O–CO_{2} mixtures and silicate melt in the system Mg_{2}SiO_{4}–SiO_{2}.

## Activity–composition relations

*a*–

*x*relations, the activity coefficient for an end member

*l*in a phase with

*n*(independent) end members, is given by

*q*

_{i}=1−

*p*

_{i}when

*i*=

*l*and

*q*

_{i}=−

*p*

_{i}when

*i*≠

*l*. The

*W*

_{ij}are macroscopic interaction energies. A proportion,

*p*

_{i}, is the macroscopic fraction of

*i*in the phase, for the particular set of end members used to represent the phase. Although \(\sum\nolimits_{k = 1}^n {p_k } = 1\), individual proportions may be negative. For a given phase, the proportions will change, in general, if a different independent set of end members is chosen. For phases with order–disorder, the number of independent end members,

*n*, is the number of independent macroscopic composition variables plus the number of independent order parameters (Holland and Powell 1996a, 1996b).

*α*

_{i}, such that, when the size parameters are different for the end members in a binary, asymmetry is introduced into that binary. The ASF expressions equivalent to Eq. (1) have the same form

*α*:

*q*

_{i}=1−

*φ*

_{i}when

*i*=

*l*, and

*q*

_{i}=−

*ϕ*

_{i}when

*i*≠

*l*, with

*φ*

_{i}effectively a size parameter-adjusted proportion

*W*

_{ij}* is a size parameter-adjusted interaction energy,

*α*values for all the end members are the same, Eq. (2) reduces to Eq. (1). In addition, the excess Gibbs energy of mixing is given by

Substituting \( A_{12} = {\textstyle{{2W_{12} } \over {V_1 + V_2 }}}\), this reduces to the activity coefficient expression derived from Eq. (2) for a binary. In relation to the form given for the van Laar model in Powell (1974, 1978), the* A*_{1} term therein is equivalent to \({\textstyle{{2V_1 W_{12} } \over {V_1 + V_2 }}}\), and* A*_{2} to \({\textstyle{{2V_2 W_{12} } \over {V_1 + V_2 }}}\). The advantage of the reformulation of the van Laar model in Eq. (2), in contrast to that in Prausnitz et al. (1986) and Powell (1974, 1978), lies in its ease of extension to multicomponent phases, and its clear separation of the interaction energy and size parameter terms.

*or*) in ternary feldspar is presented. In the SF it is

*A*site, and mixing of Al and Si on two

*T*sites,

This points to a problem with microscopic models used for activity modelling, as in the case of* W*_{aban}, that individual terms are not accessible experimentally. So, for example, the charge difference between Ca^{++} and Na^{+} means that no binary involving just this exchange can be investigated. In the macroscopic approach it remains useful to interpret a part of the overall mixing energy in terms of such microscopic interactions, while recognising that a macroscopic strain energy may make an important, even dominant, contribution.

*or*) in the ASF is

The above interpretation of the interaction energies for the SF carry over to the macroscopic interaction energies in the ASF with, additionally, such strain energy contributions from mixing cations or groups of cations of differing size or charge controlling the asymmetry. This is often expressed in large differences in molar volume between end members, although this effect may be masked in structures which have sufficiently flexible frameworks that the overall volume differences are relatively unaffected by substituting cations of different size or charge.

The application of the ASF both to fluids (e.g. H_{2}O–CO_{2}) and to solid solutions and silicate melt is straightforward. In the case of fluids, the molar volumes at the* PT* of interest may be used for the size parameters, whereas for solids and silicate melt the size parameters are treated as adjustable parameters which reflect some combination of the local size of interacting atoms or groups of atoms and the resulting strain effects on the mineral lattice. Although the equations are written such that one size parameter is assigned to each end member, the number of size parameters in an* n*-component solution is* n*−1 because it is the ratio of the size parameters which matters, and one of the size parameters may be arbitrarily set (say, at unity).

*x*=0.5 for a solution with site multiplicity of

*n*), the relationships can be shown graphically, as in Fig. 1. As the asymmetry increases, the critical temperature rises slightly, and the critical composition of the solvus top is nearly linearly related to the ratio of the size parameters. Thus, for moderately asymmetrical solvi, a value of

*W*may be approximately estimated from the regular solution expression, and the ratio of the size parameters found from the composition of the solvus crest.

*υ*

_{2}=0.6 at the critical temperature, but with varying temperature-dependence for

*υ*

_{2}, are drawn to show the wide variety of solvus shapes which can be accommodated using the van Laar model. Even more variability may be introduced by making

*W*a function of temperature as well.

In the petrological literature the subregular model has been much favoured (following Thompson and Waldbaum 1968a, 1968b) as a vehicle for representing asymmetry in solid solutions. This has been a convenient device for binary systems but becomes rather cumbersome when extended to ternary and higher-order solutions. In comparison to the subregular model, the number of overall parameters required in the ASF becomes significantly fewer as* n*, the number of independent end members in the phase, increases. The subregular model may also need extra higher-order terms which are not properties of the constituent individual binary solutions, these being a logical outcome of the third-order polynomial formulation of* G*_{ex}. The ASF, by contrast, is essentially a quadratic (second-order) model and does not include such ternary terms. This property of building up a complex model solely from its binary properties is a particularly attractive and useful aspect of the ASF model in multicomponent solutions.

## Examples

### Alkali (and ternary) feldspars

The first example involves the binary KAlSi_{3}O_{8}–NaAlSi_{3}O_{8} (sanidine–high albite) alkali feldspar join in which an asymmetric solvus has been experimentally determined. Although both albite and K-feldspar undergo Al–Si ordering, first onto two tetrahedral sites and, with further ordering, of Al onto a single favoured site at low temperatures, the effects of ordering on the solvus will not be taken into account here, the focus being on the high-temperature sanidine–albite solvus. Thompson and Waldbaum (1969) used the experimental data of Bowen and Tuttle (1950), Orville (1963) and Luth and Tuttle (1966) to construct a third-order Margules model (i.e. the subregular model) for the alkali feldspar solvus, and were able to reproduce the experimental data up to 10 kbar. Later experiments at high pressures (from 9 to 15 kbar) were performed by Goldsmith and Newton (1974) who found that the Thompson and Waldbaum model extrapolated remarkably well to the conditions of their experiments. We now (1) demonstrate that the ASF described above can reproduce the experimental data faithfully, and (2) use the binary model as a platform for the construction of a ternary feldspar solution model involving the additional anorthite component.

*α*

_{ab}and

*α*

_{san}) and the interaction energy

*W*

_{absan}for the binary. Because it is only the ratio of the two size parameters which determines the thermodynamics, one is set at unity (here

*α*

_{san}is chosen), leaving the other as the adjustable parameter. The two equilibrium equations relating compositions of coexisting feldspars are

*RT*ln

*X*

_{i}

*γ*

_{i}for each end member at each temperature for coexisting pairs, and taking experimental values for the

*X*

_{i}, the values for the unknown parameters may be found by non-linear regression, using the following equations for the activity coefficients

*T*in K and

*P*in kbar):

The size parameters for sanidine (1.0) and albite (0.643) are more disparate than would be suggested from the molar volumes of these end members (10.9 and 10.1 J bar^{−1} respectively) but are much closer to the relative ionic radii of K^{+} and Na^{+} (1.33 vs. 0.97). Presumably the size parameters are governed in part by factors which include the cation size mismatch and by the resulting local strain effects in the feldspar structure, whereas the molar volume differences between albite and sanidine are moderated by the flexibility of the Al–Si framework.

The KAlSi_{3}O_{8}–NaAlSi_{3}O_{8} solvus serves as a starting point in calibrating the coexisting plagioclase and alkali feldspar in the ternary feldspar system KAlSi_{3}O_{8}–NaAlSi_{3}O_{8}–CaAl_{2}Si_{2}O_{8}. This system has been investigated extensively by Ghiorso (1984), Green and Usdansky (1986), Fuhrmann and Lindsley (1988) and Elkins and Grove (1990), using the Margules formulation for mixing energetics. Elkins and Grove (1990) analysed their experimental results in the range 700–900 °C, with a one-site entropy of mixing and a subregular solution. The tie lines between calcic plagioclase and alkali feldspars show a clear change in slope relative to the tie lines between sodic plagioclase and alkali feldspars, this change occurring around the* an*_{50} composition. We have reinvestigated the system, using the Elkins and Grove experimental data, separating the plagioclase feldspars into an albite-rich* C*1̄ solid solution and an anorthite-rich* I*1̄ solid solution, using the Darken's quadratic formalism (DQF, Powell 1987) approach used by Holland and Powell (1992). Plagioclase compositions more anorthite-rich than given by the* C*1–*I*1̄ boundary, \( X_{an}^b = 0.12 + 0.00038\;T\left( {\rm{K}} \right) \) (Carpenter and McConnell 1984) are taken as belonging to the* I*1̄ solid solution.

*C*1̄ for albite and sanidine, and

*I*1̄ for anorthite. Thus, for the

*C*1̄ solid solutions, a Gibbs energy increment (the DQF parameter

*I*

_{an}) is required to convert the anorthite Gibbs energies to that of a fictive anorthite with the

*C*1̄ structure. Similarly, a Gibbs energy increment for albite in the

*I*1̄ structure feldspar is required. The model was fitted to the coexisting plagioclase and alkali feldspars from the experimental dataset of Elkins and Grove (1990) in two stages. Firstly, the

*C*1̄ coexisting feldspars (i.e. the pairs involving plagioclase with less than 50% anorthite content) were regressed to give the values for the additional parameters \(W_{ansan}^{C\overline 1 } \), \(W_{anab}^{C\overline 1 } \), and \(\alpha _{an}^{C\overline 1 } \). The resulting fits to the data are shown in Fig. 4 where the slopes of the tie lines and calculated compositions agree well with the original experimental data. The parameter values from the regression are

*I*1̄ plagioclase, of which there are only three experimental brackets, all at 900 °C. With so few brackets, it was decided to keep the values for \(W_{ansan}^{I\overline 1 } \) and \(\alpha _{an}^{I\overline 1 } \) the same as for the

*C*1̄ solutions, and solve for the value of \(W_{anab}^{I\overline 1 } \). The values for the DQF parameters

*I*

_{an}and

*I*

_{ab}are constrained from the position of the

*C*1–

*I*1̄ transition in plagioclase feldspars, as determined by Carpenter and McConnell (1984) as follows: for C1̄ plagioclase,

*I*1̄ plagioclase,

*RT*ln

*γ*at the composition of the phase boundary \(X_{an} = X_{an}^b \) gives expressions for

*I*

_{ab}and

*I*

_{an}

*W*parameters are known, the two

*I*parameters may be calculated as a function of temperature for the

*C*1–

*I*1̄ boundary, giving the additional model parameters used to generate Fig. 5:

The tie-line slopes for calcic plagioclase–orthoclase pairs in Fig. 5 do not match the experimental data perfectly. This may be in part due to the simplifications introduced here (taking a one-site entropy of mixing model, a simplified DQF plagioclase model, and the assumption that* W*_{ansan} is the same in* C*1̄ and* I*1̄ feldspars). Nevertheless, the model works remarkably well overall and, in addition, produces activity–composition relations for plagioclase feldspars which are very similar to those in Holland and Powell (1992).

### Calcite–dolomite–magnesite

*cc*, Ca

_{2}(CO

_{3})

_{2}), dolomite (

*dol*, CaMg(CO

_{3})

_{2}) and magnesite (

*mag*, Mg

_{2}(CO

_{3})

_{2}). Order–disorder is treated by assigning a free energy (Δ

*G*

_{R}) to the internal equilibrium relation

*N*moles of Ca to move from the

*M*1 to the

*M*2 site (and of Mg to the

*M*2 from the

*M*1 site), such that the site fractions become

*X*can be written as \( X = {\textstyle{1 \over 2}}x_{{\rm{Mg}}}^{M1} + {\textstyle{1 \over 2}}x_{{\rm{Mg}}}^{M2} \). The proportions (or mole fractions) of the end members are

*c*=

*cc*,

*d*=

*dol*, and

*m*=

*mag*)

The size parameters are dimensionless and normalised to unit value for magnesite. The degree of asymmetry exhibited by the calcite–dolomite solvus required a temperature-dependence for* α*_{cc}. The parameters have been chosen for the purposes of illustrating the model, but may be subject to revision when a fuller treatment of the calcite–magnesite–siderite system is attempted. We wish to emphasize that the miscibility gaps are not treated as two separate binary models, but that the single set of parameters listed above generates the complete phase diagram, including the changing state of order in the carbonates. In this respect, the state of order in the dolomite is predicted to be high even at the highest temperatures shown on the diagram.

### Forsterite–enstatite–silica melting

_{2}composition in the forsterite–silica (Mg

_{2}SiO

_{4}–SiO

_{2}) binary system. Symmetric models are inadequate for representing the phase equilibria, and we present an analysis of this system using the ASF model in Fig. 7, where the temperatures and compositions of the univariant equilibria forsterite–protoenstatite–liquid, protoenstatite–cristobalite–liquid, and cristobalite–two liquids are remarkably well reproduced by the model. The compositional binary is approximated by a fictive ternary made up of the end members forsterite liquid (

*foL*, Mg

_{4}Si

_{2}O

_{8}), enstatite liquid (

*enL*, Mg

_{8/3}Si

_{8/3}O

_{8}) and silica liquid (

*qL*, SiO

_{8}), all expressed in eight-oxygen units. These units are assumed to mix ideally in terms of the entropy of mixing, but the ASF is used to express the non-ideal enthalpic contribution. In this interpretation of the thermodynamics, the mole proportion of the protoenstatite "molecule" in solution is analogous to an order parameter, taking on values from zero for a disordered solution to unity for a fully ordered solution at the protoenstatite bulk composition.

*enL*composition is given by \( enL = {\textstyle{{\rm{2}} \over {\rm{3}}}}foL + {\textstyle{{\rm{1}} \over {\rm{3}}}}qL \) and, using the two variables

*x*and

*y*to represent bulk silica and order parameter respectively, the following relations hold:

*φ*values are calculated using the end-member proportions as derived above. The Gibbs free energy of the internal equilibrium reaction

*W*

_{foL enL},

*W*

_{foL qL}and

*W*

_{qL enL}as well as the size parameters

*α*

_{foL}and

*α*

_{qL}(with

*α*

_{enL}set at unity) were found by least-squares fitting to the experimental data of Bowen and Anderson (1914) and Greig (1927).

The temperatures and compositions of the calculated univariant equilibria in Fig. 7 are within ±2 °C and ±0.04 respectively of the experimental determinations. The univariant between two liquids and cristobalite is calculated at 1,707 °C, higher than the data of Greig (1927) but in better agreement with the more recent data of Ol'shanskii (1951).

The success of this model has been made possible by three assumptions about the mixing. Firstly, the use of an order–disorder model involving the* enL* component acting as a dominant species has allowed the thermodynamics of the* foL*–*qL* binary to behave like two nearly decoupled subsystems (*foL*–*enL* and* enL*–*qL*). Secondly, the use of eight-oxygen units gives some entropic asymmetry to the effective* enL*–*qL* sub-binary, and allows the liquid immiscibility gap to move to more silica-rich compositions. Thirdly, the van Laar model allows enough additional asymmetric behaviour to fit the phase diagram features quantitatively.

### H_{2}O–CO_{2}

Aranovich and Newton (1999) performed experiments to determine the mixing properties for binary H_{2}O–CO_{2} mixtures and used a van Laar expression to fit their results. They showed that, as well as fitting their results extremely well, the resulting activities were in close agreement with the Kerrick and Jacobs (1981)version of the modified Redlich Kwong (MRK) equation of state for these mixtures.

*h*=H

_{2}O,

*c*=CO

_{2}), and taking the size parameters to be the molar volumes of the end members at

*P*and

*T*gives

*W*

_{hc}as a function of pressure and temperature which, although fitting the data over the

*PT*range of their experiments, does not agree well with the MRK predictions at very high pressure. By comparing the value of

*W*

_{hc}as obtained by fitting to the MRK equation of state (Kerrick and Jacobs 1981), it is clear that

*W*

_{hc}appears to vary nearly linearly with the inverse molar volume of the mixture, in particular the product \(W_{hc} {\textstyle{{V_h V_c } \over {V_h + V_c }}}\) remaining virtually constant over the range 0.5–20 kbar, 400–1,000 °C. Thus, for H

_{2}O–CO

_{2}mixtures we have \(W_{hc} = a_{hc} {\textstyle{{V_h + V_c } \over {V_h V_c }}}\), with

*a*

_{hc}around 12.0 kJ

^{2}kbar

^{−1}. The van Laar expressions for Gibbs energy and activity coefficients can therefore be simplified considerably to

With this convenient functional form for the* PT* dependence of the van Laar* W* parameter, we refitted the experiments of Aranovich and Newton (1999), along with all the dehydration and decarbonation experiments used in building the dataset of Holland and Powell (1998), to find the optimum value for the van Laar interaction energy parameter given by the phase equilibrium data. The best fit obtained by least squares was* a*_{hc}=10.5±0.5 kJ^{2}kbar^{−1}. This value is slightly smaller than that derived from the MRK equation of state (12 kJ^{2}kbar^{−1}), suggesting that the latter slightly overestimates the non-ideality in H_{2}O–CO_{2} mixtures at the higher pressures and temperatures of the experiments. Following the same procedure, using the standard subregular model produced a less good fit of the data.

_{2}O–CO

_{2}system at a variety of pressures and temperatures. They are very similar to, but slightly more ideal than the Kerrick and Jacobs (1981) MRK activities over the range of

*PT*(0.5–20 kbar, 400–1,000 °C). The one-parameter ASF model presented here is only in semi-quantitative agreement with the precise molar volumes and activities determined by Blencoe et al. (1999) at 400 °C. It is probable that the parameter

*a*

_{hc}fails to remain constant once pressures and temperatures approach those of the critical point of H

_{2}O. This is not surprising, given that Blencoe et al. (1999) needed to use three separate and considerably more complex equations to reproduce their data as a function of pressure for just the 400 °C isotherm. It is, nevertheless, quite remarkable that a single adjustable parameter can fit the H

_{2}O–CO

_{2}activity data well over such a large

*PT*range, covering most of the facies in metamorphic petrological applications. H

_{2}O and CO

_{2}become immiscible at low temperatures, and Fig. 9 shows the calculated critical temperature and composition for immiscibility in the H

_{2}O–CO

_{2}binary as a function of pressure, and it suggests that unmixing may be common at conditions within the blueschist and low-temperature eclogite facies. The solvus becomes extremely asymmetric at low pressures where the molar volumes of H

_{2}O and CO

_{2}diverge markedly. The 1-kbar solvus crest at 273 °C and

*X*

_{CO2}=0.17 is in good agreement with the experimental determinations of Todheide and Franck (1963) and Takeneuchi and Kennedy (1964).

*a*

_{ij}parameters can represent the H

_{2}O–CO

_{2}activities well over a large range of

*PT*conditions, then it is likely that the approach will be useful in multicomponent fluids in the COH system. When values for van Laar

*W*

_{ij}from pairs of gases in the system H

_{2}O–CO

_{2}–CO–CO

_{2}–H

_{2}–CH

_{4}are determined using the Holloway-Flowers MRK equation of state (Holloway 1977; Flowers 1979), the following

*a*

_{ij}parameters (kbar

^{−1}) are obtained.

This table brings out the fact that mixing of H_{2}O with other species is very non-ideal whereas the other gases mix nearly ideally among themselves, a feature stemming from the polar nature of the H_{2}O molecule. With the* a*_{ij} values above, the* W*_{ij} may be found at any desired* PT* and the activity coefficients determined for complex gas mixtures through Eq. (2).

In summary, the examples above illustrate both the simplicity and the utility of the ASF model in tackling petrological problems with multicomponent solid and fluid solutions, especially where several constituent binary joins are quite asymmetric. The model should find particular application in the development of multicomponent melt models, as well as in increasing the flexibility in accounting for order–disorder in solid solutions by extending the SF approach of Holland and Powell (1996b).

## Acknowledgements

We thank Frank Spear, James Blencoe and Jamie Connolly for their valuable comments which led to improvement of the manuscript. Any errors remaining are ours.