On parameterizing thermodynamic descriptions of minerals for petrological calculations

On parameterizing thermodynamic descriptions of minerals forpetrological calculations

R. POWELL,1 R.W. WHITE,2 E .C.R. GREEN,2 T. J .B . HOLLAND3 AND J.F .A. DIENER4

1School of Earth Sciences, The Univeristy of Melbourne, Melbourne, Vic., 3010, Australia ([email protected])2Institute of Geoscience, University of Mainz, D-55099, Mainz, Germany3Department of Geological Sciences, University of Cambridge, CB2 3EQ, Cambridge, UK4Department of Geological Sciences, University of Cape Town, 7701, Rondebosch, South Africa

ABSTRACT A new regularization approach, termed micro-/, is outlined for parameterizing activity–composition(a–x) relations and other aspects of the thermodynamic descriptions of minerals for petrological cal-culations. In the context of the symmetric formalism, a formulation of a–x relations that is easilygeneralizable to multi-component minerals, parameterization with micro-/ extends from where thereare good data available to constrain, for example, interaction energies, to where there are little or nodata. This involves decomposing the interaction energies, which are macroscopic between end-mem-bers, into their microscopic components involving interactions between elements on sites. Micro-/involves heuristics to simplify and approximate these microscopic interaction energies prior to theirreassembly as macroscopic interaction energies. Micro-/ also allows parameterization of Fe2+–Mgorder–disorder between the sites of minerals. Application of micro-/ is illustrated by making a ther-modynamic description of chlorite in FeO–MgO–Al2O3–SiO2–H2O–O, FMASHO. Micro-/ has beendeveloped for use with the significant upgrade of the Holland & Powell internally consistent thermo-dynamic data set which required the re-evaluation of a–x relations of several common rock-formingminerals.

Key words: a–x relations; ferromagnesian silicates; Holland & Powell data set; micro-/.

INTRODUCTION

To perform thermodynamic calculations on rocks andto forward model metamorphic processes, thermody-namic descriptions of minerals are required inchemical systems that approximate nature. Bythermodynamic description is meant the combinationof the properties of the end-members and the a–xrelations, representing the energetics of how theend-members are combined to make solid solutions.A widely used family of mineral thermodynamicdescriptions (e.g. White et al., 2007; Diener & Powell,2012) was devised to work with the internally consis-tent data set of thermodynamic properties formineral, fluid and melt end-members (Holland &Powell, 1998).

The data set has recently been updated; the latestversion (Holland & Powell, 2011) supercedes, and isa significant advance on, that of Holland & Powell(1998) in its commonly used 2003 form (as ds55). Itincorporates new methodologies and much newexperimental and calorimetric data. The a–x relationsfor rock-forming minerals that had been used withsuccess with ds55 will not generally work successfullywith the new data set (now at ds62) because of themagnitude of the data set changes. Moreover, as dis-cussed further in the companion paper, the existingds55a–x relations are an ad hoc collection built up

since 1990, evolving as needed to meet currentrequirements, and as more realistic formulations wereadopted. This evolution culminated in White et al.(2007), at least for minerals in rocks of pelitic compo-sition.Although appropriate-looking phase equilibria

could be calculated with ds55 and the White et al.(2007) a–x relations, the incoherence of the a–x rela-tions was hardly satisfactory. The development of theHolland & Powell (2011) data set provides the impe-tus, and the opportunity, to generate a consistentlyformulated set of a–x relations for the ferromagne-sian minerals using a new set of principles for para-meterizing a–x relations. This new set of principles isthe focus of this paper, and a set of a–x relationsgenerated with it is the concern of the companionpaper.Most minerals are more or less complex multi-

component solid solutions, for which typically thethermodynamics are only experimentally well con-strained in a few compositional dimensions. For exam-ple, for silicates that lie dominantly in the systemKFMASHO (K2O–MgO–FeO–Al2O3–SiO2–H2O–O,with O representing Fe2O3), adequate experimentalconstraints commonly occur only in KMASH.Regardless of the weakness of the constraints, forexample in extending from KMASH to KFMASHO,useful petrological calculations cannot be undertaken

© 2013 John Wiley & Sons Ltd 245

J. metamorphic Geol., 2014, 32, 245–260 doi:10.1111/jmg.12070

without making such extensions. This is done by tak-ing an approach, referred to here as ‘regularization’,where various heuristics are used to determine parame-ters that are needed but are unknown, or not con-strained well enough. Such heuristics, experience-based rules of thumb or educated guesses, may benumerical values, or a relationship amongst values.The aim is that, in a complete range from detailedinformation to no information, regularization shouldallow a complete transition from a value being fixed byavailable data to a value being fixed by a heuristic.Regularization, being based on a perception of whatwe think we know, amounts to use of prior knowledgein a Bayesian sense (e.g. Robert, 2007).

The idea of regularization is certainly not new. Forexample, a time-honoured approach for inferringmaterial properties is to establish correlationsbetween measured properties so that the full set ofproperties of an unknown material can be estimatedfrom just some of its properties (e.g. Davies & Nav-rotsky, 1983). Such correlations might be empiricalor be based on some approximate underlying physi-cochemical relationship. In fact, a key part of thephilosophy behind the generation of the internallyconsistent thermodynamic data set (Holland & Pow-ell, 1985 1990, 1998, 2011) has been regularization.In data set generation, entropy, volume, etc. of theend-members are taken as known, and the enthalpiesare solved for by weighted least squares using avail-able calorimetric and experimental data. For manyend-members, not all of the ‘known’ properties havebeen well enough measured or measured at all, yet itis necessary for such a data set to include these end-members to provide wide enough compositional cov-erage for useful petrological calculations. Heuristicsof various types are used to adopt values for poorlyconstrained or unconstrained parameters.

For example, commonly, the entropy of an end-member is not well known (see the S data-source col-umn in Holland & Powell, 2011, appendix 1), inwhich case a value is estimated using a heuristic thatconsists of a volume-adjusted oxide additivity ofentropy (Holland, 1989). The calculated enthalpies offormation for end-members will be strongly corre-lated with their estimated entropies via the experi-mental data used in the data set generation, so,unless the entropy estimation is quite incorrect, thecombination of the entropy and the enthalpy of for-mation together will tend to be of reasonable size.Such mutually compensating correlations can con-tribute greatly to the chance of successful regulariza-tion. Heuristics are also used to infer values in theequations of state used in ds55 and in ds62 wherethese cannot be extracted from volume measure-ments. For example, in the former, the first pressurederivative of the bulk modulus, j00, was derived fromvolume measurements when they were available, butthe heuristic j00 ¼ 4 was used when there were littleor no data (see discussion in Holland & Powell,

1998). This gave an appropriate shape to the P–Tdependence of G when data were insufficient to esti-mate j00 directly.Whereas the algebraic expression – formulation –

of the a–x relations for ds62 have the same structureas for the a–x relations used with ds55, the values,for example, of the interaction energies – parameteri-zation – are made in a new way here. Heuristics havebeen used before in the parameterization of a–x rela-tions, although commonly on an ad hoc basis, withlittle formal discussion. The a–x relations for usewith ds62 retain some of the previously used heuris-tics while incorporating a major new approach tomake a coherent regularization framework, referredto as micro-/. This new approach involves a recast-ing of macroscopic interaction energies that are pair-wise between end-members, to be equivalentlyexpressed as interactions between the cations thatwould mix on the sites in a binary between theseend-members. Simplifications and approximations atthis level then lead to the assignment of values to themacroscopic interaction energies. Through thisapproach, parallels can be drawn between differentmineral groups to aid parameterization where dataare inadequate or missing.To put micro-/ into context, a general formulation

of the thermodynamics of minerals is presentedbefore the new parameterization is developed.Throughout, the various aspects of theoretical discus-sion are illustrated using the example of chlorite inFMASHO. The thermodynamic properties of theend-members and the a–x relations in MASH forchlorite are reasonably well-established (Hollandet al., 1998; Holland & Powell, 2011). However,information to allow extension into FMASH is notstrong, in the absence of experimental constraints onend-member properties or a–x relations. There is stillless information to extend into FMASHO, althoughchlorite commonly contains significant Fe3+ (e.g.Dyar et al., 2002). As a consequence, chlorite is agood mineral to illustrate the use of micro-/. Theheuristics used in micro-/ for chlorite are then usedfor other ferromagnesian minerals as found inrocks of pelitic composition in the companion paper(White et al., 2014). There may be found the currenta–x relations for use in THERMOCALC with ds62, avail-able now on the Mainz web-site (http://www.metamorph.geo.uni-mainz.de/thermocalc/).

FORMULATION

Systems, components and end-members of minerals

Most minerals show significant solid solution, andeven those minerals like quartz that are commonlytaken to be pure phases in petrological calculationscontain minor solid solution, e.g. quartz involvingTiO2, as exploited in Ti-in-quartz thermometry (e.g.Wark & Watson, 2006). Solid solution at some level

© 2013 John Wiley & Sons Ltd

246 R . POWELL ET AL .

involves all oxides (or indeed elements) in all miner-als, but, in practice, we can only consider those com-positional dimensions that involve measureable solidsolution, and for which we can know the thermody-namics. The practical limit in the consideration ofsolid solution is that the compositional dimensionsthat need to be included are the ones that directlyaffect the macroscopic behaviour of the mineral inequilibria. So, for example, unless the TiO2 contentof quartz is of direct interest, and given that its pres-ence in quartz barely affects quartz stability inquartz-bearing equilibria, TiO2 would not normallybe included in the thermodynamic description ofquartz used in petrological calculations (e.g. P–Tpseudosections).

In the context here of a focus on ferromagnesiansilicates, the compositional dimensions of the miner-als are represented in terms of particular linear com-binations of oxides. The oxides are then thecomponents used to represent the systems for miner-als, as well as the overall system in which calculationsare performed (e.g. FMASHO). As noted above, thenumber of composition dimensions of a real mineralis very much greater than the dimensions of the‘model’ mineral, and the number of compositiondimensions of a system is commonly greater thanthat of any one model mineral within it.

The particular linear combinations of oxides usedto represent the composition of a (model) mineral arecalled its end-members. They have a fixed composi-tion, and have the stoichiometry of the phase. Forexample, CaMgSi2O6 (diopside) and Ca1

2AlSi2O6

(Ca-Eskola molecule) are end-members of clinopy-roxene (cpx). That Ca-Eskola molecule has pyroxenestoichiometry can be seen when it is written asCa1

2h1

2AlSi2O6. It is necessary to make a distinction

among three sorts of end-members, referred to hereas microscopic, macroscopic and compositional.

Microscopic end-members (or, in later sections, justend-members) are those for which the internal organi-zation of the end-member is completely specified, aswell as its structure being that of the mineral that theend-member belongs to. So, considering ilmenite, suchend-members are ordered ilmenite FeATiBO3 (oilm)and disordered ilmenite FeA1

2

TiA12

FeB12

TiB12

O3 (dilm), in

the ilmenite structure (the superscript denotes the sitethat the element occupies). Microscopic end-membersplay a central role in the formulation of the thermody-namics of minerals.

Macroscopic end-members are those for which theinternal organization of the end-member is the equi-librium one, in a specified structure. So ilmeniteFeTiO3 can be a macroscopic end-member if it is inits equilibrium state of order, in a specified structure.If this ilmenite is considered to be made of the micro-scopic end-members oilm and dilm, the equilibriumstate can be determined from the internal equilib-rium, oilm = dilm, and in general, this will be a func-

tion of P–T. The end-members in the Holland &Powell data set are macroscopic end-members (seebelow).Compositional end-members are those for which

the composition is specified, but not its internal orga-nization or structure. So FeTiO3 can be a composi-tional end-member of ilmenite solid solution,indicating only a composition, with the equilibriumstate of order and the proportions of oilm and dilmit contains being unspecified. Compositional end-members are used to represent mineral composition.In the Holland & Powell data set, the intention is

that all end-members of minerals are macroscopicend-members, the data set containing the informationto give equilibrium properties as a function of P–T.Measured properties on which the data set is basedare those of the macroscopic end-member, if the end-member is in its equilibrium state at the P–T involved(and at its specified fixed composition). In fact, themajority of end-members of minerals in the data sethave no discernible internal organization, like order–disorder, to be considered, and they are each equiva-lent to one microscopic end-member. Handlingmeasured properties in terms of a set of microscopicend-members, for example, ilmenite by oilm and dilm,involves an interpretation of the internal organizationof the macroscopic end-member. When THERMOCALC

(or another program) interrogates the data set to docalculations with a macroscopic end-member, or printits properties, it must first calculate its equilibriumstate of order at the specified P–T from the informa-tion provided in the data set if the end-memberinvolves order–disorder. In the case of ilmenite, theorder–disorder information is actually parameterizedin terms of Landau theory in the data set, but, forother macroscopic end-members, it is in terms of thesymmetric formalism (SF), and an equilibrium rela-tionship equivalent to oilm = dilm is used explicitly.While macroscopic end-members are central to the

idea of a data set, with or without order–disorder, insolid solutions, they need to be considered in termsof their constituent microscopic end-members forcombination with other microscopic end-members.For example, the macroscopic end-member, ilmenite,when considered in the context of a ilmenite-hematitesolid solution, needs to be represented in terms ofoilm and dilm first, before combining with hem. Inthe context of minerals that are solid solutions,macroscopic end-members do not feature directly inthe thermodynamic development below, in that theyare always considered in terms of their constituentmicroscopic end-members. In fact, macroscopic end-members can easily be considered as a special case ofsolid solutions as they are just fixed-compositionstoichiometric phases (see Holland & Powell, 1996a).The idea of an independent set of end-members is

essential in expressing the thermodynamics of minerals.From the discussion above, it is apparent that two


PARAMETER IZ ING THERMODYNAMIC DESCR I PT IONS 247

types of independent set are of interest. The first is anindependent set sufficient to represent the compositionof the mineral, irrespective of its internal organization,numbering n compositional end-members. The secondis an independent set sufficient to represent not onlythe composition of the mineral but also its internalorganization, numbering n + s microscopic end-mem-bers. There will then be n � 1 independent composi-tion parameters, and s order parameters, requiringsolving of internal equilibria to determine them.

There are commonly several independent sets ofcompositional and also microscopic end-membersthat can be written for the same solid solution. Lin-ear transforms relate the thermodynamic descriptionsas expressed in different sets (Powell & Holland,1999). In choosing independent sets, there are someguidelines that make the resulting thermodynamicdevelopment simpler. First the idea of ‘base level’ inthe thermodynamic description of a mineral needs tobe introduced. This is the G-plane defined by the Gof n microscopic end-members (Gmech as definedbelow), on which the mixing properties of the mineralare built (Gmix as defined below). The first n of themicroscopic end-members in the independent set ischosen to be that defining the base level. There arethree categories of microscopic end-members that canbe involved in defining the base level.

First, there are data set end-members without order–disorder that can therefore be treated as microscopicend-members. This category accounts for the majorityof the end-members in the data set. A superficial diffi-culty arises when there are dependent end-members ofa mineral in the data set, end-members that are a lin-ear combination of other data set end-members, all iso-structural. Note that the linearity also relates to thestate of order–disorder, so clinochlore, which is anordered end-member, is not a linear combination ofAl-free chlorite and amesite, whereas disordered cli-nochlore would be (see below). Only independent end-members can be used to define the base level, the dataset information for those end-members omitted beingignored in the thermodynamic description.

Second, there are data set end-members thatinvolve order–disorder. As discussed above, thesemacroscopic end-members need to be considered interms of their constituent microscopic end-memberswhen combined with other end-members. Just one ofthese can be used as part of the base level, the other(s) potentially being part of the independent set ofmicroscopic end-members, representing the state oforder. Considering binary ilmenite–hematite, the baselevel may be defined in terms of dilm–hem, whileoilm represents the state of order (see White et al.,2000, appendix 1, and the axfile on the Mainz THER-

MOCALC website).Third, there are end-members not in the data set

(non-data set end-members). Such microscopic end-members are constructed as a linear combination ofend-members from the first two categories to gener-

ate the composition of the new end-member.Although a new end-member has the correct compo-sition, making the same linear combination of the Gof the end-members will not provide the correct G.This has to be estimated, as discussed later (in THER-

MOCALC coding this G difference between the linearcombination and the actual G is referred to as aDQF). This category includes most ferric end-membersof ferromagnesian silicates and they are dealt withspecifically in a later section.The compositions of the n microscopic end-

members involved in defining the base level then givethe independent set of compositional end-members.The independent set of microscopic end-members iscompleted with s microscopic end-members thatrepresent the complete range of order–disorder thatthe mineral may show.Following the above guidelines, the binary system

Fe2+–Mg orthopyroxene, with n = 2, involves thedata set end-members, enstatite and ferrosilite, sothen as the microscopic end-members, en(MgM1MgM2Si2O6) and fs (FeM1FeM2Si2O6), theyconstitute the first two end-members of the indepen-dent set of microscopic end-members. As Mg2Si2O6

and Fe2Si2O6, they make up the independent set ofcompositional end-members. The independent set ofmicroscopic end-members is completed with anordered end-member fm, MgM1FeM2Si2O6, account-ing for the internal organization, with order–disorderinvolving Fe2+–Mg partitioning between M1 and M2.Note that the simple linear combination of en and fswould give the disordered microscopic end-memberMgM1

12

FeM112

MgM212

FeM212

Si2O6.

In a solid solution, the proportions of the micro-scopic and compositional end-members in theirrespective independent sets are needed in representingthe thermodynamics, noting that the proportions aredependent on the choice of independent set. InFig. 1a for an enstatite–ferrosilite solid solution of agiven composition, the proportions of the composi-tional end-members, Pen and Pfs, are indicated abovethe box, and the proportions of the microscopic end-members, pen, pfs and pfm, below the box. Given thatPen+Pfs = 1 and pen+pfs+pfm = 1, by inspection,Pen ¼ pen þ 1

2 pfm, Pfs ¼ pfs þ 12 pfm. Block diagrams

like Fig. 1a are a useful device for representing sim-ple solid solutions, and to see the relationshipbetween the proportions in the compositional andmicroscopic independent sets. However note that, ingeneral, proportions of compositional and micro-scopic end-members may be negative.Considering binary ilmenite–hematite, n = 2, bring-

ing together the details considered above, the data setincludes ilmenite as a macroscopic end-memberinvolving order–disorder. The first two end-membersin the independent set of microscopic end-membersare one of oilm and dilm, say dilm, and hem. Thenthe independent set of compositional end-members is



FeTiO3 and Fe2O3. The remaining microscopic end-member of the macroscopic ilmenite end-member,oilm, completes the independent set of microscopicend-members (n + s = 3). In Fig. 1, by inspection,Pilm = poilm+pdilm and Phem = phem.

These examples involve order–disorder, i.e. withs > 0. In the simple case, where s = 0, the composi-tional and microscopic end-members can be thesame, and Pi = pi. For example, for binary enstatite–ortho-diopside, with Ca only on M2, then Pen = penand Podi = podi.

Setting up a chlorite thermodynamic description

Chlorite is used as an illustrative example below.Starting in FMASH, chlorite is a reciprocal solution

between amesite [ames, Mg4Al4Si2O10(OH8) and itsFe2+-equivalent (fame, Fe4Al4Si2O10(OH8)], and Al-free chlorite (afchl, Mg6Si4O10(OH8)) and its Fe2+-equivalent, fafchl, Fig. 2a. It is ternary (n = 3), so thefirst n of the n+s microscopic end-members need tocorrespond to end-members in the data set if theguidelines above are to be followed. The chloriteend-members in the data set are afchl, ames, clin (cli-nochlore, ordered Mg5Al2Si4O10(OH8), see below),and daph (daphnite, the Fe2+-equivalent of clinoch-lore). Three end-members are needed to define abase level for chlorite, and these are then

M1

M2

Mg Fe

en

en

fs

fsfm

A

B

Fe3

Mg Fe

P P

p p p

Fe3 Fe Ti

Ti Fe

P P

p p p

hem

hem

ilm

oilmdilm

(a)

(b)

Fig. 1. Site distributions for a given composition of (a) binaryFe2+-Mg orthopyroxene and (b) binary ilmenite-hematite,showing the proportions of compositional and microscopicend-members (see text).

M4

M1

Al

Al Mg

Mg

P P

p p p

ames

ames

afchl

afchlclin

(b)

x

y

afchlochl4ochl1fafchl

clindaph

AmesFame

0

1

(a)

Fig. 2. (a) The compositions of the end-members of chloritein FMASH represented in terms of x ¼ Fe

FeþMg (increasing tothe left) and y ¼ net Tschermaks ¼ 1

2 ðxAlM1 þ xAlM4Þ(increasing downwards). Filled black circles mark the chosenindependent set of microscopic end-members, filled light greycircles are compositional end-members and unfilled circles aredependent microscopic end-members. (b) The proportions ofthe compositional and microscopic end-members for a givenchlorite composition in MASH.



afchl+ames+daph given that clin is in fact a ‘hard-wired’ ordered microscopic end-member of chlorite(see below). These first n microscopic end-members inthe independent set are given in the top half of thetable:

M1 M23 M4 T2

Mg Fe Al Mg Fe Mg Fe Al Si Al

afchl 1 0 0 4 0 1 0 0 2 0

ames 0 0 1 4 0 0 0 1 0 2

daph 0 1 0 0 4 0 0 1 1 1

clin 1 0 0 4 0 0 0 1 1 1

ochl1 1 0 0 0 4 0 1 0 2 0

ochl4 0 1 0 4 0 1 0 0 2 0

The independent set of compositional end-membersfor chlorite is then given by the compositions of af-chl, ames and daph, and they are sufficient to repre-sent the composition of any chlorite in FMASH.Once the independent set has been chosen, othercompositional end-members, such as fame and fafchlin this example, become dependent compositionalend-members. The compositions of the end-membersare shown in Fig. 2a. The proportions of all three ofthe chosen compositional end-members are positivewithin the shaded triangle, but with the proportionof ames negative outside it to low Al compositions,and afchl negative to high Al compositions (Fig. 2a).

The internal organization within the solid solutioninvolves order–disorder relating to distribution ofMg, Fe2+ and Al between the octahedral sites. Cli-nochlore (clin), an Mg–Al ordered end-member (Hol-land et al., 1998), is chosen to represent the Mg–Alordering between M1 and M4, not least because it isa data set end-member. As Fe2+ and Mg are distrib-uted across three different sites, two order parame-ters are required to uniquely specify the Fe2+–Mgdistribution at a given macroscopic composition.This entails two ordered Fe2+–Mg end-members, cho-sen to be ochl1 and ochl4. Thus s = 3, and n + s = 6.The compositions and internal organization of theordered end-members are given in the lower part ofthe table, and their compositions shown in Fig. 2a.Note that with ds55 the Fe2+–Mg substitution inchlorite was handled by equipartition, in which theFe2+/Mg ratio was taken to be the same across thesites in which Fe2+ and Mg reside. However, this wasshown to be an inappropriate approach in Holland& Powell (2006), thus requiring the explicit handlingof Fe2+–Mg order–disorder. The compositional andmicroscopic proportions in MASH are shown in Fig.2b. The corresponding relationships in FMASH arenot shown as they are too difficult to representgraphically. Instead, the algebraic expressions for theproportions of the end-members, as well as the sitefractions, are given in the Appendix, along with thedefinitions of the composition parameters and orderparameters used in them.

Gibbs energy of a mineral

The molar Gibbs energy of a mineral, G, can be writ-ten in terms of its n + s independent microscopicend-members:

G ¼Xnþs

i¼1

pili (1)

in which li is the chemical potential of i, and pi is theproportion of the end-member i in the mineral, withPnþs

i¼1 pi ¼ 1. Note that the values of pi depend onthe choice of the independent set of end-members(and some of the pi may be negative). The li aredefined in the usual way by:

li ¼@mG

@mi

� �mkðk 6¼iÞ

with pi ¼ mi

m; and m ¼

Xnþs

j¼1

mj

(2)

where mi is the number of moles of i and m is thetotal number of moles (e.g. Holland & Powell,1996b, appendix 1).Setting aside the special case of G at equilibrium

for a later section, the chemical potentials in (1) maybe expanded:

G ¼Xnþs

i¼1

pili ¼Xnþs

i¼1

pi Gi þ RT ln aið Þ

¼Xnþs

i¼1

piGi þXnþs

i¼1

piRT ln ai

(3)

in which Gi is the Gibbs energy of pure end-memberi in the structure of the mineral, and ai is its activity.Then expanding the terms in the first sum of the finalpart of (3) that involve the s end-members represent-ing order–disorder:

Xnþs

i¼nþ1

piGi �Xnþs

i¼nþ1

piGmakei þ

Xnþs

i¼nþ1

piDGi (4)

noting that these disappear (i.e are zero) if s = 0.Considering one of these end-members, h, Gmake

h isthe G on the base level at the composition of h. It isthe linear combination of the G of the first n end-members in the independent set in the same combina-tion as required to give the composition of h. In (4),DGh gives the displacement of Gh from the G baselevel, so Gh ¼ Gmake

h þ DGh. DGh represents theenergy involved in reaching the required state oforder in h from the state of order of the simple end-member combination used in constructing Gmake

h . Forthe enstatite–ferrosilite example above, Gmake for theordered end-member, fm (MgM1FeM2Si2O6), is givenby Gmake

fm ¼ 12Gen þ 1

2Gfs. Given that en and fscombine to give the disordered end-member,



MgM112

FeM112

MgM212

FeM212

Si2O6 (dfm), whereas the

ordered end-member fm is wanted, then DGfm =Gfm � Gdfm ¼ Gfm � ð12Gen þ 1

2GfsÞ; representing theenergetics of Fe2+–Mg ordering on M1 and M2 infm.

Expanding (3) using (4) and grouping terms gives

G ¼Xni¼1

piGi þXnþs

i¼nþ1

piGmakei

!

þXnþs

i¼nþ1

piDGi þXnþs

i¼1

piRT ln ai

!:

(5)

The terms are grouped to separate the contributionsto the thermodynamic description that relate to thebase level (first bracket), and the contributions fromthermodynamic mixing and order–disorder (secondbracket). In the first bracket, the Gmake terms forthe ordered end-members can be written in terms ofbase-level end-members, analogous to Gmake

fm ¼ 12Gen

þ 12Gfs: The combination of the pi that remains after

the expansion is equal to the proportions of thecompositional end-members, Pi. The first bracket isthe Gmech (Gibbs energy of mechanical mixing)term, and would be calculated from the data set.The second bracket contains the contribution madeto G by the activities of the end-members ai, andalso by the displacements DGi of the ordered end-members from the G of the base level. The activi-ties, ai, are obtained via the a–x relations sensustricto, but, sensu lato, the a–x relations mustinclude the DGi. The second bracket is then Gmix.Summarizing,

Gmech ¼Xni¼1

piGi þXnþs

i¼nþ1

piGmakei ¼

Xni¼1

PiGi (6)

Gmix ¼Xnþs

i¼nþ1

piDGi þXnþs

i¼1

piRT ln ai (7)

For s = 0, the first sum in Gmix is zero, and the sec-ond sum can be written in terms of Pi (given thatpi = Pi when s = 0), so in this caseGmix ¼ Pn

i¼1 PiRT ln ai.

Gibbs energy of chlorite

The development above is illustrated for the chloriteexample. First the Gmake are:

Gmakeclin ¼ 1

2Gafchl þ 1

2Games

Gmakeochl1 ¼

1

2Gafchl � 1

2Games þ Gdaph

Gmakeochl4 ¼

9

10Gafchl � 1

10Games þ 1

5Gdaph

Then Gmech from (6) is

Gmech ¼Xni¼1

piGi þXnþs

i¼nþ1

piGmakei

¼ pafchlGafchl þ pamesGames þ pdaphGdaph

þ pclin1

2ðGafchl þ GamesÞ

� �

þ pochl11

2Gafchl � 1

2Games þ Gdaph

� �

þ pochl49

10Gafchl � 1

10Games þ 1

5Gdaph

� �

¼ pafchl þ 1

2pclin þ 1

2pochl1 þ 9

10pochl4

� �Gafchl

þ pames þ 1

2pclin � 1

2pochl1 � 1

10pochl4

� �Games

þ pdaph þ pochl1 þ 1

5pochl4

� �Gdaph

¼ PafchlGafchl þ PamesGames þ PdaphGdaph

and substituting for pi using the compositional vari-ables from the Appendix, gives

Pafchl ¼ pafchl þ 1

2pclin þ 1

2pochl1 þ 9

10pochl4

¼ 1� y� 1

5xð3� yÞ

Pames ¼ pames þ 1

2pclin � 1

2pochl1 � 1

10pochl4

¼ y� 1

5xð3� yÞ

Pdaph ¼ pdaph þ pochl1 þ 1

5pochl4 ¼ 2

5xð3� yÞ

These Pi are the proportions of the compositionalend-members in chlorite, for example, in Fig. 2a.From these equations, it can be seen that the linesbounding the shaded area, where Pames = 0 andPafchl = 0, are slightly curved, with y ¼ 3x

5þx andy ¼ 5�3x

5�x respectively. Gmech defines the base level forG of chlorite, independent of the ordered end-mem-bers and the order parameters. Gmix from (7) is:

Gmix ¼Xnþs

i¼nþ1

piDGi þXnþs

i¼1

piRT ln ai

¼ pafchlRT ln aafchl þ pamesRT ln aames

þ pdaphRT ln adaph

þ pclinðDGclin þ RT ln aclinÞþ pochl1ðDGochl1 þ RT ln aochl1Þþ pochl4ðDGochl4 þ RT ln aochl4Þ

which includes the G difference of the ordered end-members from the chlorite base level, DGclin, DGochl1

and DGochl4.



Gibbs energy of a mineral at equilibrium

To find the Gibbs energy of a mineral at equilibriuminvolves solving for the equilibrium state of order andsubstituting it into a general expression for G, such as(3). To solve for the state of order, the followingmethod can be used (see also appendix 1 of each ofHolland & Powell, 1996a,b). Differentiating G withrespect to each order parameter, Qi, at constant compo-sition, the equilibrium state of order can be found from:

@G

@Qi

� �Qjðj 6¼iÞ

¼ 0 for i ¼ 1; 2; . . .; s (8)

More convenient – because the result is in terms ofchemical potentials – is to substitute for G using (1),and then differentiate with respect to Qi using thechain rule:

@G

@Qi

� �Qjðj 6¼iÞ

¼Xnþs

j¼1

@pj@Qi

� �Q‘ð‘ 6¼iÞ

lj þXnþs

j¼1

pj@lj@Qi

� �Q‘ð‘ 6¼iÞ

¼Xnþs

j¼1

@pj@Qi

� �Q‘ð‘ 6¼iÞ

lj ¼ 0

(9)

given that the second sum in the middle expression isidentically zero at equilibrium by Gibbs–Duhem. Theresult, for i = 1,2,…s, generates an independent set ofn equations that can each be written as Dl = 0, i.e.equilibrium relations. They can be organized to cor-respond to reactions between each of the s orderedend-members and the first n end-members in theindependent set. These may be more convenient forcalculating the equilibrium state of order than using(9) directly, even though it involves solving a set ofsimultaneous non-linear equations.

With the equilibrium state of order, Gequil can beformed from (1) by substituting the equilibrium Q,and then Gequil can be written as:

Gequil ¼Xni¼1

Pilequili (10)

in which the sum is only over n end-members, withPi the proportion of the compositional end-member i,and lequili being the chemical potential of end-mem-ber, i, at equilibrium. On the one hand, lequili mightbe derived directly from measurement on a phase atequilibrium. On the other, li can be formulated as in(2) and (3), including its dependence on order–disor-der. Then (li)equil is li with the equilibrium state oforder substituted. When at the composition of i thereis just one microscopic end-member in the indepen-dent set of microscopic end-members, thenlequili ¼ ðliÞequil: When there is more than one end-member at the composition of i, as in ilmenite–hema-tite with microscopic end-members oilm–dilm–hem,then lequililm ¼ ðldilmÞequil ¼ ðloilmÞequil; as shown in

Holland & Powell (1996b, appendix 1).

Gibbs energy of chlorite at equilibrium

Calculating the equilibrium state of order involvessolving an independent set of internal equilibria forthe order parameters, QAl, Q1 and Q4, as defined inthe Appendix. A particular choice of independent setinvolves each of the s end-members with the first nend-members:

lclin ¼ 1

2lafchl þ

1

2lames

lochl1 ¼1

2lafchl �

1

2lames þ ldaph

lochl4 ¼9

10lafchl �

1

10lames þ

1

5ldaph

(11)

These are then associated with the Gibbs energies ofreaction, DGclin, DGochl1 and DGochl4 respectively. Atequilibrium

Gequil ¼ Pafchllequilafchl þ Pamesl

equilames þ Pdaphl

equildaph

with the chemical potentials being evaluated at theequilibrium state of order. This form for Gequil bringsout that chlorite is ternary (n = 3) macroscopically,at equilibrium. Alternatively, substituting the internalequilibria (11) in (1) gives the reduction at equilib-rium from n + s terms of form pili, to n terms ofform Pili directly.

Entropic and enthalpic contributions in a–x relations

A characteristic of effectively all current a–x relationsof minerals for petrological calculations is the separa-tion of the entropic and enthalpic parts of the ener-getics, without a feedback between the two (e.g.Powell & Holland, 1993). The entropic part is writtenin terms of ideal mixing on sites, while the enthalpicpart is written in terms of the SF (macroscopic regu-lar model) involving pair-wise interaction energies (ora variant of this model). This is physically unrealisticas in reality there must be feedbacks between the twocontributions (the more interaction energies departfrom zero, the more the entropic contribution willdepart from that of ideal mixing on sites). The reasonfor this approximation of actual energetics is thatshort-range order cannot be modelled in general. Forsmall departures from ideality – much less than isneeded to generate a solvus, for example – theso-called quasichemical model can be used (Guggen-heim, 1952). Although this can be generalized in vari-ous ways (e.g. Powell, 1983), it is not clear that thegains merit the additional computational complexity.Writing the entropic and enthalpic parts separately issimple, and easy to generalize to multi-componentminerals, which may also involve order–disorder (e.g.Powell & Holland, 1993; Holland & Powell, 1996b),so this approach continues to be adopted here. Tak-ing his series expansions to the level written, thedevelopment of Thompson (1969) is formally identi-cal to that followed here.



The activities in (3) are separated into entropic(configurational) and enthalpic (non-ideal) terms, rep-resented by the ideal-mixing activity of i, aideali , andthe activity coefficient of i, ci:

RT ln ai ¼ RT ln aideali ci ¼ RT ln aideali þRT ln ci (12)

The ideal-mixing activity, aideali , is made from a prod-uct of site fraction terms

aideali ¼Q

j xhsmj;s

j;sQj x

hsmj;s

j;s jpure i(13)

with the products being taken over the elements inthe formula of end-member, i, that mix. In (13), xj,sis the site fraction of j on site s, and mj,s is the multi-plicity of j on s. The multiplicity is just the numberof atoms of j on site s in the formula of i. In circum-stances where it is known or is suspected that short-range order is important on a site, a device used is tomultiply the multiplicity of the site by a reducing fac-tor, hs. For example, h ¼ 1

2 was used for Al–Mg mix-ing in spinel (Holland & Powell, 1996a) and h ¼ 1

4 isused for Al–Si mixing on tetrahedral sites in amphi-bole (Diener et al., 2007; Diener & Powell, 2012) andpyroxene (Green et al., 2012) to reduce the configura-tional contribution. Whereas there is an increasingrange of spectroscopic methods being brought tobear on minerals to investigate the state of order(e.g. Mottana et al., 1999; Harrison et al., 2000;S�anchez-M~unoz et al., 2013), such a relatively crudereduction is what is currently available to be used.The reciprocal of the denominator in (13) is referredto as the normalization constant of the ideal-mixingactivity expression, and is equal to 1 unless the for-mula of the end-member involves more than one ele-ment on any site. The normalization constant has theeffect of ensuring that aideali ¼ 1 when applied topure i.

In (12), the non-ideal term is written using the SF(Powell & Holland, 1993; Holland & Powell, 1996ab), the focus here being on symmetric interactions(see the Discussion for comments on asymmetry):

RT ln ci ¼Xnþs

j¼1ðj 6¼iÞpjð1� piÞWij �

Xnþs�1

j¼1ðj6¼iÞ

Xnþs

k¼jþ1ðk6¼iÞpjpkWjk

(14)

in which Wjk is the (macroscopic) interaction energyfor the binary between the j and k end-members ofthe mineral.

SF stems from expressing non-ideality in terms ofthe pair-wise (microscopic) interactions between ele-ments on each site (same-site interactions) andbetween elements on different sites (cross-site interac-tions). The major advance in Powell & Holland(1993) was the realization that these microscopicinteractions could be combined to make a smallernumber of macroscopic interactions, as used in (14).

The reduction in number of interaction energiesneeded becomes more marked, the more complex themineral.

Chlorite a–x relations

The ideal mixing activity formulation for chloriteinvolves no scaling factor (h = 1) for all sites, so, forexample, the clinochlore activity is:

aidealclin ¼ 4xMgM1x4MgM23xAlM4xSiT2xAlT2

noting that the normalization constant is 4, arisingfrom the fact that in pure clinochlore there is Si andAl on T2 with xSiT2 ¼ xAlT2 ¼ 1

2. The non-ideal partof aidealclin just involves application of (14)

RTlncclin¼ pafchlð1�pclinÞWafchl;clin

þpamesð1�pclinÞWames;clin

þpdaphð1�pclinÞWdaph;clin

þpochl1ð1�pclinÞWclin;ochl1

�pafchlpamesWafchl;ames�pafchlpdaphWafchl;daph

�pafchlpochl1Wafchl;ochl1�pafchlpochl4Wafchl;ochl4

�pamespdaphWames;daph�pamespochl1Wames;ochl1

�pamespochl4Wames;ochl4�pdaphpochl1Wdaph;ochl1

�pdaphpochl4Wdaph;ochl4�pochl1pochl4Wochl;ochl4

PARAMETERIZATION

Over the years, the a–x relations used in THERMOCALC

files have been created and evolved incrementally,and additions/corrections have appeared across manypapers. For example, the a–x relations for cordieritehave remained unchanged since 1990 (Powell & Hol-land, 1990), when all minerals were taken to be idealsolid solutions. This situation is not inherently prob-lematical because for many solid solutions, the a–xrelations are independent of the thermodynamic dataused, and for these, the user is free to implement themost reasonable and justifiable model available.However, for some solid-solutions (e.g. en–mgts inopx), the enthalpies depend on the a–x relations usedin their derivation and these a–x relations must beused in calculations to maintain internal consistency.In other solid solutions, the properties of the non-data set end-members depend on the a–x relationssensu stricto, so if the a–x relations change, then thenon-data set end-members need to be reassessed.Within the freedom allowable in the general case,there is scope for improvement by making thermo-dynamic mixing models, which use a consistentapproach to non-ideality when little or no experimen-tal information is available. In this section, anattempt is made to devise and implement a coherentstrategy for making such thermodynamic descriptionsof minerals, specifically the a–x relations sensu lato,



with base-level end-member properties being takenfrom the Holland & Powell (2011) data set.

A fundamental problem with generating thermody-namic descriptions is that there are almost alwaysinsufficient data to calibrate them fully. Formulationsgenerally involve more parameters than can be esti-mated from measurement, commonly many more.Whereas for the enstatite end-member of orthopyrox-ene, for example, it is feasible to measure its DfH, S,and so on, this is not true for end-members likeortho-diopside, which cannot be made, so its proper-ties cannot be measured. Yet, end-members likeortho-diopside are key constituents of minerals (inthis case Ca-bearing opx). One answer to the prob-lem of lack of information is regularization, in thesense outlined in the Introduction, an example beingextension of the chlorite thermodynamic descriptionfrom MASH to FMASH for the a–x relations, andfor handling the Fe2+–Mg order–disorder. This kindof approach is typically needed for all of the ferro-magnesian silicates of interest.

Even the simplest substitution Fe2+ for Mg, betweenlike-sized and same-charge cations, appears to involvenon-ideality of around wFeMg � 4 kJ per cationexchange (e.g. Wiser & Wood, 1991). It is clearlydesirable to use regularization, incorporating infor-mation of this sort into a–x relations, and further-more to ensure that exchanges between more-unlikecations are modelled as still less ideal, even if themagnitudes of the relevant cation exchanges in aparticular mineral have not been measured. Outlinedin the next section is a new regularization approachfor parameterizing interaction energies, which seeksto capitalize on our expectations about the relativeenergetics of cation exchanges. As the quality of dataused to constrain a property decreases, the value ofthe property becomes more uncertain. If the uncer-tainty on the value overlaps the heuristic, then it isappropriate to adopt the heuristic value. Formally,this can be done with Bayesian methods, using priorexpectations about the likely magnitude of para-meters. Use of heuristics in this way may avoid thedanger of over-fitting the data, particularly when thedata fitting needs to allow extrapolation, not justinterpolation within the realm of the data. Heuristicsare only valuable if they maintain a reasonable matchbetween the ‘shape’ of the model a–x relations inP–T–x space and the shape of the underlying thermo-dynamics and, ultimately, their quality can onlybe assessed by the compatibility of petrological calcu-lations with experimental and natural phase relations.

Explicit handling of order–disorder of Fe2+ andMg between different octahedral sites in ferromagne-sian silicates cannot be avoided, given the stricturesof Holland & Powell (2006). Equipartition, the normuntil then, was shown to be a thermodynamicallyinconsistent way of handling the partitioning betweensites. A suggested regularization for Fe2+–Mg order–disorder is also given below.

For most ferromagnesian silicates covered in thedata set, there are data for a Fe2+ end-member, ifonly arising from a natural assemblage calibration,as, for example, in chlorite. Given the likelihood thatregularization of a–x relations or of order–disorder isnot perfect, it is appropriate to consider the possibil-ity that the enthalpy of this end-member be modified.Generally, the information on which this can bebased is little more than knowledge about petrologi-cal phase relations and mineral chemistry. In the caseof chlorite, adding ochl1 and ochl4 to af-chl+ames+clin means adding 12 different W, plus theFe2+–Mg order–disorder information. If these can behandled by regularization, then it leaves just oneremaining parameter, the modification of theenthalpy of the Fe2+ end-member, to be addressed.This is a substantial simplification in parameterizingthermodynamic descriptions.

Parameterizing a–x relations

The starting point in regularizing a–x relations is totake apart macroscopic W, relating to mixing betweenend-members, into their pair-wise microscopic interac-tions w, relating to mixing between elements on sites.A crucial advantage of this is that such on-site termsmay be transferable from one mineral phase toanother, perhaps with incorporation of a scaling fac-tor. Following simplifications and approximations atthat level, the macroscopic W are then reassembled asin Powell & Holland (1993). The aim is to involve rela-tively few parameters, for example particular w, whosevalues can be suggested by analogy with the equivalentw in thermodynamically better known minerals.An example of expressing a macroscopic W in terms

of its microscopic w constituents is Wames,clin = wMgAl,

M1+ 14wSiAl;T2 þ 1

2wAlAlMgSi;M1T2 in chlorite. The w termsreflect the mixing that occurs between the elements onthe sites for mixing of ames and clin as would occur inthe ames–clin binary. The meanings of the same-siteand cross-site microscopic w in terms of pair-wiseinteractions between the elements on the sites areexplained in Powell & Holland (1993). In this equiva-lence, the same-site wMgAl,M1 concerns Mg–Al mixingon the M1 site, wSiAl,T2 concerns Al–Si mixing on theT2 site, and the cross-site wAlAlMgSi,M1T2 concerns Al–Mg–Si interactions between the M1 and T2 sites,where wAlAlMgSi,M1T2 represents the energy of the reci-procal exchange

AlAlþMgSi ¼ MgAlþAlSi

or, in terms of chlorite end-member cations on M1and T2 sites,

AlM1ðAlAlÞT2 þMgM1ðSiAlÞT2 ¼ MgM1ðAlAlÞT2

þAlM1ðSiAlÞT2:In this reaction, the two terms on the left-hand sidecorrespond to the two stable tschermak-related end-



members ames + clin, whereas the two pairs on theright-hand side correspond to unstable hypotheticalend-members, which do not exist naturally andprobably could not be synthesized, thus a positiveenergy for this reaction (and hence positivewAlAlMgSi,M1T2) is to be expected. In general, if wijkℓ ispositive, the subscripts say that the pairs ij and kℓ(rather than iℓ and jk) are the more stable, lowerenergy, configurations. Another way of viewing this isto note that a positive wAlAlMgSi,M1T2 implies thatAlM1–AlT2 and MgM1–SiT2 are more favoured as nearneighbours than AlM1–SiT2 and MgM1–AlT2 pairs.

Same-site contributions can be scaled by the sitemultiplicity; but, for the cross-site energies, the scal-ing is not always so obvious, as discussed in Powell& Holland (1993). However, by inspection of the rep-resentative reciprocal reaction, the scaling may oftenbe discerned. For example, if there are twice as manysites (e.g. in amphibole M4-T exchange) such that thecorresponding reciprocal reaction becomesAl2Al2 + Mg2Si2 = Al2Si2 + Mg2Al2, then the valueof wAlAlMgSi,M1T2 would be doubled.

The core use of the approach concerns the han-dling of a–x relations in analogous subsystems of amineral, most obviously the Fe2+-subsystem in rela-tion to the Mg-subsystem. This idea is not new in thesense that it was used, for example, in Dale et al.(2005) for amphibole. However, there, it was appliedto macroscopic W, assuming that the Fe2+-subsystemis less non-ideal by a factor, k, than the equivalentMg-subsystem. So, there, for example, the Fe2+-equivalent to Wtr,ts, Wact,fts, was taken to be equal tokWtr,ts (with tr = tremolite, act = actinolite,ts = tschermakite and fts = ferro-tschermakite). How-ever, the Fe2+-equivalent to Wts,gl, Wfts,fgl, is identi-cally equal to Wts,gl showing that, in general, amacroscopic k approach breaks down (withgl = glaucophane, fgl = ferro- glaucophane). Logi-cally, a macroscopic k is only likely to be effective ifa significant proportion of the non-ideality relates todifferences between Mg- and Fe2+-interactions, whichis unlikely to be the case if, for example, the muchlarger contributions from Na–Ca or Na–Al coupledsubstitutions are also involved. In fact, in the case ofWts,gl = Wfts,fgl, Mg- and Fe2+-interactions are notinvolved at all.

Some general principles are now outlined for sim-plifying and approximating interaction energies. Thefirst is the idea of a proportional change of thenon-ideality being applied at the microscopic level tosame-site interactions. Thus, for Fe2+–Mg interac-tions in chlorite, wFeAl,M1 is taken to be equal to/wMgAl,M1, or in general wFeX,site=/wMgX,site (withFe2+ in subscripts of w written as Fe). Experiencesuggests that / < 1, and by analogy with the quad-rilateral (CFMS) pyroxenes, / is taken to be 0.7.Also, regarding same-site interactions on the octahe-

dral sites, following on from Wiser & Wood (1991)for olivine, and from the fitting of the Fe2+–Mg sitedistribution data for orthopyroxene (Holland &Powell, 1996b) and cummingtonite-grunerite (Dieneret al., 2007), wMgFe,oct is taken to be 4 kJ perexchange. Another key octahedral same-site w iswMgAl,oct, which is taken to be 10 kJ per exchange.For example, considering sodic clinopyroxene, whereWjd,di is larger than Wjd,hed from Green et al.(2007), then using W–w relations, Wjd,di = wo +w2 � 26 kJ, and Wjd,hed = /wo + w2 � 24 kJ, withwo = wMgAl,M1 and w2 = wNaCa,M2�wAlCaFeNa,M1M2,gives wMgAl,M1 � 7 kJ. Here, we assume the heuristicthat wAlCaFeNa,M1M2 is numerically equivalent towAlCaMgNa,M1M2, as explained in the next paragraph.The cross-site interactions that involve crystallo-

chemically similar exchanges are expected to benumerically equivalent and are therefore set to beidentical, unless information to the contrary isavailable. Others are expected to be small, andtherefore as a heuristic are set to zero. Consideringcross-site interactions involving just Fe2+ and Mg,the site distribution fitting referred to above sug-gested that these are small and positive or negative,and so they are set to zero. Interactions of theform whhhAl,M1M4 between octahedral sites (usingchlorite site naming), where h is Fe or Mg, are setto zero, as are whAlhSi,M1T2 between octahedral andtetrahedral sites. Interactions like wAlAlhSi,M4T2 areset equal to each other and are likely to be posi-tive, as discussed above for the similar M1–T2interactions. Interactions like wAlAlhh,M1M4 are setequal to each other and are likely to be negative,due to Al-avoidance.As already indicated above, the idea of using regu-

larization in the parameterization of a–x relations isonly appropriate when there are little or no data toconstrain interaction energies. Commonly, this willbe the case for just some compositional dimensionsor element exchanges in a mineral. So, if, for exam-ple, MASH relations are well constrained by mea-surement, the corresponding interaction energies aregenerated from the data. If there are no, or insuffi-cient, measurements in FMASH, then the regulariza-tion approach is adopted to provide the additionalinteraction energies. Such a thermodynamic descrip-tion is defensibly better than the bias-introducingscheme of assuming ideality in the unconstrained partof the a–x relations.

Parameterizing chlorite a–x relations

The W for chlorite in MASH is taken as known in thedata set generation (Holland & Powell, 2011), withWames,clin = Wafchl,clin = 17 kJ and Wafchl,ames = 20 kJ,so this provides a starting point for looking at applica-tion of the approach. The W–w relations are:



Wafchl;ames ¼ wAlAlMgMg;M1M4 þ wAlAlMgSi;M1T2

þ wAlAlMgSi;M4T2 þ wMgAl;M1 þ wMgAl;M4

þ wSiAl;T2

Wafchl;clin ¼ 1

2wAlAlMgSi;M4T2 þ wMgAl;M4 þ 1

4wSiAl;T2

Wames;clin ¼ 1

2wAlAlMgSi;M1T2 þ wMgAl;M1 þ 1

4wSiAl;T2

(15)

The first important step in regularization is to notethat these W–w relations can be simplified bysubstituting wcrt = wAlAlMgSi,M1T2 = wAlAlMgSi,M4T2,treating M1 and M4 as similar, and alsosubstituting wo = WMgAl,M1, wt = WAlSi,T2 andwcro = wAlAlMgMg,M1M4. So:

Wafchl;ames ¼2woþ wcroþ 2wcrtþ wt

Wafchl;clin ¼woþ wcrt

2þ wt

4

Wames;clin ¼woþ wcrt

2þ wt

4

With the only knowns being Wclin,ames = Wclin,af-

chl = 17 kJ and Wames,afchl = 20 kJ, the system is un-derdetermined. Given that wcro and wcrt are likelyto have opposite signs, and taking the same-siteterms to be positive and of similar magnitude, wo islikely to be < 15 kJ, and so is consistent with thechosen value of the heuristic, 10 kJ.

Now, with the heuristics as outlined above, withw = wFeMg, site, extending from MASH into FMASH:

Wafchl;daph ¼ 5wþ 1

2wcrtþ woþ 1

4wt

Wafchl;ochl1 ¼ 5w

Wafchl;ochl4 ¼ w

Wames;daph ¼ 4wþ 1

2wcrtþ /woþ 1

4wt

Wames;ochl1 ¼ 4wþ wcroþ 2wcrtþ woþ /woþ wt

Wames;ochl4 ¼ wcroþ 2wcrtþ woþ /woþ wt

Wdaph;clin

Wdaph;ochl1 ¼ wþ 1

2wcrtþ /woþ 1

4wt

Wdaph;ochl4 ¼ 4wþ 1

2wcrtþ woþ 1

4wt

Wclin;ochl1 ¼ 4wþ 1

2wcrtþ /woþ 1

4wt

Wclin;ochl4 ¼ wþ 1

2wcrtþ woþ 1

4wt

Wochl1;ochl4 ¼ 6w

Substituting 2wo + wcro + 2wcrt + wt = 20 kJ andwoþ 1

2 wcrtþ 14 wt ¼ 17 kJ; with w = 4 kJ,

wo ¼ 10 kJ and / = 0.7, and adopting a strictlyupper triangular matrix form for the result:

W kJ ames daph clin ochl1 ochl4

afchl 16 37 17 20 4

ames 10 17 29 13

daph 20 18 33

clin 30 21

ochl1 24

giving a complete parameterization of the a–x rela-tions for chlorite. It is worth noting that applicationof micro-/, as here, means that the dependent end-member formalism of Powell & Holland (1999) isautomatically obeyed. In other words, starting withthis matrix and transforming to a new independentset of end-members gives the same matrix as applyingmicro-/ directly.

Fe2+–Mg order–disorder

Parameterization of Fe2+–Mg order–disorder betweenoctahedral sites would be simple to achieve with mea-sured site distribution information, but there are nodata for chlorite, and few data for the other ferro-magnesian minerals of interest. Therefore, a heuristicis sought. The main minerals for which Fe2+–Mg sitedistributions have been measured are orthopyroxene,cummingtonite-grunerite and olivine. In search of apattern that can be used in regularization, first theFe2+–Mg element exchange needs to be expressed ina standard form. This is chosen to beFepsite + Mgosite = Feosite + Mgpsite, with psite thepreferred site for Fe2+, and osite the less preferredsite, with one element of each involved. So, for exam-ple, for opx, en + fs = 2fm is already in this formwhen written as FeM2 + MgM1 = FeM1 + MgM2,given that Fe2+ prefers M2 to M1. Energetically, thecorresponding relation can be written as:

DGpsite;osite ¼ GFe;osite þ GMg;psite � GMg;osite � GFe;psite

in the manner of Kr€oger et al. (1959), where G ratherthan e is used here for the energy of the ‘structureelement’, an element on a site. From the dataanalysis in Holland & Powell (1996b),DGopx

M2;M1 ¼ 13:2 kJ, using wFeMg = 4 kJ to be consis-tent with the heuristics used here. Reducing the cum-mingtonite-grunerite data to this DG form is morecomplicated because Fe2+–Mg is partitioned amongthree sites. The data analysis of Diener et al. (2007)gives the DG of the ordered Fe2+–Mg end-members,camo1 and camo2, to be DGcamo1 = �66.2 kJ andDGcamo2 = �81.2 kJ, where these DG have the samemeaning as those in (5). Writing out the form of theordered end-members in the above way givesDGcamo1 ¼ �6DGamph

M4;M13 þ 6DGamphM13;M2 and DGcamo2 ¼

�4DGamphM4;M13�10DGamph

M13;M2. Solving gives DGamphM4;M13 ¼

13:6 kJ, and DGamphM13;M2 ¼ 2:7 kJ, also using

wFeMg = 4 kJ. Olivine gives a small DG from Redfern



et al. (2000), noting the complications discussedthere. The implication of the data is that Fe2+, in pre-ferring the larger and/or more distorted site that Cacan also enter (M4 in amphibole, M2 in opx), willinvolve a large DG with other sites, while DG betweensites that involves Al with a site that just involvesFe2+ and Mg, will be relatively small. The heuristicadopted is DG = 3 kJ for such partitioning in ferro-magnesian minerals.

For chlorite in FMASH, Fe2+ and Mg are distrib-uted between M1, M4 and M23, with two additionalend-members (ochl1 and ochl4) and two orderparameters (Q1 and Q4) used to represent the distri-bution. Two internal equilibria, (11), can be used tocalculate values of the order parameters at givencomposition and temperature. For this, not only thea–x relations involving ochl1 and ochl4 are neededbut also the DG of the internal equilibria. With theinternal equilibria written as in (11), these are justDGochl1 and DGochl4, the G displacement of ochl1 andochl4 from the base level defined by afchl–ames–daph. These are taken to be independent of P–T inthe absence of information to the contrary. UsingDG = 3 kJ for partitioning between M1, M23 andM4, then DGochl1 = DG = 3 kJ and DGochl4 ¼45DG ¼ 2:4 kJ:Having used the a–x and DG heuristics, it remains

to assess whether anything needs to be consideredadditionally. Although daphnite is in the data set, itis constrained only by natural assemblage Fe2+–Mgexchange, not by experimentally determined phaseequilibria, and so is less well constrained than theother chlorite end-members in the data set. Further-more, if in fact the heuristics have not worked as wellas intended, it is feasible that modifying an enthalpyon the basis of petrological data and/or insight willimprove the behaviour of the thermodynamic model.It is logical that such a modification be applied todaphnite, the least well-constrained end-member. Ingeneral, modifying the enthalpy of an end-memberfor which there are experimental constraints has theimplication that the experiments are flawed in someway (unless the data are modified within their uncer-tainties). In fact, in this case, the data set enthalpy ofdaphnite appears not to require modification. Even ifit did, note that a problem that involves finding 12W and 2 DG in extending chlorite from MASH toFMASH, has become one just involving the oneenthalpy.

Adding Fe3+ to minerals in FMASH

As emphasized above, for petrological calculations ingeologically realistic systems, end-member data forend-members additional to those in the data set arecommonly required, as well as the additional a–xrelations to incorporate them in their minerals. Anobvious example involves Fe3+ end-members in sili-cates, with the exception of those of garnet and epi-

dote for which data set information is available.Although ferric contents of rocks and their constitu-ent minerals are not commonly well known, the orderof partitioning of Fe3+ among coexisting phases iscommonly discernible, and Fe3+ is certainly not acomponent of rocks and minerals that should ever beignored.The approach followed is to define /3 to be the

proportion by which a Fe3+ subsystem is different innon-ideality from that of the corresponding Al sub-system, assuming that Fe3+ occupies the same octa-hedral site as Al. A value of wFe3Al = 2 per exchangeis adopted, following the use of this value in the dataset generation for andradite in andr–gr garnet andalso for epidote (Holland & Powell, 2011), with Fe3used in the w subscript for Fe3+. So wMgFe3 =/3wMgAl, and wFeFe3 = //3wMgAl. As implied byacm–jd relations in clinopyroxene (Diener & Powell,2012), /3 is taken to be 0.8.In extending chlorite from FMASH to FMASHO,

Fe3+ is assumed to reside only on M4, rather thanallowing Fe3+ to be partitioned between M1 and M4(as reflected in QAl for Al), thus reducing both thenumber of order parameters and end-membersrequired by one. Using /3 = 0.8, with the Fe3+ end-member being ferri-clinochlore (f3clin) gives Wafchl,

f3clin = 15 kJ, Wames,f3clin = 19 kJ, Wdaph,f3clin = 22 kJ,Wclin,f3clin = 2 kJ, Wf3clin,ochl1 = 28.6 kJ, and Wf3clin,

ochl4 = 19 kJ. The end-member properties of f3clinare made using the ‘reaction’, f3clin ¼clin þ 1

2 ðandr � grÞ, with the enthalpy change (DQF)to be found such that calculated chlorites in phaseequilibria calculations have Fe3+ contents in agree-ment with, for example, Dyar et al. (2002). The cur-rent preferred value is Gf3clin ¼ Gclin þ 1

2 ðGandr

�GgrÞ þ 2 kJ.

DISCUSSION AND CONCLUSIONS

Thermodynamic calculations for petrological applica-tion require a–x relations for a wide range of phasesin large chemical systems. Regularization is requiredto extend these relations from where there are datafor calibration, to where there are little or none.Under the moniker of ‘micro-/’, a set of heuristics ispresented to simplify and approximate a microscopicpair-wise representation of macroscopic regular-model (SF) W, based on analogies with better knownminerals, when insufficient data are available for cali-bration. This approach is then applied in the com-panion paper, the aim there being to build acomprehensive set of mutually consistent a–x rela-tions that are of the right shape and allow observedphase relations in metapelitic rocks to be reproduced.It remains to be discovered whether the various

aspects of micro-/ work well beyond the examplespresented in the companion paper. It would be fool-hardy to say anything other than that this is just astart, but we believe that the approach has real



potential. Along with validation through application,the approach is open to refinement and extension,either by modifying current heuristic values or byadding new heuristic relationships. It goes withoutsaying that experimental work that allows direct cali-bration of thermodynamic descriptions would bevaluable if it can be used to constrain a–x relationsdirectly and/or to inform the choice of values to usein heuristics.

It is important to observe that the micro ? macroapproach in Powell & Holland (1993) involved thecombination of pair-wise element-on-site microscopicinteraction energies (w) into macroscopic interactionenergies (W) that are pair-wise between end-members.Micro-/ outlined above then builds on this. But thisshould not obscure the fact that intrinsically macro-scopic effects, like strain, can contribute to macro-scopic W. If the latter are significant, then seekingregularization by micro-/ will work less well. Moresignificantly, micro-/ cannot currently accommodateasymmetry as asymmetric models (for example, theasymmetric formalism of Holland & Powell, 2003),are intrinsically macroscopic and cannot be repre-sented microscopically. It is only at the microscopiclevel that it makes sense to use the simplificationsand approximations of micro-/. Work is currently inprogress to resolve this difficulty.

An aspect of micro-/ adopted here that has notbeen discussed above is that, in the absence of infor-mation to the contrary, W (and w) are taken to beindependent of pressure and temperature, i.e. ifW = a + bT + cP, the heuristic is adopted thatb = c = 0. Whereas this is the simplest assumption inthe absence of information, when there have beendata that could be fitted to extract pressure or tem-perature dependence, we have chosen to assume alack of dependence if data-fitting has produced band c values that are within error of zero. Such anassessment is, of course, fraught with difficulty,requiring a nuanced appraisal of experimental uncer-tainties, recognition of outliers, etc. But given thatthe a–x relations are commonly to be used well awayfrom the P–T conditions of the data, adopting thisapproach is least likely to cause difficulties. Such anapproach in data fitting (taking b = c = 0) is part ofa drive to bring attention to the dangers of overfit-ting in the development of a–x relations.

Having applied micro-/ to obtain values for W, itis important to note that a full thermodynamicdescription for a phase still requires values for theproperties of the non-data set end-members, and theproperties of end-members in the data set that aredeemed to be not well constrained. In terms of chlo-rite, the Fe3+ end-member, f3clin, is an example ofthe former, and the Fe2+ end-member, daph, is anexample of the latter, as discussed above. Althoughthere is the possibility of circularity of argument, ulti-mately inclusion of petrological information isrequired. This can be in the form of the relation

between mineral assemblages in rocks and the min-eral compositions involved, and the predictions madeby pseudosections. A fruitful line of enquiry will beto discover how best to extract information frommineral assemblages in rocks directly, maybe by anadaption of the avPT approach to thermobarometry.

ACKNOWLEDGEMENTS

RP acknowledges support from ARC DP0451770and DP0987731.

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Appendix Detai ls for the thermodynamics ofchlor i te

The chosen composition parameters and orderparameters for chlorite are:

x¼ Fe

FeþMg

¼ xFeM1þ4xFeM23þxFeM4

xFeM1þ4xFeM23þxFeM4þxMgM1þ4xMgM23þxMgM4

y¼netTschermaks¼ xAlM1þxAlM4

2

QAl¼ xAlM4�xAlM1

2

Q1¼x� xFeM1

xFeM1þxMgM1

Q4¼x� xFeM4

xFeM4þxMgM4

in which, for example, xFeM1 is the site fraction ofFe2+ on M1. So, x = 0 is for chlorite in MASH,x = 1 is for FASH, y = 0 is for Al-free chlorite, andy = 1 is for maximally Al-substituted chlorite. Chlo-rite is known to be relatively strongly (Fe,Mg)–Alordered with Al preferring M4 to M1, with QAl

approaching y for y� 12, and 1�y for y[ 1

2. Clin aty ¼ 1

2, is completely ordered, so has QAl ¼ 12. Chlo-

rite is taken to be weakly Mg–Fe2+ ordered betweenM1, M4 and M23 (see above), so Q1 and Q4 takesmall values. The end-members ochl1 and ochl4 areat y = 0 and x ¼ 5

6, and y = 0 and x ¼ 16, respec-

tively. The proportions of the end-members and thesite fractions in terms of these composition and orderparameters are given below.Using the chosen composition variables, the site

fractions are:

xMgM1¼1�x�yþxyþQ1�yQ1þQAl�xQAlþQ1QAl

xFeM1¼x�xy�Q1þyQ1þxQAl�Q1QAl

xAlM1¼y�QAl

xMgM23¼1�x�Q1

4þyQ1

4�Q4

4þyQ4

4�Q1QAl

4þQ4QAl

4

xFeM23¼xþQ1

4�yQ1

4þQ4

4�yQ4

4þQ1QAl

4�Q4QAl

4

xMgM4¼1�x�yþxyþQ4�yQ4�QAlþxQAl�Q4QAl

xFeM4¼x�xy�Q4þyQ4�xQAlþQ4QAl

xAlM4¼yþQAl

xSiT2¼1�y

xAlT2¼y

and the proportions of the end-members are:



pafchl ¼ 1� 2x� yþ 3xyþ 5Q1

4� 5yQ1

4þ 9Q4

4� 9yQ4

4

�QAl þ xQAl þ 5Q1QAl

4� 9Q4QAl

4

pames ¼ y�QAl

pdaph ¼ xyþQ1

4� yQ1

4þ 5Q4

4� 5yQ4

4þ xQAl

þQ1QAl

4� 5Q4QAl

4

pclin ¼ �xy�Q1

4þ yQ1

4� 5Q4

4þ 5yQ4

4þ 2QAl

� xQAl �Q1QAl

4þ 5Q4QAl

4

pochl1 ¼ x� xy�Q4 þ yQ4 � xQAl þQ4QAl

pochl4 ¼ x� 2xy� 5Q1

4þ 5yQ1

4� 5Q4

4þ 5yQ4

4

� 5Q1QAl

4þ 5Q4QAl

4

These relations are produced using the axe attack setof MathematicaTM functions, specifically axsolver(Powell, unpublished data), allowing error-free gener-ation of the algebraic expressions. The relations pro-duced by axsolver are always in expanded form, butfor aesthetics, simplification is sometimes possible,for example pochl1 = (Q4 � x)(y + QAl � 1).

Received 7 August 2013; revision accepted 19 December 2013.



On parameterizing thermodynamic descriptions of minerals for petrological calculations

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