an automated strategy for calculation of phase diagram sections and

An automated strategy for calculation of phase diagramsections and retrieval of rock properties as a functionof physical conditions

J . A. D. CONNOLLY AND K. PETRINIEarth Sciences Department, Swiss Federal Institute of Technology, 8092 Zurich, Switzerland([email protected])

ABSTRACT We formulate an algorithm for the calculation of stable phase relations of a system with constrainedbulk composition as a function of its environmental variables. The basis of this algorithm is theapproximate representation of the free energy composition surfaces of solution phases by inscribedpolyhedra. This representation leads to discretization of high variance phase fields into a continuousmesh of smaller polygonal fields within which the composition and physical properties of the phases areuniquely determined. The resulting phase diagram sections are useful for understanding the phaserelations of complex metamorphic systems and for applications in which it is necessary to establish thevariations in rock properties such as density, seismic velocities and volatile-content through ametamorphic cycle. The algorithm has been implemented within a computer program that is generalwith respect to both the choice of variables and the number of components and phases possible in asystem, and is independent of the structure of the equations of state used to describe the phases of thesystem.

Key words: phase diagram; pseudosection; rock properties; seismic velocity: thermodynamics.

INTRODUCTION

Traditionally petrologists have relied on phase dia-gram projections that implicitly show phase relationsfor all possible compositions of a system as an explicitfunction of environmentally determined thermody-namic potentials (see Appendix for nomenclature).The utility of this projection, i.e. a petrogenetic grid,that normally represents only one-dimensional uni-variant phase fields, is that it constrains the maximumstability of any possible phase assemblage irrespectiveof the bulk composition of the system. Unfortunately,interpretation of such diagrams is complex in thatthere may be many assemblages that are stable withinany given region of the diagram, and only a smallsubset of the depicted phase relations may be relevantfor a specified composition. Moreover, if the phasesof the system are solutions, it is to be expected thatfor a given composition, high variance fields ratherthan the univariant fields of the conventional petro-genetic grid will limit phase stabilities. Phase diagramsections computed for a specified bulk compositionoffer an alternative to projections that avoid thesecomplexities. Following recent geological usage (e.g.Powell, 1978), we designate such sections as pseudo-sections. In contrast to projections, there is only onestate represented by any point within a pseudosection,so that at each point the composition and proportionsof the phases and the thermodynamic properties of

the system are uniquely determined. While the valueof pseudosections is recognized in the geological lit-erature, techniques for the construction of pseudo-sections are laborious. In this paper we present astrategy for the construction of pseudosections fromthermodynamic data. The essence of this strategy isthe discretization of the continuous fields of apseudosection by a polygonal mesh. The phase rela-tions of the polygonalized section are easily inter-preted, and, because properties such as volatilecontents or density are associated with the polygons,the diagrams provide a simple means of retrievingrock and mineral properties as a function of environ-mental conditions.

The value of pseudosections as a tool for interpret-ing petrological equilibria has been amply demon-strated (e.g. Powell & Holland, 1988; Dymoke &Sandiford, 1992; Xu et al., 1994). Because pseudosec-tions provide a map of both mineral chemistry andmodes as a function of environmental variables theyhave application in thermobarometry (e.g. Stuewe &Powell, 1995); where, compared to conventionalinverse modelling thermobarometric techniques, phasediagram based methods have the advantage of ther-modynamic consistency (e.g. Connolly et al., 1994).The efficacy of phase diagram methods has been lim-ited by the requirement of a complete thermodynamicmodel for the solution phases of interest; however,efforts to expand data compilations (e.g. Berman &

J. metamorphic Geol., 2002, 20, 697–708

� Blackwell Science Inc., 0263-4929/02/$15.00 697Journal of Metamorphic Geology, Volume 20, Number 7, 2002

Aranovich, 1996; Gottschalk, 1997; Holland & Powell,1998) have made substantial progress toward over-coming this limitation. A fundamental caveat in theapplication of pseudosections to petrological problemsis that it is necessary to define an effective thermo-dynamic bulk composition (e.g. Stuewe, 1997). Inmany natural systems, there is no meaningful effectivebulk composition because disequilibrium fractiona-tion continuously modifies the composition of theequilibrated portion of the rock in response toenvironmental factors. Where it can be argued thatdisequilibrium is not important, a second complicationin the use of pseudosections is that their geometries canbe sensitive to uncertainties in the composition ofinterest. Thus, thermobarometric applications requirean efficient means by which this sensitivity can beexplored; the method discussed here is a tool for thisexploration.

Perhaps the most important application ofpseudosections is that they provide a practical modelfor the average behaviour of rocks in metamorphicsystems. While such models may appear crude topetrologists, they are essential for understandingmany geological processes. Past applications of thisilk have focused on physical models of the meta-morphic process (e.g. Trommsdorff & Connolly,1996; Stuewe & Powell, 1997; Connolly, 1997;Kerrick & Connolly, 1998; Guiraud et al., 2001) orthe influence of phase transitions on geophysical orgeodynamic models (e.g. Saxena & Eriksson, 1983;

Wood & Holloway, 1984; Gubbins et al., 1994;Sobolev & Babeyko, 1994; Bina, 1998). Recentapplications of both types (e.g. Jull & Kelemen, 2001;Kerrick & Connolly, 2001a,b; Petrini et al., 2001;Lucassen et al., 2001; Muntener et al., 2001; Cesareet al., 2002) demonstrate the effectiveness of thecomputer method presented here.

Several completely or partially automated com-puter strategies for the calculation of petrologicalphase equilibria are widely used by geologists. Webegin by reviewing these strategies to clarify themerits of the techniques and to provide the motiva-tion for our modification. Then our strategy ispresented in detail and concludes with a demon-stration of its computer implementation, in whichthe phase relations of a metapelite are calculated asa function of pressure (P) and temperature (T). Forsimplicity, we adopt terminology appropriate for theanalysis of an isobaric-isothermal closed chemicalsystem. In this case the Gibbs energy is the ther-modynamic function that is minimized during phaseequilibrium calculations, the compositional variablesdescribe the proportions of the different kinds ofmass that may vary among the phases of the system,and the environmental variables are pressure andtemperature. However, our computational strategy isgeneral and can be used for compositions that definethe thermal and mechanical properties and environ-mental variables that relate to the chemistry(Connolly, 1990).

Fig. 1. (a) Schematic isobaric-isothermal G� X diagram for a binary system with solution phases (a, b, c, d, e). Minimization strategies(e.g. Saxena & Eriksson, 1983; Wood & Holloway, 1984; De Capitani & Brown, 1987; Eriksson & Hack, 1990) determine thestable assemblage for a specified composition, for example for Xsystem the stable assemblage would be b + c. Phase equilibriumcalculators compute the equilibrium compositions (e.g. Powell et al., 1998; Spear, 1988) of a specified set of coexisting phases. If theassemblage d + c is specified, an equilibrium calculator determines the compositions (filled circles) of d and c that are tangent to acommon G� X plane. If these compositions span that of the system, the assemblage is a possible equilibrium in terms of mass balance,but, as illustrated, the assemblage may be metastable. Connolly & Kerrick (1987) developed a strategy that simultaneously determinesall the assemblages on the systems minimum G� X surface (heavy solid curve). This strategy is efficient if pseudocompounds are usedto define the possible compositions of solutions, e.g. in (b) b is represented by pseudocompounds b1…b8, the accuracy of thisrepresentation is determined by the pseudocompound spacing.

6 98 J . A . D . CO NN OL LY & K . PET R I N I

REVIEW OF EXISTING STRATEGIES

Four basic strategies have been employed for the cal-culation of pseudosections for petrological applicationsinvolving solution phases. The most straightforwardstrategy is to use a nonlinear free energy minimizationtechnique to map the phase relations of a system as afunction of the section variables (e.g. Saxena & Eriks-son, 1983; Wood & Holloway, 1984; DeCapitani &Brown, 1987; Sobolev & Babeyko, 1994; Bina, 1998).With this method all the variables of the system arespecified and the phase assemblage that minimizes itsfree energy is computed. To illustrate this procedureconsider a molar free-energy (G) composition (X) dia-gram for a binary system (Fig. 1a). For the specifiedcomposition (Xsystem), minimization establishes theidentities, amounts, and compositions of the stablephases (c + b). By varying pressure or temperature itwould then be possible to find conditions at which eitherone of these phases disappears or a new phase appears,thus defining a pseudosection phase field boundary.Although the calculation of any individual equilibriumwith this strategy is done by computer, integration ofthe results to obtain a pseudosection is time consuming.Attempts to automate this integration have met withsome success (Eriksson & Hack, 1990; DeCapitani,pers. comm., 1994), but are not always well suited forthe types of problems common in petrology. A technicaldifficulty in nonlinear minimization problems is con-vergence to local (false) minima; while many algorithmsreduce the probability of this, absolute certainty cannotbe assured for systems with nonideal phases.

A second strategy makes use of a phase equilibriumcalculator (Powell & Holland, 1988; Powell et al.,1998; see also Hillert, 1981). This strategy is distin-guished from free-energy minimization in that thephases of an equilibrium are specified, rather than thevariables of the system. Returning to our illustration(Fig. 1a), by this methodology the user might specifythe metastable assemblage c + d. The calculatorcomputes the equilibrium compositions of the coex-isting phases, but in contrast to a minimization tech-nique does not test the stability of the assemblage.The calculator can also determine whether theassemblage is possible for a composition of the sys-tem, and if it is, the environmental conditions atwhich one phase in the equilibrium vanishes. Theseconditions may define a phase field boundary in apseudosection. Because phase field boundaries can belocated directly, rather than by the iterative proce-dures, the technique offers some advantages over free-energy minimization. Spear (1988) advocates a thirdstrategy by which the changes in the phases of a stableequilibrium are determined as a function of environ-mental variables by application of the Gibbs–Duhemrelation. As with the Holland and Powell strategy, thismethodology can be used to determine conditionswhen a phase disappears from a system due tohomogeneous equilibration in response to changing

environmental conditions. Because these techniquesdo not directly establish the stability of equilibria,construction of a phase diagram section by thesemethods is labour intensive and requires a prioriknowledge of phase stabilities. The recognition ofphase immiscibility also creates technical complica-tions for equilibrium calculators because the calcula-tors solve systems of equations that are formulated interms of phase species, rather the phases themselves.

We focus on a fourth strategy in which the systemsminimum free energy surface is determined directly(i.e. the heavy solid curve in Fig. 1a; Connolly &Kerrick, 1987; Greiner, 1988). The result of such acomputation is identical to that which would beobtained by energy minimization computationsrepeated for all possible compositions of a system, oralternatively by establishing the stability of all possibleequilibria in a system through the use of a phaseequilibrium calculator. Once these phase relations areknown, mass balance constraints can be used toestablish which equilibrium is relevant for a particularcomposition, and the phase relations of this equilibri-um can be monitored as a function of environmentalvariables to obtain the desired phase diagram section.It is not to be expected that this strategy will in generalbe superior to iterative application of a minimizationtechnique. However, the efficiency of this strategyincreases enormously if the continuous compositionalvariation of solutions is represented by a series ofdiscrete compositions, i.e. discretized (Connolly &Kerrick, 1987). The initial implementation of thisstrategy as a component of a computer program calledVERTEXVERTEX (Connolly, 1990) has had application for thecalculation of pseudosections (e.g. Gubbins et al.,1994; Trommsdorff & Connolly, 1996; Connolly, 1997;Kerrick & Connolly, 1998), but was complicated bytwo problems that we resolve here. First, to maximizeefficiency, Connolly & Kerrick (1987) made use ofgeometric constraints that required substantial com-puter memory and thereby limited the dimension andresolution of the compositional space that could betreated. Here a simpler and more general algorithm isintroduced that overcomes these limitations. The sec-ond problem is that VERTEX defines phase diagrams byidentifying the conditions at which changes in thestable phase assemblages occur rather than by identi-fying the assemblages themselves. While this is desir-able for the construction of petrogenetic grids wherethe stability fields of phase assemblages may overlap, itresults in a loss of information in the context of apseudosection. To recover this information we adoptan algorithm whereby pseudosections are subdividedinto polygonal regions that correspond to phase fields.

COMPOSITIONAL DISCRETIZATION:PSEUDOCOMPOUND APPROXIMATION

The free energy surface of a solution phase is a non-linear function of its composition and consequently the

C A L C U L A T I O N O F P H A S E D I A G R AM S E C T I O N S 69 9

numerical solution of phase equilibrium problems iscomplicated. To circumvent such complications,Connolly & Kerrick (1987) suggested an approxima-tion by which the compositional variation of a solutionis represented by a series of compounds defined suchthat each compound has the thermodynamic proper-ties of the solution at an arbitrarily chosen composi-tion. To distinguish these compounds from truephases, the compounds are designated pseudo-compounds. This approximation is illustrated for abinary system with three possible phases (Fig. 1b),where a and c are true compounds, and b is a solutionrepresented by the pseudocompounds b1…b8. Afterthe approximation the true curvilinear minimumG� X surface the system, which defines both the stablephase assemblages and the possible compositions ofthe stable phases, becomes a piecewise linear hulldefined by the vertices a, b2,…,b6, c (the heavy seg-mented curve in Fig. 1b). More generally, in ac-component system, at arbitrarily specified environ-mental conditions, the approximated minimum G� Xsurface of the system is a convex faceted c ) 1dimensional surface, each facet of which is defined by ccompounds that stably coexist. The true variance ofthe assemblage corresponding to any facet can beestablished by counting the number of phases repre-sented by the compounds. Thus, a + b2 and b6 + c aretwo-phase assemblages, whereas assemblages such asb2 + b3 represent a portion of the homogeneous one-phase region of solution b. Immiscibility is recognizedby testing whether two, or more, coexisting pseudo-compounds of a solution bound the composition of ametastable pseudocompound of the same solution. Forexample, stability of the assemblage b2 + b7 wouldimply the existence of a solvus in b spanning thecompositions b3, …, b6.

The distinction between apparent and true varianceof phase assemblages approximated by pseudocom-pounds is also useful in characterizing changes in thephase relations of a system as a function of environ-mental variables. Returning to our illustration(Fig. 1b), if the solution b is destabilized with respectto the compounds a and c in response to a change insome environmental variable, then this destabilizationmust be manifest by a discontinuous reaction of theform b2 ¼ b3 + a or b6 ¼ b5 + c. The equilibriumof such a reaction has an apparent variance of one (i.e.the equilibrium is pseudounivariant), but because thesame phase appears twice in the reaction the truevariance of the phase assemblage represented by thereaction is two. In a phase diagram projection or sec-tion, the conditions of these pseudounivariant equi-libria define isopleths of the solution b in the divariantfields of the assemblages a + b or b + c. In the contextof this example, the ultimate stability of the solution bwould be associated with a reaction of the formb4 ¼ a + c, this reaction is recognized as univariantbecause each compound represents a distinct phase. Incontrast to the true univariant field, for which the

associated reaction stoichiometry changes continu-ously, the variation in the reaction stoichiometry forthe approximated univariant field occurs discontinu-ously at pseudoinvariant fields defined by an assem-blage such as b4 + b5 + a + c.

The essence of the pseudocompound approximationis that in a pseudosection defined by two environ-mental variables, any two-dimensional phase fieldmust represent the stability field of an assemblageconsisting of c compounds and therefore has anapparent variance of two. Pseudodivariant phase fieldsare separated by one-dimensional univariant fields thatintersect to define zero-dimensional invariant fields.The true variance of these fields is determined bycounting the number of phases represented by thefields, or, in the special case of phase immiscibility, byapplying a geometric test. These procedures can bedone within the same computer program that is used toestablish the phase fields. Thus, pseudocompounds arean internal representation and both input and outputto programs based on the algorithm can be expressedin terms of continuous solution phases and true vari-ance. The accuracy of the approximation is determinedby the compositional spacing of the pseudocom-pounds, given current computational resources this isnot an important limitation for geological problems.Various schemes (e.g. Connolly & Kerrick, 1987) forthe internal generation of pseudocompounds aredescribed in the program documentation.

STABLE PHASE ASSEMBLAGES AS A FUNCTIONOF COMPOSITION

In this section, an abbreviated combinatorial algo-rithm to establish the stable phase assemblages of asystem as a function of its composition at constant,and arbitrarily chosen, environmental conditions isdeveloped. For the purposes of the algorithm there isno distinction between true compounds and pseudo-compounds, so discussion is simplified by theassumption that all possible phases are true com-pounds. Gibbs’ conditions require that for the equi-librium of p compounds the chemical potentials ofeach component, in every compound in which itoccurs, must be equal. From the definition of theGibbs energy, this condition is expressed

Rc

j¼1Xi

jlj ¼ Gi; i ¼ 1; . . . ; p ð1Þ

where i indexes the properties of the compounds. If theGibbs energies and compositions of the compounds areknown, then eq. (1) is a system of p linear equations in cunknowns. It follows, that all, and only, assemblages ofp ¼ c compounds with linearly independent compo-sitions uniquely define the equilibrium states of thesystem. Such assemblages are stable if the condition:

Gi � R

c

j¼1Xi

jlj � 0 i ¼ 1; . . . ; p ð2Þ


is true for the p compounds possible in the system.Based on these constraints, we formulate an algorithmconsisting of an initial step in which a stable com-pound assemblage is identified. New assemblagesinvolving a subset of c – 1 compounds of the assem-blage(s) already known to be stable are identified in thesecond step. This step is repeated using the assem-blages identified in the previous iteration as seeds untilno new assemblages are identified and the compositiondiagram is completely determined. In detail, thestructure of each step is:

Step 1. The stable phase in a one-component sub-composition is found. Components are progressivelyadded to the subcomposition, and after each addition,the compound that stably coexists with the compoundsknown to be stable from the previous step is deter-mined. When all c components have been included, theassemblage of c compounds is a stable divariantassemblage that defines a corner on the simplicial hullof the systems minimum G� X surface.

Step 2. Each c – 1 compound permutation of thec-compound assemblages identified in the previousstep is examined to determine if the permutation is notpresent in any other stable assemblage identified in thecurrent or previous step. If this condition is met, then acompound is sought that lies on the opposite side ofthe c – 1 dimensional plane spanning the compositionsof the permutation from the compound already knownto coexist with the permutation. If such a compoundexists, then there is an additional stable assemblageinvolving the permutation. The compound is thentaken as an initial guess for the compound that coexistswith the permutation and eq. (1) is solved. The stabilityof this new assemblage is tested by eq. (2), if theassemblage is found to be metastable with respect tothe jth phase, the guess for the stable phase is updatedto be the jth phase, and eq. (2) is evaluated for theremaining p – j phases. The assemblage identified inthis manner is stable, but may duplicate an assemblagealready identified in the same iteration. Thus theassemblage must be compared to a list of assemblagesidentified in the current step.

To illustrate this procedure consider the ternarysystem shown in Fig. 2. In the initial step of thealgorithm the stable compound, a, in the subcompo-sition defined by component 1 is identified; the sub-composition is expanded to include component 2, andthe stable compound, c, that coexists with a in thebinary, is established. The subcomposition is thenaugmented to the full ternary composition, and thecompound, b3, that coexists with the a + c is found.For the second step, the assemblage is a + c + b3 isnow known to be stable; therefore there are two stablec – 1 phase permutations to be considered a + b3 andc + b3 that may coexist with a third compound (a + cis excluded a priori because it lies in a subcompositionand therefore must be on the edge of the compositionspace). Beginning with a + b3, the compounds b4, b5

and b6 lie on the opposite side of the line (i.e. a c – 1

dimensional plane) spanning the compositions ofa + b3 from the compound c and therefore maycoexist with a + b3. One of these compounds is chosento form a trial configuration, such as a + b3 + b6, forwhich eq. (1) is solved. The stability of b4 and b5 isthen tested relative to the trial configuration witheq. (2), if either of these compounds is found to bestable relative to the trial configuration the compoundreplaces b6 in the trial configuration, and the stabilityof the new trial configuration is tested against theremaining compounds. The procedure for the secondpermutation, c + b3, is identical except that thisassemblage may only coexist with the compounds b4,b5, b6 and d. If a trial configuration is not metastablewith respect to any of the compounds with which itmay coexist, then it is accepted as a stable assemblage.

Fig. 2. Schematic isobaric-isothermal G� X diagram for aternary system illustrating the unconstrained minimizationstrategy used in VERTEX. From eq. (1), any assemblage of p ¼ ccompositionally nondegenerate compounds defines a G� Xplane. The assemblage is stable if there are no compoundswith G� X co-ordinates below this plane (eq. 2) is true as isthe case for the assemblage a + c + b3 (open circles indicatecompounds that lie above the a + c + b3 G� X plane). In theinitial step of the algorithm the stable compound, a, in thesubcomposition defined by component 1 is identified; the sub-composition is expanded to include component 2, and the stablecompound, c, which coexists with a in the binary, is established.With the addition of the third component to the compositionspace, the stable compound b3 that coexists with the binaryassemblage a + c is identified. In the second step, phases aresought that coexist with every permutation of c–1 compoundsfrom the assemblage(s) identified in the previous step.


If at the conclusion of this step, the stable configura-tions identified in this manner are a + b3 + b4 andc + b3 + b4, then in the next iteration of this stepthere are again two stable c – 1 phase permutations tobe considered a + b4 and c + b5 (here b3 + b4 isexcluded a priori because it is common to two stableassemblages). The calculation is concluded when allthe stable c – 1 phase assemblages identified are eithercommon to two assemblages or cannot coexist withany of the compounds possible in the system (i.e. theyare on the edge of the portion of the compositionspace that can be physically realised given the com-pounds defined for the problem).

As a demonstration of the practicality of the fore-going algorithm, Fig. 3 shows a calculated �AFM�diagram (Thompson, 1957) in which the compositionsof two ternary and nine binary solutions are resolvedto an accuracy of one percent. The computer timerequired to calculate and draft the diagram is less thanfive seconds on a computer with a clock speed of200 MHz.

Extension to constrained bulk composition

The algorithm outlined above determines all the stablecompound assemblages of a system irrespective of itscomposition. Excepting cases of pathological degen-eracy, only one of these assemblages will be stable for agiven bulk composition. Discounting the caveat, thisassemblage is identified by determining the propor-tions ai of the compounds in each assemblage neces-sary to obtain the systems composition by solving themass balance constraint:

Rp

j¼1Xi

jai ¼ Xsystemj ; j ¼ 1; . . . ; c ð3Þ

The stable compound assemblage is that assemblagefor which the proportions of the compounds aregreater than zero.

For most purposes, pseudosections are used forproblems in which the amounts of all the componentsare specified, i.e. the rank of Eqs. 2 and 3 is c. How-ever, under some circumstances it may be desirable to

constrain only a subset of the systems components, e.g.a situation where the Fe:Mg ratio of a system isknown, but the proportions of the remaining compo-nents cannot be defined. Alternatively, it may bedesirable to reduce the dimension of the compositionspace by specifying that one or more components arepresent in excess. In VERTEX, multiple component sat-uration constraints are implemented by a componentsaturation hierarchy; a scheme that preserves thermo-dynamic consistency and permits thermodynamicprojections through solution phases (Connolly, 1990).In these cases, the rank of eq. (3) is greater than that ofeq. (1). As there is no algorithmic constraint on therank of eq. (3), these situations are easily dealt withusing the foregoing algorithm.

POLYGONALIZED PHASE DIAGRAM SECTIONS

As a consequence of the pseudocompound approxi-mation, the maximum apparent variance of any phaseassemblage is two and thus the stability of theseassemblages can only be limited by either univariantfields or invariant fields where univariant fieldsintersect. To establish the conditions for these lowvariance fields we employ the following algorithm,illustrated in Fig. 4, for a system with a fullyconstrained composition; for partially constrainedcompositions we employ the algorithm outlined byConnolly (1990).

Step 1. The stable divariant assemblage is deter-mined, by the procedure outlined previously, at anarbitrary point along the boundary of the pseudosec-tion co-ordinate frame (a + c at point a, Fig. 4).

Step 2. One co-ordinate frame variable is incre-mented until the divariant assemblage becomes meta-stable with respect to exactly one compound (b1 atconditions immediately beyond point b, Fig. 4). Thiscompound together with the initial assemblagecomprises the assemblage of the univariant field. Theunivariant field is located on the edge of the co-ordi-nate frame by solving for the equilibrium condition ofthe relevant univariant reaction (a + c ¼ b1 at pointb, Fig. 4; see also Eqs. A1 and A2, Appendix).

Fig. 3. AFM diagram illustrating thecapacity of the program VERTEX to providehigh resolution of compositional phaserelations, thermodynamic data as discussedin reference to Fig. 5, mineral notation afterKretz (1983) with stoichiometric phasesindicated by lower case abbreviations. Insetshows phase relations in terms of thepseudocompounds specified for the calcula-tion. Calculation and drafting, whichrequired < 5 s on a 200-MHz computer, wasdone using >20000 pseudocompounds.Reconstruction of true phase relations isdone automatically by the graphics programPSVDRAWPSVDRAW.


Step 3. The univariant field is traced within theco-ordinate frame, by incrementing one section vari-able and solving for the other, until it either becomesmetastable at an invariant field (a + c + b1 + b2 atpoint c, Fig. 4) or intersects the edge of the co-ordinateframe. In the former case, the additional stable uni-variant fields that emanate from the invariant field(a + c + b2 and a + b1 + b2) are established bydetermining the c + 1 phase permutations of theinvariant assemblage that span the bulk composition(eq. 3) and these are traced in the same manner. This stepis repeated to determine all the univariant fields thatconnect to the initial univariant field via invariant fields.

If the foregoing procedure is repeated for everydivariant field identified on the perimeter of the sec-tions co-ordinate frame, then all the univariant fieldsrelevant to the bulk composition are established. Eachof these fields separates two divariant fields, the iden-tities of which are found by determining the twoc phase permutations that span the bulk composition(eq. 3). This information completes the definition ofthe pseudosection, but the interpretation of thepseudosection is complicated by the fact the output ofthe algorithm defines the edges of the two-dimensionalregions of the pseudosection rather than the

two-dimensional regions themselves. To eliminatethis difficulty we make use of the fact there is aunique divariant assemblage in every two-dimensionalfield of the pseudosection. Each of these fields is apolygon, the vertices of which correspond to the pointsalong the bounding univariant fields and associatedinvariant fields or intercepts with the co-ordinate framewhere these univariant fields terminate. The programPOLYGONPOLYGON assembles this polygonal mesh from the out-put created by VERTEX and associates a divariantassemblage with each element of the mesh. For ourexample (Fig. 4), the output from POLYGON would bea mesh consisting of six elements representing thedivariant fields a + c, c + b1, c + b2, b2 + b3, a + b3

and a + b4. If it is acknowledged that b1…b4 representa single solution b, then four true phase fields can bereconstructed from the internal pseudocompoundrepresentation: the divariant field a + c; the one phasetrivariant field of b represented by the pseudodivariantassemblage b2 + b3; and the divariant fields of c + band a + b represented by divariant assemblages{c + b1, c + b2} and {a + b3, a + b4}, respectively.For graphical purposes the true phase relations areobtained with the program PSVDRAW. In its defaultmode PSVDRAW shows all true univariant fields

Fig. 4. Schematic pseudosection for a binary system illustrating the tracing algorithm used by VERTEX. Arrows indicate the path tracedby VERTEX, with G� X diagrams at points along the path as discussed in the text. In the G� X diagrams, the vertical dashed lineindicates the systems composition. One-dimensional univariant fields are traced by locating conditions at which c + 1 phases aresimultaneously tangent to a G� X plane subject to the condition that there are no other compounds below the plane, as in (b);univariant fields may terminate at zero-dimensional invariant fields located by conditions at which an additional compound becomestangent to the G� X plane of the univariant field. The divariant fields separated by each univariant field are established by findingthe two c-phase permutations of the univariant assemblage that span the composition of the system (eq. 3). The univariant fields dividethe P–T space of the section into a polygonal mesh, the program POLYGON assembles this mesh from the output generated by VERTEX

and associates a divariant assemblage with each polygon, e.g. the field b1 + b2 (shaded) constitutes one element of the mesh. Truephase field variance is determined by counting the number of phases represented by the compounds; thus, truly univariant phase fieldssuch as a + c ¼ b1 are distinguished from pseudounivariant b1 + c ¼ b2 fields. Likewise, the programs recognize that two ormore elements of the polygonal mesh, such as c + b1 and c + b2, may represent a single true phase field.


(e.g. a + b1 + c) and any pseudounivariant fields(c + b2 + b3 and a + b2 + b3) that define the limitof a true phase region, and suppresses pseudounivari-ant fields that correspond to boundaries betweenpolygons that represent the same true phase assem-blage (c + b1 + b2 and a + b3 + b4).

To illustrate the tractability of geological problemsby our method, the computer implementation of thisstrategy was applied to the calculation of the phaserelations for Shaw’s (1956) average metapelite compo-sition (atomic proportions 6.32 Na: 7.12 K: 2.21 Ca:5.73 Mg: 9.05 Fe: 24.00 Al: 210.14 Si). To constrainthe proportions of H2O, CO2 and O2, the system isassumed to be in equilibrium with a fluid formed byequilibration of water with graphite, a model argued toprovide a reasonable model for metapelite devolatil-ization (Connolly & Cesare, 1993). The calculation wasmade with the thermodynamic data of Holland &Powell (1998), taking into account the crystalline solu-tions as detailed in Table 1. Computer time required tocalculate and draft the pseudosection is c. 10 min on a200-MHz computer. The raw output from VERTEX thatforms the basis of the polygonalized pseudosection isshown in Fig. 5(a); the section after processing withPOLYGONPOLYGON is shown in Fig. 5(b), with labelling sup-pressed for legibility; and the final redrafted section isshown in Fig. 5(c). The computed phase relations areconsistent with petrographic observation (e.g. Bucher& Frey, 1994), an agreement that provides some

assurance that current thermodynamic models areadequate for modelling natural systems. Discretizationof solution composition limits the resolution of certainpseudosection features, most notably in Fig. 5(c) this ismanifest in the narrow high variance fields that sepa-rate phase fields of lower variance and uncertainty as tothe arrangement of high variance phase fields about theends of univariant fields. Examples of this uncertaintyare the small triangular trivariant fields that terminatethe univariant field that limits the stability of albite withrespect to plagioclase (at c. 450 �C, 0.7 GPa in Fig. 5c).While the existence of these fields cannot be rejected onthe basis of topological arguments, their small size leadsto the suspicion that they are an artefact of composi-tional discretization. The validity of such features canusually be verified by refining the discretization scheme,but in some cases unambiguous resolution requiresnumerically exact computational procedures (e.g. thoseof Powell et al., 1998 or Spear, 1989).

Once a polygonalized pseudosection has been com-puted, chemical and thermophysical properties at anycondition within the section are retrieved by the pro-gram WERAMI. WERAMI determines the polygonal ele-ment spanning the condition of interest and reportsthe requested properties, a program with similarstructure is used by Spear (1999) to recover phasediagram information. Because of the pseudocompoundapproximation, mineral chemistry changes only acrossthe boundaries of the polygonal elements of thepseudosection, whereas thermophysical properties varyboth continuously through a polygonal element anddiscontinuously across its boundary. These propertiescan be visualized directly as a false colour image byassigning representative values to each element of thepolygonal mesh. Alternatively, property values can besampled on a grid or path, defined by an arbitrarypolynomial, through the pseudosection and theninterpolated to provide a continuous model for theproperties. This latter mode of representation is usedto depict modal variation of the mineralogy and watercontent (Fig. 5d) and plagioclase and garnet compo-sitions (Fig. 5e) for our model pelite as a function ofpressure and temperature. The latter plot (Fig. 5e)demonstrates that when the use of pseudosections isjustified, mineral solution chemistry provides anextraordinarily simple thermobarometric method. Theirregular trajectories of the mineral isopleths are due toabrupt changes in the P–T dependence of mineralcomposition across phase field boundaries.

PRACTICAL IMPLEMENTATIONOF THE ALGORITHM

The programs VERTEX, POLYGON and WERAMI are com-ponents of a collection of FORTRAN computer programsfor the calculation and graphical representation ofphase equilibria. This package, including documenta-tion, can be copied via Internet (http://www.perplex.ethz.ch). This web site also includes a tutorial

Table 1. Mineral solution notation, formulae and model sources(1 ¼ Holland & Powell, 1998; 2 ¼ modified from Anovitz &Essene, 1987; 3 ¼ Holland et al., 1998; 4 ¼ modified frommodified from Gasparik, 1985 and Holland & Powell, 1998;5 ¼ Thompson & Waldbaum, 1969; 6 ¼ modified fromChatterjee & Froese, 1975 and Holland & Powell, 1998; 7 ¼Newton et al., 1980). The compositional variables w, x, y, andz may vary between zero and unity and are determined as afunction of pressure and temperature by free-energyminimization.

Symbol Solution Formula Source

Amph amphibole Ca2)2zNazMgxFe(1)x)Al3w+4ySi8)2w)2yO22(OH)2 1

Bio biotie KMg(3)y)xFe(3)y)(1)x)Al1+2ySi3)yO10(OH)2 1

Cal calcite Ca1)x)yMgxFeyCO3 2

Chl chlorite Mg(5)y+z)xFe(5)y+z)(1)x)Al2(1+y)z)Si3)y+zO10(OH)8 3

Cph carpholite MgxFe1)xAl2Si2O6(OH)4; 1

Cpx clinopyroxene Na1)yCayMgxyFe(1)x)yAlySi2O6 4

Crd cordierite Mg2xFe2)2xAl4Si5O18• (H2O)y 1

Ctd chloritoid MgxFe1)xAl2SiO5(OH)2 1

Dol dolomite CaMgxFe1)x(CO3)2 1

Ep epidote Ca2Al3)2xFe2xSi3O12OH 1

Gl glaucophane Ca2)2zNazMgxFe1)xAl3w+4ySi8)2w)2yO22(OH)2 1

Grt garnet Fe3xCa3yMg31)x)yAl2Si3O12 1

Kfs alkali feldspar NaxKyAlSi3O8 5

Mgs magnesite MgxFe1)xCO3 1

Ms muscovite KxNa1)xMgywFe(1)y)wAl3)2wSi3+wO10(OH)2 6

Ol olivine MgxFe1)xSiO4 1

Opx orthopyroxene Mgx(2)y)Fe(1)x)(2)y)Al2ySi2)yO6 1

Pl plagioclase NaxCa1)xAl2)xSi2+xO8 7

Pg paragonite KxNa1)xMgywFe(1)y)Al3)2wSi3+wO10(OH)2 6

Prg amphibole Ca2NawMgxFe1)xAl3w+4ySi8)2w)2yO22(OH)2 1

Sa sanidine NaxKyAlSi3O8 5

Sp spinel MgxFe1)xAlO3 1

St staurolite Mg4xFe4)4xAl18Si7.5O48H4 1


outlining the steps necessary for the computationillustrated by Fig. 5. The programs can be used withmost recent geological thermodynamic databases (e.g.Johnson et al., 1992; Berman & Aranovich, 1996;

Gottschalk, 1997; Holland & Powell, 1998). Becausethe FORTRAN sources of the programs are available, theprograms can be modified to accommodate mineralequations of state that have not been anticipated in the

Fig. 5. Computed pseudosection for Shaw’s (1956) average (subaluminous) metapelite composition, solution models as in Table 1,compositions were resolved with minimum accuracy of 3 mol percentage. Calculation and drafting required c. 10 min on a 200-MHzcomputer. Raw output from VERTEX (a) consists of pseudounivariant curves; this output is processed by Polygon to obtain the pseudo-section (b, labels omitted for legibility), which is redrafted in (c). Univariant fields shown by heavy solid lines, divariant fields by white fill,and higher variance fields by progressively darker shading. All fields include quartz in addition to the indicated mineralogy. Numberedfields in (c) correspond to: (1) ab + Chl + Ms + wrk (wairakite), (2) ab + Cal + Chl + Ms + Pg, (3) Chl + Ep + Ms + Pl,(4) Bt + Chl + Grt + Ms + Pl, (5) Bt + Grt + Ms + Pl + St, (6) Bt + Ms + Pl + sil, (7) Bt + Grt + Pl + Sa + sil,(8) Bt + Crd + Grt + Pl + sil, (9) Amph + Bt + Grt + Ms + Pl, (10) Bt + Crd + Ol + Pl + Sa, (11) Chl + Crp (carpho-lite) + lws + Ms + Pg, (12) Chl + Cpx + Crp + lws + Ms, (13) Chl + Cpx + lws + Ms + Pg, (14) Amph + Cal + Chl +Ms + Pg. Unlabelled phase fields can be deduced from the rule that adjacent phase fields differ by only one phase. (d) Modal variations,along path indicated by heavy dashed curve in (c). (e) Contour plot of the mole fractions of the anothite end-member in plagioclase and thepyrope end-member in garnet, heavy solid curves indicate stability fields of plagioclase and garnet. Both (d) and (e) constructed fromthe polygonalized pseudosection by WERAMI.


present code or for other specialized purposes.Graphical output from the package is written ininterpreted PostScript that can be imported intocommercial graphical editors.

DISCUSSION

The advance accomplished by this work is technicalin that we have developed a robust and generalmethod for computing pseudosections that requiresno a priori knowledge of the phase relations inquestion. Our method eliminates several difficultiesendemic to numerically exact nonlinear optimizationstrategies; these problems include convergence tolocal (metastable) free energy minima; numericalinstability; and, in the case of equilibrium calculators,complications caused by phase separation. Addition-ally, the approach advocated here offers a morerational means of representing the computed phaserelations than can be obtained by conventional energyminimization strategies (e.g. Saxena & Eriksson, 1983;Wood & Holloway, 1984; DeCapitani & Brown,1987; Eriksson & Hack, 1990). In contrast to themethods outlined by Powell et al. (1998) and Spear(1988), the approach has the advantage that of beingfully automated. The feasibility of our strategy restson the discretization of the continuous compositionalvariation of solution phases with pseudocompounds(Connolly & Kerrick, 1987). While discretizationresults in finite accuracy, it is unlikely that this isa major limitation given present computationalresources and accuracy of mineralogical solutionmodels. In cases where this assumption proves untrue,our approach may serve as a complement to numer-ically exact procedures.

ACKNOWLEDGEMENTS

We are grateful to F. Spear and R. White for construc-tive reviews and to R. Powell for his editorial commentsand suggestions. Y. Podladchikov, A. Barnicoat andD. Kerrick provided the impetus for this work, whichwas supported by US National Science FoundationMARGINS program.

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Received 9 May 2001; revision accepted 6 April 2002.

APPENDIX: PHASE DIAGRAM NOMENCLATURE

Although much of current petrological phase diagram jargon isgenerally understood, the meaning of some terms is vague. Toavoid the possibility that our language gives rise to any ambiguitywe review some aspects of phase diagram terminology. The inde-pendent variables of phase diagrams can be related to a ratio ofextensive properties (e.g. chemical composition), or to an intensivestate function (P, T, l1,…,lc). Both kinds of variables are inten-sive and to distinguish them they are designated potential (Hillert,1985; Palatnik & Landau, 1964) and compositional (Connolly,1990) variables, respectively. It is convenient to consider thesevariables to be the representative co-ordinates of a system com-posed of c independently variable kinds of matter. A real phys-icochemical system is open with respect to some extensive propertyif the property may vary through interactions with the environ-ment of the system. These properties must be characterized bypotentials, referred to as environmental variables, because inequilibrium thermodynamics the nature of such interactionscan only be defined if these potentials have the same valueboth within and beyond the boundaries of the system. The asso-ciation of the variables of a diagram with the determinative

physicochemical properties of a real system is of course unspecifiedbecause it is impossible to prescribe the way a phase diagram willbe used.

A diagram is a phase diagram only if its geometry defines therepresentative co-ordinates of both a system and its stable phases(e.g. Schmalzreid & Pelton, 1973; Hillert, 1985; Palatnik & Landau,1964). From the definition of composition variables, it follows thatsuch a diagram must be c + 1 dimensional (i.e. only c + 1 of theintensive variables can be independent). If all c + 1 variables of thisdiagram are compositions, then diagram is composed of c + 1dimensional phase fields that define the loci of conditions for which aspecific phase assemblage of p phases is in equilibrium. Definingphase field variance as f ” c + 2 ) p, if v composition variables ofthe phase diagram are replaced by their conjugate potentials, thenthe dimension of any phase field with f < v is reduced by v ) fdimensions, because of the equality of potentials among coexistingequilibrium phases.

In general the information in phase diagrams must be presented intwo dimensions by sectioning or projection to be of practical value.Phase diagram sections may be defined by potential or compositionsectioning variables, and are designated potential and compositionsections, respectively (Hillert, 1985). Potential sections are themselves


phase diagrams because the section defines the state of both thesystem and its phases. In contrast, composition sections are notphase diagrams because the section defines neither the compositionsnor the amounts of the phases. In this respect, usage of the termpseudosection to designate composition sections, as implicit in recentgeological literature (e.g. Powell et al., 1998), is not inappropriate.We employ the term in this paper as short-hand for compositionsections made on a two-dimensional co-ordinate frame defined bytwo potentials. In this context, c + 1 dimensional univariant fieldsand c dimensional invariant fields appear as curves and points,respectively. It is always possible to write a mass balance relation-ship, i.e. a reaction, among the phases of a univariant field of thegeneral form:

Rcþ1

i¼1mi~uui ¼ 0 ðA:1Þ

where ~uui is a vector representing the composition of the ith phase andmi is its reaction coefficient. If the potentials of interest are pressureand temperature, a necessary condition for the stability of theunivariant assemblage is

Rcþ1

i¼1miG

i ¼ DG ¼ 0; ðA:2Þ

the sufficient condition being that eq. (1) is true for all nondegeneratepermutations of c phases of the univariant assemblage. The reactioncoefficients of one, or more, phases in Eq. (A.1) may be zero, thusthe reaction equation can be written among a subset of the equilib-rium phases, e.g. as would be the case for a polymorphic phasetransformation in a system with c > 1. We do not follow the com-mon petrological practice of identifying univariant fields by reactionsto emphasize that within a univariant field the state of the system canonly be defined by all c + 1 phases of the fields, regardless of thenumber of phases involved in the associated reaction. Likewise, wedo not follow the practice of identifying univariant and invariantfields as univariant curves and invariant points for two reasons: (i)curves and points represent, by definition, univariant and invariantconditions, and therefore the adjectives are superfluous; and (ii) weseek language that allows concise statement of phase diagram prin-ciples such as the Maessing-Palatnik rule, which states that adjacentphase fields differ by exactly one phase (e.g. Hillert, 1985).


an automated strategy for calculation of phase diagram sections and

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