Are reversed Fe-Mg exchange and solid solution experiments

American Mineralogist, Volume 79, pages 938-950, 1994

Are reversed Fe-Mg exchange and solid solution experiments really reversed?

D.l,vro R. M. P,lrrrsoNDepartment of Geology and Geophysics, University of Calgary, Calgary, Alberta T2N 1N4, Canada

Ansru,cr

Numerous experimental studies investigating Fe-Mg exchange and other solid solutionvariations in minerals have claimed to have reversed their results because of apparentconvergence to the equilibrium composition from opposing compositional directions. Thesereversals are actually apparent compositional brackets that, in many cases, cannot bereversed. Furthermore, it is likely that many if not all apparent compositional brackets arenot strictly limiting, in the sense that they may not unequivocally constrain a composi-tional interval within which the equilibrium composition must lie. A true compositionalbracket is obtained most confidently if a single compositional parameter (e.g., Fe-Mgexchange) is isolated and a continuous difi.rsionally controlled compositional gradientfrom the starting to the final composition is established. The gradient indicates the direc-tion of the equilibrium composition required by the concept of bracketing. In contrast, anequilibration mechanism involving multicomponent solution and precipitation typicallyinvolves variation in several rather than one compositional parameter and may result incompositional discontinuities between initial and final compositions, opening the possi-bility ofjumping to metastable compositions. In such situations, the identification of thedirection to equilibrium from the final compositions becomes ambiguous, and the logicalbasis for bracketing is lost. Analysis of the large, apparently reversed experimental dataset of Pattison and Nefion (1989) on garnet * clinopyroxene Fe-Mg exchange shows firstthat the experiments were not truly reversed and second that the equilibration mechanismwas mainly multicomponent solution and precipitation with negligible diffirsion, with theresult that the experiments were not unequivocally bracketed. Examination of several otherFe-Mg exchange and solid solution experimental studies suggests a similar conclusion.Calculated diffusion-penetration distances suggest that diffirsion should not be expected tocause measurable compositional changes in normal experiment times. Unreversed (and inmany cases probably not strictly bracketed) solid solution experiments using crystallinestarting materials yield consistent compositional trends as a function of pressure, temper-ature, and other intensive parameters, which are necessary, if not sufficient, indicationsthat equilibrium was closely approached. The lack of true reversals nevertheless influencesthe way in which thermodynamic data are extracted from such experiments.

Irrnoouc"noN lution minerals in various assemblages for purposes ofReversibility is regarded as an essential demonstration geothermometry, geobarometry, and determining activi-

that equilibrium has been attained in phase equilibrium ty-composition relations (e.g., diopside-enstatite solid so-experiments (Holloway and Wood, 1988, p. ll). Unre- lutions: Lindsley and Dixon, 1976; Nickel et al., 1985;versed or unbracketed experiments are open to criticism Al solubility of pyroxenes in equilibrium with garnet:becauseofthepossibilitiesthatequilibriumcompositions Perkins and Newton, 1980; Lane and Ganguly, 1980;were not attained or that the phase assemblage or phase Perkins et al., 1981; Ca in olivine in equilibrium withcompositions were misinterpreted. clinopyroxene: Adams and Bishop, 1986; calcium-iron-

Several studies have calibrated Fe-Mg exchange ther- magnesium-manganese garnet compositions in a varietymometers on the basis of what are claimed to be reversed of assemblages: Koziol and Newton, 1989; Koziol , 1990;experiments using crystalline starting materials (e.g., gar- Koziol and Bohlen, 1992).In all these experimental stud-net + biotite: Ferry and Spear, 1978; garnet + clinopy- ies, crystalline starting materials were used to avoid theroxene: Pattison and Newton, 1989; garnet + olivine: potential problems of nucleation and the persistence ofHackler and Wood, 1989; garnet + orthopyroxene: Eck- metastable crystalline compositions from disordered, highert and Bohlen, 1992; clinopyroxene + olivine: Perkins free-energy starting materials such as glass or gels. Theand Vielzeuf, 1992). Many other experimental studies also other theoretical advantage is that changes in the com-have claimed to have reversed compositions of solid so- positions of crystalline starting materials might be used

ooo3-004x/94l0910-0938$02.00 938

to indicate the direction to equilibrium, even if equilib-rium compositions themselves were not attained.

The purpose of this study is to evaluate whether solidsolution experiments using crystalline starting materialsprovide either reversals or limiting brackets of equilib-rium compositions. First, a distinction is made betweena reversal and a compositional brackel I argue that whatare referred to as solid solution reversals are actually ap-parent compositional brackets that, in many cases, can-not be reversed. Second, I argue that a true compositionalbracket, defining a compositional interval within whichone is certain the equilibrium composition must lie, canonly be obtained confidently ifthe phases equilibrated bya diffi.rsionally controlled mechanism rather than by amulticomponent solution and precipitation mechanism,the latter ofwhich introduces several theoretical and an-alytical complications. Several published experimentalstudies, including Pattison and Newton's (1989) appar-ently reversed study on the Fe-Mg exchange between cli-nopyroxene and garnet, are examined to evaluate reac-tion mechanisms and the extent to which the experimentalresults were either reversed or unequivocally bracketed.Experimental diffusion data are used to calculate the dif-fusion penetration distances for typical experiment timesto see whether diffusional exchange by itselfis fast enoughto allow significant compositional change. Although theresults of these investigations suggest that most solid so-lution experiments involving crystalline starting materi-als equilibrate by a multicomponent solution and precip-itation mechanism and consequently do not provide ei-ther true reversals or unequivocal brackets ofequilibriumcompositions, several lines of evidence suggest that theexperimental results represent a close approach to equi-librium and are superior to synthesis experiments usingglass or gel starting materials. Nevertheless, these consid-erations carry implications for the techniques used to ex-tract thermodynamic data from such experiments.

Rnvnnsr. vs. coMposITroNAL BRAcr(r'TrNGEXPERIMENTS

Reversal experiments

Reversibility and equilibrium bracketing are common-ly taken to be synonymous, even though there are subtlediferences between the two. A general definition of re-versibility is the ability to reverse the direction of a re-action by changing P, Z, or the composition of the system(Holloway and Wood, 1988, p. ll). This is most easilyunderstood in the context of a univariant reaction in-volving phases of fixed composition such as

3CaAlrSirOr: CarAlrSirO,, + 2AlrSiOs + SiO,anorthite (An) grossular (Gr) kymite (Ky) quntz (Qlz)

(1 )

in the system CaO-AlrOr-SiO, (e.g., Koziol and Newton,1988). The equilibrium compositions of the minerals inthe mineral assemblage are known in advance (stoichio-metric end-members) and do not change significantly over

939

the P-T range of the experiments, so they can be synthe-sized individually in advance and added together to makemineral mixes. An experimental reversal between phase

assemblages A (An) and B (Gr + Ky + Qtz) involves twoexperiments at different P-Iconditions containing a mix-ture of the two phase assemblages (i.e, An + Gr + Ky *

Qtz). At one P-T condition, phase assemblage A (An)grows at the expense of phase assemblage B (Gr + Ky +

Qtz), from which one can conclude that the free energy,G, of assemblage A is less than that of assemblage B, i.e.,G* ( Gc.**"*oo (Gordon, 1973). At the second P-T con-dition, phase assemblage B (Gr + Ky + Qtz) grows atthe expense of phase assemblage A (An), from which onecan conclude that G6"11raqo 1 G^. In both cases, thedriving force for the reaction is the minimization of thetotal free energy of the system. The equilibrium P- 7 con-dition for the univariant equilibrium, at which G* :

G6.*1"*qo, must lie between these two P-?" conditions,thereby establishing the concept of equilibrium bracket-ing. The sense of reversal comes from phase assemblageA (An) forming from phase assemblage B (Gr + Ky +

Qtz) at the first P-7 condition and vice versa at the sec-ond P-Z condition, such that the two assemblages arealternately products and reactants for each other (if oneassumes that no metastable intermediate reaction prod-

ucts formed).In experiments involving phases whose compositions

can vary (i.e., show solid solution), reversibility and equi-librium bracketing may not be equivalent' Taking Fe-Mgexchange between garnet and clinopyroxene as an ex-ample, a theoretical reversal of their Fe-Mg ratios mightinvolve a pair of experiments such as illustrated in Figure

lA, with the assumptions that only Fe and Mg are ex-changing, that Fe-Mg diffusion is fast enough in the twominerals that the reacting bulk composition is the sameas the inserted bulk composition, and that there are nokinetic barriers to the attainment of equilibrium' At tem-perature I,, garnet and clinopyroxene are assumed to havethe equilibrium compositions Grt' and Cpx'. When tem-perature is changed from ?", to Tr, the total free energyof the system, G, is lowered by the Fe-Mg ratios of Grt'and Cpx, changing to those of Grt, and Cpxr, such thatG1*r*.n^r) { G16n1+cpxj' If temperature changes from ?"'back to 2,, then Grt, and Cpx, change in Fe-Mg ratioback to Grt, and Cpx,, such that at Ir, G(c.r+cp,r) (

G16n2+cp*).

Although the above theoretical example satisfies therequirements of a reversal in the sense that the directionofthe reaction is reversed and that each phase assemblageis alternately a reactant or product for the other, thereare several problems in its practical implementation. Oneis that starting materials comprising coexisting mineralcompositions in chemical equilibrium at I' must some-how be synthesized. Otherwise, Grt, and Cpx, would beunlikely to change in composition back to Grtr and Cpx'when the temperature was changed from T, to Zr, andthe criterion of reversibility would be lost. However, be-cause this information cannot be known a priori, the above

PATTISON: ARE REVERSED EXPERIMENTS REALLY REVERSED?

940 PATTISON: ARE REVERSED EXPERIMENTS REALLY REVERSED?

-0.6 0.8 1.0

XMgCP'

--+0 . 0 0 2 0 4

theoretical experiment would be very difficult to createin practice. One possibility might be to use as startingmaterials coexisting minerals crystallized in synthesis ex-periments and to employ an iterative approach involvingseveral experiments such as described above until theminerals repetitively change to the same compositions,going back and forth from Z, to Zr.

Even if the first problem were overcome, there are twomore problems common to all solid solution experi-ments: (l) how can one be sure that the phases in factchange to the equilibrium (most stable) compositionsrather than to more stable, but possibly metastable, com-

Fig. l. Schematic representatior oftie-line rotations involv-ing X* t: M/@g + Fe)l of garnet and clinopyroxene, if oneassumes infinitely fast diffirsion. Labeled tie lines correspond tolabeled points in Fig. 2. (A) Stable tie lines for two temperatures.(B) Initial and final tie lines for two pairs ofgarnet and clino-pyroxene in equal modal abundance at one temperature. Dottedlines represent tie lines connecting compositions that are initiallyout of equilibrium. Dash-dot lines are tie lines connecting finalequilibrium compositions. Arrows show direction of composi-tional change of garnet and clinopyroxene going from initial tofinal compositions. (C) Garnet present in overwhelming abun-dance relative to clinopyroxene. Solid lines are tie lines con-necting equilibrium compositions. The dashed line represents atie line connecting equilibrium compositions given no change ingarnet composition during the experiment. Other symbols arethe same as in B.

€-

positions and (2) even ifthe phases do change to equilib-rium compositions, how can one be sure that the equili-brated portions are coffectly identified and measured?

Compositional bracketing experiments

The solution has been to use an apparent convergencetechnique, popularly known as a compositional reversal,which uses the principle of bracketing equilibrium com-positions from opposing compositional directions (e.9.,Ferry and Spear, 1978; Pattison and Newton, 1989; Ko-ziol and Newon, 1989). The approach, although straight-forward in theory, is difficult in practice. First, as pointedout by Perkins and Newton (1980) and others, it is nec-essary to isolate the variation in a single compositionalparameter or exchange vector (e.g., Fe-Mg exchange),which may be difficult in phases that can vary in severalcompositional parameters. If more than one composi-tional parameter varies in the experiment, one cannot besure to what extent the variation in the desired compo-sitional parameter is controlled by variation in the otherparameters.

Second, achieving a convergence requires a special ex-perimental design. Again taking Fe-Mg exchange betweengarnet and clinopyroxene as an example, the approach isto allow the Fe-Mg composition of either garnet or clino-pyroxene to vary in response to changes in either P or Tor one other compositional parameter. A convergence inMgl(Mg + Fe) would be obtained by inserting two cli-nopyroxenes with Mgl(Mg * Fe) ratios that are out ofequilibrium, such that, as the minerals changed compo-sitions to lower the total free energy of the system, thefinal equilibrium composition would be constrained bythe approach from the Fe-rich and Mg-rich directions inthe two experiments. Even if the final compositions didnot coincide, the equilibrium composition in theory mustlie between them, thereby establishing a compositionalbracket.

Although it establishes a bracket, such an experimentstill might not be a true reversal because it might never

o 2 0.4 0.6 0.8 1.0-

-0.2 0.4 0.6 0.8 1.0

xMgcrl

xMgcPt

Starting Grt

1 0xMgcrr

Starting Cpx\ .. I Finat Cpx\ :

' \ f \ / t r i I i . r+

0.0 0.2 0.4

Stable tie line at T2

PATTISON: ARE REVERSED EXPERIMENTS REALLY REVERSED? 941

be possible to recreate the starting compositions that areinitially out of equilibrium from the final mineral com-positions (i.e., reverse the experiment) simply by chang-ing P, T, or one other compositional parameter. For ex-ample, recreating a starting garnet + clinopyroxene pairin which garnet is more magnesian than clinopyroxene isimpossible because garnet is always less magnesian thancoexisting clinopyroxene in nature. For convergence ex-periments of this kind, the term compositional reversalshould be abandoned in favor of compositional bracket.

A third requirement for such an experiment, implicitin the concept of convergence to equilibrium, is that thedirection to equilibrium can be demonstrated. This is asubstantial problem because sluggish intragranular dif-fusion rates inhibit the attainment of equilibrium in ex-periment times (see below).

Isor.lrroN oF oNE coMposrrroNAl pARAMETER

In the above example of Fe-Mg exchange between gar-net and clinopyroxene, the isolation of Fe-Mg change inonly one of garnet or clinopyroxene is impossible becausethe Fe-Mg composition of one can change only throughexchange with the other. In Figures lB and 2, if gametand Fe-rich clinopyroxene are inserted in equal modalabundance into an experiment and equilibrate, Fe-Mgexchange between both phases gives a tieJine rotation inFigure lB that corresponds to vector B, in Figure 2. Ifthe same experiment is repeated with an Mg-rich clino-pyroxene, vector B, results. The two experiments do notbracket any one point on the equilibrium isotherm (otherthan in a very loose way) because both the garnet andclinopyroxene changed in Mg/(Mg * Fe) during the ex-periments, which result in vectors that diverge rather thanconverge (see Fig. 2). Although each individual vectorprobably represents an approach to equilibrium, one can-not be sure that a longer experiment time or more anal-yses of the rim compositions wouldn't produce a differentfinal composition (and hence a change in length or direc-tion of the vector) for one or other of the two minerals.

Ferry and Spear (1978) were the first to propose amethod that overcame this problem in their calibrationof the garnet + biotite Fe-Mg exchange thermometer.They reasoned that ifan overwhelming abundance ofoneof the exchanging Fe-Mg minerals (garnet) were insertedinto the experiments, then, from the mass balance, theother mineral (biotite) could change in composition sub-stantially, with negligible change in the composition ofthe garnet (see Figs. lC and 2). In principle, this approachhad many advantages: (l) it efectively reduced the num-ber of minerals changing composition in the experimentfrom two to one, thereby eliminating the need to analyzeone of the minerals (garnet) as a potential source of un-certainty (2) it seemed to provide a kinetic advantage inthat the extent of the Fe-Mg exchange of garnet, relativelythe more sh'ggish of the two minerals to participate inFe-Mg exchange, would be minimized; and (3) it allowedan effective (although not strict) bracketing of the Mg/(Mg + Fe) composition of biotite in equilibrium with a

oo 02 04 xMgct t og 10

Fig. 2. Schematic Fe/Mg distribution isotherm at constanttemperature and pressure. Curve is the equilibrium isotherm.Solid circles represent starting compositions, corresponding tothe dotted tie lines in Fig. lB and lC. Labeled arrowheads andsqrEres represent final compositions, corresponding to the la-beled tie lines in Fig. l. Solid, dashed, and dash-dot vectorsrepresent the direction of change from initial to final composi-tion, corresponding to tieJine rotations in Fig. 1. Squares arethe final compositions for diferent modal proportions of startingcomposition D. Squares and dotted lines rather than arrows areused to show that there may not be a continuous compositionalgradient from initial to frnal compositions (see text).

single garnet by convergence to an intermediate compo-sition from Fe-rich and Mg-rich starting compositions(see tie lines C, and C, in Fig. lA and correspondingvectors in Fig. 2). An underlying assumption made byFerry and Spear was that the only compositional changeinduced in garnet and biotite in the experiments was Fe-Mg exchange. Pattison and Newton (1989), Hackler andWood (1989), Eckert and Bohlen (1992), and Perkins andVielzeuf (1992) applied the same principles in their at-tempts to bracket Fe-Mg exchange between garnet * cli-nopyroxene, garnet + olivine, garnet + orthopyroxene,and clinopyroxene + olivine, respectively. This approachappeared to achieve its aim: the exchanging minerals,which were in small modal abundances in pairs of ex-periments (e.g., biotite in Ferry and Spear; clinopyroxenein Pattison and Newton), afterward had intermediate Fe-Mg compositions that generally agreed quite closely.

In contrast to Fe-Mg exchange experiments, Koziol andNewton's (1989) and Koziol's (1990) experiments ongrossular activity-composition relationships in calcium-iron-magnesium garnets were based on the displacementof the equilibrium Gros : An + Ky + Qtz for garnets ofdiferent composition. These experiments involved only

'I 0

xao

t=x

0 4

o 2

B1


one solid solution phase, garnet, so that apparent bracketsof garnet compositions could be obtained without thecomplications of compositional variations in the otherphases.

IonxtmlcluoN oF THE DIREcrroNTO EQUILIBRIUM:

Rn.c.crroN MEcHANTsMs

Diffusion vs. solution and precipitation

An important underlying assumption in the above ap-proach is that starting compositions that were initiallyout of equilibrium converged to the final equilibriumcomposition, thereby establishing the direction to equi-librium required by the concept ofbracketing. In exper-iments in which the inserted out-of-equilibrium compo-sitions are preserved in portions of the mineral grains(e.g., clinopyroxene in Pattison and Newton; olivine inHackler and Wood), evidence for convergence to equilib-rium is best provided in theory by a continuous, diffu-sionally controlled compositional gradient from the un-reacted seed composition to the equil ibrated rimcomposition (Figs. 34' and 4A). Each successive spot downthe gradient represents a lower free energy (more stable)composition, such that even if the final equilibrium com-position is not attained or cannot be measured, the di-rection to equilibrium is unambiguous (Figs. 3A and 4A).Moreover, as discussed by Goldsmith and Newton ( I 974,p. 341-343 and their Fig. l), it is thermodynamicallyimpossible to overstep the equilibrium by this mecha-nism because this would involve forming a composilionwith relatively higher free energy at the expense of a com-position with lower free energy. Consequently, conver-gence from the opposing compositional directions estab-lishes an unequivocal bracket of the equilibriumcomposition.

In contrast, the development of newly formed com-positions by recrystallization or solution and precipita-tion violates the requirement for bracketing of equilibri-um, because these mechanisms may result in jumps tocompositions that for kinetic reasons may overshoot orundershoot the stable equilibrium composition (see Figs.38 and 4B). Such a process is thermodynamically per-missible in the sense that the new compositions are morestable, i.e., have a lower free energy, than their respectivestarting compositions, even if they are not the most stablecompositions (see the discussion in Goldsmith and New-ton, 1974). Several authors have observed this effect insome of their experiments, including Bachinski and MiiLl-ler (1971) in experiments on the alkali feldspar solvus,Lindsley and Dixon (1976) in experiments on the ensta-tite-diopside system, Perkins and Newton (1980) in ex-periments on the AI solubility of orthopyroxene in equi-librium with garnet, and Pattison and Newton (1989) ingarnet + clinopyroxene Fe-Mg exchange experiments. Theresult is apparent compositional overlaps in pairs of ex-periments starting from opposing compositional direc-tions. In such a situation, there is no guarantee that the

newly formed compositions will subsequently approachequilibrium from the same compositional direction as thestarting material (see Fig. 4B). If knowledge of the direc-tion to equilibrium from the final compositions is lost,the basis for establishing an unequivocal bracket of theequilibrium is also lost. For example, in experiments inwhich compositional overlaps are observed, one cannotbe certain whether the equilibrium composition lies with-in the range of the compositional overlap or outside therange. Even if nonoverlapping apparent brackets frompairs of experiments are obtained, there is no guaranteethat the equilibrium composition lies within the range ofthe apparent bracket (see Fig. 4B).

These considerations indicate that to achieve unequiv-ocal compositional brackets, the equilibration mecha-nism must be controlled by intragranular ditrusion of theelements of interest. In contrast, a multicomponent so-lution and precipitation mechanism involves the partic-ipation of an intermediate phase, typically an intergran-ular fluid or melt, into which the inserted compositionsdissolve and from which new, more stable (but not nec-essarily equilibrium) compositions precipitate either ontopreexisting seed grains or as new crystals (Fig. 4B). Thisprocess is unidirectional in the sense that all the startingmaterial will eventually be consumed, given enough timeand the absence of kinetic barriers, and that in manycases it will be impossible to remake the starting com-positions from the final compositions. Strictly speaking,such experiments are a kind of synthesis experiment, inwhich the starting materials are crystalline materials rath-er than glass or gels. Moreover, in experiments involvingphases that show solid solution on more than one crys-tallographic site, a solution and precipitation mechanismusually results in changes in more than one compositionalparameter, further eroding the criterion of strict brack-etmg.

The effect of fluid

The presence or absence of a fluid phase has a majoreffect on experimental reaction rates and reaction mech-anisms (Rubie and Thompson, 1985). Fyfe and Verhoo-gen (1958) estimated that the presence of even smallamounts of HrO increased reaction rates by 8-10 ordersof magnitude, most likely because of solution and precip-itation. A compilation of rate constants for dissolutionand phase equilibrium experiments in the presence offluid by Wood and Walther (1983) shows that the averagelogarithm to the base l0 ofthe rate constant at 800'C isabout -9.5 cm2ls, compared with interdiffusion coefr-cients for Fe-Mg exchange in garnet and clinopyroxeneat 800 t of -15 to -17 cm2/s and for Al diffusion inclinopyroxene at 800'C of -20 cm'zls (see the referenceslisted at the bottom of Table l). These data support theconclusions of Rubie (1986) and Liittge and Metz (1993)that, in the presence of HrO-bearing fluid, reactions pro-ceed mainly by a solution and precipitation mechanism.


A. DIFFUSION

edge of crystal

\initial seed composition

quenched diffusion profile

G O . G S c G 4 < G 3 < G 2 < G 1

B. SOLUTION.PRECIPITATION

Fig. 3. Schematic illustration of contrasting core-rim com-positional profiles arising from diffirsion and solution and pre-cipitation equilibration mechanisms. With reference to Fig. 4,the mineral illustrated is clinopyroxene. Numbered circles cor-respond with those in Fig. 4. (A) Diftrsion: the open circle (l): starting seed composition, solid circles (2-6) : individual pointson the quenched diftrsion profile. From the starting compositiontoward the rim, the free energy of each successive quenchedpoint on the profile is lower than the previous one. (B) Solutionand precipitation: the open circle (1) : starting seed composi-tion, the solid circle (2) : composition of precipitated rim. Anabrupt compositional discontinuity may occur between the pre-cipitated rim and the seed crystal. The free energy of the rimcomposition, whether stable or metastable, is nevertheless lowerthan the starting composition.

equilibrium comocition is not bracketed

Mg/(Mg+Fe)

Fig. 4. Schematic free energy (G) vs. Me/(Mg + Fe) plots,showing contrasts between diffusionally controlled and solutionand precipitation equilibration mechanisms. Garnet is assumedto be present in overwhelming abundance so that only clinopy-roxene shows observable compositional change. Numbers cor-respond to those in Fig. 3. (A) Diffusion: open circles : twostarting clinopyroxene compositions, one more Fe-rich and theother Mg-rich than the equilibrium composition. Numbered sol-id circles show the lowering of free energy of the clinopyroxene(and hence of the system) as it changes composition going downa diffirsional profile from the initial Mg-rich seed material to-ward the Fe-richer rim. If the Fe-rich starting clinopyroxene de-velops a complementary Mg-richer rim by diftrsion, the twoexperiments, even if they do not attain the equilibnum com-position, nevertheless provide an unequivocal bracket of theequilibrium composition. @) Solution and precipitation: start-ing compositions as in A. The equilibration mechanism involvesa two-step process (arrows): first, solution of clinopyroxene andgarnet into an intergranular fluid or melt (dashed curve), thecomposition of which is unknown (?), and second, precipitationof rims or new crystals of clinopyroxene, the cornpositions ofwhich are ofa lower free energy than the starting clinopyroxene,but for kinetic reasons may not necessarily be the equilibriumcomposition (2?). The equilibrium composition cannot thereforebe bracketed using the final compositions.

G

A. DIFFUSION

equilibrium compcitions

cpxMg/(Mg+Fe)

B. SOLUTION.PRECIPITATION

intergranular fluid or meh

su@ssively more stableCpx @mpGitions preserved

in ditfirsion profile

III

equilibrium @mposition is brac*eted

oll.+ED=CD=

vG

pr*ipitation ot rims tromintergranular fluid or meh

III

oIl+CD=ED=

.<-

' ) . t

initial seed composition

compositional discontinuity betweenrim and unreacted seed

newly precipitated rim


TaBLE 1. Calculated diffusion penetration distances in minerals in experiment times

Expt. conditions Diffusion penetration distance (rlm), based on X: V4Dt

Al Cpx Fe-Mg Ol Fe-Mg BtCa Grt(slow) Fe-Mg Px'

Fe-Mg Grt Fe-Mg Grt(fast) (slow)

P T t(kbaO rc) (d)

1 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 5

500500500550550550600600600700700700800800800900900900900

10001000100010001 1001 1001 100120012001 2001300130014001400

0.010.01

<0.010.030.o20.010.050.040.030.280.200.140.760.530.362.281.611 . 1 00.724.092.791.831.066.184.052.U8.05o.c /4.65

12.O8.46

20.314.4

<0.01<0.01<0.01<0.01<0.01<0.01

0.01<0.01<0.01

0.050.030.020.150.110.070.550.390.260.171 . 1 50.780.510.301.971.290.752.882.351.664.723.348.746.18

<0.01<0.01<0.01<0.01<0.01<0.01<0.01<0.01<0.01

0.030.020.020.110.080.0s0.39o.270.19o.120.810.550.360.211.400.910.532.O31.661 . 1 73.342.366.184.37

<0.01<0.01<0.01

0.01<0.01<0.01

0.010.010.010.080.060.040.28o.200.130.990.700.480.312.051.40o.920.533.522.311.385.104.',t72.958.34s.90

15.410.9

<0.01<0.01<0.01<0.01<0.01<0.01<0.01<0.01<0.01<0.01<0.01<0.01<0.01<0.01<0.01

0.010.010.01

<0.010.030.020.010.010.050.030.020.070.060.040.120.080.220.16

o.260 .190.090.740.s30.261.320.930.666.254.423.12

15.71 1 . 17.57

44.831.721.614.276.752.434.319.8

1 1 173.142.2

1411 1 581.1

203144385237

0.&t0.590.292.351.660.8i14 .172.9s2.08

19.814.09.88

49.535.023.9

14110068.444.8

24216610962.7

3532311 Sril4443tr!257il2454

1060750

120601 5

120601 560301 560301 5301 57

301 573

1 57317313212'|21

Nofe.' D(Fe-Mg)crt, fast : Cygan and Lasaga (1985) with pressure @rrection of Chakraborty and Ganguly (1991). D(F+Mg)Grt, sbw: Chakrabortyand Ganguly (1991). D(Ca)crt : ca. 0.5.D(Mg Grt), Chakraborty and Ganguly (1991). D(Fe-Mg)Ol : Misener (1974); Mg' : 0.3. D(Fe-Mg)Opx :

Ganguly andTazzoli (1994); D(F+Mg cpx) = D(Fe-Mg Opx) : Ganguly in Ghose and Ganguly (1982). D(Al)Cpx: Jaoul et al. (1991). D(FeMg)Bt:ca. 10.D(Fe-Mg Ol) (see text).

- Fe-Mg Cpx = Fe-Mg Opx.

Thus, in experiments in which HrO has been added tocatalyze the reaction, it is likely that the increased rate ofreaction represents a transition from a reaction mecha-nism controlled by lattice diffusion to a much faster in-terface-controlled solution and precipitation mechanisminvolving rapid diffusion through the fluid. At the hightemperatures at which HrO reacts with the solid phasesto make a melt phase, the melt may also act as an effec-tive intergranular medium for solution and precipitation(e.g., Boettcher and Wyllie, 1968). The effectiveness ofeven a small amount of moisture as a catalyst is shownby the increase in reaction progress when an experimentalmineral mix is exposed to a humid atmosphere or isbreathed on (e.9., Boettcher and Wyllie, 1968; Holland,1980). Vielzeuf and Clemens (1992) noted that tenacious-ly adsorbed moisture, even in specially dried experimen-tal materials, may be very difficult to drive off entirely,leading to the possibility of some d€gree of fluxing evenin ostensibly unfluxed experiments. In summary, it maybe that equilibration by solution and precipitation is moreprevalent in experimental solid solution studies than pre-viously thought.

GlnNnr + cLrNopyRoxENE Fe-Mg ExCHANGEEXPERIMENTS

Below, the apparently reversed experimental data setof Pattison and Newton (1989), reanalyzed by Aranovichand Pattison (1994), on the Fe-Mg exchange between gar-net and clinopyroxene is examined in detail to evaluatethe extent to which Fe-Mg exchange was the only com-positional parameter that varied in the experiments andto assess the relative importance of multicomponent so-lution and precipitation and diffusion in the experiments(see Tables 1-3 of Pattison and Newton).

Experimental details

The starting clinopyroxene compositions in Pattisonand Newton (1989) were synthesized along the diopside-hedenberyite join, Ca(Fe,Mg)SirOu. Microprobe analyses(EDS analysis in Pattison and Newton; WDS analysis inAranovich and Pattison, 1994) confirmed the expectedstoichiometric compositions, with minor heterogeneity(typically <50/o). Inserted garnets were synthesized(Ca,Fe,Mg)rAl2Si3Or2 solid solutions. Microprobe analy-


ses from both the Pattison and Newton and Aranovichand Pattison studies confirmed that the garnets were closeto the expected stoichiometric compositions, althougheach showed up to l0o/o heterogeneity.

Garnet and clinopyroxene crystals were inserted as loosegrains, 10-50 pm in size, in a garnet: clinopyroxene massratio of 95:5 (Method A) or 90:10 (Method B). In hydro-thermal experiments between 700 and 1000 "C, the ex-perimental materials were mixed with a small amount ofHrO and oxalic acid hydrate and sealed in Ag capsules.In contrast to the inserted loose mixture of starting ma-terials, the experimental products emerged as variablycemented aggregates of crystals. The extent of cementa-tion increased with temperature. Backscattered electronimages of experimental products (Fig. I of Aranovich andPattison, 1994) invariably showed evidence ofcorrosionand rounding ofboth garnet and clinopyroxene grains. Inexperiments at I100 and 1200 oC, the mineral mixes weredried at I l0'C prior to insertion into graphite capsules.Although no flux was added, the experimental productsemerged as cemented pellets, most likely because of asmall amount of grain boundary melt that developed inthe samples (Aranovich and Pattison, 1994). The sourceof moisture for the melt may have been atmosphericmoisture that entered the mineral mix during insertioninto the containers, possibly augmented by residual mois-ture in the containers or by the penetration of pressuremedium material (especially Na).

Compositions of experirnental products

Comparison of averaged microprobe analyses of garnetrims and cores (Pattison and Newton. 1989: Aranovichand Pattison, 1994) showed that, within some scatter,garnet crystals underwent subtle Fe-Mg compositionalchanges in their rims [AMe/(Me + Fe) (core-rim) < 0.03],generally consistent with mass balance constraints (Table2 of Pattison and Newton and Figs. 2-4 of Aranovichand Pattison). In contrast, clinopyroxene crystals devel-oped rims 5-10 pm wide, even in low temperature ex-periments (800 "C), showing substantial compositionalchanges in all elements including Fe-Mg (Table 2 of Pat-tison and Newton and Table I of Aranovich and Patti-son). Final compositions were complex subcalcic clino-pyroxene solid solutions bearing Al t N4 contrastingwith the starting material of Al-absent, Ca-saturated cli-nopyroxene.

The textural and compositional observations providestrong evidence that the equilibration mechanism in Pat-tison and Newton's experiments involved primarily themulticomponent solution and precipitation of both gar-net and clinopyroxene, possibly in addition to meltingand crystallization processes and minor diffirsion in theexperiments at 1100 and 1200'C (see below). The onlysignificant source for Al in the equilibrated clinopyroxenewas the volumetrically abundant garnet in the sample.The source of the small and variable amounts of Na de-tected in the equilibrated clinopyroxenes may have beentrace amounts of pressure-medium glass mixed in with

the starting garnet, or possibly Na that diffirsed throughthe container material into the experimental sample (Pat-tison and Newton, 1989; Aranovich and Pattison, 1994).The subtle but generally consistent Fe-Mg shifts in gar-net-rim compositions, consistent with mass balance con-straints, suggest that clinopyroxene also dissolved andtherefore participated in the multicomponent solution andprecipitation mechanism.

Evidence for a minor amount of pure Fe-Mg diffusionin clinopyroxene was found only in the experiments atthe highest temperature by examining contrasting core-rim profiles of Fe/(Fe + Mg) vs. Al in individual grainscontaining an Al-bearing rim that mantles an unreacted,Al-free core (Fig. 5). Interpretation of these profiles iscomplicated by convolution effects (Ganguly et al., 1988),where the excitation volume from microprobe analysis islarge compared with the scale of compositional profilesor discontinuities. With the assumption of a finely fo-cused beam diameter of I pm and an accelerating voltageof l5 keV, the diameter of the excitation volume for Fe,Mg, and Al for 99olo of the X-ray production in clinopy-roxene is about 3.0, 4.4, and 4.3 pm, respectively (equa-tions on p.200-201of Reed, 1993). Analysis points with-in about 2.2 p.m of the interface between the rims ald theseed material (half the diameter of the excitation volumefor Mg) may show some smoothing that resembles a dif-fusion profile but that instead may be due to an overlapof the microprobe excitation volume with material oneither side ofa sharp discontinuity.

In Figure 5, the solid lines are the inferred locations ofthe interface between the rim and seed material, locatedin the interval of maximum slope of the profiles in Al(Ganguly et al., 1988), the slowest diffirsing of the ele-ments (see below). The dashed lines 2.2 rrm from eitherside of the solid lines are the limits beyond which theeffects ofconvolution across a sharp discontinuity shouldcease to be important. In the experiment at 800 "C shownin Figure 5A, all clinopyroxene compositions showingsignificant Fe-Mg change, compared with the starting Fe-Mg composition, contain significant Al (Fig. 5A). Beyondthe limits of the dashed lines, there is no clear evidenceof an Fe-Mg diffusional gradient between the seed ma-terial and rim, although a very small amount of diffusion(<2 ttm) cannot be ruled out. In contrast, in the experi-ment at 1200 'C shown in Figure 58, there is evidencefor Fe-Mg gradients in Al-free clinopyroxene 3-4 pm in-ward from the Al-bearing clinopyroxene rim, suggestingsome purely diftrsional Fe-Mg exchange between the rimand the seed. This approximate penetration distance is ingood agreement with the calculated diftrsional profilesdiscussed below.

In summary, the experiments of Pattison and Newton(1989) provide neither reversals nor unequivocal brack-ets of the equilibrium clinopyroxene compositions. Thecriterion of reversibility is undermined in that there is noway to vary pressure or temperature in such a way thatthe equilibrated grains of subcalcic, Al-bearing, inter-mediate Fe-Mg clinopyroxene can be returned to their

A. Rlm-rim profiles of Mg/(Mg+Fe) and Al catlons in Cpx

800'C; Diggi Mg/(Mg+Fe)(Grt)=0.29


I

^ o ' 9o

L

B o "=E o t=

o.6

o 0 2co

E o to( s

O 2 a 6 8 b l 2 l , l l 6 B r c

Distance in pm

B. Rim-rlm proflles ot Mg/(Mg+Fe) and Al catlons in Cpx

1200'C; Digs; M9/(M9+FeXGn)=0.39

o 2' 6 " Stoi '"*t l" f i

ts n 22 21 6 n

Fig. 5. Rirn-to-rim step-scan profrles across two clinopyrox-ene grains consisting of a central core of unreacted or little-re-acted starting material (central part of profiles) mantled by rimsof Al-bearing material (marginal parts of profiles) (from Ara-novich and Pattison, 1994). The solid lines represent the inferredlocation ofthe interface between rim and core, whereas the dashedlines 2.2 pm to either side of the solid lines represent the limitsofconvolution effects ifthere v/ere an abrupt compositional dis-continuity at the interface. See the text for discussion of theprofiles. (A) Expt. 76b(l): I 5 kbar, 800 'C, 484 h; starting Cpx-Diro; starting Gfi-XG: 0.2, Mg/(Mg + Fe) : O.43. @) Expt.88q 15 kbar, 1200 qC, 60 h; starting Cpx-Diro; starting Grt-XG: 0.25, Ml(Me + Fe) : 0.33.

starting compositions, especially those that are more Fe-rich than the starting garnet. The criterion ofbracketingis undermined for two reasons: first, although there isgenerally an apparent convergence (more correctly a closecoincidence) in clinopyroxene Fe-Mg compositions inpairs of experiments using Fe-rich and Mg-rich startingmaterials, the sense of change in Ca and Al content isunidirectional (the Ca content is always lower, and theAl content is always higher, than in the starting materials,

so that there is no approach from the Ca-poor, Al-richdirection that would be required to provide a bracket);second, the dominance of a solution and precipitationmechanism over a purely diffusional mechanism resultsin the loss of certainty of the direction to equilibriumfrom the final measured compositions. This last point isemphasized by some experiments that resulted in com-positional overlaps.

CrrurarroN oF DIFFUSToN PENETRATIoNDISTANCES IN EXPERIMENT TIMES

Table I shows calculated diffusion penetration distanc-es for Fe-Mg interdiffusion and some other elements inseveral minerals for a range of temperatures and experi-ment times. The diftrsion penetration distances were cal-culated using the expression X : \An (Crank, 1975),where X is the diftrsion penetration distance, D is thediftrsion coefficient, and I is time. Appendix I shows asample calculation.

Sources for the experimentally determined diffirsiondata are listed at the bottom ofTable l. For garnet, theFe-Mg calculations are based on Mg tracer diftrsion databecause Mg diffusion is the rate-limiting step in the cou-pled exchange (Cygan and Lasaga, 1985). The Fe-Mg cal-culations for slow and fast garnet are based, respectively,on Chakraborty and Ganguly (1991) and Cygan and La-saga (1985), the latter study incorporating the pressurecorrection of Chakraborty and Ganguly (1991). Diffusionpenetration distances for orthopyroxene are based on theFe-Mg interdiftrsion data of Ganguly and Tazzoli(199D1'Fe-Mg interdiffusion in clinopyroxene is expected to besimilar to or a little slower than in orthopyroxene (Ghoseand Ganguly, 1982; Ganguly, 1993 personal communi-cation). Although there are no experimental Fe-Mg dif-fusion data for biotite, it is likely from theoretical con-siderations (Dowty, 1980) that Fe-Mg diftrsion is fasterin biotite than in olivine, the next-fastest mineral, a viewsupported by the observation that biotite tends to be un-zoned even when olivine in the same rock may be zoned(Ganguly, 1993 personal communication). Thus, biotiteFe-Mg interdiffusion is arbitrarily assumed to be one or-der of magnitude faster than in olivine.

The calculated values in Table I represent the approx-imate distances into a mineral grain that the diftrsionfront reaches and should not be confused with the widthof a measurable equilibrated rim. Because of convolutioneffects (Ganguly et al., 1988), measurement of the truerim composition from single-microprobe spot analysesrequires a diffusion penetration distance substantiallylarger than the ca. 4-1rm diameter of the microprobe ex-citation volume. Ganguly et al. (1988) outlined an ap-proach in which step scans across a difiilsion profile canbe deconvolved to provide a reasonable estimate of truecompositions even when convolution affects individualanalyses along the gradient. They noted that, for a stepsize of 0.5 rrm and normal microprobe operating condi-tions [similar to those used in the studies of Pattison andNewton (1989) and Aranovich and Pattison (1994)1, con-

o-a

o3

I

09

o

t o .oE o :oE o s

o a

0 3

2, 02o

3 0 1< o


volution effects cease to become significant once the dif-fusion profile reaches about 15 pm in width.

The calculated values in Table I suggest that below1300 "C for typical experiment times, Fe-Mg diffirsion isunlikely to yield a wide enough equilibrated rim in gar-net, clinopyroxene, and orthopyroxene to be analyzed withsingle spot analyses without incorporating a substantialproportion of partially reacted or unreacted material. Forolivine and biotite, the corresponding temperaturethreshold is about 800 'C. The calculated diftrsion pen-etration distances for clinopyroxene are in good agree-ment with the measured profiles in Figure 5.

The negligible observed and calculated extents of dif-fusion for the garnet + clinopyroxene experiments of Pat-tison and Newton (1989) support the conclusion that theprocess of multicomponent solution and precipitation wasthe dominant equilibration mechanism in their experi-ments. Even in their experiments at 800 "C, rim com-positions were wide enough to obtain numerous repro-ducible spot analyses from the edges ofsingle grains. Closecoincidence of final compositions from pairs of experi-rnents with substantially different starting Fe-Mg com-positions indicates that there was little to no incorpora-tion of unreacted starting material in these analyses,indicating a minimum width of equilibrated material ofabout 4 pm. On the basis of the calculations in Table l,such widths would not be expected by Fe-Mg diffirsionalone.

Pattison and Newton (1989) interpreted the range ofcompositions between unreacted starting clinopyroxeneand equilibrated rims to represent diftrsionally con-trolled convergence toward equilibrium, the basis for theirclaim of reversal (i.e., bracketing) of Fe-Mg (but not otherelements). Althoueh this explanation is consistent withdiffi.rsion, the above considerations suggest that it is morelikely that the range of compositions they obtained rep-resent instead mixed analyses from microprobe excita-tion volumes that incorporated variable proportions ofunreacted starting clinopyroxene and equilibrated rims orzones.

Evlt,ulTroN oF orHER ExpERTMENTAL sruDrosExchange and solid solution experiments

Below, other apparently reversed Fe-Mg exchange andsolid solution experiments involving crystalline startingmaterials are examined to see if equilibration was alsoachieved by a multicomponent solution-precipitationmechanism and whether the results are truly reversed orunambiguously bracketed. The purpose of this assess-ment is not to undermine the value of the experiments(see discussion below) but merely to see if the experi-ments satisfu the strict reversal and bracketing criteriadiscussed above.

Experiments by Eckert and Bohlen (1992) on the Fe-Mg exchange between garnet and orthopyroxene wereconducted using a similar approach to that of Pattisonand Newton (1989), with overwhelming abundance ofgarnet relative to orthopyroxene. In addition to a close

coincidence in Fe-Mg ratio in orthopyroxene from pairsof experiments using Fe-rich and Mg-rich starting com-positions, the initially Al-free orthopyroxene took on Alduring the experiments. Although Eckert (1994 personalcommunication) noted complexities in textural and com-positional variations in their experimental products thatmade identification of an equilibration mechanism onobservational grounds equivocal, the addition of Al tothe initially Al-free orthopyroxene seems to dictate somecomponent of solution and precipitation involving thedissolution ofgarnet.

Ferry and Spear (1978), in their study ofgarnet + bi-otite Fe-Mg exchange, obtained a generally close coinci-dence of final biotite compositions from paired experi-ments using Fe-rich and Mg-rich starting compositionsin HrO-fluxed experiments from 550 to 800 "C, whichthey interpreted as evidence for reversal (i.e., composi-tional bracketing). They did not analyze for any elementsother than Fe and Mg in biotite, so evidence for multi-component solution and precipitation in their experi-ments is difficult to demonstrate. However, the closeagreement between pairs of experiments at temperaturesas low as 550 oC suggests a solution and precipitationmechanism because diffirsion alone at 550 "C for theirexperiment durations should result in minimal change inbiotite and especially garnet (see Table l). In experimentsof similar P- Z conditions and duration to those of Ferryand Spear, Seifert (1970) noted that stoichiometric phlog-opite inserted into HrO-fluxed experiments with alumi-nous phases (AlrSiOs, cordierite) emerged as Al-bearingeastonitic mica after the experiments, indicating the ef-fectiveness of solution and precipitation in mica recrys-tallization at these temperatures.

Phase equilibrium experiments by Koziol and Newton(1989) and Koziol (1990) on solution properties ofcal-cium-iron-magnesium garnet, in which a LiMoOo fluxwas used to increase reaction rates, resulted in closelyagreeing final garnet compositions from experiments withsignificantly different starting compositions, which theytook as evidence for convergence to equilibrium. Theyreported equilibrated rims up to l0 pm in width in ex-periments at 900-1000 "C, for 2-10 d, from which re-peated spot analyses gave reproducible results. Exami-nation of Table I suggests that, even for fast garnetdiffusion, such reproducibility and agreement betweenexperiments using different starting compositions wouldnot be expected from a diftrsionally controlled equilibra-tion mechanism alone but more likely are due to solutionand precipitation.

Similar arguments apply to the results of Perkins andVielzeuf (1992) on Fe-Mg exchange between olivine andclinopyroxene. Examination of their Table 4 shows thatinitially Ca-free olivine took on Ca during the experi-ments, in addition to changing in Fe/Mg, whereas ini-tially Ca-saturated clinopyroxene became Ca-undersatu-rated, in addition to changing in Fe/Mg, suggestive of amulticomponent solution and precipitation equilibrationprocess. According to Table l, their experiments of 3-4d at 1000'C would not appear to be long enough to


develop measurable equilibrated rims in clinopyroxene ifthe equilibration mechanism was Fe-Mg diffusion alone.

Several studies on AlrO, solubility in orthopyroxene inequilibrium with garnet (Perkins and Newton, 1980; Laneand Ganguly, 1980; Perkins et al., 1981) obtained goodagreement from HrO-fluxed experiments using Al-rich andAl-poor starting compositions over the temperature range900-1600 oC and using experimental times ranging from6 d at 900 "C to 3 h at 1600'C. The negligible calculatedAl-diffusion penetration distances for clinopyroxene (as-sumed to be similar to orthopyroxene) listed in Table Isuggest that the equilibration mechanism in these studiesinvolved solution precipitation, a conclusion supportedby the occurrence of compositional overlaps in severalpairs of experiments (see the discussion above).

In experiments involving phases that show solid solu-tion in only one site (e.g., Na-K exchange between alkalifeldspars), the effects of solution precipitation vs. diff.r-sion may be more difficult to distinguish. Goldsmith andNewton (1974), in HrO-fluxed experiments on the alkalifeldspar solvus, obtained close coincidence among exper-iments whose initial compositions lay inside and outsidethe solvus over the temperature range 300-600 "C. Onthe basis of changes in X-ray diffraction patterns, theynoted increasing degrees ofconvergence in progressivelylonger experiments at the same P-Z conditions, whichthey ascribed to continuous compositional change towardthe equilibrium composition, and hence evidence for re-versal (compositional bracketing). Given the very lowtemperatures of some of their apparent convergences,however, diffirsion rates (Yund, 1983) would most likelybe too slow to effect these changes. An alternate interpre-tation is that the final equilibrated compositions grew atthe expense of the starting compositions that were out ofequilibrium by a solution and precipitation mechanism,with the increasing degrees ofapparent convergence withtime representing increasing degrees of reaction progress.

Stable isotopic exchange experiments

The issue of equilibrium bracketing in stable isotopicexchange experiments has been discussed by Chacko(1993) and O'Neil (1986). Chacko noted that, for the samereasons discussed in this paper, strict isotopic reversalsusing a convergence approach (similar to that describedabove for cation exchange experiments) are unlikely be-cause ofthe negligible expected diffusion penetration dis-tances for the experiment times involved. Both authorsnoted that experiments involving recrystallization ofcrystalline starting materials in fluxed experiments arepreferable to synthesis experiments from gels or oxidemixes, although neither method provides an unequivocalbracket of the equilibrium isotopic fractionation (bothprocesses involve either solution or precipitation or both).

Iprpr,rclrroNs FoR soLID soLUTroN EXPERTMENTS

The above survey represents a sampling of solid solu-tion experiments using crystalline starting materials. Re-versals (meaning apparent compositional convergences)

were inferred from the close agreement among expen-ments using different starting materials. Because theequilibration mechanism in most, if not all, cases prob-

ably involved a multicomponent solution and precipita-tion mechanism, sometimes with no possibility of re-turning the final compositions to the starting compositionsand typically involving variation in several composition-al parameters, the experiments, strictly speaking, cannotbe regarded as truly reversed or bracketed.

These conclusions, although seemingly negative, are notintended to undermine the value of solid solution exper-iments that use crystalline starting materials or as justi-

fication for crystallization and synthesis experiments thatuse disordered, high free-energy starting materials suchas glass, gels, or oxide mixes. First, in experiments usingcrystalline starting materials, the inserted phases are ofthe same structure and generally of similar compositionto the final products. This results in a smaller free-energydifference between the starting material and the finalproducts than between glass, gels, and oxides and finalproducts, which in turn reduces the possibility ofjumpingto grossly metastable final compositions compared withsynthesis experiments (see the discussion in Chacko, 1993,p.362).

Second, there are many reasons to suggest that exper-iments using crystalline starting materials, although pos-

sibly unreversed and unbracketed, result in a close ap-proach to equilibrium and provide high-quality data.Excellent agreement and generally minimal overlap be-tween pairs of experiments that started from opposingcompositional directions suggest that the equilibriumcomposition lies within the range bounded by the mea-sured final compositions and their analytical uncertain-ties. Smooth, consistent compositional variations in pres-sure, temperature, and other compositional parametersare suggestive of a close approach to equilibrium. Highlyreproducible results from the same starting materials fordifferent experiment times and using different experimen-tal techniques (e.g., Pattison and Newton, 1989; Perkinsand Vielzeuf, 1992) further suggest a close approach toequilibrium. Finally, consistent thermodynamic data havebeen extracted from redundant sets ofexperiments: Ber-man et al. (1994) extracted thermodynamic data for gar-net, clinopyroxene, and olivine that sadsry individualstudies on garnet * clinopyroxene, olivine * clinopyrox-ene, and garnet + olivine.

Nevertheless, use of the terms reversal and composi-tional bracket to describe apparent convergence experi-ments that equilibrated by a solution and precipitationmechanism, especially those involving variation in morethan one compositional parameter, should be abandonedin favor of the terms consistency experiment or redun-dancy experiment, which emphasize the increased degreeof confidence in an experimental result when it is achievedusing different starting materials or starting composi-tions. In diagrams showing compositional changes in ex-periments, the use of solid arrows connecting startingcompositions to final compositions like B and C in Figure2, which give the impression of continuous change, should


be replaced with nondirectional symbols connecting ini-tial and final compositions like D in Figure 2.

Iupr,rcltroNs FoR THERMoDyNAMTc DATAEXTRACTION

The above considerations carry implications for tech-niques for extracting thermodynamic data and mixingproperties from solid solution experiments involvingcrystalline starting materials. The free-energy inequalitytechniques of Gordon (1973) and Berman et al. (1986)rely on free-energy brackets to constrain possible solu-tions. This approach, initially developed to treat reversalbrackets of univariant equilibria involving phases of fi xedcomposition, can be extended to solid solution experi-ments, provided that unequivocal compositional brack-ets are obtained and are assumed to be equivalent toreversal brackets.

However, because most solid solution data are proba-bly not truly reversed or bracketed, many apparent half-brackets, strictly speaking, can no longer be treated aslimiting. In pairs of experiments starting from opposingbompositional directions, the equilibrium compositionsmust have lower free energy than the starting composi-tions and as such must lie between compositional brack-ets defined by the starting compositions. In contrast, thefinal compositions themselves cannot be interpreted aslimiting compositional brackets, even though they mayindeed represent a close approach to equilibrium (see thediscussion above and Fig. aB).

One solution to this problem is to assume that, in pairsof experiments starting from opposing compositional di-rections, the equil ibrium composition probably l iessomewhere in the range defined by the frnal measuredcompositions plus their analytical uncertainties, allowingthe assigrrment of effective (but not strictly limiting)brackets (Berman et al., 1994). Nevertheless, without un-ambiguous knowledge of the direction to equilibrium fromthe final compositions, least-squares regression of (as-sumed) equilibrium compositions (e.g., Powell and Hol-land, 1985) is an equally justifiable means of treating ex-perimental solid solution data. Both techniques requirequalitative judgment in the rejection of outliers. Perkinsand Yielzeuf (1992), in applying both techniques to theirexperiments on Fe-Mg exchange between clinopyroxeneand olivine, achieved essentially identical results, sug-gesting that both approaches are satisfactory in extractingthermodynamic data from solid solution experiments.

AcxNowr,nncMENTs

This research was supported by NSERC operating grant 037233. I thankT. Chacko for providing the impetus to calculate difftrsion penetrationdistances for experiment times and T.M. Gordon for clarifring my think-ing on reversibility vs. compositional bracketing. The paper was irn-provod by the thoughtful reviews of R.G. Berman, E.D. Ghent, A.M.Koziol, and especially J.O. Eckert and D. Perkins III. In addition to thosenamed above, I acknowledge stimulating discussions with L.Ya. Arano-vich, J. Farquhar, J. Ganguly, S.L. Harley, R. Joesten, and R.C. Newtonon solid solution experiments. All errors or misconceptions are my own.

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PATTISON: ARE REVERSED EXPERIMENTS REALLY REVERSED?

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MaNuscnrsr RlcErvED Sesrnmen 21, 1993MeNuscnnr AccEPTED Mev 5, 1994

Apprxorx l. Srvrpr,n cALcuLATroN oF ADIFFUSION PENETRATION DISTANCE

The difrrsion penetration distance for Fe-Mg interdifrrsion ingarnet at 1000'C for a 7-d experiment is calculated below usingthe experimental data for Mg tracer diftrsion of Chakrabortyand Ganguly (1991). Their expression for the calculation ofthediftrsion coemcient is as follows:

D*(P): Do'expl l-Q + (P - 1) 'Al l+l/Rn

where D*(P) is the tracer diffusion coefiicient at pressure P (bars)and temperature f (K), D" is the preexponential factor, Q is theactivation energy at I bar, and AZ* is the activation volume.From Table I of Chakraborty and Ganguly, the values for theabove parameters for Mg2+ are Do: 2.79 x l0 a cm2/s; Q :

65457 cal/mol; AV+ : 0.078 caV(bar'mol); R : 1.98726 caU(mol.K); and T : IOOO + 273 : 1273 K. This results in a valueofD*(P) at 1000'C and 15 kbar (15000 bars) of2.57 x l0-r5cm2ls, or a value of log,oD*(P) of -14.6 cm2/s.

The expression for diffirsion penetration distance (Crank, 1975)ts

x: vqnwhere X is the penetration distance, D is the diftrsion coefrcientcalculated above, and / is time. For a 7-d experiment, t : 7 x24 x 6O x 60 : 6.05 x 105 s. This results in a value for X of7.8 x l0-t cm, or 0.78 rrm.

Are reversed Fe-Mg exchange and solid solution experiments

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