American Mineralogist, Volume 74, pages 100U1015, 1989 Thermodynamic properties in a multicomponent solid solution involving cation disorder: FerOo-MgFerOo-FeAltOo-MgAltOo spinels Jonmr Nnr,r-* Brnulnp J. Wooo* Department of Geological Sciences, Northwestern University, Evanston, Illinois 60208, U.S.A. AesrRAcr The spinel quaternary FerOo-FeAlrOo-MgFerOo-MgAlrOo is petrologically important but difficult to deal with thermodynamically because of complex order-disorder relations. We have used our recent measurements of Fe2+and Fe3* site occupancies together with measured activity-composition relations, interphasecation distributions, and solvi to de- velop an internally consistentthermodynamic model for this system.The model is based on a second-degree Taylor series expansionofthe vibrational part ofthe Gibbs free energy in terms of order and compositional parameters.It can readily be related to the familiar Margules parameters, Wg, and reciprocal interactions commonly used to representactiv- ities in multisite solid solutions. With appropriate simplifications, the model reduces to the Navrotsky-Kleppa (-RT ln Ko is constant)or O'Neill-Navrotsky (-RZln Ko is a function of order parameters) models of octahedral-tetrahedral disorder. Although neither of these simpler models provides a complete description of cation distributions in the quaternary,the O'Neill-Navrotsky for- malism works well over wide ranges of composition. INtnonucrroN Spinels are important petrogenetic indicators in ig- neousand metamorphic rocks of both crustal and mantle origin. The wide compositional ranges of multivalent cat- ions in spinelsmay, in principle, be used to constrain the valuesofthe intensivevariablesthat operated during their formation. Spinel chemistry is particularly sensitive to oxygen fugacity and temperature (e.g., Buddington and Lindsley, 1964;Irvine, 1965; Mattioli and Wood, 1988), and the FerOo contents of both aluminous spinels and titanomagnetites are commonly used for oxygenbarome- try in natural systems.In general, however, application of experimental calibrations to more complex natural systemsrequires either projection of the mineral com- positions into simple systemsor application of an activ- ity-composition model to the multicomponent phases. Because ofits generality, the latter approachis preferable, but is difficult to implement because of the lack of infor- mation on both macroscopic thermodynamic and micro- scopic order-disorder properties of the solid solutions. The purpose of this paper is to addressthe microscopic and macroscopicproperties of spinels in the quaternary FerOo-MgFerOo-FeAlrOo- MgAlrOo. Experimental calibrations of spinel equilibria mainly involve measurement of cation partitioning betweenthis phaseand others such as rhombohedral oxide (Budding- ton and Lindsley, 1964),silicate(Jamieson and Roeder, 1984), and fluid (Lehmann and Roux, 1986). In addition, * Present address:Department of Geological Sciences, Uni- versity of Bristol, Bristol BS8 1zu, England. 0003-o04x/89/09 l 0-l 000$02.00 there are direct activity-composition measurements for binaries such as FerOo-MgAlrOo(Mattioli and Wood, 1988), FerOo-FeAlrOo (Petric et al., l98l), and FerOo- MgFerOo (Shishkov et al., 1980).The activity data may be extended in temperature and composition spaceand integrated with the phaserelations and cation-partition- ing data in order to derive a reasonable entropy model for spinel solid solutions. The task is complicated, how- ever, by the varying degrees of disorder of the major cat- ions (Fe2*, Fe3+, Mg2+, Al3+)between octahedral and tet- rahedral spinel sites and the fact that thesedistributions are extremely difficult to quench from high temperature (Wood et al., 1986).Recentlya thermopowerand con- ductivity technique (Wu and Mason, l98l; Mason, 1987) has been developedfor the in situ determination of Fe2*- Fe3*disorder in spinels at high temperature.Use of this technique to characlerize the temperature and composi- tion dependence of octahedral-tetrahedral partitioning provides most of the information necessary to link to- getheractivity-composition and cation-distribution data. In a previous paper (Nell et al., 1989) we employed the thermopower-conductivity technique to measurein- tersite cation distributions in FerOo-MgFerOo, FerOo- FeAlrOo, and FerOo-MBAlrOo solid solutions and usedthe data to evaluatethe applicability of commonly used cat- ion-distribution models to spinels. We found that the join FerOo-MgFerOo fits reasonably well to either the Navrot- sky-Kleppa (1967) model (both -Rf ln K["2--r"r' und -RZ ln Kge-r"r* constant) or to the O'Neill-Navrotsky (1983, 1984)model in which the intersite cation distri- butions (-R?"ln Ko) are linear functionsof the degree of inversion of the solid solution. In Fe.Oo-FeAlrOo solid I 000
Transcript
American Mineralogist, Volume 74, pages 100U1015, 1989
Thermodynamic properties in a multicomponent solid solution involving cation disorder:FerOo-MgFerOo-FeAltOo-MgAltOo spinels
Jonmr Nnr,r-* Brnulnp J. Wooo*Department of Geological Sciences, Northwestern University, Evanston, Illinois 60208, U.S.A.
AesrRAcr
The spinel quaternary FerOo-FeAlrOo-MgFerOo-MgAlrOo is petrologically importantbut difficult to deal with thermodynamically because of complex order-disorder relations.We have used our recent measurements of Fe2+ and Fe3* site occupancies together withmeasured activity-composition relations, interphase cation distributions, and solvi to de-velop an internally consistent thermodynamic model for this system. The model is basedon a second-degree Taylor series expansion ofthe vibrational part ofthe Gibbs free energyin terms of order and compositional parameters. It can readily be related to the familiarMargules parameters, Wg, and reciprocal interactions commonly used to represent activ-ities in multisite solid solutions.
With appropriate simplifications, the model reduces to the Navrotsky-Kleppa (-RT lnKo is constant) or O'Neill-Navrotsky (-RZln Ko is a function of order parameters) modelsof octahedral-tetrahedral disorder. Although neither of these simpler models provides acomplete description of cation distributions in the quaternary, the O'Neill-Navrotsky for-malism works well over wide ranges of composition.
INtnonucrroN
Spinels are important petrogenetic indicators in ig-neous and metamorphic rocks of both crustal and mantleorigin. The wide compositional ranges of multivalent cat-ions in spinels may, in principle, be used to constrain thevalues ofthe intensive variables that operated during theirformation. Spinel chemistry is particularly sensitive tooxygen fugacity and temperature (e.g., Buddington andLindsley, 1964;Irvine, 1965; Mattioli and Wood, 1988),and the FerOo contents of both aluminous spinels andtitanomagnetites are commonly used for oxygen barome-try in natural systems. In general, however, applicationof experimental calibrations to more complex naturalsystems requires either projection of the mineral com-positions into simple systems or application of an activ-ity-composition model to the multicomponent phases.Because ofits generality, the latter approach is preferable,but is difficult to implement because of the lack of infor-mation on both macroscopic thermodynamic and micro-scopic order-disorder properties of the solid solutions.The purpose of this paper is to address the microscopicand macroscopic properties of spinels in the quaternaryFerOo-MgFerOo-FeAlrOo- MgAlrOo.
Experimental calibrations of spinel equilibria mainlyinvolve measurement of cation partitioning between thisphase and others such as rhombohedral oxide (Budding-ton and Lindsley, 1964), silicate (Jamieson and Roeder,1984), and fluid (Lehmann and Roux, 1986). In addition,
* Present address: Department of Geological Sciences, Uni-versity of Bristol, Bristol BS8 1zu, England.
0003-o04x/89/09 l 0-l 000$02.00
there are direct activity-composition measurements forbinaries such as FerOo-MgAlrOo (Mattioli and Wood,1988), FerOo-FeAlrOo (Petric et al., l98l), and FerOo-MgFerOo (Shishkov et al., 1980). The activity data maybe extended in temperature and composition space andintegrated with the phase relations and cation-partition-ing data in order to derive a reasonable entropy modelfor spinel solid solutions. The task is complicated, how-ever, by the varying degrees of disorder of the major cat-ions (Fe2*, Fe3+, Mg2+, Al3+) between octahedral and tet-rahedral spinel sites and the fact that these distributionsare extremely difficult to quench from high temperature(Wood et al., 1986). Recently a thermopower and con-ductivity technique (Wu and Mason, l98l; Mason, 1987)has been developed for the in situ determination of Fe2*-Fe3* disorder in spinels at high temperature. Use of thistechnique to characlerize the temperature and composi-tion dependence of octahedral-tetrahedral partitioningprovides most of the information necessary to link to-gether activity-composition and cation-distribution data.In a previous paper (Nell et al., 1989) we employedthe thermopower-conductivity technique to measure in-tersite cation distributions in FerOo-MgFerOo, FerOo-FeAlrOo, and FerOo-MBAlrOo solid solutions and used thedata to evaluate the applicability of commonly used cat-ion-distribution models to spinels. We found that the joinFerOo-MgFerOo fits reasonably well to either the Navrot-sky-Kleppa (1967) model (both -Rf ln K["2--r"r' und-RZ ln Kge-r"r* constant) or to the O'Neill-Navrotsky(1983, 1984) model in which the intersite cation distri-butions (-R?"ln Ko) are linear functions of the degree ofinversion of the solid solution. In Fe.Oo-FeAlrOo solid
I 000
NELL AND WOOD: THERMODYNAMICS OF SPINEL SOLID SOLUTIONS 1001
TABLE 1. "Fictive" end-member components of normal and inverse spinels in Fe.Oo-MgFerOo-FeAlrOo-MgAlrOo solid solutionsexpressed in terms of compositional and order parameters
Mg(Al),O4 Al(MgAl)Oo Fe2*(Fe3*),O4 Fe3*(Fe2*Fe3*)O4 Fe,*(Al),Ol Al(Fe,*Al)O4 Mg(Fe3*LO4 FeF.(MgFe3*)O4P a r a m e t e r 1 2 3 4 5 6 7 8
Note. Species in parentheses reside on octahedral sites. The number below each component is the number by which that component is identified ineouations.
'I'I
000
11'I
00
10000
10010
01001
01100
r1
s2s3
00010
00001
solutions, however, we found that the compositional de-pendence of -RZ ln K5.'-ru is more complex than thatgiven by either of these simple models. Representationof the properties of spinels in the quaternary Fe3O4-MgFerOo-FeAlrO4-MgAlrO4 requires consideration of thiscomplexity and generation of an internally consistentthermodynamic model that describes and predicts bothmicroscopic and macroscopic phenomena.
This paper presents a model that uses a Taylor seriesexpansion ofthe vibrational part ofthe Gibbs free energyof the solid solution in terms of both order and compo-sitional parameters. This approach was originally intro-duced by Thompson (1969) and has subsequently beenextensively applied to the description of the thermody-namic properties of solid solutions (e.g., Sack, 1982; Hiltand Sack, 1987; Andersen and Lindsley, 1988).
Fonlrur,arroN
The composition of any spinel in the system Fe.Oo-MgFerOo-FeAlrOo-MgAlrOo can be described in terms ofthe fictive normal and inverse components given in Table1. These end-members will serve as reference compo-nents that will be used to derive the mixing properties ofFerOo-MgFerOo-FeAlrOo-MgAlrOo solid solutions. Thereare two independent compositional exchange vectors(Fez+ffig-r and Fe3*Al-t) in the system of interest, and twocompositional parameters are thus required to expressthe bulk chemical composition of any given solid solu-tion. These are labeled r, arrd r, and are defined as fol-lows:
r r : l - ( X # s + 2 X W
r.: t/z(X,;i + 2XK), (l)
where Xffi;, etc., refers to the atomic fraction of Mg onthe tetrahedral site. In addition there are four intersitecation-exchange reactions between the reference end-member components, namely:
Fef,i + Alo", : Felj * A1,",
F e l i + F e i J : F e l l + F e f l " f
M9,", * A1"".: Mg*, * A1,",
andM&", + Fe3.1 : Mg.., * Fe,3j. (2)
Only three of these reactions are independent, and we usethe following order parameters (s,, sr, and s3) to charac-
terize the cation distributions in the solid solution:
' s r : X i i ' - V z X f ; i
sr :X95. - t /zXf l t -
sr :2XP;. (3)
There are therefore a total offive independent parametersthat are needed to completely characterize any solid so-lution in the FerOo-MgFerOo-FeAlrO4-MgAlrO4 system.Values of these five parameters in the pure fictive end-members are given in Table l, and species concentrationsper structural formula unit are given in Table 2. The con-figurational entropy of a solid solution (&."J may then beexpressed as a function of order and compositional pa-rameters through the equation
,S: -R ) ) n.X,,,ln(X,."), (4)
where n, is the numbe. or rn", (a) per formula unit and-(, is the mole fraction of component I on site a.
The Gibbs free energy of a solid solution (G) is ob-tained by combining Equation 4 with an expression forthe vibrational part of the Gibbs free energy (G*) throughthe relation
G: G* - IS"onr. (5)
We formulate the vibrational part of the Gibbs free en-ergy using a Taylor series expansion in terms of the se-lected composition and order parameters. The first-de-gree terms in such an expansion describe equilibriumconditions in an ideal solid solution, and nonideality mustbe taken care of by higher-order terms (Thompson, 1969).
Second-ilegree Taylor series expansion of G*
We obtained the vibrational part of the Gibbs free en-ergy from a second-degree Taylor series expansion:
where the & parameters are the series expansion coeffi-cients. The second-degree expansion allows for two types
1002
Tesle 2. Species concentrations per structural formula unit ex-pressed in terms of composition parameters (r, and rr)and order parameters (s,, s2, and s3).
Tetrahedral site Octahedral site
NELL AND WOOD: THERMODYNAMICS OF SPINEL SOLID SOLUTIONS
MgAIFe2*Fe3*Sum
531 2 + s rl - D 1 - D 2 - D 3
1 + E - 1 2
of term to describe the excess free energy of mixing. Theseare (1) symmetric binary interaction parameters and (2)reciprocal reaction terms. Reciprocal reactions are pres-ent in all but 4 (binaries l-5, l-7,3-5, and 3-7) of thepossible 28 binary solid solutions between the eight ref-erence end-member components. Denoting octahedralsites with parentheses, we write the following general re-ciprocal reactions between two fictive end-member com-ponents B3+(A2+B3+)O o (a) and D3*(C'z*D3*)Oo (D):
The standard state (pure phases at P and Z of interest)free energy ofthe reciprocal reaction (AG!r) is defined as
AG}o: Lro, -f tro- - t4 - r4, (8)
where rr2 and y.o- are used to denote standard-state chem-ical potentials oflinearly dependent or independent "fic-tive" end-member comoonents derived from reference
components a and b through any one of the reciprocalrelations in Reactions 7. In all, an additional 26 linearlydependent fictive end-member components are impliedby relations such as those given in Reactions 7;20 ofthese components are charge unbalanced. The use ofcharge-unbalanced end-member components is valid giv-en the additional constraint of equal amounts of oppo-sitely charged species in order to satisfy electroneutrality(Wood and Nicholls, 1978).
Symmetric binary interaction parameters between thereference end-member components may be interpretedin terms of excess on-site microscopic interaction param-eters. For a solid solution containing mole fractions xand (1 - x) of components B3+(A2+B3+)O. @) andD3+(C2+D3+)O, (b), respectively, we express the regularsymmetric interaction parameter (Wf) as follows:
/ . \ / \w 6 b : ( x ) ( 1 - x ) w E L . , ( - ) ( ; )
/ \ ' /
1 - 1 1 - s 3
h - S tr 1 + s 1 + s r + s 3 - 1' l - s ,
h1
1 - f ,
2r"11
2 - 2 r 2e
. lw#+ w#+ wff i+ wtr t l
+ z0'wr; * ,(t--t)'r*- 1 _ v- ) rn- i rn
:(x)(r - a1w.ry.ry.ry.ry-ry-ry)' (e)
where Zfi, etc., refers to the excess free energy of mixingof 1 mol of B3* and I mol of D3* on tetrahedral sites and
TABLE 3, Thermodynamic coefficients in a second-degree Taylor series expansion for the vibrational part of the Gibbs free energyof Fe.Oo-MgFerO4-FeAl2O4-MgAlrO4 solid solutions
9",",: -wzu - aGlLgqe: -Wt - LG!,grr: wLu + w8 - w3 - wE - w'3 - wE + LG16+ AGgs - AG?z- AGlu + AG38 - AG3.gn,": Wt + Wg - W6l Wt5,.",: WE - WE - ilr'26 + AG?6 - AG3"9,,",: wE - w3 - wt + LGl, - AG8ogn^: wT + wuE - 2wE . w"3 - wE + w'n' - w13 + LG84 + aGSs + aG?6 - AGS8 - AG9' - AGgsgrr: w"3 + w"ot - wE + LGBB + AG86 - AGgsgr*: wt - w"3 - t/Y34 + aG86 - AGSr - AGS.9 r : w 3 + w " 3 - w 3 - w 39",.: Wg + Wt - W3 - Wt + AG38 - aG!, - aGSug",*:w'3 + w3 - wE - w'3 - w"3 w"3 + aG?6 + aG38 - AG9" + AGS8 - AG8. - AGS69 , :WE + WE - WE - WE - WE - WE + WE - Wt + aG?6 + aG3 , - AG9 , - ac l s+AG8 ' -AG8E +ac9s -AG81
NELL AND WOOD: THERMODYNAMICS OF SPINEL SOLID SOLUTIONS 1003
the factor 2 takes account of the two octahedral sites performula unit.
The coefficients in the second-degree Taylor series ex-pansion were determined by sequentially setting the com-position of the solid solution to each of the 28 binaryjoins between the eight reference end-member compo-nents defined in Table l. This gives, in each case, a valuefor G* in terms of the coefficients in the power series. Theexpressions may then be equated with an equivalent re-lationship for G* in terms of AG! o and W"! that may thenbe used to solve for the power-series coefficients. For ex-ample, the free energy along the join 4-6IFe3+(Fe2+Fs:t)Oo-Al(Fe,*Al)O"] is given by
G* : Bo * &, * g,2r2 + g,,,, I g,r,rr2, t g,vzrz: rzQt\ - u.2 + W2: + LG36)
The excess free energy contributions to the resultant so-lution set (Table 3) are not uniquely defined, however,since the system ofequations used to solve for the valuesof the coefficients is overdetermined. Reciprocal excessfree-energy terms are related to one another within theframework of possible reciprocal reactions that exist forthe binary solid solutions. In solid solutions with multiplereciprocal relations (for example the join 4-6 as shownabove), AGlu must be independent of the way in whichthe possible reciprocal reactions between components 4a:nd b are formulated, thus implying that pf; I po- : t"?,* pfl, where n, m, n', and m' are end-members formedthrough reciprocal relations between components a andb. The symmetric nature of the expansion furthermorerequires that AG96 : AGlr: -AG2u: -AG% and thatWffiL^,: WffiLr.,. : W{*-o,: Wy}-r"t*.
Given these relationships between the excess free-en-ergy terms, the solution set of the thermodynamic coef-ficients in the second-degree Taylor series expansion (Ta-ble 3) is consistent with the free-energy expressions ofall28 binary solid solutions between the end-member com-ponents defined in Table l.
Equilibrium conditions
Equilibrium in FerO.-MgFerOr-FeAlrOo-MgAlrOo sol-id solutions is established when at fixed temperature,pressure, and composition the following conditions aresatisfied:
Substituting Equations 4 and 6 into 5 and diferentiatinggives the following equilibrium conditions:
l dG \t;l
:&, * zg"r"r'tr + &,",r1 * gn",rz I g",,zsz\ " o l l i l P , r r . n . s z . s t
/tr"11XA1.",)+ s....s. + Rf ln =:--orrrr-r
(Fej.i)(A1,.,)
: 0 ( r2 )
/ dG \t;/_
: 8,rl 28"2"2s2 + 8a,2r1 I 8,r"rrz + 8"r"2st\ - ' 2 / t - r , r l r 2 . s l . s l
* 8.,us, + Rrln S+?f:P' (tse6.1)(rei.l.,
: 0 ( 1 3 )
and
iac\t*/_
:&, * 28'"3,3s3 + 8,sr1 * 8o*/z * 9",*J'\ " ' 5 / t . r , r l . t 2 . s l . s 2
-f &:,:s. -r- Rrln g&4{:tO"'(Mg,.,)(Felg)
(14)
Application of equilibrium conditions
Equations 12 to 14 are applicable to the calculation ofintersite cation distributions in Fe.Oo-MgFerOo-FeAlrO.-MgAlrO4 solid solutions. First we applied the equilibriumconditions to pure magnetite, hercynite, magnesioferrite,and spinel. From Equation 12 it follows that for pureFeAlrOo,
(Fe3"1)(A1,.,)-RIln ff i:
-(&, t 8,,,, t 8,,",)
- ) o ( .- o r l r l " l
: 0 ' 2 - t 4 - W ; - A G % )
+ 2(W5E + AG86)s,:0.r2- t4- W\le"z.
- Wfff,,r,- AG%)
+ 2(W,;iF.,- + Wfrh"2.
+ AG!u)s,. (15)
1004
Similarly, Equation l3 gives the equilibrium conditionfor pure FerOo:
The equilibrium conditions for spinel and magnesiofer-rite were obtained from linear combinations of Equations14 and 12 and Equations 14 and 13, respectively. ForMgAlrO4 and MgFerOo, respectively, we find that
A comparison of Equations l5 to l8 with the O'Neill-Navrotsky (1983, 1984) formalism (-Rr h[(,a3"1xB:"ly(,4aiXB:J)l : -RT ln K" : a + 208tri) shows that forpure end-member spinels the O'Neill-Navrotsky modelis equivalent to a second-degree Taylor series expansionof the vibrational free energy [note that for the end-mem-ber components, s, and .r2 are equal to (1 - ,Bli)]. Thethermodynamic interpretations of the a and B energy pa-
NELL AND WOOD: THERMODYNAMICS OF SPINEL SOLID SOLUTIONS
rameters used in the O'Neill-Navrotsky model are thusas follows:
As a first approximation, O'Neill and Navrotsky (1983,1984) suggested that the B parameter could be treatedas constant for 2-3 spinels with a value of about -20
kJ/mol. Neglecting the reciprocal terms in Equations 19,a constant B parameter would imply that Wf;le"z- *Wnl"u : Wf!z*s"t* I Wp!*r"t. : Wfi".c, t WffiL", :
llff"r"t- + W?1LF&-: 20 kJ/mol. In binary and more com-plex solid solutions, the O'Neill-Navrotsky formalism isnot, as will be shown below, equivalent to the second-degree expansion unless simplifications are made.
The equilibrium conditions for end-member spinels(Eqs. 15 to 18) are reduced to the Navrotsky-Kleppa( I 967) formalism ( -R 7" ln Ko : constant) when all excessfree energy of mixing parameters are set to zero. Second-degree expansion terms are thus eliminated, and first-order expansion coefficients are simplified to includechemical potential terms only. Intersite cation distribu-tion coefficients for hercynite, magnetite, spinel, andmagnesioferrite, respectively, are then given by the fol-lowing equations in terms of simplified first-order expan-sion coefficients (si, s! and s!):
The constants in the Navrotsky-Kleppa formalism aretherefore equal to the chemical-potential differences be-tween inverse and normal spinel end-members.
The equilibrium conditions (Eqs. 12 to 14) may nowbe applied to calculate cation distributions in binary spi-nel solid solutions. For examples we use the FerOo-MgFerOo and FerOo-FeAlrOo systems as these have beenpreviously dealt with using the O'Neill-Navrotsky for-malism (Nell et a1., 1989). For FerOo-MgFerOo solid so-lutions, it follows from Equations 13 and l4 through the
Equilibria rn quaternary FerOo-MgFerOo-FeAlrOo-MgAlrO4 solid solutions are obtained from Equations 12,13, and 14 as follows:
-Rr rn H*]"+ : &, r 2s,,,s,(f e6JXAl,.,)
-F &,,,(X.*oo I Xr,nroo)
* 8,r,r(X*"orroo * Xr"o,roo)
* &,"rs, + &,,rJ, (25)
/Fe2+\/FAi + \-RZ ln j f f i :8 ,2* 25" , , , s ,(re6clXtseiel)
t 8,,,r(Xr"roo I xr"^roo)
* 8,r,t(X*norroo I Xr.orroo)
* &,"rst + &2"rJ. (26)
and
-Rrlnffi:&, f 28",",s,* &,*(X.*oo I Xr"^,roo)
I 8,r,r(X*"orroo * Xr.orroo)
+ &,",sr * &r,,Jr. (27)
Equations 25 to 27 may also be used to calculate intersitecation distributions in FerOo-MgAlrOo and FeAlrOo-MgFerO, solid solutions, these binary systems being de-generate srmplifications of quaternary solid solutions inwhich r , : | - rz : Xr , . ,q .
The Navrotsky-Kleppa (19 67) and O'Neill-Navrotsky
1005
Activity-composition relations
There are three second-order composition coefficientsavailable to describe activity-composition relations onthe six binary joins in the quaternary FerOo-FeAlrOo-MgFerOo-MgAlrOo. Excess free-energy contributions forFe2*-Mg mixing in both FerOo-MgFerOo and MgAlrO4-FeAlrOo solid solutions are required to be identical invalue and are modeled with the &r,r coemcient. A similarsituation applies to Fe3*-Al mixing in FerOo-FeAlrOo andMgFerOo-MgAlrOo solid solutions where the 9,r,, coeffi-cient is used. Mixing in Fe.Oo-MgAlrOo and MgFerOo-FeAlrOo solid solutions is a function of the g,,,,, g,2,2, vr,d&,,, coefficients where the latter coefficient serves as athird adjustable parameter.
Substituting Equations 4 and 6 into Equation 5 anddifferentiating with respect to r, and r, gives
NELL AND WOOD: THERMODYNAMICS OF SPINEL SOLID SOLUTIONS
substitution of appropriate values for the order and com- formalisms for the functional form of the intersite distri-position parameters that bution coefficients in binary and quaternary spinel solid
-RZ ln flei'l]!5"1) - ^ , - - olutions are readilv derived from Equatior$ 2l to 27 bv
1r";;1r.gj : B, I 29,,,,s, introducing appropriate simplifications. Navrotsky-
Kleppa expressions for the intersite distribution coeffi-* g,1,2Xrqoa + &r"rJ3 (21) cients are obtained when all excess energy of mixing terms
and are zero. Intersite distribution coemcients in solid solu-tions are thus identical to those in pure end-members as
(Mg.",XFeil) given by Equations 20.-fi7 ln 114;,ffi
: &, * 2g,"rJ, + g,14xr4oa o'Neill-Navrotsky expressions for the intersite cation-distribution coemcients in complex solid solutions are
* &."s, . (22) obtained when the combined order and composition ex-Similarly, for FerOo-FeAlrOo solid solutions, we find from pansion coefficients (8,r,, 8,,"r, 9,1q, 8,2"1, 8,2"2,a.Dd 8rr) areEquations 12 and 13 that set to zero. The distribution coefficients are simplified to
be functions of order parameters only and this formalism-RZtn (FeiiXAl"",)
: o -r o -L )o " is therefore equivalent to a second-degree Taylor series--- " '(Fei.;)(A1,",) orr ' 6/rrr ' -6rrrrJr expansion in which combined order and composition
+ o y_ - + o c /)t\ coefficients are neglected. The expressions for the cross-\'J' order coefficients (g,,,r, S,1,3, ord &"r) in the Taylor series
and expansion (Table 3) are, ofcourse,different from the en-/trc2+\/FA3+\ ergy terms in the O'Neill-Navrotsky model (Nell et al.,
-RIln ffi{)#:8",t E,,"rt 2g"r",s2 1989), the latter being linearly dependent combinations\r !oct'r\r Lter 'r of pure end-member B energy parameters (Eqs. 19) rather
* E,r,rXr.orroo * &,,rJ,. (24) than uniquely defined series expansion coefficients.
(**),,,,,",.",,",: s" * 2s""r' t g"'r'
+ &,,,,t, + g,r"rs2
+ 8,,,,s, * nr tn (ug*J@ (2g)
: 8,, * 2g,,,,r, t g,,,rr,
+ g,r,,s, I g,z"zsz + g,r,rs,
+ Rrrn 9-f=)e!'4 . Qs)(tseielxtseil1)
and
t#)lP,r1,s1,s2,s3
1006 NELL AND WOOD: THERMODYNAMICS OF SPINEL SOLID SOLUTIONS
TABLE 4, Chemical potentials of reference end-member components
& v s l a h l ) o . : 9 o t 9 , " 1 - g s j - q , , , , r i - 9 " " r 2 - 9 q " , s ? - 9 r r 4 - 9 " * s 3 - g q q f r s z - 9 q q f , s s - 9 q q s z s g+ (2g,*,t g,"",)r" + (9,,,"+ g,,",)r,-t (9,,., + 29",q)s,-t (g& r g",*)s,
& r e z * ( r e g * e e s ' ) o . : 9 0 + 9 , , - t g " " - 9 n n r ? - 9 r r r t - 9 , * s ? - 9 r r s 7 - 9 * * s € + ( 2 9 n n + 9 , , r ) r r + ( 9 , , , , + 9 , , " " ) r ,+ ( 9 n " , + 9 " , * ) s t . | - ( 9 , , q + 2 g q q ) s 2 + ( 9 , , * * 9 " J 4 - 9 q r y r r r z - g n q t r s r - 9 r " s , r z s r - Q , , r r 1 s 2-
9,"""t252 -
9",eS,Sz -
9**SrS. -
9rqfrSe -
9aqfzSs -
9"'""StS.+ RrIzIn(Fei j) + In(Feet) - 2tn2l
t h 1 r e , * a r p o : 9 o t 9 , . + g , " - g , , , , r ? - 9 , " r r Z - 9 " , " , s ? - 1 q r s B - 9 * " " s 3 + ( 2 9 n n * Q , , , " ) r t+ (2g""+ g,,,,)r" + (9,,", + 9"",)s, + (9,,r+ g,"r)s" * (9n* + 9,,*)s. - 9,,,,412- 9,.".rtst- gqqfrS" -
9r""GS3 -
9,"",1251 -
9nqlzSz- 9nqfzSs -
9",eSt9z -
9",qSrSg -
9""""S2S"
+ Ff!n(FeAlXAl*,XA1,.,) - 2ln2l
t F e 2 + ( A l A l ) o 4 : 9 o + 9 , , + 9 , , + 9 " , - 9 n n r ? - 9 r r r Z - 9 , q s ? - q * r s Z - 9 * + s 3 + ( 2 9 n n + 9 , , h + 9 , ' s , ) r 1+ (2gr.t gn"t g,"",)rz+ e5.,".+ g,q + g.,",)sr + (9,.""+ grrt grr)s" + (9,,* + 9"* * g","")ss- gqafrf" 9a",frs,
L r F e s + ( F e p + M s ) o 4 : 9 o * 9 q - 9 n n r ? - 9 , , a r Z - 9 " , " , s ? - 9 " " " " s 7 - 9 * * s 3 l 9 n * f , 1 9 " * l r * 9 * * 5 t f 9 * * s ,
+29 " . " " s3 -gq ry f r t " - g , . " , \ s r - 9 , , " " r 1s2 -gq r " f r ss -g , " " , f 2s1 -9a " " f zs r -9aq f zss -9 ,q i sz -9n * iQ -9 " r r " s rQ
+ Fr[n(Fel;i)(FeS)(M9.",) - 2tn2]
&vsqre3*reo*y6. :9ot gq- gnnr? 9"rrT- 9" ," ,s? -
9qqs3- 9**s3* gqqrr l 9r" , r "* 9*qsr * 2g*rs"+ g**s"-gna f r f r - 9 , , " . f 15 r -gqqnSz -9 r+ l rSg -9 , " " r f 251 -9aq f rS " -9a " " f "Ss -9 " ' qS tS r -9 " , " " s r sg -9 " " " "QSs
+ RTI?In(Fe*1) + ln(Mgi .J - 2 ln2l
The chemical potential of any component is obtainedby partial differentiation of the free energy of the solidsolution with respect to each of the order and composi-tional parameters:
MgAlrOo, FerOo, FeAlrO4, and MgFe2O 4atthe conditionsofinterest. In order to obtain chemical potentials relativeto partially disordered "real" MgAlrOo, FerOo, FeAlrOo,and MgFerO4 at the conditions of interest, it is necessaryto insert appropriate values of the respective order pa-rameters into Equation 30. The activities of MgAlrOo,FerOo, FeAlrO4, and MgFe2Oo relative to standard statesof the disordered pure end-members at the temperatureand pressure of interest are then given by the followingexpresslons:
RT ln a-^rroo : - g,rnn - 8,2a(r2 - l)2- & r . r [ ' s r ( s r - 2 + 2 s r )
- g,r,r(r, - lXst + s! - l)- g""s2(r2 - l)- +o,r(t, - lXs. - s3)- 8"'"'s'(st + s9 - l)- & , , , ( s . - ^ 4 X s r + 4 + l )- &'"tsz(s' - 5'3;
t t . u .a . : G + 1a - , , , ( f ) . " " , , " , , ,
+ ( b -
+ ( c -
+ ( d -
+ ( e -
D(+\\dr:/ rr.,1.,1.,2,,.
,,,(f)..'",*,,,(,T)..."",-,r(H)..,,.",-(30)
The coefficients d to e in Equation 30 refer to the coor-dinates of the component of interest in rr-l2-Jr-Jz-srspace. We present in Table 4 the results for each of thereference end-member components. The reader shouldnote that these reference components are perfectly or-dered or inverse and hence do not refer to pure "real"
^- I ̂ ^,- KAI,.,XAI".XMg"",)l* I+nrlsetni f f i l *J t l t tRTln ar,roo: -9a,,(r1 - l) t - g,r,r f t - g,r, ,s1
- &"'(s' - s9)'- & r r s 3 - g , r , r J r ( r r - l )
- gn,r(r, - lXs, - ,9 - g.r,3J3(rr - l)-
8'2'1t251 -
8'r,rfz(52 - So,)
- g''"'r's' - g""s1(s' - 5'3;-
&t".S,S, -
&r.rSl(S, -
4;
+ nr{,q[z h ggi)-, * h Era:l]t1 (Fe3.1)-, -(Feil)*, l j
where the superscripts ss, sp, mt, hc, and mf refer, re-spectively, to solid solutions, pure spinel, pure magnetite,pure hercynite, and pure magnesioferrite, whereas the su-perscript 0 on the order parameters sr, sr, and s, refers tothe values of these parameters in pure disordered end-member spinels.
CoNsrn,q.rNr oF MODEL PARAMETERS
Values of the order and combined order and compo-sition coefficients in the Taylor series expansion (Table5) were obtained from least-squares fits to available cat-ion-distribution data for magnetite (Wu and Mason,1981), spinel (Wood et al., 1986), and magnesioferrite(Pauthenet and Bochirol, 195 I ; Kriessman and Harrison,1956; Epstein and Frackiewicz, 1958; Mozzi and Pala-dino, 1963; Blasse, 1964; Tell ier, 1967) and from theFerOo-MgFerOo and Fe.Oo-FeAlrOo solid solutions mea-sured by Nell et al. (1989). Cation disordering in hercyn-ite is poorly constrained (Bohlen et al., 1986), and as aninitial approximation, we assumed that Fe'z*-Al disorder-ing in FeAl,Oo is identical to Mg-Al disordering inMgAlrOo (Wood et al., 1986). Such disordering behaviorin hercynite is in agreement with measured Fe,*-Al dis-tributions in FerOo-FeAlrOo solid solutions (Nell et a1.,1989), whereas the resultant small absolute value of the8",", coefficient facilitates fitting of the macroscopic ther-modynamic measurements in FeAlrOo-MgAlrOo, FerOo-MgFerOo, and Fe.Oo-MgAlrOo solid solutions.
First, cation-distribution data for magnetite, magnesio-ferrite, and spinel were fitted to Equations 16 to 18, re-spectively, thus fixing the value of the 9,r", coeffioient andconstraining the values of the quantities (g", * &,o), (&,I g,r,, * 2g,r", - g,1 - 8,2,y - g",,3), (&,., - &,,, - 8,r",)'(&, - &, I 2g,t,t - g,r,,), and (9"r,, - 8"t"2 - &r,r), whereasthe assumed disordering of hercynite implied that g,,,. :
NELL AND WOOD: THERMODYNAMICS OF SPINEL SOLID SOLUTIONS
+ nr{rr - sll
[,,,ffi. 9",9q
9c9nq
9nn9"'q9+q9*r9qa9,,",9nc9rrs,,",9a+9+q9qq9q+9qe
1008 NELL AND WOOD: THERMODYNAMICS OF SPINEL SOLID SOLUTIONS
Fig. l. (A) Cation distributions on the join FerOo-MgFerOo at 1000 "C (Nell et al., 1989) plotted as a function of mole fraction
of MgFerOo (X-J. Solid curves are values calculated from the model (see text). (B) Activity-composition relations for the FerOo-
MgFerOo join at 1000 'C. Solid curves are values calculated from the model.
1 . 00.9o.8o.7
1 .0
0 .9
0.8
,:' o.o=
(5
E 0 .5(tt
o" 0.4o
L(u0.3
,+Fei"t
O S h i s h k o v e t a l ( 1 9 8 0 )
O Tr ine l -Dufour and Per ro t (1977)
NELL AND WOOD: THERMODYNAMICS OF SPINEL SOLID SOLUTIONS 1009
T= 1300 'C
T\tiI
, , / calculated with Wor = 4.5 KJ/mol
. , ' - ca lcu la ted w i th Wo ' = 5 .5 KJ /mol
,. . /calculated with Wol = 3.5 KJ/mol
ThisStudy
- /second-degree po lynomia l f i t (Jamieson and Roeder , 1984)
\
0 0.1 0.2 0.3 o.4 0.5 0.6Xmt
Fig.2. Calculated values of ln Kgr-Fse [Kd : (,fT::b) (Xf,]stu 5o)) /((Xmr",oJ (XFl.,o,o))l compared to the measurements of Jamiesonand Roeder (1984) at 1300'C. The calculations take account of the nonideality in olivine (Wron a l-atom basis) and the probableuncertainty in the regular-solution parameter. Direction of approach to equilibrium in the experimental measurements is indicatedby arrows on the error bars.
ooTLIo o
Yc
2.3
2 .2
2 .1
2.O
1.9
1 .8
1 .7
.6 1o.
1o.7 0.8 0.9 1.0
9,3,3 and &,"1 : g,,,3 - 8", - 8,2"7 - 2g"r"r, Next, Fe2*-Mgdistributions in FerOo-MgFerOo were fitted to Equation22 to obtah values for g,,"3, g"3, &1,3, and g,r",. Consistencywith the constraints imposed by the fits to magnetiteand magnesioferrite then allowed the calculation of g"r,r,&, and &,r. Similarly, Fe2*-Al distributions in FerOo-FeAlrOo solid solutions were fitted to Equation 23, fixing8,2q, ?t7d &, * &,,,, and g,r"rwas obtained from the Fe3*-Fe'?* distribution in FerOo-FeAlrOo solid solutions throughEquation 24. The values of all but three (g",, 9,24, vnd&,,,) of the Taylor series expansion coefficients in Equa-tions 12 to 14 were thus determined. The 8", and 9,",coefficients could be obtained by fitting Equation 12 toFe2*-Al distributions in FeAl.Or-MgAlrOo, FeAlrOo-MgFerOo, or FerOo-MgAlrOo solid solutions. The Fe'z*-Alintersite distributions in these solid solutions are, how-ever, not amenable to direct measurement, and in orderto estimate the values of 9,, and &1"1, we fitted Equation12 Io an assumed linear relationship between the Al-Fe,*distribution coemcient and composition in FeAIrOo-MgAlrOo solid solutions.
Values of the composition parameters in the Taylorseries expansion were determined from activity-compo-sition relations and phase-equilibrium experiments inFerOo-FeAlrOo, FerOo-MgFerOo, FerOo-MgAlrOo, andMgAlrOo-FeAlrO. spinels. The value of the g,,. parame-
ter was determined from the experimentally observedsolvus in Fe.Oo-FeAlrOo solid solutions, which has a con-solute temperature of 860 + l5'C (Turnock and Eugster,1962). The &r,r parameter was determined from activity-composition measurements in FerOo-MgFerOo solid so-Iutions at 1000 "C (Shishkov et al., 1980; Trinel-Dufourand Perrot, 1977) and Fe2*-Mg partitioning experimentsbetween olivine and spinel solid solutions at 1300 'C
(Jamieson and Roeder, 1984) and a chloride aqueous so-lution and spinel at 800'C (Lehmann and Roux, 1986).Finally, the value of g.,o was constrained from activity-composition relations in FerOo-MgAlrOo solid solutionsat 1000'C (Mattioli and Wood, 1988).
The coefficients &, &,, and g,rdetermine absolute val-ues of end-member chemical potentials and are not re-quired for activity-composition calculations (Eqs. 3l to34). These parameters were therefore set to zero. Finally,the derived values of the order, compositional, and com-bined order and composition parameters were tested andrefined by comparing calculated cation-site fractions, ac-tivity-composition relations, and partitioning coefficientswith experimentally measured cation distributions, activ-ity-composition relations, and partitioning data in FerOo-MgFerOo, FerOo-FeAlrOo, MgAlrOo-FeAlrOo, and FerOo-MgAlrO4 solid solutions. The refined values for the ex-pansion coefficients and their estimated uncertainties are
l 0 l 0
reported in Table 5. The values are not unique since theexpansion coefficients are highly correlated. The estimat-ed uncertainties are not intended to reflect these corre-lations but rather to indicate ranges over which the re-fined values may be varied while preserving a satisfactoryfit to the experimental data.
Appllc,qrroN oF THE MODEL ToFerOo-FeAlrOo-MgFerOo-MgAlrO.
SOLID SOLUTIONS
Theoretical cation distributions and activity-com-position relations in Fe.Oo-MgFerOo, FerOo-FeAlrOo,MgAlrOo-FeAlrOo, and FerOo-MgAlrOo solid solutionswere calculated using the values of the expansion coeffi-cients in Table 5. Cation distributions were calculatedfrom Equations 2l to 27. The nonlinear simultaneousequations were solved using a Newton-Raphson method(e.g., Gerald and Wheatley, 1984, p. 133-159), and theresults are presented as solid lines in Figures lA, 3A,, and4,A.. Activities of spinel, magnetite, hercynite, and mag-nesioferrite were calculated from Equations 3l to 34, re-spectively, and the results are presented as solid lines inFigures lB, 38, and 48.
FerOo-MgFerOo solid solutions
Calculated cation distributions at 1000 "C (Fig. lA) arein excellent agreement with experimental results obtainedfrom the combined thermopower and conductivity tech-nique (Nell et al., 1989). Activity-composition relationsat 1000'C display a positive deviation from ideality in-termediate between the data of Shishkov et al. (1980) andTrinel-Dufour and Perrot (1977) (Fig. lB). Error bars onthe data points in Figure lB represent an estimated un-certainty of +l0o/o in the measured activities. Jamiesonand Roeder (1984) performed Fe2*-Mg exchange ex-periments between olivine and FerO.-MgFerOo solidsolutions at high temperatures. We used their data to con-strain FerOo and MgFerOo activities at 1300 oC. The Fe2*-Mg exchange equilibrium between olivine (ol) and FerOo-MgFerOo solid solutions (Fesp) is given by
NELL AND WOOD: THERMODYNAMICS OF SPINEL SOLID SOLUTIONS
where Kor-Fesp is the equilibrium constant for the olivine-spinel exchange reaction, ,lT::"o. is the mole fraction ofmagnetite in the spinel solid solution, ?re3oa (: ar.roo/lT::aJ is the activity coefficient of magnetite in spinelsolid solution, and Wo, is the regular-solution parameterfor Fe-Mg olivine. Theoretical K$-r""o 'out.r"t at fixed spi-nel compositions were calculated from Equation 35 usingyr"3oa iIDd ̂ yugre2o4 values obtained from our calculated
activity-composition relations. Assuming a value of 4.5+ 1.0 kJlmol on a l-atom basis for the regular-solutioninteraction parameter (Wo) in olivine (Nafziger and Muan,1967; O'Neill and Wall, 1987), the mean value for lnKof'F6e over the compositional range 0.1 < XMsF"ro o < 0.15was estimated to be 1.80. As can be seen from Figure 2,the calculated ln Kgt-r""p values are in good agreement withthe experimental measurements within the range of likelyvalues of I;Z.,.
FerOn-FeAlrOo solid solutions
Phase-equilibrium experiments at 2-kbar pressure(Turnock and Eugster, 1962) indicate the presence of asolvus at a composition of 55 molo/o FeAlrOo and a con-solute temperature of 860 + 15 'C in this system. Onlythese data were used as constraints on the mixing prop-erties on this join, and our model calculates a solvus inalmost perfect agreement with the experimental data.Calculated cation distributions and activity-compositionrelations at 1300 oC are presented in Figures 3A and 38,respectively. There is excellent agreement with the mea-sured cation distributions, and the predicted activity-composition relations are within the uncertainties in thedata of Petric et al. (1981). Petric et al. determined activ-ities by equilibrating spinel with Pt-Fe solid solutions atfixed Por. Their data give an uncertainty of +0.06 logunits in log a." in the metal solution, and this uncertaintyresults in the error bars given in Figure 38.
Fe.Oo-MgAlrOo solid solutions
The combined thermopower and conductivity tech-nique used to determine cation distributions in FerOo-MgFerOo and FerOo-FeAlrOo solid solutions (Nell et al.,1989) does not provide information on the intersite dis-tributions of Mg and Al in FerOo-MgAlrOo solid solu-tions. In our earlier paper (Nell et al., 1989), we com-bined thermopower-conductivity measurements on FerOo-MgAlrO4 solid solutions with the O'Neill-Navrotskymodel to estimate the amount of tetrahedral Al. In thepresent study we opted to use the Gibbs-Duhem equationto calculate partial molar spinel entropies from our mea-sured partial molar entropies of FerOo. These were thensolved to obtain a model-independent concentration ofAl on tetrahedral sites. Uncertainties in the calculatedvalues of A1,", arising from the integration of the Gibbs-Duhem equation were assessed by calculating Al
", in
FerOo-FeAlrOo solid solutions through the use of a simi-lar procedure and then by comparing the result with theactual measured values of A1
",. The results and uncer-
tainties at 1000 oC are shown in Figure 4,{ together withtheoretical cation-distribution curves. Cation distribu-tions in this system were not used to constrain the valuesof the expansion coefficients, and the calculated cationdistributions are therefore entirely predicted.
Calculated activity-composition relations are com-pared to the data of Mattioli and Wood (1988) in Figure48. The calculated solvus has a consolute temperature ofabout 1025 "C at a composition of 56 molo/o FerOo. The
NELL AND WOOD: THERMODYNAMICS OF SPINEL SOLID SOLUTIONS
E x o l a n a t i o n
l 0 1 l
cJ
sE
(g
o
at,
q)o.c.9
coocooo.9oooatt
2.O
1 . 8
1 . 6
1 . 4
i o n
F e2*
F e 3 *
AI
t e t . s i t e o c t . s i t e
ooD I
oaT=13OO"C
F e o c t1 . 2
1 . 0
0.8
0.6
o.4
o.2
0.00.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Xn"0.9 1 . 0o.8
o
ou
6!tc(It
ooq
(!
1 . 0
0.9
0.8
o.7
0.6
0.5
o.4
0.3
o.2
0 .1
o .o ' ' 'o.o 0.1 o.2 0.4 0.5 0.6 0.7 0.8
Xn"
Fig. 3. (A) Calculated cation distributions on the join Fe3Oo-FeAIrOn at 1300 'C plotted as a function of mole fraction of FeAlrOo(Xo"). Experimental data are from Nell et al. (1989). (B) Predicted activity-composition relations in FerOo-FeAlOn spinels at 1300'C compared to the measurements of Petric et al. (1981).
1 .00.90.3
T=13oo"G B
1 . 6
1 . 4
1 . 2
=cJ
(U
Eo(sL
F()f
(t
oo.c.9(U
coocoIU'.9oooo
2 .O
1 .8
1 . 0
o.
0.6
o.4
o.2
0.0
t on
F e2*
Fe 3 *
A I
M g
Exp lanat ion
te t . s i te oc t . s i te
ooDI
oaAA
o
T=1OOO G_ 3 +F 9 o " t
A l o " t
0.8 0.90 .0 0 .1 o.2 0.3 0.4 0.5 0.6 0.7X"p
1 .0
d
(U!tco
ootL
(!
1 .0
0.9
0.8
o.7
0.6
0.5
0.4
0.3
o.2
0 .1
0.5 0.6Xsp
1 .0
e J
Fei"t
r"?J,
T= l OOO'G
o.o v-0.0 0 . 1 o.2 0.3 o.4 o.7 0.8 0.9
NELL AND WOOD: THERMODYNAMICS OF SPINEL SOLID SOLUTIONS 1 0 1 3
0.40
0.30
o.20
0 .10
o.00
-0 .1 0
-0.20
-0.300.50 0.60 0.70 0.80 0.90
X " o
Figure 5. Calculated values of ln KoLAbe U(oLAse : [(Xfif,,")(XF!.,0,o)]/[(XA:n,o)(X$,r,0,o,)l] for the join FeAlrOo-MgAl,On com-pared to the measurements of Jamieson and Roeder (1984) at 1300'C. Calculated distribution coefficients favor a value of about3.5 kJ/mol on a l-atom basis for the regular solution interaction parameter in olivine (lZ",). Direction of approach to equilibriumin the experimental measurements is indicated by arrows on the error bars.
o
O oYc
solvus predicted by our model is a compromise betweenthe model of Lehmann and Roux (1986), who calculateda wider miscibility gap aI 1000 'C, and the results ofMattioli and Wood (1988), who estimated the consolutetemperature to be between 900 and 1000.C.
MgAlrOo-FeAlrOn solid solutions
Activity-composition relations were estimated fromFe2*-Mg exchange experiments between olivine and spi-nel (Jamieson and Roeder, 1984) at 1300'C and aqueouschloride solution and spinel at 800 "C (Lehmann andRoux, 1986). The equilibrium between olivine (ol) andMgAlrOo-FeAlrOo solid solutions (Alsp) is expressed as
Following the procedure outlined above for FerOo-MgFerOo solid solutions, we used a value of W", of 4.5 !1.0 kJlmol on a 1-atom basis and found a mean value ofln Kor-Arse over the composition range 0.5 < XMsAr2o4 < 0.9to be -0.02. Calculated values of ln K3t-rt"o u." tt excellentagreement with the measured values of Jamieson andRoeder (198a) (Fig. s).
Equilibrium between an ideal chloride aqueous solu-tion (fl) and MgAlrO.-FeAlrOo solid solutions (Alsp) isgiven by the follo'ildng relationship:
tffi#:J:rnKnA'Isp
lnKg-^"o:"ffi#* 1n
7r"rea:oo'Y
r.l.l2oa(37)
The agreement between our calculated activities and thedata of Lehmann and Roux (1986) is excellent (Fig. 6).The mean value of ln 61-et"o ie calculated to be - 1.4.
CoNcr,usroNs
(16.t A second-degree Taylor series expansion ofthe vibra-tional part of the Gibbs free energy was used to model
: ln Kol-Alsp * tn Te"mzor
TMgAt2oa
w"t , 2w"t , .^ ,- Rf
t TT^Fi. 'orol
Fig. 4. (A) Predicted cation distributions on the join FerOo-MgAlrOo at 1000 "C plotted as a function of mole fraction of MgAlrOo(X"o).Fe'* and Fe3* site-occupancy dataarc from Nell et al. (1989). Al and Mg intersite distributions were calculated by using theGibbs-Duhem method (see text). (B) Calculated activity-composition relations for the FerOo-MgALO. join at 1000 "C. Experimentalmeasurements are from Mattioli and Wood (1988).
T= 1300"G
-/ calculated with W^, =4.5 KJ/mol I
--" ' catcutated with w",= u.u KJ/mol I
fnis Study
- . / ' ca lcu la ted w i th Wor =3 .5 KJ /mol J
- - / regress ion l ine (Jamieson and Roeder , 1984)
1014 NELL AND WOOD: THERMODYNAMICS OF SPINEL SOLID SOLUTIONS
o.o
- 1 . 0
-2.O
-3.0
-4.O0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Xn"
Fig. 6. Calculated values of ln Ka-Abe {Kn-Nse : [(X$]Kr".Xxfl,"or)l/(X#;furo.Xxg*rr)l] for the join FeAlrOo-MgAlrOo compared tothe measurements of Lehmann and Roux (1986) at 800 "C.
2.O
1 . 0
o
I
Y
1 . O0.9
cation distributions and activity-composition relations inFerOo-MgFerOo-FeAlrOo-MgAlrOo solid solutions. Theresulting expression for the Gibbs free energy is symmet-ric with respect to order, composition, and combined or-der and composition parameters. Activity-compositionrelations and intersite cation distributions are thus treat-ed with symmetric excess mixing energies. The Taylorseries expansion coeftcients are overdetermined, and asa result, certain relationships exist between the excess free-energy terms in the solution set of expansion coefficients.The Navrotsky-Kleppa (1967) model (-RZln Ko :
"oo-stant) for intersite cation distributions is obtained fromthe Taylor series expansion when all excess mixing ener-gies are set to zero. The constants used in this formalismare shown to be differences between chemical potentialsof reference end-member components. The O'Neill-Nav-rotsky (1983, 1984) model (-RZln Ko: a + 2BB3$)forintersite cation distributions in pure end-member spinelsis identical to the second-degree Taylor series expansionwith the B energy parameter given by the regular sym-metric interaction parameter between divalent and tri-valent cations on tetrahedral and octahedral sites. Inbinary and quaternary solid solutions, the O'Neil l-Navrotsky formalism corresponds to a second-degree se-ries expansion in which the combined order and com-position parameters are equal to zero.
The values of the series expansion coefficients wereconstrained from measured cation distributions and ac-tivity-composition relations in FerOo-MgFerOo, FerOo-FeAlrOo, MgAlrOo-FeAl2O4, MgFerOo-MgAlrOo, andFerOo-MgAlrOo solid solutions. The resultant data set al-lows calculation of internally consistent cation distri-butions and activity-composition relations in Fe.Oo-MgAIrOo-MgFerOo-FeAlrOo solid solutions.
Application of the model to calculate cation distri-butions and activity-composition relations in FerOo-
MgFerOo, Fe.Oo-FeAlrOo, MgAlrOo-FeAlrOo, and FerOo-MgAlrOo solid solutions shows good agreement betweenthe calculated values and experimental measurements.Regular symmetric interaction parameters adequately de-scribe mixing in FerOo-FeAlrO4, MgAlrOo-FeAlrOo, FerOo-MgFerOo, and Fe.O.-MgAlrOo solid solutions, and asolvus with consolute temperature of about 1025 "C isgenerated in the latter system.
AcxNowr-nncMENTS
This research was made possible in part by financial support from theCouncil for Mineral Technology (MINTEK) to J.N. Support by NSF GrantEAR-8416793 to B.J.W. is also acknowledged. The manuscnpt also ben-efited from constructive reviews bv A. Navrotsky and R. O. Sack.
Rnrnnnucns crrno
Andersen, D.J., and Lindsley, D.H. (1988) Intemally consistent models
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