Thermodynamics of Formation of Some Compgunds with The

A merican M ineralogist, Volume60, pages249-256, 1975

Thermodynamicsof Formationof SomeCompgunds with the Pseudobrookite-Structureand of the FeTirOu-TisOrSolid SolutionSeries

AtexnNone Nnvnorsrv Departmentof Chemistry,Arizona State Uniuersity, Tempe,Arizona 8528i,

Abstract

The enthalpyof formationof pseudobrookite,FezTiO,, from FezOaand TiO, is found by oxidemelt solutioncalorimetry to be *1.99 * 0.15kcal mol-r (*8.33 + 0.63KJ mol-'). Thermochemicaldata for other compoundswith the pseudobrookitestructure are summarized.Pseudobrookites are "entropy- stabilized"high temperaturephases with partiallydisordered cation distributions. Their decompositionat low temperaturesto assemblagesofbinary oxides(for,{rg+Ba+Ou pseudobrookites) or to assemblagesof phasesofilmenite and rutile structure(for A'+ Bzn+Oopseudobrookites) reflects the balancebetween en- dothermicenthalpies of formationfrom theseassemblages and the substantialpositive entropies of formation arisingfrom the substitutionaldisorder. A simplecation distribution model is appliedto the distribution of iron between4c (A) and8/(A) sitesin the FeTizOu-TisO'system, from which high Ti'Oo is predictedto havethe cation distribution(Ti'+0 o6Tie+0 go)A(Ti'+o.e6Ti3+' o.)eOu. Interchange enthalpies (or freeenergies) for the reactions:Mg'+r * Tin+e= Mg'*g + Tio+A,Fe'+A + Til+B = Fe2+s* Tia+r,and Tis+A+ Ti'+a : Tio+B+ Ti'*a are +8.6, + 13,and * l0 kcalmol-r respectively(36.0, 54, and 42 KJ mol-', respectively).MgTtOc exhibitssomewhat disordered pseudobrookite structures derived from a "normal" structurestable at low temperature,while TirOoand probablyalso FezTiOu, AlrTios, and GazTiOuhave somewhatdisordered structures derived from an "inversestructure stable at low temperature.Thus Ti'+ appearsto havea strongpreference for the 8/(A) sitesin all thesecompounds ofpseudobrookite structure.

Introduction with 4 A82o' formula units in the unit cell, all the cationsare in octahedralcoordination. There are two Studyof lunar basaltshas resulted in considerable kinds of octahedralsites, with fourfold (4c or,4 sites) interestin mineralswith the pseudobrookitestruc- and eightfold (8/ or B sites)multiplicity. Three oc- ture. These include FezTiOs (pseudobrookite), tahedra share edgesto form units which are then FeTLOu(ferropseudobrookite), MgTirOu (karrooite), linkedtogether into doublechains as building blocks (Mg,Fe)TirOu(armalcolite), and solid solutionsin the for the structure. In monoclinic pseudobrookites FerTiOu-TirOuseries. The purposeof this paper,is such as TirOuand the titanium-richmembers of the threefold:(a) to presentnew high temperaturesolu- TirOu-FeTizOusolid solutionseries (Grey and Ward tion calorimetricdata for the enthalpyof formation (1973),the 8/ (B) site is further split into two non- of FezTiOs,(b) to reviewthe existingthermochemical equivalentsites. For armalcolite,Lind and Housley data for ternary oxides with the pseudobrookite (1972)give the following metal oxygendistances for structure, and (c) to correlatethe thermodynamic the 4c and 8/ octahedra-8f, 2.061, 1.845,l'993, and structural data and to apply a simple thermo- 2.172,1.938, and 1.938A;4c, 1.960,2.191,l'960, dynamic model to the cation distributions in 2.191.2.030.and 2.030A. Thus,both octahedraare MgTirOo, FeTizOu,and the FeTizOu-TirOusolid distorted.The averagebond lengthin 8/is 1.992+ solution series. 0.1l4(s.d.) A; thatin 4c is 2.072t 0'096A' Thusthe The PseudobrookiteStructure ,4 site is somewhatlarger and slightly lessdistorted Early structure determinationswere by Pauling than the B site. For the presentpaper' the most im- (1930) and Hamelin (1958). Recently,Lind and portant feature of the pseudobrookitestructure is Housley(1972) have performed structure refinements that the two sitesare sufficiently similar to allow con- on syntheticsingle crystalsof karrooite, MgTirOu, siderablesubstitutional disorder of the cationsoc- and armalcolite,Mgo.uFeo.6Ti2ou. In the orthorhom- cupyingthem. As in the spinels,we canrecognize two bic pseudobrookitestructure, spacegroup Bbmm, limiting distributions,the normal,(A)4"(B)BtOo and 249 250 ALEXANDRA NAVROTSKY

TnsI-e l. Enthalpy of Solution of Pseudobrookite in of FezTiOu,although its molar enthalpy of solution 3NarO.4MoO, at 692"C was fortuitously very close to zero, so that the heat effects actually measured were always less than 0. 1 FerTiO, callobs I H"oa(kcal mol-}) calorie in magnitude.The resultsof five solution ex- (in mg) periments(Table 1)give a meanof -0.10 t 0.10kcal mol-1. One may then calculate,for 46.07 -0. 050 -o.26 the reaction: 86.06 -0. 020 -0. 06 FerO, (hematite)+ TiO" (rutile) : Fe"TiOs LL4.46 !0. 000 r0. 00 (pseudobrookite) (l) 43.60 -0. 015 -0. 08 49,25 -0. 0r9 -0. 09 AHo"uu= +1.99+ 0.15kcal mol-'

Discussionof the Stability of FerTiOu AV = -0.10!0.10 and other Compoundswith the PseudobrookiteStructure the inverse,(B)k(AB)ErOu, and we cananticipate that, Fe"TiOu especially at high temperature,intermediate distributions will be the rule rather than the exception. The calorimetric data show the enthalpy of forma- pseudobrookite CalorimetricDetermination of the tion of from hematite plus rutile to positiue. Enthalpy of Formation of be This suggests that the ternary oxide is Pseudobrookiteil 692 + 2"C stabilized at high temperatures by a positive entropy of formation and should becomeunstable relative to FerTiOu was prepared by ignition of a stoi- the binary oxides at low temepratures. Such decom- chiometricmixture of dried BakerAnalyzed Reagent position has indeed been found by Haggerty and FerO, (hematite) and TiO, (rutile) for a total of Lindsley (1970),with an equilibrium temperatureof 72 hours in platinum a cruciblein air at 1350.C, 565 + 15"C. We can calculatethe entropy of the for- with four intermediategrindings of the sample.At mation reaction since at 838 K its free energy change the end of eachfiring, the samplewas cooled relative- is zero. ly rapidly in air. The powder X-ray diffraction +=tllo pattern of the product showedit to be singlephase ASo: : +2.4 cal K-' mol-' (2) pseudobrookite. 838 The high temperatureCalvet-type twin micro- The free energy of formation of FerTiOu from calorimeter and the sample assemblyfor measur- FerO, and TiO, is then given by the equation: ing the heat of solutionof a solid oxide samplein a molten oxide solvent have been describedpre- AGo: + 1990- 2.4T (T inK) (3) viously (Navrotskyand Kleppa, 1968).Navrotsky and Kleppa(1967a,1968) have shown that a sodium Thisequation gives AG' with an estimatederror of t molybdatemelt of thecomposition 3NarO-4MoO, is 300cal in the range700-1200 K, sincethe difference a suitable solvent for the calorimetricstudy of in heat capacity between reactantsand products anatase,rutile, Tia+-containingspinels, FerOr, and shouldbe small at high temperaturesand one can ZnFerOn.A similar calorimeterand essentiallyiden- neglect any changes in cation distribution with tical conditionswere usedin the presentwork. In temperature.To our knowledge,no enthalpyor free eachsolution experiment, 50-100 mg of solutewas energy of formation data have been reported for dissolvedin l0- I 5 g of solventat 692L 2oC.Two ex- FerTiOr. Heat content and standardentropy values periments each on a-Fe2Osand TiO, (rutile) fell aretabulated by Robieand Waldbaum (1968) and by within experimentalerror of the valuesof the heatof Kelley(1960). From thesetabulations, the entropyof solution of these oxides reported previously by formation of pseudobrookitefrom hematite and Navrotsky and Kleppa (1968).Accordingly, those rutile is given in Table 2. The agreementwith our values,*2.38 kcal mol-l and -0.49 + 0.05 kcal value at 858 K is excellentbut, we believe,may be mol-'were used in the presentwork for the fortuitous. The heat capacityand heat contentdata enthalpiesof solution of rutile and hematite,respec- are to be questionedon two counts. First, the tively.rNo difficulty wasencountered in the solution calculatedentropy of formation of pseudobrookite varies rather strongly with temperature,which is somewhatsurprising. If this is a real effect,it may be I Throughoutthis paper,heat effects are given in calories,where relatedto the anomalouslysmall thermal expansion I cal = 4.184ioule. coefficientsreported for pseudobrookite-typecom- FORMATION OF PSEUDOBROOKITES AND FeTizOu-TisOuSOLID SOLUTIONS 251 pounds(Bayer, l97l), or to changesin thecation dis- Trsts 2. Lattice Entropies(Calorimetric Entropies) of FezTiO,and MgTirOufrom the Oxides tribution (seebelow). The datafor MgTirOu(Table 2) Formationof no anomaly,though the valuesof AS" show such -'-2 otgo" Ferore FerTlora MeTtrOrb scattersomewhat. In addition, the tabulatedvalues c ^o -o -o TK -T -T -T -Tco -"7 si os; do not include any contribution for the con- o figurationalentropy arising from an "inverse"or partially disorderedcation distributionof Fe3+and Tia+ 298 r2.o4 6.44 20.89 37.40 +4.47 30.40 -0,12 800 27.85 15.98 5r.64 81.75 +2.26 7L.65 -1.03 over nonequivalent octahedral sites. Waldbaum 1200 34.98 27.93 56.59 LO2,45 +0.78 91.33 -0.56 1600 40.24 25.53 76.52 117.60 +0.84 105.71 -O.70 (1973)has discussed this point. If the cationdistribu-

tion in FerTiO,is indeedinverse as Lind and Housley a, source of date, R, A. Roble and D. R. Waldbes, U's. Geol. suney Bull. 1259 (1968); (1972) suggest,then, as Waldbaum (1973) rec- Soc. 71' 4569 b. Data f or ugTt ro< f r@ s. s. Todd, J. Aner. Ch6. ommends,one should add 2Rln2 : 2.76 cal K-r (1952) and x..-ll orr and J. P. CouthllE, J. Amer' Ch@. soc. 21' 3186 G952), mol-' to the standardentropy of pseudobrookite. cal K-I nol-l This would make the entropy of formation of FerTiOucalculated from the correctedheat capacity data more positiveby about 3 cal K-' mol-' than pseudobrookites,GarTiOu and AlrTiOu,are even less the entropy we calculate at the decomposition stablethan FerTiOu;their decompositions,observed temperature,838 K. Assumingour calorimetricdata by differential thermal analysis, occur at higher to be correct, this discrepancymay have several temperaturewith markedly exothermiceffects. It is causes.FerTiOu may indeedhave the normal cation hard to envision that lattice effects alone could distribution,since the X-ray data are not totally con- produce enough entropy to stabilize these ener- vincing and one does not really know the site getically unfavorablephases, so these data argue preferenceof Fe8+compared to Tia+.With an inverse for configurationaldisorder in thosepseudobrookites distribution at high temperatures,the Fe3+ and as well. Quantitativecalorimetric study of thesecom- Tia+ions may possiblyorder on their sublatticeat poundsis planned. lower temperatures,producing local order or even M2+Ti2O5 a superstructure.If this ordering occurs gradually over a rangeof temperature,it might not be detected The W+TirOu compoundspresent an analogous as such in the heat capacitymeasurements, but the picture; thermodynamic data are summarizedin solidwould not havemuch zeropoint entropy.The Table 3. CoTizOu,FeTirOu, and MgTizOuare stable plus kineticsof suchan orderingprocess probably cannot only at high temperature,the assemblagerutile be estimatedon the basisof currentknowledge. Last- MTiOs being stable at low temperature.The com- ly, a systematicerror is possiblein the third law en- pounds NiTirOu and MnTizO6are apparentlynever tropy of FezTiOu(or evenof FezOgor TiOr). The heat stablewith respectto MTiOsand TiOr, but estimates capacityof FezTiOuwas measured only down to 50 K of their freeenergies of formationhave been obtained (Todd and King, 1953),and the extrapolationto 0 K from the thermodynamicsof extensivesolid solutions (Evans and may produce greater uncertainty than previously in some ternary ,4O-BO-TiO2systems probably thought (Waldbaum, 1974, personal commu- Muan, l97la, b). The instabilityof NiTirOu nication). In addition, problems associatedwith stems from the loss of ligand field stabilization the magneticentropy of oxidescontaining transition of Ni2+ in the distorted octahedral sites of the metalions with partially filled d shellsmay causethis pseudobrookitestructure; a similarfactor is probably discrepancy. partly responsiblefor the instability of NiSiO' and Regaidlessof the detailsof the third law entropyof NiGeOgpyroxenes (Navrotsky, l97lb). Theinstabili- FerTiOu,this work has clearlyestablished a positive ty of MnTirOusuggests that Mn2+may be somewhat enthalpyof formation for pseudobrookitefrom the too large for sitesin the pseudobrookitestructure. MgTLOu oxides,which is compensatedby a positiveentropy of The existenceof CoTirOu,FeTizOs, and with formation,resulting in the stabilityof FerTiO' at high as high temperaturephases only is consistent temperatureonly. positive entropies of formation from the oxides, coupledwith enthalpiesof formationclose to thoseof M3+2TiO5 the corresponding ilmenite analogs. Since both The known thermodynamic data for some other FeTizOuand MgTizOuhave somewhat disordered cat- compounds with the pseudobrookite structure are ion distributions(Lind and Housley,1972; also, see summarized in Table 3. Two other M3+zTiOu below),and sincethe equilibriumdegree of disorder 252 ALEXANDRA NAVROTSKY

Test-B3. Thermodynamicsof Formationof Compoundswith Pseudobrookite Structurefrom the ComponentOxides

ac; (8t K) an! (at r) ^s; (et K)

kcal nol-l kcal ool-1 cel K-l nol-l Deconpoeltlon

Fe2TlO5 o (838 ! 15)8 +1.99 ! 0.15 (96s)b + 2.4c Fe2O3 + TlO2 below very aoalll 838 K8

FeTlrO, -4.3 ! o.l (1s23)d PcTlO3 + T1O2 below 1413 r l0 Ka

hlsh Tlro, -18.0 r r.5 (1304)e low f13O5,T,500 Kf ileT12oS -6.E ! 1.0 (r573)l t4.45 (298)rh r+ 1.5r1 HgTlO3 + T1O2 bclo -7.3 t 1.0 (1673)J ^,823 K

CoTtrO, -3.9 (1473)r! CoT103 + TLO2 belou 14130 c 5 Ka

A12T1O5 endothernlcl a-Al2O3.+ fl02 at L273 Kr

G82T105 eadothernlcl B-Ga2O3-+TtO2 at L273 KI

(Nlr12o5) -2.0 ! 0.4 (1673)J rmetable reletlve to NlTlO,r + T1O, at 811 TJ

(hrr2os) -6.3 r 0.8 (1523)k rrnstable relatlve to l{nTlO3 + T1O2 at all TE e. s. Heggcaty and D. yearbook LLndalcy, caraegte roet. $, 247 (L97o). s. Kcsaoa and D. 8. Ltadaley, pcrsooal [email protected]. b. Thla york. c. C€lculeted fro a. aod b., AS? - +1990/838. d. R. E. Johnco!, E. tloernaon, ald A. lruan, ancr. Jour. sci. 27!,279 (1971). e. Calculated fron data of R. R. lterlltt and B. G. Eydc, Phtf-Tienr. noy. 3oc. Loadon L214,627 (1973) for reactl.on TIOZ + TlrO, - Tt3O5. f. G. Asbrtnk and A. Magoell, Acta-Chen.-Scaod. 21, l,9l7 (L967). g. B. Brezny and A. !ft,rao, Therrochb. Acta e. 10f(197f). h. K._K. Belley, s. s. Todd, and E. c. Krnglu.s. Bur. ld.aea Repr. rnv. 5059 (1954). Eydrof luoric acl.d ealortnatry. 1. Eatlnated fron g. and h., asE - r-4450-(-6s00'Dll573, Bs. t€xt for dl.gcuaalon. L. G. Evane J. ad A. !ftr8n, Ttrcnochtl. Acta ?, tZI- (1971). L. L. G. Evans and A. !tuaa, Ther.uochh. Acts 2,277 (1971). 1. G. Bayet, J. Le3s Comonl{etala 24, L29 (1t7f), deconporl,tton tcnperaturea do not represeot rgvertala. m. B. Brezay and A. ltuan, J. ftrorg. ouc1. Chen. 3.L, 649 (1969). is expectedto increasewith increasingtemperature, it disorderin this material,then of unknownstructure, seems clear that these materials are "entropy could causeits stabilizationat high temperature. stabilized" in the samesense as are copper spinels We haveseveral independent means for estimating (Navrotskyand Kleppa, 1968). the entropychange, ASo, associatedwith the decom- In their early calorimetricwork on titanates,Kel- position: ley,Todd, and King (1954)found that the formation MTizo (pseudobrookitestructure) of MgTirOu from TiO, and MgTiO, is endothermic s by 1.0 kcal at 298 K. They noted that structural - MTiOs(ilmenite structure) + TiO, (rutile) (4) FORMATION OF PSEUDOBROOKITES AND FeTi,Ou-ThOuSOLID SOLUTIONS 253

The preliminarydata of Kessonand Lindsley(1974, Cation Distributionsin ABzOoCompounds unpublished)for the decompositionof karrooite to and in the FeTirOu-TisOs geikelite and rutile gave a pressure-temperature Solid SolutionSeries slope,0Pl0 7" of about 23.4bar/deg. SinceAV for FezTiOu,Al2TiO u, GazTiO, this transitionis -5.39 cc/mol at atmosphericcon- Re-evaluationof the crystallographicdata favors ditions (volumedata from Robie and Waldbaum, an inversecation distributionfor FezTiOo(Lind and 1968,and Lind and Housley,1972),we can estimate Housley, 1972).The positiue entha$ies of forma- from the Clausius-Clapeyronequation that A^So= lion of these compounds and their instability at 23.4x -5.39 X 0.0242: -3.0 cal K-'. Evansand low temperatureis consistentwith substitutional Muan (l97la) have determinedthe standardfree disorder. Very probably these materials have energiesof formation from the oxides of MgTizO" intermediatecation distributions, which would and MgTiO3to be -7.3 + 0.8and -5.1 + 0.5kcal be temperature dependent.The configuration of mol-1 at 1400"C,respectively. Thus for decomposi- greatest entropy, the random distribution (Mr,t- tion Reaction(4), AGo : 0 at 550'C and t2.2 kcal T\/s)&(M4/sTi273)6;O6,may be approachedat high at l400oc. Since(OAG'/07), = -ASo. we can temperature. This point has been discussedin calculateASo = 2200/850: -2.6 cal K-'. Lastly, detail for spinels (Navrotsky and Kleppa, 1967b; the calorimetricdata of Kelley, Todd, and King Waldbaum,1973), and analogous arguments hold for (1954)give an enthalpyfor Reaction(4) at 823K of the pseudobrookitestructure. Al2TiOu may be a -5.85 + 4.55: - 1.30kcal. Then the entropy change better candidate for crystallographicstudy than is -1300/823: -1.6 cal K-'. This lastcalculation FezTiOubecause of the larger differencebetween the does not take into accountany configurationalen- two cationsin scatteringfactors for X-rays' tropy. Similar calculationsare possiblefor CoTizOu.The MgTi2Ou,CoTizOa, and FeTizO' decompositiontemperature at one atmosphereoc- X-ray diffractionstudy of MgTizOequenched from cursnear I135"C, and Breznyand Muan (1969)give 1500"C(Lind and Housley,1972) has given a cat- valuesof free energiesof formation such that for ion distribution (Mgo..roTio.rr.)n"(Mgo.r,"Ti,.rrn)rrOr. Reaction(4), AG" : -0.1 kcalat ll00'C and *0.8 In terms of the simpledisordering model appliedto -900/200 -4.5 kcal at 1300'C.This givesASo : : spinelsby Navrotskyand Kleppa(1967b),.this cor- cal K-t. respondsto a degreeof disorder,x, of0.3l6 (x : 0 Johnson,Woermann, and Muan (1971)give the for a normal,2/3 for a random,and I for an inverse freeenergies of formation from the oxidesfor FeTiOg cationdistribution). The interchangeenthalpy for the andfor FerTiOuas -4.3 t 1.0kcal mol-1at 1300'C. reaction Since decompositionto ilmenite plus rutile occurs (Mg)n"+(Ti).r: (Mg)61+(Ti)." (5) below I140"Cin thissystem (Haggerty and Lindsley, given 1970),a value close to zero for Reaction (4) at is then by per- 1300"Cis reasonable.However, the datado not -AHiat - (6) mit an estimateof the entropychange for this reac- RT tion. with Allr,t : *8.6 kcal mol-r. This value has -1.6 The calculatedvalues of ASo rangefrom to previously been calculated by Waldbaum (1973).'z -4.5 cal K-r for decompositionReaction (4) for different metals.In view o[ the large uncertainties '!The model usedby Navrotskyand Kleppa(1967b) for cation arising both from uncertaintiesin the equilibrium distributionsin spinelsand that discussedby Waldbaum(1973) for pseudobrookites,and measurementsand calorimetricdata and from the nonconvergentdisordering in melilites, spinels,are essentiallyequivalent if one assumesthe "lattice en- assumptionsin the calculations,this scatteris not at tropies"of Navrotskyand Kleppa( = the "calorimetricentropies" all surprising.Rather, the data are quite consistent of Waldbaum)to be independentof the degreeof order. Wald- and point to a probablevalue of ASo of -3 t I cal baum'sorder parameter,Z, is equalto l-x, where.x is our degree K-'. This valuelooks very reasonable in light of the of disorder.His calculationof the differencein enthalpybetween distributionsfor probable configurationaldisorder in the pseudo- normal and inverse(ordered and antiordered) MgTirOu from the site population of a samplequenched from brookite structure, although it is not altogether | 500"Cis entirelyequivalent conceptually to our calculationof in- too large in magnitudeto be causedby lattice terchangeenthalpies for somethirteen spinels from their cation vibrational effectsalone. distributions,each known for one temperature. 254 ALEXANDRA NAVROTSKY

For FeTirOs Quenchedfrom l400oC, Grey and However, the bulk composition, charge balance, and Ward(1973) give a distribution,determined by M6ss- site balance require that bauer spectroscopy,(Feo.rroTio.rro)r"(Feo.rroTir.rr)rrOu. Fe2+o+Fe2+s:l-x (11) If FeTirOuhas all iron presentas Fe2+and all titanium as Tia+,which seemsreasonable, then Afl1o1 Tie+A+ Tig+":2t (r2) : 9.2 kcal mol-', catculatedfrom Equation (5). Ti4+r + 7.ir+" -_ 2 - x (13) However,Grey and Ward suggestthat the true per- centof total iron in 4c sitesmay begreater than their and estimate because of difficulties in fitting the Fe2+o*Tia+A+Ti4+A+1 (14) M6ssbauerspectra. This would make AIIr,t more positivefor FeTirOr.This point is discussedfurther Fe2+6*Ti8+B+Tin+s=2 (15) below. Thus, for a given composition, x, there are only two Solid Solutionsin the Fer-,Ti2*,Ou Series independent site population parameters,swhich we The distribution of iron betweenA(4c) and B(8/) may pick as - siteshas been studied by Grey and Ward (1973)using F"'*, Xp",r^ and Ti3* a : X,1i"*^ M6ssbauerspectroscopy. Their data canbe subjected and we may re-write our equilibrium constantsas to thermodynamicanalysis using the simplemodel (l - x - Fe'*,n;11- Fe'*, - tit*r) appliedby Navrotskyand Kleppa(1967b) to spinels n':---F]ffir;) (16) and by Navrotsky(l97la) to (Mg,Fe)SiO,pyroxene solid solutions. and Considerthe solid solutionseries FeTirO6-TisOb. If (2x- Ti'*rXl_ r"'*a_:_lt'a) oneassumes that all iron is presentas Fe2+, and that x'- :_ -616 trz) the reducedtitanium is presentas Ti3+only, thenthe -l 1 rFl=-TrT solid solution seriesmay be written as Fe'+r-, The Mdssbauerdata of Greyand Ward (1973) give Tl3+2,Ti4+2-,Ou,where the mole fraction of TirOuis Fe'*n as a function of composition, x, for given by x. Callingthe 4c sitesA andthe8/sites.B, we FeTirOu-TirOssolid solutionsamples quenched from have, nominally, population six site parameters: 1400"C.For pure FeTirOu,Fez+, : 0.72from the Fe'* Ti3+A,Til+i, Fe'*r, Ti3+3,Tio+s, where each o, Mdssbauerdata and Fe2+t : 0.75from a neutron symbol standsfor the numberof molesof the given diffractionstudy. This gives, from Equation(16) with ion on the appropriatesite. For the distribution of (Ti'*o) : 0, and x : 0, Kr : 0.06and K, : 0.05, the threecations. Fe2+. Ti'+. and Ti8+over the two respectively.Knowing Kr, x, and Fe'*r, Equations sites,I and B, we can write two independentex- (16) and (17) can be treated as simultaneous change reactions which lead to two independent equationsin two unknowns,Ti3+, and Kr, andsolved equilibrium constants.For the first exchangereac- at eachvalue of x for a valueof Ti8+a(which will vary tions, with x) and of K, (which will remainconstant if the

Fe2+1 * Tit+a : Fe2+s * Tin+,a, 0\ modelholds and the dataare accurate). Because Grey and Ward experiencedsome difficulty in fittirrg their with M6ssbauerdata at compositionsnear FeTi2Ou,they suggestthat their value of Fe2+1for pure FeTi2Ou X'""* t X'"' ^ K' : (8) may be somewhat underestimated(leading to an Xp.or^Xai.r" overestimateof Kr). (From their data a graph of the where each X is the mole fraction of a given ion on percentageof the total iron on the 4c sitesuJ com- the specified site, and, in the simple model, ap- positionshows a maximum;but a plot of the fraction proximates the activity. For the second exchange of the 4c sitesoccupied by Fe us compositiondoes reaction, not; seeFig. l.) Thus,some uncertainty exists in the valuechosen for Kr; consequently,we havecalculated Ti8+, + Tir+a : Ti3+B+ Ti4+A, (9) the valuesof Tis+and of Kz for valuesof K, from 0.06 with to 0.01.The resultsof someof thesecalculations are shown in Table 4. Whereasthe calculatedvalues of (10) 3 Of the 5 equations I l-15, only 4 are independent FORMATION OF PSEUDOBROOKITESAND FeTizOu-ThOESOLID SOLUTIONS 2s5

K, show some scatter,they do not appearto vary systematicallywith composition,x. We may then take the averagecalculated value of Kr, and use it with the assumedvalue of K, to then calculateback valuesof Fe2+,a(and Ti8+r) as functions of composition, x. Using Equations(16) and (17), curvesof Fe'+aus x for severalvalues of K, (and Kr) aregiven in Figure l, togetherwith the experimentaldata points. In addition,the limits correspondingto a completelyordered distribution of iron corresponding to "normal" FeTirOuand of a randomdistribu- tion, with one third of the iron on the 4c site,are shown.We canconclude the following.On the iron- poor side (x)-0.5), the calculateddistribution of iron is rather insensitiveto the actualvalues chosen for K'. To discriminateamong valuesof Kt which vary by a factorof six (from 0.01to 0.06),one must look at the pointswith x (0.5. For thesedata, and allowingthat the point at.rr= 0 shouldlie somewhat 00 0.2 0.4 0.6 0I 1.0 higher than 0.72, one seesthat curvesfor K, from FIc l. Variation of fraction of4c sites occupied by iron (Fe'+a) 0.04to 0.01approximate the data.We do not believe with mol fraction TirOu, x. Curves, in order of increasing Fe'+1 that a statisticalfit to the datawould add muchinfor- are for: (l) random distribution, K, = 1; (2) Kr:0'060, K, = mation,and would sayrather that the experimental 0.187;(3)K1:0.050,,(z:0.135; (4)K' = 0.040,K,= 0.093;(5)(' : 0.030,K, : 0.061;(6) K' : 0.020,K2: 0.035;(7) K': 0.010,r(, data are consistentwith a value of & of 0.025 (no -0.02) : 0.015;(8) completelyordered distribution Fe'+ on 8/sites), (+ andK": 0.055(+0.04). Using these values Kr : 0. Filled circles are cation distributions determined from and Equations(16) and (17), one can calculate Mmagnetite;that is, are

Trslr 4. CalculatedValues of EquilibriumConstant for Exchange Reactionsin FeTLOu-TisOuSolid Solutions

K, = 0.06 K, = 0.04 K, = 0.02 )+ ?+ (rii+)calc (r1;')calc v (rli+) r,2 (rei' ) ''2 "2 calc "*n o.27 0.534 0.223 0.232 0.296 0.089 0,377 0.024 0.34 0.488 0.27r 0.258 0 .343 0.112 0.423 0.035 0.40 0.453 0.297 0.313 0.371_ 0.L42 0.454 0.047 0.s0 0.361 o.437 0.201 0 .498 0.105 0.565 0 .040

0.68 0.201 0.678 0.102 0.715 0.051 0.756 a.027 0.75 0.170 0.689 0.150 o.732 0.089 0.779 0.039 0.80 0.132 0.743 0.134 0.782 0.081 0.823 0.036 0.90 0.061 0.845 0.106 0.874 0.066 0.906 0.031

Krav = Krav= Krav= 0.187 t 0.093r 0.035t 0.o77 0.026 0 .007 256 ALEXANDRA NAVROTSKY

the Ti3+and Tin+ions distinguishable, and can TirOu -, AND_ (1971b)Activity_composition relations of solid haveanomalously high electricalconductivity due to solutions and stabilitiesof manganeseand nickel titanates the hopping of electronsover sitesin the 8/ sublat- at 1250'C as derived from equilibria in the systems MnO-CoO-TiOz and MnO-NiO-TiOr. Thermochim.Acta, 2, tice? 277-292. = With Kr 0.025and Kz: 0.055,we calculate,for Gnny,I. 8., lNo J. Weno (1973)An X-rayand Mdssbauer study the exchangeReactions (l) and (2), A(': +13.0 of the FeTi,O6-TirOusystem. J. Solid StateChem.7,300-30'7. (f 5) kcal and AG": +10.2(+5) kcal.In thesimple Hlccrnrv, S. E., elo D. H. Lrroslrv (1970)Stability of the pseudobrookite modelof Navrotskyand Kleppa(1967b) these values (Fe2TiOr)-ferropseudobrookite(FeTirOu) series.Carnegie Inst. llash. YearBook,6t,24'l-249. arejustthe interchange enthalpies for theabove reac- Hlnnr-tN, M. (1958) The structure of the composition, tions. TiOr-AlrOs,in comparisonwith the pseudobrookite.Bull. Soc. The modelused has assumed (a) that themolar in- Chim.France, 195E, 1559-1566. terchangeenthalpy is independentof composition JouNsol,R. 8., E. Wornumtu,ero A. MulN (1971)Equilibrium andof degreeof disorder,and (b) that theentropy studiesin the system MgO-"FeO"-TiO,. Am. J. Sci. 271, of 278-292. forming the solid solutionis simply the changein KELLEv,K. K. (1960)Contributions to the data on theoretical configurationalentropy, calculated from the cation metallurgyXIII, high temperatureheat content and entropy distribution.aIn particular,this neglects effects of the data for the elementsand inorganiccompounds. U. S. Bur. changefrom orthorhombicto monoclinicsymmetrv Mines Bull.5M. _, with increasingTi3Ou content in the solid solution ANDE. G. KrNc (1961)Contriburions to the data on theoreticalmetallurgy series. SiV; Entropiesof the elementsand in- Sincethis structuralchange causes changes in organic compounds. U.,S.Bur. Mines Bull 592. the metal-oxygenbond lengths and angles (Grey and -, S. S. Tooo, lr.ro E. G. KrNc (1954)Heat and freeenergy Ward, 1973),its neglectin thethermodynamic model datafor titanatesof iron andthe alkalineearth metals. U.S. Bur. is clearlyan oversimplification.However, the present Mines Rep. Inuest. 5059. data do not seemto warrant equationsusing a larger Lno, M. D., lNo R. M. Housr-sy(1972) Crystallization studies of lunar igneousrocks; crystalstructure of syntheticarmalcolite. numberof parameters,and the modelused appears Science,115,521-523. adequateas a first approximation. Nevnorsrv, A. (1971a)The intracrystallin€cation distribution and the thermodynamicsof solid solution formation in the Acknowledgments systemFeSiO3-MgSiOr. Am. Mineral. 56, 201-211. Nlvnorsxv, A. (l97lb) Thermodynamicsof formationof the This work wassupported by NSF Grant GH 39767. Stimulating silicatesand germanatesof somedivalent transition metals and discussionswith S. Kesson,D. Lindsley,and D. Waldbaum areap- of magnesium.J. Inorg. Nucl. preciated. Chem.33,4035-4050. -, ANDO. J. Kr_rppe(1967a) Enthalpy of the anatase-rutile transformation.J. Am. Ceram Soc. 50r 626. References -, AND- (1967b)The thermodynamicsof cation dis- Beven,G, (1971)Thermal expansion characteristics and stability tributionsin simplespinels. "/. Inorg. Nucl. Chem.29,270l-2'114. of pseudobrookite-typecompounds, Me.O.. ,/. Less Common -, AND- (1968)Thermodynamics of formationof sim- Metals. 24. 129-138. ple spinels.J. Inorg. Nucl. Chem. 30, 479498. BREzNy,8., ,qNoA. MueN (1969)Phase relations and stabilitiesof Prur-txc, L. (1930)The crystalstructure of pseudobrookite.Z. compoundsin the systemCoO-TiOr. J. Inorg. Nucl. Chem3l, Kristallogr. 73, 97-l 12. 649-655. Rosrr, R. A., lNo D. R. WAr-osluu (1968)Thermodynamic EvrNs, L. G., rNo A. MurN (l97la) Activity-composition propertiesof minerals and related substancesat 298.15'K relationsand stabilitiesof end-membercompounds in the system (25.0"C)and oneatmosphere (L013 bars)pressure and at higher MgO-NiO-TiO, in contactwith metallicnickel at 1400"C.Ther- temperatures.U.S, Geol. Suru. Bull. 1259, mochim.Acta, 2. l2l-134. Tono, S. S., rlo E, G. KrNa (1953)Heat capacitiesat low temperaturesand entropiesat 298.16"Kof titanomagnetiteand ferric titanate.J, Am. Chem.Soc. 75. 4547-4551. r This model is again cssentiallyidentical to that used by Werouuu, D. R. (1973) The configurationalentropies of Waldbaum(1973) for the melilitesolid solutionseries, when the CarMgSiOr-CazSiAlrOzmelilites and relatedminerals. Contrib. assumptionthat the as : 0 is used. waldbaum wrote his Mineral petol. 39, 33-54. equationsfor the freeenergy of a phaserelative to the elementsor the oxides;we write ours for the freeenergy ofmixing betweenend Manuscript receiued,June 10, 1974; accepted members. for publication,Nouember 8, 1974.