American Mineralogist, Volume 79, pages485-496, 1994 Hematite-ilmenite (FerOr-FeTiO3) solid solutions:The effectsof cation ordering on the thermodynamics of mixing N.clqcy E. BnowN,* Ar,nxnNona Navnorsry Department of Geological and GeophysicalSciences, Princeton University, Guyot Hall, Princeton, New Jersey08544-1003, U.S.A. Ansrucr Enthalpies of reaction from lead borate drop solution calorimetry at 1057 K show that endothermic mixing dominates hematite-ilmenite solid solutions with compositions from x,,- : 0 to 0.65, whereas exothermic mixing dominates the compositional regionfrom x,,- : 0.65 to l The measuredenthalpy of mixing is interpreted as arising from two contri- butions: a positive enthalpy of mixing due to repulsive interaction energies (presumably within a hexagonallayer) and a negative enthalpy of mixing due to attractive interaction energies betweenlayers (i.e., the driving force for ordering). Enthalpiesof reaction have also been measured on compositions from x,,- : 0.6 to 0.85 that have different measuredcation distributions. Within the resolution of the measure- ments (+3 kJ/mol), the enthalpies of isocompositional sampleswith varying cation dis- tributions are indistinguishable. This observation supports significant short-rangeorder. As the driving force for ordering increases with increasingilmenite content, progression from short-rangeorder in the more Ti-poor solid solutions to long-range order in the Ti- rich compositions occurs. The progression from short-range to long-range order is ex- pressed by increasinglynegative enthalpiesof mixing. Free energiesof mixing have been determined independently from the measured tie lines betweennonstoichiometric spinel and the sesquioxide phase at 1573 K. This requires modeling the activity of Fe2Ooin the nonstoichiometric spinel solid solutions coexisting with hematite-ilmenite solid solutions. Combination of these free energies of mixing and the experimentallydeterminedenthalpies of mixing suggest that the entropy of mixing is far lesspositive than that predicted by the maximum configurational entropy implied by the measuredsite occupancies. These results also support significant short-rangeorder. Irvrnooucrrox Iron titanium oxides of the FerO,-FeTiO, solid solu- tion seriesimpact the evolution of igneousand meta- morphic rocks because of the largenumber of subsolidus equilibria in which they are involved (Haggerty,1976). Their ubiquitous occurrence with Fe.Oo-FerTiOo solid solutions allows petrologiststo estimate the temperature and fo, conditions under which these mineral assem- blagesformed. In the classical approach for determining equilibrium T and fo. conditions, activity-composition relations of Fe.Oo-FerTiOo and FerO.-FeTiO. solid so- lutions are approximated by subregular solution (Mar- gules) formalisms. Data for selected chemicalequilibrium reactions at a number of temperatures,including ilmen- ite-hematiteand magnetite-ulviispinel pairs, are then used to determine empirically the Margules parameters (Pow- ell and Powell, 1977; Buddington and Lindsley, 1964; Spencer and Lindsley,l98l; Andersen and Lindsley, 1988; Ghiorso, 1990). Because of the empirical nature of this approach, the most common complications affecting the thermody- namic properties of these solid solutions are ignored. These complications include cation ordering in both FerOo-FerTiOo and FerOr-FeTiO. solid solutions,non- stoichiometry in FerOo-FerTiOo solid solution at T > I173 K (Taylor, 1964; Schmalzried, 1983; Webster and Bright, 196l; Senderov et al., 1993), and chemical im- purities. Thus, the calculated parameters have little phys- ical meaning with regard to the atomic interactions that causemacroscopicthermodynamic properties to vary as a function of P, T, and X. The most rigorous attempt to include the effects of cation ordering within the con- straints of a simple chemical model has been made by Ghiorso (1990). His resultsshow the importanceof in- cluding suchbehavior. Microscopically basedmodels, on the other hand, take into account the apparently cooperativeand higher order nature ofthe order-disorder process. The most general of such approachesinvokes the cluster variation method (CVM), in which interactions out to several nearest neighbors are considered (Burton and Davidson, 1988; Burton, 1985; Burton,1984;Burton and Kikuchi, 1984; * present address: Norton Company,I New Bond Street Kikuchi, 1977). In this way, structural and thermody- MS420-601, worcester, Massachusetts 01615-0008, U.S.A. namic parameters are linked rigorouslyto the free ener- 0003-o04x/94l0s06-{485$02.00 485
Transcript
American Mineralogist, Volume 79, pages 485-496, 1994
Hematite-ilmenite (FerOr-FeTiO3) solid solutions: The effects of cation ordering on thethermodynamics of mixing
N.clqcy E. BnowN,* Ar,nxnNona NavnorsryDepartment of Geological and Geophysical Sciences, Princeton University, Guyot Hall, Princeton, New Jersey 08544-1003, U.S.A.
Ansrucr
Enthalpies of reaction from lead borate drop solution calorimetry at 1057 K show thatendothermic mixing dominates hematite-ilmenite solid solutions with compositions fromx,,- : 0 to 0.65, whereas exothermic mixing dominates the compositional region from x,,-: 0.65 to l The measured enthalpy of mixing is interpreted as arising from two contri-butions: a positive enthalpy of mixing due to repulsive interaction energies (presumablywithin a hexagonal layer) and a negative enthalpy of mixing due to attractive interactionenergies between layers (i.e., the driving force for ordering).
Enthalpies of reaction have also been measured on compositions from x,,- : 0.6 to 0.85that have different measured cation distributions. Within the resolution of the measure-ments (+3 kJ/mol), the enthalpies of isocompositional samples with varying cation dis-tributions are indistinguishable. This observation supports significant short-range order.As the driving force for ordering increases with increasing ilmenite content, progressionfrom short-range order in the more Ti-poor solid solutions to long-range order in the Ti-rich compositions occurs. The progression from short-range to long-range order is ex-pressed by increasingly negative enthalpies of mixing.
Free energies of mixing have been determined independently from the measured tielines between nonstoichiometric spinel and the sesquioxide phase at 1573 K. This requiresmodeling the activity of Fe2Oo in the nonstoichiometric spinel solid solutions coexistingwith hematite-ilmenite solid solutions. Combination of these free energies of mixing andthe experimentally determined enthalpies of mixing suggest that the entropy of mixing is farless positive than that predicted by the maximum configurational entropy implied by themeasured site occupancies. These results also support significant short-range order.
Irvrnooucrrox
Iron titanium oxides of the FerO,-FeTiO, solid solu-tion series impact the evolution of igneous and meta-morphic rocks because of the large number of subsolidusequilibria in which they are involved (Haggerty, 1976).Their ubiquitous occurrence with Fe.Oo-FerTiOo solidsolutions allows petrologists to estimate the temperatureand fo, conditions under which these mineral assem-blages formed. In the classical approach for determiningequilibrium T and fo. conditions, activity-compositionrelations of Fe.Oo-FerTiOo and FerO.-FeTiO. solid so-lutions are approximated by subregular solution (Mar-gules) formalisms. Data for selected chemical equilibriumreactions at a number of temperatures, including ilmen-ite-hematite and magnetite-ulviispinel pairs, are then usedto determine empirically the Margules parameters (Pow-ell and Powell, 1977; Buddington and Lindsley, 1964;Spencer and Lindsley, l98l; Andersen and Lindsley, 1988;Ghiorso, 1990).
Because of the empirical nature of this approach, the
most common complications affecting the thermody-namic properties of these solid solutions are ignored.These complications include cation ordering in bothFerOo-FerTiOo and FerOr-FeTiO. solid solutions, non-stoichiometry in FerOo-FerTiOo solid solution at T >I173 K (Taylor, 1964; Schmalzried, 1983; Webster andBright, 196l; Senderov et al., 1993), and chemical im-purities. Thus, the calculated parameters have little phys-ical meaning with regard to the atomic interactions thatcause macroscopic thermodynamic properties to vary asa function of P, T, and X. The most rigorous attempt toinclude the effects of cation ordering within the con-straints of a simple chemical model has been made byGhiorso (1990). His results show the importance of in-cluding such behavior.
Microscopically based models, on the other hand, takeinto account the apparently cooperative and higher ordernature ofthe order-disorder process. The most general ofsuch approaches invokes the cluster variation method(CVM), in which interactions out to several nearestneighbors are considered (Burton and Davidson, 1988;Burton, 1985; Burton,1984; Burton and Kikuchi, 1984;
* present address: Norton Company, I New Bond Street Kikuchi, 1977). In this way, structural and thermody-MS420-601, worcester, Massachusetts 01615-0008, U.S.A. namic parameters are linked rigorously to the free ener-
0003-o04x/94l0s06-{485$02.00 485
486 BROWN AND NAVROTSKY: HEMATITE-ILMENITE SOLID SOLUTIONS
Trele 1. Site-occupancy data for specified compositions andannealing temperatures
Anneal- Moleing f fraction(K) FeTiO" Ntn ," Nr*
" Ne*. e N-,.- o Nre-,a Nr*-^
tion FeTiO.) were annealed at various temperatures be-low the order-disorder transition to vary their cationdistributions. The composition of each phase at each an-nealing temperature was determined by WDS micro-probe analyses, lattice parameters, and wet chemicalanalyses. The results showed the compositions to be ac-curate to within + 1.0 mo10/0. The saturation magnetiza-tion extrapolated to 0 K was used to determine cationdistributions for each annealing temperature (Table l).More detailed discussion of the synthesis and analysisprocedures can be found in Brown et al. (1993).
Calorimetric protocol
The purpose of the calorimetric study was to devise athermodynamic protocol that converted a sample ofknown structure and oxidation state into a well-defineddissolved final state in the lead borate solvent commonlyused for oxide-melt solution calorimetry (Navrotsky,1977). Because both the degree of order and oxidationcan change rapidly in the solid phase at high temperature,the starting state was taken as the sample at room tem-perature, which was then dropped directly into the sol-vent at high temperature. In such a drop-solution calori-metric experiment, the total measured enthalpy containscontributions from heat content, dissolution, and any ox-idation or reduction in the solvent. No high-temperatureequilibration is required, and thus the ambiguity ofthe sam-ples' oxidation state prior to dissolution is eliminated.
The appropriate calorimetric solvent was determinedfrom preliminary drop-solution experiments with ilmen-ite in three solvents: sodium molybdate (Navrotsky, 1977);alkali borate (Takayama-Muromachi and Navrotsky,1988); and lead borate (Navrotsky, 1977). The sodiummolybdate solvent showed incomplete dissolution after120 min at 973 K. The alkali borate solvent showed pre-cipitation of a sodium titanate by optical examinationsand X-ray powder diffraction. Experiments in lead borateat 1057 K showed complete dissolution, and X-ray pow-der diffraction and optical examinations indicated noPbTiO3 precipitation, which had been a problem in ear-lier experiments using TiO, at 973 K (Navrotsky, 1977).
The final state of Fe after dissolution in 2PbO-BrO.solvent was determined by weight gain experiments withsolid solutions and mechanical mixtures having compo-sitions between FerO, (all p.:+; and FeTiO, (all Fe'?* )under conditions of both static air and flowing Ar. TheAr was purified by passing the gas through a Ti purifier(R. D. Mathis GP-100 inert gas purifier) that has beenshown to produce an fo. of < 10-6. Each weight gain ex-periment was carried out under conditions that mimicthe calorimetric experiments as closely as possible. Thefollowing precautions were taken: the surface area of themelt-air interface was kept the same; sample to solventratios were restricted to the same range as for a series ofcalorimetric experiments; experiment durations were 90min for experiments in air and 70 min for experimentsunder flowing Ar (as in calorimetry); sample mass wasmaintained within the same ranse as that used in the
Note.'errors in site occupancies range from +0 01 to +0.024. They havebeen determined by including the errors arising from the linear extrapo-lations of the J" vs. f data to 0 K and the 1 mol7" errors in composition.Data are from Brown et al. (1993).
gies of mixing. There are, however, two drawbacks tothese types of models. The first is that they are mathe-matically cumbersome and do not lead to closed-formequations for the mixing parameters. The second is thattheir energetic parameters cannot be uniquely evaluatedfrom phase equilibria alone, and other direct estimates ofthe energetics ofordering are not available.
Thus, a major gap remains between the two approach-es. The formalism based on simple polynomials and em-pirical calibration, though petrologically useful, is hard toreconcile with the microscopic complexity of order-dis-order and may lead to difficulties when extrapolating out-side the P, T, and X rcnge of its determination. The mi-croscopic approach, without more detailed structural andenergetic data to constrain its parameters, remains morea conceptual framework than a useful description. Thiswork is a first step in bridging that gap.
In a previous paper (Brown et al., 1993), we reportedthe preparation and magnetic characleization of a seriesof ilmenite-hematite solid solutions, which leads to a de-scription of the degree of long-range order of quenchedsamples. This work reports a drop-solution calorimetricstudy of the same samples, which provides enthalpies ofmixing at room temperature of the same quenched sam-ples. Combination of the structural and calorimetric datawith phase equilibrium studies leads to insights into theshort- and long-range atomic interactions that affect thethermodynamic mixing properties of hematite-ilmenitesolid solutions.
PnocrnunrsSamples
The phases used in this calorimetric study spanned theFeTiOr-FerO. solid solution series in composition andwere synthesized under a controlled atmosphere aL l573K. Relevant compositions (i.e., solid solutions with com-positions from Jr,,- : 0.6 to 0.85, where x,,- : mole frac-
BROWN AND NAVROTSKY: HEMATITE-ILMENITE SOLID SOLUTIONS 487
calorimetric experiments (5-15 mg); and temperature waskept at the calorimeter temperature of 1057 K. Pt cruci-bles containing lead borate solvent were brought to con-stant weight at 1057 K under the appropriate atmospher-ic conditions. Samples were also brought to constantweight at 383 K and ranged from well sintered to poorlysintered.
Figure I shows the percent weight gain vs. compositionfor all experiments. The dashed line indicates the calcu-lated weight gain anticipated for complete oxidation ofFe2+ to Fe3* in the solvent for each composition acrossthe solid solution series. The percent ofoxidation ofFe2*to Fe3* in the solvent is then calculated relative to thismaximum weight gain line. Each data point representsone dissolution experiment, and the error bars representthe propagated cumulative weighing errors (- +0.06 mg).
For the solid solution experiments conducted in staticair (solid circles in Fig. I ), the total range in the calculatedpercent oxidation lies between 7l and 930/0, with the ma-jority of experiments falling between 84 and 930/0. Muchofthis variation in the calculated percent oxidation arisesfrom the extreme sensitivity of this percentage to verysmall (+0.01 mg) cumulative weighing errors (e.g., - +60/0at xir- : 0.2; -+1.5o/o at 4'- : 0.85). A l ine fitted to thestatic air data for compositions between x,,- : 0.2 and0.85 indicates that, after a 90 min experiment, 88 + 5.20loof the Fe'?* in the sample oxidizes to Fe3*. For solid so-lutions with x,,- = 0.85, however, the degree of oxidationdoes not reach this limit in 90 min. For ilmenite, thepercent oxidation reached after 90 min is only -650/o.By-300 min, the percent oxidation reached -880/0. We in-fer from this that the oxidation process is controlled bythe diffusion ofO into the lead borate solvent at the sol-vent-air interface. Similar conclusions were reached byZhou et al. (1993) for the Cu+ + Cu2* reaction.
Mechanical mixtures (open circles in Fig. l) of hema-tite and ilmenite indicate a final oxidation state reflectinga weighted average ofthe oxidation states ofthe two end-members. The difference between the final oxidation stateof ilmenite (i.e., 650/o oxidation of Fe2* to Fe3* after 90min) and that for solid solution with x'* < 0.85 (i.e.,88o/o oxidation of Fe2+ to Fe3* after 90 min) causes thefinal oxidation state for the mechanical mixture to bedifferent from that for solid solutions. Thus the amountof oxidation occurring during a calorimetric experimentis not the same for solid solutions and mechanicalmixtures, making interpretation of the calorimetric re-sults from experiments in static air dependent on the en-thalpy of oxidation. For solid solutions of a differentcomposition, there is no inherent reason why this en-thalpy of oxidation should be constant for all composi-tions, and the large errors associated with the amount ofoxidation determined by weight gain analysis results inuncertainties in the enthalpy of oxidation of - l5-20o/o.
The experiments in Ar (solid stars in Fig. l), on theother hand, show no change in weight for all composi-tions across the solid solution series after 70 min. How-ever, this is only true when the lead borate solvent has
0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0
MOL FRACTION FeTiO3Fig. 1. Percent weight gained in lead borate solvent at 1057
K because of oxidation of FerOr-FeTiOr solid solutions and me-chanical mixtures. The dashed line represents the percent weightgain expected for complete oxidation ofFe2* to Fer* in the sol-vent. The solid circles represent 90-min experiments with FerO.-FeTiO, solid solutions in static air. The open circles represent90-min experiments with mechanical mixtures of Fe.O. andFeTiO, in static air. The solid stars represent 70-min experi-ments with FerO.-FeTiO, solid solutions in flowing Ar.
been equilibrated for - 15 h under flowing Ar at the cal-orimeter temperature prior to dissolution of the samples.Calorimetric experiments conducted without this equili-bration time show small parasitic exothermic reactionsand a very slow return to the base line, suggesting thatthe sample was oxidized slowly. This suggests that thereis a small solubility of O in the solvent in equilibriumwith air that is greatly reduced when the solvent is equil-ibrated with flowing Ar. Once this O is removed fromthe solvent, no oxidation or reduction ofFe in the solventoccurs during the calorimetric experiment.
As a result of the above considerations, drop-solutioncalorimetric experiments were completed in a Calvet-typemicrocalorimeter (Navrotsky,1977) at 1057 K in 2PbO-BrO. solvent under conditions of flowing Ar, with over-night preequilibration of the solvent under Ar gas flow.Each experiment was 70 min in length. Solid solutionexperiments involved dropping small chunks (7-25 mg)of FerO.-FeTiO. solid solutions. Mechanical mixture ex-periments involved dropping two small chunks of theend-members together. This procedure eliminated the ex-tra heat effect attributable to any capsule. The samplesdissolved rapidly and completely. The high temperature(1057 instead of 973 K) and intimate contact of thedropped sample with the solvent (no Pt sample capsule)may have aided rapid dissolution and avoided localPbTiO3 saturation.
. No errors are indicated for the mechanical mixtures because eachvalue represents one experiment in which a chunk of both Fe2o3 andFeTiO. were dropped simultaneously. Thus, compositions are calculatedfrom the relative weights of each chunk.
*r Error represents 2 sd of the mean.1 Number in parentheses indicates the number of exDeriments.
C^ll-onrvrnrRrc RESuLTS
Observed enthalpies
The enthalpies of the drop solution of mechanicalmixtures and FerOr-FeTiO. solid solutions with varyingcompositions and degrees of cation order are listed inTable 2. For compositions from x,* : 0.6 to 0.85 withdiffering cation distribution, no measurable difference inreaction enthalpy was found for isocompositional sam-ples (Table 2); hence, only the average values are plottedin Figure 2. Excess enthalpies are determined by subtract-ing the measured enthalpies of the solid solutions fromthose of the mechanical mixtures. This difference repre-sents the enthalpy of mixing at 298 K of solid solutionshaving a structural state assumed to be characteristic oftheir quench temperatures. The enthalpies of disorderingare determined by subtracting the total measured enthal-pies for two solid solutions of the same composition butwith differing thermal hislories.
Enthalpies of mixing
Excess enthalpies for each solid solution are shown inFigure 3. Solid solutions with compositions from x,,- :
0 to 0.65 show endothermic enthalpies of mixing, where-as solid solutions with compositions from x'- : 0.65 toI show exothermic enthalpies of mixing. These resultsare consistent with the model of Burton and Davidson(1988), which incorporates dominantly positive (i.e., re-pulsive) intralayer Fe2+-Ti4+ interactions in hematite-richsolid solutions and dominantly negative (i.e., attractive,ordenng) interlayer Fe2+-Ti4+ interactions in ilmenite-richsolid solutions. However, other interpretations are notruled out; e.g., size effects result in the repulsive forcesthat drive phase separation, whereas attractive interlayerforces drive ordering. The data obtained can be used to
o o . 2 0 . 4 0 . 6 0 . 8 1 . 0
MOL FRACTION FeTiO3Fig. 2. Reaction enthalpy (kJ/mol) from drop-solution ex-
periments at 1057 K in flowing Ar vs. mole fraction ilmenite.The open circles represent the reaction enthalpies for individualexperiments with mechanical mixtures of ilmenite and hematite,and the solid circles represent the reaction enthalpies measuredfor the FerOr-FeTiO, solid solutions. The reaction enthalpy in-cludes contributions from the heat content and dissolution. Theerror bars represent 2 sd ofthe mean.
constrain parameters in microscopically based modelssuch as CVM, but doing so is beyond the scope of thispaper.
The data can be fitted by a subregular (two-parameter)solution model
The change in sign in the enthalpy of mixing is reflectedin the opposite signs of the two parameters. We stressthat the above polynomial, though convenient, has littlephysical significance because of the complex variation ofboth short- and long-range order with composition andtemperature (see below).
Enthalpies of disordering
In order to measure the enthalpy of disordering, com-positions for x,,- : 0.6 to 0.85 were annealed at temper-atures below the order-disorder transition and their cat-ion distributions measured (Table I of this paper; Brownet al., 1993). The order parameter determined from sat-uration magnetization measurements, OM,, varies directlywith the crystallographic site occupancies of Tia+, Fe3+,and Fe2*. From a crystallographic perspective, the threecations would be distributed over the A and B sublatticessuch that charge balance over the structure would bemaintained and cation to cation repulsion across theshared octahedral face would be minimized. Minimiza-tion of the highly repulsive Ti4+-Ti4+ pairs in both or-
f-
-
i1>-
70
T = 9 5 7 KFLOWING ARGON
. SOLID SOLUTIONSO MECHANICAL MIXTURES
o . 2 0 . 4 0 . 6 0 . 8
MOL FRACTION FeTiO3Fig. 3. Enthalpies of mixing at 1057 K for FerO.-FeTiO.
solid solutions (solid circles) calculated from the fit to the datain Fig. 2 and Table 3. The error bars represent the propagated 2sd of the mean. The line with long dashes represents ideal be-havior. The curves with short dashes represent one set of posi-tive (upper dashed curve) and negative (lower dashed curve)enthalpies of mixing (see text) that would sum to give the mea-sured enthalpies of mixing.
dered and disordered solid solutions of all compositionsoccurs when Fer+ is distributed equally in the A and Bsites. Therefore, assuming that Fe3* is equally distributedover both sublattices, we need only be concerned withthe ordering ofFe2* and Tia+ on the A and B sublattices.The order parameter, Or., is related to the site occupan-cies by the general definit ion
Q., : (Xo..- o - XF"'t.)/(Xp",*.a * Xe"'*,g)
-- (xrn*.o - xr,,-.)/(xr1n*.a -F x1;n*,s) (2)
where Xo.,*.o is the sublattice mole fraction of Fe2* in theA sublattice (Brown et al., 1993).
Table 3 shows that as the composition approaches il-menite, the range in the order parameter for different an-nealing temperatures becomes smaller (i.e., for compo-sition with x,,- : 0.6 the order parameter varies from 0to 0.66, for x,* : 0.7 it varies from 0.53 to 0.66, and forx,,- :0.85 the range is only 0.58 to 0.61). This decreasein variation of the order parameter as compositions be-come more ilmenite rich suggests that the high-temper-ature disordered state becomes from difficult to quench(Brown et al., 1993).
Enthalpies of drop solution (Table 2) measured on thesesamples are indistinguishable within the resolution of thecalorimetric experiments (+3 kJlmoD. These results sug-gest that significant differences in the cation distributionsin this compositional range do not result in significantenthalpy changes. This is permissible if short-range order
489
Tre|-e 3. Entropies of mixing calculated with S"."',,,. : 3.3 J/(mol'K) and S-",.'m : 0 J/(mol ' K) for the Fe2 03 -FeTiO3 solidsolutions
Anneal- Moleing f fraction AS..'(K) FeTiO" [J/(mol K)]
t The AS., calculated with S-.,,,. : 3.3 J/(mol K).t. The as-,, calculated with s-",,,. : 0 J/(mol' K).t The errors represent the uncertainty in S-".*, calculated assuming
the maximum error (i.e., 0.024) in the Ti site-occupancy data, adjustingthe other site occupancies according to compositional constraints' andassuming that Fe3* is distributed equally on the A and B sublattices (seetext for discussion).
dominates and supports our interpretation ofthe changein sign of the enthalpy of mixing curve (above). Thus,short-range order is suggested by both the increasinglynegative enthalpies of mixing in the intermediate com-positional range and the insignificant enthalpy changesfor compositions with large changes in degree of long-range order (e.g., xu- : 0.6). As long-range order beginsto dominate (close to ilmenite compositions), the enthal-py of mixing becomes exothermic.
Cs,cuurroNs FRoM PHAsE EQUTLTBRTA
The purpose of this section is to estimate free eneryiesof mixing in hematite-ilmenite solid solutions for theequilibrium between sesquioxide and spinel phases. Thesefree eneryies are then combined with our measured en-thalpies of mixing to estimate entropies of mixing in he-matite-ilmenite solid solutions and to provide evidencefor the diminution of configurational entropy due to bothshort- and long-range order.
NoNsrorcrrroMETRrc SPTNEL solrD soLUTroNs
Calculation of activity and free energy oftransformation of Fee,rO. (a-'y)
The data of Taylor (1964) showed that at 1573 K asignificant range of nonstoichiometry exists in the ulvij-spinel-magnetite solid solution series (dashed line in Fig.4). Schmalzried (1983) has investigated the range in non-stoichiometry in Fe.Oo at temperatures between ll73 and1673 K. His results are listed in Table 4 in terms of molefraction Fee,,Oo and FerOo. Combining the data of Taylor(1964\ andSchmalzried (1983), the limit of nonstoichi-ometry for FerOo-FerTiOo solid solutions has been esti-mated (dashed line in Fig. 4) and used to estimate thecompositions of the nonstoichiometric spinel solid solu-
BROWN AND NAVROTSKY: HEMATITE-ILMENITE SOLID SOLUTIONS
490 BROWN AND NAVROTSKY: HEMATITE-ILMENITE SOLID SOLUTIONS
TABLE 4. Solubility data for Fer"Oo in Fe"Oo coexisting with FerO"at temperatures between 1173 and 1573
r(K) o/Fe' 4" Xr".o,** xcey,oo
FerTiOE-FeTirO6 Join
Fe203-FeTiO3
Fer0a-FerTiO{ Join
FeO o .0Fe2O3
Fig. 4. Ternary phase diagram for the FeO-TiOr-Fe,O, sys-tem (modified from Taylor, 1964). The composition in weightfraction oxides (solid circles) is plotted for each of four Iog /o.(-0.68, -3.43, -6.0, -8.0) from the experimental data of Tay-lor (1964). The dashed line represents the limit of nonstoichiom-etry in the spinel solid solutions assumed in our model (see text).The open circle represents the composition of Fe.On in equilib-rium with Fe,O. at 1573 K from Schmalzried (1983). TheFe.TiOr-FeTi.O,, FerO.-FeTiO., and FerOo-FerTiOo joins areshown for reference.
tions coexisting with ilmenite-hematite solid solutions(Table 5). These results are in good agreement with theoxidation boundary calculated by Aragon and Mc-Callister (1982). If enough fo, dala were distributed inthe Fe-Ti-O ternary, one could calculate the activities ofall oxide components by appropriate ternary Gibbs-Du-hem integrations. After several attempts in this direction,we concluded that the data are too sparse at any onetemperature for a meaningful calculation independent ofsimplifying assumptions. We chose to apply a simplemixing model to the spinel solid solutions. We could thenutilize the coexisting pairs at 1573 K (our data and thoseof Taylor, 1964) to determine activity-composition rela-tions along the pseudobinary FerO.-FeTiO. join. Our ap-proach to calculating the free energies of mixing in boththe titanomagnetite and hematite-ilmenite solid solutionseries differs from that of Arag6n and McCallister (1982)rn two respects. For the titanomagnetite series, we haveboth assumed athermal mixing; however, Arag6n andMcCallister assumed mixing over only one site. In ourmodel, we have taken into account cation ordering (i.e.,mixing over two sites). In addition, their approach forthe hematite-ilmenite series was to assume a simple mix-ing model for the excess free energies. In our model, wemake no assumptions for the free energies of mixing inthe hematite-ilmenite solid solution series (see below).
With the assumption that mixing in 7-Fee,.O.-FerTiO4-FerOo solid solutions is athermal and govi:rned by theconfigurational entropy, it is possible to calculate the
. Estimated from Fig 4 in Schmalzried (1983).*. Calculated as[X'*or."n, - x.".or."no,rJ/[x.q."nr xrqreq.o.rl : 100/0.333 : 300.003
300.003(Ax,.,", n*,) : AX..o, - n*,where xr".o,,."n, (: 100) and Xr".o4r.eo.) (: 0) are the mole fractions of Fe3O4in pure Fe3O4 and pure Fer.Oo, re'spectively. The AxFq-"s, and AXFqF.qo.)are the mole fractions of Fe in pure Fe.Oo and pure Fe%O4, respec-tivelv
change in free energy relative to hematite (FerO, :
a-Fey.O), magnetite (FerOo), and ulvospinel (FerTiOo) as
AG* : yL,G,^ "(a-1)
+ AG-,. (3)
where y is the mole fraction of 7-Fee,.O. in the ternaryspinel sotid solution and AG,*".(a-7) is the free energy oftransformation for the reaction
a-FeyrOo :7-FeBr.Oo. (4)
This transformation free energy can be calculated fromthe binary solid solubility data of Schmalzried (1983)(Table 4). For an Feq,Oo-saturated magnetite phase co-existing with hematite, &r.,,o, rnust be equal in both phasesat equilibrium. In hematite, r/r.e.o, is the free energy of for-mation per four O atoms mole of FerO.. In the saturated7 phase,
ItFey.oa: 1r9.9.o"., + RI ln clF.troa,t (5)
where
F?orroo.r: &P.rr,o.,. * AG,,"'r(a-7). (6)
At equilibrium, this gives
AG,..""(a-7) : -RZlrr er",,.oo., Q)
where
R?" ln ao.r.on." : - ?"Asr.r,oo,". (8)
In this equation, ASr..non is the partial molar entropy ofFer,O+ in the 7 phase (see below). Computation yields avalue for AG,."""(a-7) : 14.6 + 0.6 kJ/mol. The error inAG,.,." has been calculated assuming a compositional er-ror of 2 molo/o Feu.Oo in the solubility data of Schmalz-ried (1983) (Table'4). These data can also be used toestimate the enthalpy and entropy of the transition Af/,,"n,and A,St.,.", respectively, where
dac,,"""/dT: -a.s,*,". (9)
The AG,.,.. is calculated as described above. The transi-tion temperature where pure a-FeqOo transforms to pure
4.fA.-4,
lq157314731 37312731173
1.3641 3571 3491.3411 337
29322 9482.9662.9812.992
0 7950 8450.8990.9470.974
0.2050.1 550.1 010.0530.026
WEIGHT FRACTION FerO3
BROWN AND NAVROTSKY: HEMATITE-ILMENITE SOLID SOLUTIONS 491
TABLE 5. Compositions of the nonstoichiometric spinel solid so-lutions and coexisting ilmenite-hematite solid solutionsestimated from Taylor (1964)
TleLe 6. Values of aFer.oo.,, aFer,oo.", an.- , and.4,. calculatedfor the coexistihg nonstciichiometric spinel and ilmen-ite-hematite solid solutions, as discussed in the text
In fo. Elrey"o,t a rev"oo ' ?m 4t^
-25.33 9.10 x 10 6-23.03 1.79 x 10-"-20.72 6.52 x 10 4*18'42 2'25 x 1o-"_16.12 7.96 x 10 3_14.97 1.53 x 10 ,_1g.B2 2.90 x 10 ,
11 .51 8 .75 x 10 ' ?-7.90 0.199-3.71 0.326
2 . 9 x 1 0 55 . 5 x 1 0 02 . 0 x 1 0 36 . 9 x 1 0 32 . 4 x 1 0 24 . 7 x 1 0 29 . 9 x 1 0 ,0.2680 6 1 11.000
4 . 0 x 1 0 0 0 . 9 6 3 93 . 6 x 1 0 . 0 . 9 0 0 39.5 x 10-" 0.84622 . 4 x 1 0 ' z 0 . 7 7 1 16 . 1 x 1 0 ' 0 . 6 6 4 20.100 0.58440.163 0.48690.371 0.29300.691 0.14531.000 0
occupancies of the end-members are defined as ([ ] :
The reaction describing the solid solution can be writtenas follows, assuming that the solid solution maintains thesite preferences of the end-members (i.e., Ti in octahedralsile only, tr distributed randomly, etc.)
x (F e|| F e1,i )[Fei. Fei* I O.
+ y(Feflfrnlo ,,,)[Fe]irrtro rrrlOo
+ { l - y - x}(Fe'?+)[Fe2*Ti4*]Oo
: (Fefl.{rr,*ryrFef 1, ',rrE6,,,,)
' [Fe]17s,*o.orFe11 y,%[Jo222yTii1,, ]O4 (13)
where x, y, and (l - x - y) are the mole fractions of
FerOo, FeyrOo, and FerTiOo, respectively. The change in
configurational entropy, AS-"r, for this reaction is given by
Nofe.'converted from weight fraction FeO, TiOr, and FerO3 (from Taylor,1 964).
7-FeqaO. can be estimated by extrapolating the FerOo-Fery,Oo solid solubility data (Table 4) to pure "y-FeqOo. Itis important to remember, however, that the limit of thesolubility data is 0.205 mole fraction Feq,,Oo. Thus, theextrapolated temperature of 3135 K must be taken withcaution. In any case, it is above the actual decompositionand melting temperatures. This transition temperaturecan be combined with the transition entropy determinedby Equation 9 at equilibrium where
A.I1,*.. : fAS,.""". (10)
Values of A.FI,*." and A,S,."". are 36.1 kJlmol and ll.5J/(mol.K), respectively. The free energy of transition canthen be calculated to be 17 .9 kJlmol at 157 3 K from thesevalues. This value is in reasonable agreement with thevalue determined with Equations 7 and 8, when one con-siders the uncertainty in the transition temperature at pureFeqOo. Because of this uncertainty, we have chosen tousqthe value determined from Equations 7 and 8 in thecalculations below.
For the Ti-bearing solid solutions,
Itree,.o'o: lr9..yroo,. + RZ ln aF.troo,o. (l l)
Therefore, at equilibrium between a nonstoichiometricspinel solid solution and an ilmenite-hematite solid so-lution,
The data of Taylor (1964) were used to estimate the com-positions of the nonstoichiometric spinel solid solutionsin equilibrium with hematite-ilmenite solid solutions(Table 5). Values for er.r,,o*o zlrd er.*,,o'., are listed inTable 6.
Calculations of the entropy of mixing in thespinel phase
Configurational entropies and the correspondingly sta-tistically ideal entropies of mixing fory-FeqOo-FerOo-FerTiOo solid solutions are calculated as follows. The site
492 BROWN AND NAVROTSKY: HEMATITE-ILMENITE SOLID SOLUTIONS
fi
- 0 . 6>t
P
,o 0 .4
T = 1 5 7 3 K
0 0 . 2 o . 4 0 . 6 0 . 8 1 . 0
MOL FRACTION FeTiO3Fig. 5. Activity-composition relations at 1573 K in Fe.O,-
FeTiO. solid solutions. The activity of FerO, (open circles andsolid curve) has been calculated with our model for nonstoichio-metric spinel solid solutions and the data of Taylor (1964) forcoexisting spinel and ilmenite solid solutions (see text). The ac-tivity of FeTiO. (solid circles, solid curve) has been calculatedby Gibbs-Duhem integration (see text). In addition, the activitiesof FerO. and FeTiO. (curves with small dashes) have also beencalculated from the empirical formulation of Ghiorso (1990).The lines with large dashes indicate the condition a, : x,, wherei is either FerO, or FeTiOr.
S"o,r.n".oo : -R{2 ln % + lfl t4} (16)
Sconr,ne,r;on : -R{2 ln %} (17)
Sconr , r .s , ,oo : -R{0 . l l l l n 0 . l l l + 0 .889 In0 .889
+ 1 . 7 7 8 I n 0 . 8 8 9 + 0 . 2 2 2 1 n 0 . 1 1 1 ) .( l 8 )
By combining terms, this becomes
,S." . r : -R{0.333y ln y - ( l - 0 .1I ly - x) ln 3
+ (0.889y +'),/,) ln(2.667y + 2x)
+ (l - y -,1,)ln(3 - 3y - 2x)
+ (1.778y + a'l)ln(5.334y + 4x)
+ (l + 0.778y - x)ln 6
+ ( l - y - l ) r n ( 3 - 3 y - x )
+ ( 1 - x - y ) l n ( l - x - y )
+ (l - ,1, - y)ln2 - al. ln4
- 2.667y ln 0.889) . (19)
This equation can be written in terms of the partial molarentropres as
+ l .778ln(5.334y + 4x) - t .778In6-2.667 ln 0.8891
+ zf-ln 3 + ln(3 - 3y - 2x)
- l n 6 + l n ( 3 - 3 y - x )
+ ln(l - x - y) + ln 2l). (2r)
The partial molar entropies can then be written as
Asp..oo : -R ln[(2.667y + 2x)4(3 - 3y - 2x)/,
.(5.334y t 4x)Y.
'(3 - 3Y - x)"(2)-t.(4)-t, l Q2)
Asr",ooo : -R ln[(y).333(3) 088e(2.667y + 2y1oa*
' (5.334Y I 4x) t t tz($)- t t ts
. (0.889; '* '1 Q3)
ASp":r io, : -R ln[(3)- '(3 - 3y - 2x)(6) t
' ( 3 - 3 y - r x l - x - y ) ( 2 ) ) . ( 2 4 )
We believe that for the high temperature of equilibri-um involved (1573 K), the above model, which assumesmaximum randomness and only {6lTi4+, is a good ap-proximation. Other models incorporating Fez+-Fer+ site-preference equilibria (e.g., O'Neill and Navrotsky, 1984)would give slightly different results. In view of the im-probability of completely quenching the high-tempera-ture structural state and the ambiguity in even distin-guishing Fe2+ and Fe3+ at high temperature, this simplestmodel is probably the best approach.
Cor,xrsrrNc sprNEL-sESeuroxrDE pArRS
Activities of FerO. and FeTiO. in the sesquioxide phase
The activity of FerO, in the ilmenite-hematite solidsolution series, 4n"-, can be calculated from the activityof FeyrOo in the spinel solid solution series (Eq. 12) andthe relation
aF.y .o*o : @n ) \ . (2s)
Values for activities of FerO. are shown in Table 6 andFigure 5. No polynomial form could be found to fit theactivity-composition relations, such that the a-x relationsreduce to Raoult's law as the mole fraction approaches1.0. Such complex behavior is not surprising, since themixing behavior results from the interplay between neg-ative and positive interactions that will vary as a functionof temperature and composition.
0 .2 0 .4
BROWN AND NAVROTSKY: HEMATITE-ILMENITE SOLID SOLUTIONS 493
The activity of FeTiOr, eih, earr be calculated by Gibbs-Duhem integration, where
ln 4'-: - , [ , : : ."- . ,(xn"-/x,,-)d
ln 4n.-.
Trau 7. Calculated free energies and entropies of mixing
Phase equilibriaand calorimetry Site-occupancy data
(26) Molefraction AG-,,FeTiOs (kJ/mol)
MoleAS.,,' fraction AS.',-" t AS.',t f
U/(mol.K)l FeTiO. F/(mol'K)l [J/(mol'K)]
In this equation, xn.- and x'm are the mole fractions ofFerO. and FeTiO., respectively, in the ilmenite solid so-lution coexisting with the spinel solid solution. The aboveintegration was carried out by plotting xn ̂ / xu^ vs. ln 4n"-and fitting these data with exponential equations suchthat ln an"- would tend to negative infinity as xn"-/x,,-approached zero, and xn" /xu^would tend to positive in-finity as ln an"- approached one. Calculated results areshown in Figure 5 and Table 6. In addition to the activ-ities calculated from our microscopically based model forthe spinel solid solution series, activities calculated usingthe empirical model of Ghiorso (1990) are also shown forcomparison. Both sets of activities show negative devia-tions from Raoult's Law and agree moderately well witheach other. The integral free energies of mixing agree quitewell (see below).
The free energies of mixing can be calculated from theactivities of FerO. and FeTiO, as
AG-,. : RZ{x,,-ln a,'. -+ (l - x,*)ln an"-} (27)
where R is the gas constant, I is the temperature (in K),and a,,- and an"* are the activities of ilmenite and he-matite, respectively, determined from our model of theactivity of FeqOo in the nonstoichiometric spinel solidsolution. Valuris for the free energy of mixing are calcu-lated in Table 7.
ENrnoprrs oF MDilNG rN FerOr-FeTiO,
Maximum configurational entropies of mixing fromsite occupancies
Calculation of the entropies of mixing in the FerO.-FeTiO, system requires explicit definition of the config-urational entropies in the end-member components. Forhematite, this definition is obvious with S."".n.. : 0J/(mol.K). For ilmenite, cation ordering has been studiedextensively both in situ at high temperatures and pres-sures and on samples quenched from high temperatures(e.g., Wechsler and Prewitt, 1984; Shirane er al., 1962).The neutron diffraction data of Shirane et al. (1962) onquenched samples have shown ilmenite to have (x,'- -
x) : 0.95 [i.e., (x,,- - x): degree of order : sublatticemole fraction Ti in the B sitel, although no error barshave been given for this result (S"."ru- : 3.30). The dif-fraction data of Wechsler and Prewitt (1984) have shownthat no appreciable disorder occurs at high temperatures(S..".u- : 0). Our determination of the degree of order inilmenite from saturation magnetization measurements to4 K (Brown et al., 1993) agree with the data of Wechslerand Prewitt (1984).
Configurational entropies ofthe solid solutions can becalculated from measured site occupancies (Table 1), fol-lowing the formulation of Thompson (1969, 1970)
* Errors (in parentheses) in AS-,, have been calculated from the erroron the fit to aH-,,.
". Calculated AS-,, with site-occupancy data at 1573 K.t The errors (in parentheses) represent the uncertainty in S"-r"., cal-
culated assuming the maximum error (i.e., 0 024) in the Ti site-occupancydata, adiusting the other site occupancies according to compositional con-straints, and assuming that Fe3 | is distributed equally on the A and Bsublattices (see text for discussion).
+ S*,,,- : 3.30 J/(mol K) with site occupancy data at 1573 K and S-.',,,.: 0 J/(mol' K).
S." . r : -R ) ) n,x , , lnx, , (28)
where R is the gas constant, n, is the number of sites, s,pfu, and x,, is the mole fraction of atom I on site s. It isimportant to note that these configurational entropiesrepresent maximum values in the sense that any short-range order on the A and B sublattices is neglected. Theentropy of mixing is then defined as the difference be-tween the configurational entropy of the solid solution'S.""r,., and the weighted sum of the standard-state config-urational entropies of the end-members (i.e., S.""rn.- andS.""t',.)
where x,,- is the mole fraction of FeTiOt in the solidsolution. Calculations of the entropies of mixing at var-ious temperatures have been made for both standard states
[Table 3 for S.."ru- : 0, Table 3 and Figure 6 for S.".r',-: 3.3 J/(mol'K)1. Figure 6 shows the change in the en-tropy of mixing as a function of annealing temperatureand reflects the increase in order with decreasing anneal-ing temperature for solid solutions with compositions fromx'- : 0.4 to 0.85.
Calculation of entropies of mixing fromAG-,* and AII-'-
The free energies of mixing determined from the phase
equilibria data can then be used to calculate the entropyof mixing, AS-t-, with the relation
(-aG-* + LH^)/T: aS-,.. (30)
In this equation, the enthalpies of mixing are those de-termined experimentally (Eq. l). Calculated entropies ofmixing are listed in Table 7 and Figure 7. Values for AS-'^
494 BROWN AND NAVROTSKY: HEMATITE-ILMENITE SOLID SOLUTIONS
- 1 5
r-
\ r oh
.I
- T - 1 5 7 3 K- - - T - t 2 7 3 K
I = 9 9 8 KT = 9 2 3 K
0 . 2 0 . 4 0 . 6 0 . 8
MOL FRACTION FeTiO3Fig. 6. Entropies of mixing calculated from site-occupancy
data vs. mole fraction FeTiO, for different annealing tempera-tures and S.""r,,- : 3.3 J/(mol.K). The decreasing entropy ofmixing with decreasing temperature expresses the tncrease tncation order (see Table l). The solid line represents data onsamples quenched from 1573 K. The dashed lines represent en-tropies of mixing calculated for lower temperature anneals from12'73,993, and 923 K (dash size decreases with decreasing tem-perature) (see Table 3).
(dashed curve in Fig. 7) have also been calculated usingAG-* (Ghiorso, 1990) and our experimentally deter-mined enthalpies of mixing. The agreement between thetwo calculated entropies of mixing shows that determi-nation of the free energies of mixing in FerOr-FeTiO,from our microscopically based model for the spinel solidsolutions is in reasonable agreement with Ghiorso's em-pirical fit to the existing data in the hematite-ilmenitesolid solution series. Ghiorso's empirical fit includes ex-perimental data on both the order-disorder transition andthe position of the solvus in the FerOr-FeTiO. solid so-lution series. In addition, both models show that the cal-culated entropies of mixing are significantly less positivethan those inferred from measured cation distributions,especially for compositions with ,r,,- = 0.3. This obser-vation is also consistent with significant short-range orderand may imply clustering of Ti in the Ti-rich B sublattice.
DrscussroN
Solid solutions in the FerO.-FeTiO3 system are char-acteized by two varieties of ordering: antiferromagneticordering of Fe in the temperature range of -60-953 K,and chemical ordering of Fe2* and Ti4+ between adjacentbasal planes at temperatures between 873 and 1350 K(Ishikawa, 1958; Ishikawa and Akimoro. 1957: Ishikawaand Syono, 1963; Shirane et al., 1962). Magnetic orderingresults from ferromagnetic intralayer magnetic interac-
T- t573K
0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0
MOL FRACTION FeTiO3
Fig. 7. Solid circles represent the entropies of mixing calcu-lated from site-occupancy data for S...,,,- : 3.30 J/(mol K). Theerrors are represented by the symbol size. The open circles rep-resent the entropies of mixing calculated with the free energiesof mixing from our nonstoichiometric spinel model and themeasured enthalpies of mixing. The error bars represent the un-certainty in this calculated entropy due to the error in the fit tothe enthalpy of mixing. The open triangles represent the entro-pies of mixing, calculated using free energies of mixing from theempirical formulation of Ghiorso ( I 990), and our measured en-thalpies of mixing. Errors bars for these data have been elimi-nated for clarity, but are the same as for the open circles.
tions (i.e., magnetic moments within each sublattice arealigned parallel) and antiferromagnetic interlayer inter-actions between Fe atoms on the A and B sublattices (i.e..the magnetic moments of alternating A and B layers pointin opposite directions). Chemical ordering arises from re-pulsive intralayer interaction and attractive interlayer in-teractions between Fe2+ and Ti4*. In attempting to modelthe thermodynamic mixing properties by subregular andregular solution parameterizations, one loses the natureof these complex microscopic interactions that give riseto long-range and short-range ordering. CVM (clustervariation method) calculations (Burton, 1984; Burton andKikuchi, 1984; Burton, 1985; BurtonandDavidson, 1988)include the signs and magnitudes of these interactionsand thus allow for more atomistically based descriptionsof the thermodynamic properties of this solid solutionseries. They have been applied to carbonates (Capobian-co et al., 1987) and to FerO,-FeTiO. (Burton, 1984; Bur-ton, I 985; Burton and Davidson, I 988). The present datawill be useful to constrain their parameters better, butsuch calculations are outside the scope ofthe current study.
Evidence for short-range ordering can be found for bothmagnetic and chemical ordering reactions. For example,
. I 'ROM MEASURED CATION DISTRIBUTIONSO FROM CALCULATED AdM' AND MEASURED AI/V,.a FRoM Mu (cHtoRSo.1090) lNn MEISURED-'l iru,,
- l o-d
\ r zF-
x Bd
Vl
0 . 6 0 . 8
N FeT iO3
Gronvold and Samuelsen (1975) and Holland (1989) re-ported that hematite exhibits a long tail in the tr heat-capacity anomaly associated with magnetic ordering(transition temperature, 7., : 955 K), suggesting signifi-cant incipient disorder in the magnetic spins even at 298K. This disorder amounts to l7o/o of the maximum en-tropy that would be gained at f. Evidence of short-rangechemical order can be seen for the range of compositionsbetween x,,- : 0.4 and 0.8. During quenching from tem-peratures above the order-disorder transition, both nat-ural and synthetic solid solutions attempt to order. Localordered regions nucleate and grow together. When twoimpinging ordered regions are in phase with one another,one large ordered region is formed. When they are out ofphase, a compositionally distinct, cation-disordered twin-domain boundary (TDB) separates the two orderedregions. Such microscopic chemical domains and domainboundaries represent local variations in chemical order.These domains and domain boundaries, in turn, pro-foundly affect the magnetic ordering of solid solutions inthis compositional range. Both reversed magnetization(Nord and Lawson, 1989,1992:- Hoffman, 1992; Lawsonet al., l98l) and high coercivities (Brown et al., 1993) areintimately related to the development of TDBs duringquenching through the order-disorder transition. The twindomain boundaries provide a second, more Fe-rich, mag-netic phase that has a higher Curie temperature and caus-es the reversed magnetization seen in quenched naturaland synthetic samples (Lawson, et al., l98l; Nord andLawson. 1989: Nord and Lawson, 1992). In addition,Nord and Lawson (1992) proposed that when the chem-ical domain size is on the order of the magnetic domainsize, the magnetic domain walls become pinned in thechemical domain walls. That gives rise to high coercivi-ties in intermediate compositions and locally complexmagnetlc structure.
Evidence for short-range order can also be seen in thethermodynamic mixing properties of the FeTiOr-FerO,solid solution series. Measured enthalpies of mixing,A11-*, can be interpreted as follows. The enthalpy of mix-ing for each solid solution composition arises from thesum of the atomic interaction energies in the crystal. Ina solid solution series where both cation ordering andexsolution occur at varying temperatures, the measuredmixing enthalpy can be broken down into two contribu-tions: a positive mixing curve that will drive unmixingand a negative mixing curve that will drive ordering (Fig.3). The functional form of the two opposite componentsofthe total enthalpy depends on the strength ofthe pos-itive and negative atomic interaction energies as a func-tion of temperature and composition. One possible set ofcurves describing the two contributions to the measuredenthalpy of mixing in FerOr-FeTiO, solid solutions isshown in Figure 3 (small dashed curves). As drawn, thesecurves suggest that in hematite-rich solid solutions, pos-itive interaction energies dominate during initial substi-tution of FeTiO. into FerO.. Such behavior could arisefrom an increase in cation to cation repulsion across the
495
shared octahedral face due to increasing Ti4* content andaccompanying local displacements of the O ions to pro-vide more shielding across the shared face. In ilmenite-rich solid solutions, negative ordering interactions dom-inate and reduce cation to cation repulsion across theshared face by placing Fe2+ and Tia* on separate sublat-tices. In the intermediate compositional range, as the Tia*content increases, the driving force for ordering increases,resulting in a progression from short-range ordering inthe more Ti-poor solid solution to long-range ordering inthe more Ti-rich solid solutions.
The implications of this work are important in termsof the development of thermodynamic models used topredict the activity-composition relationships at geolog-
ically relevant temperatures (-673-1173 K). This tem-perature range is precisely the range in which one wouldexpect significant short-range and long-range ordering(both magnetic and chemical) to affect the thermodynam-ic properties. Paramagnetic to antiferromagnetic transi-tions in this temperature range occur for compositionsfrom x,,- : 0 to 0.4. Chemical ordering also occurs inthis temperature range for compositions from x,,- : 0.5to 0.65 and possibly from as low as x,,- : 0.2. This lowcompositional estimate is suggested by the high coerciv-ities found in composition x'- : 0.2 arrd 0.4 (Brown etal., 1993). The complex temperature and compositionaldependence of both the chemical and magnetic orderingreactions support use of such models as CVM, which in-clude the atomic interactions that govern this complexZ- and X-dependent behavior.
At first thought, one might be tempted to construct anempirical activity-composition model based on our en-thalpies of mixing (Eq. l), incorporating our excess en-tropies of mixing (open circles in Fig. 7). Algebraically'that could be done, but we believe it would have little,and only accidental, validity. Because the site occupan-cies and degree of short-range order depend strongly ontemperature, it is highly likely that the entropy of mixinglikewise has a strong temperature dependence. It seemsunlikely that the activities depend on temperature andcomposition in a way that is simple enough to be mod-eled reliably by simple polynomials, especially i\the T-Xregion ofthe order-disorder transition. The present study,in our interpretation, counsels caution in the indiscrim-inate use of ilmenite-spinel geothermobarometry for rocks,especially for lower grades of metamorphism, since ex-trapolation of higher temperature (> I100 K) activity datato the 700-1000 K range may be complicated by order-disorder phenomena.
The present data furnish a beginning to the quantifi-
cation of these interactions, by providing constraints onboth the enthalpies and entropies of mixing. However'the detailed temperature dependence of both long- andshort-range order needs further study, and the further de-velopment of CVM or other structurally based models isnecessary. The present calorimetric data refer to the en-thalpy of mixing at room temperature of samples with adegree of short-range order and long-range order char-
BROWN AND NAVROTSKY: HEMATITE-ILMENITE SOLID SOLUTIONS
496 BROWN AND NAVROTSKY: HEMATITE-ILMENITE SOLID SOLUTIONS
acteristic of some higher temperature, but probably notpreserving all the disorder of the high-temperature state.Thus, although the change in sign of All-* with compo-sition and the smaller than configurational AS-,. are verysuggestive ofmajor short-range order, the data are insuf-ficient to constrain quantitatively a microscopic mixingmodel as a function of both temperature and composi-tion. The development of a complete microscopicallybased thermodynamic model for this system would re-quire data on the in-situ state of ordering for each com-position, heat-capacity data for each composition, and in-situ volume data as a function of composition. Once thesedata have been constrained, such a model could be con-structed, and its free energies of mixing might then beused to parameterize more convenient empirical equa-tions that would have the correct composition and tem-perature dependence.
AcrNowr,rncMENTs
Funding for this research was provided by the Mineralogical Society ofAmerica through the M.S.A. Petrology grant for 1989 (to N.E.B.) and bythe National Science Foundarion (grant DMR-8912549 and 9215802 toA. Navrotsky) The authors would like to thank B.P. Burton, p.M. Da-vidson, and M.S. Ghiorso for their reviews of the manuscript.
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