American Mineralogist, Volume 79, pages 485-496, 1994
Hematite-ilmenite (FerOr-FeTiO3)solid solutions:The effects of cation ordering on the
thermodynamicsof 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, whereasexothermicmixing dominatesthe compositionalregionfrom x,,: 0.65 to l The measuredenthalpy of mixing is interpreted as arising from two contributions: a positive enthalpy of mixing due to repulsive interaction energies(presumably
within a hexagonallayer) and a negative enthalpy of mixing due to attractive interaction
energiesbetween layers (i.e., the driving force for ordering).
Enthalpiesof reaction have also been measuredon compositions from x,,- : 0.6 to 0.85
that have different measuredcation distributions. Within the resolution of the measurements (+3 kJ/mol), the enthalpies of isocompositional sampleswith varying cation distributions are indistinguishable. This observation supports significant short-range order.
As the driving force for ordering increaseswith increasingilmenite content, progression
from short-rangeorder in the more Ti-poor solid solutions to long-rangeorder in the Tirich compositions occurs. The progression from short-range to long-range order is expressedby increasinglynegative enthalpiesof mixing.
Free energiesof mixing have been determined independently from the measured tie
lines betweennonstoichiometric spinel and the sesquioxidephaseat 1573 K. This requires
modeling the activity of Fe2Ooin the nonstoichiometric spinel solid solutions coexisting
with hematite-ilmenite solid solutions. Combination of these free energiesof mixing and
the experimentallydeterminedenthalpiesof mixing suggestthat 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.
most common complications affecting the thermodyproperties of these solid solutions are ignored.
namic
Iron titanium oxides of the FerO,-FeTiO, solid soluThese complications include cation ordering in both
tion seriesimpact the evolution of igneousand metaFerOo-FerTiOoand FerOr-FeTiO. solid solutions,nonmorphic rocks becauseof the largenumber of subsolidus
stoichiometry in FerOo-FerTiOosolid solution at T >
equilibria in which they are involved (Haggerty,1976).
K (Taylor, 1964;Schmalzried,1983;Websterand
Their ubiquitous occurrencewith Fe.Oo-FerTiOosolid I173
Bright,
196l; Senderovet al., 1993),and chemicalimsolutions allows petrologists to estimate the temperature
purities. Thus, the calculatedparametershave little physand fo, conditions under which these mineral assemical meaning with regard to the atomic interactions that
blagesformed. In the classicalapproach for determining
properties to vary as
equilibrium T and fo. conditions, activity-composition causemacroscopicthermodynamic
X. The most rigorous attempt to
function
P,
T,
and
a
of
relations of Fe.Oo-FerTiOoand FerO.-FeTiO. solid soinclude the effects of cation ordering within the conlutions are approximated by subregular solution (Marstraints of a simple chemical model has been made by
gules)formalisms. Data for selectedchemicalequilibrium
Ghiorso (1990). His resultsshow the importance of inreactions at a number of temperatures,including ilmencluding
suchbehavior.
ite-hematiteand magnetite-ulviispinelpairs, are then used
Microscopically basedmodels, on the other hand, take
to determine empirically the Margules parameters(Powinto account the apparently cooperative and higher order
ell and Powell, 1977; Buddington and Lindsley, 1964;
nature ofthe order-disorderprocess.The most generalof
Spencerand Lindsley,l98l; Andersenand Lindsley,1988;
such approachesinvokes the cluster variation method
Ghiorso, 1990).
(CVM), in which interactions out to several nearest
Becauseof the empirical nature of this approach, the
neighborsare considered(Burton and Davidson, 1988;
Burton, 1985;Burton,1984;Burton and Kikuchi, 1984;
* presentaddress:Norton Company,I New Bond Street Kikuchi, 1977). In this way, structural and thermodyworcester,
MS420-601,
Massachusetts
01615-0008,
U.S.A.
namic parametersare linked rigorouslyto the free enerIrvrnooucrrox
0003-o04x/94l0s06-{485$02.00
485
486
BROWN AND NAVROTSKY: HEMATITE-ILMENITE
Trele 1. Site-occupancydata for specifiedcompositionsand
annealingtemperatures
Anneal- Mole
ing f
fraction
(K)
FeTiO"
1573
1126
977
1573
1273
1023
1573
1073
998
923
1573
1573
0.8630
0.8570
0.8595
0.7001
0.7023
0.7029
0.6094
0.5993
0.6013
0.5929
0.4080
0.2024
Ntn ,"
Nr*
0.698
0.677
0.691
0.536
0.574
0.586
0.311
0.406
0.467
0.493
0.206
0.102
0.166
0.180
0.169
0.164
0.128
0 120
0.299
0.194
0.135
0.100
0.202
0.097
"
Ne*. e
N-,.-o
Nre-,a Nr*-^
0.137
0.143
0.140
0.300
0.298
0.297
0.391
0.401
0.399
0.407
0.593
0.802
0.166
0.180
0.169
0.164
0.128
0.120
0.299
0.194
0.135
0.100
0.202
0.097
0.698
0.677
0.691
0.536
0.574
0.586
0.311
0.406
0.467
0.493
0.206
0.102
0.137
0.143
0.140
0.300
0.298
o.297
0.391
0.401
0.399
o.407
0.593
0.802
Note.'errorsin site occupanciesrange from +0 01 to +0.024. They have
been determined by including the errors arising from the linear extrapolations 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 to
these types of models. The first is that they are mathematically cumbersome and do not lead to closed-form
equations for the mixing parameters.The secondis that
their energeticparameterscannot be uniquely evaluated
from phaseequilibria alone, and other direct estimatesof
the energeticsofordering are not available.
Thus, a major gap remains betweenthe two approaches.The formalism basedon simple polynomialsand empirical calibration, though petrologically useful, is hard to
reconcile with the microscopic complexity of order-disorder and may lead to difficulties when extrapolating outside the P, T, and X rcnge of its determination. The microscopicapproach,without more detailed structural and
energeticdata to constrain its parameters,remains more
a conceptual framework than a useful description. This
work is a first step in bridging that gap.
In a previous paper (Brown et al., 1993),we reported
the preparation and magnetic characleization of a series
of ilmenite-hematitesolid solutions,which leadsto a description of the degreeof long-range order of quenched
samples.This work reports a drop-solution calorimetric
study of the same samples,which provides enthalpies of
mixing at room temperature of the same quenchedsamples. Combination of the structural and calorimetric data
with phase equilibrium studies leads to insights into the
short- and long-rangeatomic interactions that affect the
thermodynamic mixing properties of hematite-ilmenite
solid solutions.
Pnocrnunrs
Samples
The phasesused in this calorimetric study spannedthe
FeTiOr-FerO. solid solution seriesin composition and
were synthesizedunder a controlledatmosphereaL l573
K. Relevantcompositions(i.e.,solid solutionswith compositionsfrom Jr,,-: 0.6 to 0.85, wherex,,- : mole frac-
SOLID SOLUTIONS
tion FeTiO.) were annealedat various temperaturesbelow the order-disorder transition to vary their cation
distributions. The composition of each phaseat each annealing temperature was determined by WDS microprobe analyses, lattice parameters, and wet chemical
analyses.The results showed the compositions to be accurate to within + 1.0 mo10/0.
The saturation magnetization extrapolated to 0 K was used to determine cation
distributions for each annealing temperature (Table l).
More detailed discussion of the synthesis and analysis
procedurescan be found in Brown et al. (1993).
Calorimetric protocol
The purpose of the calorimetric study was to devise a
thermodynamic protocol that converted a sample of
known structure and oxidation state into a well-defined
dissolved final state in the lead borate solvent commonly
used for oxide-melt solution calorimetry (Navrotsky,
1977). Becauseboth the degreeof order and oxidation
can changerapidly in the solid phaseat high temperature,
the starting state was taken as the sample at room temperature, which was then dropped directly into the solvent at high temperature. In such a drop-solution calorimetric experiment, the total measuredenthalpy contains
contributions from heat content, dissolution, and any oxidation or reduction in the solvent. No high-temperature
equilibration is required, and thus the ambiguity ofthe samples' oxidation state prior to dissolution is eliminated.
The appropriate calorimetric solvent was determined
from preliminary drop-solution experimentswith ilmenite in three solvents:sodium molybdate(Navrotsky, 1977);
alkali borate (Takayama-Muromachi and Navrotsky,
1988);and lead borate (Navrotsky, 1977).The sodium
molybdate solvent showed incomplete dissolution after
120 min at 973 K. The alkali borate solventshowedprecipitation of a sodium titanate by optical examinations
and X-ray powder diffraction. Experimentsin lead borate
at 1057 K showed complete dissolution, and X-ray powder diffraction and optical examinations indicated no
PbTiO3 precipitation, which had been a problem in earlier experimentsusing TiO, at 973 K (Navrotsky, 1977).
The final state of Fe after dissolution in 2PbO-BrO.
solvent was determined by weight gain experimentswith
solid solutions and mechanical mixtures having compositions betweenFerO, (all p.:+; and FeTiO, (all Fe'?*)
under conditions of both static air and flowing Ar. The
Ar was purified by passingthe gas through a Ti purifier
(R. D. Mathis GP-100 inert gas purifier) that has been
shown to producean fo. of < 10-6. Eachweight gain experiment was carried out under conditions that mimic
the calorimetric experiments as closely as possible. The
following precautions were taken: the surfacearea of the
melt-air interface was kept the same; sample to solvent
ratios were restricted to the same range as for a seriesof
calorimetric experiments; experiment durations were 90
min for experiments in air and 70 min for experiments
under flowing Ar (as in calorimetry); sample mass was
maintained within the same ranse as that used in the
BROWN AND NAVROTSKY: HEMATITE-ILMENITE
calorimetric experiments(5-15 mg); and temperaturewas
kept at the calorimetertemperatureof 1057 K. Pt crucibles containing lead borate solvent were brought to constant weight at 1057 K under the appropriate atmospheric conditions. Samples were also brought to constant
weight at 383 K and ranged from well sintered to poorly
sintered.
Figure I showsthe percentweight gain vs. composition
for all experiments. The dashedline indicates the calculated weight gain anticipated for complete oxidation of
Fe2+to Fe3* in the solvent for each composition across
the solid solution series.The percentofoxidation ofFe2*
to Fe3* in the solvent is then calculated relative to this
maximum weight gain line. Each data point represents
one dissolution experiment, and the error bars represent
the propagatedcumulativeweighingerrors(- +0.06 mg).
For the solid solution experiments conducted in static
air (solid circles in Fig. I ), the total rangein the calculated
with the mapercentoxidation lies between7l and 930/0,
jority of experiments falling between 84 and 930/0.Much
ofthis variation in the calculatedpercentoxidation arises
from the extreme sensitivity of this percentageto very
small (+0.01 mg) cumulativeweighingerrors(e.g.,- +60/0
at xir- : 0.2; -+1.5o/oat 4'- : 0.85).A line fittedto the
static air data for compositionsbetweenx,,- : 0.2 and
0.85 indicatesthat, after a 90 min experiment,88 + 5.20lo
of the Fe'?*in the sampleoxidizesto Fe3*. For solid solutions with x,,- = 0.85, however,the degreeof oxidation
does not reach this limit in 90 min. For ilmenite, the
percentoxidation reachedafter 90 min is only -650/o.By
-300 min, the percentoxidation reached-880/0.We infer from this that the oxidation processis controlled by
the diffusion ofO into the lead borate solventat the solvent-air interface. Similar conclusions were reached by
Zhou et al. (1993) for the Cu+ + Cu2* reaction.
Mechanical mixtures (open circles in Fig. l) of hematite and ilmenite indicate a final oxidation statereflecting
a weighted averageofthe oxidation statesofthe two endmembers.The differencebetweenthe final oxidation state
of ilmenite (i.e., 650/ooxidation of Fe2* to Fe3* after 90
min) and that for solid solution with x'* < 0.85 (i.e.,
88o/ooxidation of Fe2+to Fe3* after 90 min) causesthe
final oxidation state for the mechanical mixture to be
different from that for solid solutions. Thus the amount
of oxidation occurring during a calorimetric experiment
is not the same for solid solutions and mechanical
mixtures, making interpretation of the calorimetric results from experimentsin static air dependenton the enthalpy of oxidation. For solid solutions of a different
composition, there is no inherent reason why this enthalpy of oxidation should be constant for all compositions, and the large errors associatedwith the amount of
oxidation determined by weight gain analysis results in
uncertaintiesin the enthalpy of oxidation of - l5-20o/o.
The experimentsin Ar (solid stars in Fig. l), on the
other hand, show no changein weight for all compositions acrossthe solid solution seriesafter 70 min. However, this is only true when the lead borate solvent has
487
SOLID SOLUTIONS
r SOLID SOLUTIONS - STATIC AIR
o MECHANICAL MIXTURES - STATIC AIR
* S O L I D S O L U T I O N S- F L O W I N GA R G O N
z
e
(J s.o
F
lTl
ll
z.o
=
in
6\
0
0.2
0.4
0.6
0.8
1.0
N F
FeTiO3
eTiO3
M OL FRACTION
Fig. 1. Percentweightgainedin leadboratesolventat 1057
solidsolutionsandmeK because
of oxidationof FerOr-FeTiOr
the percentweight
chanicalmixtures.Thedashedline represents
gainexpected
for completeoxidationofFe2* to Fer* in the solwith FerO.90-minexperiments
vent.The solidcirclesrepresent
FeTiO, solid solutionsin staticair. The opencirclesrepresent
90-min experimentswith mechanicalmixturesof Fe.O. and
70-minexperiFeTiO,in staticair. The solidstarsrepresent
in flowingAr.
solidsolutions
mentswith FerO.-FeTiO,
been equilibrated for - 15 h under flowing Ar at the calorimeter temperatureprior to dissolutionof the samples.
Calorimetric experiments conducted without this equilibration time show small parasitic exothermic reactions
and a very slow return to the base line, suggestingthat
the sample was oxidized slowly. This suggeststhat there
is a small solubility of O in the solvent in equilibrium
with air that is greatly reducedwhen the solvent is equilibrated with flowing Ar. Once this O is removed from
the solvent, no oxidation or reduction ofFe in the solvent
occurs during the calorimetric experiment.
As a result of the above considerations,drop-solution
calorimetric experimentswere completedin a Calvet-type
microcalorimeter(Navrotsky,1977)at 1057K in 2PbOBrO. solvent under conditions of flowing Ar, with overnight preequilibration of the solvent under Ar gas flow.
Each experiment was 70 min in length. Solid solution
experiments involved dropping small chunks (7-25 mg)
of FerO.-FeTiO.solid solutions.Mechanicalmixture experiments involved dropping two small chunks of the
end-memberstogether.This procedureeliminated the extra heat effect attributable to any capsule. The samples
dissolved rapidly and completely. The high temperature
(1057 instead of 973 K) and intimate contact of the
dropped sample with the solvent (no Pt sample capsule)
may have aided rapid dissolution and avoided local
PbTiO3 saturation.
488
BROWN AND NAVROTSKY: HEMATITE-ILMENITE
SOLID SOLUTIONS
TleLe 2. Drop-solutionreactionenthalpiesat 1057 K in lead
borate solvent under conditionsof flowing Ar for solid
solutionsand mechanicalmixtures
Solid solutions
Annealing f
(K)
Mole
fraction
FeTiO3
1573
1573
1573
1573
1073
998
1573
1273
1023
1126
977
1338
0
0.1985
0.4068
0.6094
0.5993
0.6013
0.7001
0.7023
07029
0 8570
0.8595
1
Mechanicalmixtures
Total H
(kJ/mol)
177.1+
166.0+
1s8.0+
1s7.s*
157.7+
156.2+
160.1+
158.1+
162.8+
157.4+
157.4+
148.9+
2.s(s)'-I
4.3(5)
4.2(5)
3 s(5)
33(5)
3.0(s)
2.7(s)
4 2(5)
4 1(5)
0.4(5)
2.6(5)
3.4(5)
Mole
fraction
FeTiO3
Total H'
(kJ/mol)
0.0800
0.2758
0.4653
0.5648
0.8905
176.5
170.6
1 6 59
1 6 1I
156.s
. No errors are indicated for
the mechanical mixtures because each
value represents one experiment in which a chunk of both Fe2o3 and
FeTiO. were dropped simultaneously.Thus, compositionsare calculated
from the relative weights of each chunk.
*r Error represents2 sd the mean.
of
1 Number in parenthesesindicatesthe number of exDeriments.
C^ll-onrvrnrRrc RESuLTS
Observedenthalpies
The enthalpies of the drop solution of mechanical
mixtures and FerOr-FeTiO. solid solutions with varying
compositions and degreesof cation order are listed in
Table 2. For compositionsfrom x,* : 0.6 to 0.85 with
differing cation distribution, no measurabledifferencein
reaction enthalpy was found for isocompositional samples (Table 2); hence,only the averagevalues are plotted
in Figure 2. Excessenthalpiesare determined by subtracting the measuredenthalpiesof the solid solutionsfrom
those of the mechanicalmixtures. This differencerepresentsthe enthalpy of mixing at 298 K of solid solutions
having a structural state assumedto be characteristic of
their quench temperatures.The enthalpiesof disordering
are determined by subtracting the total measuredenthalpies for two solid solutions of the same composition but
with differingthermalhislories.
Enthalpies of mixing
T=957K
FLOWINGARGON
f-
-
i1
>-
70
. SOLID SOLUTIONS
O MECHANICALMIXTURES
o
o.2
0.4
0.6
0.8
1.0
MOL FRACTIONFeTiO3
Fig. 2. Reactionenthalpy(kJ/mol) from drop-solutionexperimentsat 1057K in flowingAr vs. mole fractionilmenite.
The opencirclesrepresent
the reactionenthalpies
for individual
with mechanical
mixturesof ilmeniteandhematite,
experiments
measured
and the solidcirclesrepresent
the reactionenthalpies
for the FerOr-FeTiO,solid solutions.The reactionenthalpyincludescontributionsfrom the heatcontentanddissolution.The
2 sd ofthe mean.
errorbarsrepresent
constrain parameters in microscopically based models
such as CVM, but doing so is beyond the scopeof this
paper.
The data can be fitted by a subregular(two-parameter)
solution model
AI1-,*: x,*(1 - x,,-)(79.3788- 120.843x,,-).(1)
The changein sign in the enthalpy of mixing is reflected
in the opposite signs of the two parameters. We stress
that the above polynomial, though convenient, has little
physical significancebecauseof the complex variation of
both short- and long-range order with composition and
temperature (seebelow).
Enthalpies of disordering
Excessenthalpies for each solid solution are shown in
Figure 3. Solid solutionswith compositionsfrom x,,- :
In order to measurethe enthalpy of disordering, com0 to 0.65 show endothermicenthalpiesof mixing, where- positionsfor x,,- : 0.6 to 0.85 were annealedat temperas solid solutionswith compositionsfrom x'- : 0.65 to atures below the order-disorder transition and their catI show exothermic enthalpiesof mixing. These results ion distributions measured(Table I of this paper; Brown
are consistent with the model of Burton and Davidson et al., 1993).The order parameterdeterminedfrom sat(1988),which incorporatesdominantly positive (i.e., re- uration magnetizationmeasurements,OM,,varies directly
pulsive) intralayer Fe2+-Ti4+interactionsin hematite-rich with the crystallographicsite occupanciesof Tia+, Fe3+,
solid solutionsand dominantly negative(i.e., attractive, and Fe2*. From a crystallographic perspective,the three
ordenng) interlayer Fe2+-Ti4+interactionsin ilmenite-rich cations would be distributed over the A and B sublattices
solid solutions. However, other interpretations are not such that charge balance over the structure would be
ruled out; e.g.,size effectsresult in the repulsiveforces maintained and cation to cation repulsion across the
that drive phaseseparation,whereasattractive interlayer shared octahedral face would be minimized. Minimizaforces drive ordering. The data obtained can be used to tion of the highly repulsive Ti4+-Ti4+pairs in both or-
BROWN AND NAVROTSKY: HEMATITE-ILMENITE
489
SOLID SOLUTIONS
: 3.3 J/(mol'
Tre|-e3. Entropiesof mixingcalculatedwith S"."',,,.
K) and S-",.'m: 0 J/(mol' K) for the Fe203 -FeTiO3solid
solutions
'-'6,0
d.
-1
F-
2.0
5
X
llr
\
-2.0
o.2
0.4
0.6
0.8
1.0
M O L FR A C T IONF e T i O3
Fig. 3. Enthalpiesof mixing at 1057K for FerO.-FeTiO.
solid solutions(solidcircles)calculatedfrom the fit to the data
2
the propagated
in Fig. 2 andTable3. The errorbarsrepresent
idealbesd of the mean.The line with long dashesrepresents
onesetof posihavior.The curveswith shortdashesrepresent
tive (upperdashedcurve)and negative(lowerdashedcurve)
enthalpiesof mixing(seetext)that wouldsumto givethe meaof mixing.
suredenthalpies
Annealing f
(K)
Mole
fraction
FeTiO"
AS..'
[J/(mol K)]
1338
1573
1126
977
1573
1273
1023
1573
1073
998
923
1573
1573
1 0000
0 8630
0.8570
0.8595
0.7001
0.7023
0.7029
0.6094
0.5993
0.6013
0.5929
0.4084
0.2024
0
10.81(63)t
11.32(56)
10.98(61)
14.18(51)
13.36(64)
13 . 11 ( 7 5 )
16.14(s0)
1s.49(34)
14.53(56)
13.75(70)
14.59(55)
9.91(27)
as-,,"
tJ/(mol K)l
0
13.66(63)
14.1s(56)
13 82(61)
16.49(51)
15.67(64)
15.43(75)
18.15(50)
17.47(34)
16.50(56)
15.71(70)
15.94(55)
10.58(27)
(rMs
1000
0.616
0.s80
0.607
0.531
0 635
0.660
0.020
0.353
0.558
0 663
0.005
0
t The AS., calculatedwith S-.,,,. : 3.3 J/(mol K).
t. The as-,, calculatedwith s-",,,. : 0 J/(mol' K).
t The errors represent the uncertainty in S-".*, calculated assuming
the maximumerror (i.e.,0.024)in the Ti site-occupancydata, adjusting
the other site occupancies according to compositional constraints' and
assuming that Fe3* is distributed equally on the A and B sublattices(see
text for discussion).
dominates and supports our interpretation ofthe change
in sign of the enthalpy of mixing curve (above).Thus,
short-range order is suggestedby both the increasingly
negative enthalpies of mixing in the intermediate compositional range and the insignificant enthalpy changes
for compositions with large changesin degree of longdered and disorderedsolid solutionsof all compositions rangeorder (e.g.,xu- : 0.6). As long-rangeorder begins
occurs when Fer+ is distributed equally in the A and B to dominate (closeto ilmenite compositions),the enthalsites.Therefore,assumingthat Fe3* is equally distributed py of mixing becomesexothermic.
over both sublattices, we need only be concerned with
FRoM PHAsE EQUTLTBRTA
Cs,cuurroNs
the orderingofFe2* and Tia+ on the A and B sublattices.
The order parameter, Or., is related to the site occupanThe purpose of this section is to estimate free eneryies
ciesby the generaldefinition
of mixing in hematite-ilmenite solid solutions for the
equilibrium betweensesquioxideand spinel phases.These
Q., : (Xo..-o XF"'t.)/(Xp",*.a* Xe"'*,g)
free eneryiesare then combined with our measured en-- (xrn*.o - xr,,-.)/(xr1n*.a-F x1;n*,s)
(2) thalpies of mixing to estimate entropies of mixing in hematite-ilmenite solid solutions and to provide evidence
where Xo.,*.ois the sublattice mole fraction of Fe2* in the
for the diminution of configurational entropy due to both
A sublattice(Brown et al., 1993).
short- and long-rangeorder.
Table 3 shows that as the composition approachesilmenite, the rangein the order parameter for different anNoNsrorcrrroMETRrc SPTNELsolrD soLUTroNs
nealing temperaturesbecomessmaller (i.e., for compoCalculation of activity and free energy of
sition with x,,- : 0.6 the order parameter varies from 0
transformation of Fee,rO.(a-'y)
to 0.66, for x,* : 0.7 it variesfrom 0.53 to 0.66, and for
x,,- :0.85 the rangeis only 0.58 to 0.61).This decrease The data of Taylor (1964) showed that at 1573 K a
in variation of the order parameter as compositions be- significant range of nonstoichiometry exists in the ulvijcome more ilmenite rich suggeststhat the high-temper- spinel-magnetitesolid solution series(dashedline in Fig.
ature disordered state becomesfrom difficult to quench 4). Schmalzried(1983)hasinvestigatedthe rangein non(Brownet al., 1993).
stoichiometry in Fe.Ooat temperaturesbetween ll73 and
Enthalpiesof drop solution (Table 2) measuredon these 1673K. His resultsare listed in Table 4 in terms of mole
and FerOo.Combining the data of Taylor
samplesare indistinguishablewithin the resolution of the fraction Fee,,Oo
calorimetric experiments(+3 kJlmoD. Theseresults sug- (1964\ andSchmalzried (1983),the limit of nonstoichigestthat significant differencesin the cation distributions ometry for FerOo-FerTiOosolid solutions has been estiin this compositional range do not result in significant mated (dashed line in Fig. 4) and used to estimate the
enthalpy changes.This is permissible if short-rangeorder compositions of the nonstoichiometric spinel solid solu-
490
BROWN AND NAVROTSKY: HEMATITE-ILMENITE
SOLID SOLUTIONS
TABLE
4. Solubilitydata for Fer"Oo
in Fe"Oocoexistingwith FerO"
at temperaturesbetween 1173 and 1573
4.
-4,
r(K)
o/Fe'
1573
1473
1373
1273
1173
1.364
1 357
1 349
1.341
1 337
Xr".o,**
4"
xcey,oo
fA.
lq
FerTiOE-FeTirO6 Join
Fe203-FeTiO3
Fer0a-FerTiO{
Join
F e Oo .
W E I G H T F R A C T I O NF e r O 3
0
Fe2O3
Fig. 4. Ternary phase diagram for the FeO-TiOr-Fe,O, system (modified from Taylor, 1964).The composition in weight
fraction oxides (solid circles) is plotted for each of four Iog /o.
(-0.68, -3.43, -6.0, -8.0) from the experimentaldata of Taylor (1964).The dashedline represents
the limit of nonstoichiometry in the spinel solid solutions assumedin our model (seetext).
The open circle representsthe composition of Fe.Onin equilibrium with Fe,O. at 1573 K from Schmalzried(1983). The
joins are
Fe.TiOr-FeTi.O,, FerO.-FeTiO., and FerOo-FerTiOo
shown for reference.
tions coexisting with ilmenite-hematite solid solutions
(Table 5). These results are in good agreementwith the
oxidation boundary calculated by Aragon and McCallister (1982). If enough fo, dala were distributed in
the Fe-Ti-O ternary, one could calculatethe activities of
all oxide components by appropriate ternary Gibbs-Duhem integrations.After severalattempts in this direction,
we concluded that the data are too sparse at any one
temperature for a meaningful calculation independent of
simplifying assumptions. We chose to apply a simple
mixing model to the spinel solid solutions. We could then
utilize the coexistingpairs at 1573K (our data and those
of Taylor, 1964) to determine activity-composition relations along the pseudobinaryFerO.-FeTiO. join. Our approach to calculating the free energiesof mixing in both
the titanomagnetite and hematite-ilmenite solid solution
seriesdiffers from that of Arag6n and McCallister (1982)
rn two respects.For the titanomagnetite series,we have
both assumed athermal mixing; however, Arag6n and
McCallister assumed mixing over only one site. In our
model, we have taken into accountcation ordering (i.e.,
mixing over two sites). In addition, their approach for
the hematite-ilmeniteserieswas to assumea simple mixing model for the excessfree energies.In our model, we
make no assumptions for the free energiesof mixing in
the hematite-ilmenite solid solution series(seebelow).
With the assumption that mixing in 7-Fee,.O.-FerTiO4FerOo solid solutions is athermal and govi:rned by the
configurational entropy, it is possible to calculate the
2932
2 948
2.966
2.981
2.992
0 795
0 845
0.899
0.947
0.974
0.205
0.155
0.101
0.053
0.026
. Estimatedfrom Fig 4 in Schmalzried(1983).
*. Calculatedas
: 100/0.333 : 300.003
[X'*or."n, x.".or."no,rJ/[x.q."nr xrqreq.o.rl
:
300.003(Ax,.,",n*,) AX..o, - n*,
(: 100) and Xr".o4r.eo.)
(: 0) are the mole fractionsof Fe3O4
where xr".o,,."n,
in pure Fe3O4and 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,respectivelv
change in free energy relative to hematite (FerO, :
a-Fey.O), magnetite(FerOo),and ulvospinel (FerTiOo)as
(3)
AG* : yL,G,^
+ AG-,.
"(a-1)
where y is the mole fraction of 7-Fee,.O.in the ternary
spinel sotid solution and AG,*".(a-7) is the free energyof
transformation for the reaction
a-FeyrOo:7-FeBr.Oo.
(4)
This transformation free energy can be calculated from
the binary solid solubility data of Schmalzried(1983)
(Table 4). For an Feq,Oo-saturatedmagnetite phase coexistingwith hematite,&r.,,o,rnust be equal in both phases
at equilibrium. In hematite, r/r.e.o,is the free energyof formation per four O atoms mole of FerO.. In the saturated
7 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
: - ?"Asr.r,oo,".
R?" ln ao.r.on."
(8)
In this equation, ASr..non
is the partial molar entropy of
Fer,O+in the 7 phase(seebelow). Computation yields a
value for AG,."""(a-7): 14.6 + 0.6 kJ/mol. The error in
AG,.,." has been calculated assuminga compositional error of 2 molo/oFeu.Ooin the solubility data of Schmalzried (1983) (Table'4). These data can also be used to
estimatethe enthalpy and entropy of the transition Af/,,"n,
and A,St.,.",respectively,where
(9)
dac,,"""/dT: -a.s,*,".
The AG,.,.. is calculated as described above. The transition temperature where pure a-FeqOotransforms to pure
BROWN AND NAVROTSKY: HEMATITE-ILMENITE
491
SOLID SOLUTIONS
an.-, and.4,. calculated
aFer,oo.",
TABLE5. Compositionsof the nonstoichiometricspinel solid so- TleLe 6. Valuesof aFer.oo.,,
spineland ilmenfor the coexistihgnonstciichiometric
lutionsand coexistingilmenite-hematite
solidsolutions
ite-hematitesolid solutions,as discussedin the text
estimatedfrom Taylor (1964)
Nonstoichiometric
spinel solid solution
llmenite-hematite
solid solution
In fo.
-25.33
-23.03
-20.72
-18.42
- 16.12
-14.97
- 13.82
-11.51
-7.90
-3.7'l
Mole
fraction
FeTiO.
0.982
0.957
0.923
0.887
0.828
0.751
0.701
0.547
0.333
0
Mole
fraction
Fe,O.
0.018
0.043
o.o77
0.113
o.172
0.249
0.299
0.453
0.667
1.000
Mole
fraction
Fe.Oo
Mole
fraction
Fe,TiO4
Mole
fraction
Feq.On
0.009
0.038
0.o74
0.125
0.209
0.269
0.334
0.501
0.664
0.796
0.976
0 930
0.886
o.822
0.722
0.648
0.564
0.361
0.157
0.015
0.032
0.040
0.053
0.070
0.083
0.102
0.138
0.179
o.204
a rev"oo'
Elrey"o,t
In fo.
-25.33
-23.03
-20.72
*18'42
_16.12
_14.97
_1g.B2
11.51
-7.90
-3.71
9.10 x
1.79x
6.52 x
2'25 x
7.96 x
1.53x
2.90 x
8.75x
0.199
0.326
10 6
10-"
10 4
1o-"
10 3
10 ,
10 ,
10 '?
?m
2.9x105
5.5x100
2.0x103
6.9x103
2.4x102
4.7x102
9.9x10,
0.268
0 611
1.000
4t^
4.0x100
3.6x10.
9.5 x 10-"
2.4x10'z
6.1x10'
0.100
0.163
0.371
0.691
1.000
0.9639
0.9003
0.8462
0.7711
0.6642
0.5844
0.4869
0.2930
0.1453
0
occupanciesof the end-members are defined as ([ ] :
octahedral site, ( ) : tetrahedral site, fl : vacancies):
Nofe.'convertedfrom weight fraction FeO,TiOr, and FerO3(from Taylor,
1964).
FerOo: random spinel at high temperature
(Feif Fe'z,u)F el| Fe|| )o'
7-FeqaO.can be estimated by extrapolating the FerOoFery,Oo
solid solubility data (Table 4) to pure "y-FeqOo.It
Fe",Oo: random spinel at high temperature
is important to remember, however, that the limit of the
Thus, the
solubility data is 0.205 mole fraction Feq,,Oo.
(Fefij'n.o,, ,)[Fefirrlo ,rr]Oo
extrapolated temperature of 3135 K must be taken with
FerTiOo : inverse spinel at high temperature
caution. In any case,it is above the actual decomposition
and melting temperatures. This transition temperature
(Fe'z*)[Fe'z*Tio*
]Oo.
can be combined with the transition entropy determined
by Equation 9 at equilibrium where
The reaction describing the solid solution can be written
(10) as follows, assumingthat the solid solution maintains the
A.I1,*..: fAS,.""".
site preferencesof the end-members(i.e., Ti in octahedral
Values of A.FI,*."and A,S,."".are 36.1 kJlmol and ll.5
sile only, tr distributed randomly, etc.)
J/(mol.K), respectively.The free energyof transition can
x (Fe||Fe1,i)[Fei.Fei*IO.
then be calculatedto be 17.9 kJlmol at 1573 K from these
with
values. This value is in reasonableagreement
the
+ y(Feflfrnlo ,,,)[Fe]irrtro rrrlOo
value determined with Equations 7 and 8, when one considersthe uncertainty in the transition temperatureat pure
+ {l - y - x}(Fe'?+)[Fe2*Ti4*]Oo
FeqOo. Becauseof this uncertainty, we have chosen to
: (Fefl.{rr,*ryrFef1, ',rrE6,,,,)
usqthe value determined from Equations 7 and 8 in the
calculations below.
'[Fe]17s,*o.orFe11
y,%[Jo222yTii1,,]O4 (13)
For the Ti-bearing solid solutions,
Itree,.o'o:
lr9..yroo,.+
RZ
ln aF.troo,o.
(l l)
Therefore, at equilibrium between a nonstoichiometric
spinel solid solution and an ilmenite-hematite solid solution,
RZ ln cr.ro,.. : AG,.","(a-7)+ R7" Irr ar"q,oo.,. (12)
The data of Taylor (1964) were usedto estimatethe compositions of the nonstoichiometric spinel solid solutions
in equilibrium with hematite-ilmenite solid solutions
(Table 5). Values for er.r,,o*ozlrd er.*,,o'.,are listed in
Table 6.
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
AS".*:
S"onr.r. -
XS"onip.:or
-./S"o'tn"yroa
-
ZSconlFeut'or
(14)
wherez:(l -x-y)and
S"o.r"": -R{(0.889y + Vr)ln(0.889y+'%)
+(l-y-'1lr)lrr(l-y-'4)
+ ( 0 . 1I l y l n 0 . 1I l y )
+ (1.778y+ a4)ln(1.778y+
Calculations of the entropy of mixing in the
spinel phase
+(l-y-4)ln(I-y-1,)/2
Configurational entropies and the correspondinglystatistically ideal entropies of mixing fory-FeqOo-FerOoFerTiOo solid solutions are calculatedas follows. The site
+ 0 . 2 2 2 yl n 0 . 1I l y
-x-y)/2
+(l -x-y)ln(l
"'/,)/2
(15)
492
BROWN AND NAVROTSKY: HEMATITE-ILMENITE
SOLID SOLUTIONS
This gives
T=1573K
S..",: -R{xl/.ln(2.667y
+ 2x) + t/rln(3 - 3y - 2x)
+ alln(5.334y + 4x) + lln(3
- %ln2 - l,ln 4l
fi
- 3y - x)
+ y [ 0 . 3 3 3l n y - 0 . 8 8 9l n 3
- 0.6
>t
P
+ 0.889 ln(2.667y + 2x)
,
+ l . 7 7 8 l n ( 5 . 3 3 4 y+ 4 x ) - t . 7 7 8 I n 6
-2.667 ln 0.8891
o 0.4
+ zf-ln 3 + ln(3 - 3y - 2x)
-ln6+ln(3-3y-x)
+ ln(l - x - y) + ln 2l).
0
0.2
o.4
0
0.6
0.8
1. 0
MOL FRACTION FeTiO3
Fig. 5. Activity-compositionrelationsat 1573 K in Fe.O,FeTiO. solid solutions. The activity of FerO, (open circles and
solid curve) has beencalculatedwith our model for nonstoichiometric spinel solid solutionsand the data of Taylor (1964) for
coexisting spinel and ilmenite solid solutions (seetext). The activity of FeTiO. (solid circles, solid curve) has been calculated
by Gibbs-Duhem integration (seetext). In addition, the activities
of FerO. and FeTiO. (curves with small dashes)have also been
calculatedfrom the empirical formulation of Ghiorso (1990).
The lines with large dashesindicate the condition a, : x,, where
i is eitherFerO,or FeTiOr.
: -R{2
S"o,r.n".oo
: -R{2
Sconr,ne,r;on
ln % + lfl t4}
(16)
ln %}
(17)
S c o n r , r . s , , o- R
o :{ 0 . l l l
ln 0.lll
+ 0.889 In0.889
+ 1 . 7 7 8 I n0 . 8 8 9+ 0 . 2 2 2 1 n
0.111).
(l8)
By combining terms, this becomes
,S.".r: -R{0.333y ln y - (l - 0.1Ily - 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)rn(3-3y-x)
+(1-x-y)ln(l-x-y)
(2r)
The partial molar entropies can then be written as
: -R ln[(2.667y + 2x)4(3 - 3y - 2x)/,
Asp..oo
.(5.334yt 4x)Y.
'(3 - 3Y - x)"(2)-t.(4)-t,l
Q2)
088e(2.667y
: -R ln[(y).333(3)
Asr",ooo
+ 2y1oa*
' (5.334Y I
4x)t ttz($)-t tts
.(0.889;'*'1
Q3)
ASp":rio,: -R ln[(3)-'(3 - 3y - 2x)(6) t
'(3 -3y -rxl
- x-y)(2)).
(24)
We believe that for the high temperature of equilibrium involved (1573 K), the above model, which assumes
maximum randomnessand only {6lTi4+,is a good approximation. Other modelsincorporatingFez+-Fer+sitepreferenceequilibria (e.g.,O'Neill and Navrotsky, 1984)
would give slightly different results. In view of the improbability of completely quenching the high-temperature structural state and the ambiguity in even distinguishingFe2+and Fe3+at high temperature,this simplest
model is probably the best approach.
sprNEL-sESeuroxrDE pArRS
Activities of FerO. and FeTiO. in the sesquioxidephase
Cor,xrsrrNc
The activity of FerO, in the ilmenite-hematite solid
solution series,4n"-, can be calculated from the activity
of FeyrOoin the spinel solid solution series(Eq. 12) and
the relation
aF.y.o*o:
@n )\.
(2s)
Values for activities of FerO. are shown in Table 6 and
Figure 5. No polynomial form could be found to fit the
activity-composition relations, such that the a-x relations
- 2 . 6 6 7 yl n 0 . 8 8 9 ) .
( 1 9 ) reduce to Raoult's law as the mole fraction approaches
1.0. Such complex behavior is not surprising,since the
This equation can be written in terms of the partial molar
mixing behavior results from the interplay between negentropresas
ative and positive interactions that will vary as a function
AS.o"r: xASo.,o, -F j/AJ.",,,o. * zAJ."rIo..
(20)
of temperatureand composition.
+ (l - ,1, - y)ln2 - al.ln4
BROWN AND NAVROTSKY: HEMATITE-ILMENITE
The activity of FeTiOr, eih, earrbe calculatedby GibbsDuhem integration, where
ln 4'-:
-,[,::.".,(xn"-/x,,-)d
ln 4n.-.
(26)
In this equation, xn.- and x'm are the mole fractions of
FerO. and FeTiO., respectively,in the ilmenite solid solution coexistingwith the spinel solid solution. The above
integration was carried out by plotting xn ^/ xu^vs. ln 4n"and fitting these data with exponential equations such
that ln an"- would tend to negative infinity as xn"-/x,,approachedzero, and xn" /xu^would tend to positive infinity as ln an"- approached one. Calculated results are
shown in Figure 5 and Table 6. In addition to the activities calculatedfrom our microscopically basedmodel for
the spinel solid solution series,activities calculatedusing
the empirical model of Ghiorso (1990)are also shownfor
comparison. Both sets of activities show negative deviations from Raoult's Law and agreemoderately well with
eachother. The integral free energiesof mixing agreequite
well (seebelow).
The free energiesof mixing can be calculated from the
activities 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 hematite, respectively, determined from our model of the
activity of FeqOo in the nonstoichiometric spinel solid
solution. Valuris for the free energy of mixing are calculated in Table 7.
ENrnoprrs oF MDilNG rN FerOr-FeTiO,
Maximum configurational entropies of mixing from
site occupancies
Calculation of the entropiesof mixing in the FerO.FeTiO, system requires explicit definition of the configurational entropies in the end-member components. For
hematite, this definition is obvious with S."".n.. : 0
J/(mol.K). For ilmenite, cation orderinghasbeenstudied
extensively both in situ at high temperatures and pressures and on samplesquenched from high temperatures
(e.g.,Wechslerand Prewitt, 1984; Shiraneer al., 1962).
The neutron diffraction data of Shirane et al. (1962) on
quenched samples have shown ilmenite to have (x,'- x) : 0.95 [i.e., (x,,- - x): degreeof order : sublattice
mole fraction Ti in the B sitel, although no error bars
have been given for this result (S"."ru- : 3.30). The diffraction data of Wechslerand Prewitt (1984) have shown
that no appreciabledisorder occurs at high temperatures
(S..".u- : 0). Our determination of the degreeof order in
ilmenite from saturation magnetizationmeasurementsto
4 K (Brown et al., 1993)agreewith the data of Wechsler
and Prewitt (1984).
Configurational entropies ofthe solid solutions can be
calculatedfrom measuredsite occupancies(Table 1), following the formulation of Thompson (1969, 1970)
493
SOLID SOLUTIONS
Trau 7. Calculatedfree energiesand entropiesof mixing
Phaseequilibria
and calorimetry
Mole
fraction
FeTiOs
1
0.982
0.957
0.923
0.887
0 828
0.751
0.701
o.547
0.330
0
AS.,,'
AG-,,
(kJ/mol) U/(mol.K)l
Site-occupancydata
Mole
AS.',t f
fraction AS.',-" t
FeTiO. F/(mol'K)l [J/(mol'K)]
100
00
-2.314
1.310(0.2) 0.8630
,4.478
1.877(0.s) 0 7 0 0 1
2.789(0.8) 0 . 6 0 9 4
6 70s
-8.527
3.629(0.9) 0.4084
-10.723 4 924(1.61 0.2402
-12.775 6.751(2.5) 0 0 0
- 13.691 7.97s(3.1
)
-14.644 11.384(3.6)
-11629 12.903(4.3)
00
10.81(63)
14.18(s1)
16.14(s0)
14.59(55)
9.91(27)
13.66(63)
16.49(51)
18.15(50)
15.94(55)
10.s8(27)
* Errors (in parentheses)in AS-,, have been calculatedfrom the error
on the fit to aH-,,.
". CalculatedAS-,, with site-occupancydata at 1573 K.
t The errors (in parentheses)represent the uncertainty in S"-r"., calculatedassumingthe maximumerror (i.e.,0 024) in the Ti site-occupancy
data, adiustingthe other site occupanciesaccordingto compositionalconstraints, and assuming that Fe3| is distributed equally on the A and B
sublattices (see text for discussion).
+ S*,,,- : 3.30 J/(mol K) with site occupancydata 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 is
important to note that these configurational entropies
represent maximum values in the sensethat any shortrange order on the A and B sublatticesis neglected.The
entropy of mixing is then defined as the difference between the configurational entropy of the solid solution'
S.""r,.,and the weighted sum of the standard-stateconfigurational entropiesof the end-members(i.e.,S.""rn.-and
S.""t',.)
- (l - x,r-)S.-r.n"- (29)
AS-" : S.*r.* - x'msconr.'m
where x,,- is the mole fraction of FeTiOt in the solid
solution. Calculations of the entropies of mixing at various temperatureshave beenmade for both standardstates
:
[Table 3 for S.."ru- 0, Table 3 and Figure 6 for S.".r',: 3.3 J/(mol'K)1. Figure 6 showsthe changein the entropy of mixing as a function of annealing temperature
and reflectsthe increasein order with decreasingannealing temperaturefor solid solutionswith compositionsfrom
x'- : 0.4 to 0.85.
Calculation of entropies of mixing from
AG-,* and AII-'The free energiesof mixing determined from the phase
equilibria data can then be used to calculatethe entropy
of mixing, AS-t-, with the relation
(-aG-* + LH^)/T:
aS-,..
(30)
In this equation, the enthalpies of mixing are those determined experimentally (Eq. l). Calculated entropies of
mixing are listed in Table 7 and Figure 7. Values for AS-'^
494
BROWN AND NAVROTSKY: HEMATITE-ILMENITE
SOLID SOLUTIONS
T-t573K
. I'ROM MEASURED CATION DISTRIBUTIONS
O FROM CALCULATED AdM' AND MEASURED AI/V,.
(cHtoRSo.1090) lNn MEISURED-'liru,,
a FRoM Mu
-15
r-
-lo
-
d
\ro
\rz
h
F-
.I
xB
d
-T-1573K
---T-t273K
I=998K
T=923K
0.2
0.4
0.6
Vl
0.8
M O L F RA C T ION F e T i O3
0
0.2
0.4
0.6
0.8
MOL FRACTION
N F
FeTiO3
eTiO3
1.0
Fig. 6. Entropiesof mixing calculatedfrom site-occupancy
datavs. mole fractionFeTiO, for differentannealingtemperaFig. 7. Solidcirclesrepresent
the entropiesof mixingcalcuturesand S.""r,,-: 3.3 J/(mol.K).The decreasing
entropyof latedfromsite-occupancy
datafor S...,,,-: 3.30J/(molK). The
mixing with decreasing
temperatureexpresses
the tncreasetn errorsarerepresented
by the symbolsize.The opencirclesrepcation order (seeTable l). The solid line represents
data on resentthe entropiesof mixing calculatedwith the freeenergies
quenched
samples
from 1573K. Thedashed
linesrepresent
en- of mixing from our nonstoichiometric
spinel model and the
tropiesof mixingcalculated
for lowertemperature
annealsfrom measured
enthalpies
of mixing.The errorbarsrepresent
the un12'73,993,
and923K (dashsizedecreases
with decreasing
tem- certaintyin this calculated
entropydueto the errorin the fit to
perature)
(seeTable3).
the enthalpyof mixing.The opentrianglesrepresentthe entropiesof mixing,calculatedusingfreeenergies
of mixingfrom the
( I 990),and our measured
empirical
formulation
of
Ghiorso
en(dashedcurve in Fig. 7) have also been calculated using
thalpiesof mixing. Errorsbarsfor thesedatahavebeenelimi(Ghiorso,
AG-*
1990) and our experimentally deter- natedfor clarity,but arethe sameas for
the opencircles.
mined enthalpies of mixing. The agreementbetween the
two calculated entropies of mixing shows that determination of the free energies of mixing in FerOr-FeTiO, tions (i.e., magnetic moments within each sublattice are
from our microscopically basedmodel for the spinel solid aligned parallel) and antiferromagnetic interlayer intersolutions is in reasonableagreementwith Ghiorso's em- actionsbetweenFe atoms on the A and B sublattices(i.e..
pirical fit to the existing data in the hematite-ilmenite the magnetic moments of alternatingA and B layerspoint
solid solution series.Ghiorso'sempirical fit includesex- in opposite directions). Chemical ordering arisesfrom reperimental data on both the order-disordertransition and pulsive intralayer interaction and attractive interlayer inthe position of the solvus in the FerOr-FeTiO.solid so- teractions betweenFe2+and Ti4*. In attempting to model
lution series.In addition, both models show that the cal- the thermodynamic mixing properties by subregularand
culated entropies of mixing are significantly less positive regular solution parameterizations,one loses the nature
than those inferred from measured cation distributions, of these complex microscopic interactions that give rise
especiallyfor compositionswith ,r,,- = 0.3. This obser- to long-range and short-range ordering. CVM (cluster
vation is also consistentwith significant short-rangeorder variation method) calculations(Burton, 1984;Burton and
and may imply clusteringof Ti in the Ti-rich B sublattice. Kikuchi, 1984;Burton, 1985;BurtonandDavidson,1988)
include the signs and magnitudes of these interactions
DrscussroN
and thus allow for more atomistically baseddescriptions
Solid solutionsin the FerO.-FeTiO3systemare char- of the thermodynamic properties of this solid solution
acteized by two varieties of ordering: antiferromagnetic series.They have been applied to carbonates(Capobianordering of Fe in the temperaturerangeof -60-953 K, co et al., 1987)and to FerO,-FeTiO.(Burton, 1984;Burand chemical ordering of Fe2* and Ti4+ betweenadjacent ton, I 985; Burton and Davidson, I 988).The presentdata
basal planes at temperaturesbetween873 and 1350 K will be useful to constrain their parameters better, but
(Ishikawa, 1958;Ishikawaand Akimoro. 1957:Ishikawa suchcalculationsare outsidethe scopeofthe current study.
and Syono, 1963; Shiraneet al., 1962).Magnetic ordering
Evidencefor short-rangeordering can be found for both
results from ferromagnetic intralayer magnetic interac- magnetic and chemical ordering reactions. For example,
BROWN AND NAVROTSKY: HEMATITE-ILMENITE
Gronvold and Samuelsen(1975) and Holland (1989)reported that hematite exhibits a long tail in the tr heatcapacity anomaly associated with magnetic ordering
(transition temperature, 7., : 955 K), suggestingsignificant incipient disorder in the magnetic spins even at 298
K. This disorder amounts to l7o/oof the maximum entropy that would be gained at f. Evidenceof short-range
chemical order can be seenfor the rangeof compositions
betweenx,,- : 0.4 and 0.8. During quenchingfrom temperatures above the order-disorder transition, both natural and synthetic solid solutions attempt to order. Local
ordered regions nucleate and grow together. When two
impinging ordered regionsare in phasewith one another,
one large ordered region is formed. When they are out of
phase,a compositionally distinct, cation-disorderedtwindomain boundary (TDB) separates the two ordered
regions.Suchmicroscopicchemicaldomainsand domain
boundaries representlocal variations in chemical order.
These domains and domain boundaries,in turn, profoundly affect the magnetic ordering of solid solutions in
this compositional range. Both reversed magnetization
(Nord and Lawson, 1989,1992:-Hoffman, 1992;Lawson
et al., l98l) and high coercivities(Brown et al., 1993)are
intimately related to the development of TDBs during
quenchingthrough the order-disordertransition. The twin
domain boundariesprovide a second,more Fe-rich, magnetic phasethat has a higher Curie temperatureand causes the reversed magnetization seen in quenched natural
and syntheticsamples(Lawson,et al., l98l; Nord and
Lawson. 1989: Nord and Lawson, 1992). In addition,
Nord and Lawson (1992)proposedthat when the chemical domain size is on the order of the magnetic domain
size, the magneticdomain walls become pinned in the
chemical domain walls. That gives rise to high coercivities in intermediate compositions and locally complex
magnetlc structure.
Evidence for short-rangeorder can also be seenin the
thermodynamic mixing propertiesof the FeTiOr-FerO,
solid solution series. Measured enthalpies of mixing,
A11-*, can be interpreted as follows. The enthalpy of mixing for each solid solution composition arises from the
sum of the atomic interaction energiesin the crystal. In
a solid solution series where both cation ordering and
exsolution occur at varying temperatures,the measured
mixing enthalpy can be broken down into two contributions: a positive mixing curve that will drive unmixing
and a negative mixing curve that will drive ordering (Fig.
3). The functional form of the two opposite components
ofthe total enthalpy dependson the strengthofthe positive and negative atomic interaction energiesas a function of temperatureand composition.One possibleset of
curves describing the two contributions to the measured
enthalpy of mixing in FerOr-FeTiO, solid solutions is
shown in Figure 3 (small dashedcurves).As drawn, these
curves suggestthat in hematite-rich solid solutions, positive interaction energiesdominate during initial substitution of FeTiO. into FerO.. Such behavior could arise
from an increasein cation to cation repulsion acrossthe
SOLID SOLUTIONS
495
sharedoctahedralfacedue to increasingTi4* content and
accompanyinglocal displacementsof the O ions to provide more shielding acrossthe shared face. In ilmeniterich solid solutions, negative ordering interactions dominate and reduce cation to cation repulsion across the
shared face by placing Fe2+and Tia* on separatesublattices.In the intermediatecompositionalrange,as the Tia*
content increases,the driving force for ordering increases,
resulting in a progression from short-range ordering in
the more Ti-poor solid solution to long-rangeorderingin
the more Ti-rich solid solutions.
The implications of this work are important in terms
of the developmentof thermodynamic models used to
predict the activity-composition relationships at geologically relevant temperatures(-673-1173 K). This temperature range is preciselythe range in which one would
expect significant short-range and long-range ordering
(both magneticand chemical) to affectthe thermodynamic properties. Paramagneticto antiferromagnetic transitions in this temperaturerange occur for compositions
from x,,- : 0 to 0.4. Chemical ordering also occurs in
this temperature range for compositions from x,,- : 0.5
to 0.65 and possiblyfrom as low as x,,- : 0.2. This low
by the high coercivcompositionalestimateis suggested
ities found in composition x'- : 0.2 arrd0.4 (Brown et
al., 1993).The complex temperatureand compositional
dependenceof both the chemical and magnetic ordering
reactions support use of such models as CVM, which include the atomic interactions that govern this complex
Z- and X-dependent behavior.
At first thought, one might be tempted to construct an
empirical activity-compositionmodel basedon our enthalpies of mixing (Eq. l), incorporatingour excessentropies of mixing (open circlesin Fig. 7). Algebraically'
that could be done, but we believe it would have little,
and only accidental, validity. Becausethe site occupancies and degreeof short-range order depend strongly on
temperature, it is highly likely that the entropy of mixing
likewise has a strong temperaturedependence.It seems
unlikely that the activities depend on temperature and
composition in a way that is simple enoughto be modeledreliably by simplepolynomials,especiallyi\the T-X
region ofthe order-disordertransition. The presentstudy,
in our interpretation,counselscaution in the indiscriminate useof ilmenite-spinelgeothermobarometryfor rocks,
especiallyfor lower gradesof metamorphism,since extrapolation of higher temperature(> I100 K) activity data
to the 700-1000 K rangemay be complicatedby orderdisorder phenomena.
The present data furnish a beginning to the quantification of theseinteractions,by providing constraintson
both the enthalpies and entropies of mixing. However'
the detailed temperature dependenceof both long- and
short-rangeorder needsfurther study, and the further development of CVM or other structurally basedmodels is
necessary.The present calorimetric data refer to the enthalpy of mixing at room temperature of sampleswith a
degree of short-range order and long-range order char-
496
BROWN AND NAVROTSKY: HEMATITE-ILMENITE
acteristic of some higher temperature, but probably not
preservingall the disorder of the high-temperaturestate.
Thus, although the changein sign of All-* with composition and the smaller than configurational AS-,. are very
suggestiveofmajor short-rangeorder, the data are insufficient to constrain quantitatively a microscopic mixing
model as a function of both temperature and composition. The development of a complete microscopically
based thermodynamic model for this system would require data on the in-situ state of ordering for each composition,heat-capacitydata for eachcomposition,and insitu volume data as a function of composition. Once these
data have been constrained,such a model could be constructed, and its free energiesof mixing might then be
used to parameterize more convenient empirical equations that would have the correct compositionand temperature dependence.
AcrNowr,rncMENTs
Funding for this researchwas provided by the Mineralogical Societyof
America through the M.S.A. Petrologygrant for 1989 (to N.E.B.) and by
the National ScienceFoundarion (grant DMR-8912549 and 9215802 to
A. Navrotsky) The authors would like to thank B.P. Burton, p.M. Davidson, and M.S. Ghiorso for their reviews of the manuscript.
RnrnnnNcns crrED
Andersen, D.J., and Lindstey, D.H. (1988) Internally consistentsolution
models for Fe-Mg-Mn-Ti oxides:Fe-Ti oxides.American Mineralogist,
73,714-726.
Arag6n, R., and McCallister, R H. (1982) Phaseand point defectequilibria in the titanomagnetite solid solution. Physics and Chemistry of
Minerals, 8, ll2-120.
Brown, N E, Navrotsky,A., Nord, G.L., Jr., and Banedee,S.K (1993)
Hematite-ilmenite (Fe.O.-FeTiO,) solid solutions: Determinations of
Fe-Ti order from magnetic properties. American Mineralogist, 78,94195l.
Buddington, A.F., and Lindsley, D.H. (1964) Iron-titanium oxide minerals and synthetic equivalents.Journal ofPetrology, 5, 310-357.
Burton, B.P. (1984) Thermodlmamic analysis of the system Fe,O.-FeTiO.. Physicsand Chemistry of Minerals, ll, 132-139.
-(1985)
Theoretical analysisof chemical and magnetic ordering in
the systemFe=O.-FeTiO,.American Mineralogist, 70, 1027-1035.
Burton, B.P., and Davidson, P.M. (1988) Multicritical phaserelations in
minerals. In S. Ghose, J.M.D. Coey, and E. Salje, Eds., Structural and
magneticphasetransitionsin minerals,p. 60-90. Springer-Verlag,New
York.
Burton, B.P., and Kikuchi, R. (1984) The antiferromagnetictransition in
FerO. in the singleprism approximation of the cluster variation method. Physicsand Chemistryof Minerals, I l, 125-l3l
Capobianco,C., Burton, 8.P., Davidson,P.M., and Navrotsky,A. (1987)
Structural and Calorimetric studies of order-disorder in CdMg(CO)r.
Journal of Solid State Chemistry, 71,214-223.
Ghiorso, M S. (1990) Thermodynamic properties of hematite-ilmenitegeikielite solid solutions. Contributions to Mineralogy and petrology,
104,645-667.
Gronvold, F., and Samuelsen,E.J. (1975) Heat capacity and thermodynamic propertiesof a-Fe,O, in the region 300 to 1050K: Antiferromagnetic transition. Journal of the Physics and Chemistry of Solids, 36,
249-256.
Haggerty,S.E. (1976) Opaque mineral oxides in terrestrial igneousrocks.
In Mineralogical Societyof America Reviews in Mineralogy, 3, HGIHGl00.
SOLID SOLUTIONS
Hoffman, K.A. (1992) Self-reversalof thermoremanentmagnetizationin
the ilmenite-hematite system: Order-disorder, symmetry, and spin
alignment. Journal of GeophysicalResearch,97, 10883- 10E95.
Holland, T.J.B. (1989) Dependenceofentropy on volume for silicateand
oxide minerals: A review and a predictive model. American Mineralogist,74,5-13.
Ishikawa, Y. (l 958) An order-disordertransformation phenomenain the
FeTiO,-Fe.O. solid solution series Journal of the Physical Society of
Japan,13, 828-837.
Ishikawa, Y., and Akimoto, S. (1957) Magnetic propertiesof the FeTiO.Fe,O, solid solution series.Journal of the Physical Society of Japan,
12,1083-1098.
Ishikawa, Y, and Syono, Y. (1963) Order-disorder transformation and
reverse thermo-remanent magnetism in the FeTiO,-Fe2Or system.
Journalofthe Physicsand ChemistryofSolids, 24,517-528.
Kikuchi, R. (1977) The cluster variation method. Journal de Physique,
colloqueC7, suppl. 12, tome 38, 307-313
Lawson,C.A., Nord, G.L., Jr., Dowty, E., and Hargraves,R.B (1981)
Antiphase domains and reversethermoremanentmagnetismin ilmenite-hematite minerals. Science,2 I 3, 1372-137 4.
Navrotsky, A. (1977) Progressand new directions in high temperature
calorimetry Physicsand Chemistry of Minerals, 2, 89-104.
Nord, G.L., Jr., and Lawson, C.A (1989) Order-disorder transition-induce twin domains and magnetic properties in ilmenite-hematite.
American Mineralogist, 74, 160-176.
-(1992)
Magnetic properties of ilmenite,o-hematitero:Effect of
transforrnation-inducedtwin boundarieson magneticstability and selfreversal.Journal of GeophysicalResearch,97, I 0897- I 09 I 0.
O'Neill, H St.C.,and Navrotsky, A. (1984) Cation distributions and thermodynamic propertiesof binary spinel solid solutions.American Min'1
eralogist,69, 33-7 53.
Powell, R., and Powell, M. (1977) Geothermometry and oxygen barometry using coexistingiron-titanium oxides:A reappraisal.Mineralogical
Magazine,41,257-263.
Schmalzned, H. (1983) Thermodynamics of compounds with narrow
rangesof nonstoichiometry.Berichteder Bunsen-Gesellschaft
liir physikafischeChemie,87, 726-7 33.
Senderov,E , Dogan, A.U., and Navrotsky, A. (1993) Nonstoichiometry
of magnetite-ulvtispinelsolid solutions quenchedfrom 1300t. American Mineralogist, 78, 565- 573.
Shirane,G., Cox, D E., Takei, W.J., and Ruby, S.L. (1962)A study of
the magnetic properties of the FeTiO.-Fe,Or system by neutron diffraction and the Miissbauer effect Journal of the Physical Society of
J a p a n1
, 7 ,1 5 9 8 - l 6 l l .
Spencer,K. J , and Lindsley, D.H. (l 98 l) A solution model for coexisting
iron-titanium oxides.AmericanMineralogist,66, I 189-1201.
Takayama-Muromachi, E., and Navrotsky, A. (1988) Energeticsof compounds (A':+Ba+Or)with the perovskitestructure.Journal ofSolid State
Chemistry,72,244-256.
Taylor, R.W. (1964) Phase equilibria in the system FeO-FerO,-TiO, at
1300"C. AmericanMineralogist,49,l0l6-1030
Thompson, J.B , Jr. ( 1969)Chemical reactionsin crystals American Mineralogist,54,341-375
-(1970)
Chemical reactions in crystals: Corrections and clarifications. American Mineralogist, 55, 528-532.
Webster,A H, and Bright, N F.H. (1961)The systemiron-titanium-oxygen at 1200 .C and partial oxygen pressuresbetween I atm and 2 x
l0 11atm. Journalof the AmericanCeramicSociety,44,ll0-l16.
Wechsler, B.A., and Prewitt, C.T (1984) Crystal structure of ilmenite
(FeTiO.) at high temperature and at high pressure American Minera l o g i s t , 6 9 ,1 7 6 - 1 8 5 .
Zhon,Z, Navrotsky,A., and McClure, D.S. (1993)Oxidation statesof
copper in lead borate (2PbO BrOr) glass. Physics and Chemistry of
Glasses.34. 251-254.
Mrxuscnrrr REcErwDSemsusen 24, 1992
Mexuscnrrr AccEmEDJam;npy 12. 1994