Introduction

A recent experimental investigation of the significanceof Ti substitution into forsterite (Mg2SiO4) in the systemMgO-SiO2-TiO2 (Hermann et al in press) has raised theneed for accurate thermodynamic data for Mg2TiO4 spinel(the mineral qandilite) At present there exist some calori-metric data for this substance (reviewed by Eriksson ampPelton 1993) but these data are insufficient to give the freeenergy of formation as Mg2TiO4 is an inverse spinel whichis expected to have configurational entropy from mixing ofMg and Ti on the octahedral sites of the spinel structure(OrsquoNeill et al 2003b) Full development of this configu-rational entropy would contribute 115 JKmol inMg2TiO4 but this may be reduced if there is short-rangeordering Since Mg2TiO4 with the usual cubic spinel struc-ture (Fd3

ndashm) is known to undergo a phase transition on

cooling at ~ 1000ordmC to a tetragonal modification (P4122)in which Mg and Ti are ordered onto distinct octahedralsites (Wechsler amp Navrotsky 1984 Millard et al 1995)some short-range ordering is a possibility

Akimoto amp Syono (1967) found that Mg2TiO4 breaksdown to MgTiO3 (geikielite) plus MgO (periclase) withincreasing pressure according to the univariant reaction

Mg2TiO4 = MgTiO3 + MgO (1)Since this reaction occurs in an experimentally easilyaccessible region of P-T space and since there arecomplete calorimetric data for MgTiO3 and MgO (whichare both nearly pure stoichiometric phases in the systemMgO-TiO2) this reaction provides an excellent means of

determining the free energy of formation of Mg2TiO4which moreover can be done over an adequate temperaturespan to test whether the full configurational entropy of Mg-Ti mixing is developed However the study of Akimoto ampSyono (1967) is really only reconnaissance work in thattheir results do not bracket the reaction particularly snugly(see Fig 1) Hence the purpose of this study is to reinves-tigate reaction (1) applying modern piston-cylindermethods to obtain reversals that bracket the reaction withbetter precision than that achieved by Akimoto amp Syono(1967) and probably also with better accuracy as regardstemperature and pressure measurement

Experimental

Starting materials consisted of synthetic Mg2TiO4(material from the study of OrsquoNeill et al 2003b) andMgTiO3 synthesised similarly to Mg2TiO4 from a stoi-chiometric mixture of MgO and TiO2 ground underacetone in an agate mortar pressed into pellets and reactedat 1400degC in air in a box furnace MgO was from BDH(Analar grade) fired at 1200degC in air before use A startingmixture consisting of all three phases in the molar ratio of1 Mg2TiO4 to 1 MgTiO3 to 2 MgO (ie MgO in excess ofthat required by the stoichiometry of reaction 1) was thenprepared

Experiments at atmospheric pressure were made in avertical tube furnace in air with a type B thermocouplepositioned directly above the sample ensuring that the

Eur J Mineral2005 17 315-323

The free energy of formation of Mg2TiO4 (synthetic qandilite) an inversespinel with configurational entropy

HUGH ST C OrsquoNEILL and DEAN R SCOTT

Research School of Earth Sciences Australian National University Canberra ACT 0200 Australia

Abstract Synthetic qandilite (Mg2TiO4) is an inverse spinel in which random mixing of Mg and Ti on the octahedral site shouldgive rise to a configurational entropy of 115 J K-1 mol-1 The free energy of formation of Mg2TiO4 has been determined from thelocation of the univariant reaction MgTiO3 + MgO = Mg2TiO4 in pressure-temperature space between 0 and 20 kbar together withliterature data for MgTiO3 and MgO the results give So

(298K)= 1112 plusmn 06 J K-1 mol-1 for Mg2TiO4 This entropy is only 76 plusmn 09J K-1 mol-1 more than the calorimetrically determined value of 1036 plusmn 07 J K-1 mol-1 The discrepancy implies either consider-able short-range order of Mg and Ti in Mg2TiO4 or perhaps an error in the high-temperature heat-capacity data for MgTiO3 causedby disordering in this substance

Key-words spinel thermodynamic data order-disorder phase equilibria

0935-1221050017-0315 $ 405copy 2005 E Schweizerbartrsquosche Verlagsbuchhandlung D-70176 StuttgartDOI 1011270935-122120050017-0315

E-mail hughoneillanueduau

H St C OrsquoNeill D R Scott

temperature of the experiment was measured accurately toplusmn 1ordmC The thermocouple was checked against the meltingpoint of gold with results within 05ordmC of the recom-mended value of 106418ordmC (ITS-90) Previous work onMg2TiO4 has shown that the kinetics of reaction (1) are toosluggish for any perceptible reaction to occur at atmo-spheric pressure in the simple system MgO-TiO2 (Wechsleramp Navrotsky 1984) hence a flux of Na2B4O7 (dehydratedborax) was added The charge was run in a Pt capsule (5mm diameter) sealed at one end and crimped closed at theother to minimize loss of Na2O by volatilization This fluxproved less than completely ideal as it reacts with MgO toproduce a Mg-Ti-borate phase thus the direction of reac-tion cannot be inferred simply from a change in propor-tions of the phases Nevertheless the results are stronglysuggestive in that abundant euhedral Mg2TiO4 is observedat 1100degC but is completely absent at 1070degC Theformula of the Mg-Ti-borate is Mg3TiB2O8 from electronmicroprobe determination of MgO and TiO2 with B2O3estimated by difference Trials of a number of otherpossible fluxes (78 wt BaO + 22 wt B2O3 K2CO3Na2WO4) were unsuccessful due to their incompatibilitywith one or more of Mg2TiO4 MgTiO3 or MgO

Experiments at high pressure (6 to 20 kbar) wereconducted either in a 30 mm (at 12 kbar) or a 58rdquo (otherpressures) piston-cylinder apparatus with a NaCl-pyrexassembly surrounding a cylindrical graphite heater Thelengths of the cells were 60 mm for the 30 mm diametercell and 15rdquo (38 mm) for the 58rdquo cell and both cells useda graphite heater of 95 mm OD by 7 mm ID In order toallow initial friction to decay the run was first taken to800degC and the pressure of interest where it was held for~24 hours see Bose amp Ganguly (1995) for a discussion offriction decay in the piston-cylinder apparatus with NaCl asthe pressure medium The run was then heated to the finaldesired temperature and the pressure adjusted to the finaldesired value Samples were held in sealed Pt capsuleswith dimensions 23 mm OD 17 mm ID 5 mm longTemperature was controlled and measured with a type B

thermocouple positioned directly above the capsuleThermocouples were made from the same batches of wireas those used in the 1-bar experiments Temperatures in thepiston-cylinder runs were not corrected for any effect ofpressure apart from this the reported temperatures whichwere controlled to plusmn 1degC are thought to be accurate to plusmn5degC at T le 1400degC the uncertainty being caused by smalldeviations in sample position within the thermal profile ofthe piston-cylinder cell The thermal profile of the 30 mmcells should have a larger constant-temperature zone less-ening this source of error Temperatures are probably moreuncertain at T gt 1400degC due to greater temperature gradi-ents and also to several other factors to do with thermo-couple calibration and performance Pressures which arereported without any correction for friction are precise toplusmn 02 kbar and probably to plusmn 01 kbar in the 30 mmassembly

Following the quenching of each run the sample wasmounted in epoxy sectioned and polished for examinationusing the electron microprobe (CAMECA SX100) forback-scattered electron imaging and quantitative analysisby wavelength dispersive spectrometry The starting mate-rial and selected run products were also examined bypowder X-ray diffraction using a STOE STADIP diffrac-tometer in the transmission mode with monochromaticCoKα1 radiation (λ = 178897 Aring) scanning from 10 to130degC An internal standard of NIST 640c Si (a0 =54312 Aring) was used and the lattice parameters wereextracted by Rietveld refinement of the whole pattern usingthe program LHPM-Rietica (Hunter amp Howard 2000)

Results

Results are reported in Table 1 and the definingbrackets are plotted in Fig 1 Experiments at ge 10 kbar areat a sufficiently high temperature that the reaction takesplace rapidly and no flux is needed The results are straight-forward reversals which bracket the reaction to within a

316

Fig 1 Pressure-temperature brackets on theunivariant reaction MgTiO3 + MgO =Mg2TiO4 from this study and from Akimotoamp Syono (1967) Only the results from theexperiments defining the curve are shownThe solid curve is an empirical 4th orderpolynomial drawn through the brackets ofthis study (note that this univariant reactionis not linear in P-T space) The dashed line isthe equation quoted by Akimoto amp Syono(1967) P (kbar) = ndash18 + 0019 T (degC)Although their equation is in poor agreementwith the results of this study only one oftheir defining half-brackets (MT22 at 10kbar) violates the curve of this study

nominal 10 K at 10 and 15 kbar and 20 K at 20 kbar At12 kbar in the 30 mm pressure assembly where the controlof both temperature and pressure is sufficiently good towarrant the attempt at greater precision the bracket is only5 K in width

There is a remarkable asymmetry in the texturesproduced on either side of the reaction in these fluxlessexperiments On the high temperature side (Mg2TiO4stable) there is extensive recrystallization to the typicalgranular texture expected for textural equilibrium althoughin several runs close to the reaction boundary residualgrains of MgTiO3 persist but completely surrounded byMg2TiO4 and isolated from MgO (Fig 2a) By contrast onthe low temperature side of the reaction there is littlerecrystallization of the original MgTiO3 and MgO grainswhile what were originally grains of Mg2TiO4 have decom-

posed to a eutectoid-like intergrowth of MgTiO3 and MgO(Fig 2b) Despite this textural disequilibrium the extent ofreaction is complete in all cases with no residual Mg2TiO4persisting in any experiment

For runs at 6 kbar a small pinch (20 wt) of oxalic acidwas added to the first run (C1922 at 1250degC) A consider-able amount of MgO dissolved in the CO2-H2O fluidproduced by the decomposition of the oxalic acid drivingthe starting composition off the target stoichiometry whichmakes the result in terms of reaction direction difficult toascertain The MgTiO3 contains carbonatevapour ldquoholesrdquowhile the Mg2TiO4 does not (Fig 2c) which may perhapsindicate that Mg2TiO4 is stable alternatively this texturecould be due to growth of the MgTiO3 trapping the inclu-sions Hence we regard this run as completely ambiguousFor the next run at 6 kbar and 1250ordmC (D362) some extra

The free energy of formation of Mg2TiO4 spinel 317

Table 1 Experimental results

a +50 Na2B4O7 (dehydrated borax) b +20 Na2B4O7 c +20 oxalic acid d +10 oxalic acid e 30 mm cell all other high pressure runs in 15875 mm cells e no Mg2TiO4 instarting mix

Run P T Time Result(kbar) (degC) (h)

C19503a 00001 1050 048 MgTiO3 MgO Mg-Ti-borate quenched flux many crystals of MgTiO3 contain MgOinclusions

C8404b 00001 1070 046 MgTiO3 MgO Mg-Ti-borate quenched flux

C21703b 00001 1080 024 Euhedral Mg2TiO4 MgTiO3 corroded MgO Mg-Ti borate quenched flux

C26304be 00001 1088 066 Large euhedral Mg2TiO4 poikiliticallyenclosing some euhedral MgTiO3 and MgO MgTiO3 and corroded MgO Mg-Ti-borate quenched flux

C26503a 00001 1090 0235 Mg2TiO4MgTiO3MgO Mg-Ti-boratequenched flux reaction direction ambiguous

C14703b 00001 1100 048 Euhedral Mg2TiO4 MgTiO3 and Mg-Ti-borate corroded MgO quenched flux

D366 060 1240 168 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoC1922c 060 1250 006 Mg2TiO4 MgTiO3 with holes quench carbonateD362d 060 1250 048 No discernible reactionD369 060 1260 192 MgO + Mg2TiO4 100 reactionD350 100 1340 050 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoC1918 100 1360 006 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoD354 100 1370 006 MgO + Mg2TiO4 some unreacted MgTiO3

D343 100 1380 006 MgO + Mg2TiO4 100 reactionD342 100 1420 006 MgO + Mg2TiO4 100 reactionD340 100 1460 006 MgO + Mg2TiO4 100 reactionD338 100 1500 007 MgO + Mg2TiO4 100 reactionR204e 120 1405 006 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoR193e 120 1410 020 MgO + Mg2TiO4 some unreacted MgTiO3

D361 150 1480 006 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoD364 150 1490 006 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoD358 150 1500 006 MgO + Mg2TiO4 some unreacted MgTiO3

D363 200 1580 003 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoD360 200 1600 006 MgO + Mg2TiO4 few grains unreacted MgTiO3

D359 200 1640 003 MgO + Mg2TiO4 100 reaction

H St C OrsquoNeill D R Scott

MgO was added and the amount of oxalic acid reduced to10 This run produced no discernible reaction directionand the use of oxalic acid was abandoned The two subse-quent runs at 6 kbar produced an unambiguous bracketbetween 1240 and 1260degC

The bracket at atmospheric pressure is placed betweenthe run at 1070degC (C8404) which produced euhedralMgTiO3 plus MgO with no Mg2TiO4 (Figure 3a) and at1080degC (C21703) which contained Mg2TiO4 andMgTiO3 with MgO but with the Mg2TiO4 showing goodcrystal faces while the MgO appears somewhat corroded(Fig 3b) However no reaction direction could be inferredfrom a run at 1090degC in which all three phases werepresent but none show good crystal faces An additionalrun at 1088degC (C26304) which used a starting mix ofMgTiO3 MgO and flux only (ie no Mg2TiO4) producedlarge euhedrally faceted crystals of Mg2TiO4 poikiliticallyenclosing MgTiO3 and the Mg-Ti-borate phase with occa-sional small poorly faceted grains of MgO interpreted tobe residual (Fig 3c) but the unambiguous growth ofMg2TiO4 in this run implies clearly that it is stable at thistemperature Run C19503 which at 1050degC is well belowthe breakdown temperature shows many grains of MgTiO3enclosing MgO blobs (Fig 3d) which is somewhat remi-niscent of the ldquoeutectoid-likerdquo textures of the high-pressureexperiments and like for these runs we interpret thesegrains to be due to decomposition of original Mg2TiO4grains

The results are compared to the data of Akimoto ampSyono (1967) in Fig 1 At first sight there appears to be alarge difference between the results if ours are compared tothe equation for the univariant curve provided by Akimotoand Syono which is P (kbar) = -18 + 0019 T (degC) In factonly one experiment of Akimoto and Syono is actuallydiscrepant namely their run MT22 in which Mg2TiO4 isreported to break down to MgTiO3 and MgO at 10 kbar and1440degC (cf our bracket between 1360 and 1370degC at10 kbar) This is far outside any likely experimental uncer-tainty in pressure or temperature measurement and wesuggest an ad hoc experimental malfunction in run MT22such as piston seizure or thermocouple extrusion

Stoichiometry and composition of the run products

Like many spinels Mg2TiO4 might be anticipated toshow some non-stoichiometry by solid solution towardsMgTiO3 at high temperatures (ie like MgAl2O4 towardsγ-Al2O3) However electron microprobe analysis showedalmost negligible deviations in all runs from Mg2TiO4 stoi-chiometry (Table 2) It is probable that at atmospheric pres-sure the temperature at which reaction (1) occurs (ie1075 plusmn 10degC) is not high enough for deviations from stoi-chiometry to be significant while higher pressuressuppress the non-stoichiometry due to the relatively highpartial molar volume of the defect spinel componentSimilarly MgTiO3 was found to be stoichiometric withinanalytical error in all runs The observation of smallamounts of TiO2 (~ 06 wt corresponding to a mole frac-tion of 0003) in MgO is likely at least in part an analyticalartefact due to the ldquostray electron effectrdquo (this is an issue of

318

Fig 2 Electron back-scattered images of run products from high-pressure experiments The lighter the area the higher the meanatomic number Contrast has been adjusted to illuminate optimallythe textures discussed in the text a) run R193 showing residualMgTiO3(Gk) surrounded by Mg2TiO4 (Q) with no contact withMgO (P) b) run D364 showing typical ldquoeutectoid-likerdquo inter-growths of MgTiO3 + MgO (labelled ldquoEurdquo) replacing originalMg2TiO4 plus original grains of MgTiO3 (Gk) and MgO (P) c)Run C1922 showing MgTiO3 (Gk) trapping vapour (or quenchcarbonate) now present as holes the Mg2TiO4 (Q) does not containthese holes What this means for the direction of reaction is equiv-ocal

experimental interest that is discussed in more detail inHermann et al in press) However the jump to 09 wtTiO2 in the highest temperature run analysed (D3631580degC 20 kbar) looks real as the MgO grain size in thisrun is larger than in the analysed runs with 06 wt TiO2Pelton et al (1998) report somewhat similar amounts of Tiin MgO in equilibrium with Mg2Ti4+O4-MgTi2

3+O4 spinelsin the system MgO-Ti-O at 1500degC and atmospheric pres-sure the amounts increasing with decreasing oxygen

fugacity suggesting that the valence state of the substi-tuting Ti is mainly Ti3+

The accuracy with which lattice parameters can bemeasured can often make these data sensitive indicators ofpurity and stoichiometry especially for phases with highcrystallographic symmetry such as the three of interesthere The downside is that it is difficult to interpret thecause of a change in lattice parameter unambiguously alsoit is possible that important changes in purity and stoi-

The free energy of formation of Mg2TiO4 spinel 319

Fig 3 Electron back-scattered images of run products from experiments at atmospheric pressure using Na2B2O4 flux All runs containcrystals of a Mg-Ti-borate phase (generally sub-acicular in habit slightly lighter shade of grey to MgO) and quenched melt rich in Na2Oand B2O3 (dark areas but generally indistinguishable from holes in these micrographs) The euhedral MgTiO3 crystals in these runs are~ 10 microm in diametera) C8404 (1270degC) showing euhedral MgTiO3 (Gk) and MgO (P) with no sign of any Mg2TiO4 b) C21703 (1280degC) showing euhe-dral Mg2TiO4 (Q) and MgTiO3 with somewhat corroded MgO c) C26304 (1288degC) showing a large crystal of Mg2TiO4 which hasgrown to enclose subhedral to euhedral MgTiO3 and some Mg-Ti-borate with the occasional microblob of MgO d) C19503 (1050degC)in which some but not all MgTiO3 contains many inclusions of MgO ndash cf Fig 2b

Table 2 Electron microprobe analysis of selected run products to check for the stoichiometry of the phases

Analytical conditions WDS 15 kV 20 nA MgO and TiO2 standardsTAP and LPET crystals

Run P(kbar) T(degC) wt TiO2 in molar MgTi molar MgTiMgO geikielite quandilite

D362 06 1250 066(3) 1020(6) 1982(3)C1922 06 1250 060(6) 1021(7) 1991(9)D369 06 1260 064(1) - 1980(8)R204 12 1495 069(4) 1007(3) -D364 15 1490 068(6) 1018(6) 1971(8)D363 20 1580 092(9) 1009(10) -

H St C OrsquoNeill D R Scott

chiometry either do not affect the lattice parameter or thatmore than one type change occurs with a net cancellingeffect Here the lattice parameters of MgO in the run prod-ucts are consistent with the pure MgO in the starting mate-rial (Table 3) except for D350 a run with theldquoeutectoid-likerdquo intergrowth of MgTiO3 + MgO here theslightly larger lattice parameter may be caused by strainassociated with this texture The effect on the lattice param-eter of MgO from the substitution of Ti is not known butby analogy with magnesiowuumlstite in the system MgO-FeO-FeO15 (eg OrsquoNeill et al 2003a) we expect this substitu-tion to lower the lattice parameter both because Ti4+ is asmaller cation than Mg2+ and because charge-balance ismaintained in heterovalent-substituted MgO by cationvacancies which decrease the lattice parameter

For Mg2TiO4 the lattice parameters in the run productsare ~ 0001 Aring larger than in the starting mix (Table 3)Since solid solution towards a defect spinel with cationvacancies would lower the lattice parameter this is possiblydue to the presence of Ti3+ in the form of solid solutiontowards MgTi2

3+O4 in the moderately low fO2 environmentof the piston-cylinder apparatus (cf Pelton et al 1998)The lattice parameter of end-member MgTi2

3+O4 spinel isinferred to be about 8505 Aring (Hohl et al 1996) hence theobserved increase of 0001 Aring corresponds to a mole frac-tion of 0015 MgTi2

3+O4 This would imply a molar MgTiratio of 196 in reasonable agreement with the ratio

observed by electron microprobe analysis 198 plusmn 001(Table 2) For MgTiO3 the lattice parameters of run D350are identical to those of the starting material and also tothose found by Wechsler amp von Dreele (1989) who give ao= 50548 (3) Aring co = 138992 (7) Aring V = 30756 Aring3

Thermodynamic evaluation

The equilibrium condition can be expressed as

∆rG(T P) = 0 = ∆rGO(T 1 bar) + int

P

1 ∆rV

O(T P)dP + RT ln K (2)

where

ailmMgTiO3

aoxMgO

K = mdashmdashmdashmdashmdash (3)asp

Mg2TiO4

Following the discussion above on the stoichiometryand purity of MgO MgTiO3 and Mg2TiO4 in the run prod-ucts we assume that K = 1 with a probable uncertainty ofonly plusmn 001 since such minor deviations from stoichiom-etry and purity that are expected in the run products willtend to cancel out across the reaction The experimentalhalf-brackets define points in free energy - temperature -pressure space at which ∆rG(T P) ge 0 (low temperature half-bracket) and ∆rG(T P) le 0 (high temperature half-bracket)Splitting ∆rG

O(T 1 bar) into entropy and the enthalpy terms

gives

∆rGO(T 1 bar) = ∆rH

O(298K) ndash T∆rS

O(298K) +

T

int298

∆rCPO(T)dT

ndash TT

int298

[∆rmdashCP

O

mdashTmdashmdash

(T)]dT (4)

Available data for high temperature heat capacities(Table 4) can be used to evaluate the last two terms in eqn

(3) (ie the two terms in ∆rCPO) For the

P

int1∆rVO(T P)dP

term in eqn 2 we have adopted the equation of staterecommended by Holland et al (1996) and used byHolland amp Powell (1998) For MgO and MgTiO3 the heatcapacities molar volumes thermal expansivities and bulk

320

Table 3 Lattice parameters (in Aring)Sample MgO Mg2TiO4 MgTiO3

a c V(Aring3)Starting mix 42118 84418 50551 138987 30758D338 42119 84430 - - -D343 42122 84426 - - -D350 42128 - 50552 138977 30757D360 42120 84433 5057 13895 3077

Only c1 MgTiO3 in run productsmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdash

Estimated uncertainties (one standard deviation) are 00001 Aringfor MgO 00002 Aring for Mg2TiO4 and 001 Aring3 for the unit cellvolume (V) of MgTiO3 (except D360)

Phase ∆fHo(298K1 bar) Sordm(298 K 1 bar) Cp = a + bT + cT-2 + d T-12 (in J K-1mol-1) Vordm(298K1 bar) αo κ

mdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdash(J bar-1) (K-1) (kbar)

a b c d(x102)

MgO 0-60165 0269 0605 0362 0-535800 -2992 1125 62 1650periclaseMgTiO3 -156742 0746 1510 - -1890400 -6522 3086 495 1770geikieliteMg2TiO4 -215741 1118a 1617b 03286 -2382200 -2786 4529c 548c 1890d

Qandilite

Table 4 Thermodynamic data

a calorimetrically measured value is 1036 J K-1 mol-1from Todd (1952) b Cp data from Orr and Coughlin (1952) andTodd (1952) fitted in this study c OrsquoNeill et al (2003b) d value for Fe2TiO4 from Holland and Powell (1998)mdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashData for MgO and MgTiO3 are from Holland amp Powell (1998) for Mg2TiO4 from this study except where noted

moduli are taken from Holland amp Powell (1998) assummarized in Table 4 For Mg2TiO4 the heat capacityequation is our fit to the heat content measurements of Orramp Coughlin (1952) which extend to 1820 K (31 data)Temperature was corrected from IPTS-48 to ITS-90 andthe equation was anchored at low temperature using theadiabatic calorimetric measurements of Todd (1952) in therange 250 to 300 K (5 data) Volume and thermal expansionat 1 bar are from OrsquoNeill et al (2003b) refitted to theequation for thermal expansion of Holland et al (1996)The bulk modulus of Mg2TiO4 has not yet been measuredto our knowledge so we adopt the value listed for Fe2TiO4by Holland amp Powell (1998)

With these data the experimental half-brackets can beused define maximum and minimum values of the quantity(∆rH

O(298K) ndash T∆rS

O(298K)) at each experimental pressure

These values are plotted in Fig 4 against experimentaltemperature To allow for experimental uncertainties inboth temperature and pressure measurement in the piston-cylinder apparatus 5 K has been subtracted from and02 kbar added to the low temperature half-brackets and 5K added to and 02 kbar subtracted from the high tempera-ture half-brackets For example the half-brackets nomi-nally at 10 kbar and 1360degC and 1370degC are treated asbeing at 1355degC and 102 kbar and 1375degC and 98 kbarrespectively For the experiments at atmospheric pressurean uncertainty of 5 K in the half-brackets was also adopted

The resulting array of limiting values of (∆rHO(298K) ndash

T∆rSO(298K)) defines a set of lines of slope -∆rS

O(298K) and

intercept ∆rHO(298K) By inspection we locate the two

limiting values (one from a high-temperature half-bracketthe other from a low-temperature half-bracket) that definethe line of maximum slope and similarly the two thatdefine the line of minimum slope These lines givemaximum and minimum values of ∆rS

O(298K) of 1010 and

934 J K-1 mol-1 respectively with corresponding values of∆rH

O(298K) of 1227 and 1105 kJ mol-1 For a best estimate

of ∆rSO(298K) we take the mean of these maximum and

minimum slopes 972 J K-1 mol-1 with an uncertaintygiven by half their difference plusmn 038 J K-1 mol-1 Likewise∆rH

O(298K) = 1166 plusmn 061 kJ mol-1Using the values of ∆f elHO

(298K) and SO(298K) for MgTiO3

and MgO from Holland amp Powell (1998) in Table 4 givesfor Mg2TiO4 ∆f elH

O(298K) = ndash 215741 plusmn 135 kJ mol-1 and

SO(298K) = 1112 plusmn 06 J K-1 mol-1 This latter value is 76 plusmn

09 J K-1 mol-1 more than the calorimetrically determinedvalue of 1036 plusmn 07 J K-1 mol-1 (Todd 1952) The differ-ence may be ascribed to zero-point entropy but is some-what less than the theoretical amount of 2Rln2 or 115 JK-1mol-1 expected from the random mixing of Mg and Tion the octahedral site of Mg2TiO4 spinel

One possibility is some short-range ordering of Mg andTi which seems plausible given the long-range ordering ofMg and Ti onto distinct octahedral sites at the phase transi-tion from to Fd

ndash3m P4122 at ~ 1000degC Unfortunately as

with many hypotheses regarding short-range order incrystal chemistry this explanation is not really practicableto test with existing experimental methods It is thereforenecessary to eliminate other possibilities such as error inthe calorimetric data for the other phases in reaction (1)ie MgTiO3 andor MgO For MgO it is reasonable toassume that the calorimetric data are secure not onlybecause they have been measured many times but alsobecause MgO figures in so many other reactions that havebeen subsumed into the compilation of the Holland ampPowell (1998) database The value SO

(298K) of for MgTiO3also seems well known since the low temperature heatcapacity measurements of Robie et al (1989) are in goodagreement with the earlier measurements of Shomate(1946) However at high temperatures the heat-contentmeasurements of MgTiO3 (Naylor amp Cook 1946) leave

The free energy of formation of Mg2TiO4 spinel 321

Fig 4 The quantity ∆rHO(298K) ndash T∆rS

O(298K) vs T (K) for the defining half-brackets plotted to allow for experimental uncertainties as

discussed in the text The lines with maximum and minimum slopes are drawn

H St C OrsquoNeill D R Scott

some room for doubt as follows Firstly the measurementsextend only to 1720 K with only three data above 1410 KRefitting the data of Naylor amp Cook (1946) to the four-term heat capacity polynomial used by Holland and Powell(ie Cp = a + bT + cT2 + dT-12) gives a result very muchthe same as that listed by Holland amp Powell (1998) butadding an extra fifth term to the polynomial (ie the termeT2) produces a better fit but one which leads to signifi-cantly higher computed values of Cp above ~ 1400K As anillustration Cp calculated at 1800 K with the four-term fitis 1356 J K-1 mol-1 whereas that with the five-term fit is1589 J K-1 mol-1 This would be a large enough change toaccount for the missing residual entropy in Mg2TiO4Whether this increase in the heat capacity of MgTiO3 isreal is difficult to judge since the extended Maier-Kelleyequation for heat capacities is an entirely empiricaldevice for representing the experimental data in which theinclusion of the term in eT2 is well known to lead to ambi-guities in the extrapolation of the equation to highertemperatures The value for the four-term fit at 1800 K isin excellent agreement with the Dulong-Petit value(Cp = 3nR + TVα2κ) which is calculated to be 136 J K-1

mol-1 at 1800 K using the data in Table 4 Excess heatcapacity could result from disordering of Mg and Ti as theilmenite structure in which Mg and Ti are ordered intolayers transforms towards the corundum structure(R

ndash3 rarr R

ndash3c) This transition has been observed in NiTiO3

by powder neutron diffraction at ~ 1560 K (Boysen et al(1995) and is inferred at ~ 1673 K in FeTiO3 from extrap-olation of the transition in FeTiO3-Fe2O3 solid solutions(Harrison et al 2000) Harrison et al (2000) show thatFeTiO3 is ~ 10 disordered at 1473 K The occurrence ofthis transition in MgTiO3 has been discounted by Reynardamp Guyot (1994) from their investigation by Raman spec-troscopy to 1820 K on the grounds that the symmetrychange would reduce the number of Raman active modesfrom ten to seven In fact the ten modes observable atroom temperature could only be tracked to ~ 1400 K due toline broadening and thermal emission so the possibility ofsymmetry reduction at higher temperatures remainsopen as does the possibility of significant Mg-Ti disor-dering within the ilmenite structure at temperatures belowthe putative transition as has been shown to occur inFeTiO3 by Harrison et al (2000) The question of excessentropy in MgTiO3 could be addressed by more precisehigh-temperature calorimetry (which would give the dataneeded to resolve the question of interest here directly)or other methods such as high-temperature powder neutrondiffraction

Acknowledgments We thank Mark Ghiorso andDominique Lattard for helpful reviews

References

Akimoto S amp Syono Y (1967) High-pressure decomposition ofsome titanate spinels J Chem Phys 47 1813-1817

Bose K amp Ganguly J (1995) Quartz-coesite transition revisited -reversed experimental-determination at 500-1200-degrees-C

and retrieved thermochemical properties Am Mineral 80231-238

Boysen H Frey F Lerch M Vogt T (1995) A neutron powderinvestigation of the high-temperature phase-transition inNiTiO3 Zeitschr fuumlr Kristallogr 210 328-337

Eriksson G amp Pelton AD (1993) Critical evaluation and opti-mization of the thermodynamic properties and phasediagrams of the MnO-TiO2 MgO-TiO2 FeO-TiO2 Ti2O3-TiO2Na2O-TiO2 and K2O-TiO2 systems Metall Trans B 24795-805

Harrison RJ Redfern SAT Smith RI (2000) In-situ study ofthe R

ndash3 to R

ndash3c phase transition in the ilmenite-hematite solid

solution using time-of-flight neutron powder diffraction AmMineral 85 194-205

Hermann J OrsquoNeill HStC Berry AJ (in Press) Titanium solu-bility in olivine in the system TiO2-MgO-SiO2 no evidence foran ultra-deep origin of Ti-bearing olivine Contrib MineralPetrol

Holland TJB amp Powell R (1998) An internally consistent ther-modynamic data set for phases of petrological interest J meta-morphic Geol 16 309-343

Holland TJB Redfern SAT Pawley AR (1996) Volumebehaviour of hydrous minerals at high pressure and tempera-ture 2 Compressibilities of lawsonite zoisite clinozoisite andepidote Am Mineral 81 341-348

Hohl H Kloc C Bucher E (1996) Electrical and magnetic prop-erties of spinel solid solutions Mg2-xTi1+xO4 0 le x le 1 J SolidState Chem 125 216-223

Hunter BA amp Howard CJ (2000) LHPM a computer programfor Rietveld analysis of X-ray and neutron powder diffractionpatterns Australian Nuclear Science and TechnologyOrganization Lucas Heights Research Laboratory MenaiNSW 2234 Australia

Millard RL Peterson RC Hunter BK (1995) Study of thecubic to tetragonal transition in Mg2TiO4 and Zn2TiO4 spinelsby 17O MAS NMR and Rietveld refinement of X-ray diffractiondata Am Mineral 80 885-896

Naylor BF amp Cook OA (1946) High-temperature heat contentsof the metatitanates of calcium iron and magnesium J AmerChem Soc 68 1003-1005

OrsquoNeill HStC Pownceby MI McCammon CA (2003a) Themagnesiowuumlstite iron equilibrium and its implications for theactivity-composition relations of (MgFe)2SiO4 olivine solidsolutions Contrib Mineral Petrol 146 308-325

ONeill HStC Redfern SAT Kesson S Short S (2003a) Anin situ neutron diffraction study of cation disordering insynthetic qandilite Mg2TiO4 at high temperatures AmMineral 88 860-865

Orr RL amp Coughlin JP (1952) High temperature heat contentsof magnesium orthotitanate and magnesium dititanate J AmerChem Soc 74 3186-3187

Pelton AD Eriksson G Krajewski D Goumlbbels M WoermannE (1998) Measurement and thermodynamic evaluation ofphase equilibria in the Mg-Ti-O system Zeitschr fuumlr PhysikalChem 207 163-180

Reynard B amp Guyot F (1994) High-temperature properties ofgeikielite (MgTiO3-ilmenite) from high-temperature high-pres-sure Raman-spectroscopy ndash some implications for MgSiO3-ilmenite Phys Chem Min 21 441-450

Robie RA Haselton Jr HT Hemingway BS (1989) Heatcapacities and entropies at 29815 K of MgTiO3 (geikielite) ZnO(zincite) and ZnCO3 (smithsonite) J Chem Thermodynamics21 743-749

322

H St C OrsquoNeill D R Scott

temperature of the experiment was measured accurately toplusmn 1ordmC The thermocouple was checked against the meltingpoint of gold with results within 05ordmC of the recom-mended value of 106418ordmC (ITS-90) Previous work onMg2TiO4 has shown that the kinetics of reaction (1) are toosluggish for any perceptible reaction to occur at atmo-spheric pressure in the simple system MgO-TiO2 (Wechsleramp Navrotsky 1984) hence a flux of Na2B4O7 (dehydratedborax) was added The charge was run in a Pt capsule (5mm diameter) sealed at one end and crimped closed at theother to minimize loss of Na2O by volatilization This fluxproved less than completely ideal as it reacts with MgO toproduce a Mg-Ti-borate phase thus the direction of reac-tion cannot be inferred simply from a change in propor-tions of the phases Nevertheless the results are stronglysuggestive in that abundant euhedral Mg2TiO4 is observedat 1100degC but is completely absent at 1070degC Theformula of the Mg-Ti-borate is Mg3TiB2O8 from electronmicroprobe determination of MgO and TiO2 with B2O3estimated by difference Trials of a number of otherpossible fluxes (78 wt BaO + 22 wt B2O3 K2CO3Na2WO4) were unsuccessful due to their incompatibilitywith one or more of Mg2TiO4 MgTiO3 or MgO

Experiments at high pressure (6 to 20 kbar) wereconducted either in a 30 mm (at 12 kbar) or a 58rdquo (otherpressures) piston-cylinder apparatus with a NaCl-pyrexassembly surrounding a cylindrical graphite heater Thelengths of the cells were 60 mm for the 30 mm diametercell and 15rdquo (38 mm) for the 58rdquo cell and both cells useda graphite heater of 95 mm OD by 7 mm ID In order toallow initial friction to decay the run was first taken to800degC and the pressure of interest where it was held for~24 hours see Bose amp Ganguly (1995) for a discussion offriction decay in the piston-cylinder apparatus with NaCl asthe pressure medium The run was then heated to the finaldesired temperature and the pressure adjusted to the finaldesired value Samples were held in sealed Pt capsuleswith dimensions 23 mm OD 17 mm ID 5 mm longTemperature was controlled and measured with a type B

thermocouple positioned directly above the capsuleThermocouples were made from the same batches of wireas those used in the 1-bar experiments Temperatures in thepiston-cylinder runs were not corrected for any effect ofpressure apart from this the reported temperatures whichwere controlled to plusmn 1degC are thought to be accurate to plusmn5degC at T le 1400degC the uncertainty being caused by smalldeviations in sample position within the thermal profile ofthe piston-cylinder cell The thermal profile of the 30 mmcells should have a larger constant-temperature zone less-ening this source of error Temperatures are probably moreuncertain at T gt 1400degC due to greater temperature gradi-ents and also to several other factors to do with thermo-couple calibration and performance Pressures which arereported without any correction for friction are precise toplusmn 02 kbar and probably to plusmn 01 kbar in the 30 mmassembly

Following the quenching of each run the sample wasmounted in epoxy sectioned and polished for examinationusing the electron microprobe (CAMECA SX100) forback-scattered electron imaging and quantitative analysisby wavelength dispersive spectrometry The starting mate-rial and selected run products were also examined bypowder X-ray diffraction using a STOE STADIP diffrac-tometer in the transmission mode with monochromaticCoKα1 radiation (λ = 178897 Aring) scanning from 10 to130degC An internal standard of NIST 640c Si (a0 =54312 Aring) was used and the lattice parameters wereextracted by Rietveld refinement of the whole pattern usingthe program LHPM-Rietica (Hunter amp Howard 2000)

Results

Results are reported in Table 1 and the definingbrackets are plotted in Fig 1 Experiments at ge 10 kbar areat a sufficiently high temperature that the reaction takesplace rapidly and no flux is needed The results are straight-forward reversals which bracket the reaction to within a

316

Fig 1 Pressure-temperature brackets on theunivariant reaction MgTiO3 + MgO =Mg2TiO4 from this study and from Akimotoamp Syono (1967) Only the results from theexperiments defining the curve are shownThe solid curve is an empirical 4th orderpolynomial drawn through the brackets ofthis study (note that this univariant reactionis not linear in P-T space) The dashed line isthe equation quoted by Akimoto amp Syono(1967) P (kbar) = ndash18 + 0019 T (degC)Although their equation is in poor agreementwith the results of this study only one oftheir defining half-brackets (MT22 at 10kbar) violates the curve of this study

nominal 10 K at 10 and 15 kbar and 20 K at 20 kbar At12 kbar in the 30 mm pressure assembly where the controlof both temperature and pressure is sufficiently good towarrant the attempt at greater precision the bracket is only5 K in width

There is a remarkable asymmetry in the texturesproduced on either side of the reaction in these fluxlessexperiments On the high temperature side (Mg2TiO4stable) there is extensive recrystallization to the typicalgranular texture expected for textural equilibrium althoughin several runs close to the reaction boundary residualgrains of MgTiO3 persist but completely surrounded byMg2TiO4 and isolated from MgO (Fig 2a) By contrast onthe low temperature side of the reaction there is littlerecrystallization of the original MgTiO3 and MgO grainswhile what were originally grains of Mg2TiO4 have decom-

posed to a eutectoid-like intergrowth of MgTiO3 and MgO(Fig 2b) Despite this textural disequilibrium the extent ofreaction is complete in all cases with no residual Mg2TiO4persisting in any experiment

For runs at 6 kbar a small pinch (20 wt) of oxalic acidwas added to the first run (C1922 at 1250degC) A consider-able amount of MgO dissolved in the CO2-H2O fluidproduced by the decomposition of the oxalic acid drivingthe starting composition off the target stoichiometry whichmakes the result in terms of reaction direction difficult toascertain The MgTiO3 contains carbonatevapour ldquoholesrdquowhile the Mg2TiO4 does not (Fig 2c) which may perhapsindicate that Mg2TiO4 is stable alternatively this texturecould be due to growth of the MgTiO3 trapping the inclu-sions Hence we regard this run as completely ambiguousFor the next run at 6 kbar and 1250ordmC (D362) some extra

The free energy of formation of Mg2TiO4 spinel 317

Table 1 Experimental results

a +50 Na2B4O7 (dehydrated borax) b +20 Na2B4O7 c +20 oxalic acid d +10 oxalic acid e 30 mm cell all other high pressure runs in 15875 mm cells e no Mg2TiO4 instarting mix

Run P T Time Result(kbar) (degC) (h)

C19503a 00001 1050 048 MgTiO3 MgO Mg-Ti-borate quenched flux many crystals of MgTiO3 contain MgOinclusions

C8404b 00001 1070 046 MgTiO3 MgO Mg-Ti-borate quenched flux

C21703b 00001 1080 024 Euhedral Mg2TiO4 MgTiO3 corroded MgO Mg-Ti borate quenched flux

C26304be 00001 1088 066 Large euhedral Mg2TiO4 poikiliticallyenclosing some euhedral MgTiO3 and MgO MgTiO3 and corroded MgO Mg-Ti-borate quenched flux

C26503a 00001 1090 0235 Mg2TiO4MgTiO3MgO Mg-Ti-boratequenched flux reaction direction ambiguous

C14703b 00001 1100 048 Euhedral Mg2TiO4 MgTiO3 and Mg-Ti-borate corroded MgO quenched flux

D366 060 1240 168 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoC1922c 060 1250 006 Mg2TiO4 MgTiO3 with holes quench carbonateD362d 060 1250 048 No discernible reactionD369 060 1260 192 MgO + Mg2TiO4 100 reactionD350 100 1340 050 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoC1918 100 1360 006 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoD354 100 1370 006 MgO + Mg2TiO4 some unreacted MgTiO3

D343 100 1380 006 MgO + Mg2TiO4 100 reactionD342 100 1420 006 MgO + Mg2TiO4 100 reactionD340 100 1460 006 MgO + Mg2TiO4 100 reactionD338 100 1500 007 MgO + Mg2TiO4 100 reactionR204e 120 1405 006 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoR193e 120 1410 020 MgO + Mg2TiO4 some unreacted MgTiO3

D361 150 1480 006 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoD364 150 1490 006 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoD358 150 1500 006 MgO + Mg2TiO4 some unreacted MgTiO3

D363 200 1580 003 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoD360 200 1600 006 MgO + Mg2TiO4 few grains unreacted MgTiO3

D359 200 1640 003 MgO + Mg2TiO4 100 reaction

H St C OrsquoNeill D R Scott

MgO was added and the amount of oxalic acid reduced to10 This run produced no discernible reaction directionand the use of oxalic acid was abandoned The two subse-quent runs at 6 kbar produced an unambiguous bracketbetween 1240 and 1260degC

The bracket at atmospheric pressure is placed betweenthe run at 1070degC (C8404) which produced euhedralMgTiO3 plus MgO with no Mg2TiO4 (Figure 3a) and at1080degC (C21703) which contained Mg2TiO4 andMgTiO3 with MgO but with the Mg2TiO4 showing goodcrystal faces while the MgO appears somewhat corroded(Fig 3b) However no reaction direction could be inferredfrom a run at 1090degC in which all three phases werepresent but none show good crystal faces An additionalrun at 1088degC (C26304) which used a starting mix ofMgTiO3 MgO and flux only (ie no Mg2TiO4) producedlarge euhedrally faceted crystals of Mg2TiO4 poikiliticallyenclosing MgTiO3 and the Mg-Ti-borate phase with occa-sional small poorly faceted grains of MgO interpreted tobe residual (Fig 3c) but the unambiguous growth ofMg2TiO4 in this run implies clearly that it is stable at thistemperature Run C19503 which at 1050degC is well belowthe breakdown temperature shows many grains of MgTiO3enclosing MgO blobs (Fig 3d) which is somewhat remi-niscent of the ldquoeutectoid-likerdquo textures of the high-pressureexperiments and like for these runs we interpret thesegrains to be due to decomposition of original Mg2TiO4grains

The results are compared to the data of Akimoto ampSyono (1967) in Fig 1 At first sight there appears to be alarge difference between the results if ours are compared tothe equation for the univariant curve provided by Akimotoand Syono which is P (kbar) = -18 + 0019 T (degC) In factonly one experiment of Akimoto and Syono is actuallydiscrepant namely their run MT22 in which Mg2TiO4 isreported to break down to MgTiO3 and MgO at 10 kbar and1440degC (cf our bracket between 1360 and 1370degC at10 kbar) This is far outside any likely experimental uncer-tainty in pressure or temperature measurement and wesuggest an ad hoc experimental malfunction in run MT22such as piston seizure or thermocouple extrusion

Stoichiometry and composition of the run products

Like many spinels Mg2TiO4 might be anticipated toshow some non-stoichiometry by solid solution towardsMgTiO3 at high temperatures (ie like MgAl2O4 towardsγ-Al2O3) However electron microprobe analysis showedalmost negligible deviations in all runs from Mg2TiO4 stoi-chiometry (Table 2) It is probable that at atmospheric pres-sure the temperature at which reaction (1) occurs (ie1075 plusmn 10degC) is not high enough for deviations from stoi-chiometry to be significant while higher pressuressuppress the non-stoichiometry due to the relatively highpartial molar volume of the defect spinel componentSimilarly MgTiO3 was found to be stoichiometric withinanalytical error in all runs The observation of smallamounts of TiO2 (~ 06 wt corresponding to a mole frac-tion of 0003) in MgO is likely at least in part an analyticalartefact due to the ldquostray electron effectrdquo (this is an issue of

318

Fig 2 Electron back-scattered images of run products from high-pressure experiments The lighter the area the higher the meanatomic number Contrast has been adjusted to illuminate optimallythe textures discussed in the text a) run R193 showing residualMgTiO3(Gk) surrounded by Mg2TiO4 (Q) with no contact withMgO (P) b) run D364 showing typical ldquoeutectoid-likerdquo inter-growths of MgTiO3 + MgO (labelled ldquoEurdquo) replacing originalMg2TiO4 plus original grains of MgTiO3 (Gk) and MgO (P) c)Run C1922 showing MgTiO3 (Gk) trapping vapour (or quenchcarbonate) now present as holes the Mg2TiO4 (Q) does not containthese holes What this means for the direction of reaction is equiv-ocal

experimental interest that is discussed in more detail inHermann et al in press) However the jump to 09 wtTiO2 in the highest temperature run analysed (D3631580degC 20 kbar) looks real as the MgO grain size in thisrun is larger than in the analysed runs with 06 wt TiO2Pelton et al (1998) report somewhat similar amounts of Tiin MgO in equilibrium with Mg2Ti4+O4-MgTi2

3+O4 spinelsin the system MgO-Ti-O at 1500degC and atmospheric pres-sure the amounts increasing with decreasing oxygen

fugacity suggesting that the valence state of the substi-tuting Ti is mainly Ti3+

The accuracy with which lattice parameters can bemeasured can often make these data sensitive indicators ofpurity and stoichiometry especially for phases with highcrystallographic symmetry such as the three of interesthere The downside is that it is difficult to interpret thecause of a change in lattice parameter unambiguously alsoit is possible that important changes in purity and stoi-

The free energy of formation of Mg2TiO4 spinel 319

Fig 3 Electron back-scattered images of run products from experiments at atmospheric pressure using Na2B2O4 flux All runs containcrystals of a Mg-Ti-borate phase (generally sub-acicular in habit slightly lighter shade of grey to MgO) and quenched melt rich in Na2Oand B2O3 (dark areas but generally indistinguishable from holes in these micrographs) The euhedral MgTiO3 crystals in these runs are~ 10 microm in diametera) C8404 (1270degC) showing euhedral MgTiO3 (Gk) and MgO (P) with no sign of any Mg2TiO4 b) C21703 (1280degC) showing euhe-dral Mg2TiO4 (Q) and MgTiO3 with somewhat corroded MgO c) C26304 (1288degC) showing a large crystal of Mg2TiO4 which hasgrown to enclose subhedral to euhedral MgTiO3 and some Mg-Ti-borate with the occasional microblob of MgO d) C19503 (1050degC)in which some but not all MgTiO3 contains many inclusions of MgO ndash cf Fig 2b

Table 2 Electron microprobe analysis of selected run products to check for the stoichiometry of the phases

Analytical conditions WDS 15 kV 20 nA MgO and TiO2 standardsTAP and LPET crystals

Run P(kbar) T(degC) wt TiO2 in molar MgTi molar MgTiMgO geikielite quandilite

D362 06 1250 066(3) 1020(6) 1982(3)C1922 06 1250 060(6) 1021(7) 1991(9)D369 06 1260 064(1) - 1980(8)R204 12 1495 069(4) 1007(3) -D364 15 1490 068(6) 1018(6) 1971(8)D363 20 1580 092(9) 1009(10) -

H St C OrsquoNeill D R Scott

chiometry either do not affect the lattice parameter or thatmore than one type change occurs with a net cancellingeffect Here the lattice parameters of MgO in the run prod-ucts are consistent with the pure MgO in the starting mate-rial (Table 3) except for D350 a run with theldquoeutectoid-likerdquo intergrowth of MgTiO3 + MgO here theslightly larger lattice parameter may be caused by strainassociated with this texture The effect on the lattice param-eter of MgO from the substitution of Ti is not known butby analogy with magnesiowuumlstite in the system MgO-FeO-FeO15 (eg OrsquoNeill et al 2003a) we expect this substitu-tion to lower the lattice parameter both because Ti4+ is asmaller cation than Mg2+ and because charge-balance ismaintained in heterovalent-substituted MgO by cationvacancies which decrease the lattice parameter

For Mg2TiO4 the lattice parameters in the run productsare ~ 0001 Aring larger than in the starting mix (Table 3)Since solid solution towards a defect spinel with cationvacancies would lower the lattice parameter this is possiblydue to the presence of Ti3+ in the form of solid solutiontowards MgTi2

3+O4 in the moderately low fO2 environmentof the piston-cylinder apparatus (cf Pelton et al 1998)The lattice parameter of end-member MgTi2

3+O4 spinel isinferred to be about 8505 Aring (Hohl et al 1996) hence theobserved increase of 0001 Aring corresponds to a mole frac-tion of 0015 MgTi2

3+O4 This would imply a molar MgTiratio of 196 in reasonable agreement with the ratio

observed by electron microprobe analysis 198 plusmn 001(Table 2) For MgTiO3 the lattice parameters of run D350are identical to those of the starting material and also tothose found by Wechsler amp von Dreele (1989) who give ao= 50548 (3) Aring co = 138992 (7) Aring V = 30756 Aring3

Thermodynamic evaluation

The equilibrium condition can be expressed as

∆rG(T P) = 0 = ∆rGO(T 1 bar) + int

P

1 ∆rV

O(T P)dP + RT ln K (2)

where

ailmMgTiO3

aoxMgO

K = mdashmdashmdashmdashmdash (3)asp

Mg2TiO4

Following the discussion above on the stoichiometryand purity of MgO MgTiO3 and Mg2TiO4 in the run prod-ucts we assume that K = 1 with a probable uncertainty ofonly plusmn 001 since such minor deviations from stoichiom-etry and purity that are expected in the run products willtend to cancel out across the reaction The experimentalhalf-brackets define points in free energy - temperature -pressure space at which ∆rG(T P) ge 0 (low temperature half-bracket) and ∆rG(T P) le 0 (high temperature half-bracket)Splitting ∆rG

O(T 1 bar) into entropy and the enthalpy terms

gives

∆rGO(T 1 bar) = ∆rH

O(298K) ndash T∆rS

O(298K) +

T

int298

∆rCPO(T)dT

ndash TT

int298

[∆rmdashCP

O

mdashTmdashmdash

(T)]dT (4)

Available data for high temperature heat capacities(Table 4) can be used to evaluate the last two terms in eqn

(3) (ie the two terms in ∆rCPO) For the

P

int1∆rVO(T P)dP

term in eqn 2 we have adopted the equation of staterecommended by Holland et al (1996) and used byHolland amp Powell (1998) For MgO and MgTiO3 the heatcapacities molar volumes thermal expansivities and bulk

320

Table 3 Lattice parameters (in Aring)Sample MgO Mg2TiO4 MgTiO3

a c V(Aring3)Starting mix 42118 84418 50551 138987 30758D338 42119 84430 - - -D343 42122 84426 - - -D350 42128 - 50552 138977 30757D360 42120 84433 5057 13895 3077

Only c1 MgTiO3 in run productsmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdash

Estimated uncertainties (one standard deviation) are 00001 Aringfor MgO 00002 Aring for Mg2TiO4 and 001 Aring3 for the unit cellvolume (V) of MgTiO3 (except D360)

Phase ∆fHo(298K1 bar) Sordm(298 K 1 bar) Cp = a + bT + cT-2 + d T-12 (in J K-1mol-1) Vordm(298K1 bar) αo κ

mdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdash(J bar-1) (K-1) (kbar)

a b c d(x102)

MgO 0-60165 0269 0605 0362 0-535800 -2992 1125 62 1650periclaseMgTiO3 -156742 0746 1510 - -1890400 -6522 3086 495 1770geikieliteMg2TiO4 -215741 1118a 1617b 03286 -2382200 -2786 4529c 548c 1890d

Qandilite

Table 4 Thermodynamic data

a calorimetrically measured value is 1036 J K-1 mol-1from Todd (1952) b Cp data from Orr and Coughlin (1952) andTodd (1952) fitted in this study c OrsquoNeill et al (2003b) d value for Fe2TiO4 from Holland and Powell (1998)mdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashData for MgO and MgTiO3 are from Holland amp Powell (1998) for Mg2TiO4 from this study except where noted

moduli are taken from Holland amp Powell (1998) assummarized in Table 4 For Mg2TiO4 the heat capacityequation is our fit to the heat content measurements of Orramp Coughlin (1952) which extend to 1820 K (31 data)Temperature was corrected from IPTS-48 to ITS-90 andthe equation was anchored at low temperature using theadiabatic calorimetric measurements of Todd (1952) in therange 250 to 300 K (5 data) Volume and thermal expansionat 1 bar are from OrsquoNeill et al (2003b) refitted to theequation for thermal expansion of Holland et al (1996)The bulk modulus of Mg2TiO4 has not yet been measuredto our knowledge so we adopt the value listed for Fe2TiO4by Holland amp Powell (1998)

With these data the experimental half-brackets can beused define maximum and minimum values of the quantity(∆rH

O(298K) ndash T∆rS

O(298K)) at each experimental pressure

These values are plotted in Fig 4 against experimentaltemperature To allow for experimental uncertainties inboth temperature and pressure measurement in the piston-cylinder apparatus 5 K has been subtracted from and02 kbar added to the low temperature half-brackets and 5K added to and 02 kbar subtracted from the high tempera-ture half-brackets For example the half-brackets nomi-nally at 10 kbar and 1360degC and 1370degC are treated asbeing at 1355degC and 102 kbar and 1375degC and 98 kbarrespectively For the experiments at atmospheric pressurean uncertainty of 5 K in the half-brackets was also adopted

The resulting array of limiting values of (∆rHO(298K) ndash

T∆rSO(298K)) defines a set of lines of slope -∆rS

O(298K) and

intercept ∆rHO(298K) By inspection we locate the two

limiting values (one from a high-temperature half-bracketthe other from a low-temperature half-bracket) that definethe line of maximum slope and similarly the two thatdefine the line of minimum slope These lines givemaximum and minimum values of ∆rS

O(298K) of 1010 and

934 J K-1 mol-1 respectively with corresponding values of∆rH

O(298K) of 1227 and 1105 kJ mol-1 For a best estimate

of ∆rSO(298K) we take the mean of these maximum and

minimum slopes 972 J K-1 mol-1 with an uncertaintygiven by half their difference plusmn 038 J K-1 mol-1 Likewise∆rH

O(298K) = 1166 plusmn 061 kJ mol-1Using the values of ∆f elHO

(298K) and SO(298K) for MgTiO3

and MgO from Holland amp Powell (1998) in Table 4 givesfor Mg2TiO4 ∆f elH

O(298K) = ndash 215741 plusmn 135 kJ mol-1 and

SO(298K) = 1112 plusmn 06 J K-1 mol-1 This latter value is 76 plusmn

09 J K-1 mol-1 more than the calorimetrically determinedvalue of 1036 plusmn 07 J K-1 mol-1 (Todd 1952) The differ-ence may be ascribed to zero-point entropy but is some-what less than the theoretical amount of 2Rln2 or 115 JK-1mol-1 expected from the random mixing of Mg and Tion the octahedral site of Mg2TiO4 spinel

One possibility is some short-range ordering of Mg andTi which seems plausible given the long-range ordering ofMg and Ti onto distinct octahedral sites at the phase transi-tion from to Fd

ndash3m P4122 at ~ 1000degC Unfortunately as

with many hypotheses regarding short-range order incrystal chemistry this explanation is not really practicableto test with existing experimental methods It is thereforenecessary to eliminate other possibilities such as error inthe calorimetric data for the other phases in reaction (1)ie MgTiO3 andor MgO For MgO it is reasonable toassume that the calorimetric data are secure not onlybecause they have been measured many times but alsobecause MgO figures in so many other reactions that havebeen subsumed into the compilation of the Holland ampPowell (1998) database The value SO

(298K) of for MgTiO3also seems well known since the low temperature heatcapacity measurements of Robie et al (1989) are in goodagreement with the earlier measurements of Shomate(1946) However at high temperatures the heat-contentmeasurements of MgTiO3 (Naylor amp Cook 1946) leave

The free energy of formation of Mg2TiO4 spinel 321

Fig 4 The quantity ∆rHO(298K) ndash T∆rS

O(298K) vs T (K) for the defining half-brackets plotted to allow for experimental uncertainties as

discussed in the text The lines with maximum and minimum slopes are drawn

H St C OrsquoNeill D R Scott

some room for doubt as follows Firstly the measurementsextend only to 1720 K with only three data above 1410 KRefitting the data of Naylor amp Cook (1946) to the four-term heat capacity polynomial used by Holland and Powell(ie Cp = a + bT + cT2 + dT-12) gives a result very muchthe same as that listed by Holland amp Powell (1998) butadding an extra fifth term to the polynomial (ie the termeT2) produces a better fit but one which leads to signifi-cantly higher computed values of Cp above ~ 1400K As anillustration Cp calculated at 1800 K with the four-term fitis 1356 J K-1 mol-1 whereas that with the five-term fit is1589 J K-1 mol-1 This would be a large enough change toaccount for the missing residual entropy in Mg2TiO4Whether this increase in the heat capacity of MgTiO3 isreal is difficult to judge since the extended Maier-Kelleyequation for heat capacities is an entirely empiricaldevice for representing the experimental data in which theinclusion of the term in eT2 is well known to lead to ambi-guities in the extrapolation of the equation to highertemperatures The value for the four-term fit at 1800 K isin excellent agreement with the Dulong-Petit value(Cp = 3nR + TVα2κ) which is calculated to be 136 J K-1

mol-1 at 1800 K using the data in Table 4 Excess heatcapacity could result from disordering of Mg and Ti as theilmenite structure in which Mg and Ti are ordered intolayers transforms towards the corundum structure(R

ndash3 rarr R

ndash3c) This transition has been observed in NiTiO3

by powder neutron diffraction at ~ 1560 K (Boysen et al(1995) and is inferred at ~ 1673 K in FeTiO3 from extrap-olation of the transition in FeTiO3-Fe2O3 solid solutions(Harrison et al 2000) Harrison et al (2000) show thatFeTiO3 is ~ 10 disordered at 1473 K The occurrence ofthis transition in MgTiO3 has been discounted by Reynardamp Guyot (1994) from their investigation by Raman spec-troscopy to 1820 K on the grounds that the symmetrychange would reduce the number of Raman active modesfrom ten to seven In fact the ten modes observable atroom temperature could only be tracked to ~ 1400 K due toline broadening and thermal emission so the possibility ofsymmetry reduction at higher temperatures remainsopen as does the possibility of significant Mg-Ti disor-dering within the ilmenite structure at temperatures belowthe putative transition as has been shown to occur inFeTiO3 by Harrison et al (2000) The question of excessentropy in MgTiO3 could be addressed by more precisehigh-temperature calorimetry (which would give the dataneeded to resolve the question of interest here directly)or other methods such as high-temperature powder neutrondiffraction

Acknowledgments We thank Mark Ghiorso andDominique Lattard for helpful reviews

References

Akimoto S amp Syono Y (1967) High-pressure decomposition ofsome titanate spinels J Chem Phys 47 1813-1817

Bose K amp Ganguly J (1995) Quartz-coesite transition revisited -reversed experimental-determination at 500-1200-degrees-C

and retrieved thermochemical properties Am Mineral 80231-238

Boysen H Frey F Lerch M Vogt T (1995) A neutron powderinvestigation of the high-temperature phase-transition inNiTiO3 Zeitschr fuumlr Kristallogr 210 328-337

Eriksson G amp Pelton AD (1993) Critical evaluation and opti-mization of the thermodynamic properties and phasediagrams of the MnO-TiO2 MgO-TiO2 FeO-TiO2 Ti2O3-TiO2Na2O-TiO2 and K2O-TiO2 systems Metall Trans B 24795-805

Harrison RJ Redfern SAT Smith RI (2000) In-situ study ofthe R

ndash3 to R

ndash3c phase transition in the ilmenite-hematite solid

solution using time-of-flight neutron powder diffraction AmMineral 85 194-205

Hermann J OrsquoNeill HStC Berry AJ (in Press) Titanium solu-bility in olivine in the system TiO2-MgO-SiO2 no evidence foran ultra-deep origin of Ti-bearing olivine Contrib MineralPetrol

Holland TJB amp Powell R (1998) An internally consistent ther-modynamic data set for phases of petrological interest J meta-morphic Geol 16 309-343

Holland TJB Redfern SAT Pawley AR (1996) Volumebehaviour of hydrous minerals at high pressure and tempera-ture 2 Compressibilities of lawsonite zoisite clinozoisite andepidote Am Mineral 81 341-348

Hohl H Kloc C Bucher E (1996) Electrical and magnetic prop-erties of spinel solid solutions Mg2-xTi1+xO4 0 le x le 1 J SolidState Chem 125 216-223

Hunter BA amp Howard CJ (2000) LHPM a computer programfor Rietveld analysis of X-ray and neutron powder diffractionpatterns Australian Nuclear Science and TechnologyOrganization Lucas Heights Research Laboratory MenaiNSW 2234 Australia

Millard RL Peterson RC Hunter BK (1995) Study of thecubic to tetragonal transition in Mg2TiO4 and Zn2TiO4 spinelsby 17O MAS NMR and Rietveld refinement of X-ray diffractiondata Am Mineral 80 885-896

Naylor BF amp Cook OA (1946) High-temperature heat contentsof the metatitanates of calcium iron and magnesium J AmerChem Soc 68 1003-1005

OrsquoNeill HStC Pownceby MI McCammon CA (2003a) Themagnesiowuumlstite iron equilibrium and its implications for theactivity-composition relations of (MgFe)2SiO4 olivine solidsolutions Contrib Mineral Petrol 146 308-325

ONeill HStC Redfern SAT Kesson S Short S (2003a) Anin situ neutron diffraction study of cation disordering insynthetic qandilite Mg2TiO4 at high temperatures AmMineral 88 860-865

Orr RL amp Coughlin JP (1952) High temperature heat contentsof magnesium orthotitanate and magnesium dititanate J AmerChem Soc 74 3186-3187

Pelton AD Eriksson G Krajewski D Goumlbbels M WoermannE (1998) Measurement and thermodynamic evaluation ofphase equilibria in the Mg-Ti-O system Zeitschr fuumlr PhysikalChem 207 163-180

Reynard B amp Guyot F (1994) High-temperature properties ofgeikielite (MgTiO3-ilmenite) from high-temperature high-pres-sure Raman-spectroscopy ndash some implications for MgSiO3-ilmenite Phys Chem Min 21 441-450

Robie RA Haselton Jr HT Hemingway BS (1989) Heatcapacities and entropies at 29815 K of MgTiO3 (geikielite) ZnO(zincite) and ZnCO3 (smithsonite) J Chem Thermodynamics21 743-749

322

nominal 10 K at 10 and 15 kbar and 20 K at 20 kbar At12 kbar in the 30 mm pressure assembly where the controlof both temperature and pressure is sufficiently good towarrant the attempt at greater precision the bracket is only5 K in width

There is a remarkable asymmetry in the texturesproduced on either side of the reaction in these fluxlessexperiments On the high temperature side (Mg2TiO4stable) there is extensive recrystallization to the typicalgranular texture expected for textural equilibrium althoughin several runs close to the reaction boundary residualgrains of MgTiO3 persist but completely surrounded byMg2TiO4 and isolated from MgO (Fig 2a) By contrast onthe low temperature side of the reaction there is littlerecrystallization of the original MgTiO3 and MgO grainswhile what were originally grains of Mg2TiO4 have decom-

posed to a eutectoid-like intergrowth of MgTiO3 and MgO(Fig 2b) Despite this textural disequilibrium the extent ofreaction is complete in all cases with no residual Mg2TiO4persisting in any experiment

For runs at 6 kbar a small pinch (20 wt) of oxalic acidwas added to the first run (C1922 at 1250degC) A consider-able amount of MgO dissolved in the CO2-H2O fluidproduced by the decomposition of the oxalic acid drivingthe starting composition off the target stoichiometry whichmakes the result in terms of reaction direction difficult toascertain The MgTiO3 contains carbonatevapour ldquoholesrdquowhile the Mg2TiO4 does not (Fig 2c) which may perhapsindicate that Mg2TiO4 is stable alternatively this texturecould be due to growth of the MgTiO3 trapping the inclu-sions Hence we regard this run as completely ambiguousFor the next run at 6 kbar and 1250ordmC (D362) some extra

The free energy of formation of Mg2TiO4 spinel 317

Table 1 Experimental results

a +50 Na2B4O7 (dehydrated borax) b +20 Na2B4O7 c +20 oxalic acid d +10 oxalic acid e 30 mm cell all other high pressure runs in 15875 mm cells e no Mg2TiO4 instarting mix

Run P T Time Result(kbar) (degC) (h)

C19503a 00001 1050 048 MgTiO3 MgO Mg-Ti-borate quenched flux many crystals of MgTiO3 contain MgOinclusions

C8404b 00001 1070 046 MgTiO3 MgO Mg-Ti-borate quenched flux

C21703b 00001 1080 024 Euhedral Mg2TiO4 MgTiO3 corroded MgO Mg-Ti borate quenched flux

C26304be 00001 1088 066 Large euhedral Mg2TiO4 poikiliticallyenclosing some euhedral MgTiO3 and MgO MgTiO3 and corroded MgO Mg-Ti-borate quenched flux

C26503a 00001 1090 0235 Mg2TiO4MgTiO3MgO Mg-Ti-boratequenched flux reaction direction ambiguous

C14703b 00001 1100 048 Euhedral Mg2TiO4 MgTiO3 and Mg-Ti-borate corroded MgO quenched flux

D366 060 1240 168 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoC1922c 060 1250 006 Mg2TiO4 MgTiO3 with holes quench carbonateD362d 060 1250 048 No discernible reactionD369 060 1260 192 MgO + Mg2TiO4 100 reactionD350 100 1340 050 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoC1918 100 1360 006 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoD354 100 1370 006 MgO + Mg2TiO4 some unreacted MgTiO3

D343 100 1380 006 MgO + Mg2TiO4 100 reactionD342 100 1420 006 MgO + Mg2TiO4 100 reactionD340 100 1460 006 MgO + Mg2TiO4 100 reactionD338 100 1500 007 MgO + Mg2TiO4 100 reactionR204e 120 1405 006 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoR193e 120 1410 020 MgO + Mg2TiO4 some unreacted MgTiO3

D361 150 1480 006 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoD364 150 1490 006 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoD358 150 1500 006 MgO + Mg2TiO4 some unreacted MgTiO3

D363 200 1580 003 MgTiO3 + MgO 100 reaction ldquoeutectoidrdquoD360 200 1600 006 MgO + Mg2TiO4 few grains unreacted MgTiO3

D359 200 1640 003 MgO + Mg2TiO4 100 reaction

H St C OrsquoNeill D R Scott

MgO was added and the amount of oxalic acid reduced to10 This run produced no discernible reaction directionand the use of oxalic acid was abandoned The two subse-quent runs at 6 kbar produced an unambiguous bracketbetween 1240 and 1260degC

The bracket at atmospheric pressure is placed betweenthe run at 1070degC (C8404) which produced euhedralMgTiO3 plus MgO with no Mg2TiO4 (Figure 3a) and at1080degC (C21703) which contained Mg2TiO4 andMgTiO3 with MgO but with the Mg2TiO4 showing goodcrystal faces while the MgO appears somewhat corroded(Fig 3b) However no reaction direction could be inferredfrom a run at 1090degC in which all three phases werepresent but none show good crystal faces An additionalrun at 1088degC (C26304) which used a starting mix ofMgTiO3 MgO and flux only (ie no Mg2TiO4) producedlarge euhedrally faceted crystals of Mg2TiO4 poikiliticallyenclosing MgTiO3 and the Mg-Ti-borate phase with occa-sional small poorly faceted grains of MgO interpreted tobe residual (Fig 3c) but the unambiguous growth ofMg2TiO4 in this run implies clearly that it is stable at thistemperature Run C19503 which at 1050degC is well belowthe breakdown temperature shows many grains of MgTiO3enclosing MgO blobs (Fig 3d) which is somewhat remi-niscent of the ldquoeutectoid-likerdquo textures of the high-pressureexperiments and like for these runs we interpret thesegrains to be due to decomposition of original Mg2TiO4grains

The results are compared to the data of Akimoto ampSyono (1967) in Fig 1 At first sight there appears to be alarge difference between the results if ours are compared tothe equation for the univariant curve provided by Akimotoand Syono which is P (kbar) = -18 + 0019 T (degC) In factonly one experiment of Akimoto and Syono is actuallydiscrepant namely their run MT22 in which Mg2TiO4 isreported to break down to MgTiO3 and MgO at 10 kbar and1440degC (cf our bracket between 1360 and 1370degC at10 kbar) This is far outside any likely experimental uncer-tainty in pressure or temperature measurement and wesuggest an ad hoc experimental malfunction in run MT22such as piston seizure or thermocouple extrusion

Stoichiometry and composition of the run products

Like many spinels Mg2TiO4 might be anticipated toshow some non-stoichiometry by solid solution towardsMgTiO3 at high temperatures (ie like MgAl2O4 towardsγ-Al2O3) However electron microprobe analysis showedalmost negligible deviations in all runs from Mg2TiO4 stoi-chiometry (Table 2) It is probable that at atmospheric pres-sure the temperature at which reaction (1) occurs (ie1075 plusmn 10degC) is not high enough for deviations from stoi-chiometry to be significant while higher pressuressuppress the non-stoichiometry due to the relatively highpartial molar volume of the defect spinel componentSimilarly MgTiO3 was found to be stoichiometric withinanalytical error in all runs The observation of smallamounts of TiO2 (~ 06 wt corresponding to a mole frac-tion of 0003) in MgO is likely at least in part an analyticalartefact due to the ldquostray electron effectrdquo (this is an issue of

318

Fig 2 Electron back-scattered images of run products from high-pressure experiments The lighter the area the higher the meanatomic number Contrast has been adjusted to illuminate optimallythe textures discussed in the text a) run R193 showing residualMgTiO3(Gk) surrounded by Mg2TiO4 (Q) with no contact withMgO (P) b) run D364 showing typical ldquoeutectoid-likerdquo inter-growths of MgTiO3 + MgO (labelled ldquoEurdquo) replacing originalMg2TiO4 plus original grains of MgTiO3 (Gk) and MgO (P) c)Run C1922 showing MgTiO3 (Gk) trapping vapour (or quenchcarbonate) now present as holes the Mg2TiO4 (Q) does not containthese holes What this means for the direction of reaction is equiv-ocal

experimental interest that is discussed in more detail inHermann et al in press) However the jump to 09 wtTiO2 in the highest temperature run analysed (D3631580degC 20 kbar) looks real as the MgO grain size in thisrun is larger than in the analysed runs with 06 wt TiO2Pelton et al (1998) report somewhat similar amounts of Tiin MgO in equilibrium with Mg2Ti4+O4-MgTi2

3+O4 spinelsin the system MgO-Ti-O at 1500degC and atmospheric pres-sure the amounts increasing with decreasing oxygen

fugacity suggesting that the valence state of the substi-tuting Ti is mainly Ti3+

The accuracy with which lattice parameters can bemeasured can often make these data sensitive indicators ofpurity and stoichiometry especially for phases with highcrystallographic symmetry such as the three of interesthere The downside is that it is difficult to interpret thecause of a change in lattice parameter unambiguously alsoit is possible that important changes in purity and stoi-

The free energy of formation of Mg2TiO4 spinel 319

Fig 3 Electron back-scattered images of run products from experiments at atmospheric pressure using Na2B2O4 flux All runs containcrystals of a Mg-Ti-borate phase (generally sub-acicular in habit slightly lighter shade of grey to MgO) and quenched melt rich in Na2Oand B2O3 (dark areas but generally indistinguishable from holes in these micrographs) The euhedral MgTiO3 crystals in these runs are~ 10 microm in diametera) C8404 (1270degC) showing euhedral MgTiO3 (Gk) and MgO (P) with no sign of any Mg2TiO4 b) C21703 (1280degC) showing euhe-dral Mg2TiO4 (Q) and MgTiO3 with somewhat corroded MgO c) C26304 (1288degC) showing a large crystal of Mg2TiO4 which hasgrown to enclose subhedral to euhedral MgTiO3 and some Mg-Ti-borate with the occasional microblob of MgO d) C19503 (1050degC)in which some but not all MgTiO3 contains many inclusions of MgO ndash cf Fig 2b

Table 2 Electron microprobe analysis of selected run products to check for the stoichiometry of the phases

Analytical conditions WDS 15 kV 20 nA MgO and TiO2 standardsTAP and LPET crystals

Run P(kbar) T(degC) wt TiO2 in molar MgTi molar MgTiMgO geikielite quandilite

D362 06 1250 066(3) 1020(6) 1982(3)C1922 06 1250 060(6) 1021(7) 1991(9)D369 06 1260 064(1) - 1980(8)R204 12 1495 069(4) 1007(3) -D364 15 1490 068(6) 1018(6) 1971(8)D363 20 1580 092(9) 1009(10) -

H St C OrsquoNeill D R Scott

chiometry either do not affect the lattice parameter or thatmore than one type change occurs with a net cancellingeffect Here the lattice parameters of MgO in the run prod-ucts are consistent with the pure MgO in the starting mate-rial (Table 3) except for D350 a run with theldquoeutectoid-likerdquo intergrowth of MgTiO3 + MgO here theslightly larger lattice parameter may be caused by strainassociated with this texture The effect on the lattice param-eter of MgO from the substitution of Ti is not known butby analogy with magnesiowuumlstite in the system MgO-FeO-FeO15 (eg OrsquoNeill et al 2003a) we expect this substitu-tion to lower the lattice parameter both because Ti4+ is asmaller cation than Mg2+ and because charge-balance ismaintained in heterovalent-substituted MgO by cationvacancies which decrease the lattice parameter

For Mg2TiO4 the lattice parameters in the run productsare ~ 0001 Aring larger than in the starting mix (Table 3)Since solid solution towards a defect spinel with cationvacancies would lower the lattice parameter this is possiblydue to the presence of Ti3+ in the form of solid solutiontowards MgTi2

3+O4 in the moderately low fO2 environmentof the piston-cylinder apparatus (cf Pelton et al 1998)The lattice parameter of end-member MgTi2

3+O4 spinel isinferred to be about 8505 Aring (Hohl et al 1996) hence theobserved increase of 0001 Aring corresponds to a mole frac-tion of 0015 MgTi2

3+O4 This would imply a molar MgTiratio of 196 in reasonable agreement with the ratio

observed by electron microprobe analysis 198 plusmn 001(Table 2) For MgTiO3 the lattice parameters of run D350are identical to those of the starting material and also tothose found by Wechsler amp von Dreele (1989) who give ao= 50548 (3) Aring co = 138992 (7) Aring V = 30756 Aring3

Thermodynamic evaluation

The equilibrium condition can be expressed as

∆rG(T P) = 0 = ∆rGO(T 1 bar) + int

P

1 ∆rV

O(T P)dP + RT ln K (2)

where

ailmMgTiO3

aoxMgO

K = mdashmdashmdashmdashmdash (3)asp

Mg2TiO4

Following the discussion above on the stoichiometryand purity of MgO MgTiO3 and Mg2TiO4 in the run prod-ucts we assume that K = 1 with a probable uncertainty ofonly plusmn 001 since such minor deviations from stoichiom-etry and purity that are expected in the run products willtend to cancel out across the reaction The experimentalhalf-brackets define points in free energy - temperature -pressure space at which ∆rG(T P) ge 0 (low temperature half-bracket) and ∆rG(T P) le 0 (high temperature half-bracket)Splitting ∆rG

O(T 1 bar) into entropy and the enthalpy terms

gives

∆rGO(T 1 bar) = ∆rH

O(298K) ndash T∆rS

O(298K) +

T

int298

∆rCPO(T)dT

ndash TT

int298

[∆rmdashCP

O

mdashTmdashmdash

(T)]dT (4)

Available data for high temperature heat capacities(Table 4) can be used to evaluate the last two terms in eqn

(3) (ie the two terms in ∆rCPO) For the

P

int1∆rVO(T P)dP

term in eqn 2 we have adopted the equation of staterecommended by Holland et al (1996) and used byHolland amp Powell (1998) For MgO and MgTiO3 the heatcapacities molar volumes thermal expansivities and bulk

320

Table 3 Lattice parameters (in Aring)Sample MgO Mg2TiO4 MgTiO3

a c V(Aring3)Starting mix 42118 84418 50551 138987 30758D338 42119 84430 - - -D343 42122 84426 - - -D350 42128 - 50552 138977 30757D360 42120 84433 5057 13895 3077

Only c1 MgTiO3 in run productsmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdash

Estimated uncertainties (one standard deviation) are 00001 Aringfor MgO 00002 Aring for Mg2TiO4 and 001 Aring3 for the unit cellvolume (V) of MgTiO3 (except D360)

Phase ∆fHo(298K1 bar) Sordm(298 K 1 bar) Cp = a + bT + cT-2 + d T-12 (in J K-1mol-1) Vordm(298K1 bar) αo κ

mdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdash(J bar-1) (K-1) (kbar)

a b c d(x102)

MgO 0-60165 0269 0605 0362 0-535800 -2992 1125 62 1650periclaseMgTiO3 -156742 0746 1510 - -1890400 -6522 3086 495 1770geikieliteMg2TiO4 -215741 1118a 1617b 03286 -2382200 -2786 4529c 548c 1890d

Qandilite

Table 4 Thermodynamic data

a calorimetrically measured value is 1036 J K-1 mol-1from Todd (1952) b Cp data from Orr and Coughlin (1952) andTodd (1952) fitted in this study c OrsquoNeill et al (2003b) d value for Fe2TiO4 from Holland and Powell (1998)mdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashData for MgO and MgTiO3 are from Holland amp Powell (1998) for Mg2TiO4 from this study except where noted

moduli are taken from Holland amp Powell (1998) assummarized in Table 4 For Mg2TiO4 the heat capacityequation is our fit to the heat content measurements of Orramp Coughlin (1952) which extend to 1820 K (31 data)Temperature was corrected from IPTS-48 to ITS-90 andthe equation was anchored at low temperature using theadiabatic calorimetric measurements of Todd (1952) in therange 250 to 300 K (5 data) Volume and thermal expansionat 1 bar are from OrsquoNeill et al (2003b) refitted to theequation for thermal expansion of Holland et al (1996)The bulk modulus of Mg2TiO4 has not yet been measuredto our knowledge so we adopt the value listed for Fe2TiO4by Holland amp Powell (1998)

With these data the experimental half-brackets can beused define maximum and minimum values of the quantity(∆rH

O(298K) ndash T∆rS

O(298K)) at each experimental pressure

These values are plotted in Fig 4 against experimentaltemperature To allow for experimental uncertainties inboth temperature and pressure measurement in the piston-cylinder apparatus 5 K has been subtracted from and02 kbar added to the low temperature half-brackets and 5K added to and 02 kbar subtracted from the high tempera-ture half-brackets For example the half-brackets nomi-nally at 10 kbar and 1360degC and 1370degC are treated asbeing at 1355degC and 102 kbar and 1375degC and 98 kbarrespectively For the experiments at atmospheric pressurean uncertainty of 5 K in the half-brackets was also adopted

The resulting array of limiting values of (∆rHO(298K) ndash

T∆rSO(298K)) defines a set of lines of slope -∆rS

O(298K) and

intercept ∆rHO(298K) By inspection we locate the two

limiting values (one from a high-temperature half-bracketthe other from a low-temperature half-bracket) that definethe line of maximum slope and similarly the two thatdefine the line of minimum slope These lines givemaximum and minimum values of ∆rS

O(298K) of 1010 and

934 J K-1 mol-1 respectively with corresponding values of∆rH

O(298K) of 1227 and 1105 kJ mol-1 For a best estimate

of ∆rSO(298K) we take the mean of these maximum and

minimum slopes 972 J K-1 mol-1 with an uncertaintygiven by half their difference plusmn 038 J K-1 mol-1 Likewise∆rH

O(298K) = 1166 plusmn 061 kJ mol-1Using the values of ∆f elHO

(298K) and SO(298K) for MgTiO3

and MgO from Holland amp Powell (1998) in Table 4 givesfor Mg2TiO4 ∆f elH

O(298K) = ndash 215741 plusmn 135 kJ mol-1 and

SO(298K) = 1112 plusmn 06 J K-1 mol-1 This latter value is 76 plusmn

09 J K-1 mol-1 more than the calorimetrically determinedvalue of 1036 plusmn 07 J K-1 mol-1 (Todd 1952) The differ-ence may be ascribed to zero-point entropy but is some-what less than the theoretical amount of 2Rln2 or 115 JK-1mol-1 expected from the random mixing of Mg and Tion the octahedral site of Mg2TiO4 spinel

One possibility is some short-range ordering of Mg andTi which seems plausible given the long-range ordering ofMg and Ti onto distinct octahedral sites at the phase transi-tion from to Fd

ndash3m P4122 at ~ 1000degC Unfortunately as

with many hypotheses regarding short-range order incrystal chemistry this explanation is not really practicableto test with existing experimental methods It is thereforenecessary to eliminate other possibilities such as error inthe calorimetric data for the other phases in reaction (1)ie MgTiO3 andor MgO For MgO it is reasonable toassume that the calorimetric data are secure not onlybecause they have been measured many times but alsobecause MgO figures in so many other reactions that havebeen subsumed into the compilation of the Holland ampPowell (1998) database The value SO

(298K) of for MgTiO3also seems well known since the low temperature heatcapacity measurements of Robie et al (1989) are in goodagreement with the earlier measurements of Shomate(1946) However at high temperatures the heat-contentmeasurements of MgTiO3 (Naylor amp Cook 1946) leave

The free energy of formation of Mg2TiO4 spinel 321

Fig 4 The quantity ∆rHO(298K) ndash T∆rS

O(298K) vs T (K) for the defining half-brackets plotted to allow for experimental uncertainties as

discussed in the text The lines with maximum and minimum slopes are drawn

H St C OrsquoNeill D R Scott

some room for doubt as follows Firstly the measurementsextend only to 1720 K with only three data above 1410 KRefitting the data of Naylor amp Cook (1946) to the four-term heat capacity polynomial used by Holland and Powell(ie Cp = a + bT + cT2 + dT-12) gives a result very muchthe same as that listed by Holland amp Powell (1998) butadding an extra fifth term to the polynomial (ie the termeT2) produces a better fit but one which leads to signifi-cantly higher computed values of Cp above ~ 1400K As anillustration Cp calculated at 1800 K with the four-term fitis 1356 J K-1 mol-1 whereas that with the five-term fit is1589 J K-1 mol-1 This would be a large enough change toaccount for the missing residual entropy in Mg2TiO4Whether this increase in the heat capacity of MgTiO3 isreal is difficult to judge since the extended Maier-Kelleyequation for heat capacities is an entirely empiricaldevice for representing the experimental data in which theinclusion of the term in eT2 is well known to lead to ambi-guities in the extrapolation of the equation to highertemperatures The value for the four-term fit at 1800 K isin excellent agreement with the Dulong-Petit value(Cp = 3nR + TVα2κ) which is calculated to be 136 J K-1

mol-1 at 1800 K using the data in Table 4 Excess heatcapacity could result from disordering of Mg and Ti as theilmenite structure in which Mg and Ti are ordered intolayers transforms towards the corundum structure(R

ndash3 rarr R

ndash3c) This transition has been observed in NiTiO3

by powder neutron diffraction at ~ 1560 K (Boysen et al(1995) and is inferred at ~ 1673 K in FeTiO3 from extrap-olation of the transition in FeTiO3-Fe2O3 solid solutions(Harrison et al 2000) Harrison et al (2000) show thatFeTiO3 is ~ 10 disordered at 1473 K The occurrence ofthis transition in MgTiO3 has been discounted by Reynardamp Guyot (1994) from their investigation by Raman spec-troscopy to 1820 K on the grounds that the symmetrychange would reduce the number of Raman active modesfrom ten to seven In fact the ten modes observable atroom temperature could only be tracked to ~ 1400 K due toline broadening and thermal emission so the possibility ofsymmetry reduction at higher temperatures remainsopen as does the possibility of significant Mg-Ti disor-dering within the ilmenite structure at temperatures belowthe putative transition as has been shown to occur inFeTiO3 by Harrison et al (2000) The question of excessentropy in MgTiO3 could be addressed by more precisehigh-temperature calorimetry (which would give the dataneeded to resolve the question of interest here directly)or other methods such as high-temperature powder neutrondiffraction

Acknowledgments We thank Mark Ghiorso andDominique Lattard for helpful reviews

References

Akimoto S amp Syono Y (1967) High-pressure decomposition ofsome titanate spinels J Chem Phys 47 1813-1817

Bose K amp Ganguly J (1995) Quartz-coesite transition revisited -reversed experimental-determination at 500-1200-degrees-C

and retrieved thermochemical properties Am Mineral 80231-238

Boysen H Frey F Lerch M Vogt T (1995) A neutron powderinvestigation of the high-temperature phase-transition inNiTiO3 Zeitschr fuumlr Kristallogr 210 328-337

Eriksson G amp Pelton AD (1993) Critical evaluation and opti-mization of the thermodynamic properties and phasediagrams of the MnO-TiO2 MgO-TiO2 FeO-TiO2 Ti2O3-TiO2Na2O-TiO2 and K2O-TiO2 systems Metall Trans B 24795-805

Harrison RJ Redfern SAT Smith RI (2000) In-situ study ofthe R

ndash3 to R

ndash3c phase transition in the ilmenite-hematite solid

solution using time-of-flight neutron powder diffraction AmMineral 85 194-205

Hermann J OrsquoNeill HStC Berry AJ (in Press) Titanium solu-bility in olivine in the system TiO2-MgO-SiO2 no evidence foran ultra-deep origin of Ti-bearing olivine Contrib MineralPetrol

Holland TJB amp Powell R (1998) An internally consistent ther-modynamic data set for phases of petrological interest J meta-morphic Geol 16 309-343

Holland TJB Redfern SAT Pawley AR (1996) Volumebehaviour of hydrous minerals at high pressure and tempera-ture 2 Compressibilities of lawsonite zoisite clinozoisite andepidote Am Mineral 81 341-348

Hohl H Kloc C Bucher E (1996) Electrical and magnetic prop-erties of spinel solid solutions Mg2-xTi1+xO4 0 le x le 1 J SolidState Chem 125 216-223

Hunter BA amp Howard CJ (2000) LHPM a computer programfor Rietveld analysis of X-ray and neutron powder diffractionpatterns Australian Nuclear Science and TechnologyOrganization Lucas Heights Research Laboratory MenaiNSW 2234 Australia

Millard RL Peterson RC Hunter BK (1995) Study of thecubic to tetragonal transition in Mg2TiO4 and Zn2TiO4 spinelsby 17O MAS NMR and Rietveld refinement of X-ray diffractiondata Am Mineral 80 885-896

Naylor BF amp Cook OA (1946) High-temperature heat contentsof the metatitanates of calcium iron and magnesium J AmerChem Soc 68 1003-1005

OrsquoNeill HStC Pownceby MI McCammon CA (2003a) Themagnesiowuumlstite iron equilibrium and its implications for theactivity-composition relations of (MgFe)2SiO4 olivine solidsolutions Contrib Mineral Petrol 146 308-325

ONeill HStC Redfern SAT Kesson S Short S (2003a) Anin situ neutron diffraction study of cation disordering insynthetic qandilite Mg2TiO4 at high temperatures AmMineral 88 860-865

Orr RL amp Coughlin JP (1952) High temperature heat contentsof magnesium orthotitanate and magnesium dititanate J AmerChem Soc 74 3186-3187

Pelton AD Eriksson G Krajewski D Goumlbbels M WoermannE (1998) Measurement and thermodynamic evaluation ofphase equilibria in the Mg-Ti-O system Zeitschr fuumlr PhysikalChem 207 163-180

Reynard B amp Guyot F (1994) High-temperature properties ofgeikielite (MgTiO3-ilmenite) from high-temperature high-pres-sure Raman-spectroscopy ndash some implications for MgSiO3-ilmenite Phys Chem Min 21 441-450

Robie RA Haselton Jr HT Hemingway BS (1989) Heatcapacities and entropies at 29815 K of MgTiO3 (geikielite) ZnO(zincite) and ZnCO3 (smithsonite) J Chem Thermodynamics21 743-749

322

H St C OrsquoNeill D R Scott

MgO was added and the amount of oxalic acid reduced to10 This run produced no discernible reaction directionand the use of oxalic acid was abandoned The two subse-quent runs at 6 kbar produced an unambiguous bracketbetween 1240 and 1260degC

The bracket at atmospheric pressure is placed betweenthe run at 1070degC (C8404) which produced euhedralMgTiO3 plus MgO with no Mg2TiO4 (Figure 3a) and at1080degC (C21703) which contained Mg2TiO4 andMgTiO3 with MgO but with the Mg2TiO4 showing goodcrystal faces while the MgO appears somewhat corroded(Fig 3b) However no reaction direction could be inferredfrom a run at 1090degC in which all three phases werepresent but none show good crystal faces An additionalrun at 1088degC (C26304) which used a starting mix ofMgTiO3 MgO and flux only (ie no Mg2TiO4) producedlarge euhedrally faceted crystals of Mg2TiO4 poikiliticallyenclosing MgTiO3 and the Mg-Ti-borate phase with occa-sional small poorly faceted grains of MgO interpreted tobe residual (Fig 3c) but the unambiguous growth ofMg2TiO4 in this run implies clearly that it is stable at thistemperature Run C19503 which at 1050degC is well belowthe breakdown temperature shows many grains of MgTiO3enclosing MgO blobs (Fig 3d) which is somewhat remi-niscent of the ldquoeutectoid-likerdquo textures of the high-pressureexperiments and like for these runs we interpret thesegrains to be due to decomposition of original Mg2TiO4grains

The results are compared to the data of Akimoto ampSyono (1967) in Fig 1 At first sight there appears to be alarge difference between the results if ours are compared tothe equation for the univariant curve provided by Akimotoand Syono which is P (kbar) = -18 + 0019 T (degC) In factonly one experiment of Akimoto and Syono is actuallydiscrepant namely their run MT22 in which Mg2TiO4 isreported to break down to MgTiO3 and MgO at 10 kbar and1440degC (cf our bracket between 1360 and 1370degC at10 kbar) This is far outside any likely experimental uncer-tainty in pressure or temperature measurement and wesuggest an ad hoc experimental malfunction in run MT22such as piston seizure or thermocouple extrusion

Stoichiometry and composition of the run products

Like many spinels Mg2TiO4 might be anticipated toshow some non-stoichiometry by solid solution towardsMgTiO3 at high temperatures (ie like MgAl2O4 towardsγ-Al2O3) However electron microprobe analysis showedalmost negligible deviations in all runs from Mg2TiO4 stoi-chiometry (Table 2) It is probable that at atmospheric pres-sure the temperature at which reaction (1) occurs (ie1075 plusmn 10degC) is not high enough for deviations from stoi-chiometry to be significant while higher pressuressuppress the non-stoichiometry due to the relatively highpartial molar volume of the defect spinel componentSimilarly MgTiO3 was found to be stoichiometric withinanalytical error in all runs The observation of smallamounts of TiO2 (~ 06 wt corresponding to a mole frac-tion of 0003) in MgO is likely at least in part an analyticalartefact due to the ldquostray electron effectrdquo (this is an issue of

318

Fig 2 Electron back-scattered images of run products from high-pressure experiments The lighter the area the higher the meanatomic number Contrast has been adjusted to illuminate optimallythe textures discussed in the text a) run R193 showing residualMgTiO3(Gk) surrounded by Mg2TiO4 (Q) with no contact withMgO (P) b) run D364 showing typical ldquoeutectoid-likerdquo inter-growths of MgTiO3 + MgO (labelled ldquoEurdquo) replacing originalMg2TiO4 plus original grains of MgTiO3 (Gk) and MgO (P) c)Run C1922 showing MgTiO3 (Gk) trapping vapour (or quenchcarbonate) now present as holes the Mg2TiO4 (Q) does not containthese holes What this means for the direction of reaction is equiv-ocal

experimental interest that is discussed in more detail inHermann et al in press) However the jump to 09 wtTiO2 in the highest temperature run analysed (D3631580degC 20 kbar) looks real as the MgO grain size in thisrun is larger than in the analysed runs with 06 wt TiO2Pelton et al (1998) report somewhat similar amounts of Tiin MgO in equilibrium with Mg2Ti4+O4-MgTi2

3+O4 spinelsin the system MgO-Ti-O at 1500degC and atmospheric pres-sure the amounts increasing with decreasing oxygen

fugacity suggesting that the valence state of the substi-tuting Ti is mainly Ti3+

The accuracy with which lattice parameters can bemeasured can often make these data sensitive indicators ofpurity and stoichiometry especially for phases with highcrystallographic symmetry such as the three of interesthere The downside is that it is difficult to interpret thecause of a change in lattice parameter unambiguously alsoit is possible that important changes in purity and stoi-

The free energy of formation of Mg2TiO4 spinel 319

Fig 3 Electron back-scattered images of run products from experiments at atmospheric pressure using Na2B2O4 flux All runs containcrystals of a Mg-Ti-borate phase (generally sub-acicular in habit slightly lighter shade of grey to MgO) and quenched melt rich in Na2Oand B2O3 (dark areas but generally indistinguishable from holes in these micrographs) The euhedral MgTiO3 crystals in these runs are~ 10 microm in diametera) C8404 (1270degC) showing euhedral MgTiO3 (Gk) and MgO (P) with no sign of any Mg2TiO4 b) C21703 (1280degC) showing euhe-dral Mg2TiO4 (Q) and MgTiO3 with somewhat corroded MgO c) C26304 (1288degC) showing a large crystal of Mg2TiO4 which hasgrown to enclose subhedral to euhedral MgTiO3 and some Mg-Ti-borate with the occasional microblob of MgO d) C19503 (1050degC)in which some but not all MgTiO3 contains many inclusions of MgO ndash cf Fig 2b

Table 2 Electron microprobe analysis of selected run products to check for the stoichiometry of the phases

Analytical conditions WDS 15 kV 20 nA MgO and TiO2 standardsTAP and LPET crystals

Run P(kbar) T(degC) wt TiO2 in molar MgTi molar MgTiMgO geikielite quandilite

D362 06 1250 066(3) 1020(6) 1982(3)C1922 06 1250 060(6) 1021(7) 1991(9)D369 06 1260 064(1) - 1980(8)R204 12 1495 069(4) 1007(3) -D364 15 1490 068(6) 1018(6) 1971(8)D363 20 1580 092(9) 1009(10) -

H St C OrsquoNeill D R Scott

chiometry either do not affect the lattice parameter or thatmore than one type change occurs with a net cancellingeffect Here the lattice parameters of MgO in the run prod-ucts are consistent with the pure MgO in the starting mate-rial (Table 3) except for D350 a run with theldquoeutectoid-likerdquo intergrowth of MgTiO3 + MgO here theslightly larger lattice parameter may be caused by strainassociated with this texture The effect on the lattice param-eter of MgO from the substitution of Ti is not known butby analogy with magnesiowuumlstite in the system MgO-FeO-FeO15 (eg OrsquoNeill et al 2003a) we expect this substitu-tion to lower the lattice parameter both because Ti4+ is asmaller cation than Mg2+ and because charge-balance ismaintained in heterovalent-substituted MgO by cationvacancies which decrease the lattice parameter

For Mg2TiO4 the lattice parameters in the run productsare ~ 0001 Aring larger than in the starting mix (Table 3)Since solid solution towards a defect spinel with cationvacancies would lower the lattice parameter this is possiblydue to the presence of Ti3+ in the form of solid solutiontowards MgTi2

3+O4 in the moderately low fO2 environmentof the piston-cylinder apparatus (cf Pelton et al 1998)The lattice parameter of end-member MgTi2

3+O4 spinel isinferred to be about 8505 Aring (Hohl et al 1996) hence theobserved increase of 0001 Aring corresponds to a mole frac-tion of 0015 MgTi2

3+O4 This would imply a molar MgTiratio of 196 in reasonable agreement with the ratio

observed by electron microprobe analysis 198 plusmn 001(Table 2) For MgTiO3 the lattice parameters of run D350are identical to those of the starting material and also tothose found by Wechsler amp von Dreele (1989) who give ao= 50548 (3) Aring co = 138992 (7) Aring V = 30756 Aring3

Thermodynamic evaluation

The equilibrium condition can be expressed as

∆rG(T P) = 0 = ∆rGO(T 1 bar) + int

P

1 ∆rV

O(T P)dP + RT ln K (2)

where

ailmMgTiO3

aoxMgO

K = mdashmdashmdashmdashmdash (3)asp

Mg2TiO4

Following the discussion above on the stoichiometryand purity of MgO MgTiO3 and Mg2TiO4 in the run prod-ucts we assume that K = 1 with a probable uncertainty ofonly plusmn 001 since such minor deviations from stoichiom-etry and purity that are expected in the run products willtend to cancel out across the reaction The experimentalhalf-brackets define points in free energy - temperature -pressure space at which ∆rG(T P) ge 0 (low temperature half-bracket) and ∆rG(T P) le 0 (high temperature half-bracket)Splitting ∆rG

O(T 1 bar) into entropy and the enthalpy terms

gives

∆rGO(T 1 bar) = ∆rH

O(298K) ndash T∆rS

O(298K) +

T

int298

∆rCPO(T)dT

ndash TT

int298

[∆rmdashCP

O

mdashTmdashmdash

(T)]dT (4)

Available data for high temperature heat capacities(Table 4) can be used to evaluate the last two terms in eqn

(3) (ie the two terms in ∆rCPO) For the

P

int1∆rVO(T P)dP

term in eqn 2 we have adopted the equation of staterecommended by Holland et al (1996) and used byHolland amp Powell (1998) For MgO and MgTiO3 the heatcapacities molar volumes thermal expansivities and bulk

320

Table 3 Lattice parameters (in Aring)Sample MgO Mg2TiO4 MgTiO3

a c V(Aring3)Starting mix 42118 84418 50551 138987 30758D338 42119 84430 - - -D343 42122 84426 - - -D350 42128 - 50552 138977 30757D360 42120 84433 5057 13895 3077

Only c1 MgTiO3 in run productsmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdash

Estimated uncertainties (one standard deviation) are 00001 Aringfor MgO 00002 Aring for Mg2TiO4 and 001 Aring3 for the unit cellvolume (V) of MgTiO3 (except D360)

Phase ∆fHo(298K1 bar) Sordm(298 K 1 bar) Cp = a + bT + cT-2 + d T-12 (in J K-1mol-1) Vordm(298K1 bar) αo κ

mdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdash(J bar-1) (K-1) (kbar)

a b c d(x102)

MgO 0-60165 0269 0605 0362 0-535800 -2992 1125 62 1650periclaseMgTiO3 -156742 0746 1510 - -1890400 -6522 3086 495 1770geikieliteMg2TiO4 -215741 1118a 1617b 03286 -2382200 -2786 4529c 548c 1890d

Qandilite

Table 4 Thermodynamic data

a calorimetrically measured value is 1036 J K-1 mol-1from Todd (1952) b Cp data from Orr and Coughlin (1952) andTodd (1952) fitted in this study c OrsquoNeill et al (2003b) d value for Fe2TiO4 from Holland and Powell (1998)mdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashData for MgO and MgTiO3 are from Holland amp Powell (1998) for Mg2TiO4 from this study except where noted

moduli are taken from Holland amp Powell (1998) assummarized in Table 4 For Mg2TiO4 the heat capacityequation is our fit to the heat content measurements of Orramp Coughlin (1952) which extend to 1820 K (31 data)Temperature was corrected from IPTS-48 to ITS-90 andthe equation was anchored at low temperature using theadiabatic calorimetric measurements of Todd (1952) in therange 250 to 300 K (5 data) Volume and thermal expansionat 1 bar are from OrsquoNeill et al (2003b) refitted to theequation for thermal expansion of Holland et al (1996)The bulk modulus of Mg2TiO4 has not yet been measuredto our knowledge so we adopt the value listed for Fe2TiO4by Holland amp Powell (1998)

With these data the experimental half-brackets can beused define maximum and minimum values of the quantity(∆rH

O(298K) ndash T∆rS

O(298K)) at each experimental pressure

These values are plotted in Fig 4 against experimentaltemperature To allow for experimental uncertainties inboth temperature and pressure measurement in the piston-cylinder apparatus 5 K has been subtracted from and02 kbar added to the low temperature half-brackets and 5K added to and 02 kbar subtracted from the high tempera-ture half-brackets For example the half-brackets nomi-nally at 10 kbar and 1360degC and 1370degC are treated asbeing at 1355degC and 102 kbar and 1375degC and 98 kbarrespectively For the experiments at atmospheric pressurean uncertainty of 5 K in the half-brackets was also adopted

The resulting array of limiting values of (∆rHO(298K) ndash

T∆rSO(298K)) defines a set of lines of slope -∆rS

O(298K) and

intercept ∆rHO(298K) By inspection we locate the two

limiting values (one from a high-temperature half-bracketthe other from a low-temperature half-bracket) that definethe line of maximum slope and similarly the two thatdefine the line of minimum slope These lines givemaximum and minimum values of ∆rS

O(298K) of 1010 and

934 J K-1 mol-1 respectively with corresponding values of∆rH

O(298K) of 1227 and 1105 kJ mol-1 For a best estimate

of ∆rSO(298K) we take the mean of these maximum and

minimum slopes 972 J K-1 mol-1 with an uncertaintygiven by half their difference plusmn 038 J K-1 mol-1 Likewise∆rH

O(298K) = 1166 plusmn 061 kJ mol-1Using the values of ∆f elHO

(298K) and SO(298K) for MgTiO3

and MgO from Holland amp Powell (1998) in Table 4 givesfor Mg2TiO4 ∆f elH

O(298K) = ndash 215741 plusmn 135 kJ mol-1 and

SO(298K) = 1112 plusmn 06 J K-1 mol-1 This latter value is 76 plusmn

09 J K-1 mol-1 more than the calorimetrically determinedvalue of 1036 plusmn 07 J K-1 mol-1 (Todd 1952) The differ-ence may be ascribed to zero-point entropy but is some-what less than the theoretical amount of 2Rln2 or 115 JK-1mol-1 expected from the random mixing of Mg and Tion the octahedral site of Mg2TiO4 spinel

One possibility is some short-range ordering of Mg andTi which seems plausible given the long-range ordering ofMg and Ti onto distinct octahedral sites at the phase transi-tion from to Fd

ndash3m P4122 at ~ 1000degC Unfortunately as

with many hypotheses regarding short-range order incrystal chemistry this explanation is not really practicableto test with existing experimental methods It is thereforenecessary to eliminate other possibilities such as error inthe calorimetric data for the other phases in reaction (1)ie MgTiO3 andor MgO For MgO it is reasonable toassume that the calorimetric data are secure not onlybecause they have been measured many times but alsobecause MgO figures in so many other reactions that havebeen subsumed into the compilation of the Holland ampPowell (1998) database The value SO

(298K) of for MgTiO3also seems well known since the low temperature heatcapacity measurements of Robie et al (1989) are in goodagreement with the earlier measurements of Shomate(1946) However at high temperatures the heat-contentmeasurements of MgTiO3 (Naylor amp Cook 1946) leave

The free energy of formation of Mg2TiO4 spinel 321

Fig 4 The quantity ∆rHO(298K) ndash T∆rS

O(298K) vs T (K) for the defining half-brackets plotted to allow for experimental uncertainties as

discussed in the text The lines with maximum and minimum slopes are drawn

H St C OrsquoNeill D R Scott

some room for doubt as follows Firstly the measurementsextend only to 1720 K with only three data above 1410 KRefitting the data of Naylor amp Cook (1946) to the four-term heat capacity polynomial used by Holland and Powell(ie Cp = a + bT + cT2 + dT-12) gives a result very muchthe same as that listed by Holland amp Powell (1998) butadding an extra fifth term to the polynomial (ie the termeT2) produces a better fit but one which leads to signifi-cantly higher computed values of Cp above ~ 1400K As anillustration Cp calculated at 1800 K with the four-term fitis 1356 J K-1 mol-1 whereas that with the five-term fit is1589 J K-1 mol-1 This would be a large enough change toaccount for the missing residual entropy in Mg2TiO4Whether this increase in the heat capacity of MgTiO3 isreal is difficult to judge since the extended Maier-Kelleyequation for heat capacities is an entirely empiricaldevice for representing the experimental data in which theinclusion of the term in eT2 is well known to lead to ambi-guities in the extrapolation of the equation to highertemperatures The value for the four-term fit at 1800 K isin excellent agreement with the Dulong-Petit value(Cp = 3nR + TVα2κ) which is calculated to be 136 J K-1

mol-1 at 1800 K using the data in Table 4 Excess heatcapacity could result from disordering of Mg and Ti as theilmenite structure in which Mg and Ti are ordered intolayers transforms towards the corundum structure(R

ndash3 rarr R

ndash3c) This transition has been observed in NiTiO3

by powder neutron diffraction at ~ 1560 K (Boysen et al(1995) and is inferred at ~ 1673 K in FeTiO3 from extrap-olation of the transition in FeTiO3-Fe2O3 solid solutions(Harrison et al 2000) Harrison et al (2000) show thatFeTiO3 is ~ 10 disordered at 1473 K The occurrence ofthis transition in MgTiO3 has been discounted by Reynardamp Guyot (1994) from their investigation by Raman spec-troscopy to 1820 K on the grounds that the symmetrychange would reduce the number of Raman active modesfrom ten to seven In fact the ten modes observable atroom temperature could only be tracked to ~ 1400 K due toline broadening and thermal emission so the possibility ofsymmetry reduction at higher temperatures remainsopen as does the possibility of significant Mg-Ti disor-dering within the ilmenite structure at temperatures belowthe putative transition as has been shown to occur inFeTiO3 by Harrison et al (2000) The question of excessentropy in MgTiO3 could be addressed by more precisehigh-temperature calorimetry (which would give the dataneeded to resolve the question of interest here directly)or other methods such as high-temperature powder neutrondiffraction

Acknowledgments We thank Mark Ghiorso andDominique Lattard for helpful reviews

References

Akimoto S amp Syono Y (1967) High-pressure decomposition ofsome titanate spinels J Chem Phys 47 1813-1817

Bose K amp Ganguly J (1995) Quartz-coesite transition revisited -reversed experimental-determination at 500-1200-degrees-C

and retrieved thermochemical properties Am Mineral 80231-238

Boysen H Frey F Lerch M Vogt T (1995) A neutron powderinvestigation of the high-temperature phase-transition inNiTiO3 Zeitschr fuumlr Kristallogr 210 328-337

Eriksson G amp Pelton AD (1993) Critical evaluation and opti-mization of the thermodynamic properties and phasediagrams of the MnO-TiO2 MgO-TiO2 FeO-TiO2 Ti2O3-TiO2Na2O-TiO2 and K2O-TiO2 systems Metall Trans B 24795-805

Harrison RJ Redfern SAT Smith RI (2000) In-situ study ofthe R

ndash3 to R

ndash3c phase transition in the ilmenite-hematite solid

solution using time-of-flight neutron powder diffraction AmMineral 85 194-205

Hermann J OrsquoNeill HStC Berry AJ (in Press) Titanium solu-bility in olivine in the system TiO2-MgO-SiO2 no evidence foran ultra-deep origin of Ti-bearing olivine Contrib MineralPetrol

Holland TJB amp Powell R (1998) An internally consistent ther-modynamic data set for phases of petrological interest J meta-morphic Geol 16 309-343

Holland TJB Redfern SAT Pawley AR (1996) Volumebehaviour of hydrous minerals at high pressure and tempera-ture 2 Compressibilities of lawsonite zoisite clinozoisite andepidote Am Mineral 81 341-348

Hohl H Kloc C Bucher E (1996) Electrical and magnetic prop-erties of spinel solid solutions Mg2-xTi1+xO4 0 le x le 1 J SolidState Chem 125 216-223

Hunter BA amp Howard CJ (2000) LHPM a computer programfor Rietveld analysis of X-ray and neutron powder diffractionpatterns Australian Nuclear Science and TechnologyOrganization Lucas Heights Research Laboratory MenaiNSW 2234 Australia

Millard RL Peterson RC Hunter BK (1995) Study of thecubic to tetragonal transition in Mg2TiO4 and Zn2TiO4 spinelsby 17O MAS NMR and Rietveld refinement of X-ray diffractiondata Am Mineral 80 885-896

Naylor BF amp Cook OA (1946) High-temperature heat contentsof the metatitanates of calcium iron and magnesium J AmerChem Soc 68 1003-1005

OrsquoNeill HStC Pownceby MI McCammon CA (2003a) Themagnesiowuumlstite iron equilibrium and its implications for theactivity-composition relations of (MgFe)2SiO4 olivine solidsolutions Contrib Mineral Petrol 146 308-325

ONeill HStC Redfern SAT Kesson S Short S (2003a) Anin situ neutron diffraction study of cation disordering insynthetic qandilite Mg2TiO4 at high temperatures AmMineral 88 860-865

Orr RL amp Coughlin JP (1952) High temperature heat contentsof magnesium orthotitanate and magnesium dititanate J AmerChem Soc 74 3186-3187

Pelton AD Eriksson G Krajewski D Goumlbbels M WoermannE (1998) Measurement and thermodynamic evaluation ofphase equilibria in the Mg-Ti-O system Zeitschr fuumlr PhysikalChem 207 163-180

Reynard B amp Guyot F (1994) High-temperature properties ofgeikielite (MgTiO3-ilmenite) from high-temperature high-pres-sure Raman-spectroscopy ndash some implications for MgSiO3-ilmenite Phys Chem Min 21 441-450

Robie RA Haselton Jr HT Hemingway BS (1989) Heatcapacities and entropies at 29815 K of MgTiO3 (geikielite) ZnO(zincite) and ZnCO3 (smithsonite) J Chem Thermodynamics21 743-749

322

experimental interest that is discussed in more detail inHermann et al in press) However the jump to 09 wtTiO2 in the highest temperature run analysed (D3631580degC 20 kbar) looks real as the MgO grain size in thisrun is larger than in the analysed runs with 06 wt TiO2Pelton et al (1998) report somewhat similar amounts of Tiin MgO in equilibrium with Mg2Ti4+O4-MgTi2

3+O4 spinelsin the system MgO-Ti-O at 1500degC and atmospheric pres-sure the amounts increasing with decreasing oxygen

fugacity suggesting that the valence state of the substi-tuting Ti is mainly Ti3+

The accuracy with which lattice parameters can bemeasured can often make these data sensitive indicators ofpurity and stoichiometry especially for phases with highcrystallographic symmetry such as the three of interesthere The downside is that it is difficult to interpret thecause of a change in lattice parameter unambiguously alsoit is possible that important changes in purity and stoi-

The free energy of formation of Mg2TiO4 spinel 319

Fig 3 Electron back-scattered images of run products from experiments at atmospheric pressure using Na2B2O4 flux All runs containcrystals of a Mg-Ti-borate phase (generally sub-acicular in habit slightly lighter shade of grey to MgO) and quenched melt rich in Na2Oand B2O3 (dark areas but generally indistinguishable from holes in these micrographs) The euhedral MgTiO3 crystals in these runs are~ 10 microm in diametera) C8404 (1270degC) showing euhedral MgTiO3 (Gk) and MgO (P) with no sign of any Mg2TiO4 b) C21703 (1280degC) showing euhe-dral Mg2TiO4 (Q) and MgTiO3 with somewhat corroded MgO c) C26304 (1288degC) showing a large crystal of Mg2TiO4 which hasgrown to enclose subhedral to euhedral MgTiO3 and some Mg-Ti-borate with the occasional microblob of MgO d) C19503 (1050degC)in which some but not all MgTiO3 contains many inclusions of MgO ndash cf Fig 2b

Table 2 Electron microprobe analysis of selected run products to check for the stoichiometry of the phases

Analytical conditions WDS 15 kV 20 nA MgO and TiO2 standardsTAP and LPET crystals

Run P(kbar) T(degC) wt TiO2 in molar MgTi molar MgTiMgO geikielite quandilite

D362 06 1250 066(3) 1020(6) 1982(3)C1922 06 1250 060(6) 1021(7) 1991(9)D369 06 1260 064(1) - 1980(8)R204 12 1495 069(4) 1007(3) -D364 15 1490 068(6) 1018(6) 1971(8)D363 20 1580 092(9) 1009(10) -

H St C OrsquoNeill D R Scott

chiometry either do not affect the lattice parameter or thatmore than one type change occurs with a net cancellingeffect Here the lattice parameters of MgO in the run prod-ucts are consistent with the pure MgO in the starting mate-rial (Table 3) except for D350 a run with theldquoeutectoid-likerdquo intergrowth of MgTiO3 + MgO here theslightly larger lattice parameter may be caused by strainassociated with this texture The effect on the lattice param-eter of MgO from the substitution of Ti is not known butby analogy with magnesiowuumlstite in the system MgO-FeO-FeO15 (eg OrsquoNeill et al 2003a) we expect this substitu-tion to lower the lattice parameter both because Ti4+ is asmaller cation than Mg2+ and because charge-balance ismaintained in heterovalent-substituted MgO by cationvacancies which decrease the lattice parameter

For Mg2TiO4 the lattice parameters in the run productsare ~ 0001 Aring larger than in the starting mix (Table 3)Since solid solution towards a defect spinel with cationvacancies would lower the lattice parameter this is possiblydue to the presence of Ti3+ in the form of solid solutiontowards MgTi2

3+O4 in the moderately low fO2 environmentof the piston-cylinder apparatus (cf Pelton et al 1998)The lattice parameter of end-member MgTi2

3+O4 spinel isinferred to be about 8505 Aring (Hohl et al 1996) hence theobserved increase of 0001 Aring corresponds to a mole frac-tion of 0015 MgTi2

3+O4 This would imply a molar MgTiratio of 196 in reasonable agreement with the ratio

observed by electron microprobe analysis 198 plusmn 001(Table 2) For MgTiO3 the lattice parameters of run D350are identical to those of the starting material and also tothose found by Wechsler amp von Dreele (1989) who give ao= 50548 (3) Aring co = 138992 (7) Aring V = 30756 Aring3

Thermodynamic evaluation

The equilibrium condition can be expressed as

∆rG(T P) = 0 = ∆rGO(T 1 bar) + int

P

1 ∆rV

O(T P)dP + RT ln K (2)

where

ailmMgTiO3

aoxMgO

K = mdashmdashmdashmdashmdash (3)asp

Mg2TiO4

Following the discussion above on the stoichiometryand purity of MgO MgTiO3 and Mg2TiO4 in the run prod-ucts we assume that K = 1 with a probable uncertainty ofonly plusmn 001 since such minor deviations from stoichiom-etry and purity that are expected in the run products willtend to cancel out across the reaction The experimentalhalf-brackets define points in free energy - temperature -pressure space at which ∆rG(T P) ge 0 (low temperature half-bracket) and ∆rG(T P) le 0 (high temperature half-bracket)Splitting ∆rG

O(T 1 bar) into entropy and the enthalpy terms

gives

∆rGO(T 1 bar) = ∆rH

O(298K) ndash T∆rS

O(298K) +

T

int298

∆rCPO(T)dT

ndash TT

int298

[∆rmdashCP

O

mdashTmdashmdash

(T)]dT (4)

Available data for high temperature heat capacities(Table 4) can be used to evaluate the last two terms in eqn

(3) (ie the two terms in ∆rCPO) For the

P

int1∆rVO(T P)dP

term in eqn 2 we have adopted the equation of staterecommended by Holland et al (1996) and used byHolland amp Powell (1998) For MgO and MgTiO3 the heatcapacities molar volumes thermal expansivities and bulk

320

Table 3 Lattice parameters (in Aring)Sample MgO Mg2TiO4 MgTiO3

a c V(Aring3)Starting mix 42118 84418 50551 138987 30758D338 42119 84430 - - -D343 42122 84426 - - -D350 42128 - 50552 138977 30757D360 42120 84433 5057 13895 3077

Only c1 MgTiO3 in run productsmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdash

Estimated uncertainties (one standard deviation) are 00001 Aringfor MgO 00002 Aring for Mg2TiO4 and 001 Aring3 for the unit cellvolume (V) of MgTiO3 (except D360)

Phase ∆fHo(298K1 bar) Sordm(298 K 1 bar) Cp = a + bT + cT-2 + d T-12 (in J K-1mol-1) Vordm(298K1 bar) αo κ

mdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdash(J bar-1) (K-1) (kbar)

a b c d(x102)

MgO 0-60165 0269 0605 0362 0-535800 -2992 1125 62 1650periclaseMgTiO3 -156742 0746 1510 - -1890400 -6522 3086 495 1770geikieliteMg2TiO4 -215741 1118a 1617b 03286 -2382200 -2786 4529c 548c 1890d

Qandilite

Table 4 Thermodynamic data

a calorimetrically measured value is 1036 J K-1 mol-1from Todd (1952) b Cp data from Orr and Coughlin (1952) andTodd (1952) fitted in this study c OrsquoNeill et al (2003b) d value for Fe2TiO4 from Holland and Powell (1998)mdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashData for MgO and MgTiO3 are from Holland amp Powell (1998) for Mg2TiO4 from this study except where noted

moduli are taken from Holland amp Powell (1998) assummarized in Table 4 For Mg2TiO4 the heat capacityequation is our fit to the heat content measurements of Orramp Coughlin (1952) which extend to 1820 K (31 data)Temperature was corrected from IPTS-48 to ITS-90 andthe equation was anchored at low temperature using theadiabatic calorimetric measurements of Todd (1952) in therange 250 to 300 K (5 data) Volume and thermal expansionat 1 bar are from OrsquoNeill et al (2003b) refitted to theequation for thermal expansion of Holland et al (1996)The bulk modulus of Mg2TiO4 has not yet been measuredto our knowledge so we adopt the value listed for Fe2TiO4by Holland amp Powell (1998)

With these data the experimental half-brackets can beused define maximum and minimum values of the quantity(∆rH

O(298K) ndash T∆rS

O(298K)) at each experimental pressure

These values are plotted in Fig 4 against experimentaltemperature To allow for experimental uncertainties inboth temperature and pressure measurement in the piston-cylinder apparatus 5 K has been subtracted from and02 kbar added to the low temperature half-brackets and 5K added to and 02 kbar subtracted from the high tempera-ture half-brackets For example the half-brackets nomi-nally at 10 kbar and 1360degC and 1370degC are treated asbeing at 1355degC and 102 kbar and 1375degC and 98 kbarrespectively For the experiments at atmospheric pressurean uncertainty of 5 K in the half-brackets was also adopted

The resulting array of limiting values of (∆rHO(298K) ndash

T∆rSO(298K)) defines a set of lines of slope -∆rS

O(298K) and

intercept ∆rHO(298K) By inspection we locate the two

limiting values (one from a high-temperature half-bracketthe other from a low-temperature half-bracket) that definethe line of maximum slope and similarly the two thatdefine the line of minimum slope These lines givemaximum and minimum values of ∆rS

O(298K) of 1010 and

934 J K-1 mol-1 respectively with corresponding values of∆rH

O(298K) of 1227 and 1105 kJ mol-1 For a best estimate

of ∆rSO(298K) we take the mean of these maximum and

minimum slopes 972 J K-1 mol-1 with an uncertaintygiven by half their difference plusmn 038 J K-1 mol-1 Likewise∆rH

O(298K) = 1166 plusmn 061 kJ mol-1Using the values of ∆f elHO

(298K) and SO(298K) for MgTiO3

and MgO from Holland amp Powell (1998) in Table 4 givesfor Mg2TiO4 ∆f elH

O(298K) = ndash 215741 plusmn 135 kJ mol-1 and

SO(298K) = 1112 plusmn 06 J K-1 mol-1 This latter value is 76 plusmn

09 J K-1 mol-1 more than the calorimetrically determinedvalue of 1036 plusmn 07 J K-1 mol-1 (Todd 1952) The differ-ence may be ascribed to zero-point entropy but is some-what less than the theoretical amount of 2Rln2 or 115 JK-1mol-1 expected from the random mixing of Mg and Tion the octahedral site of Mg2TiO4 spinel

One possibility is some short-range ordering of Mg andTi which seems plausible given the long-range ordering ofMg and Ti onto distinct octahedral sites at the phase transi-tion from to Fd

ndash3m P4122 at ~ 1000degC Unfortunately as

with many hypotheses regarding short-range order incrystal chemistry this explanation is not really practicableto test with existing experimental methods It is thereforenecessary to eliminate other possibilities such as error inthe calorimetric data for the other phases in reaction (1)ie MgTiO3 andor MgO For MgO it is reasonable toassume that the calorimetric data are secure not onlybecause they have been measured many times but alsobecause MgO figures in so many other reactions that havebeen subsumed into the compilation of the Holland ampPowell (1998) database The value SO

(298K) of for MgTiO3also seems well known since the low temperature heatcapacity measurements of Robie et al (1989) are in goodagreement with the earlier measurements of Shomate(1946) However at high temperatures the heat-contentmeasurements of MgTiO3 (Naylor amp Cook 1946) leave

The free energy of formation of Mg2TiO4 spinel 321

Fig 4 The quantity ∆rHO(298K) ndash T∆rS

O(298K) vs T (K) for the defining half-brackets plotted to allow for experimental uncertainties as

discussed in the text The lines with maximum and minimum slopes are drawn

H St C OrsquoNeill D R Scott

some room for doubt as follows Firstly the measurementsextend only to 1720 K with only three data above 1410 KRefitting the data of Naylor amp Cook (1946) to the four-term heat capacity polynomial used by Holland and Powell(ie Cp = a + bT + cT2 + dT-12) gives a result very muchthe same as that listed by Holland amp Powell (1998) butadding an extra fifth term to the polynomial (ie the termeT2) produces a better fit but one which leads to signifi-cantly higher computed values of Cp above ~ 1400K As anillustration Cp calculated at 1800 K with the four-term fitis 1356 J K-1 mol-1 whereas that with the five-term fit is1589 J K-1 mol-1 This would be a large enough change toaccount for the missing residual entropy in Mg2TiO4Whether this increase in the heat capacity of MgTiO3 isreal is difficult to judge since the extended Maier-Kelleyequation for heat capacities is an entirely empiricaldevice for representing the experimental data in which theinclusion of the term in eT2 is well known to lead to ambi-guities in the extrapolation of the equation to highertemperatures The value for the four-term fit at 1800 K isin excellent agreement with the Dulong-Petit value(Cp = 3nR + TVα2κ) which is calculated to be 136 J K-1

mol-1 at 1800 K using the data in Table 4 Excess heatcapacity could result from disordering of Mg and Ti as theilmenite structure in which Mg and Ti are ordered intolayers transforms towards the corundum structure(R

ndash3 rarr R

ndash3c) This transition has been observed in NiTiO3

by powder neutron diffraction at ~ 1560 K (Boysen et al(1995) and is inferred at ~ 1673 K in FeTiO3 from extrap-olation of the transition in FeTiO3-Fe2O3 solid solutions(Harrison et al 2000) Harrison et al (2000) show thatFeTiO3 is ~ 10 disordered at 1473 K The occurrence ofthis transition in MgTiO3 has been discounted by Reynardamp Guyot (1994) from their investigation by Raman spec-troscopy to 1820 K on the grounds that the symmetrychange would reduce the number of Raman active modesfrom ten to seven In fact the ten modes observable atroom temperature could only be tracked to ~ 1400 K due toline broadening and thermal emission so the possibility ofsymmetry reduction at higher temperatures remainsopen as does the possibility of significant Mg-Ti disor-dering within the ilmenite structure at temperatures belowthe putative transition as has been shown to occur inFeTiO3 by Harrison et al (2000) The question of excessentropy in MgTiO3 could be addressed by more precisehigh-temperature calorimetry (which would give the dataneeded to resolve the question of interest here directly)or other methods such as high-temperature powder neutrondiffraction

Acknowledgments We thank Mark Ghiorso andDominique Lattard for helpful reviews

References

Akimoto S amp Syono Y (1967) High-pressure decomposition ofsome titanate spinels J Chem Phys 47 1813-1817

Bose K amp Ganguly J (1995) Quartz-coesite transition revisited -reversed experimental-determination at 500-1200-degrees-C

and retrieved thermochemical properties Am Mineral 80231-238

Boysen H Frey F Lerch M Vogt T (1995) A neutron powderinvestigation of the high-temperature phase-transition inNiTiO3 Zeitschr fuumlr Kristallogr 210 328-337

Eriksson G amp Pelton AD (1993) Critical evaluation and opti-mization of the thermodynamic properties and phasediagrams of the MnO-TiO2 MgO-TiO2 FeO-TiO2 Ti2O3-TiO2Na2O-TiO2 and K2O-TiO2 systems Metall Trans B 24795-805

Harrison RJ Redfern SAT Smith RI (2000) In-situ study ofthe R

ndash3 to R

ndash3c phase transition in the ilmenite-hematite solid

solution using time-of-flight neutron powder diffraction AmMineral 85 194-205

Hermann J OrsquoNeill HStC Berry AJ (in Press) Titanium solu-bility in olivine in the system TiO2-MgO-SiO2 no evidence foran ultra-deep origin of Ti-bearing olivine Contrib MineralPetrol

Holland TJB amp Powell R (1998) An internally consistent ther-modynamic data set for phases of petrological interest J meta-morphic Geol 16 309-343

Holland TJB Redfern SAT Pawley AR (1996) Volumebehaviour of hydrous minerals at high pressure and tempera-ture 2 Compressibilities of lawsonite zoisite clinozoisite andepidote Am Mineral 81 341-348

Hohl H Kloc C Bucher E (1996) Electrical and magnetic prop-erties of spinel solid solutions Mg2-xTi1+xO4 0 le x le 1 J SolidState Chem 125 216-223

Hunter BA amp Howard CJ (2000) LHPM a computer programfor Rietveld analysis of X-ray and neutron powder diffractionpatterns Australian Nuclear Science and TechnologyOrganization Lucas Heights Research Laboratory MenaiNSW 2234 Australia

Millard RL Peterson RC Hunter BK (1995) Study of thecubic to tetragonal transition in Mg2TiO4 and Zn2TiO4 spinelsby 17O MAS NMR and Rietveld refinement of X-ray diffractiondata Am Mineral 80 885-896

Naylor BF amp Cook OA (1946) High-temperature heat contentsof the metatitanates of calcium iron and magnesium J AmerChem Soc 68 1003-1005

OrsquoNeill HStC Pownceby MI McCammon CA (2003a) Themagnesiowuumlstite iron equilibrium and its implications for theactivity-composition relations of (MgFe)2SiO4 olivine solidsolutions Contrib Mineral Petrol 146 308-325

ONeill HStC Redfern SAT Kesson S Short S (2003a) Anin situ neutron diffraction study of cation disordering insynthetic qandilite Mg2TiO4 at high temperatures AmMineral 88 860-865

Orr RL amp Coughlin JP (1952) High temperature heat contentsof magnesium orthotitanate and magnesium dititanate J AmerChem Soc 74 3186-3187

Pelton AD Eriksson G Krajewski D Goumlbbels M WoermannE (1998) Measurement and thermodynamic evaluation ofphase equilibria in the Mg-Ti-O system Zeitschr fuumlr PhysikalChem 207 163-180

Reynard B amp Guyot F (1994) High-temperature properties ofgeikielite (MgTiO3-ilmenite) from high-temperature high-pres-sure Raman-spectroscopy ndash some implications for MgSiO3-ilmenite Phys Chem Min 21 441-450

Robie RA Haselton Jr HT Hemingway BS (1989) Heatcapacities and entropies at 29815 K of MgTiO3 (geikielite) ZnO(zincite) and ZnCO3 (smithsonite) J Chem Thermodynamics21 743-749

322

H St C OrsquoNeill D R Scott

chiometry either do not affect the lattice parameter or thatmore than one type change occurs with a net cancellingeffect Here the lattice parameters of MgO in the run prod-ucts are consistent with the pure MgO in the starting mate-rial (Table 3) except for D350 a run with theldquoeutectoid-likerdquo intergrowth of MgTiO3 + MgO here theslightly larger lattice parameter may be caused by strainassociated with this texture The effect on the lattice param-eter of MgO from the substitution of Ti is not known butby analogy with magnesiowuumlstite in the system MgO-FeO-FeO15 (eg OrsquoNeill et al 2003a) we expect this substitu-tion to lower the lattice parameter both because Ti4+ is asmaller cation than Mg2+ and because charge-balance ismaintained in heterovalent-substituted MgO by cationvacancies which decrease the lattice parameter

For Mg2TiO4 the lattice parameters in the run productsare ~ 0001 Aring larger than in the starting mix (Table 3)Since solid solution towards a defect spinel with cationvacancies would lower the lattice parameter this is possiblydue to the presence of Ti3+ in the form of solid solutiontowards MgTi2

3+O4 in the moderately low fO2 environmentof the piston-cylinder apparatus (cf Pelton et al 1998)The lattice parameter of end-member MgTi2

3+O4 spinel isinferred to be about 8505 Aring (Hohl et al 1996) hence theobserved increase of 0001 Aring corresponds to a mole frac-tion of 0015 MgTi2

3+O4 This would imply a molar MgTiratio of 196 in reasonable agreement with the ratio

observed by electron microprobe analysis 198 plusmn 001(Table 2) For MgTiO3 the lattice parameters of run D350are identical to those of the starting material and also tothose found by Wechsler amp von Dreele (1989) who give ao= 50548 (3) Aring co = 138992 (7) Aring V = 30756 Aring3

Thermodynamic evaluation

The equilibrium condition can be expressed as

∆rG(T P) = 0 = ∆rGO(T 1 bar) + int

P

1 ∆rV

O(T P)dP + RT ln K (2)

where

ailmMgTiO3

aoxMgO

K = mdashmdashmdashmdashmdash (3)asp

Mg2TiO4

Following the discussion above on the stoichiometryand purity of MgO MgTiO3 and Mg2TiO4 in the run prod-ucts we assume that K = 1 with a probable uncertainty ofonly plusmn 001 since such minor deviations from stoichiom-etry and purity that are expected in the run products willtend to cancel out across the reaction The experimentalhalf-brackets define points in free energy - temperature -pressure space at which ∆rG(T P) ge 0 (low temperature half-bracket) and ∆rG(T P) le 0 (high temperature half-bracket)Splitting ∆rG

O(T 1 bar) into entropy and the enthalpy terms

gives

∆rGO(T 1 bar) = ∆rH

O(298K) ndash T∆rS

O(298K) +

T

int298

∆rCPO(T)dT

ndash TT

int298

[∆rmdashCP

O

mdashTmdashmdash

(T)]dT (4)

Available data for high temperature heat capacities(Table 4) can be used to evaluate the last two terms in eqn

(3) (ie the two terms in ∆rCPO) For the

P

int1∆rVO(T P)dP

term in eqn 2 we have adopted the equation of staterecommended by Holland et al (1996) and used byHolland amp Powell (1998) For MgO and MgTiO3 the heatcapacities molar volumes thermal expansivities and bulk

320

Table 3 Lattice parameters (in Aring)Sample MgO Mg2TiO4 MgTiO3

a c V(Aring3)Starting mix 42118 84418 50551 138987 30758D338 42119 84430 - - -D343 42122 84426 - - -D350 42128 - 50552 138977 30757D360 42120 84433 5057 13895 3077

Only c1 MgTiO3 in run productsmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdash

Estimated uncertainties (one standard deviation) are 00001 Aringfor MgO 00002 Aring for Mg2TiO4 and 001 Aring3 for the unit cellvolume (V) of MgTiO3 (except D360)

Phase ∆fHo(298K1 bar) Sordm(298 K 1 bar) Cp = a + bT + cT-2 + d T-12 (in J K-1mol-1) Vordm(298K1 bar) αo κ

mdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdash(J bar-1) (K-1) (kbar)

a b c d(x102)

MgO 0-60165 0269 0605 0362 0-535800 -2992 1125 62 1650periclaseMgTiO3 -156742 0746 1510 - -1890400 -6522 3086 495 1770geikieliteMg2TiO4 -215741 1118a 1617b 03286 -2382200 -2786 4529c 548c 1890d

Qandilite

Table 4 Thermodynamic data

a calorimetrically measured value is 1036 J K-1 mol-1from Todd (1952) b Cp data from Orr and Coughlin (1952) andTodd (1952) fitted in this study c OrsquoNeill et al (2003b) d value for Fe2TiO4 from Holland and Powell (1998)mdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashmdashData for MgO and MgTiO3 are from Holland amp Powell (1998) for Mg2TiO4 from this study except where noted

moduli are taken from Holland amp Powell (1998) assummarized in Table 4 For Mg2TiO4 the heat capacityequation is our fit to the heat content measurements of Orramp Coughlin (1952) which extend to 1820 K (31 data)Temperature was corrected from IPTS-48 to ITS-90 andthe equation was anchored at low temperature using theadiabatic calorimetric measurements of Todd (1952) in therange 250 to 300 K (5 data) Volume and thermal expansionat 1 bar are from OrsquoNeill et al (2003b) refitted to theequation for thermal expansion of Holland et al (1996)The bulk modulus of Mg2TiO4 has not yet been measuredto our knowledge so we adopt the value listed for Fe2TiO4by Holland amp Powell (1998)

With these data the experimental half-brackets can beused define maximum and minimum values of the quantity(∆rH

O(298K) ndash T∆rS

O(298K)) at each experimental pressure

These values are plotted in Fig 4 against experimentaltemperature To allow for experimental uncertainties inboth temperature and pressure measurement in the piston-cylinder apparatus 5 K has been subtracted from and02 kbar added to the low temperature half-brackets and 5K added to and 02 kbar subtracted from the high tempera-ture half-brackets For example the half-brackets nomi-nally at 10 kbar and 1360degC and 1370degC are treated asbeing at 1355degC and 102 kbar and 1375degC and 98 kbarrespectively For the experiments at atmospheric pressurean uncertainty of 5 K in the half-brackets was also adopted

The resulting array of limiting values of (∆rHO(298K) ndash

T∆rSO(298K)) defines a set of lines of slope -∆rS

O(298K) and

intercept ∆rHO(298K) By inspection we locate the two

limiting values (one from a high-temperature half-bracketthe other from a low-temperature half-bracket) that definethe line of maximum slope and similarly the two thatdefine the line of minimum slope These lines givemaximum and minimum values of ∆rS

O(298K) of 1010 and

934 J K-1 mol-1 respectively with corresponding values of∆rH

O(298K) of 1227 and 1105 kJ mol-1 For a best estimate

of ∆rSO(298K) we take the mean of these maximum and

minimum slopes 972 J K-1 mol-1 with an uncertaintygiven by half their difference plusmn 038 J K-1 mol-1 Likewise∆rH

O(298K) = 1166 plusmn 061 kJ mol-1Using the values of ∆f elHO

(298K) and SO(298K) for MgTiO3

and MgO from Holland amp Powell (1998) in Table 4 givesfor Mg2TiO4 ∆f elH

O(298K) = ndash 215741 plusmn 135 kJ mol-1 and

SO(298K) = 1112 plusmn 06 J K-1 mol-1 This latter value is 76 plusmn

09 J K-1 mol-1 more than the calorimetrically determinedvalue of 1036 plusmn 07 J K-1 mol-1 (Todd 1952) The differ-ence may be ascribed to zero-point entropy but is some-what less than the theoretical amount of 2Rln2 or 115 JK-1mol-1 expected from the random mixing of Mg and Tion the octahedral site of Mg2TiO4 spinel

One possibility is some short-range ordering of Mg andTi which seems plausible given the long-range ordering ofMg and Ti onto distinct octahedral sites at the phase transi-tion from to Fd

ndash3m P4122 at ~ 1000degC Unfortunately as

with many hypotheses regarding short-range order incrystal chemistry this explanation is not really practicableto test with existing experimental methods It is thereforenecessary to eliminate other possibilities such as error inthe calorimetric data for the other phases in reaction (1)ie MgTiO3 andor MgO For MgO it is reasonable toassume that the calorimetric data are secure not onlybecause they have been measured many times but alsobecause MgO figures in so many other reactions that havebeen subsumed into the compilation of the Holland ampPowell (1998) database The value SO

(298K) of for MgTiO3also seems well known since the low temperature heatcapacity measurements of Robie et al (1989) are in goodagreement with the earlier measurements of Shomate(1946) However at high temperatures the heat-contentmeasurements of MgTiO3 (Naylor amp Cook 1946) leave

The free energy of formation of Mg2TiO4 spinel 321

Fig 4 The quantity ∆rHO(298K) ndash T∆rS

O(298K) vs T (K) for the defining half-brackets plotted to allow for experimental uncertainties as

discussed in the text The lines with maximum and minimum slopes are drawn

H St C OrsquoNeill D R Scott

some room for doubt as follows Firstly the measurementsextend only to 1720 K with only three data above 1410 KRefitting the data of Naylor amp Cook (1946) to the four-term heat capacity polynomial used by Holland and Powell(ie Cp = a + bT + cT2 + dT-12) gives a result very muchthe same as that listed by Holland amp Powell (1998) butadding an extra fifth term to the polynomial (ie the termeT2) produces a better fit but one which leads to signifi-cantly higher computed values of Cp above ~ 1400K As anillustration Cp calculated at 1800 K with the four-term fitis 1356 J K-1 mol-1 whereas that with the five-term fit is1589 J K-1 mol-1 This would be a large enough change toaccount for the missing residual entropy in Mg2TiO4Whether this increase in the heat capacity of MgTiO3 isreal is difficult to judge since the extended Maier-Kelleyequation for heat capacities is an entirely empiricaldevice for representing the experimental data in which theinclusion of the term in eT2 is well known to lead to ambi-guities in the extrapolation of the equation to highertemperatures The value for the four-term fit at 1800 K isin excellent agreement with the Dulong-Petit value(Cp = 3nR + TVα2κ) which is calculated to be 136 J K-1

mol-1 at 1800 K using the data in Table 4 Excess heatcapacity could result from disordering of Mg and Ti as theilmenite structure in which Mg and Ti are ordered intolayers transforms towards the corundum structure(R

ndash3 rarr R

ndash3c) This transition has been observed in NiTiO3

by powder neutron diffraction at ~ 1560 K (Boysen et al(1995) and is inferred at ~ 1673 K in FeTiO3 from extrap-olation of the transition in FeTiO3-Fe2O3 solid solutions(Harrison et al 2000) Harrison et al (2000) show thatFeTiO3 is ~ 10 disordered at 1473 K The occurrence ofthis transition in MgTiO3 has been discounted by Reynardamp Guyot (1994) from their investigation by Raman spec-troscopy to 1820 K on the grounds that the symmetrychange would reduce the number of Raman active modesfrom ten to seven In fact the ten modes observable atroom temperature could only be tracked to ~ 1400 K due toline broadening and thermal emission so the possibility ofsymmetry reduction at higher temperatures remainsopen as does the possibility of significant Mg-Ti disor-dering within the ilmenite structure at temperatures belowthe putative transition as has been shown to occur inFeTiO3 by Harrison et al (2000) The question of excessentropy in MgTiO3 could be addressed by more precisehigh-temperature calorimetry (which would give the dataneeded to resolve the question of interest here directly)or other methods such as high-temperature powder neutrondiffraction

Acknowledgments We thank Mark Ghiorso andDominique Lattard for helpful reviews

References

Akimoto S amp Syono Y (1967) High-pressure decomposition ofsome titanate spinels J Chem Phys 47 1813-1817

Bose K amp Ganguly J (1995) Quartz-coesite transition revisited -reversed experimental-determination at 500-1200-degrees-C

and retrieved thermochemical properties Am Mineral 80231-238

Boysen H Frey F Lerch M Vogt T (1995) A neutron powderinvestigation of the high-temperature phase-transition inNiTiO3 Zeitschr fuumlr Kristallogr 210 328-337

Eriksson G amp Pelton AD (1993) Critical evaluation and opti-mization of the thermodynamic properties and phasediagrams of the MnO-TiO2 MgO-TiO2 FeO-TiO2 Ti2O3-TiO2Na2O-TiO2 and K2O-TiO2 systems Metall Trans B 24795-805

Harrison RJ Redfern SAT Smith RI (2000) In-situ study ofthe R

ndash3 to R

ndash3c phase transition in the ilmenite-hematite solid

solution using time-of-flight neutron powder diffraction AmMineral 85 194-205

Hermann J OrsquoNeill HStC Berry AJ (in Press) Titanium solu-bility in olivine in the system TiO2-MgO-SiO2 no evidence foran ultra-deep origin of Ti-bearing olivine Contrib MineralPetrol

Holland TJB amp Powell R (1998) An internally consistent ther-modynamic data set for phases of petrological interest J meta-morphic Geol 16 309-343

Holland TJB Redfern SAT Pawley AR (1996) Volumebehaviour of hydrous minerals at high pressure and tempera-ture 2 Compressibilities of lawsonite zoisite clinozoisite andepidote Am Mineral 81 341-348

Hohl H Kloc C Bucher E (1996) Electrical and magnetic prop-erties of spinel solid solutions Mg2-xTi1+xO4 0 le x le 1 J SolidState Chem 125 216-223

Hunter BA amp Howard CJ (2000) LHPM a computer programfor Rietveld analysis of X-ray and neutron powder diffractionpatterns Australian Nuclear Science and TechnologyOrganization Lucas Heights Research Laboratory MenaiNSW 2234 Australia

Millard RL Peterson RC Hunter BK (1995) Study of thecubic to tetragonal transition in Mg2TiO4 and Zn2TiO4 spinelsby 17O MAS NMR and Rietveld refinement of X-ray diffractiondata Am Mineral 80 885-896

Naylor BF amp Cook OA (1946) High-temperature heat contentsof the metatitanates of calcium iron and magnesium J AmerChem Soc 68 1003-1005

OrsquoNeill HStC Pownceby MI McCammon CA (2003a) Themagnesiowuumlstite iron equilibrium and its implications for theactivity-composition relations of (MgFe)2SiO4 olivine solidsolutions Contrib Mineral Petrol 146 308-325

ONeill HStC Redfern SAT Kesson S Short S (2003a) Anin situ neutron diffraction study of cation disordering insynthetic qandilite Mg2TiO4 at high temperatures AmMineral 88 860-865

Orr RL amp Coughlin JP (1952) High temperature heat contentsof magnesium orthotitanate and magnesium dititanate J AmerChem Soc 74 3186-3187

Pelton AD Eriksson G Krajewski D Goumlbbels M WoermannE (1998) Measurement and thermodynamic evaluation ofphase equilibria in the Mg-Ti-O system Zeitschr fuumlr PhysikalChem 207 163-180

Reynard B amp Guyot F (1994) High-temperature properties ofgeikielite (MgTiO3-ilmenite) from high-temperature high-pres-sure Raman-spectroscopy ndash some implications for MgSiO3-ilmenite Phys Chem Min 21 441-450

Robie RA Haselton Jr HT Hemingway BS (1989) Heatcapacities and entropies at 29815 K of MgTiO3 (geikielite) ZnO(zincite) and ZnCO3 (smithsonite) J Chem Thermodynamics21 743-749

322

moduli are taken from Holland amp Powell (1998) assummarized in Table 4 For Mg2TiO4 the heat capacityequation is our fit to the heat content measurements of Orramp Coughlin (1952) which extend to 1820 K (31 data)Temperature was corrected from IPTS-48 to ITS-90 andthe equation was anchored at low temperature using theadiabatic calorimetric measurements of Todd (1952) in therange 250 to 300 K (5 data) Volume and thermal expansionat 1 bar are from OrsquoNeill et al (2003b) refitted to theequation for thermal expansion of Holland et al (1996)The bulk modulus of Mg2TiO4 has not yet been measuredto our knowledge so we adopt the value listed for Fe2TiO4by Holland amp Powell (1998)

With these data the experimental half-brackets can beused define maximum and minimum values of the quantity(∆rH

O(298K) ndash T∆rS

O(298K)) at each experimental pressure

These values are plotted in Fig 4 against experimentaltemperature To allow for experimental uncertainties inboth temperature and pressure measurement in the piston-cylinder apparatus 5 K has been subtracted from and02 kbar added to the low temperature half-brackets and 5K added to and 02 kbar subtracted from the high tempera-ture half-brackets For example the half-brackets nomi-nally at 10 kbar and 1360degC and 1370degC are treated asbeing at 1355degC and 102 kbar and 1375degC and 98 kbarrespectively For the experiments at atmospheric pressurean uncertainty of 5 K in the half-brackets was also adopted

The resulting array of limiting values of (∆rHO(298K) ndash

T∆rSO(298K)) defines a set of lines of slope -∆rS

O(298K) and

intercept ∆rHO(298K) By inspection we locate the two

limiting values (one from a high-temperature half-bracketthe other from a low-temperature half-bracket) that definethe line of maximum slope and similarly the two thatdefine the line of minimum slope These lines givemaximum and minimum values of ∆rS

O(298K) of 1010 and

934 J K-1 mol-1 respectively with corresponding values of∆rH

O(298K) of 1227 and 1105 kJ mol-1 For a best estimate

of ∆rSO(298K) we take the mean of these maximum and

minimum slopes 972 J K-1 mol-1 with an uncertaintygiven by half their difference plusmn 038 J K-1 mol-1 Likewise∆rH

O(298K) = 1166 plusmn 061 kJ mol-1Using the values of ∆f elHO

(298K) and SO(298K) for MgTiO3

and MgO from Holland amp Powell (1998) in Table 4 givesfor Mg2TiO4 ∆f elH

O(298K) = ndash 215741 plusmn 135 kJ mol-1 and

SO(298K) = 1112 plusmn 06 J K-1 mol-1 This latter value is 76 plusmn

09 J K-1 mol-1 more than the calorimetrically determinedvalue of 1036 plusmn 07 J K-1 mol-1 (Todd 1952) The differ-ence may be ascribed to zero-point entropy but is some-what less than the theoretical amount of 2Rln2 or 115 JK-1mol-1 expected from the random mixing of Mg and Tion the octahedral site of Mg2TiO4 spinel

One possibility is some short-range ordering of Mg andTi which seems plausible given the long-range ordering ofMg and Ti onto distinct octahedral sites at the phase transi-tion from to Fd

ndash3m P4122 at ~ 1000degC Unfortunately as

with many hypotheses regarding short-range order incrystal chemistry this explanation is not really practicableto test with existing experimental methods It is thereforenecessary to eliminate other possibilities such as error inthe calorimetric data for the other phases in reaction (1)ie MgTiO3 andor MgO For MgO it is reasonable toassume that the calorimetric data are secure not onlybecause they have been measured many times but alsobecause MgO figures in so many other reactions that havebeen subsumed into the compilation of the Holland ampPowell (1998) database The value SO

(298K) of for MgTiO3also seems well known since the low temperature heatcapacity measurements of Robie et al (1989) are in goodagreement with the earlier measurements of Shomate(1946) However at high temperatures the heat-contentmeasurements of MgTiO3 (Naylor amp Cook 1946) leave

The free energy of formation of Mg2TiO4 spinel 321

Fig 4 The quantity ∆rHO(298K) ndash T∆rS

O(298K) vs T (K) for the defining half-brackets plotted to allow for experimental uncertainties as

discussed in the text The lines with maximum and minimum slopes are drawn

H St C OrsquoNeill D R Scott

some room for doubt as follows Firstly the measurementsextend only to 1720 K with only three data above 1410 KRefitting the data of Naylor amp Cook (1946) to the four-term heat capacity polynomial used by Holland and Powell(ie Cp = a + bT + cT2 + dT-12) gives a result very muchthe same as that listed by Holland amp Powell (1998) butadding an extra fifth term to the polynomial (ie the termeT2) produces a better fit but one which leads to signifi-cantly higher computed values of Cp above ~ 1400K As anillustration Cp calculated at 1800 K with the four-term fitis 1356 J K-1 mol-1 whereas that with the five-term fit is1589 J K-1 mol-1 This would be a large enough change toaccount for the missing residual entropy in Mg2TiO4Whether this increase in the heat capacity of MgTiO3 isreal is difficult to judge since the extended Maier-Kelleyequation for heat capacities is an entirely empiricaldevice for representing the experimental data in which theinclusion of the term in eT2 is well known to lead to ambi-guities in the extrapolation of the equation to highertemperatures The value for the four-term fit at 1800 K isin excellent agreement with the Dulong-Petit value(Cp = 3nR + TVα2κ) which is calculated to be 136 J K-1

mol-1 at 1800 K using the data in Table 4 Excess heatcapacity could result from disordering of Mg and Ti as theilmenite structure in which Mg and Ti are ordered intolayers transforms towards the corundum structure(R

ndash3 rarr R

ndash3c) This transition has been observed in NiTiO3

by powder neutron diffraction at ~ 1560 K (Boysen et al(1995) and is inferred at ~ 1673 K in FeTiO3 from extrap-olation of the transition in FeTiO3-Fe2O3 solid solutions(Harrison et al 2000) Harrison et al (2000) show thatFeTiO3 is ~ 10 disordered at 1473 K The occurrence ofthis transition in MgTiO3 has been discounted by Reynardamp Guyot (1994) from their investigation by Raman spec-troscopy to 1820 K on the grounds that the symmetrychange would reduce the number of Raman active modesfrom ten to seven In fact the ten modes observable atroom temperature could only be tracked to ~ 1400 K due toline broadening and thermal emission so the possibility ofsymmetry reduction at higher temperatures remainsopen as does the possibility of significant Mg-Ti disor-dering within the ilmenite structure at temperatures belowthe putative transition as has been shown to occur inFeTiO3 by Harrison et al (2000) The question of excessentropy in MgTiO3 could be addressed by more precisehigh-temperature calorimetry (which would give the dataneeded to resolve the question of interest here directly)or other methods such as high-temperature powder neutrondiffraction

Acknowledgments We thank Mark Ghiorso andDominique Lattard for helpful reviews

References

Akimoto S amp Syono Y (1967) High-pressure decomposition ofsome titanate spinels J Chem Phys 47 1813-1817

Bose K amp Ganguly J (1995) Quartz-coesite transition revisited -reversed experimental-determination at 500-1200-degrees-C

and retrieved thermochemical properties Am Mineral 80231-238

Boysen H Frey F Lerch M Vogt T (1995) A neutron powderinvestigation of the high-temperature phase-transition inNiTiO3 Zeitschr fuumlr Kristallogr 210 328-337

Eriksson G amp Pelton AD (1993) Critical evaluation and opti-mization of the thermodynamic properties and phasediagrams of the MnO-TiO2 MgO-TiO2 FeO-TiO2 Ti2O3-TiO2Na2O-TiO2 and K2O-TiO2 systems Metall Trans B 24795-805

Harrison RJ Redfern SAT Smith RI (2000) In-situ study ofthe R

ndash3 to R

ndash3c phase transition in the ilmenite-hematite solid

solution using time-of-flight neutron powder diffraction AmMineral 85 194-205

Hermann J OrsquoNeill HStC Berry AJ (in Press) Titanium solu-bility in olivine in the system TiO2-MgO-SiO2 no evidence foran ultra-deep origin of Ti-bearing olivine Contrib MineralPetrol

Holland TJB amp Powell R (1998) An internally consistent ther-modynamic data set for phases of petrological interest J meta-morphic Geol 16 309-343

Holland TJB Redfern SAT Pawley AR (1996) Volumebehaviour of hydrous minerals at high pressure and tempera-ture 2 Compressibilities of lawsonite zoisite clinozoisite andepidote Am Mineral 81 341-348

Hohl H Kloc C Bucher E (1996) Electrical and magnetic prop-erties of spinel solid solutions Mg2-xTi1+xO4 0 le x le 1 J SolidState Chem 125 216-223

Hunter BA amp Howard CJ (2000) LHPM a computer programfor Rietveld analysis of X-ray and neutron powder diffractionpatterns Australian Nuclear Science and TechnologyOrganization Lucas Heights Research Laboratory MenaiNSW 2234 Australia

Millard RL Peterson RC Hunter BK (1995) Study of thecubic to tetragonal transition in Mg2TiO4 and Zn2TiO4 spinelsby 17O MAS NMR and Rietveld refinement of X-ray diffractiondata Am Mineral 80 885-896

Naylor BF amp Cook OA (1946) High-temperature heat contentsof the metatitanates of calcium iron and magnesium J AmerChem Soc 68 1003-1005

OrsquoNeill HStC Pownceby MI McCammon CA (2003a) Themagnesiowuumlstite iron equilibrium and its implications for theactivity-composition relations of (MgFe)2SiO4 olivine solidsolutions Contrib Mineral Petrol 146 308-325

ONeill HStC Redfern SAT Kesson S Short S (2003a) Anin situ neutron diffraction study of cation disordering insynthetic qandilite Mg2TiO4 at high temperatures AmMineral 88 860-865

Orr RL amp Coughlin JP (1952) High temperature heat contentsof magnesium orthotitanate and magnesium dititanate J AmerChem Soc 74 3186-3187

Pelton AD Eriksson G Krajewski D Goumlbbels M WoermannE (1998) Measurement and thermodynamic evaluation ofphase equilibria in the Mg-Ti-O system Zeitschr fuumlr PhysikalChem 207 163-180

Reynard B amp Guyot F (1994) High-temperature properties ofgeikielite (MgTiO3-ilmenite) from high-temperature high-pres-sure Raman-spectroscopy ndash some implications for MgSiO3-ilmenite Phys Chem Min 21 441-450

Robie RA Haselton Jr HT Hemingway BS (1989) Heatcapacities and entropies at 29815 K of MgTiO3 (geikielite) ZnO(zincite) and ZnCO3 (smithsonite) J Chem Thermodynamics21 743-749

322

H St C OrsquoNeill D R Scott

some room for doubt as follows Firstly the measurementsextend only to 1720 K with only three data above 1410 KRefitting the data of Naylor amp Cook (1946) to the four-term heat capacity polynomial used by Holland and Powell(ie Cp = a + bT + cT2 + dT-12) gives a result very muchthe same as that listed by Holland amp Powell (1998) butadding an extra fifth term to the polynomial (ie the termeT2) produces a better fit but one which leads to signifi-cantly higher computed values of Cp above ~ 1400K As anillustration Cp calculated at 1800 K with the four-term fitis 1356 J K-1 mol-1 whereas that with the five-term fit is1589 J K-1 mol-1 This would be a large enough change toaccount for the missing residual entropy in Mg2TiO4Whether this increase in the heat capacity of MgTiO3 isreal is difficult to judge since the extended Maier-Kelleyequation for heat capacities is an entirely empiricaldevice for representing the experimental data in which theinclusion of the term in eT2 is well known to lead to ambi-guities in the extrapolation of the equation to highertemperatures The value for the four-term fit at 1800 K isin excellent agreement with the Dulong-Petit value(Cp = 3nR + TVα2κ) which is calculated to be 136 J K-1

mol-1 at 1800 K using the data in Table 4 Excess heatcapacity could result from disordering of Mg and Ti as theilmenite structure in which Mg and Ti are ordered intolayers transforms towards the corundum structure(R

ndash3 rarr R

ndash3c) This transition has been observed in NiTiO3

by powder neutron diffraction at ~ 1560 K (Boysen et al(1995) and is inferred at ~ 1673 K in FeTiO3 from extrap-olation of the transition in FeTiO3-Fe2O3 solid solutions(Harrison et al 2000) Harrison et al (2000) show thatFeTiO3 is ~ 10 disordered at 1473 K The occurrence ofthis transition in MgTiO3 has been discounted by Reynardamp Guyot (1994) from their investigation by Raman spec-troscopy to 1820 K on the grounds that the symmetrychange would reduce the number of Raman active modesfrom ten to seven In fact the ten modes observable atroom temperature could only be tracked to ~ 1400 K due toline broadening and thermal emission so the possibility ofsymmetry reduction at higher temperatures remainsopen as does the possibility of significant Mg-Ti disor-dering within the ilmenite structure at temperatures belowthe putative transition as has been shown to occur inFeTiO3 by Harrison et al (2000) The question of excessentropy in MgTiO3 could be addressed by more precisehigh-temperature calorimetry (which would give the dataneeded to resolve the question of interest here directly)or other methods such as high-temperature powder neutrondiffraction

Acknowledgments We thank Mark Ghiorso andDominique Lattard for helpful reviews

References

Akimoto S amp Syono Y (1967) High-pressure decomposition ofsome titanate spinels J Chem Phys 47 1813-1817

Bose K amp Ganguly J (1995) Quartz-coesite transition revisited -reversed experimental-determination at 500-1200-degrees-C

and retrieved thermochemical properties Am Mineral 80231-238

Boysen H Frey F Lerch M Vogt T (1995) A neutron powderinvestigation of the high-temperature phase-transition inNiTiO3 Zeitschr fuumlr Kristallogr 210 328-337

Eriksson G amp Pelton AD (1993) Critical evaluation and opti-mization of the thermodynamic properties and phasediagrams of the MnO-TiO2 MgO-TiO2 FeO-TiO2 Ti2O3-TiO2Na2O-TiO2 and K2O-TiO2 systems Metall Trans B 24795-805

Harrison RJ Redfern SAT Smith RI (2000) In-situ study ofthe R

ndash3 to R

ndash3c phase transition in the ilmenite-hematite solid

solution using time-of-flight neutron powder diffraction AmMineral 85 194-205

Hermann J OrsquoNeill HStC Berry AJ (in Press) Titanium solu-bility in olivine in the system TiO2-MgO-SiO2 no evidence foran ultra-deep origin of Ti-bearing olivine Contrib MineralPetrol

Holland TJB amp Powell R (1998) An internally consistent ther-modynamic data set for phases of petrological interest J meta-morphic Geol 16 309-343

Holland TJB Redfern SAT Pawley AR (1996) Volumebehaviour of hydrous minerals at high pressure and tempera-ture 2 Compressibilities of lawsonite zoisite clinozoisite andepidote Am Mineral 81 341-348

Hohl H Kloc C Bucher E (1996) Electrical and magnetic prop-erties of spinel solid solutions Mg2-xTi1+xO4 0 le x le 1 J SolidState Chem 125 216-223

Hunter BA amp Howard CJ (2000) LHPM a computer programfor Rietveld analysis of X-ray and neutron powder diffractionpatterns Australian Nuclear Science and TechnologyOrganization Lucas Heights Research Laboratory MenaiNSW 2234 Australia

Millard RL Peterson RC Hunter BK (1995) Study of thecubic to tetragonal transition in Mg2TiO4 and Zn2TiO4 spinelsby 17O MAS NMR and Rietveld refinement of X-ray diffractiondata Am Mineral 80 885-896

Naylor BF amp Cook OA (1946) High-temperature heat contentsof the metatitanates of calcium iron and magnesium J AmerChem Soc 68 1003-1005

OrsquoNeill HStC Pownceby MI McCammon CA (2003a) Themagnesiowuumlstite iron equilibrium and its implications for theactivity-composition relations of (MgFe)2SiO4 olivine solidsolutions Contrib Mineral Petrol 146 308-325

ONeill HStC Redfern SAT Kesson S Short S (2003a) Anin situ neutron diffraction study of cation disordering insynthetic qandilite Mg2TiO4 at high temperatures AmMineral 88 860-865

Orr RL amp Coughlin JP (1952) High temperature heat contentsof magnesium orthotitanate and magnesium dititanate J AmerChem Soc 74 3186-3187

Pelton AD Eriksson G Krajewski D Goumlbbels M WoermannE (1998) Measurement and thermodynamic evaluation ofphase equilibria in the Mg-Ti-O system Zeitschr fuumlr PhysikalChem 207 163-180

Reynard B amp Guyot F (1994) High-temperature properties ofgeikielite (MgTiO3-ilmenite) from high-temperature high-pres-sure Raman-spectroscopy ndash some implications for MgSiO3-ilmenite Phys Chem Min 21 441-450

Robie RA Haselton Jr HT Hemingway BS (1989) Heatcapacities and entropies at 29815 K of MgTiO3 (geikielite) ZnO(zincite) and ZnCO3 (smithsonite) J Chem Thermodynamics21 743-749

322

Shomate CH (1946) Heat capacities at low temperatures of themetatitanates of iron calcium and magnesium J Am ChemSoc 68 964-966

Todd SS (1952) Low temperature heat capacities and entropies at29816 K of magnesium orthotitanate and magnesium diti-tanate J Am Chem Soc 74 4669-4670

Wechsler BA amp von Dreele RB (1989) Structure refinement ofMg2TiO4 MgTiO3 and Mg2Ti2O5 by time-of-flight neutronpowder diffraction Acta Cryst B 45 542-549

Wechsler BA amp Navrotsky A (1984) Thermodynamics andstructural chemistry of compounds in the system MgO-TiO2 JSolid State Chem 55 165-180

Received 12 May 2004Modified version received 18 October 2004Accepted 11 November 2004

The free energy of formation of Mg2TiO4 spinel 323