American Mineralogist, Volume 97, pages 1330–1338, 2012

0003-004X/12/0809–1330$05.00/DOI: http://dx.doi.org/10.2138/am.2012.4076 1330

Thermodynamics of the magnetite-ulvöspinel (Fe3O4-Fe2TiO4) solid solution

Kristina i. LiLova,1 CaroLyn i. PearCe,2 ChristoPher GorsKi,3 Kevin M. rosso,2 and aLexandra navrotsKy1,*

1Peter A. Rock Thermochemistry Laboratory and NEAT ORU, University of California at Davis, Davis, California 95616, U.S.A.2Pacific Northwest National Laboratory, Richland, Washington 99352, U.S.A.

3Eawag, Ueberlandstrasse 133, 8600 Duebendorf, Switzerland

abstraCt

The thermodynamics of mixing and its dependence on cation distribution in the Fe3O4–Fe2TiO4 (magnetite-ulvöspinel) spinel solid solution were studied using high-temperature oxide melt solu-tion calorimetry and a range of structural and spectroscopic probes. The enthalpies of formation of ilmenite and ulvöspinel from the oxides and from the elements were obtained using oxidative drop solution calorimetry at 973 K in molten sodium molybdate. The enthalpy of mixing, determined from the fit to the measured enthalpies of drop solution calorimetry, is endothermic and represented by a quadratic formalism, ∆Hmix = (22.60 ± 8.46)x(1 – x) kJ/mol, where x is the mole fraction of ulvöspinel. The entropies of mixing are more complex than those for a regular solution and have been calculated based on average measured and theoretical cation distributions. Calculated free energies of mixing show evidence for a solvus at low temperature in good agreement with that observed experimentally.

Keywords: Titanomagnetite, magnetite-ulvöspinel solid solution, enthalpies of mixing, calorimetry

introduCtion

Because of their prospectively accessible reactive Fe2+ con-tent, spinel-type iron oxides such as magnetite (Fe3O4) are key phases that can participate in heterogeneous electron transfer reactions at the mineral-water interface in sediments. These include direct transfer of electron equivalents to the oxide-water interface through contact with dissolved oxygen or redox-active contaminants, or indirectly to aqueous solution species through acidic dissolution of Fe2+. In sediments derived from weath-ered basalt, magnetites typically contain titanium impurities (Zachara et al. 2007a). Lattice Fe3+ replacement by Ti4+ yields a proportional increase in the molar Fe2+ content to maintain charge neutrality, resulting in the preponderance of so-called titanomagnetites (Fe3–xTixO4). This increase in Fe2+ relative to Fe3+ is complicated by partitioning into either tetrahedral (A site) or octahedral (B site) cationic sublattices, resulting in a range of magnetic, electronic, and structural properties that have inspired extensive investigation. For example, increased tetrahedral Fe2+ content leads to enhanced magnetostriction and therefore a high coercive force, thus titanomagnetites play a major role in paleomagnetic properties of the Earth’s crust and are of related interest for technological applications (Kąkol et al. 1991; Banerjee 1991; O’Reilly 1984).

Stoichiometric magnetite Fe3O4 has an inverse spinel struc-ture with one Fe3+ per formula unit on tetrahedral sites, and Fe2+ and the remaining Fe3+ in a 50:50 ratio on the octahedral sites: Fe3+[Fe2+Fe3+]O4 (Bragg 1915; Nishikawa 1915; Verwey and de Boer 1936) with fast electron hopping between Fe2+ and Fe3+ on the octahedral sites at room temperature (Bauminger et al. 1961; Mackay 1961; Della Giusta et al. 1987; Vandenberghe

and De Grave 1989; Okudera et al. 1996). A complete solid solution exists along the binary join from magnetite (x = 0) to ulvöspinel (x = 1) as replacement of Fe3+ by Ti4+ yields titano-magnetites (Fe3–xTixO4). Ulvöspinel also has the inverse spinel structure (Barth and Posnjak 1932; Gorter 1957; Basta 1959; MacChesney and Muan 1959; Rossiter and Clarke 1965; Forster and Hall 1965). The magnetite-ulvöspinel solid solution has been reviewed recently by Pearce et al. (2010), including a range of analytical measurements that elucidate the series in terms of atomic structure, magnetic properties, and cation distribution.

The thermodynamics of the magnetite-ulvöspinel solid solu-tion are of interest for understanding the temperature-dependent extent of mixing, and also for providing a basis to define effective half-cell reduction potentials for heterogeneous electron transfer reactions with environmentally relevant species. The “tunable” solid-state Fe2+/Fe3+ ratio provides an ideal parameter to system-atically examine the relationship between the thermodynamic reduction potential of the solid and the reduction rate and extent of key environmentally relevant contaminants such as synthetic organic dyes, e.g., methylene blue (Yang et al. 2009), nitroben-zene (Gorski and Scherer 2010), and the polyvalent radionuclides 99Tc7+ and 238U6+ (Zachara et al. 2007b; Peretyazhko et al. 2008; Ilton et al. 2010; Liu et al. 2012). Despite exhaustive previous research on the thermodynamic properties of magnetite, ulvöspi-nel, and compositionally relevant ilmenite (FeTiO3) (Shomate et al. 1946; Schmahl et al. 1960; MacChesney and Muan 1961; Verhoogen 1962; Westrum and Groenvold 1969; Groenvold and Sveen 1974; Levitskii et al. 1976; Carel and Vallet 1981; Anovitz et al. 1985; Holland and Powell 1998; Berman 1988; O’Neill et al. 1988; Hemingway 1990; Sundman 1991; Kale and Jacob 1992; Senderov et al. 1993; Brown and Navrotsky 1994; Takai et al. 1994; Mehta et al. 1994; Stølen et al. 1996; Eriksson et al. * E-mail: anavrotsky@ucdavis.edu

LILOVA ET AL.: THERMODYNAMICS OF THE MAGNETITE-ULVÖSPINEL 1331

1996; Gottschalk 1997; Fabrichnaya and Sundman 1997; Linton et al. 1998; Xirouchakis and Lindsley 1998; Itoh 1999; Cornell and Schwertmann 2003; Itoh et al. 2003; Lykasov et al. 2006; Sauerzapf et al. 2008), debate remains regarding characteristics of the solvus and associated cation distributions for intermediate compositions. These studies also do not include direct calorimet-ric determination of enthalpies of formation.

The first evidence of a miscibility gap at lower tempera-ture in the titanomagnetites was noted by Mogensen (1946). A phase diagram with a critical temperature of 1023 K and x = 0.35 Fe2TiO4 was proposed by Kawai et al. (1954) and then recalculated (Kawai 1956) as 873 K and x = 0.42. After heating at different temperatures and compositions Vincent et al. (1957) confirmed Tc = 873 K and an asymmetrical solvus. Rumble (1977), using thermodynamic calculations, predicted an asym-metric solvus with a critical temperature at 923 K and x = 0.65. Using experimental results from natural titanomagnetites, Price (1981) suggested a lower critical temperature (728 K), which was not confirmed by Lindsley (1981) in hydrothermal experiments. Trestman-Matts et al. (1983) used the cation distribution in the titanomagnetite solid solutions, determined from thermoelectric measurements, and O’Neill and Navrotsky model (1984) to cal-culate a solvus with a critical temperature of 783 K and x = 0.53.

The purpose of the present study is to obtain experimental thermodynamic data for the Fe3O4-Fe2TiO4 solid solution, verify-ing the existence of the miscibility gap and the critical temperature and composition of the predicted phase separation at low tempera-ture. We use high-temperature oxide melt solution calorimetry, a procedure with advantages for the present system as described in detail in Lilova et al. (2011). We include detailed analytical, diffraction, synchrotron-based X-ray magnetic circular dichroism (XMCD), and Mössbauer characterization along the series, and interpret the collective data in terms of cation distribution.

exPeriMentaL ProCedures

Sample preparationStoichiometric magnetite and all titanomagnetite solid solutions were prepared

as described in Pearce et al. (2010). Briefly, stoichiometric mixtures of TiO2, Fe2O3, and Fe metal were ground under acetone, loaded into silver foil and heated at 1170 K in evacuated, sealed quartz tubes for seven days. Similar synthesis methods have been employed for both ilmenite and ulvöspinel. Barth and Posnjak (1932) included rutile, hematite, and iron as starting materials at 1170 K, which is just below the melting temperature of silver (Wechsler et al. 1984) and, in the absence of silver foil, at 1373 K, which is the limiting temperature for evacuated silica tubes (O’Reilly and Banerjee 1965; Forster and Hall 1965; Jensen and Shive 1973; Fujino 1974; Hill and Sack 1987).

During our experiments, the use of hematite and rutile as starting materials resulted in formation of a two phase product: nonstoichiometric ulvöspinel as a major phase (around 80 wt%) and ilmenite as the second phase. According to the FeO-TiO2 pseudobinary phase diagram, the stoichiometry range of both ulvöspi-nel and ilmenite around 1273 K and at atmospheric pressure is narrow (several mole percent) (Eriksson and Pelton 1996). Therefore, in this work ilmenite was synthesized from a stoichiometric mixture of rutile (TiO2), hematite (Fe2O3), and iron powder (Alfa Aesar and Aldrich) with purities of 99.95, 99.999, and 99.95%, respectively. The initial materials were ground under ethanol, formed into pellets in a glove box under argon, wrapped in silver foil and annealed at 1173 K for two weeks in an evacuated sealed silica glass tube (oxygen fugacity between 2.76·10−5 and 2.96·10−6 atm). Ulvöspinel was prepared from a stoichiometric mixture of ana-tase (TiO2), iron powder, and a more reactive magnetite-maghemite solid solution (Fe2.80O4) with smaller particle size (around 200 nm). A single phase product was formed after five days of annealing at 1173 K.

CharacterizationStructure and chemical composition. The titanomagnetite solid solutions

were characterized in terms of structure and chemical composition by X-ray powder diffraction (XRD), electron probe microanalysis (EPMA), and chemical analysis, as described in Pearce et al. (2010) (Table 1).

Synchrotron-based X-ray magnetic circular dichroism. X-ray absorption spectra (XAS) for titanomagnetite samples with x = 0, 0.171, 0.2, 0.295, 0.46, 0.6, 0.77, 0.915 and 1.0 were collected on beamline 4.0.2 at the Advanced Light Source (ALS) in Berkeley, California, using the eight-pole magnet end station (Arenholz and Prestemon 2005). The XAS signal was monitored in total electron yield (TEY) mode, giving an effective probing depth of ∼4.5 nm. At each energy point, the XAS was measured for two opposite magnetization directions by reversing the applied field of 0.6 T. The XAS spectra of the two magnetization directions were normalized to the incident beam intensity and subtracted from each other to obtain the XMCD spectrum (Pattrick et al. 2002). The measured Fe L2,3 XMCD was used to extract the site occupancies and oxidation states of the cations in the spinel structure of the titanomagnetite. For titanomagnetite, the XMCD has four main peaks in the Fe L3 edge with positive, negative, positive, and negative signals, which are related to the amounts of Fe d 6 Td (∼709 eV), d 6 Oh (710 eV), d5 Td (∼711 eV), and d5 Oh (∼712 eV), respectively. To obtain the relative amounts of Fe on the four sites, the experimental spectra were fitted by means of a nonlinear least-squares analysis, using calculated spectra for each of the Fe sites. Details of the calculations are described in Pearce et al. (2010) and van der Laan and Thole (1991).

Mössbauer spectroscopyMössbauer spectra were recorded for titanomagnetite samples with x = 0,

0.2, 0.295, and 0.46 at room temperature with a FAST ComTek 1024-multichan-nel analyzer system using a constant acceleration drive (RT, γ-source ∼50 mCi 57Co/Rh matrix). An Ar-Kr proportional counter was used to detect the radiation transmitted through the holder, and the counts were stored in a multichannel scalar (MCS) as a function of energy (transducer velocity) using a 1024 channel analyzer. Data were folded to 512 channels to give a flat background and a zero-velocity. Calibration spectra were obtained with a 25–30 µm-thick α-Fe(m) foil (Amersham, England). The Mössbauer data were modeled with the Recoil software using an extended-Voigt based fiting (xVBF) approach (Gorski and Scherer 2010; Lagarec and Rancourt 1997) using Recoil Software (University of Ottawa, Ottawa, Canada). The room-temperature Mössbauer spectrum of stoichiometric magnetite (x = 0) contains two distinct hyperfine components. The smaller sextet (1/3 of the spectral area) with 490 kG hyperfine magnetic field (HMF) and 0.30 mm/s center shift (CS) corresponds to the terahedral (A-site) Fe3+ ions. The larger sextet with HMF = 460 kG and IS = 0.65 mm/s is related to the octahedral (B-site) Fe2.5+ cations, which arises from the rapid electron exchange between octahedral Fe3+ and Fe2+ (Verwey and Haayman 1941). Titanomagnetites contain an additional site (i.e., the C site), which manifests as a sextet with the CS of approximately 1 mm/s, and very broad peaks (Hamdeh et al. 1999). The large CS (∼1 mm/s) is characteristic of Fe2+ that could be present in tetrahedral or octahedral sites. Using this three component fit, the A-site fractional area (f) can be accurately measured, and by invoking the charge balance, the cation distribution can be expressed by:

Fe3+f(3–x)Fe2+

1–f(3–x)[Fe3+2(1–x)–f(3–x)Fe2+

x+f(3–x)Ti4+x ]O2–

4. (1)

Table 1. Site occupancies for the titanomagnetite solid solutionx Site occupancies used in thermodynamic calculations Fe2+

tet Fe2+oct Fe3+

tet Fe3+oct

0 –0.05 1.09 1.02 0.940.1805* 0.00 1.15 0.96 0.720.205† 0.118 1.062 0.882 0.7380.295† 0.152 1.095 0.849 0.6010.46† 0.308 1.157 0.692 0.3880.603* 0.47 1.07 0.55 0.330.77* 0.82 0.94 0.32 0.160.9075* 0.90 0.85 0.19 0.14Note: x is the mole fraction of Fe2TiO4 with ±2% error.* Site occupancies obtained by XMCD, Pearce et al. (2010).† The values obtained from XMCD are 0.12, 1.085, 0.845, 0.755 for x = 0.205; 0.15, 1.145, 0.775, 0.625 for x = 0.295; and 0.31, 1.185, 0.665, 0.38 for x = 0.46. The values obtained from Mössbauer spectroscopy are: 0.2356, 0.9644, 0.7644, 0.8356 for x = 0.205; 0.2926, 1, 0.7074, 0.6926 for x = 0.295; and 0.5053, 0.9447, 0.4947, 0.6053 for x = 0.46. Averages are used for the thermodynamic calculations

LILOVA ET AL.: THERMODYNAMICS OF THE MAGNETITE-ULVÖSPINEL1332

Differential scanning calorimetryTo verify the stoichiometry of the titanomagnetite solid solutions, thermo-

gravimetric and differential thermal analysis (TG/DSC) were conducted using a Setaram Labsys Evo. A pellet weighting between 30 and 40 mg was packed in a standard platinum crucible with a lid. During the first run, the loaded crucible was placed in the DSC and heated from room temperature to 973 K at 10 K/min under argon flow to get the weight change associated with water loss. The second run was performed to 1173 K in oxygen and the weight gain of the sample was registered, which was used to calculate the initial Fe2+/Fe3+ ratio.

High-temperature calorimetryHigh-temperature oxide melt solution calorimetry was performed using a

Tian Calvet twin calorimeter described in detail by Navrotsky (1977, 1997). In the drop solution calorimetry experiment, samples in the form of pellets (around 5 mg) were dropped from room temperature (298 K) into the molten sodium mo-lybdate (3Na2O⋅4MoO3) solvent at 973 K and into molten lead borate (2PbO⋅B2O3) at 1073 K (ilmenite) in a platinum crucible. Oxygen gas was flushed over the solvent at 90 mL/min and bubbled through it at 5 mL/min. The calorimeters were calibrated using the heat content of 5 mg α-Al2O3 pellets. The optimization of calorimetric technique for iron-containing samples has been discussed in detail recently (Lilova et al. 2011).

resuLts and disCussion

Titanomagnetite stoichiometry determined by EPMA and TG/DSC

The Fe3–xTixO4 titanomagnetite sample stoichiometries were determined by TG/DSC measurement. When compared with the EPMA results from Pearce et al. (2010), the TG results showed a higher Fe2TiO4 mole fraction for the x = 0.17 and 0.20 samples and a lower Fe2TiO4 mole fraction for the x = 0.915 sample. The differences in the other compositions are within measurement error. As discussed by Lilova et al. (2011), the reproducibility of runs on the same sample suggests precision in the ulvöspinel mole fraction (x) of about ±1%, for each characterization, but the true uncertainty is larger due to the assumptions inherent in each technique. The average error in x from both EPMA and TG/DSC measurements is estimated on the order of ±2%.

Iron site occupancy determined using XMCD and Mössbauer spectroscopy

Several theoretical models have been proposed for iron site occupancy in magnetite-ulvöspinel solid solutions. Akimoto (1954) proposed that the concentration of individual cations in both sites varies linearly according to

Fe3+1–xFe2+

x [Fe3+1–xFe2+Ti4+

x ]O2–4 . (2)

Néel (1955) and Chevallier et al. (1955) proposed a dif-ferent model that takes into account the preference of Fe3+ for tetrahedral sites. Thus Ti4+ and Fe2+ substitute for Fe3+ on octahedral sites only until all Fe3+ in octahedral sites is used up, after which Fe2+ substitutes for Fe3+ in tetrahedral sites, so that the two substitution intervals are given by

Fe3+(Fe2+1+xFe3+

1–2xTi4+x )O2–

4 for x ≤ 0.5 (3)

and

Fe3+2–2xFe2+

2x–1(Fe2+2–xTi4+

x )O2–4 for x ≥ 0.5. (4)

Several iron site occupancy models have also been deter-mined experimentally for titanomagnetites using a range of techniques including XRD, magnetometry, Mössbauer spec-troscopy, and XMCD, as reviewed in Pearce et al. (2010). These experimental models are in reasonably good agreement for tit-anomagnetite samples with x > 0.5, but for lower x values there are discrepancies, with measured iron site occupancies falling between the two theoretical models discussed above (Eq. 2 vs. Eqs. 3 and 4). For the present study, XMCD is used to provide direct information about the oxidation state and site occupancy of the Fe ions, as described in Pearce et al. (2010). To charac-terize discrepancies in iron site occupancies for samples with x > 0.5 determined using different techniques, titanomagnetite samples with x = 0, 0.2, 0.295, and 0.46 were measured using both XMCD, which is measured on a femtosecond timescale with an effective sampling depth of 4.5 nm, and room-temperature Mössbauer spectroscopy, which is a bulk technique and has a measurement timescale of 10−8 s (Pearce et al. 2010). Mössbauer and XMCD spectra for titanomagnetite samples with x = 0, 0.2, 0.295, and 0.46, along with the modeled data from which peak areas and subsequently site occupancies were determined, are shown in Figure 1. The somewhat discordant results for the site occupancies likely arise from these differences in timescale and sampling depth. Because it is beyond the scope of this study to scrutinize the physical causes of the differences between these two approaches, we instead take the average of the iron site occupancy values obtained using these two techniques. These averages compare very well with previous work, for example, the measured tetrahedral Fe2+ site occupancies are shown in Figure 2 and compared with Bosi et al. (2009), Kakol et al. (1991), and Hamdeh et al. (1999) data. We use these average measured values in all further calculations (Table 1).

High-temperature calorimetryThe methodology and the previous studies of Fe(II) contain-

ing compounds were described by Lilova et al. (2011). Average drop solution enthalpies and the enthalpies of formation from oxides obtained for ilmenite FeTiO3, stoichiometric magnetite Fe3O4, ulvöspinel Fe2TiO4 and Fe3–xTixO4 solid solutions are given in Table 2. In general, oxidation is very exothermic and with increasing Fe2+ content becomes the dominant factor in the energetics. As a result, the heat effects for the solid solutions at the magnetite-rich side are less negative than those at the ulvöspinel-rich side.

The thermodynamic cycles used to calculate the enthalpies of formation from oxides (∆Hf,ox) of ilmenite and ulvöspinel at 298 K, are shown in Table 3. The only prior data on the enthalpy of formation from oxides at 298 K, obtained from calorimetry is given by Levitskii et al. (1976) as ∆Hf,ox (FeTiO3) = –25.10 ± 2.09 kJ/mol, which is in good agreement with our value –24.15 ± 1.34 kJ/mol. The enthalpy of formation of Fe2TiO4 is calculated using EMF data (Levitskii et al. 1976), and is much more negative at –40.71 ± 2.09 kJ/mol than our measured value –17.62 ± 2.12 kJ/mol.

The standard enthalpy of the reaction FeTiO3 + Fe0.947O + 0.053Fe = Fe2TiO4 is positive: 6.54 ± 2.76 kJ/mol. The entropy of formation, 8.16 ± 0.51 J/(mol⋅K), is a sum of the standard

LILOVA ET AL.: THERMODYNAMICS OF THE MAGNETITE-ULVÖSPINEL 1333

Energy (eV)

x = 0.00

x = 0.20

x = 0.30

x = 0.45

A

B

FiGure 1. Mössbauer (a) and XMCD (b) spectra of samples x = 0.00, x = 0.20, x = 0.30, and x = 0.45. The fits are also shown.

entropy of the reaction and the configurational entropy (Brown and Navrotsky 1994). The temperature at which the Gibbs free energy becomes zero and the system is in equilibrium is 801 ± 150 K. Despite the large uncertainty, this temperature is in the same range as critical temperatures reported in the literature (Schmahl et al. 1960; Taylor and Schmalzriedh 1964; Simons and Woermann 1978; O’Neill et al. 1988; Eriksson et al. 1996), which themselves vary significantly.

Hence ulvöspinel is a high-temperature, entropy-stabilized phase, i.e., it becomes stable with increase of the entropy term –T∆S, i.e., its positive enthalpy of formation needs substantial

positive entropy of formation to stabilize it, with such stability occurring only above a certain temperature.

The enthalpies of formation from elements for ilmenite and ulvöspinel are calculated using the oxidative drop solution en-thalpies in molten sodium molybdate, as –1235.71 ± 2.93 and –1496.73 ± 4.93 kJ/mol, respectively. These values are in very good agreement with the Robie and Hemingway (1995) tabulated data of –1232.0 ± 2.5 and –1493.80 ± 2.00 kJ/mol. To verify the results for ilmenite, an additional high-temperature oxide melt solution calorimetry experiment in lead borate at 1073 K and oxygen was performed. An enthalpy of formation from elements

LILOVA ET AL.: THERMODYNAMICS OF THE MAGNETITE-ULVÖSPINEL1334

FiGure 2. Tetrahedral site occupancies of Fe2+, obtained in this work = circles, compared with Bosi et al. (2009) = empty squares, Kakol et al. (1991) = empty triangles, and Hamdeh et al. (1999) = empty star data. The solid curve represents the distribution, calculated at room temperature using Equations 6–8.

Table 2. Average drop solution enthalpies (ΔHds) at 973 K and en-thalpies of mixing (ΔHmix) of Fe3–xTixO4

x ΔHds (kJ/mol) ΔHmix (kJ/mol)0 15.17 ± 0.96 (7)* 00.1805 –19.84 ± 1.34 (10) 4.22 ± 2.830.205 –21.54 ± 0.69 (7) 4.40 ± 2.750.295 –33.2 ± 0.74 (7) 4.75 ± 2.740.46 –56.46 ± 0.66 (8) 3.61 ± 2.760.603 –80.27 ± 0.89 (6) 6.28 ± 2.730.77 –104.59 ± 0.89 (7) 5.91 ± 2.800.9075 –120.66 ± 1.16 (9) 1.65 ± 2.901 –132.69 ± 1.14 (8) 0FeTiO3 –20.59 ± 1.14 (8) TiO2 60.82 ± 0.85 Fe2O3 94.46 ± 0.93 (8)* Fe –384.05 ± 2.23 (8)* Fe0.947O –85.21 ± 0.92(8)* Note: Uncertainties are two standard deviations of the mean. The drop solution enthalpy of TiO2 is an average of the values measured over five years in the Peter A. Rock Thermochemistry Laboratory. x is the mole fraction of Fe2TiO4

with ±2% error, and the number in parentheses is the number of performed measurements.* Lilova et al. (2011).

–1238.49 ± 4.27 kJ/mol was obtained, which is consistent with the results in sodium molybdate at 973 K and the literature data. The enthalpies of formation from elements, calculated from the drop solution enthalpies of Linton et al. (1998) and Brown and Navrotsky (1994), obtained in argon at 1073 K and lead borate are less negative than our values, which could be an indication of a partly oxidized sample during their experiment, leading to a less positive enthalpy of drop solution (since oxidation is exother-mic) and thus an apparently less negative enthalpy of formation.

Enthalpies of mixingThe enthalpies of mixing of the Fe3O4-Fe2TiO4 solid-solution

series are determined from the fit to the measured enthalpies of drop solution (Fig. 3). There is a negative deviation of the enthalpy of drop solution from the ideal-mixing straight line

connecting the two end-members, which implies a positive enthalpy of mixing. The drop solution enthalpy data can be fit-ted as a quadratic function. This gives, for the regular solution

∆Hmix = Wx(1–x) (5)

where x is the mole fraction of Fe2TiO4 and W = 22.60 ± 8.46 kJ/mol.

Experimental values of the heat of mixing and the calculated curve are shown in Figure 4.

Magnetite-ulvöspinel mixing propertiesThe cation distributions and free energies of mixing in spi-

nel solid solutions such as this one can be calculated using the O’Neill and Navrotsky (1984) thermodynamic model. Three factors are considered in that model: cation distribution, size mismatch, and electron exchange reactions.

The model was formulated by minimizing the free energy of disorder, in which disordering enthalpy varies linearly with the inversion parameter (z) and disordering entropy is related to configurational entropy (Sconf). According to the model, both the composition and the cation arrangement of a solid solution can be described using a set of parameters (O’Neill and Navrotsky 1984). As the substitution occurs on both cation sites, the distri-bution parameters can be defined as

Tet Oct SumFe2+ 1–z x+z 1+xFe3+ z 2–2x–z 2–2xTi4+ 0 x xSum 1 2 3.

The inversion parameter is given by z. Using these param-eters, the solid solution can be written in the form

−+ +

++

− −+ +(Fe Fe )[Fe Fe Ti )Oz z x z x z x1

2 3 22 23 4

4.

Table 3. Thermodynamic cycles for FeTiO3 and Fe2TiO4

Enthalpy of formation from oxidesFeTiO3(s, 298 K) + 0.25O2 (g, T K) → 0.5Fe2O3 (soln, T K) + TiO2 (soln, T K) ΔH1

TiO2(s, 298 K) → TiO2 (dis, T K) ΔH2

Fe (s, 298 K) + 0.75 O2 (g, T K) → 0.5 Fe2O3 (soln, T K) ΔH3

Fe0.947O (s, 298 K) + 0.25 O2 (g, T K) → 0.5 Fe2O3 (soln, T K) ΔH4

Fe0.947O (s, 298 K) + 0.053Fe (s, 298 K) + TiO2 (s, 298 K) → FeTiO3 (s, 298 K) ΔH5

ΔH5 = –ΔH1 + ΔH2 + 0.053ΔH3 + ΔH4 Fe2TiO4(s, 298 K) + 0.5O2 (g, T K) → Fe2O3 (soln, T K) + TiO2 (soln, T K) ΔH6

TiO2(s, 298 K) → TiO2 (soln, T K) ΔH2

Fe (s, 298K) + 0.75 O2 (g, T K) → 0.5 Fe2O3 (soln, T K) ΔH3

Fe0.947O (s, 298 K) + 0.25 O2 (g, T K) → 0.5 Fe2O3 (soln, T K) ΔH4

2Fe0.947O (s, 298 K) + 0.106Fe (s, 298 K) + TiO2 (s, 298 K) → Fe2TiO4 (s, 298 K) ΔH7

ΔH7 = –ΔH6 + ΔH2 + 0.106ΔH3 + 2ΔH4 Enthalpy of formation from elementsFeTiO3(s, 298 K) + 0.25O2 (g, T K) → 0.5Fe2O3 (soln, T K) + TiO2 (soln, T K) ΔH1

orFe2TiO4(s, 298 K) + 0.5O2 (g, T K) → Fe2O3 (soln, T K) + TiO2 (soln, T K) ΔH6

TiO2(s, 298 K) → TiO2 (soln, T K) ΔH2

Fe2O3 (s, 298 K) → Fe2O3 (soln, T K) ΔH8

Fe (s, 298 K) + 0.75 O2 (g, T K) → 0.5 Fe2O3 (soln, T K) ΔH3

Ti (s, 298 K) + O2 (g, T K) → TiO2 (soln, T K) ΔH9

Fe (s, 298 K) + Ti (s, 298 K) + 1.5O2 (g, 298 K) → FeTiO3(s, 298 K) ΔH10

ΔH10 = –ΔH1 + ΔH2 + 0.5ΔH8 + 0.5 ΔH3 + ΔH9

or2Fe (s, 298 K) + Ti (s, 298 K) + 2O2 (g, 298 K) → Fe2TiO4(s, 298 K) ΔH11

ΔH10 = –ΔH6 + ΔH2 + ΔH8 + ΔH3 + ΔH9

LILOVA ET AL.: THERMODYNAMICS OF THE MAGNETITE-ULVÖSPINEL 1335

The cation distribution equation is

− +− − −

= α + β−+ +RT z x z

z x zzln ( )

(1 )(2 2 )2Fe Fe2 3 . (6)

The α and β interchange enthalpy parameters are expected to be comparable in magnitude but opposite in sign. Trestman-Matts et al. (1983) found that both parameters are temperature dependent

αFe2+–Fe3+ = –9.61 + 0.0256T kJ (7)βFe2+–Fe3+ = 1.85RT J. (8)

The enthalpies of mixing can be calculated from the cation distribution, using α and β interchange enthalpy parameters as the difference between the disordering enthalpy of a mechanical mixture of end-members Fe3O4 and Fe2TiO4 and the disordering enthalpy of the solid solution. If the disordering enthalpy depends linearly on total degree of inversion z, then:

∆Hdis,ss = z(αFe2+–Fe3+ + βFe2+–Fe3+z) (9)∆Hdis,mm = z(∆Hdis,ss,Fe2TiO4) + (1 – z)(∆Hdis,ss,Fe3O4) (10)∆Hmix = ∆Hdis,ss – ∆Hdis,mm. (11)

The calculated enthalpies of mixing are shown in Figure 4. Agreement between them and the experimental value is observed within the error range.

The configurational entropies of the Fe3–xTixO4 solid solu-tions are much more complex than those resulting from simply 1 mole of ions mixing over 1 mole of sites. Thus despite the ap-proximately quadratic form of the enthalpy of mixing, magnetite-ulvöspinel cannot be described as a regular solution.

For a spinel AB2O4 or A1–xBx[Ax/2B(2–x)/2]2O4 with twice as many octahedral sites than tetrahedral and inversion parameter z, the configurational entropy can be written in the following

FiGure 3. Drop solution enthalpies of Fe3O4-Fe2TiO4 solid solutions in sodium molybdate at 973 K. The straight line connects the two end-members, the curve represents a second-degree polynomial fit of the experimental data. The estimated uncertainty of the ulvöspinel mole fraction is ±2%.

FiGure 4. Enthalpies of mixing of Fe3O4-Fe2TiO4 solid solutions. The straight line corresponds to zero heat of mixing, the curve is calculated from ∆Hmix = 22.60 ± 8.46x(1 – x) kJ/mol. The estimated uncertainty of the magnetite mole fraction is ±2%.

form, assuming that the substitution of the two ions on each sublattice is random

S R x x x x

x x xconf = − + − − +

+ −

{ ln ( ) ln( )

ln

1 1

22 2

12

−

ln }1

2x . (12)

The entropy value starts from 0 for a completely normal spinel, rises to maximum of 15.48 J/mol⋅K for a random cation distri-bution and has the value 2Rln2 or 11.53 J/(mol⋅K) for a totally inverse spinel (Navrotsky and Kleppa 1967).

For Fe3O4-Fe2TiO4 solid solutions with Fe2+ and Fe3+ ran-domly mixed on both tetrahedral and octahedral sites and Ti4+ oc-cupying octahedral sites only, the configurational entropy will be

S R z z z z

x x x zconf = − + − − +

++

{ ln ( ) ln( )

ln

1 1

22 2 2

+

+

− −

− −

ln lnx z x z x z2

2 22

2 22

}.

(13)

Then the entropy of mixing (the difference in configurational entropy of the solid solution and that of a weighted average of configurational entropy of the end-members) is given by

∆Smix = Sconf(solid solutions) – xSconf(Fe2TiO4) – (1 – x)Sconf(Fe3O4). (14)

The configurational entropy and entropy of mixing are shown on Figures 5a and 5b, compared with the entropy for a solution of two completely inverse spinels.

Gibbs free energy of mixing and calculated solvusThe Gibbs free energy of mixing can be obtained, using the

experimental enthalpies of mixing and the calculated entropies of mixing from the average measured cation distributions using

LILOVA ET AL.: THERMODYNAMICS OF THE MAGNETITE-ULVÖSPINEL1336

∆Gmix = ∆Hmix – T∆Smix. (15)

The solvus, calculated from the Gibbs free energies of mix-ing, obtained between 473 and 823 K is shown in Figure 6. It is asymmetric toward the ulvöspinel end-member with a critical temperature of 823 K and critical composition x = 0.54. The temperature falls in the interval, found from previous experi-mental measurements and theoretical calculations (Kawai 1956; Vincent et al. 1957; Lindsley 1981; Trestman-Matts et al. 1983). The difference in the experimentally obtained solvus in this work and by Vincent et al. (1957) can be explained with the natural titanomagnetite samples, used by the latter, which usually contain impurities (MgO, Al2O3). The uncertainty in the Lindsley (1981)

FiGure 5. (a) The configurational entropies calculated from Equation 9 for the Fe3–xTixO4 solid solution system. The solid curve represents the configurational entropy of a solid solution of two inverse spinels. (b) The entropies of mixing calculated from Equation 10 for the Fe3–xTixO4 solid-solution system. The solid curve represents the mixing entropy of a solid solution of two inverse spinels.

critical temperature is ±15 K (838 ± 15 K) but the uncertainty in the composition is not estimated. Trestman-Matts et al. (1983) calculated the solvus using experimentally obtained cation distributions and the O’Neill and Navrotsky model (1984) and treated the heats of mixing as a subregular solution. Their solvus shifts toward ulvöspinel with decreasing temperature, similar to the one obtained in this work.

We conclude that the enthalpies of mixing in Fe3O4-Fe2TiO4 spinel solid solution are small (less than about 5 kJ/mol in mag-nitude), positive and symmetric, and that the asymmetry of the solvus arises from the configurational entropy.

aCKnowLedGMentsWe gratefully acknowledge support from the Pacific Northwest National

Laboratory Science Focus Area (SFA) Subsurface Biogeochemical Research (SBR) Program of the U.S. Department of Energy (DOE). Calorimetry of the present system was supported by contract DEAC02-98CH10886 between PNNL and UCD. Development of a calorimetric technique for iron-bearing compounds was supported by DOE grant DEFG02-97ER14749 (UCD). We acknowledge Elke Arenholz for her assistance with XA and XMCD measurements. XA and XMCD measurements were performed at the Advance Light Source supported by the DOE Office of Science, Office of Basic Energy Sciences under contract no. DE-AC02-05CH11231. We also acknowledge Neil Telling at Keele University, U.K., for development of the q-fit program for fitting XMCD data. We thank David Vaughan and Paul Wincott at the University of Manchester, U.K., for the Mössbauer spectroscopy. We are also very grateful to Michael Henderson at the Science and Technology Facilities Council, U.K., for his invaluable contributions to titanomagnetite synthesis.

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Manuscript received deceMber 12, 2011Manuscript accepted april 10, 2012Manuscript handled by hongwu Xu