Nonstoichiometry, defects and thermodynamic properties of YFeO3, YFe2O4 and Y3Fe5O12

Nonstoichiometry, defects and thermodynamic properties of YFeO3, YFe2O4 and Y3Fe5O12

Solid State Ionics 224 (2012) 32–40 Contents lists available at SciVerse ScienceDirect Solid State Ionics journal homepage: www.elsevier.com/locate/...

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Solid State Ionics 224 (2012) 32–40

Contents lists available at SciVerse ScienceDirect

Solid State Ionics journal homepage: www.elsevier.com/locate/ssi

Nonstoichiometry, defects and thermodynamic properties of YFeO3, YFe2O4 and Y3Fe5O12 K.T. Jacob ⁎, G. Rajitha Department of Materials Engineering, Indian Institute of Science, Bangalore 560012, India

a r t i c l e

i n f o

Article history: Received 31 January 2012 Received in revised form 28 June 2012 Accepted 3 July 2012 Available online 3 August 2012 Keywords: Gibbs free energy Enthalpy Entropy Yttrium orthoferrite Yttrium iron garnet Buffer electrode

a b s t r a c t Oxygen nonstoichiometry of three ternary oxides, YFeO3 −δ, YFe2O4 − α and Y3Fe5O12 −θ, in the system Y–Fe–O was investigated as a function of oxygen partial pressure by thermogravimetry at high temperature. The defects responsible for nonstoichiometry were identified as oxygen vacancies for YFeO3 − δ and YFe2O4 − α, although the manner of variation of nonstoichiometric parameter with oxygen partial pressure for these two oxides is quite different. Cation interstitials are the predominant defects in Y3Fe5O12 −θ. Gibbs energies of formation of the three nonstoichiometric oxides were determined using solid-state electrochemical cells in the temperature range from 975 to 1475 K. YFe2O4 −α was found to be stable only above 1391 K. Gibbs energies of formation of the three stoichiometric compounds from their component binary oxides were obtained by combining information from solid state cells with results of thermogravimetric analysis using the Gibbs–Duhem relation. The results can be summarized as: O

ð1=2ÞY2 O3 þ ð1=2ÞFe2 O3 →YFeO3 ; ΔGf ðoxÞ ð250ÞðJ=molÞ ¼ −17; 126−8:263T O

ð1=2ÞY2 O3 þ FeO þ ð1=2ÞFe2 O3 →YFe2 O4 ; ΔGf ðoxÞ ð260ÞðJ=molÞ ¼ −10; 352−13:24T O

ð3=2ÞY2 O3 þ ð5=2ÞFe2 O3 →Y3 Fe5 O12 ; ΔGf ðoxÞ ð780ÞðJ=molÞ ¼ −56; 647−31:091T: © 2012 Elsevier B.V. All rights reserved.

1. Introduction Two ternary oxides in the system Y–Fe–O have interesting magnetic properties. Yttrium orthoferrite (YFeO3) having orthorhombic perovskite structure is a canted antiferromagnet with weak ferromagnetic behavior. It has recently gained prominence because of its magneto-optical properties and unusual domain wall motion. Domains in this material are wider than in the garnet (Y3Fe5O12) and domain wall velocity is the highest (up to 20 km/s) among all ferromagnets [1]. The orthoferrite has very strong magnetic uniaxial anisotropy, rectangular hysteresis loop, high coercive field and rapid demagnetization. Hence, the material is used in new devices such as fast latching optical switches and magneto-optical current sensors [2,3]. Yttrium iron garnet (Y3Fe5O12), also known as YIG, has high resistivity, narrow ferromagnetic resonance bandwidth, low dielectric loss in the microwave region, low absorption of infrared wavelengths up to 600 nm, and tunable saturation magnetization and Curie temperature. It finds application in microwave communication ⁎ Corresponding author. Tel.: +91 80 22932494; fax: +91 80 23600472. E-mail address: [email protected] (K.T. Jacob). 0167-2738/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2012.07.003

devices, circulators, isolators, gyrators and phase-shifters. YIG–collagen biocomposites are being evaluated for hypothermia applications. A third oxide, YFe2O4, containing one Fe 2+ ion is stable at high temperatures under reducing conditions. Although the three oxides are known to be oxygen deficient, the magnitude of nonstoichiometry and the defects responsible for it have not been clearly defined. Solid-state electrochemical cells [4–6] and gas equilibration techniques [7,8] have been deployed for the determination of isothermal sections of the phase diagram and thermodynamic properties of the three oxides. However, the results for YFeO3 and Y3Fe5O12were analyzed ignoring their oxygen nonstoichiometry. In this study, variation of the nonstoichiometric parameter with oxygen partial pressure is measured for the three ternary oxides. The defects responsible for nonstoichiometry are identified. Oxygen potentials corresponding to three-phase equilibrium involving the ternary oxides are measured as a function of temperature using solid-state electrochemical cells. Gibbs energies of formation of oxygen-deficient oxides are obtained from the emf of the cells. Gibbs energies of stoichiometric oxides are then calculated using the Gibbs–Duhem relation and information on the variation of nonstoichiometric parameter with oxygen

K.T. Jacob, G. Rajitha / Solid State Ionics 224 (2012) 32–40

33

partial pressure. The results provide comprehensive information on nonstoichiometry, defects and thermodynamic properties of important ternary oxides in the system Y–Fe–O.

2.3. Solid-state electrochemical measurements

2. Experimental

Pt; Fe þ Y2 O3 þ YFeO3−δ ==ðY2 O3 ÞZrO2 ==O2 ð0:1MPaÞ; Pt

2.1. Materials

Pt; YFeO3−δ þ Fe3 O4 þ Y3 Fe5 O12−θ ==ðY2 O3 ÞZrO2 ==O2 ð0:1MPaÞ; Pt

The two inter-oxide compounds YFeO3 and Y3Fe5O12 were prepared under pure oxygen gas by the solid state reaction method. The binary oxides Y2O3 and Fe2O3 of purity greater than 99.99% were dried at 573 K before use. They were then mixed thoroughly in the required stoichiometric ratio, compacted into pellets using a steel die at 150 MPa pressure and heat treated at 1673 K. The pellets were quenched, reground and repelletized after heat treatment for ~ 24 h (86 ks). The progress of the reaction was followed using XRD of quenched samples. The formation of YFeO3 having brown color was more rapid than the formation of green‐colored Y3Fe5O12. It took ~ 64 h (230 ks) to form a single phase YFeO3 and ~ 170 h (610 ks) for Y3Fe5O12. Synthesis of YFe2O4 from Y2O3 and Fe2O3 was done at 1473 K in flowing CO2 + CO gas mixture, with a ratio of CO2/CO = 1.58. Partial reduction of Fe2O3 occurs during the reaction. Although YFe2O4 is a high temperature phase, it can be quenched to room temperature. The formation of the required compounds was verified using XRD, SEM and EDS. The lattice parameters of the orthorhombic perovskite YFeO3 (space group Pnma) were a = 0.5593, b= 0.7604 and c = 0.5282 nm. Y3Fe5O12 had a cubic structure (space group Ia3d) with a = 1.238 nm. Hexagonal unit cell of YFe2O4 (space  group R 3m) had the dimensions a = 0.3518 and c = 2.4780 nm. The Y2O3 used in this study was the C-type with Ia3 cubic structure.

were measured as a function of temperature in the range from 975 to 1475 K. The right-hand electrodes are positive. Impervious yttriastabilized zirconia (YSZ) tube functioned as the solid electrolyte with predominant oxygen ion conduction (tion > 0.999) under the experimental conditions encountered in this study. Flowing pure oxygen gas at a pressure of 0.1 MPa served as the non-polarizable reference electrode. Working electrode of cell (I) consisted of a mixture of three condensed phases, Fe, Y2O3 and YFeO3−δ in the molar ratio 1:1:1.5. In cell (II) the working electrode consisted of a mixture of three coexisting oxides, YFeO3−δ, Fe3O4 and Y3Fe5O12−θ in the molar ratio 1:1:1.5. An excess of the compound that decomposed to establish equilibrium oxygen partial pressure in the apparatus was used to form the electrode. Because of trace electron (or hole) conductivity in the solid electrolyte, there is always a small electrochemical flux of oxygen from the electrode with high oxygen potential to the electrode having low oxygen potential. The flux is caused by coupled transport of oxygen ions and electrons (or holes) through the electrolyte. This flux can cause polarization of the working electrode, especially when they consist of three condensed phases. A buffer electrode, maintained at oxygen chemical potential close to that of the working electrode was introduced between the reference and the working electrodes, as shown in Fig. 1, to act as a sink for the electrochemical flux of oxygen and prevent the flux from reaching the working electrode. There was no physical contact between the buffer electrode and the solid electrolyte tube enclosing the working electrode. The two closed-end solid electrolyte tubes enclosing the working and reference electrodes were linked through an ionic bridge of the same material so that the emf between the working and the reference electrodes was determined by the chemical potentials of oxygen at the two unpolarized electrodes. The oxygen ions would not have traveled through the ionic bridge to the working electrode because of higher resistance associated with this path. The ionic current always takes the lowest resistance path. Thus, the three-electrode design of the cell prevented errors in EMF caused by polarization of the working electrode, ensuring accurate high-temperature thermodynamic measurements. The magnitude of the error caused by polarization in the conventional two‐electrode cell designs can be assessed from the difference in the EMF between the reference and the buffer on the one hand, and the reference and the working electrodes on the other hand. Since the apparatus and procedure used for high-temperature emf measurements has been described in detail earlier [9,10], only a brief account of the salient features will be given here. The apparatus consisted of three distinct compartments, separated by impervious yttria-stabilized zirconia (YSZ) tubes and an YSZ crucible. Separate

2.2. Thermogravimetric measurements A known mass (0.7 to 1 g) of each ternary oxide was kept in an YSZ crucible and suspended from a microbalance in a furnace under controlled gas atmospheres. The measurement system had a mass resolution of 1 μg and temperature control of ±1 K. Flowing mixtures of Ar + O2 and CO+ CO2 of known composition established the oxygen partial pressure in the gas phase. The oxygen chemical potential computed from the gas composition was found to agree well (±50 J/mol) with that determined at 1373 K using a solid-state sensor based on (Y2O3)ZrO2 as the electrolyte and using pure oxygen as the reference. At the start, oxygen nonstoichiometry of YFeO3 in pure oxygen gas was explored by heating the sample to different temperatures and recording mass changes at each temperature as a function of time. In pure oxygen up to ~1000 K, no significant nonstoichiometry was detected. At higher temperatures, very small oxygen loss was detected in pure oxygen. By changing the composition of the gas, oxygen partial pressure was varied in steps and the corresponding mass loss was recorded at 1373 K. From the mass loss, after applying buoyancy correction, the nonstoichiometric parameter δ in YFeO3−δ was computed. Reductive gravimetry was used to determine the oxygen content of the sample equilibrated in pure oxygen at 975 K and quenched to room temperature. After reducing the sample to Fe and Y2O3 in flowing dry hydrogen gas at 1273 K for ~7 h (25 ks), the mass change was determined using a microbalance. The mass loss confirmed that the sample had stoichiometric composition YFeO3. A similar procedure was used to study the nonstoichiometry of Y3Fe5O12. Since YFe2O4−α is stable only at high temperatures and reducing conditions, its mass loss was measured as a function of CO/CO2 ratio in the gas phase at 1473 K. Each oxide sample was equilibrated with the gas phase at the lowest oxygen partial pressure used in thermogravimetric measurement and then quenched into liquid nitrogen. The quenched sample was examined under SEM to check for the formation of secondary phases. All the three oxides were found to be single phase without any precipitates.

The reversible emfs of the solid-state electrochemical cells, ðIÞ ðIIÞ

(Y2O3)ZrO2

Reference

Working Electrode O2 (Po' ) 2

Pt

(Y2O3)ZrO2 No flux

Buffer Electrode '' ), Pt O2 (Po

2

(Y2O3)ZrO2 ← O2− Flux

Electrode O2(0.1 MPa) Pt

Po'' 2 ≈ Po' 2

Fig. 1. Schematic representation of the solid state cell with a buffer electrode between the reference and the working electrodes.

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zirconia tubes were used to contain the working and reference electrodes. The working and buffer electrodes were sealed under vacuum. The oxygen partial pressure over these electrodes was established by the decomposition of an oxide. The buffer electrode was prepared by consolidating an intimate mixture of three solid phases in the YSZ crucible, with a short Fe wire embedded in the powder. The Fe wire was spot-welded to a Pt lead. Initially the oxygen potentials of the working and buffer electrodes were the same, since the same three-phase mixture was used at the two electrodes. Electrical contact to the working electrode was made by spot welding a Pt lead to a short Fe wire embedded in the oxide mixture. Pure oxygen gas at a pressure of 0.1 MPa was flowed through the reference electrode in the YSZ tube at a rate of 3 (ml/s). The cell assembly was suspended inside a vertical resistance furnace, with the electrodes located in the even-temperature zone (± 1 K). Induced emf on cell leads from the furnace winding was minimized by connecting to earth a thick stainless steel foil placed between the furnace and the cell assembly. The furnace temperature, measured by a Pt/Pt-13pctRh thermocouple, was accurate to ±1 K. A highimpedance digital voltmeter with a sensitivity of (±0.1) mV measured the cell potentials. At the end of each experiment, the electrodes were cooled to room temperature and examined by OM, SEM, EDS and XRD. Microcoulometric titration in both directions checked the reversibility of EMF. An external D.C. potential was applied to cell terminals to pass a small current (~25 μA) for ~ 3 min (180 s) through the cell. The current caused essentially infinitesimal displacement of oxygen potential at the electrode–electrolyte interface. The open-circuit emf was then recorded as a function of time. The EMF was found to return to the original value before the titration. Reversibility was ensued when the EMF returned to the same value after titration in opposite directions. The reproducibility of the EMF on temperature cycling also confirmed correct functioning of the cell. Further, no significant effect on the EMF was caused by the change in the flow rate of oxygen gas through the reference electrode by a factor of three.

Fig. 2. Variation of the non-stoichiometric parameter (δ) of YFeO3−δ with the partial pressure of oxygen at 1373 K. P0 is the standard atmospheric pressure.

− 0.1667. The proportionality comes from the application of mass action law to the defect reaction and assuming that defect concentrations are sufficiently low so that activities of defect species are proportional to their concentration (Henry's law). The value obtained in this study (− 0.1649 ± 0.00094) is close to the model prediction, thus providing experimental confirmation of the assumed model. It would be useful to measure electrical conductivity and Seebeck coefficient as functions of oxygen partial pressure to confirm the model. As part of a study on phase relations and oxygen potentials in the system Y–Fe–O at 1373 K, Kitayama et al. [7] obtained approximate compositions of YFeO3 −δ at oxygen potentials corresponding to various three-phase equilibria. They presented a linear relation between oxygen concentration and oxygen partial pressure, without discussing defects responsible for nonstoichiometry:

NO =N YFeO3 ¼ 2:26  10

3. Results and discussion

−3

  0 −3 log P O2 =P −2:123  10

ð3Þ

3.1. YFeO3 3.1.1. Variation of non-stoichiometry of YFeO3 − δ with oxygen partial pressure The thermogravimetric measurements show that YFeO3 at high temperatures in pure oxygen is oxygen deficient. The deviation from stoichiometry in pure oxygen is very small at 1373 K, δ = 1.558 × 10 −4. The deviation increases with the decrease in oxygen partial pressure up to the phase stability limit of YFeO3 − δ. The variation of the non-stoichiometric parameter (δ) with oxygen partial pressure at 1373 K observed in this study is shown in Fig. 2. The equilibrium oxygen partial pressure corresponding to each CO + CO2 mixture was calculated from thermodynamic data. The results can be adequately represented by Eq. (1):   0 lnðδÞ ¼ −0:1649 ln P O2 =P −8:7668

ð1Þ

where P 0 is the standard atmospheric pressure. Eq. (1) suggests that oxygen deficiency may be caused by vacancy formation on the anionic sub-lattice and reduction of Fe 3+ to Fe 2+ on the cationic sub-lattice of the perovskite structure as required for charge neutrality. The electrons released by oxygen are assumed to be localized at the cationic sites. Using the Kröger–Vink notation, vacancy formation in YFeO3 − δ can be represented as: ··

0

OO þ 2FeFe →VO þ 2FeFe þ 1=2O2 :

ð2Þ

According to this model, the non-stoichiometric parameter (δ) is proportional to (PO−2 1/6) and the slope of the plot in Fig. 2 should be

where NO and NYFeO3 represent the mole fraction of oxygen and YFeO3, respectively. In pure oxygen at ambient pressure, Eq. (3) suggests a value of 2.123 × 10−3 for the nonstoichiometric parameter (δ) while the results of this study suggest a much lower value of 1.558 × 10 −4.

3.1.2. Oxygen chemical potential for three-phase equilibrium, Fe + YFeO3 − δ + Y2O3 The EMF between the reference and the buffer electrodes of cell (I) was 6 to 10 mV less than that between the reference and the working electrodes, indicating that there was a significant polarization of the three-phase buffer electrode. Variation of the unpolarized reversible EMF of cell (I), measured between the reference and the working electrodes, with temperature is shown in Fig. 3. The response time of the cell varied from ~ 6 h (22 ks) at the lower temperatures to ~ 1 h (4 ks) at the higher temperatures. However, on increasing the temperature from 1375 to 1400 K, there was an unexpected delay (~ 8 h/29 ks) in registering a constant EMF. A careful analysis of the EMF data indicates a small change in slope at ~ 1391 K. In one experiment the cell was rapidly cooled from 1450 K and the working electrode was examined by SEM, EDS and XRD. There was no trace of YFeO3−δ in the quenched electrode; instead a new phase YFe2O4 was detected. Within experimental uncertainty, the EMF can be represented by two linear functions of temperature. The linear least-squares regression analysis of the emf in the lower temperature range from 975 to 1375 K gave the expression: EðIÞ ð0:57ÞðmVÞ ¼ 1453:9−0:39503T:

ð4Þ

K.T. Jacob, G. Rajitha / Solid State Ionics 224 (2012) 32–40

Fig. 4. Comparison for the oxygen potential for the equilibrium involving the three condensed phases (Fe+Y2O3 +YFeO3−δ) measured in this study with the data reported in the literature.

Fig. 3. Variation of the EMF of cell (I) with temperature.

At higher temperatures from 1400 to 1475 K, the emf can be represented by, EðIÞ ð0:5ÞðmVÞ ¼ 1433:5−0:3804T

ð5Þ

where temperature (T) is in Kelvin. The two linear curves intersect at 1391.4 K. The open circuit potential of the cell is related to the oxygen partial pressures at the two electrodes:   r w E ¼ ðRT=4F Þ ln P O2 =P O2

ð6Þ

where POr2 and POw2 are the oxygen pressures at the reference and working electrodes respectively, F is the Faraday constant, and R the gas constant. As pure oxygen at 0.1 MPa was used as the reference electrode, the EMF directly gives the oxygen potential (ΔμO2 = RT ln PO2) at the working electrode. In the temperature range from 975 to 1391.4 K, Δμ O2 ðFe þ Y2 O3 þ YFeO3−δ Þð220ÞðJ=molÞ ¼ −561; 118 þ 152:458T: ð7Þ This oxygen potential is established by the three condensed phases at the working electrode and is defined by the reaction: Fe þ fð3−2δÞ=4gO2 þ ð1=2ÞY2 O3 →YFeO3−δ :

35

ð8Þ

The ternary system Y–Fe–O has only one degree of freedom when three solid phases coexist in equilibrium with a gas phase. Therefore, at constant temperature the dissociation pressure of oxygen is uniquely defined. The oxygen chemical potential for the three-phase equilibrium involving Fe, Y2O3 and YFeO3 − δ obtained in this study is compared with data obtained by previous investigators in Fig. 4. To clearly display the difference between the different sets of data, the difference function (ΔμO2 (others) − ΔμO2 (this study)) is plotted as a function of temperature in Fig. 4. The results of Tretyakov et al. [4], who used a conventional solid-state cell with air as reference electrode, are more positive than those obtained in this study by 2.2 to 3.9 (kJ/mol). The difference is probably caused by polarization of the three-phase electrode in the experiments of Tretyakov et al. [4], caused by the electrochemical flux of oxygen through the electrolyte from the reference side. The results obtained in this study are in good agreement with the studies of Piekarczyk et al. [5] who used a solid-state electrochemical technique with two calcia-stabilized zirconia electrolyte tubes and air as the reference electrode. In their apparatus an electrochemical pump was used to nullify the effect of oxygen permeability through the electrolyte. This probably explains the good agreement between their data and the results obtained in this study. However, air is not a very accurate oxygen potential standard since the partial pressure of oxygen is dependent on total pressure and humidity which can vary

with time depending on altitude and climatic conditions. Kitayama et al. [7] established phase equilibria in the system Y–Fe–O at 1373 K by varying the oxygen partial pressure using controlled mixtures of CO2 and H2. The oxygen partial pressure (PO2/P0) was found to be 10−13.32 at 1373 K for the equilibrium involving three condensed phases Fe, Y2O3 and YFeO3−δ. Their results, depicted in Fig. 4, are more positive by 1.7 (kJ/mol). Yamauchi et al. [6] measured the EMF of the solid electrolyte galvanic cell, Pt/Fe, Y2O3, YFeO3 //ZrO2–CaO // Ni–NiO/Pt. The oxygen potential for the three-phase equilibrium was calculated from their EMF, using values for the reference electrode from Charette and Flengas [11]: ΔμO2 (±200) (J/mol) = −574,242+ 162.73 T. Their results show deviations ranging from −1 (kJ/mol) at lower temperatures to 1.36(kJ/mol) at the higher temperatures. Because of the considerably smaller oxygen potential gradient across the electrolyte in cell employed by Yamauchi et al. [6], polarization of the working electrode is less significant. 3.1.3. Limit of non-stoichiometry of YFeO3 − δ The decomposition oxygen partial pressure of YFeO3 − δ to Fe and Y2O3 at 1373 K computed from Eq. (7) is (PO2/P 0) = 4.135 × 10 −14. The value of oxygen non-stoichiometric parameter (δ) corresponding to this oxygen partial pressure, calculated from Eq. (1), is δmax = 0.0251. This compares with a value of 0.03 indicated in text of Kitayama et al. [7] and 0.0322 calculated from their equation. 3.1.4. Gibbs free energy of formation of YFeO3 − δ and YFeO3 The standard Gibbs free energy change for the formation of nonstoichiometric YFeO3 − δ at 1373 K according to reaction (8) can now be computed from the measured oxygen potential corresponding O to three-phase equilibrium as ΔGr(8) = [(3 − 2δ) /4] ΔμO2 = −259,427 (±165) (J/mol). Assuming that the nonstoichiometry is negligible at 975 K, the temperature dependence of the Gibbs energy change for reaction (8) is given by: O

ΔGrð8Þ ð165ÞðJ=molÞ ¼ −431; 661 þ 125:44T:

ð9Þ

At 1473 K, the value of − 246.888 kJ/mol obtained from this equation is significantly higher than the value of − 250.2 (±0.84) kJ/mol suggested by Kimizuka and Katsura [8] based on phase equilibrium studies. Neglecting the nonstoichiometry of YFeO3 and assuming the high temperature three-phase equilibrium to be represented by the hypothetical reaction, Fe þ ð3=4ÞO2 þ ð1=2ÞY2 O3 →YFeO3

ð10Þ

the Gibbs energy of formation of the stoichiometric phase was computed in earlier studies [4–7] from the measured oxygen potentials.

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If this procedure is adopted, the results obtained in this study yield O a value ΔG(10) = (3/4) ΔμO2 = − 263,845 (±165) (J/mol) at 1373 K. However, this method of analysis is incorrect at the higher temperatures where the deviation from stoichiometry is significant. The second-law enthalpy and entropy of formation for YFeO3 deduced from such analysis may not be compared with calorimetric data. The error caused by the neglect of non-stoichiometry is assessed below. The difference in the Gibbs energies of formation of stoichiometric and non-stoichiometric compositions can be assessed using information on the variation of the non-stoichiometric parameter (δ) with PO2 in the stability domain of the perovskite phase. This assessment involves the use of the Gibbs–Duhem equation to connect the integral molar free energy of mixing GM for stoichiometric and non-stoichiometric compounds with the variation of activity (partial pressure) of oxygen with composition along paths of constant ratio of XY/XFe. For the system Y–Fe–O, the molar Gibbs free energy of mixing can be expressed as: M

G ¼ GY X Y þ GFe X Fe þ GO X O

ð11Þ

where Xi and Gi denote the mole-fraction and relative partial molar free energy of component i. For convenience, the symbol Δ before GM and Gi is deleted. Since the sum of the three mole fractions is equal to one, 1 − XO = XY + XFe. It is convenient to express the mole fractions XY and XFe in terms of the ratio XY/XFe. Thus, XY = (1− XO) / [1+ (XFe/XY)] and XFe = (1− XO) / [1+ (XY/XFe)]. Hence, M

G ¼ ½ð1−X O Þ=ð1 þ ðX Fe =X Y ÞÞGY þ ½ð1−X O Þ=ð1 þ ðX Y =X Fe ÞÞGFe þ GO X O :

ð12Þ In the perovskite phase, the mole fraction ratio XY/XFe is constant and only XO varies with oxygen nonstoichiometry. Therefore, differentiating G M with respect to XO, 

M

∂G =∂X O

 X Y =X Fe

¼ ½ð1−X O Þ=ð1 þ ðX Fe =X Y ÞÞð∂GY =∂X O Þ −GY =½1 þ ðX Fe =X Y Þ þ ½ð1−X O Þ=ð1 þ ðX Y =X Fe ÞÞ

ð∂GFe =∂X O Þ−GFe =½1 þ ðX Y =X Fe Þ þ GO þ ð∂GO =∂X O ÞX O :

ð13Þ

The free energy of formation of non-stoichiometric yttrium ferrite from elements, Y þ Fe þ fð3−δÞ=2gO2 →YFeO3−δ

can be obtained by combining assessed above with data for Y2O3 from Pankratz [12]. Thus, at 1373 K, for δmax = 0.0251, O ΔG(17) = − 1,014,746 (J/mol). The corresponding value of free energy of mixing for the non-stoichiometric composition is: M

O

G ¼ ΔGð17Þ =ð5−δÞ ¼ −203; 973ðJ=molÞ:

O

ð14Þ

Each differential is taken at constant value of XY/XFe. On combining Eq. (14) with Eq. (11), and invoking the Gibbs–Duhem relation i.e. ∑Xi dGi = 0 and integrating with limits XO′ and XO″ one obtains:   M G

X ″O

      M ″ ′ ¼ G 1−X O = 1−X O ′ X  O  ″ ″ X 2 þ 1−X O ∫X O′ fGO =ð1−X O Þ gdX O :

ð15Þ

O

The mole fraction of oxygen (XO) in YFeO3 − δ is related to δ: X O ¼ ð3−δÞ=ð5−δÞ:

ð16Þ

The relative partial free energy of oxygen is related to the oxygen partial pressure: GO = 0.5 ΔμO2 = 0.5RT ln PO2. The integral Gibbs energy of mixing can be evaluated as a function of composition by integrating the oxygen potential as a function of composition provided its value is known at one composition. The value of mole-fraction of oxygen (XO) corresponding to δmax = 0.0251 at 1373 K can be calculated using Eq. (15) as XO = 0.59798.

X ′O

XO)2dXO =−1766(J/mol);(GM)XO″ =−203,654(J/mol). Hence, the standard Gibbs energy of formation of the stoichiometric YFeO3 from Fe, Y2O3 and O2 according to the hypothetical Eq. (10) can be calculated as O ΔG(10) =−262,958(J/mol) at 1373 K. Thus, the error introduced by the neglect of nonstoichiometry in calculation of the standard Gibbs energy of formation of YFeO3 from the measured oxygen potential for threephase equilibrium is 887 (J/mol). However, these corrections would become less significant at lower temperatures. Assuming that the correction for nonstoichiometry is negligible at 975 K, the correct Gibbs free energy of formation of the stoichiometric perovskite according to reaction (10) can be computed as: O

ΔGrð10Þ ð200ÞðJ=molÞ ¼ −423; 011 þ 116:572T:

ð19Þ

The standard Gibbs energies of stoichiometric and nonstoichiometric YFeO3 are shown in Fig. 5 as a function of temperature. It is instructive to assess the Gibbs free energy of formation of the stoichiometric perovskite from its component binary oxides. By combining Eq. (19) with the standard Gibbs energy of formation of Fe2O3 from the assessment of Sundman [13] in the temperature range from 975 to 1500 K, 2Fe þ ð3=2ÞO2 →Fe2 O3 ΔGrð20Þ ðJ=molÞ ¼ −811; 770 þ 249:67T:

ð20Þ ð21Þ

one obtains for the reaction: ð1=2ÞY2 O3 þ ð1=2ÞFe2 O3 →YFeO3

M

dG =dX O ¼ X Y ðdGY =dX O Þ−X Y GY =ð1−X O Þ þ X Fe ðdGFe =dX O Þ−X Fe GFe =ð1−X O Þ þ GO þ ðdGO =dX O ÞX O :

ð18Þ

Based on this value of the integral Gibbs energy of mixing at δmax = 0.0251, the molar Gibbs energy of mixing corresponding to the stoichiometric compound (δ=0) can be obtained using Eq. (15) with XO′ =0.59798, XO″ =0.6, (GM)X ′ =−203,973(J/mol) and ∫ XO″GO/(1−

O

Since XO is the only variable characterizing the solid solution, (∂ Gi/∂ Xi) = dGi/dXi. Hence,

ð17Þ

O ΔGr(8)

O

ΔGrð22Þ ð250ÞðJ=molÞ ¼ −17; 126−8:263T:

ð22Þ ð23Þ

The first term of the right-side of this equation corresponds to the enthalpy of formation of the perovskite from component binary oxides O (ΔH(ox) (YFeO3) = −17.13 (±0.68) (kJ/mol)) at an average temperature of 1183 K. Unfortunately, there are no calorimetric data on enthalpy of formation for comparison. The magnitude of the second term is related to the corresponding second-law entropy change for reacO tion (22); (ΔS(ox) (YFeO3) = 8.26 (±0.58)(J/mol K)). It would be interesting to compare the entropy change obtained by the second-law method with that derived from heat capacity measurements based on third law of thermodynamics. Shen et al. [14] determined the heat capacity of single crystal YFeO3 with superconducting quantum interference device magnetometer in the temperature range from 2.5 to 300 K, and a differential scanning calorimeter in the range from 300 to 730 K. A second order phase transition from anti-ferromagnetic to paramagnetic with Neel temperature at 644.5 K is seen in the heat capacity profile. The heat capacity at temperatures below 50 K can be represented by the equation CPO = (3.72559× 10 −5) T 3 + 0.09022 T. The entropy of YFeO3 was calculated by integrating (CPO / T) as

K.T. Jacob, G. Rajitha / Solid State Ionics 224 (2012) 32–40

37

vacancies or cationic interstitials. In the Kröger–Vink notation, oxygen vacancy formation can be written as ··

OO →VO þ 2e′ þ ð1=2ÞO2 :

ð27Þ

If the electrons are free, the non-stoichiometric parameter (α) is proportional to (PO2 −1/6) which is inconsistent with the experimental results. However, if the electrons are localized on the Fe +3 ions, the defect reaction can be represented as: þ3

··

þ2

OO þ 2FeFe →VO þ 2FeFe þ ð1=2ÞO2 :

ð28Þ +2

Fig. 5. The Gibbs energies of formation of nonstoichiometric YFeO3−δ, in equilibrium with Fe and Y2O3, and of stoichiometric YFeO3 from Fe, Y2O3 and O2 gas.

function of temperature. The entropy values at 298.15 K and 730 K are 99.7 (±0.95) (J/mol K) and 210.81 (±1.3)(J/mol K), respectively. The corresponding entropy changes for the formation of YFeO3 from component oxides according to reaction (22) are 6.22 (±1.2)(J/mol K) at 298.15 K and 8.81 (±1.6)(J/mol K) at 730 K. These values are compatible with the value of 8.26 (±0.58)(J/mol K) at 1183 K obtained in this study. 3.2. YFe2O4 3.2.1. Oxygen chemical potential for three-phase equilibrium Fe + YFe2O4 − α + Y2O3 Variation of the EMF of cell (I) with temperature from 1400 to 1475 K depicted in Fig. 3 is given by Eq. (5). The three phases at the working electrode in this temperature range were identified as Fe, YFe2O4 − α and Y2O3. From the EMF, the oxygen chemical potential in the temperature range from 1400 to 1475 K established by the reaction: 2Fe þ fð5−2αÞ=4gO2 þ ð1=2ÞY2 O3 →YFe2 O4−α

ð24Þ

can be calculated as: Δμ O2 ðFe þ Y2 O3 þ YFe2 O4−α Þð193ÞðJ=molÞ

ð25Þ

¼ −553; 245 þ 146:81T:

At 1473 K, oxygen chemical potential for the three-phase equilibrium obtained in this study is 1.09 (kJ/mol) more negative than that reported by Kimizuka and Katsura [8], who used controlled mixtures of CO and CO2 to study this equilibrium.

and Since the structure has a large concentrations of both Fe Fe +3 ions on the cation sublattice, their concentrations will not be affected significantly by the defect reaction and can be considered as constant. Accordingly, the non-stoichiometric parameter (α) will be proportional to (PO−1/2 ) or the slope of the plot in Fig. 6 should 2 be − 0.5. The value obtained in this study is (− 0.48705 ± 0.00951) which is close to the model prediction, thus providing experimental confirmation of the model. If the cation interstitials are responsible for non-stoichiometry, the defect reaction would be: ···

··

···



YY þ FeFeðþ2Þ þ 9FeFeðþ3Þ þ 4OO →Yi þ Fei þ Fei þ 8FeFeð3þÞ þ 2O2 : ð29Þ One Y, two Fe and four O sites are reduced by this reaction. To avoid precipitation of a second phase, it is necessary to have both Y and Fe interstitials. For charge balance eight Fe 3+ ions on the cationic sublattice will become Fe 2+ because of localization of electrons. According to this model, the non-stoichiometric parameter (α) is proportional to (PO−2/3 ), which is inconsistent with the experimental 2 result. Because of the presence of Fe 2+ and Fe 3+ in equal numbers on the same structural site, the average charge on the lattice site is 2.5. Therefore, deviation from Kröger–Vink notation is used in Eq (29). Based on limited data, Kimizuka and Katsura [8] have suggested that the relationship between (αmax − α) and logPO2 is essentially linear. Their relation cannot be justified on any defect model. 3.2.3. Limit of non-stoichiometry of YFe2O4 − α The decomposition oxygen partial pressure of YFe2O4 − α to Fe and Y2O3 at 1473 K computed from Eq. (25) is (PO2/P 0) = 1.122 × 10 −12. The value of oxygen non-stoichiometric parameter (α) corresponding to this oxygen partial pressure, calculated from Eq. (26), is αmax = 0.0885. This compares with a value 0.095 suggested by Kimizuka and Katsura [8].

3.2.2. Variation of non-stoichiometry of YFe2O4 − α with oxygen partial pressure YFe2O4 − α is stable only over a limited range of oxygen potential at temperatures above 1391 K. The thermogravimetric measurements show mass loss with decreasing oxygen partial pressure. The variation of the non-stoichiometric parameter (α) with oxygen partial pressure at 1473 K observed in this study is shown in Fig. 6. The results can be adequately represented by the equation:   0 lnðα Þ ¼ −0:48705 ln P O2 =P −15:826:

ð26Þ

YFe2O4 has a hexagonal crystal structure composed of layered triangular nets of Y–O and Fe–O along the c-axis. On this triangular lattice equal number of Fe +2 and Fe +3 ions are distributed randomly. Oxygen deficiency in YFe2O4 − α can be caused either by oxygen

Fig. 6. Variation of the nonstoichiometric parameter (α) of YFe2O4−α with the partial pressure of oxygen at 1473 K.

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K.T. Jacob, G. Rajitha / Solid State Ionics 224 (2012) 32–40

3.2.4. Gibbs free energy of formation of YFe2O4 − α and YFe2O4 The standard Gibbs free energy change for the formation of nonstoichiometric YFe2O4−α (α=0.0885) at 1473 K according to reaction (24) can be computed from the measured oxygen potential O corresponding to three-phase equilibrium as ΔG(24) =[(5−2α)/4] ΔμO2 =−406,322 (±233)(J/mol). The variation of the nonstoichiometric parameter (α) with pressure has been determined only at 1473 K. Assuming that α does not change significantly in the temperature range from 1391.4 to 1475 K, the temperature dependence of the Gibbs energy change for reaction (24) can be expressed as: O

ΔGrð24Þ ð233ÞðJ=molÞ ¼ 667; 064 þ 177:01T:

ð30Þ

The Gibbs energy of formation of the stoichiometric YFe2O4 can be calculated by Gibbs–Duhem integration (Eq. (15)) using data on the variation of α with oxygen partial pressure. The mole fraction of oxygen (Xo) in YFe2O4 − α is given by, XO ¼ ð4−αÞ=ð7−αÞ:

ð31Þ

Following the same procedure as in the case of YFeO3, the standard Gibbs free energy of formation of stoichiometric YFe2O4 can be calculated. For the reaction, 2Fe þ ð5=4ÞO2 þ ð1=2ÞY2 O3 →YFe2 O4 O

ΔGrð32Þ ð250ÞðJ=molÞ ¼ −680; 810 þ 177:01T:

ð32Þ ð33Þ

Thus, the Gibbs energy of formation of stoichiometric YFe2O4 is 13.745 (kJ/mol) more negative than for the nonstoichiometric YFe2O4 − α in equilibrium with Fe and Y2O3. It is instructive to assess the Gibbs free energy of formation of YFe2O4 from its component binary oxides, represented by the reaction, ð1=2ÞY2 O3 þ FeO þ ð1=2ÞFe2 O3 →YFe2 O4 :

ð34Þ

By combining Eq. (33), with data for FeO and Fe2O3 [13,15], standard free energy change for the reaction is obtained as: O

ΔGrð34Þ ð260ÞðJ=molÞ ¼ −10; 352−13:24T:

ð35Þ

The first term of the right-side of this equation corresponds to the enthalpy of formation of YFe2O4 from component binary oxides O (ΔH(oxi) (YFe2O4) = −10,352 (±1400) (J/mol)) at an average temperature of 1433 K. The second term is related to the corresponding O entropy change: ΔS(ox) (YFe2O4) = 13.24 (±1) (J/mol K). The higher entropy of formation from oxides of YFe2O4 compared to YFeO3 o (ΔS(ox) (YFeO3) = 8.26(±0.58) (J/mol K)) is probably related to the entropy of mixing of Fe 2+ and Fe 3+ ions (ΔS M = 5.76 J/mol K) on the Fe site.

The uncertainty estimate is based on standard deviation from regression analysis of EMF and error in temperature measurement. At 1373 K, the oxygen partial pressure, PO2/P 0 = 4.83 × 10 −8. The value of the perovskite nonstoichiometric parameter δ under these conditions is 0.0025, which is almost one‐tenth of its maximum value. Hence, the Gibbs free energy of YFeO3 − δ at this oxygen partial pressure can be considered to be very close to that of stoichiometric YFeO3. As shown in Fig. 8, the oxygen chemical potential for the three-phase equilibrium obtained in this study is significantly more negative than values reported earlier [5,7,8]. Non-optimal experimental factors in potentiometric measurements generally result in lower emf and higher oxygen potential when measured against air or pure oxygen. 3.3.2. Variation of non-stoichiometry of Y3Fe5O12 − θ with oxygen partial pressure The variation of the non-stoichiometric parameter (θ) of YIG with oxygen partial pressure at 1373 K observed in this study is shown in Fig. 9. Regression analysis gives:   0 lnðθÞ ¼ −0:1906 ln P O2 =P −7:272

where P 0 is the standard atmospheric pressure. There are three types of cation sites in Y3Fe5O12 − θ. Yttrium is dodecahedrally coordinated by eight oxygens in an irregular cube, two Fe 3+ ions are in tetrahedrally (4-fold) coordinated sites, and three remaining Fe 3+ ions are in octahedrally (6-fold) coordinated sites. The oxygen deficiency may be caused by either oxygen vacancies or metal interstitials. In a loose structure such as that of garnets, metal interstitials may be preferred over vacancies [16]. If vacancies are the predominant defects, the defect reaction is given by Eq. (27), from which it follows that the plot of ln θ versus ln (PO2/P 0) will be a straight line with a slope of − 1/6 (− 0.1667). The same relation will hold irrespective of whether the electrons are localized on Fe 3+ ions or not, since there are no Fe 2+ ions in the structure and the concentration of Fe 2+ ions formed by electron localization would be proportional to vacancy concentration. Based on the general preference of Fe 2+ ions for octahedral coordination compared to Fe 3+, the electron localization will probably occur on the octahedral cation site. When metal interstitials are predominant, the defect formation can be represented as: ···

···

3YY þ 5FeFe þ 12OO →3Yi þ 5Fei þ 24e′ þ 6O2 :

3.3.1. Oxygen chemical potential for three-phase equilibrium YFeO3 − δ + Fe3O4 + Y3Fe5O12 − θ The EMF between the reference and the buffer electrodes of cell (II) was lower by 17 to 19 mV than that between the reference and the working electrodes, indicating that there was significant polarization of the three-phase buffer electrode. The variation of the reversible EMF of cell (II) with temperature from 975 to 1475 K, shown in Fig. 7, can be represented by the equation: ð36Þ

The oxygen potential for the three-phase equilibrium calculated from EMF is: Δμ O2 ðYFeO3−δ þ Fe3 O4 þ Y3 Fe5 O12−θ Þð240ÞðJ=molÞ ¼ −511; 448 þ 232:44T:

ð39Þ

It is necessary to have both cations in interstitial positions in the stoichiometric ratio to prevent the precipitation of a second

3.3. Y3Fe5O12

EðIIÞ ð0:21ÞðmVÞ ¼ 1325:2−0:60227T:

ð38Þ

ð37Þ Fig. 7. Variation of the EMF of cell (II) with temperature.

K.T. Jacob, G. Rajitha / Solid State Ionics 224 (2012) 32–40

39

4.83× 10−8, calculated from Eq. (38) is θmax = 0.0172. This compares with an approximate value of 0.01 suggested by Kitayama et al. [7]. 3.3.4. Gibbs free energy of formation of Y3Fe5O12 − θ and Y3Fe5O12 The standard Gibbs energy of formation of nonstoichiometric Y3Fe5O12 − θ according to the reaction, 3YFeO3−δ þ ð2=3ÞFe3 O4 þ ½ð1−3θ þ 9δÞ=6O2 →Y3 Fe5 O12−θ

ð42Þ

can be computed from the EMF of cell (II): O

ΔGrð42Þ ð40Þ ðJ=molÞ ¼ ½ð1–3θ þ 9δÞ=6Δμ O2

ð43Þ

¼ −½ð2–6θ þ 18δÞ=3FEII :

Fig. 8. Comparison for the oxygen potential for the equilibrium involving the three condensed phases (YFeO3−δ + Fe3O4 + Y3Fe5O12−θ) measured in this study with the data reported in the literature.

phase. The equilibrium constant for this defect reaction can be written as: ··· 3 ··· 5  0 24 6 0 ··· 32 6 K ¼ ½Yi  ½Fei  e P O2 ¼ k ½Yi  ∙P O2

ð40Þ

since [Fei···] and [e′] are proportional to [Yi···]. Therefore, ···



3=16

½Y i ∝θ∝P O2

:

Since values of θ and δ are known at 1373 K, the Gibbs energy O = change can be obtained exactly at this temperature; ΔGr(42) − 31,119 (J/mol). The Gibbs energy of formation of the stoichiometric compound Y3Fe5O12 at this temperature can be calculated using Gibbs–Duhem integration (Eq. (15)) along with data on the variation of nonstoichiometric parameter (θ) with oxygen partial pressure using a procedure identical to that described in Section 3.1.4 for YFeO3. For the formation of Y3Fe5O12 according to the reaction, 3YFeO3 þ ð2=3ÞFe3 O4 þ ð1=6ÞO2 →Y3 Fe5 O12

ð44Þ

O ΔGr(44)

ð41Þ

The plot of ln θ as a function of ln (PO2/P 0) will have a slope of − 3/16 or − 0.1875. The experimentally determined slope of − 0.1906 (±0.0086) is consistent with the metal interstitial model for oxygen deficiency. The presence of interstitials provides an explanation for the unusual behavior of the magnetic disaccommodation spectrum observed in YIG, where accommodation instead of disaccommodation, is detected below 130 K [17]. Metselaar and Huyberts [18] studied nonstoichiometry of doped and undoped YIG single crystal samples using thermogravimetry in a range of oxygen partial pressures 10 −5 b (PO2/P 0) b 1 at temperatures from 1223 to 1543 K. They attributed nonstoichiometry to oxygen vacancies although their data does not unambiguously support the conclusion. Their results appear to be influenced by impurities in YIG crystals grown from PbO–PbF2–B2O3 melt. 3.3.3. Limit of oxygen non-stoichiometry of Y3Fe5O12 − θ The value of oxygen non-stoichiometric parameter (θ) at 1373 K, corresponding to the decomposition oxygen partial pressure PO2/P 0 =

(±50)(J/mol) =−31,858. This value is 739(J/mol) more negative than that for the nonstoichiometric compound. The difference is small, but significant. Since nonstoichiometry is insignificant at 975 K, equations for the temperature dependence of the Gibbs energy change for reactions (42) and (44) can be written as: O

ð45Þ

O

ð46Þ

ΔGrð42Þ ð40ÞðJ=molÞ ¼ −87; 512 þ 41:073T ΔGrð44Þ ð50ÞðJ=molÞ ¼ −85; 714 þ 39:225T:

It is useful to compute the Gibbs energy of formation of Y3Fe5O12 from component binary oxides Y2O3 and Fe2O3. For the oxidation of Fe3O4 to Fe2O3 according to the reaction, ð2=3ÞFe3 O4 þ ð1=6ÞO2 →Fe2 O3

ð47Þ

the standard Gibbs energy change from Sundman [13] is: O

ΔGrð47Þ ð150ÞðJ=molÞ ¼ −80; 445 þ 45:527T:

ð48Þ

Combining reactions (22), (44) and (47), and Eqs. (23), (46) and (48) one obtains for, ð3=2ÞY2 O3 þ ð5=2ÞFe2 O3 →Y3 Fe5 O12

ð49Þ

O ΔGrð49Þ ð780ÞðJ=molÞ

ð50Þ

¼ −56; 647−31:091T:

The second-law enthalpy of formation of Y3Fe5O12 from component oxides at a mean temperature of 1225 K is −56,647(J/mol) and the corresponding entropy of formation is 31.091(J/mol.K). Invoking the Neumann–Kopp rule, the enthalpy of formation at 298.15 K can be obtained as −56.65(kJ/mol) and standard entropy as 219.17 (J/mol.K). Calorimetric measurement of the heat capacity as a function of temperature and enthalpy of formation of Y3Fe5O12 can confirm the data obtained in this study. 4. Conclusions

Fig. 9. Variation of the nonstoichiometric parameter (θ) of Y3Fe5O12 −θ with the partial pressure of oxygen at 1373 K. P0 is the standard atmospheric pressure.

Three ternary oxides YFeO3, YFe2O4 and Y3Fe5O12 were identified in the system Y–Fe–O in the temperature range from 975 to 1475 K; YFe2O4 is stable only at temperatures above 1391 K and under reducing conditions. All the three oxides are oxygen deficient at high temperature. The variation of the oxygen nonstoichiometry was mapped as function of oxygen partial pressure at 1373 K for YFeO3 − δ and

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Y3Fe5O12 − θ and at 1473 K for YFe2O4 − α using thermogravimetry. Vacancies on the anionic sublattice were identified as the predominant defects responsible for nonstoichiometry of YFeO3−δ and YFe2O4−α. Interstitial cations were found to be the major defects in Y3Fe5O12−θ. Gibbs free energies of formation of the oxygen-deficient compounds were measured in the temperature in the range from 975 to 1475 K using specially designed solid-state electrochemical cells, which prevented polarization of electrodes containing three condensed phases. Change in the slope of oxygen chemical potential for three-phase equilibrium involving Fe + YFeO3 − δ + Y2O3 as a function of temperature suggested the formation of YFe2O4 − α above 1391 K. Examination of rapidly cooled electrode by SEM, EDS and XRD confirmed the formation of YFe2O4 at high temperature. The standard Gibbs free energies of formation of stoichiometric ternary oxides were computed by coupling information on nonstoichiometric compositions from EMF data with that from thermogravimetry through the Gibbs–Duhem relation. All the three ternary oxides exhibited positive entropies of formation from their corresponding binary oxides. For YFeO3 the entropy derived in this study is consistent with calorimetric data on low and high-temperature heat capacities. The mixing of Fe 2+ and Fe 3+ on the Fe sublattice appears to result in marginally higher entropy of formation of YFe2O4. The results obtained in this study provide a major improvement in the understanding of defect chemistry of ternary oxides with important applications and refine thermodynamic data for these compounds. Acknowledgments The authors thank Mr. Sudarshan Mukherjee for computational assistance. K.T.J. is grateful to the Indian National Academy of Engineering,

New Delhi, for the award of INAE Distinguished Professorship. G.R. thanks the University Grants Commission of India for the award of Dr. D.S. Kothari Postdoctoral Fellowship.

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