Thermodynamic Assessment of the La-Cr-O System

Abstract

The La-Cr and the La-Cr-O systems are assessed using the Calphad approach. The calculated La-Cr phase diagram as well as LaO_1.5-CrO_1.5 phase diagrams in pure oxygen, air, and under reducing conditions are presented. Phase equilibria of the La-Cr-O system are calculated at 1273 K as a function of oxygen partial pressure. In the La-Cr system reported solubility of lanthanum in bcc chromium is considered in the modeling. In the La-Cr-O system the Gibbs energy functions of La₂CrO₆, La₂(CrO₄)₃, and perovskite-structured LaCrO₃ are presented, and oxygen solubilities in bcc and fcc metals are modeled. Emphasis is placed on a detailed description of the perovskite phase: the orthorhombic to rhombohedral transformation and the contribution to the Gibbs energy due to a magnetic order-disorder transition are considered in the model. The following standard data of stoichiometric perovskite are calculated: $\Updelta_{\text{f,oxides}} {^{\circ }H}_{{298{\rm K}}} ( {\text{LaCrO}}_{ 3} )= - 7 3. 7 {\text{ kJ}}\,{\text{mol}}^{-1}$, and ${}^{\circ } S_{{298\,{\text{K}}}} ( {\text{LaCrO}}_{ 3} )= 109.2{\text{ J}}\,{\text{mol}}^{ -1}\, {\text{K}}^{ -1}$. The Gibbs energy of formation from the oxides, $\Updelta_{\text{f,oxides}} {^{\circ } G} ( {\text{LaCrO}}_{ 3} )= -72.403 - 0.0034$ T (kJ mol⁻¹) (1273-2673 K) is calculated. The decomposition of the perovskite phase by the reaction ${\text{LaCrO}}_{3} \to \frac{1}{2}{\text{La}}_{2} {\text{O}}_{3} + {\text{Cr}} + \frac{3}{4}{\text{O}}_{2} ({\text{g}}) \uparrow$ is calculated as a function of temperature and oxygen partial pressure: at 1273 K the oxygen partial pressure of the decomposition, $p_{{{\text{O}}_{{\text{2}}} {\text{(decomp)}}}} = 10^{-20.97}\,{\text{Pa}}$. Cation nonstoichiometry of La_1–x CrO₃ perovskite is described using the compound energy formalism (CEF), and the model is submitted to a defect chemistry analysis. The liquid phase is modeled using the two-sublattice model for ionic liquids.

Introduction

In solid oxide fuel cells (SOFC), the thermodynamic stability of the cathode is particularly important for efficient long-term operation. Sr-doped lanthanum manganites (LSM) with the perovskite structure are used as cathode materials in SOFC. Diffusion of chromium from the metallic interconnect with high chromium content into the cathode leads to the formation of Mn(Cr,Mn)₂O₄ spinel and Cr₂O₃ along with a severe cell voltage decrease.[1-4] As the thermal expansions of LaCrO₃-based interconnect and conventional perovskite cathode materials are similar, and Cr-diffusion into the cathode from LaCrO₃-based interconnects is significantly lower than from Cr-containing metallic interconnects, recently Sr-, V-doped[5] and Zn-doped[6] La_1–x Ca _x CrO_3–δ have been considered as promising alternative interconnect materials for SOFC. Furthermore alkaline earth containing LaCrO₃ has been proposed as a cathode material in a recent study by Jiang et al.[7]

The present thermodynamic assessment of the La-Cr-O system lays the groundwork for extension to a thermodynamic La-Sr-Mn-Cr-O oxide database that is required to understand the thermodynamics of SOFC degradation by chromium. It is also a starting point for extensions to thermodynamic databases with additional components serving as dopants in LaCrO₃ for SOFC interconnect and cathode applications.

The assessment of the La-Cr-O system using the Calphad approach is based on the recently reassessed La-O[8] and Cr-O subsystems.[9] The lattice stabilities of elements are adopted from Dinsdale.[10] All available experimental phase diagram, thermodynamic, and structure-chemical data are critically assessed, aiming at minimizing the squared errors between experiments and calculation during the optimization of model parameters using the PARROT module of the Thermocalc[11] software.

Literature Review of the La-Cr System

The La-Cr system has a eutectic at 1138 K[12,13] and 3.4 at.% Cr[13] and a monotectic at 1983 K[12] or 2103 K[14] and 96 at.%[12] or 99.1 at.%[14] Cr, as well as a large liquid-liquid miscibility gap.[12,13] No intermetallic phases were found in the La-Cr system.[12,13]

Berezutskii et al.[15] determined the partial enthalpy of mixing in La-Cr liquid with infinite dilution of Cr, $\Updelta \bar{H}_{\text{Cr}}$ at 1700 K using high-temperature calorimetry.

As small additions of rare-earth metals essentially increase the high-temperature corrosion resistance of chromium,[16] modeling of the La-solubility in bcc-structured Cr, denoted as αCr_ss, is of technological interest. Small solubility of La in αCr_ss was reported,[12,14,17] whereas Cr is almost insoluble in La.[13] The solubility of La in αCr_ss was determined in investigations by Savitskii et al.[12] from 1073 K up to the melting of Cr using metallographic and micro-hardness techniques to be 2.5 at.% at 1983 K. Svechnikov et al.[14] reported a La solubility of 0.68 at.% at 2103 K, and Epstein et al.[17] found La < 0.04 at.% in αCr_ss at 1533 K. The solubility of La in αCr_ss decreases toward lower temperatures.

Literature Review of the La-Cr-O System

In the LaO_1.5-CrO_1.5 system two eutectics were found at 19 at.% Cr₂O₃ (T = 2243 K)[18] or 12 at.% Cr₂O₃ (T = 2323 ± 20 K),[19] and at 84 at.% Cr₂O₃ (T = 2248 K)[18] or 80 at.% Cr₂O₃ (T = 2473 ± 20 K)[19] in argon atmosphere on either side of the congruently melting perovskite-structured lanthanum chromite[18-20] (in this study oxides containing Cr(III) and Cr with higher valencies than three are denoted as chromite and chromate respectively). The melting temperature of lanthanum chromite in air, T _m(air) = 2773 K was determined by Foëx[21] and by Coutures et al.[20] using a thermal analysis technique described in more detail in earlier publications.[22-24] The melting temperature was measured with optical pyrometers. T _m(argon) = 2703 K was reported by Tresvjatskiy et al.,[18] but in the graphic presentation of the phase diagram in the same paper T _m(argon) ≈ 2600 K, and the exact value of the oxygen partial pressure was not specified. Experimentally determined special points in the LaO_1.5-CrO_1.5 quasibinary system reveal a considerable spread. This is not surprising as experiments are complicated due to the high investigation temperatures and evaporation predominantly of Cr.[25,26] Furthermore deviations between the data from Tresvjatskiy et al.[18] and Berjoan[19] may partly originate from differences of the oxygen partial pressure, which in both studies was not specified exactly. The peritectic phase diagram proposed by Cassedanne[27] is in gross conflict with the phase diagram data from the other groups.

Experimental oxygen solubilities in pure Cr and La were considered in thermodynamic assessments by Povoden et al.[9] and Grundy et al.,[28] but experiments on oxygen solubilities in αCr_ss are missing.

Lanthanum Chromates

The following lanthanum chromates were documented: Berjoan[19] reported that orthorhombic La₂CrO₆ forms at T > 923 K. Using differential scanning calorimetry (DSC) he determined the enthalpy change of the reaction

$$ 2{\text{La}}_{2} {\text{O}}_{3} + {\text{Cr}}_{2} {\text{O}}_{3} + \frac{3}{2}{\text{O}}_{{2({\text{g}})}} \to 2{\text{La}}_{2} {\text{CrO}}_{6} $$

(1)

at 1055 K and $p_{{{\text{O}}_{2}}}=83,000\,{\text{Pa}}$ to be −151 ± 8 kJ mol⁻¹.

The enthalpy of formation of La₂(CrO₄)₃ from the elements at 298 K was proposed by Tsyrenova et al.[29] to be −3961 ± 11.7 kJ mol⁻¹. La₂(CrO₄)₃ decomposes by

$$ {\text{La}}_{2} ({\text{CrO}}_{4} )_{3} \mathop{\rightarrow}^{{{890 - 1030{\text{K}}}}}2{\text{LaCrO}}_{3} + 0.5{\text{Cr}}_{2} {\text{O}}_{3} + 2.25{\text{O}}_{{2({\text{g}})}} \uparrow $$

(2)

An enthalpy change of 231 kJ mol⁻¹ was determined for this reaction at the average temperature of 960 K.[30]

LaCrO₄ has been interpreted as a mixed-valent intermediate decomposition product of La₂(CrO₄)₃.[30,31]

Stoichiometries and thermal stability ranges of lanthanum chromates with complex formulae were reported by Berjoan et al.[32] However these were in significant disagreement with later results obtained by the same author.[19]

The Perovskite Phase

Existing experimental data of lanthanum chromite perovskite structure,[33-45] thermodynamics,[30,33-35,43,46-53] phase stability,[54] and nonstoichiometry[55,56] along with the investigation techniques used are listed in Table 1.

Table 1 Calculated and experiment thermodynamic data of La-Cr oxides

Crystal and Magnetic Structure

LaCrO₃ is orthorhombic at room temperature and transforms to rhombohedral structure at higher temperatures.[20,33-42] Both the rhombohedral and orthorhombic forms of LaCrO₃ are slight distortions of the perovskite crystal structure. The perovskite structure is primitive cubic and with the formula ABX₃ and has an A atom at the unit cell origin, (0,0,0), a B atom at the body-centered position (½, ½, ½), and three X atoms at the face-centered positions, (½, ½, 0; ½, 0, ½, ½, 0, ½, ½). The body diagonal of the unit cell, [111], exhibits threefold rotational symmetry and has a magnitude of 3 ½ [a] where a is the lattice parameter. Any deviation of this dimension away from 3 ½ [a] retains the threefold rotational symmetry and the equality of the three lattice parameters, but reduced the angles between the lattice parameters to less than 90° for a length >3 ½ [a] or greater than 90° for a length <3 ½ [a]. Thus the symmetry in either case becomes rhombohedral. Similarly, if the perovskite until cell is distorted in such a way that one lattice parameter is shortened, a second is lengthened, and the third remains the same with retention of the 90° interaxial angles, the symmetry becomes orthorhombic. For both distortions, the atomic loci are retained with La at the cell origin, Cr at the body-centered site, and O at the three-face centered positions.[57] In the present paper, the Gibbs energy expressions for the two different forms are distinguished by superscripting with r-prv or o-prv.

The temperatures, enthalpy and entropy changes of this first-order[44] transition taken from the literature are listed in Table 2 along with the investigation techniques used. The reported transformation temperatures lie between 503 and 583 K. The determined enthalpy and entropy changes vary from 277 to 502.08 J mol⁻¹ and 0.5 to 0.96 J mol⁻¹ K⁻¹. A transformation from rhombohedral to cubic structure at a temperature close to 1300 K was reported by Ruiz et al.[37] and Momin et al.,[41] whereas Coutures et al.[20] reported 1923 K using high-temperature X-ray diffraction (HT-XRD), in agreement with Berjoan[19] (1923 ± 20 K) using dilatometry. Berjoan[19] further reported prevailing of the cubic structure at 2173 K. On the other hand Geller and Raccah[38] as well as Höfer and Kock[34] did not observe the rhombohedral to cubic transition up to T = 1873 K and T = 1823 K respectively using differential thermal analysis (DTA).

Table 2 Calculated and experimental data of the orthorhombic → rhombohedral transition of LaCrO₃

A magnetic order-disorder transition was documented to occur at T ≈ 287 K,[35] 289 K,[45] or 295 K.[46]

Enthalpy of Formation

Cheng and Navrotsky[47] determined the enthalpy of formation of LaCrO₃ by oxide melt solution calorimetry at 1078 K.

Heat Capacity and Enthalpy Increment Data

The heat capacities of LaCrO₃ were measured by Korobeinikova and Reznitskii[33] from 340 to 900 K using adiabatic calorimetry, Höfer and Kock[34] (480-610 K) and Satoh et al.[45] (150-450 K) using DSC, Satoh et al.[45] (77-280 K) using alternating current calorimetry, Sakai et al.[35] (100-1000 K) using laser-flash calorimetry, and Sakai and Stølen[43] (272-1000 K) using adiabatic shield calorimetry. Enthalpy increments of LaCrO₃ at 1090 and 1350 K were measured by Suponitskii[30] using a high-temperature heat-conducting calorimeter.

Gibbs Energy of Formation

In order to obtain the Gibbs energy of formation of LaCrO₃, Chen et al.[49] measured electromotive force (emf) of the solid oxide galvanic cell Pt/Cr, La₂O₃, LaCrO₃/MgO-stabilized ZrO₂/Cr₂O₃, Cr/Pt at 1273 K. Azad et al.,[50] Chen et al.,[51] and Dudek et al.[52] measured emf of Pt, O₂/La₂O₃, LaF₃/CaF₂/LaF₃, LaCrO₃, Cr₂O₃/O₂, Pt in pure oxygen from 855 to 1073 K, 700 to 885 K, and 1273 K respectively. Peck et al.[53] derived the Gibbs energy of formation of LaCrO₃ from the determination of the thermodynamic activity of Cr₂O₃ in LaCrO₃ for the Cr₂O₃-poor phase boundary of LaCrO₃ in the temperature range from 1887 to 2333 K using Knudsen effusion mass spectrometry.

Chemical Stability

Nakamura et al.[54] reported no weight loss of lanthanum chromite at 1273 K from pure oxygen to $p_{{{\text{O}}_{2}}}=10^{-16.1}$ Pa using thermogravimetry combined with X-ray diffraction (XRD). This means that the perovskite phase does not decompose within this oxygen partial pressure range, and its oxygen nonstoichiometry is neglectable.

Cation Nonstoichiometry and Defect Chemistry

Maximum excess Cr in single-phase La_1–x CrO₃ of 0.38 at.% in furnace-cooled LaCrO₃ annealed at 1773 K in air was reported from Khattak and Cox.[55] Single phase lanthanum chromite with 0.76-1.28 at.% excess Cr was prepared at 1773 K in a pure oxygen atmosphere.[56] Iliev et al.[56] observed an intensity decrease of the high frequency band in a Raman spectrum of lanthanum chromite measured after annealing the phase in vacuum at 1273 K. This feature was assigned to a reduced number of Cr⁴⁺ due to partial removal of oxygen during the annealing of the originally lanthanum-deficient perovskite phase.

Interpretations of the defect chemistry of the perovskite phase were made from electrical conductivity measurements: the electrical conduction in lanthanum chromite is almost purely electronic,[37,58] affirming the lack of oxygen vacancies in the structure, in line with the results from thermogravimetry.[54] Ruiz et al.[37] reported that the ionic transport number in lanthanum chromite is less than 0.05% up to 1250 K. Akashi et al.[59] measured the isothermal electrical conductivity of an equilibrated La_1–x CrO₃-Cr₂O₃ mixture with 5 vol.% excess Cr₂O₃ at 1573-1673 K from $p_{{{\text{O}}_{2}}}=1.0 \times 10^{3}\,{\text{Pa}}$ to $p_{{{\text{O}}_{2}}}=2.0 \times 10^{4}\,{\text{Pa}}$. They observed an extraordinarily slow equilibration of the samples: More than four months were required to measure the electrical conductivity at equilibrium state. The conductivity was proportional to $p_{{{\text{O}}_{{\text{2}}} }}{ ^{{{3 \mathord{\left/ {\vphantom {3 {16}}} \right. \kern-\nulldelimiterspace} {16}}}} }$, the same as reported in an earlier study.[25] On the other hand a slope of $p_{{{\text{O}}_{ 2} }}{^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}}}}$ from 700 to 1300 K and purely intrinsic conductivity >1600 K stated by Shvaiko-Shvaikovskii et al.[58] is inconsistent with the findings from Akashi et al.[58] Shvaiko-Shvaikovskii et al.[58] deduced n-type conductivity from measurements of transport number, resistivity and thermo-emf at $p_{{{\text{O}}_{2}}}=1\,{\text{Pa}}$ and $p_{{{\text{O}}_{2}}}=10^{2}\,{\text{Pa}}$, the electrical conductivity being proportional to $p_{{{\text{O}}_{2}}}^{{{{ - 3} \mathord{\left/ {\vphantom {{ - 3} 8}} \right. \kern-\nulldelimiterspace} 8}}}$. The transition from reduced to stoichiometric chromite was accompanied by a decrease of about 0.1% in weight, thus the presence of interstitial Cr in reduced chromite was proposed. However n-type conductivity was not confirmed by any further study.

Several groups[59,60] agree that the electrical neutrality is maintained by holes and lanthanum vacancies, and that the carrier is the hole in lanthanum chromite.[25,59-61] Akashi et al.[59] reported that concentrations of lanthanum vacancies and holes slightly increase from 1550 to 1700 K. In contrast to the other authors Shvaiko-Shvaikovskii et al.[58] and Meadowcroft[25] proposed the occurrence of chromium vacancies instead of lanthanum vacancies.

Thermodynamic Modeling and Optimization

Metal Phases

In order to account for the solubility of La in αCr_ss, the zeroth-order, composition-independent interaction parameter[62] ${}^{0}L_{\text{Cr,La:Va}}^{\text{bcc}}$ was given a positive value. We chose the solubility values from Svechnikov et al.[14] for its optimization, as these data are more comparable to solubilities in other rare earths-transition elements systems.

Povoden et al.[9] described the solubility of oxygen in Cr(bcc) using the model Cr(Va,O)₃. For the reasons discussed recently,[63] we reassess the oxygen-solubility in Cr(bcc) using the model (Cr)(O,Va)_1.5, and αCr_ss is then given by the two-sublattice description (La,Cr)(Va,O)_1.5. The Gibbs energy of the end-member (Cr)(O)_1.5 is defined as

$$ {}^{\circ }G_{{ ( {\text{Cr)(O)}}_{ 1. 5} }} - H_{{{\text{Cr}} }}^{\text{SER}} - \frac{3}{2}H_{\text{O}}^{\text{SER}} = {}^{\circ}G_{\text{Cr(bcc)}}{^{[10]}} + \frac{3}{4}{}^{\circ }G_{{{\text{O}}_{ 2} }}^{{{\text{gas}}[10]}} + A + BT $$

(3)

$H_{{^{x} }}^{\text{SER}}$ is the standard enthalpy of the stable state of element x at 298.15 K and 10⁵ Pa.[10] A and B are adjustable parameters; using the PARROT module of the Thermocalc software[11] A was given the fixed value 0 for the reasons discussed in an earlier paper,[9] while B and a regular interaction ${\text{parameter}}\;^{0} L_{\text{Cr:O,Va}}$were optimized with the same experimental data.[9] Due to the lack of experimental data the oxygen solubility in αCr_ss was modeled as an ideal extension of the oxygen solubilities in pure La and Cr.

Solid Oxides

Lanthanum Chromates

The Gibbs energy function of La₂CrO₆ was based on the sum of the Gibbs energy functions of La₂O₃, Cr₂O₃, and O₂ in proper stoichiometries and A + BT parameters that were fitted to the enthalpy of formation from the oxides, Eq 1 as well as thermal stability data. The thermal stability of La₂CrO₆ is slightly influenced by the thermodynamics of the intermediate, mixed-valent chromates mentioned above. In order to refine the model parameters of La₂CrO₆, it was thus necessary to consider these mixed-valent chromates in a provisional version of the thermodynamic La-Cr-O database in spite of their arguable stoichiometries, and to optimize their model parameters with phase diagram data.[19,32] The formation of chromates that contain mixed Cr valences may be explained by gradual reduction of Cr⁶⁺ in La₂CrO₆ as the temperature increases. These chromates can be interpreted as intermediate products in the scope of a sluggish decomposition of La₂CrO₆, which starts at 1153 K[19,32] and is completed at 1473 K[32] or 1523 K.[19] The simplified decomposition reaction reads

$$ {\text{La}}_{2} {\text{CrO}}_{6} \mathop{\rightarrow}^{{{\text{mixed}}{\text{-}}{\text{valent}}\,{\text{chromates}}}^{[19,32]}} \frac{{1 +x}}{2}{\text{La}}_{2} {\text{O}}_{3} + {\text{La}}_{1 - x} {\text{CrO}}_{3} + \frac{1.5 - (1.5x)}{2}{\text{O}}_{2} ({\text{g}}) \uparrow $$

(4)

Slight differences of the oxygen partial pressure during experiments may be reflected by a variable extent of Cr-reduction, and consequently ambiguous stoichiometries of mixed-valent intermediate chromates. These lanthanum chromates with conflicting stoichiometries[19,32] are not included in the presented thermodynamic database.

The Gibbs energy function of La₂(CrO₄)₃ was formulated using the same strategy as for La₂CrO₆. The model parameters were fitted to the experimental enthalpy and temperature of decomposition.[30] The enthalpy of formation from the elements[29] was not used as it is a calculated value.

We go along with the interpretation of LaCrO₄ being an intermediate reaction product during the decomposition of La₂(CrO₄)₃ by Eq 2 and do not include this phase in the modeling.

The Perovskite Phase

GVCR4O and GLCR4O stand for the Gibbs energy functions of the completely oxidized neutral endmember. GVCR4O and GLCR4O are set equal for orthorhombic and rhombohedral perovskite at the transition. Stoichiometric perovskite: The Gibbs energy function of stoichiometric rhombohedral LaCrO₃ with the sublattice formula (La³⁺)(Cr³⁺)$({\text{O}}^{2-}),\,^{\circ}G_{{{\text{LaCrO}}_{ 3} }}^{{\text{r}}{\text{-}}{\text{prv}}}$is given by

$$ \begin{aligned} {}^{\circ }G_{{{\text{LaCrO}}_{ 3} }}^{{\text{r}}{\text{-}}{\text{prv}}} - H_{\text{La}}^{\text{SER}} - H_{\text{Cr}}^{\text{SER}} - 3H_{\text{O}}^{\text{SER}}& = {}^{\circ }G_{{{\text{La}}^{{ 3 + }} : {\text{Cr}}^{{ 3 + }} : {\text{O}}^{ 2- } }} = {\text{GRPRV}} \\ &= \frac{{1}}{{2}}{}^{\circ }G_{{{\text{Cr}}_{ 2} {\text{O}}_{ 3} }}{^{[9]}} + \frac{{1}}{{2}}{}^{\circ }G_{{{\text{La}}_{ 2} {\text{O}}_{ 3} }}{^{[8]}} + G_{\text{mag}} + A + BT + CT\ln T \\ \end{aligned} $$

(5)

The parameters A, B, and C are optimized using the enthalpy of formation from Cheng and Navrotsky,[47] activity-data of Cr₂O₃ in LaCrO₃ from Peck et al.,[53] heat capacity-data obtained by adiabatic calorimetry from Sakai and Stølen,[35] and enthalpy increment-data measured at high temperatures.[30] A phase diagram with congruent melting of lanthanum chromite and two eutectics[18,19] cannot be reproduced by using the emf-experiments.[49-52] Thus these data were excluded from the optimization.

A + BT parameters of the low-temperature orthorhombic perovskite phase were optimized with those temperatures,[34,35,38,40,42] enthalpies[34,35,40] and entropies[34,35] of transition having been obtained by combined investigation techniques and being internally most consistent. The rhombohedral to cubic transformation at high temperatures is not considered in the model, as there is no existing thermodynamic data for this transition.

Cation-nonstoichiometric perovskite: to choose a proper model for nonstoichiometric perovskite the following considerations are made: the formation of interstitial Cr in lanthanum chromite proposed by Shvaiko-Shvaikovskii et al.[58] is unlikely due to the densely-packed perovskite structure, and oxygen nonstoichiometry can be excluded from thermogravimetry[54] and electrical conductivity[37,59] measurements. Thus the defects in n-type conducting[58] lanthanum chromite are ambiguous and were not considered in the model.

B-site vacancies are energetically less favored than A-site vacancies in the perovskite structure.[64,65] This means that the simplest sublattice model to describe cation nonstoichiometric La_1–x CrO₃ reads (La³⁺,Va)(Cr³⁺,Cr⁴⁺)(O²⁻)₃. While this model results in a satisfying reproduction of experimental data, irreconcilable trouble is encountered at the extension to the LaO_1.5-MnO_1.5-CrO_1.5 system required for SOFC applications due to diversities between the model descriptions of lanthanum chromite and lanthanum manganite.[66] These are solved by allowing Va on the B-site and the anion sublattice of lanthanum chromite just like in lanthanum manganite[66] leading to the appropriate sublattice formula (La³⁺,Va)(Cr³⁺,Cr⁴⁺,Va)(O²⁻,Va)₃. The optimization of selective model parameters listed in Table 3 resulted in negligible concentrations of Va on the B-site and the anion sublattice, and the perovskite formula essentially remains La_1–x CrO₃. Using the compound energy formalism (CEF)[67-69] the molar Gibbs energy of La_1–x CrO₃ reads

$$ {}^{\circ }G_{\text{m}}^{\text{prv}} = \sum\limits_{i} {\sum\limits_{j} {\sum\limits_{k} {y_{i} y_{j} y_{k} } } {}^{\circ }G_{i:j:k} } + RT\left( {\sum\limits_{i} {y_{i} \ln y_{i} } + \sum\limits_{j} {y_{j} \ln y_{j} } + \sum\limits_{k} {y_{k} \ln y_{k} } } \right) + {}^{\text{E}}G_{\text{m}}^{\text{prv}} + G_{\text{mag}} $$

(6)

where y _i is the site fraction of Va and La³⁺ on the A-sublattice, y _j is the site fraction of Cr³⁺, Cr⁴⁺ and Va on the B-sublattice, and y _k is the site fraction of O²⁻ and Va on the anion sublattice of the perovskite A_1–x BO₃. R = 8.31451 J mol⁻¹ K⁻¹. The third-last term accounts for the configurational entropy of mixing. The second-last term describes the excess Gibbs energy of mixing due to interactions of ions in the mixture. These are accounted for by the optimization of interaction parameters. The last term designates the magnetic contribution to the Gibbs energy. For the magnetic part of the Gibbs energy a magnetic ordering-model proposed by Inden[70] and simplified by Hillert and Jarl[71] was used. A short summary of this model can be found in Chen et al.[72] The magnetic parameters T _c and β were fitted to the C _p -data around the magnetic transition temperature from Sakai and Stølen.[35]

Table 3 Model descriptions and Gibbs energy functions (a)

Figure 1 is a visualization of the Cr-containing part of the model the authors use to describe the cation nonstoichiometry of lanthanum chromite. The parameters of the compound energy formalism are the Gibbs energies of the not necessarily neutral 12 end-member compounds ${}^{\circ }G_{i:j:k}$, with the 8 Cr-containing compounds being the corners of the cube. Only compounds inside the neutral plane can exist on their own.

Fig. 1

Representation of the Cr-containing part of the model for nonstoichiometric lanthanum chromite. The thin lines margin the neutral plane. The neutral compounds used for the optimization are marked by the black spots

${}^{\circ }G_{{({\text{La}}^{3 + } )({\text{Cr}}^{4 + } )({\text{O}}^{2 - } )_{3} }}^{\text{prv}}$ and ${}^{\circ }G_{{({\text{La}}^{3 + } )({\text{Cr}}^{4 + } )({\text{Va}})_{3} }}^{\text{prv}}$ are given in Table 3. These endmembers of nonstoichiometric perovskite have been fixed firmly by a sufficient number of consistent experiments in the LaO_1.5-SrO-CrO_1.5 system.[73] Thus the authors adopted ${}^{\circ }G_{{({\text{La}}^{3 + } )({\text{Cr}}^{4 + } )({\text{O}}^{2 - } )_{3} }}^{\text{prv}}$ and ${}^{\circ }G_{{({\text{La}}^{3 + } )({\text{Cr}}^{4 + } )({\text{Va}})_{3} }}^{\text{prv}}$ from Povoden.[73] The neutral Cr⁴⁺-containing endmembers

$$ \begin{aligned} {}^{\circ }G_{{{\text{VaCrO}}_{ 2} }}^{\text{prv}} - H_{\text{Cr}}^{\text{SER}} - 2H_{\text{O}}^{\text{SER}} &= \frac{2}{3}{}^{\circ }G_{{{\text{Va:Cr}}^{{ 4 + }} : {\text{O}}^{ 2- } }} + \frac{1}{3}{}^{\circ }G_{{{\text{Va:Cr}}^{{ 4 + }} : {\text{Va}}}} + 3RT\left( {\frac{2}{3}\ln \frac{2}{3} + \frac{1}{3}\ln \frac{1}{3}} \right) \hfill \\ &= {\text{GVCR4O}} = \frac{1}{2}{}^{\circ }G_{{{\text{Cr}}_{ 2} {\text{O}}_{ 3} }}^{{^{[9]} }} + \frac{1}{4}{}^{\circ }G_{{{\text{O}}_{ 2} }}^{{{\text{gas}}[10]}} + G_{\text{mag}} + A + BT \\ \end{aligned} $$

(7)

and

$$ \begin{aligned} {}^{\circ }G_{{{\text{La}}_{{{ 2\mathord{\left/ {\vphantom { 23}} \right. \kern-\nulldelimiterspace} 3}}} {\text{CrO}}_{ 3} }}^{\text{prv}} - \frac{2}{3}H_{\text{La}}^{\text{SER}} - H_{\text{Cr}}^{\text{SER}} - 3H_{\text{O}}^{\text{SER}} &= \frac{2}{3}{}^{\circ }G_{{{\text{La}}^{{ 3 + }} : {\text{Cr}}^{{ 4 + }} : {\text{O}}^{ 2- } }} + \frac{1}{3}{}^{\circ }G_{{{\text{Va:Cr}}^{{ 4 + }} : {\text{O}}^{ 2- } }} + RT\left( {\frac{2}{3}\ln \frac{2}{3} + \frac{1}{3}\ln \frac{1}{3}} \right) \\ &= {\text{GLCR4O}} = \frac{1}{2}{}^{\circ }G_{{{\text{Cr}}_{ 2} {\text{O}}_{ 3} }}^{[9]} + \frac{1}{3}{}^{\circ }G_{{{\text{La}}_{ 2} {\text{O}}_{ 3} }}^{[8]} + \frac{1}{4}{}^{\circ }G_{{{\text{O}}_{ 2} }}^{{{\text{gas}}[10]}} + G_{\text{mag}} + A \\ \end{aligned} $$

(8)

and reciprocal relations which were set zero in analogy to Grundy et al.[66] were used to obtain ${}^{\circ }G_{{({\text{Va}})({\text{Cr}}^{3 + } )({\text{O}}^{2 - } )_{3} }}^{\text{prv}}$, ${}^{\circ }G_{{({\text{Va}})({\text{Cr}}^{4 + } )({\text{O}}^{2 - } )_{3} }}^{\text{prv}}$, ${}^{\circ }G_{{({\text{Va}})({\text{Cr}}^{3 + } )({\text{Va}})_{3} }}^{\text{prv}}$, and ${}^{\circ }G_{{({\text{Va}})({\text{Cr}}^{4 + } )({\text{Va}})_{3} }}^{\text{prv}}$. The configurational entropy-term in Eq 7 describes random mixing of O²⁻ with Va on the anion sublattice. In Eq 8 it describes random mixing of La³⁺ and Va on the A-site.

The parameters A and B of Eq 7 and A of Eq 8 are optimized using experimental data of excess Cr in perovskite.[56] Furthermore the temperature dependence of lanthanum vacancy and hole concentrations from Akashi et al.[59] was considered in the optimization. As cation diffusion in La_1–x CrO₃ is extremely slow even at high temperatures, the Cr-overstoichiometry in a furnace-cooled specimen reported by Khattak and Cox[55] does most likely not represent the overstoichiometry at an intermediate temperature and was not used for the optimization.

${}^{\circ }G_{{({\text{La}}^{3 + } )({\text{Cr}}^{3 + } )({\text{Va}})_{3} }}^{\text{prv}}$results from a reciprocal relation which was set zero in analogy to Grundy et al.[66]:

$$ {}^{\circ }G_{{{\text{LaCrVa}}_{ 3} }}^{{\text{o}}{\text{-}}{\text{prv,r}}{\text{-}}{\text{prv}}} - H_{\text{La}}^{\text{SER}} - H_{\text{Cr}}^{\text{SER}} = {}^{\circ }G_{{{\text{La}}^{{ 3 + }} : {\text{Cr}}^{{ 3 + }} : {\text{Va}}}} = G_{{{\text{LaCrO}}_{ 3} }}^{{\text{o}}{\text{-}}{\text{prv,r}}{\text{-}}{\text{prv}}} - \frac{{3}}{{2}}{}^{\circ }G{(\text{O}}_{{{2({\text{g}})}}} {)}^{{{[10]}}} $$

(9)

Using Eq 5 to 8 and adopting the Gibbs energies of the remaining endmembers ${}^{\circ }G_{{({\text{La}}^{3 + } )({\text{Va}})({\text{O}}^{2 - } )_{3} }}^{\text{prv}}$, ${}^{\circ }G_{{({\text{La}}^{3 + } )({\text{Va}})({\text{Va}})_{3} }}^{\text{prv}}$,${}^{\circ }G_{{({\text{Va}})({\text{Va}})({\text{O}}^{2 - } )_{3} }}^{\text{prv}}$, and ${}^{\circ }G_{{({\text{Va}})({\text{Va}})({\text{Va}})_{3} }}^{\text{prv}}$from Grundy et al.,[66] the 12 endmembers of the compound energy formalism of the perovskite phase are defined. The introduction of positive interaction parameters ${}^{0}L_{{{\text{La}}^{{ 3 + }} , {\text{Va:Cr}}^{{ 3 + }} : {\text{O}}^{{ 2 - }} }}^{\text{prv}}$ and ${}^{0}L_{{{\text{La}}^{{ 3 + }} , {\text{Va:Cr}}^{{ 4 + }} : {\text{O}}^{{ 2 - }} }}^{\text{prv}}$ that were given the same values circumvents too high Cr⁴⁺ contents at low temperatures that would be in conflict with the experiments.

The Liquid Phase

The two-sublattice model for ionic liquids[74,75] was used for the description of the liquid phase of the La-Cr-O system. It was based on the liquid descriptions of the binary subsystems. The chromium species considered in the liquid are Cr²⁺ and Cr³⁺. Higher oxidation states are unlikely to exist in the liquid at normal oxygen partial pressures. The liquid is thus given by the model description (La³⁺,Cr²⁺,Cr³⁺) _p (O²⁻,Va^q−) _q . The experimentally determined temperatures and liquid compositions[13,14] at the eutectic and monotectic in the metallic La-Cr system and the partial enthalpy of mixing of Cr, $\Updelta \bar{H}_{\text{Cr}}$[15] in La-Cr liquid were used to optimize the temperature-dependent regular ${}^{0}L_{{{\text{Cr}}^{{ 2 + }} , {\text{La}}^{{ 3 + }} : {\text{Va}}}}^{\text{liq}}$ and subregular ${}^{1}L_{{{\text{Cr}}^{{ 2 + }} , {\text{La}}^{{ 3 + }} : {\text{Va}}}}^{\text{liq}}$ interaction parameters to account for interactions between La and Cr. Furthermore the two regular interaction parameters ${}^{0}L_{{{\text{Cr}}^{{ 3 + }} , {\text{La}}^{{ 3 + }} : {\text{O}}^{{ 2 - }} }}^{\text{liq}} = {}^{0}L_{{{\text{Cr}}^{{ 2 + }} , {\text{La}}^{{ 3 + }} : {\text{O}}^{{ 2 - }} }}^{\text{liq}}$ and the two subregular ${}^{1}L_{{{\text{Cr}}^{{ 3 + }} , {\text{La}}^{{ 3 + }} : {\text{O}}^{{ 2 - }} }}^{\text{liq}} = {}^{1}L_{{{\text{Cr}}^{{ 2 + }} , {\text{La}}^{{ 3 + }} : {\text{O}}^{{ 2 - }} }}^{\text{liq}}$ were optimized. It was assumed that the interactions between Cr²⁺-La³⁺ and Cr³⁺-La³⁺ are of the same order of magnitude in the oxide melt, thus the two regular interaction parameters were set equal to each other, as were the two sub-regular interaction parameters. Using the following data for their optimization led to the lowest error between experiments and calculation: the composition and temperature of the eutectic at the La-rich side and the composition of the eutectic at the Cr-rich side in the oxide LaO_1.5-CrO_1.5 system from Tresvjatskiy et al.,[18] the temperature of the eutectic at the Cr-rich side from Berjoan,[19] and the congruent melting temperature of the perovskite phase from Coutures et al.[20] and Foëx.[21] Berjoan[32] and Tresvjatskiy et al.[18] did not specify the value of the prevailing oxygen partial pressure during their phase diagram experiments conducted in an argon atmosphere. As a value of the oxygen partial pressure is required for the optimization, we defined $p_{{{\text{O}}_{2}}}=1\,{\text{Pa}}.$

Results and Discussion

The La-Cr System

The calculated phase diagram of the La-Cr system is presented in Fig. 2, together with experimental phase diagram data.[12-14,17]. The positive value of ${}^{0}L_{\text{Cr,La:Va}}^{\text{bcc}}$used to model the bcc phase results in a large miscibility gap between the La-rich and Cr-rich metals, which is tantamount to a small solubility of La in αCr_ss in agreement with the experiments.[14,17] The model description of the bcc phase results in a tiny solubility of Cr in La(bcc), denoted as γLa_ss, of $2\times 10^{-3}$ at.% at 1134 K, the lowest temperature of stable γLa_ss, which further decreases as a function of increasing temperature. The calculated enthalpies of mixing are shown in Fig. 3 together with the experimentally determined value[15] that is well reproduced by the calculation. Considerable deviations of the calculated liquidus from experiments at the Cr-rich side of the system can be ascribed to the problem of two different melting temperatures for Cr cited in the literature, which are 2180 and 2130 K. The higher value was favored by Dinsdale[10] and is adopted in this study, whereas the lower melting temperature was chosen by Savitskii et al.[12] and Svechnikov et al.[14]

Fig. 2

Calculated phase diagram of the La-Cr system with data from the literature included (symbols)

Fig. 3

Calculated partial enthalpies of mixing of La and Cr in La-Cr liquid, and integral enthalpies of mixing as a function of composition, with the experiment from Berezutskii et al.[15] at 1700 K included (symbol with error-bar)

A satisfying reproduction of the experimental data was obtained by considering a moderate temperature dependence of ${}^{0}L_{{{\text{Cr}}^{{ 2 + }} , {\text{La}}^{{ 3 + }} : {\text{Va}}}}^{\text{liq}}$ and ${}^{1}L_{{{\text{Cr}}^{{ 2 + }} , {\text{La}}^{{ 3 + }} : {\text{Va}}}}^{\text{liq}}$. This is unfortunately associated with an inverse liquid-liquid miscibility gap with a minimum at x(Cr) = 0.25 and T ≈ 5000 K that is of course unphysical.

The La-Cr-O System

Phase Equilibria

Calculated LaO_1.5-CrO_1.5 phase diagrams in pure oxygen at $p_{{{\text{O}}_{2}}}=10^5$ Pa, in air at $p_{{{\text{O}}_{2}}}=21,278$ Pa, and under reducing conditions at $p_{{{\text{O}}_{2}}}=1$ Pa representing the typical oxygen partial pressure in argon atmosphere are shown in Fig. 4 together with experimental data.[18-21]

Fig. 4

Calculated phase diagrams of the LaO_1.5-CrO_1.5 system in pure oxygen, air atmosphere, and under reducing conditions representing argon atmosphere at $p_{{{\text{O}}_{2}}}=1$ Pa with experimental data included (symbols)

Excess Cr in lanthanum chromite is favored at high oxygen partial pressures. A decrease of Cr⁴⁺ during annealing of an originally lanthanum-deficient perovskite phase under reducing conditions is predicted by the model, reflected by the disappearance of Cr overstoichiometry. This is in line with the interpretations of Raman spectra from Iliev et al.[56] Be it that the reported thermodynamic data of La₂CrO₆[19] and La₂(CrO₄)₃[30] are correct, lanthanum chromite is expected to be metastable at room temperature, and orthorhombic perovskite is stable only at $p_{{{\text{O}}_{ 2} }}\leq 10^2$ Pa. La₂CrO₆ is stable within a wide temperature-range in pure oxygen, whereas it does not form in air and argon atmosphere.

Due to the ambiguous oxygen partial pressure of phase diagram experiments[18,19] and the conflicting data on the melting temperature of lanthanum chromite in argon atmosphere[18] the presented liquid description is rather tentative. Under oxidizing conditions Cr³⁺ is favored over Cr²⁺ in the liquid. Analogous to Fe in the La-Fe-O system[63] this oxidation of Cr²⁺ to Cr³⁺ governs shifts of eutectic compositions and temperatures and the increase of the melting temperature of the perovskite phase on increasing the oxygen partial pressure. On the other hand a significant amount of Cr³⁺ in the ionic liquid is reduced to Cr²⁺ under reducing conditions, and the liquid stability increases considerably at the Cr-rich part of the system leading to a considerably lowered eutectic temperature. The liquid description using the two-sublattice model for ionic liquids also resulted in a significantly larger decrease of the melting temperature of lanthanum chromite at $p_{{{\text{O}}_{ 2} }}\approx 1$ Pa than the given values in argon atmosphere.[18] Despite this discrepancy we did not go for an alternative liquid model for the sake of consistency with our previously assessed systems.

In Fig. 5 calculated phase equilibria of the La-Cr-O system at 1273 K are shown as a function of oxygen partial pressure. It is obvious that no mutual solubilities of La and Cr in bcc metal in equilibrium with oxides are expected. The same oxygen solubility in Cr as in the assessment by Povoden et al.[9] was obtained using the new model description (Cr)(O,Va)_1.5. At $p_{{{\text{O}}_{2}}}=10^{-34.04}$ Pa metallic liquid forms at the lanthanum-rich side of the phase diagram.

Fig. 5

Calculated phase equilibria of the La-Cr-O system at T = 1273 K as a function of oxygen partial pressure

Thermodynamic Data

Calculated thermodynamic data of solid oxides are listed together with experimental data from the literature in Table 1. Calculated and experimental data on the orthorhombic to rhombohedral transition of LaCrO₃ are listed in Table 2. Table 3 is a compilation of the Gibbs energy functions and model descriptions of the phases in the La-Cr-O system obtained in this study.

Lanthanum Chromates

Testing an optimization of model parameters of La₂(CrO₄)₃ by using all available thermodynamic data[29,31] resulted in gross disagreement between optimized and reported values. The considerable error might be explained by experimental difficulties to reach equilibrium at the low investigation temperatures, and/or by significant deviations between the thermodynamic standard data used for the calculation of the enthalpy of formation from the elements[29] and assessed values.[8-10] Anyway the model parameters were fitted to the experimental data,[30] whereas the calculated standard enthalpy of formation from the elements[29] was rejected, bearing in mind the high degree of uncertainty of the resulting description. The perovskite phase: the calculated heat capacities of LaCrO₃ are compared with experiments from the literature in Fig. 6. The calculated C _p -curve extrapolates well to high temperatures. The use of C _p -data from Sakai and Stølen[35] along with enthalpy increment-data from Suponitskii[30] to optimize the parameter CT ln T of the Gibbs energy of stoichiometric perovskite resulted in the lowest error between experiments and calculation. As CT ln T was set equal for o-prv and r-prv, their C _p is the same. The experimentally determined C _p -peak around 545 K caused by the first-order transition o-prv ↔ r-prv is in fact a discontinuity which cannot be implemented in the model. The calculated transition temperature of 540 K is shown by the broken line in Fig. 6. The calculated C _p -peak at 290 K reflects the temperature of the magnetic order-disorder transition, the transition temperature being in agreement with the experiments. Two values for the magnetic parameter p are possible depending on the crystal structure, p = 0.28 and p = 0.4, whereby the proper p-value for structures other than bcc, fcc, and hcp is not available in the literature. The C _p -anomaly is equally well reproduced by the model[70,71] using p = 0.28 or p = 0.4. For the sake of compatibility with the recent assessment of the La-Fe-O system[63] we chose p = 0.28. Experimental enthalpy increments[30] are well reproduced by the calculation (see Table 1). Due to the consistency between both groups of calorimetric experiments[30,35] the term CT ln T is fixed firmly. A small peak which was found around 855 K can be explained most likely by the decomposition of an undetected impurity phase.[35]

Fig. 6

Calculated heat capacities of LaCrO₃ as a function of T with experimental data included (symbols). The dashed line marks the temperature of the o-prv ↔ r-prv transition

The calculated Gibbs energies of the formation of LaCrO₃ from the oxides

$$ \frac{ 1}{ 2}{\text{La}}_{ 2} {\text{O}}_{ 3} + \frac{ 1}{ 2}{\text{Cr}}_{ 2} {\text{O}}_{ 3} \to {\text{LaCrO}}_{ 3} $$

(10)

are listed as a function of temperature together with data from the literature[49-53] in Table 1. The resulting Gibbs energies of formation from emf-measurements are remarkably less negative than the Gibbs energies of formation derived from Knudsen mass spectrometry.[53] Only the use of the latter data for the optimization resulted in the proper phase diagram with congruent melting of the perovskite phase and two eutectic. It needs to be clarified why all of the emf-measurements are problematic: Azad et al.[50] stated that the Gibbs energy of formation of LaCrO₃ cannot be studied properly using the solid oxide electrolyte method due to experimental difficulties in measuring the low oxygen potentials encountered in a mixture of coexisting LaCrO₃-La₂O₃-Cr. Yet it is obvious that the CaF₂-based emf-technique is neither suitable for the determination of thermodynamic data of lanthanum chromite, as it unavoidably leads to emf that are too low. A possible explanation is found in a study by Akila and Jacob:[76] Fine precipitates of CaO can form on the surface of CaF₂ in water- or oxygen-containing atmosphere. In this case the emf depends on the activity of CaO at the electrode/electrolyte interface, and changing activity of CaO at the electrode/electrolyte interface can alter the chemical potential of fluorine at this electrode and thus the emf across the electrolyte.

Chemical Stability of the Perovskite Phase

The calculated oxygen partial pressure for the decomposition of lanthanum chromite by the reaction

$$ {\text{LaCrO}}_{3} \to \frac{1}{2}{\text{La}}_{2} {\text{O}}_{3} + \upalpha {\text{Cr}}{ + }\frac{3}{4}{\text{O}}_{2} ({\text{g}}) \uparrow $$

(11)

is $p_{{{\text{O}}_{ 2} }}=10^{-20.97}$ at 1273 K. The calculated decomposition of the perovskite phase by Eq 11 is plotted as a function of temperature and oxygen partial pressure in Fig. 7.

Fig. 7

Calculated decomposition of lanthanum chromite as a function of temperature and oxygen partial pressure

Defect Chemistry of the Perovskite Phase

Applying a defect chemistry analysis of La_1–x CrO₃ in equilibrium with Cr₂O₃ the following defect reaction for its oxidation can be written in the sublattice form, if $[{\text{Va}}_{\text{Cr}}^{\prime \prime \prime } ]$ and ${[}{\text{Va}}_{\text{O}}^{ \bullet \bullet } {]}$ are assumed to be negligible according to Akashi et al.:[59]

$$ ( {\text{La}}^{{ 3 + }} ) ( {\text{Cr}}^{{ 3 + }} ) ( {\text{O}}^{{ 2 - }} )_{ 3} + \frac{1}{4}{\text{O}}_{{ 2 ( {\text{g)}}}} \to ( {\text{La}}_{2/3}^{{ 3 + }} {\text{Va}}_{ 1 / 3} ) ( {\text{Cr}}^{{ 4 + }} ) ( {\text{O}}^{{ 2 - }} )_{ 3} $$

(12)

Using Kröger-Vink notation this defect reaction reads

$$ {\text{La}}_{\text{La}}^{x} + {\text{Cr}}_{\text{Cr}}^{x} + 3{\text{O}}_{\text{O}}^{x} + \frac{1}{4}{\text{O}}_{{ 2 ( {\text{g)}}}} \to \frac{2}{3}{\text{La}}_{\text{La}}^{x} + \frac{1}{3}{\text{Va}}_{\text{La}}^{\prime \prime \prime } + {\text{Cr}}_{\text{Cr}}^{ \bullet } + 3{\text{O}}_{\text{O}}^{x} $$

(13)

and the equilibrium constant of the oxidation reaction is

$$ K_{ox} = \frac{{[{\text{Va}}_{\text{La}}^{\prime \prime \prime } ]^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}} [{\text{La}}_{\text{La}}^{x} ]^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}}} [{\text{Cr}}_{\text{Cr}}^{ \bullet } ][{\text{O}}_{\text{O}}^{x} ]^{3} }}{{[{\text{La}}_{\text{La}}^{x} ][{\text{Cr}}_{\text{Cr}}^{x} ][{\text{O}}_{\text{O}}^{x} ]^{3} p_{{{\text{O}}_{2} }}^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}}} }} $$

(14)

For small oxidation extent $[{\text{La}}_{\text{La}}^{x} ]$, $[{\text{Cr}}_{\text{Cr}}^{x} ]$, and $[{\text{O}}_{\text{O}}^{x} ]$ can be considered to be ~1, and charge neutrality is maintained by

$$ [ {\text{Va}}_{\text{La}}^{\prime \prime \prime } ]= \frac{1}{3} [{\text{Cr}}_{\text{Cr}}^{ \bullet } ]$$

(15)

Substituting this into Eq 14 gives the proportionalities $[{\text{Va}}_{\text{La}}^{\prime \prime \prime } ], [{\text{Cr}}_{\text{Cr}}^{ \bullet } ] \propto p_{{{\text{O}}_{2} }}^{3/16}$.

The concentrations of the defects ${\text{La}}_{\text{La}}^{x} ,{\text{Va}}_{\text{La}}^{\prime \prime \prime } ,{\text{Cr}}_{\text{Cr}}^{x} ,{\text{ and Cr}}_{\text{Cr}}^{ \bullet }$ in La_1–x CrO₃ correspond to the site fractions $y_{{{}^{\text{A}}{\text{La}}^{{ 3 + }} }}^{\text{prv}}$,$y_{{{}^{\text{A}}{\text{Va}}}}^{\text{prv}}$,$y_{{{}^{\text{B}}{\text{Cr}}^{{ 3 + }} }}^{\text{prv}}$, and $y_{{{}^{\text{B}}{\text{Cr}}^{{ 4 + }} }}^{\text{prv}}$in the compound energy formalism. $y_{{{}^{\text{A}}{\text{La}}^{{ 3 + }} }}^{\text{prv}}$,$y_{{{}^{\text{A}}{\text{Va}}}}^{\text{prv}}$, $y_{{{}^{\text{B}}{\text{Cr}}^{{ 3 + }} }}^{\text{prv}}$, $y_{{{}^{\text{B}}{\text{Cr}}^{{ 4 + }} }}^{\text{prv}}$ and the tiny fractions $y_{{{}^{\text{B}}{\text{Va}}}}^{\text{prv}}$ and $y_{{{}^{\text{O}}{\text{Va}}}}^{\text{prv}}$ are plotted logarithmically as a function of $\log p_{{{\text{O}}_{2}}}$ at 1073 and 1673 K in Fig. 8 for lanthanum chromite in equilibrium with Cr₂O₃. The line for $y_{{{}^{\text{A}}{\text{La}}^{{ 3 + }} }}^{\text{prv}}$ at 1073 K cannot be seen as it is very close to 1. At 1073 K a constant slope of 3/16 of the defect concentrations $[{\text{Va}}_{\text{La}}^{\prime \prime \prime } ]{\text{ and }}[{\text{Cr}}_{\text{Cr}}^{ \bullet } ]$ shown in the triangle, is calculated from very high to very low oxygen partial pressures. This slope is fixed by the defect reaction Eq 12. At 1673 K the slope of 3/16 of $[{\text{Va}}_{\text{La}}^{\prime \prime \prime } ]{\text{ and }}[{\text{Cr}}_{\text{Cr}}^{ \bullet } ]$ is reproduced by the calculated slope using the compound energy formalism at 10⁵ > $p_{{{\text{O}}_{2}}}>10^{-8}$ Pa; hence oxidation of LaCrO₃ to La_1–x CrO₃ governs the electrical conductivity of perovskite with fixed activity of Cr₂O₃ at unity between $p_{{{\text{O}}_{2}}}=10^5$ and 10⁻⁸ Pa at this temperature. The calculated slopes of $[{\text{Va}}_{\text{La}}^{\prime \prime \prime } ]{\text{ and }}[{\text{Cr}}_{\text{Cr}}^{ \bullet } ]$ are equal to the slope of the electrical conductivity from 1573 to 1673 K between $p_{{{\text{O}}_{2}}}=1.0 \times 10^{3}\, {\text{Pa}}$ and $p_{{{\text{O}}_{2}}}=2.0 \times 10^{4}\, {\text{Pa}}$ determined by Akashi et al.[59] The conflicting data from Shvaiko-Shvaikovskii et al.[58] may be explained by problems of reaching equilibrium due to extraordinarily slow cation diffusion in lanthanum chromite. In Fig. 9 the calculated slopes of ${\text{Va}}_{\text{La}}^{\prime \prime \prime }{\text{ and Cr}}_{\text{Cr}}^{ \bullet }$ are compared with slopes of $[{\text{Va}}_{\text{La}}^{\prime \prime \prime } ]{\text{ and }}[{\text{Cr}}_{\text{Cr}}^{ \bullet } ]$ determined by Akashi et al.[59] as a function of reciprocal temperatures. The calculated concentrations agree well with the data derived from electrical conductivity measurements.[59] The calculated amount of $[{\text{Va}}_{\text{La}}^{\prime \prime \prime } ]{\text{ relative to }}[{\text{Cr}}_{\text{Cr}}^{ \bullet } ]$ is fixed by the criterion for charge neutrality, Eq 15, as calculated $[{\text{Va}}_{\text{Cr}}^{\prime \prime \prime } ]$ and $[{\text{Va}}_{\text{O}}^{ \bullet \bullet } ]$ are very small. The calculated relative defect concentrations are in line with those proposed by Akashi et al.[59]

Fig. 8

Calculated site fractions of species in La_1–x CrO₃ in the thermodynamic equilibrium with Cr₂O₃ logarithmically plotted at 1073 K and 1673 K as a function of $p_{{{\text{O}}_{ 2} }}$. The slope of 3/16 of the calculated defect concentrations is indicated in the triangle

Fig. 9

Calculated defect concentrations in La_1–x CrO₃ in the thermodynamic equilibrium with Cr₂O₃ (solid lines) logarithmically plotted as a function of reciprocal temperature along with the data from Akashi et al.[59] derived from electrical conductivity measurements (symbols with error-bars, broken lines)

The presented defect chemistry calculations are still rather tentative, as the temperature and oxygen partial pressure dependence of excess Cr in La_1–x CrO₃ has not been investigated systematically so far.

Conclusions

Model parameters of the presented thermodynamic La-Cr-O database were optimized with assessed thermodynamic and phase diagram data.

The thermodynamic descriptions of lanthanum chromates and the liquid phase are rather tentative due to humble or sketchy experimental information.

The thermodynamic modeling of lanthanum chromite was based on experimental thermodynamic data reported by Peck et al.[53] and Cheng and Navrotsky,[46] as the use of these data for the optimization of model parameters resulted in a proper reproduction of the phase equilibria derived from experiments. The orthorhombic to rhombohedral transition in lanthanum chromite and the magnetic order-disorder transformation are well reproduced by the model.

Using the new database the stability limits of lanthanum chromite in function of temperature and oxygen partial pressure can be quantified.

The proposed existence of lanthanum vacancies and holes to maintain charge neutrality in lanthanum chromite with excess Cr is reproduced by the model, and the calculated slopes of defect concentrations in function of oxygen partial pressure and temperature are in line with the slopes derived from electrical conductivity measurements. However the amounts of excess Cr in La_1–x CrO₃ used for the optimization of the cation nonstoichiometry are preliminary, and further work on the temperature dependence of excess Cr as a function of temperature and oxygen partial pressure would allow a more accurate quantification of the defect chemistry of lanthanum chromite.