Charge Compensation Research Articles

Effect of Fe2O3 on the structure and properties of Mo‐containing borosilicate glasses for nuclear waste immobilization

AbstractAs a fission product in high‐level radioactive nuclear waste, Mo has low solubility in borosilicate glass. Fe2O3 is not only a prevalent transition metal element but also a major corrosion product in high‐level radioactive nuclear waste. Against this backdrop, the effect of Fe2O3 content on the structure and chemical durability of typical molybdenum‐containing sodium borosilicate glasses for nuclear waste immobilization are studied. The results show that the samples containing more than 3.85 mol% Fe2O3, a completely homogenous amorphous glass sample is obtained. Moreover, the mechanism of the effect of Fe2O3 on the solubility of Mo is discussed in detail. In this work, a portion of Fe3+ is reduced to Fe2+ and enters into the glasses as a charge compensation ion as Fe2+O6. Concurrently, Fe3+ ions contribute to the formation of the glass networks as Fe3+O4. Iron incorporation can improve the chemical durability of the sample.

Fucoidan Cross‐Linking Polyacrylamide as Multifunctional Aqueous Binder Stabilizes LiCoO2 to 4.6 V

AbstractRaising the cutoff voltage can efficiently increase the energy density of lithium cobalt oxide (LCO). However, upon charging over 4.55 V the LCO undergoes irreversible phase transition from the pristine O3 phase to the metastable H1‐3 phases, causing serious side reactions, which results in poor cycling stability. Herein, a multifunctional aqueous composite binder derived from the cross‐linking of fucoidan (FUC) and polyacrylamide (PAM) is developed to enhance the stability of LCO cathode at 4.6 V. The cross‐linking interaction of FUC and PAM provides a uniform coating on the surface of LCO and ensures a high peel strength for the electrode, effectively mitigating irreversible phase transition and detrimental interface side reactions. More importantly, the sulfur ester and amide groups of FUC‐PAM favorably function as surface charge compensators to the high valent Co upon charging under high voltages, thus stabilizes the surface lattice of LCO and suppresses the detrimental oxygen release. As expected, the LCO with a cutoff voltage of 4.6 V exhibits a high capacity retention of 90% after 100 cycles at a current density of 110 mA g−1. The interfacial coordination effect of composite binders offers a novel strategy to enhance the stability of high‐voltage LCO for high‐energy lithium‐ion batteries.

Anion Substitution to Suppress the Voltage Hysteresis of Na3MnTi(PO4)3 as a Cathode Material for Sodium-Ion Batteries.

The Mn-based polyanion compound Na3MnTi(PO4)3 (NMTP) with a Na superionic conductor (NASICON) structure has attracted incremental attention as a potential cathode material for sodium-ion batteries. However, the occupation of Mn2+ on Na+ vacancies usually leads to severe voltage hysteresis, which in turn results in significant capacity loss, slow Na+ diffusion kinetics, and poor cycling stability. Herein, anion-substituted compounds Na3MnTi(PO4)3-x(SiO4)x (x = 0.1, 0.2, and 0.3) are synthesized. It reveals that the SiO44- substitution could induce partial oxidation of Mn2+ to Mn3+, and the latter has a lower occupancy preference on Na+ vacancies. By the proposed charge compensation strategy, the Mn2+ occupation on Na+ vacancies can be significantly suppressed. As a result, the voltage hysteresis is substantially inhibited, and greatly improved electrochemical performance is achieved. This study offers an alternative strategy to address the voltage hysteresis associated with NMTP and other Mn-based NASICON cathode materials.

Luminescence spectra and site-occupation of Eu3+ centres in lithium antimonate LiSbO3 lattice

It is a challenge to accurately determine the exact location of the rare earth ion (RE) in the lattice when there are significant differences in radius, electronegativity and valence between the lattice cation and the corresponding RE activator in a host material. This study presents the findings on the luminescence and site occupancy of Eu3+ emission centers in the lithium antimonate LiSbO3 lattice. The phase-formation, structures, elemental composition, band transition characteristics, luminescence properties, and lifetimes of the Eu3+-doped phosphors were investigated. Eu3+-doped LiSbO3 exhibits a typical 5D0→7F2 red luminescence transition at 613 nm. Site-selective excitation, luminescence, and decay properties at the 7F0→5D0 transition wavelength were measured using a tunable pulsed narrow-band dye laser. Two Eu3+ luminescence centers in LiSbO3 were identified across the temperature range from 10 K to 300 K. The doping of RE(Eu3+) in LiSbO3 consistently occupies two crystallographic positions of Li+ and Sb5+. Furthermore, the probability of RE(Eu3+) ions preferentially occupying the Li+ (M) sites is notably high. The decay curves and luminescence lifetimes of the two Eu3+ centers were determined. Drawing from the experimental data, we discuss the potential defects induced by Eu3+ doping in LiSbO3 and the mechanism of charge compensation. These results could be instrumental in the advancement of new RE-activated luminescent materials or serve as a reference for investigating the specific positioning of RE ions within a phosphor lattice.

Evidence for Morphology Control by Polyelectrolyte Templating in PEDT:PSSH

AbstractDespite decades of research, the fundamental question of structure and morphology of the poly(3,‐4‐ethylenedioxythiophene):poly(styrene sulfonate) complex (PEDT: PSS) electrically conducting polymer remains open. Here, through a systematic study of the polymerization of 3,4‐ethylenedioxythiophene (EDT) on two series of sulfonated polystyrenes, i.e., PS‐co‐SSH and PS‐co‐SSNa, with per cent sulfonation 24 ≲ x ≲ 100, this study shows that the substrate polyelectrolyte not only affords charge compensation and processability, but also templates the morphology of the growing PEDT chain—whether extended, coiled or collapsed. This determines the electrical, electronic and optical properties of the resultant PEDT: PSS interpolyelectrolyte complexes, including their work function. This study calls this the Polyelectrolyte Template model. The PEDT: PSS complex retains memory of its polymerization morphology even after subsequent ion‐exchange, for example, to the same acid form. Electrical conductivity decreases sharply with decreasing x, and depends on whether the polymerization is templated with the H+ or Na+ form of the polyelectrolyte. The work function is less sensitive to x, but exhibits a transition from low to high work function (5.2–5.3 eV) when x exceeds a threshold, which indicates formation of a polar surface driven by molecular‐scale segregation of the PSS counteranion. This surface segregation also affects the ability of these complexes to inject holes into very large‐ionization‐energy semiconductors.

Phase Transitions and Thermoelectric Properties of Charge-Compensated ZnxCu12-xSb4Se13.

In this study, we investigated the phase transitions and thermoelectric properties of charge-compensated hakite (ZnxCu12-xSb4Se13) as a function of Zn content. Based on X-ray diffraction and a differential scanning calorimetric phase analysis, secondary phases (permingeatite and bytizite) transformed into hakite depending on the Zn content, while Zn2Cu10Sb4Se13 existed solely as hakite. Nondegenerate semiconductor behavior was observed, exhibiting increasing electrical conductivity with a rising temperature. With an increase in Zn content, the presence of mixed phases of hakite and permingeatite led to enhanced electrical conductivity. However, Zn2Cu10Sb4Se13 with a single hakite phase exhibited the lowest electrical conductivity. The Seebeck coefficient exhibited positive values, indicating that even after charge compensation (electron supply) by Zn, p-type semiconductor characteristics were maintained. With the occurrence of an intrinsic transition within the measured temperature range, the Seebeck coefficient decreased as the temperature increased; at a certain temperature, Zn2Cu10Sb4Se13 exhibited the highest value. Thermal conductivity showed a low temperature dependence, obtaining low values below 0.65 Wm-1K-1. A power factor of 0.22 mWm-1K-2 and dimensionless figure of merit of 0.31 were achieved at 623 K for ZnCu11Sb4Se13.

Significantly enhanced reliability in defect-engineered BaTi1-xMgxO3 ceramics

BaTiO3-based ceramic is considered to be a fascinating dielectric material with high reliability for ultra-thin multilayer ceramic capacitors (MLCC). Here, we fabricated a series of acceptor-doped BaTiO3 (Mg: 0.2, 0.5, 1.0, 1.5, 2.0, and 5.0 mol%) ceramics with Al2O3 and SiO2 as sintering additives. With an increase in Mg concentration, the insulation resistance of the BaTi1-xMgxO3 ceramics initially increases and then decreases, in which the 0.5 mol% Mg-doped ceramic achieves superior degradation behavior. The reason is that with the incorporation of the acceptor ion Mg2+, the charge compensation mechanism in the system changes. Initially, barium vacancy is generated, which is gradually dominated by oxygen vacancy. Analysis indicates that the segregation of the acceptor state barium vacancy to the grain boundary increases the schottky barrier of the grain boundary, while the gradual increase of the donor state oxygen vacancy reduces the reliability of the ceramic. This study provides a comprehensive overview illustrating the significant role of point defects in the potential barrier of the grain boundary in acceptor-doped BaTiO3 ceramics. It identifies a pathway to achieve high reliability in BaTiO3-based MLCC.

Investigations on the EPR g factors and local structures for the orthorhombic and tetragonal Cu2+ centers in Pb[Zr0.54Ti0.46]O3.

The defect models of the orthorhombical and tetragonal Cu2+ centers in Pb[Zr0.54Ti0.46]O3 are attributed to Cu2+ ions occupying the sixfold coordinated octahedral Ti4+ site with and without charge compensation, respectively. The electron paramagnetic resonance (EPR) g factors gi (i = x, y, z) of the Cu2+ centers in Pb[Zr0.54Ti0.46]O3 are theoretically studied by using the perturbation formulas of a 3d9 ion under orthorhombically and tetragonally elongated octahedra. Based on the calculation, the impurity off-center displacements are about 0.253 and 0.162 Å for the orthorhombical and tetragonal Cu2+ centers, respectively. Meanwhile, the planar Cu2+-O2- bonds are found to experience the relative variation ΔR (≈0.102 Å) along the a- and b-axes for the orthorhombical Cu2+ center due to the Jahn-Teller (JT) effect. The theoretical EPR g factors based on the above local structures agree well with the observed values.

Constructing layered/tunnel interlocking oxide cathodes for sodium-ion batteries based on breaking Mn3+/Mn4+ equilibrium in Na0.44MnO2 via trace Mo doping

Sodium-ion batteries (SIBs) with Mn-based oxide cathodes have attracted much interest because of their cost effectiveness, minimal toxicity, and remarkable electrochemical activity. The typical tunnel Na0.44MnO2 cathode for SIBs exhibits stable cycling performance, but its low sodium content hinders the practical application. And the other NaxMnO2 cathodes with layered structures provide higher capacity but less satisfactory cycling. Therefore, it is of great significance to construct Mn-based oxide cathodes to integrate high specific capacity layered structure and high stability tunnel structure. Herein, the strategy of breaking the Mn3+/Mn4+ equilibrium in Na0.44MnO2 via doping trace amounts of Mo was proposed to promote the transformation from the original tunnel to the layered structure and to in-situ construct a layered/tunnel biphasic interlocking structure. The optimized interlocking layered/tunnel Na0.44Mn0.97Mo0.03O2 (NMO-3M) cathode integrates the advantages of layered and tunnel structures, and achieves highly reversible phase transition and excellent electrochemical performance, which delivers a high specific capacity of 178.9 mAh g−1 at 0.1 C and capacity retention of 77.8 % after 100 cycles at 1 C. The well-maintained P2 phase was revealed by the in-situ X-ray diffraction (XRD), demonstrating the highly reversible charge compensation and structural evolution. In addition, NMO-3M cathode shows a good storage performance in air over 270 days. This research designed a layered/tunnel interlocking structure induced by trace Mo doping to construct a stable and high capacity oxide cathode, providing a guideline for the preparation of excellent oxide cathodes for SIBs in the future.

Chemical Prelithiated 3D Lithiophilic/-Phobic Interlayer Enables Long-Term Li Plating/Stripping.

The up-to-date lifespan of zero-excess lithium (Li) metal batteries is limited to a few dozen cycles due to irreversible Li-ion loss caused by interfacial reactions during cycling. Herein, a chemical prelithiated composite interlayer, made of lithiophilic silver (Ag) and lithiophobic copper (Cu) in a 3D porous carbon fiber matrix, is applied on a planar Cu current collector to regulate Li plating and stripping and prevent undesired reactions. The Li-rich surface coating of lithium oxide (Li2O), lithium carboxylate (RCO2Li), lithium carbonates (ROCO2Li), and lithium hydride (LiH) is formed by soaking and directly heating the interlayer in n-butyllithium hexane solution. Although only a thin coating of ∼10 nm is created, it effectively regulates the ionic and electronic conductivity of the interlayer via these surface compounds and reduces defect sites by reactions of n-butyllithium with heteroatoms in the carbon fibers during formation. The spontaneously formed lithiophilic-lithiophobic gradient across individual carbon fiber provides homogeneous Li-ion deposition, preventing concentrated Li deposition. The porous structure of the composite interlayer eliminates the built-in stress upon Li deposition, and the anisotropically distributed carbon fibers enable uniform charge compensation. These features synergistically minimize the side reactions and compensate for Li-ion loss while cycling. The prepared zero-excess Li metal batteries could be cycled 300 times at 1.17 C with negligible capacity fading.

Nb-based AxNbOFy:Mn4+ (A=Na, Cs; x=2, 3; y=5, 6) red phosphors with excellent water resistance and enhanced luminescence by W6+/Mo6+ charge compensation

Mn4+ activated fluoride red phosphors are used in warm white light emitting diodes (WLEDs). However, their inherently low moisture resistance is a major barrier to their use in high humidity conditions. In this work, the Nb-based oxyfluoride phosphors AxNbOFy:Mn4+ (A = Na, Cs; x = 2, 3; y = 5, 6) were synthesized by the co-precipitation method. The crystal structure, morphology, and luminescence properties are investigated with composition variation. The non-equivalent replacement of Nd5+ ions by Mn4+ results in bright red emission under blue light excitation. These phosphors showed very good moisture resistant luminescence. The best performance was achieved by the Na3NbOF6:0.1%Mn4+, which retained 98 % of the initial fluorescence intensity after immersion in water for 60 min. In addition, the W6+ and Mo6+ are used to compensate for the charge difference between Mn4+ and Nd5+ in Na3NbOF6:Mn4+. The compensated phosphors have shown large enhancements of 169 % (W6+) and 168 % (Mo6+) in luminescence intensity. This strategy of non-equivalent doping approach of Mn4+ into the Nb-based oxyfluoride materials and the charge compensation by W6+/Mo6+ to enhance the luminescence intensity provides a new way to realize high performance red phosphors for WLEDs.

Eu3+ doped Ca3LiSbO6 and Eu3+, Li+ co-doped Ca3LiSbO6 phosphors for white light-emitting diodes

Novel antimonate Ca3LiSbO6: xEu3+ red phosphor was synthesised using a high-temperature solid-phase reaction. The effect of Eu3+ doping concentration on its crystal structure and luminescence performance was investigated. It was found that Eu3+ was at the inversion symmetry site in the lattice of Ca3LiSbO6. To improve the luminescence performance of Ca3LiSbO6: 0.05Eu3+ red phosphor, Ca3LiSbO6: 0.05Eu3+, xLi + red phosphor was synthesised. The doping of Li + can correct the charge difference generated by the substitution of Ca2+ by Eu3+, reduce the generation of defects, enhance the local symmetry of the luminescence centre of Eu3+, improve the structural rigidity, and thus enhance the luminous intensity of the phosphor. At 423 K, the luminous intensity of this phosphor was 68 % of the initial temperature, which was higher than that of the phosphor without the charge compensator Li+ (60 %), indicating that Li+ improved the thermal stability of the phosphor.

Unexpected Elevated Working Voltage by Na+/Vacancy Ordering and Stabilized Sodium-Ion Storage by Transition-Metal Honeycomb Ordering.

Na+/vacancy ordering in sodium-ion layered oxide cathodes is widely believed to deteriorate the structural stability and retard the Na+ diffusion kinetics, but its unexplored potential advantages remain elusive. Herein, we prepared a P2-Na0.8Cu0.22Li0.08Mn0.67O2 (NCLMO-12 h) material featuring moderate Na+/vacancy and transition-metal (TM) honeycomb orderings. The appropriate Na+/vacancy ordering significantly enhances the operating voltage and the TM honeycomb ordering effectively strengthens the layered framework. Compared with the disordered material, the well-balanced dual-ordering NCLMO-12 h cathode affords a boosted working voltage from 2.85 to 3.51 V, a remarkable ~20 % enhancement in energy density, and a superior cycling stability (capacity retention of 86.5 % after 500 cycles). The solid-solution reaction with a nearly "zero-strain" character, the charge compensation mechanisms, and the reversible inter-layer Li migration upon sodiation/desodiation are unraveled by systematic in situ/ex situ characterizations. This study breaks the stereotype surrounding Na+/vacancy ordering and provides a new avenue for developing high-energy and long-durability sodium layered oxide cathodes.

Unexpected concentration dependence of Ba2+ dominant substitution in addition to La3+ in cubic Nd3+-doped BaLaLiWO6 perovskites

There is a great need in material science to find and investigate new RE3+ ion-doped inorganic materials suitable for producing sintered ceramics of high transparency.Prospective candidates for transparent ceramics should crystalize in highly symmetric crystal systems, so BaLaLiWO6 characterized by a cubic structure (s.g. Fm3¯m) seems to be a good candidate for accomplishing this highly challenging task.A series of micro-crystalline samples activated by Nd3+ ion (0–20 mol%) was produced via the solid-state reaction. Nd3+ ion was chosen as a laser dopant and structural probe. For the first time, the BaLaLiWO6 crystal structure was solved from the single crystal.The originality of this research results from the unexpected substitution of divalent Ba2+ ions by trivalent Nd3+ ones till up to 7 mol% and above this value the replacement of probably both Ba2+ and La3+ ions. Exchange of Ba2+ by Nd3+ ions leads to the formation of cationic vacancies due to the charge compensation effect: 3Ba2+ → 2Nd3+ + □ (vacancy in La3+ cationic sublattice), well-manifested in unit cell expansion with consequences in the spectroscopic characteristics. XRD and SEM methods in combination with emission and absorption spectroscopic ones used at cryogenic temperature helped to determine the substantial properties of this perovskite which could find application in optics. Slightly different cationic positions substituted by the Nd3+ ion and its non-homogeneous arrangement in the Ba2+/La3+ site, detected by spectroscopy with selective excitation under laser at 77 K, is pointed out. The crystal disorder of the cation environment is revealed in broad absorption and emission bands even at low temperature.

Electrical conductivity of potassium polyferrite doped with doubly charged cations

To clarify the charge compensation mechanism and the way of alloying additives placement, the authors synthesised samples of potassium βʺ-polyferrites with a wide range of mole fraction of introduced doubly charged cations. For these samples, the authors measured the electronic conductivity, cationic conductivity, and performed X-ray diffraction (XRD) analysis. The authors identified the charge compensation mechanism in potassium β″-polyferrite when doped with divalent ions of calcium, strontium, magnesium, and zinc. The charge compensation mechanisms differ depending on the radius of the introduced doubly charged ion. The results of cationic conductivity measurements of potassium β″-polyferrites show the mobility reduction of large calcium and strontium cations of potassium ions. Such additives are quite promising for improving the mechanical strength and thermal stability of the catalyst granules. They also increase the chemical stability of the contact granules. Corrosion resistance of pellets is a critical parameter. It determines the period of effective functioning of the catalyst. The data on electronic conductivity allow one to conclude that the introduction of Mg2+, Zn2+ cations sharply reduces the electron exchange in the structure of potassium β″-polyferrite. This should inevitably cause deactivation of the catalyst, while Ca2+ and Sr2+ ions do not reduce the electron transfer rate. Moreover, using the proposed approach will intensify the research process.

Effect of Metal d Band Position on Anion Redox in Alkali-Rich Sulfides.

New energy storage methods are emerging to increase the energy density of state-of-the-art battery systems beyond conventional intercalation electrode materials. For instance, employing anion redox can yield higher capacities compared with transition metal redox alone. Anion redox in sulfides has been recognized since the early days of rechargeable battery research. Here, we study the effect of d-p overlap in controlling anion redox by shifting the metal d band position relative to the S p band. We aim to determine the effect of shifting the d band position on the electronic structure and, ultimately, on charge compensation. Two isostructural sulfides LiNaFeS2 and LiNaCoS2 are directly compared to the hypothesis that the Co material should yield more covalent metal-anion bonds. LiNaCoS2 exhibits a multielectron capacity of ≥1.7 electrons per formula unit, but despite the lowered Co d band, the voltage of anion redox is close to that of LiNaFeS2. Interestingly, the material suffers from rapid capacity fade. Through a combination of solid-state nuclear magnetic resonance spectroscopy, Co and S X-ray absorption spectroscopy, X-ray diffraction, and partial density of states calculations, we demonstrate that oxidation of S nonbonding p states to S2 2- occurs in early states of charge, which leads to an irreversible phase transition. We conclude that the lower energy of Co d bands increases their overlap with S p bands while maintaining S nonbonding p states at the same higher energy level, thus causing no alteration in the oxidation potential. Further, the higher crystal field stabilization energy for octahedral coordination over tetrahedral coordination is proposed to cause the irreversible phase transition in LiNaCoS2.

Achieving a Deeply Desodiated Stabilized Cathode Material by the High Entropy Strategy for Sodium-ion Batteries.

Manganese-based layered oxides are currently of significant interest as cathode materials for sodium-ion batteries due to their low toxicity and high specific capacity. However, the practical applications are impeded by sluggish intrinsic Na+ migration and poor structure stability as a result of Jahn-Teller distortion and complicated phase transition. In this study, a high-entropy strategy is proposed to enhance the high-voltage capacity and cycling stability. The designed P2-Na0.67Mn0.6Cu0.08Ni0.09Fe0.18Ti0.05O2 achieves a deeply desodiation and delivers charging capacity of 158.1 mAh g-1 corresponding to 0.61 Na with a high initial Coulombic efficiency of 98.2 %. The charge compensation is attributed to the cationic and anionic redox reactions conjunctively. Moreover, the crystal structure is effectively stabilized, leading to a slight variation of lattice parameters. This research carries implications for the expedited development of low-cost, high-energy-density cathode materials for sodium-ion batteries.

Achieving a Deeply Desodiated Stabilized Cathode Material by the High Entropy Strategy for Sodium‐ion Batteries

AbstractManganese‐based layered oxides are currently of significant interest as cathode materials for sodium‐ion batteries due to their low toxicity and high specific capacity. However, the practical applications are impeded by sluggish intrinsic Na+ migration and poor structure stability as a result of Jahn–Teller distortion and complicated phase transition. In this study, a high‐entropy strategy is proposed to enhance the high‐voltage capacity and cycling stability. The designed P2‐Na0.67Mn0.6Cu0.08Ni0.09Fe0.18Ti0.05O2 achieves a deeply desodiation and delivers charging capacity of 158.1 mAh g−1 corresponding to 0.61 Na with a high initial Coulombic efficiency of 98.2 %. The charge compensation is attributed to the cationic and anionic redox reactions conjunctively. Moreover, the crystal structure is effectively stabilized, leading to a slight variation of lattice parameters. This research carries implications for the expedited development of low‐cost, high‐energy‐density cathode materials for sodium‐ion batteries.

Altered electrochemistry of poly(3,4-ethylenedioxythiophene) after activation of the inserted cobalt ions

Cobalt ions were inserted into poly(3,4-ethylenedioxythiophene) by cyclic voltammetry in a 0.1 M CoNO32 aqueous solution (PEDOT(Co)). After activation of the inserted cobalt, the PEDOT(aCo) system was investigated by cyclic voltammetry, digital video electrochemistry, spectroelectrogravimetry, and coupled impedance techniques (ac-electrogravimetry) to elucidate the key role of inserted cobalt ions in the altered electrochemistry of PEDOT. The incorporation of Co2+ involves slow transfer of cations for charge compensation during Co2+⇄Co3+ conversion inside the PEDOT. This fact explains the enhanced charge storage showed by PEDOT(aCo) compared with pristine PEDOT at similar potentials. Finally, the stability of PEDOT(aCo) was investigated by cyclic voltammetry measuring at the same time current, mass and motional resistance variation during 100 cycles offering high stability at all times.

Oxygen potential measurement of U0.85Am0.15O2 at 1473, 1573, and 1673 K

Oxygen potential data of U0.85Am0.15O2−x were measured at 1473, 1573, and 1673 K by thermogravimetry. In U1−yAnyO2−x, where An stands for Pu or Am, and for a given value of “y” and Oxygen/Metal ratio, the oxygen potential of U1−yAmyO2−x is higher than that of U1−yPuyO2−x. The valence of cations in the hypostoichiometric region is similar to that of Nd-doped UO2. At the stoichiometric composition, it is estimated to be Am3+, U4+, and U5+ (for charge compensation of Am3+) in the stoichiometric composition. The experimental data were analyzed using a defect chemistry model, and a relationship connecting the oxygen-to-metal ratio, the temperature, and the equilibrium oxygen partial pressure was proposed.