Oxygen nonstoichiometry and thermodynamic characterization of Zr doped ceria in the 1573-1773 K temperature range

Solar-driven thermochemical conversion of H2O and CO2 into sustainable fuels

Solar-driven thermochemical conversion of H2O and CO2 into sustainable fuels, based on redox cycle, provides a promising path for alternative energy, as it employs the solar energy as high-temperature heat supply and adopts H2O and CO2 as initial feedstock. This review describes the sustainable fuels production system, including a series of physical and chemical processes for converting solar energy into chemical energy in the form of sustainable fuels. Detailed working principles, redox materials, and key devices are reviewed and discussed to provide systematic and in-depth understanding of thermochemical fuels production with the aid of concentrated solar power technology. In addition, limiting factors affecting the solar-to-fuel efficiency are analyzed; meanwhile, the improvement technologies (heat recovery concepts and designs) are summarized. This study therefore sets a pathway for future research works based on the current status and demand for further development of such technologies on a commercial scale.

A Review of Oxygen Carrier Materials and Related Thermochemical Redox Processes for Concentrating Solar Thermal Applications

Redox materials have been investigated for various thermochemical processing applications including solar fuel production (hydrogen, syngas), ammonia synthesis, thermochemical energy storage, and air separation/oxygen pumping, while involving concentrated solar energy as the high-temperature process heat source for solid-gas reactions. Accordingly, these materials can be processed in two-step redox cycles for thermochemical fuel production from H₂O and CO₂ splitting. In such cycles, the metal oxide is first thermally reduced when heated under concentrated solar energy. Then, the reduced material is re-oxidized with either H₂O or CO₂ to produce H₂ or CO. The mixture forms syngas that can be used for the synthesis of various hydrocarbon fuels. An alternative process involves redox systems of metal oxides/nitrides for ammonia synthesis from N₂ and H₂O based on chemical looping cycles. A metal nitride reacts with steam to form ammonia and the corresponding metal oxide. The latter is then recycled in a nitridation reaction with N₂ and a reducer. In another process, redox systems can be processed in reversible endothermal/exothermal reactions for solar thermochemical energy storage at high temperature. The reduction corresponds to the heat charge while the reverse oxidation with air leads to the heat discharge for supplying process heat to a downstream process. Similar reversible redox reactions can finally be used for oxygen separation from air, which results in separate flows of O₂ and N₂ that can be both valorized, or thermochemical oxygen pumping to absorb residual oxygen. This review deals with the different redox materials involving stoichiometric or non-stoichiometric materials applied to solar fuel production (H₂, syngas, ammonia), thermochemical energy storage, and thermochemical air separation or gas purification. The most relevant chemical looping reactions and the best performing materials acting as the oxygen carriers are identified and described, as well as the chemical reactors suitable for solar energy absorption, conversion, and storage.

Cation-Deficient Ce-Substituted Perovskite Oxides with Dual-Redox Active Sites for Thermochemical Applications

Identifying thermodynamically favorable and stable non-stoichiometric metal oxides is of crucial importance for solar thermochemical (STC) fuel production via two-step redox cycles. The performance of a non-stoichiometric metal oxide depends on its thermodynamic properties, oxygen exchange capacity, and its phase stability under high-temperature redox cycling conditions. Perovskite oxides (ABO_3-δ) are being considered as attractive alternatives to the state-of-the-art ceria (CeO_2-δ) due to their high thermodynamic and structural tunability. However, perovskite oxides often exhibit low entropy change compared to ceria, as they generally have one only redox active site, leading to lower mass-specific fuel yields. Herein, we investigate cation-deficient Ce-substituted perovskite oxides as a new class of potential redox materials combining the advantages of perovskites and ceria. We newly synthesized the (Ce_xSr_1-x)_0.95Ti_0.5Mn_0.5O_3-δ (x = 0, 0.10, 0.15, and 0.20; CSTM) series, with dual-redox active sites comprising Ce (at the A-site) and Mn (at the B-site). By introducing a cation deficiency (∼5%), CSTM perovskite oxides with both phase purity (x ≤ 0.15) and high-temperature structural stability under STC redox cycling conditions are obtained. Thermodynamic properties are evaluated by measuring oxygen non-stoichiometry in the temperature range T = 700-1400 °C and the oxygen partial pressure range pO₂ = 1-10^-4 bar. The results demonstrate that CSTM perovskite oxides exhibit a composition-dependent simultaneous increase of enthalpy and entropy change with increasing Ce-substitution. (Ce_0.20Sr_0.80)_0.95Ti_0.5Mn_0.5O_3-δ (CSTM20) showed a combination of large entropy change of ∼141 J (mol-O)^-1 K^-1 and moderate enthalpy change of ∼238 kJ (mol-O)^-1, thereby creating favorable conditions for thermochemical H₂O splitting. Furthermore, the oxidation states and local coordination environment around Mn, Ce, and Ti sites in the pristine and reduced CSTM samples were extensively studied using X-ray absorption spectroscopy. The results confirmed that both Ce (at the A-site) and Mn (at the B-site) centers undergo simultaneous reduction during thermochemical redox cycling.

Redox behavior of potassium doped and transition metal co-doped Ce0.75Zr0.25O2 for thermochemical H2O/CO2 splitting

CeO₂ slow redox kinetics as well as low oxygen exchange ability limit its application as a catalyst in solar thermochemical two-step cycles. In this study, Ce_0.75Zr_0.25O₂ catalysts doped with potassium or transition metals (Cu, Mn, Fe), as well as co-doped materials were synthesized. Samples were investigated by X-ray diffraction (XRD), N₂ sorption (BET), as well as by electron paramagnetic resonance (EPR) and X-ray photoelectron spectroscopy (XPS) to gain insight into surface and bulk features, which were connected to redox properties assessed both in a thermogravimetric (TG) balance and in a fixed bed reactor. Obtained results revealed that doping as well as co-doping with non-reducible K cations promoted the increase of both surface and bulk oxygen vacancies. Accordingly, K-doped and Fe-K co-doped materials show the best redox performances evidencing the highest reduction degree, the largest H₂ amounts and the fastest kinetics, thus emerging as very interesting materials for solar thermochemical splitting cycles.

Potassium doped and co-doped ceria–zirconia show improved CO₂/H₂O splitting activity. This holds huge promise for the design of high performance systems for solar thermochemical splitting cycles allowing the production of solar fuels.

Broad Applicability of Electrochemical Impedance Spectroscopy to the Measurement of Oxygen Nonstoichiometry in Mixed Ion and Electron Conductors

Oxygen nonstoichiometry is a fundamental feature of mixed ion and electron conductors (MIECs). In this work, a general electrochemical method for determining nonstoichiometry in thin film MIECs, via measurement of the chemical capacitance, is demonstrated using ceria and ceria-zirconia (Ce_0.8Zr_0.2O_2-δ) as representative materials. A.C. impedance data are collected from both materials at high temperature (750-900 °C) under reducing conditions with oxygen partial pressure (pO₂) in the range 10^-13 to 10^-20 atm. Additional measurements of ceria-zirconia films are made under relatively oxidizing conditions with pO₂ in the range 0.2 to 10^-4 atm and temperatures of 800-900 °C. Under reducing conditions, the impedance spectra are described by a simple circuit in which a resistor is in series with a resistor and capacitor in parallel, and thickness-dependent measurements are used to resolve the capacitance into interfacial and chemical terms. Under more oxidizing conditions, the impedance spectra (of Ce_0.8Zr_0.2O_2-δ) reveal an additional diffusional feature, which enables determination of the ionic resistance of the film in addition to the capacitance, and hence the transport properties. A generalized mathematical formalism is presented for recovering the nonstoichiometry from the chemical capacitance, without recourse to defect chemical models. The ceria nonstoichiometry values are in good agreement with literature values determined by thermogravimetric measurements but display considerably less scatter and are collected on considerably shorter time scales. The thermodynamic analysis of Ce_0.8Zr_0.2O_2-δ corroborates earlier findings that introduction of Zr into ceria enhances its reducibility.

Linking the Electrical Conductivity and Non-Stoichiometry of Thin Film Ce1-xZrxO2-δ by a Resonant Nanobalance Approach

Bulk ceria-zirconia solid solutions (Ce_1−xZr_xO_2−δ, CZO) are highly suited for application as oxygen storage materials in automotive three-way catalytic converters (TWC) due to the high levels of achievable oxygen non-stoichiometry δ. In thin film CZO, the oxygen storage properties are expected to be further enhanced. The present study addresses this aspect. CZO thin films with 0 ≤ x ≤ 1 were investigated. A unique nano-thermogravimetric method for thin films that is based on the resonant nanobalance approach for high-temperature characterization of oxygen non-stoichiometry in CZO was implemented. The high-temperature electrical conductivity and the non-stoichiometry δ of CZO were measured under oxygen partial pressures pO₂ in the range of 10⁻²⁴–0.2 bar. Markedly enhanced reducibility and electronic conductivity of CeO₂-ZrO₂ as compared to CeO_2−δ and ZrO₂ were observed. A comparison of temperature- and pO₂-dependences of the non-stoichiometry of thin films with literature data for bulk Ce_1−xZr_xO_2−δ shows enhanced reducibility in the former. The maximum conductivity was found for Ce_0.8Zr_0.2O_2−δ, whereas Ce_0.5Zr_0.5O_2-δ showed the highest non-stoichiometry, yielding δ = 0.16 at 900 °C and pO₂ of 10⁻¹⁴ bar. The defect interactions in Ce_1−xZr_xO_2−δ are analyzed in the framework of defect models for ceria and zirconia.

Oxygen Exchange in Dual-Phase La0.65Sr0.35MnO3-CeO2 Composites for Solar Thermochemical Fuel Production

Increasing the capacity and kinetics of oxygen exchange in solid oxides is important to improve the performance of numerous energy-related materials, especially those for the solar-to-fuel technology. Dual-phase metal oxide composites of La_0.65Sr_0.35MnO₃-x%CeO₂, with x = 0, 5, 10, 20, 50, and 100, have been experimentally investigated for oxygen exchange and CO₂ splitting via thermochemical redox reactions. The prepared metal oxide powders were tested in a temperature range from 1000 to 1400 °C under isothermal and two-step cycling conditions relevant for solar thermochemical fuel production. We reveal synergetic oxygen exchange of the dual-phase composite La_0.65Sr_0.35MnO₃-CeO₂ compared to its individual components. The enhanced oxygen exchange in the composite has a beneficial effect on the rate of oxygen release and the total CO produced by CO₂ splitting, while it has an adverse effect on the maximum rate of CO evolution. Ex situ Raman and XRD analyses are used to shed light on the relative oxygen content during thermochemical cycling. Based on the relative oxygen content in both phases, we discuss possible mechanisms that can explain the observed behavior. Overall, the presented findings highlight the beneficial effects of dual-phase composites in enhancing the oxygen exchange capacity of redox materials for renewable fuel production.

CeTi2O6-A Promising Oxide for Solar Thermochemical Hydrogen Production

A large entropy of reduction is crucial in achieving materials capable of high-efficiency solar thermochemical hydrogen (STCH) production through two-step thermochemical water splitting cycles. We have recently demonstrated that the onsite electronic entropy of reduction attains an extreme value of 4.26 k_B at 1500 K in Ce⁴⁺ → Ce³⁺ redox reactions, which explains the high performance and uniqueness of CeO₂ as an archetypal STCH material. However, ceria requires high temperatures (T > 1500 °C) to achieve a reasonable reduction extent because of its large reduction enthalpy, which is a major obstacle in practical applications. Therefore, new materials with a large entropy of reduction and lower reduction enthalpy are required. Here, we perform a systematic screening to search for Ce⁴⁺-based oxides which possess thermodynamics superior to CeO₂ for STCH production. We first search the Inorganic Crystal Structure Database (ICSD) and literature for Ce⁴⁺-based oxides and subsequently use density functional theory to compute their reduction enthalpies (i.e., oxygen vacancy formation energies). We find that CeTi₂O₆ with the brannerite structure is the most promising candidate for STCH because it possesses three essential characteristics of an STCH material: (i) a smaller reduction enthalpy compared to ceria yet large enough to split water, (ii) a high thermal stability, as reported experimentally, and (iii) a large entropy of reduction associated with Ce⁴⁺ → Ce³⁺ redox. Our proposed design strategy suggests that further exploration of Ce⁴⁺ oxides for STCH production is warranted.

A review of defect structure and chemistry in ceria and its solid solutions

Ceria and its solid solutions play a vital role in several industrial processes and devices. These include solar energy-to-fuel conversion, solid oxide fuel and electrolyzer cells, memristors, chemical looping combustion, automotive 3-way catalysts, catalytic surface coatings, supercapacitors and recently, electrostrictive devices. An attractive feature of ceria is the possibility of tuning defect-chemistry to increase the effectiveness of the materials in application areas. Years of study have revealed many features of the long-range, macroscopic characteristics of ceria and its derivatives. In this review we focus on an area of ceria defect chemistry which has received comparatively little attention - defect-induced local distortions and short-range associates. These features are non-periodic in nature and hence not readily detected by conventional X-ray powder diffraction. We compile the relevant literature data obtained by thermodynamic analysis, Raman spectroscopy, and X-ray absorption fine structure (XAFS) spectroscopy. Each of these techniques provides insight into material behavior without reliance on long-range periodic symmetry. From thermodynamic analyses, association of defects is inferred. From XAFS, an element-specific probe, local structure around selected atomic species is obtained, whereas from Raman spectroscopy, local symmetry breaking and vibrational changes in bonding patterns is detected. We note that, for undoped ceria and its solid solutions, the relationship between short range order and cation-oxygen-vacancy coordination remains a subject of active debate. Beyond collating the sometimes contradictory data in the literature, we strengthen this review by reporting new spectroscopy results and analysis. We contribute to this debate by introducing additional data and analysis, with the expectation that increasing our fundamental understanding of this relationship will lead to an ability to predict and tailor the defect-chemistry of ceria-based materials for practical applications.

Towards Solar Methanol: Past, Present, and Future

This work aims to provide an overview of producing value-added products affordably and sustainably from greenhouse gases (GHGs). Methanol (MeOH) is one such product, and is one of the most widely used chemicals, employed as a feedstock for ≈30% of industrial chemicals. The starting materials are analogous to those feeding natural processes: water, CO2, and light. Innovative technologies from this effort have global significance, as they allow GHG recycling, while providing society with a renewable carbon feedstock. Light, in the form of solar energy, assists the production process in some capacity. Various solar strategies of continually increasing technology readiness levels are compared to the commercial MeOH process, which uses a syngas feed derived from natural gas. These strategies include several key technologies, including solar-thermochemical, photochemical, and photovoltaic-electrochemical. Other solar-assisted technologies that are not yet commercial-ready are also discussed. The commercial-ready technologies are compared using a technoeconomic analysis, and the scalability of solar reactors is also discussed in the context of light-incorporating catalyst architectures and designs. Finally, how MeOH compares against other prospective products is briefly discussed, as well as the viability of the most promising solar MeOH strategy in an international context.

Thermodynamic limits of countercurrent reactor systems, with examples in membrane reactors and the ceria redox cycle

Countercurrent reactors can be utilized in chemical reaction systems which involve either a reaction between flows of different phases, or reactions between flows separated by a selective permeable membrane. This idea is quite similar in nature to a countercurrent heat exchanger, where the inlet of one participating flow is exposed to the outlet of the opposite flow. A countercurrent configuration can therefore improve the reaction conversion extent and transport properties. Here we formulate a straightforward approach in terms of an exchange coordinate, in order to determine an upper bound of species exchange in such systems, subject to the second law of thermodynamics and conservation of mass. The methodology is independent of the specifics of reactor design and can be generally applied to determine the maximum thermodynamic benefit of using a countercurrent reactor. We then demonstrate the analysis for a number of thermochemical fuel production routes; membrane thermolysis of carbon dioxide, dry methane reforming across a membrane, reverse water gas shift across a membrane, and the thermochemical ceria cycle.

Non-Stoichiometric Redox Active Perovskite Materials for Solar Thermochemical Fuel Production: A Review

Due to the requirement to develop carbon-free energy, solar energy conversion into chemical energy carriers is a promising solution. Thermochemical fuel production cycles are particularly interesting because they can convert carbon dioxide or water into CO or H2 with concentrated solar energy as a high-temperature process heat source. This process further valorizes and upgrades carbon dioxide into valuable and storable fuels. Development of redox active catalysts is the key challenge for the success of thermochemical cycles for solar-driven H2O and CO2 splitting. Ultimately, the achievement of economically viable solar fuel production relies on increasing the attainable solar-to-fuel energy conversion efficiency. This necessitates the discovery of novel redox-active and thermally-stable materials able to split H2O and CO2 with both high-fuel productivities and chemical conversion rates. Perovskites have recently emerged as promising reactive materials for this application as they feature high non-stoichiometric oxygen exchange capacities and diffusion rates while maintaining their crystallographic structure during cycling over a wide range of operating conditions and reduction extents. This paper provides an overview of the best performing perovskite formulations considered in recent studies, with special focus on their non-stoichiometry extent, their ability to produce solar fuel with high yield and performance stability, and the different methods developed to study the reaction kinetics.

Combined Ceria Reduction and Methane Reforming in a Solar-Driven Particle-Transport Reactor

We report on the experimental performance of a solar aerosol reactor for carrying out the combined thermochemical reduction of CeO₂ and reforming of CH₄ using concentrated radiation as the source of process heat. The 2 kW_th solar reactor prototype utilizes a cavity receiver enclosing a vertical Al₂O₃ tube which contains a downward gravity-driven particle flow of ceria particles, either co-current or counter-current to a CH₄ flow. Experimentation under a peak radiative flux of 2264 suns yielded methane conversions up to 89% at 1300 °C for residence times under 1 s. The maximum extent of ceria reduction, given by the nonstoichiometry δ (CeO_2-δ), was 0.25. The solar-to-fuel energy conversion efficiency reached 12%. The syngas produced had a H₂:CO molar ratio of 2, and its calorific value was solar-upgraded by 24% over that of the CH₄ reformed.

Design principles of perovskites for solar-driven thermochemical splitting of CO2

Perovskites are attractive redox materials for thermo/electrochemical fuel synthesis. To design perovskites with balanced redox energetics for thermochemically splitting CO2, the activity of lattice oxygen vacancies and stability against crystal phase changes and detrimental carbonate formation are predicted for a representative range of perovskites by electronic structure computations. Systematic trends in these materials properties when doping with selected metal cations are described in the free energy range defined for isothermal and temperature-swing redox cycles. To confirm that the predicted materials properties root in the bulk chemical composition, selected perovskites are synthesized and characterized by X-ray diffraction, transmission electron microscopy, and thermogravimetric analysis. On one hand, due to the oxidation equilibrium, none of the investigated compositions outperforms non-stoichiometric ceria - the benchmark redox material for CO2 splitting with temperature-swings in the range of 800-1500 °C. On the other hand, certain promising perovskites remain redox-active at relatively low oxide reduction temperatures at which ceria is redox-inactive. This trade-off in the redox energetics is established for YFeO3, YCo0.5Fe0.5O3 and LaFe0.5Ni0.5O3, identified as stable against phase changes and capable to convert CO2 to CO at 600 °C and 10 mbar CO in CO2, and to being decomposed at 1400 °C and 0.1 mbar O2 with an enthalpy change of 440-630 kJ mol-1 O2.

Solar thermochemical splitting of water to generate hydrogen

Solar photochemical means of splitting water (artificial photosynthesis) to generate hydrogen is emerging as a viable process. The solar thermochemical route also promises to be an attractive means of achieving this objective. In this paper we present different types of thermochemical cycles that one can use for the purpose. These include the low-temperature multistep process as well as the high-temperature two-step process. It is noteworthy that the multistep process based on the Mn(II)/Mn(III) oxide system can be carried out at 700 °C or 750 °C. The two-step process has been achieved at 1,300 °C/900 °C by using yttrium-based rare earth manganites. It seems possible to render this high-temperature process as an isothermal process. Thermodynamics and kinetics of H₂O splitting are largely controlled by the inherent redox properties of the materials. Interestingly, under the conditions of H₂O splitting in the high-temperature process CO₂ can also be decomposed to CO, providing a feasible method for generating the industrially important syngas (CO+H₂). Although carbonate formation can be addressed as a hurdle during CO₂ splitting, the problem can be avoided by a suitable choice of experimental conditions. The choice of the solar reactor holds the key for the commercialization of thermochemical fuel production.

Splitting CO2 with a ceria-based redox cycle in a solar-driven thermogravimetric analyzer

Thermochemical splitting of CO₂ via a ceria-based redox cycle was performed in a solar-driven thermogravimetric analyzer. Overall reaction rates, including heat and mass transport, were determined under concentrated irradiation mimicking realistic operation of solar reactors. Reticulated porous ceramic (RPC) structures and fibers made of undoped and Zr⁴⁺-doped CeO₂, were endothermally reduced under radiative fluxes of 1280 suns in the temperature range 1200-1950 K and subsequently re-oxidized with CO₂ at 950-1400 K. Rapid and uniform heating was observed for 8 ppi ceria RPC with mm-sized porosity due to its low optical thickness and volumetric radiative absorption, while ceria fibers with μm-sized porosity performed poorly due to its opacity to incident irradiation. The 10 ppi RPC exhibited higher fuel yield because of its higher sample density. Zr⁴⁺-doped ceria showed increasing reduction extents with dopant concentration but decreasing specific CO yield due to unfavorable oxidation thermodynamics and slower kinetics.

Tunable thermodynamic activity of La x Sr1-x Mn y Al1-y O3-δ (0 ≤ x ≤ 1, 0 ≤ y ≤ 1) perovskites for solar thermochemical fuel synthesis

Nonstoichiometric metal oxides with variable valence are attractive redox materials for thermochemical and electrochemical fuel processing. To guide the design of advanced redox materials for solar-driven splitting of CO₂ and/or H₂O to produce CO and/or H₂ (syngas), we investigate the equilibrium thermodynamics of the La _x Sr_1-x Mn _y Al_1-y O_3-δ perovskite family (0 ≤ x ≤ 1, 0 ≤ y ≤ 1) and La_0.6Ca_0.4Mn_0.8Al_0.2O_3-δ , and compare them to those of CeO₂ as the baseline. Oxygen nonstoichiometry measurements from 1573 to 1773 K and from 0.206 to 180 mbar O₂ show a tunable reduction extent, increasing with increasing Sr content. Maximal nonstoichiometry of 0.32 is established with La_0.2Sr_0.8Mn_0.8Al_0.2O_3-δ at 1773 K and 2.37 mbar O₂. As a trend, we find that oxygen capacities are most sensitive to the A-cation composition. Partial molar enthalpy, entropy and Gibbs free energy changes for oxide reduction are extracted from the experimental data using defect models for Mn⁴⁺/Mn³⁺ and Mn³⁺/Mn²⁺ redox couples. We find that perovskites exhibit typically decreasing enthalpy changes with increasing nonstoichiometries. This desirable characteristic is most pronounced by La_0.6Sr_0.4Mn_0.4Al_0.6O_3-δ , rendering it attractive for CO₂ and H₂O splitting. Generally, perovskites show lower enthalpy and entropy changes than ceria, resulting in more favorable reduction but less favorable oxidation equilibria. The energy penalties due to larger temperature swings and excess oxidants are discussed in particular. Using electronic structure theory, we conclude with a practical methodology estimating thermodynamic activity to rationally design perovskites with variable stoichiometry and valence.

Structural changes in equimolar ceria–hafnia materials under solar thermochemical looping conditions: cation ordering, formation and stability of the pyrochlore structure

Equimolar ceria–hafnia oxides form a pyrochlore Ce₂Hf₂O₇ under the reducing conditions of a solar thermochemical looping reactor for the two-step dissociation of water or carbon dioxide.

Experimental Demonstration of the Thermochemical Reduction of Ceria in a Solar Aerosol Reactor

We report on the experimental demonstration of an aerosol solar reactor for the thermal reduction of ceria, as part of a thermochemical redox cycle for splitting H₂O and CO₂. The concept utilizes a cavity-receiver enclosing an array of alumina tubes, each containing a downward gravity-driven aerosol flow of ceria particles countercurrent to an inert sweep gas flow for intrinsic separation of reduced ceria and oxygen. A 2 kW_th lab-scale prototype with a single tube was tested under radiative fluxes approaching 4000 suns, yielding reaction extents of up to 53% of the thermodynamic equilibrium at 1919 K within residence times below 1 s. Upon thermal redox cycling, fresh primary particles of 2.44 μm mean size initially formed large agglomerates of 1000 μm mean size, then sintered into stable particles of 150 μm mean size. The reaction extent was primarily limited by heat transfer for large particles/agglomerates (mean size > 200 μm) and by the gas phase advection of product O₂ for smaller particles.

Statistical thermodynamics of non-stoichiometric ceria and ceria zirconia solid solutions

The statistical mechanics of lattice configurations are used to develop an analytical model of non-stoichiometry in ceria and ceria zirconia.