# Thermodynamic Modeling of Phase Transformations and Defects: from Cobalt to Doped Cobaltate Perovskites

Open Access

- Graduate Program:
- Materials Science and Engineering
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- September 21, 2010
- Committee Members:
- Zi-Kui Liu, Committee Chair
- Long-Qing Chen, Committee Member
- Clive Randall, Committee Member
- Jorge Sofo, Committee Member
- Michael Carolan, Committee Member

- Keywords:
- ceramics
- thermodynamic modeling
- defect chemistry
- perovskites
- CALPHAD
- density functional theory

- Abstract:
- <p>Strontium-doped lanthanum cobaltate perovskite (La<sub>1-x</sub>Sr<sub>x</sub>CoO<sub>3-δ</sub>) exhibits the unique property of simultaneous electronic and ionic conductivity, which produces large fluxes of oxygen ions across an oxygen chemical potential gradient,amongst the highest known. However, the doping of LaCoO<sub>3-δ</sub> with strontium reduces the stability of the perovskite, resulting in precipitation of unwanted phases under less extreme conditions than in the undoped composition. This a barrier to widespread use of La<sub>1-x</sub>Sr<sub>x</sub>CoO<sub>3-δ</sub> as cathodes in solid oxide fuel cells and gas separation membranes. The goal of this dissertation is to develop a thermodynamic model capable of predicting the stability of La<sub>1-x</sub>Sr<sub>x</sub>CoO<sub>3-δ</sub> at arbitrary temperatures, strontium contents, oxygen partial pressures, and gas compositions. Information regarding the defect chemistry responsible for the oxygen nonstoichiometry can also be predicted from the model.</p> <p>The thermodynamic model is constructed with the CALculation of PHAse Diagrams (CALPHAD) approach, where parameterized Gibbs energy functions are fitted to experimentally measured and theoretically predicted thermochemical and phase equilibria data. Complex, multi-component systems can be efficiently and accurately described with this approach by modeling a system's constituent subsystems, where data may be more plentiful, and then extrapolating the Gibbs energy functions of the subsystems into the desired higher-order compositional space, adding more parameters as needed. The Gibbs energy of the perovskite is described with the compound energy formalism. For the case of La<sub>1-x</sub>Sr<sub>x</sub>CoO<sub>3-δ</sub> perovskite, the two ends of the solid solution, LaCoO<sub>3-δ</sub> and SrCoO<sub>3-δ</sub>, are modeled independently and are combined to describe the complete system. For LaCoO<sub>3-δ</sub>, all experimental data in the literature are used to evaluate the model parameters. For SrCoO<sub>3-δ</sub>, on the other hand, such data is not sufficient since the perovskite transforms into brownmillerite and a two-phase mixture of Sr<sub>6</sub>Co<sub>5</sub>O<sub>15</sub>+Co<sub>3</sub>O<sub>4</sub> below about 1200 K. Thermochemical data for these neighboring phases have not been measured, so first-principles calculations, based on density functional theory, are employed to predict their Gibbs energy functions. The model predictions of the combined La<sub>1-x</sub>Sr<sub>x</sub>CoO<sub>3-δ</sub> are discussed, following an extensive statistical assessment of the published and new experimental oxygen nonstoichiometry data for the perovskite.</p> <p>Although, the CALPHAD approach coupled with first-principles calculations is fast becoming the norm for metallic alloy systems, it still remains relatively untested in the case of complex oxides. To examine the applicability of these techniques this type of material, a wide range of cobalt-related systems of varying complexity are examined by first-principles density functional theory calculations and CALPHAD modeling. The simplest case is elemental cobalt, where the 0 K magnetic and structural phase transformations with pressure are predicted. The thermodynamics of the second-order magnetic Curie transformation with temperature in FCC cobalt are then examined with a partition function approach, with predictions of the heat capacity and Curie temperature.</p> <p>Since the Gibbs energy function for several complex cobaltates will be predicted for the CALPHAD modeling of the perovskite, the capability of first-principles methods to predict the vibrational contribution to the free energy for cobaltates is examined for a case where the predictions can be compared to experiments, Co<sub>3</sub>O<sub>4</sub> spinel. A scheme to efficiently predict the vibrational contribution to the free energy is developed, utilizing the Debye-GrÃ¼neisen Model and harmonic phonon calculations with the supercell approach. It is found that this scheme can predict the heat capacity and entropy of Co<sub>3</sub>O<sub>4</sub> with sufficient accuracy for CALPHAD modeling, with error on the order of 2-3 kJ/mol-atom in the Gibbs energy at around 1000 K. The heat capacity, room temperature entropy, and enthalpy of formation of Sr<sub>3</sub>Co<sub>2</sub>O<sub>5</sub> brownmillerite and Sr<sub>6</sub>Co<sub>5</sub>O<sub>15</sub> are then predicted, for use in the CALPHAD modeling of SrCoO<sub>3-δ</sub>. However, due to errors in the prediction of the brownmillerite entropy and enthalpy of formation, the stability from for the two phases is incorrect, with brownmillerite more stable at all temperatures. This error is corrected by treating the the enthalpy of formation and entropy as model parameters in CALPHAD with the experimental phase equilibria data.</p> <p>Following the results of the CALPHAD modeling of the LaCoO<sub>3-δ</sub> and SrCoO<sub>3-δ</sub> perovskites, several predictions are made. For instance, charge disproportionation of Co<sup>+3</sup> in LaCoO<sub>3-δ</sub> is on the of 40% in air. Similarly, the presence of Co<sup>+2</sup>, Co<sup>+3</sup>, and Co<sup>+4</sup> is predicted at around 1200 K. However, the combination of the LaCoO<sub>3-δ</sub> and SrCoO<sub>3-δ</sub> models, assuming ideal mixing between La<sup>+3</sup> and Sr<sup>+2</sup> ions, predicts oxygen nonstoichiometry that agrees well at lanthanum-rich compositions but gives poor agreement at strontium-rich compositions, particularly at low temperatures. It is speculated that this is due to defect-defect interactions and the formation of complex defect associates for strontium contents near SrCoO<sub>3-δ</sub>, suggested by the results of the statistical analysis of oxygen nonstoichiometry data. Several attempts are made to improve the agreement of the CALPHAD model with the experimental oxygen nonstoichiometry data by including interaction parameters to describe regular and sub-regular interactions between La<sup>+3</sup> and Sr<sup>+2</sup>. Although better agreement is achieved, a limited set of parameters capable of agreement across the entire composition space of La<sub>1-x</sub>Sr<sub>x</sub>CoO<sub>3-δ</sub> was not found. More data concerning the nature of defects at strontium-rich compositions is requested.</p>