Ab initio thermodynamics of intrinsic oxygen vacancies in ceria

Nonstoichiometric ceria(CeO${}_{2-\delta }$) is a candidate reaction medium to facilitate two-step water splitting cycles and generate hydrogen. Improving upon its thermodynamic suitability through doping requires an understanding of its vacancy thermodynamics. Using density functional theory (DFT) calculations and cluster expansion-based Monte Carlo simulations, we have studied the high-temperature thermodynamics of intrinsic oxygen vacancies in ceria. The DFT$+U$ approach was used to get the ground state energies of various vacancy configurations in ceria, which were subsequently fit to a cluster expansion Hamiltonian to efficiently model the configurational dependence of energy. The effect of lattice vibrations was incorporated through a temperature-dependent cluster expansion. Lattice Monte Carlo simulations using the cluster expansion Hamiltonian were able to detect the miscibility gap in the phase diagram of ceria. The inclusion of vibrational and electronic entropy effects made the agreement with experiments quantitative. The deviation from an ideal solution model was quantified by calculating as a function of nonstoichiometry, (a) the solid state entropy from Monte Carlo simulations, and (b) Warren-Cowley short range order parameters of various pair clusters.

Figure 1

Phonon density of states from first principles calculated under the harmonic approximation. Oxygen vacancies lead to softening of vibrational modes in CeO${}_{1.91}$ (three vacancies in a $2\times 2\times 2$ supercell) compared to CeO${}_2$.

Figure 2

ECI vs cluster diameter for a cluster expansion fit to the as calculated first principles energies (squares, denoted by CE 1). Subtracting the electrostatic energy prior to fitting and later adding it back (as an energetic correction) does not change the variation of ECIs with cluster diameter (filled triangles, CE 2). This indicates that the electrostatic interactions are captured by the cluster expansion and do not need an explicit treatment.

Figure 3

A constant $\mu$ trajectory MC simulation starting from the low-temperature ground state (CeO${}_2$) illustrating vibrational effects on the phase diagram and vacancy concentrations.

Figure 4

Miscibility gap in ceria calculated from Monte Carlo simulations. The pronounced effect of vibrations is visible from the suppression of the miscibility gap to lower temperatures and enhanced vacancy solubility in nonstoichiometric ceria. The experimental phase diagram[37] has been overlaid for comparison.

Figure 5

Entropy of CeO${}_{2-\delta }$ referenced to CeO${}_2$ computed from MC simulations at 1480 K. The deviation from ideal solution behavior (configurational disorder in Ce${}^{3+}$/Ce${}^{4+}$ and O${}^{2-}$/Vac lattice) becomes apparent even at low $\delta$. Vibrations provide entropic benefit to forming vacancies, but the marked nonideal behavior prevails.

Figure 6

Entropy change associated with forming an oxygen vacancy [see Eq. (12)] as a function of nonstoichiometry compared with experiment.[6]

Figure 7

Different vacancy pairs in the anion sublattice of a unit cell of ceria. Black boxes denote vacancies. The color scheme for the oxygen atoms corresponds with that used for plotting SRO parameters of various vacancy pairs in Fig. 8.

Figure 8

Warren-Cowley short range order parameters (${\alpha }_{lmn}$) as a function of nonstoichiometry for various pair clusters in the O${}^{2-}$/Vac sublattice at (a) 1262 K and (b) 1791 K. Of note is a strong clustering tendency along 1/2[110].