  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 a 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.
