High-concentration point-defect chemistry: Statistical-thermodynamic approach applied to nonstoichiometric cerium dioxides

Many properties of crystalline solids are controlled by the presence of point defects. Traditionally the concentrations of these defects are calculated using mass action laws that describe the reactions among the defects and with the environment. This formulation, however, is not applicable in the high-defect-concentration regime because it is based on assumptions that are valid only at the limit of dilute defect concentration. In the present work a statistical thermodynamic approach is used to develop a defect chemistry formulation that has general applicability at all levels of defect concentrations. This is accomplished by deriving an expression for the virtual chemical potential of point defects from the Gibbs free energy that includes contributions from short-range and long-range Coulombic interactions, and from a configurational entropy that incorporates generalized exclusion effects. By balancing the chemical potentials of different species and the environment as required under thermodynamic equilibrium conditions, one can derive equations that govern the concentrations of defects. At the limit of low defect concentration, the current approach is shown to be equivalent to that of the traditional theory. This formulation is demonstrated by working out an example in detail using undoped CeO_2-δ as a model system. The concentrations of different species are calculated as functions of T and PO2 using a numerical routine based on the conjugate gradient algorithm. The oxygen deficiency {δ} is obtained as the net concentrations of all oxygen vacancy containing species, and is found to be in good agreement with literature data. The effects of defect interactions and exclusions are discussed, and the thermodynamic activities of different species are calculated as functions of environmental parameters.