Analyzing the grain-boundary resistance of oxide-ion conducting electrolytes: Poisson-Cahn vs Poisson-Boltzmann theories

The diminished conductivity of pristine grain boundaries in oxide-ion conducting electrolytes, such as (Ce,Gd)O-2 and (Zr,Y)O-2, is widely interpreted with the Mott-Schottky space-charge model, or less frequently, with the Gouy-Chapman space-charge model. Although routinely applied to the entire compositional range of solid solutions, from dilute to concentrated, these models, being based on the Poisson-Boltzmann formalism, are limited in their range of validity to dilute solutions of point defects. Analyzing the grain-boundary properties of concentrated solid solutions with such models is expected to lead to errors and inconsistencies. In this study, we employ Poisson-Cahn theory to analyze literature data for the grain-boundary resistance of CeO2-Gd2O3 materials as a function of Gd concentration. Poisson-Cahn theory combines the Cahn-Hilliard theory of inhomogeneous systems with the Poisson equation of electrostatics and it is valid over the entire compositional range. We treat the realistic case of a restricted equilibrium: Gd accumulation profiles are frozen-in from sintering temperatures, while the oxygen-vacancy distributions are in equilibrium at sintering and (much lower) measurement temperatures. Data for the grain-boundary resistance are also analyzed with the standard analytical expressions from the Mott-Schottky and Gouy-Chapman models. Outside the domain of their validity, these expressions are found to perform poorly. In general, we emphasize the importance of treating the interfacial properties of concentrated solid solutions with physically appropriate theories.