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

Abstract

The diminished conductivity of pristine grain boundaries in oxide‐ion conducting electrolytes, such as (Ce,Gd)O2 and (Zr,Y)O2, 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.

The grain‐boundary resistance of CeO2 ceramics: A combined microscopy‐spectroscopy‐simulation study of a dilute solution

“…(Under very reducing conditions, singly‐charged oxygen vacancies, and even neutral oxygen vacancies, have to be taken into account, but in our study we remain in the oxidizing regime and so both these species can be safely neglected.) For the dilute regime ( < 1%), we also neglect defect–defect interactions . We note that the electrons are localized on Ce 4+ ions as small polarons, forming Ce 3+ ions.…”

“…The increased resistance of GBs compared to that of bulk ceria is attributed solely to the presence of SCLs adjacent to the GB core. That is, all other possible contributions to the grain‐boundary resistance, for example, from the GB core, are assumed to be negligible . For our continuum‐level simulations we consider a one‐dimensional grain of length , with GBs at and .…”

“…For the analytical description of the GB resistance the two extreme, simple cases, Mott–Schottky and Gouy–Chapman, are often used. The resulting ratios of the bulk to total conductivity can be expressed in terms of the Debye length , and the space‐charge potential …”

“…The analysis of the SCLs at the GBs of such concentrated solid solutions, on the other hand, has been based on the dilute‐solution approximation (Poisson–Boltzmann ansatz), in which defect–defect interactions are neglected. This discrepancy will lead necessarily to inconsistencies and errors …”

“…The attraction of the dilute‐solution approximation is that analytical solutions for the GB resistance can be derived for two extreme, simple cases: (a) the Mott–Schottky case, for which the dopant concentration is constant in the SCLs (Figure A); and (b) the Gouy–Chapman case, for which the dopant accumulation profile is in electrochemical equilibrium (Figure B). For concentrated solid solutions, neither Mott–Schottky nor Gouy–Chapman is appropriate, but even in the dilute case both approaches do not constitute a correct description of SCLs in ceria samples. Ceria is characterized, namely, by dopant cations that are mobile at high (sintering) temperatures but immobile at the (lower) temperatures of the impedance measurements.…”