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A model for ionic and electronic grain boundary transport through thin films,scales or membranes with columnar grain structure is introduced. The grainstructure is idealized as a lattice of identical hexagonal cells - a honeycombpattern. Reactions with the environment constitute the boundary conditions anddrive the transport between the surfaces. Time-dependent simulations solvingthe Poisson equation self-consistently with the Nernst-Planck flux equationsfor the mobile species are performed. In the resulting Poisson-Nernst-Plancksystem of equations, the electrostatic potential is obtained from the Poissonequation in its integral form by summation. The model is used to interpretalumina membrane oxygen permeation experiments, in which different oxygen gaspressures are applied at opposite membrane surfaces and the resulting flux ofoxygen molecules through the membrane is measured. Simulation results involvingfour mobile species, charged aluminum and oxygen vacancies, electrons, andholes, provide a complete description of the measurements and insight into themicroscopic processes underpinning the oxygen permeation of the membrane. Mostnotably, the hypothesized transition between p-type and n-type ionicconductivity of the alumina grain boundaries as a function of the appliedoxygen gas pressure is observed in the simulations. The range of validity of asimple analytic model for the oxygen permeation rate, similar to the Wagnertheory of metal oxidation, is quantified by comparison to the numericsimulations. The three-dimensional model we develop here is readily adaptableto problems such as transport in a solid state electrode, or corrosion scalegrowth.

A model for time-dependent grain boundary diffusion of ions and electrons through a film or scale, with an application to alumina