A model for ionic and electronic grain boundary transport through thin films,
scales or membranes with columnar grain structure is introduced. The grain
structure is idealized as a lattice of identical hexagonal cells - a honeycomb
pattern. Reactions with the environment constitute the boundary conditions and
drive the transport between the surfaces. Time-dependent simulations solving
the Poisson equation self-consistently with the Nernst-Planck flux equations
for the mobile species are performed. In the resulting Poisson-Nernst-Planck
system of equations, the electrostatic potential is obtained from the Poisson
equation in its integral form by summation. The model is used to interpret
alumina membrane oxygen permeation experiments, in which different oxygen gas
pressures are applied at opposite membrane surfaces and the resulting flux of
oxygen molecules through the membrane is measured. Simulation results involving
four mobile species, charged aluminum and oxygen vacancies, electrons, and
holes, provide a complete description of the measurements and insight into the
microscopic processes underpinning the oxygen permeation of the membrane. Most
notably, the hypothesized transition between p-type and n-type ionic
conductivity of the alumina grain boundaries as a function of the applied
oxygen gas pressure is observed in the simulations. The range of validity of a
simple analytic model for the oxygen permeation rate, similar to the Wagner
theory of metal oxidation, is quantified by comparison to the numeric
simulations. The three-dimensional model we develop here is readily adaptable
to problems such as transport in a solid state electrode, or corrosion scale
growth
