The distribution of mobile charge carriers in the space-charge regions at grain boundaries of ceramic materials was modeled. Delocalization effects were neglected. That is, ionic defects or polarons were considered. The calculations were performed for cubic-shaped grains of equal size. When considering the size dependence, the standard free chemical potentials of the defects rather than the specific grain-boundary charge density or the  boundary defect concentration were set to be constant in accordance with the core space-charge model. Although specific edge and corner effects were neglected in the present analysis and hence the structural potentials were invariant along grain boundaries, the accumulation or depletion of charge carriers turns out to be inhomogeneous along the grain boundary and to be particularly pronounced near to grain edges and grain corners if the grain size was smaller than four Debye lengths. Especially the accumulation near to grain edges could have a strong influence on the effective conductivity (although being a purely geometrical effect). The modeling also predicted that a contact of two grains that differ only in size, led to a redistribution of mobile ions between grains, provided that either one or both grain sizes were smaller than the double width of space-charge layers. Such a charge transfer between the grains could be viewed as so-called hetero-size charging.

Modeling of Space-Charge Effects in Nanocrystalline Ceramics - the Influence of Geometry. I.Lubomirsky, J.Fleig, J.Maier: Journal of Applied Physics, 2002, 92[11], 6819-27