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Finite Element Modelling of the Electrical Conductivity of Fluorite-Type Oxygen Ion Conductors

Until now, modelling of the electrical conductivity of these materials relies on the analytical solution of the dilute solution model [1]. This approach yields useful insights but it also has multiple drawbacks. First of all dilute solution theory is invalid at high doping percentages, which makes it impossible to predict the conductivity trend over a large range of dopant concentrations. Moreover experiments clearly show that the electrical conductivity reaches a maximum at a certain dopant concentration and declines when the doping percentage is further increased. This phenomenon is qualitatively explained through an association mechanism between dopant ions and vacancies. We observe that the implementation of such a mechanism in the dilute solution model leads however to an asymptotic maximal conductivity and not to a distinct maximum.

It is generally accepted that the grain boundary is enriched in vacancies relative to the grain interior, which leads to a potential difference between both regions. In existing models, this potential difference is imposed as a boundary condition to arrive at a closed form analytical solution. This approach is unable to treat very small grains; below a certain grain size bulk vacancy concentrations are not reached inside the bulk and thus there exists no reference level for the potential difference. The same problem arises when a bias current is applied to the sample; analytical models can only predict the electrical conductivity when the material is not conducting a net current.

We propose a model based on the linear phenomenological relations for a crystal lattice [5]. This model respects the crystalline nature of the material where only a limited number of lattice sites are available to the charge carriers, and a lattice site can only accommodate one charge carrier at a time. As a result, our model predicts a distinct maximum in the electrical conductivity when a dopant-vacancy association mechanism is implemented. We show that this maximum exists because the association not only limits the amount of available charge carriers, but also the amount of available diffusion paths. The doping percentage for which maximum conductivity is reached is dependent on the operating temperature.

The finite element method is used to solve our model on one-dimensional geometries. This numerical approach enables us to treat grains of arbitrarily small size and to predict electrical conductivities at any applied current density. We show that the ionic conductivity of a sample perpendicular to the grain boundary decreases with decreasing grain size. The grain boundary conductivity itself is independent of grain size, except for very small grains and small dopant concentrations.

The results predicted by our model are in good agreement with the available measurements in literature over a large range of dopant concentrations and temperatures. Our model is thus a good tool to interpret measurement data and to improve our understanding of these materials. The final goal of the model is designing optimal solid electrolytes.

[1] X. Guo and R. Waser, “Electrical properties of the grain boundaries of oxygen ion conductors: Acceptor-doped zirconia and ceria,” *Progress in Materials Science*, vol. 51, no. 2, pp. 151–210, 2006.

[2] J. Maier, “Ionic conduction in space charge regions,” *Progress in solid state chemistry*, vol. 23, no. 3, pp. 171–263, 1995.

[3] H. Mehrer, *Diffusion in solids: fundamentals, methods, materials, diffusion-controlled processes*, vol. 155. Springer, 2007.

[4] C. Peters, A. Weber, B. Butz, D. Gerthsen, and E. Ivers-Tiffée, “Grain-Size Effects in YSZ Thin-Film Electrolytes,” *Journal of the American Ceramic Society*, vol. 92, no. 9, pp. 2017–2024, 2009.

[5] D. Van Laethem, J. Deconinck, A. Hubin, D. Depla, “Finite element modelling of the ionic conductivity of acceptor doped ceria”, In preparation.