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The grain-size dependence of impedance spectra of electroceramics is simulated, based on a brick layer model with explicit consideration of parallel boundaries. A constant grain interior conductivity is assumed and the grain boundary (gb) conductivity is varied. Parallel and series gb are assumed isotropic, i.e. charge transport along and across gb is supposed identical. Various grain sizes between 500 and 0.5 nm are studied and impedance spectra are shown in complex plane and Bode representations.

Impedance spectra ͑IS͒ of a polycrystalline sample with macroscopic dimensions are analyzed based on the brick layer model with parallel and series grain boundaries ͑gb͒. The gb width is taken as 0.5 nm, in accordance with the classical Fisher model of gb diffusion. Three mean grain size values of 500, 50, and 5 nm are simulated, corresponding to a change from microcrystalline to nanocrystalline ceramics. A constant conductivity of the grain interior (10 Ϫ3 S/m͒ and of the series gb (10 Ϫ6 S/m͒ is assumed. Impedance spectra are simulated for parallel gb conductivity values between 10 Ϫ6 ͑blocking͒ and 10 Ϫ1 S/m ͑conducting͒. Two impedance arcs are observed in all cases, but the conventional attribution of the high-frequency arc to the grain interior and the low-frequency semicircle to the series gb is valid only when the parallel gb contribution is insignificant, i.e., for low parallel gb conductivity and/or large mean grain size. Impedance spectroscopy ͑IS͒ is a well-known and powerful technique to characterize the overall electrical conductivity ͑electronic and ionic͒ of ceramic materials. Conventional IS uses a sinusoidal electrical voltage signal of small amplitude at a given frequency and analyzes the corresponding current signal. The investigated frequency range lies typically between 10 Ϫ1 and 10 6 Hz. A first classical paper investigating zirconia ceramics was published by Bauerle in 1969, 1 who showed that the electrical response of bulk and grain boundaries ͑gb͒ is observed at different ac frequencies and can therefore be separated. The different angular frequencies of relaxation of bulk and boundaries are related to the specific electric and dielectric properties, i.e., the ratio of conductivity over dielectric permittivity.IS analysis is generally based on the brick layer model ͑BLM͒, first proposed by Burgraaf and co-workers, 2,3 where cubic grains and uniform gb are assumed. Throughout this paper, we define parallel gb as those parallel to the current flow and series gb as those perpendicular to the current flow. In microcrystalline ceramics, where the effective gb width is negligible compared to the grain size, the contribution of parallel grain boundaries ͑gbs͒ can be neglected, resulting in an equivalent circuit with two parallel resistorcapacitor elements, one for the grains and one for the gbs, connected in series. 4 Finite element calculations in microcrystalline ceramics considered deviations from the cubic grain shape, a grain size distribution, imperfect contacts between grains ͑porosity or secondary phases͒, and inhomogeneous gbs ͑conductivity distribution͒. 5 These calculations showed that the BLM is reliable for relatively narrow and homogeneous grain size and grain shape distributions. In case of a very broad or inhomogeneous grain size distribution with blocking gbs, 6 current detours through large grains and around regions with small grains are observed, leading to a change of the gb arc. Current constrictions due to pores between grains can also modify the gb arc. In these cases, the...