On the Utility of Constant Phase Elements to Characterize Heterogeneous Ceramic Grain Boundaries
A 2D electrical model of a polycrystalline ceramic conductor with individually defined grain boundary properties was developed, allowing impedance spectra to be generated for arbitrary distributions of grain boundary conductivity and/or permittivity. For the present study, the relevant properties of each grain boundary were assigned randomly according to a specified distribution. Heterogeneous conductivity values followed a log-normal distribution, while heterogeneous permittivity values followed an exponential distribution with a maximum equal to the bulk grain permittivity. Simulated impedance spectra (Fig. 1) were produced by solving Poisson’s equation and fitted to an equivalent circuit consisting of two parallel R-CPE elements in series. Grain boundary conductivity and permittivity values were calculated from the circuit parameters Rgb, Qgb, and ngb.
As expected, increasingly heterogeneous grain boundary properties correlated to increasingly depressed impedance arcs and reduced CPE exponent. When only the conductivity or the permittivity was heterogeneous, the spread of parameter values could be estimated from the CPE exponent. However, this was no longer possible when both parameters were heterogeneous. As one might hope, in all cases, the grain boundary conductivity determined from the CPE-based equivalent circuit was within ±30% of the actual mean grain boundary conductivity. Conversely, simple use of the CPE’s Qparameter as a capacitance results in severe error in calculating the mean grain boundary permittivity. A commonly used equivalent capacitance expression given by Brug  still caused the calculated permittivity to deviate by up to 60% from the actual mean value. However, a new empirical equation  allowed a more accurate estimate that differed from the mean by no more than 35% (Fig. 2). Recent extensions of this work to quantification of the impedance of the electrode–solid electrolyte interface will also be presented.
 J. Fleig, Solid State Ionics 131 (2000) 117–127.
 G.J. Brug et al., J. Electroanal. Chem. Interfacial Electrochem. 176 (1984) 275–295.