Electrochemical impedance spectroscopy (EIS) is frequently used in conjunction with simple equivalent circuit models to quantify grain boundary properties in solid ion conductors. In the case of depressed impedance arcs, which are often encountered in experimental studies, an equivalent circuit including one or more constant phase elements (CPEs) is usually employed to obtain a better fit. However, as the CPE is a mathematical abstraction with no clearly defined physical meaning, the ability to accurately estimate grain boundary properties from such a model remains suspect. Depressed grain boundary impedance arcs were previously shown to occur in samples with heterogeneous grain boundary conductivity.
Here, we use numerical modeling to investigate the utility of CPE-based equivalent circuit fitting in quantifying heterogeneous grain boundary conductivity and/or permittivity. Specifically, the relationship between the equivalent circuit parameters and the mean value and distribution of the grain boundary properties is studied.
A 2D electrical model of a geometrically ideal polycrystal 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. In each sample, the conductivity, the permittivity, or both, were heterogeneous. 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. The simulated impedance spectra were then fitted to an equivalent circuit consisting of two parallel R-CPE elements in series, and conductivity and permittivity values were calculated from the circuit parameters.
An increasingly depressed impedance arc and decreased CPE exponent were correlated, as expected, with increasingly heterogeneous grain boundary properties. When only one grain boundary parameter was heterogeneous, the range or spread of parameter values could be estimated from the CPE exponent. However, this was no longer possible when both parameters were heterogeneous. In all cases, the calculated conductivity differed from the actual mean grain boundary conductivity by no more than 30% provided a geometric correction factor was included to compensate for edge effects. Simply using the Q parameter of the CPE as a capacitance results in severe error in calculating the mean grain boundary permittivity. Using an equivalent capacitance expression commonly found in the literature, the calculated permittivity could still deviate by up to 60% from the actual mean value. However, a new empirical equation allowed a more accurate estimate which differed from the mean by no more than 35%. Recent extensions of this work to quantification of the impedance of the electrode-solid electrolyte interface will also be presented.