Numerical Simulations of Microelectrode Impedance Spectra

74 4 Results

measured on a La0.6Sr0.4CoO3-δ thin film on YSZ in a dry oxygen atmosphere at a much lower temperature of 600°C.^[161]

4.3 EIS Measurements on Thin-Film Microelectrodes 75

Table 4.5. Material properties of BiSCF at 750°C and p(O2) = 0.2 bar obtained from experiment (microelectrode EIS, (thin-film) electrical conductivity measurements, and TG experiments on bulk samples) and FEM simulations with a tip radius of 4.1 µm as described in the text.

Rs

(Ω cm²)

C^δ (kF cm^-3)

σel

(S cm^-1) exp. sim. exp. bulk sim. exp. sim.

Bi0.2Sr0.8FeO3-δ 3.8 3.8 1.0 2.8 12 12 -^**

Bi0.5Sr0.5FeO3-δ 4.6^* 4.7 0.12^* 0.43 1.0 1.3 0.46 Bi0.8Sr0.2FeO3-δ 9.7 11.2 0.015 0.18 0.10 0.71 0.07

*The values were acquired on a smaller number of samples than those in Table 4.4 and are exclusively used in this chapter. Furthermore, RHF = 42 kΩ, RIF = 0.76 Ω cm², and CIF = 54 × 10^-5 F cm^-2 was used.

**The value could not be adjusted due to the absence of an intermediate-frequency semicircle in the experimental impedance spectra.

quantitative agreement. Particularly the low-frequency (apparent chemical) capacitance deduced from the simulated spectrum of 11 F cm^-3 differs by about one order of magnitude from the low-frequency capacitance of 120 F cm^-3 from the experimental spectrum used as input estimate for the simulation. In order for CLF

(or the peak frequency of the low-frequency semicircle) from experiment and simulation to agree, the input chemical capacitance has to be increased to 1.3 kF cm^-3. The input chemical capacitance, however, represents the true chemical capacitance of the electrode material (provided that the numerical model is correct). Hence, the true chemical capacitance of Bi0.5Sr0.5FeO3-δ is by about one order of magnitude higher than the apparent chemical capacitance derived from the experimental microelectrode impedance spectrum.

To adjust the intermediate-frequency semicircle of the simulated impedance spectrum, the input values for the electrical conductivity of the electrode material and the tip radius of the probe needle were varied. As evident from Fig. 4.21a, the intermediate-frequency resistance mainly depends on the electrical conductivity.

The intermediate-frequency capacitance, in contrast, is determined by both electrical conductivity and tip radius (Fig. 4.21b). The optimum input values for the two quantities taken from Fig. 4.21 are given in Table 4.5 together with the optimum input values for the surface resistance and chemical capacitance (slightly changed from the above-mentioned values after adjustment of the intermediate-frequency semicircle). From the final microelectrode impedance spectrum simulated using these values (Fig. 4.20), RIF and CIF were estimated to 0.74 Ω cm² and 4.2 × 10^-4 F cm^-2 corresponding to 97% and 78% of the results from experiment. The exact reproduction of the experimental intermediate-frequency semicircle would require simultaneous optimization of all input parameter values which is, however, beyond the scope of this study. Furthermore, the high-frequency axis intercepts of simulated and experimental impedance spectrum do

76 4 Results

3.1 2.7 2.5

2.3 2.1 1.9 1.7 1.5

1.3 1.1

0.90 0.70

0.50

0.30

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 3

4 5 6 7 8 9

(a) 10

Tip radius (µm)

Electrical conductivity (S cm^-1)

4.0E-04 6.0E-04 8.0E-04

1.0E-03 1.2E-03 1.4E-03

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 3

4 5 6 7 8 9

(b) 10

Tip radius (µm)

Electrical conductivity (S cm^-1)

Fig. 4.21. Dependence of (a) intermediate-frequency resistance (in Ω cm²) and (b) intermediate-frequency capacitance (in F cm^-2) on the electrical conductivity and tip radius simulated for a Bi0.5Sr0.5FeO3-δ microelectrode with 60 µm in diameter on a YSZ single crystal substrate at 750°C and p(O2) = 0.2 bar.

not completely coincide (Fig. 4.20) possibly due to significant contact resistance between electrode and probe needle not accounted for in the simulation. The simulated electrical potential within a Bi0.5Sr0.5FeO3-δ microelectrode is displayed in Fig. 4.22 and shows lateral nonuniformity in particular in the high-frequency regime.

Microelectrode impedance spectra of the other Co-free BiSCF compositions were modeled according to the procedure depicted for Bi0.5Sr0.5FeO3-δ, that is, by separate adjustment of the different input parameters (but with the tip radius fixed at 4.1 µm) starting from the values in Table 4.4 and Fig. 4.10. The reconciliation of the intermediate-frequency semicircle was, however, omitted for most of the compositions except Bi0.8Sr0.2FeO3-δ, since the very small semicircle obtained via simulation was not clearly visible in the experimental spectra owing

(a)

(b)

Fig. 4.22. Electrical potential distribution (in V) at (a) 2 × ¹⁰⁴ Hz and (b) 1 × ^10-2 Hz simulated for a Bi0.5Sr0.5FeO3-δ microelectrode with 60 µm in diameter on a YSZ single crystal substrate with a tip radius of 4.1 µm at 750°C and p(O2) = 0.2 bar.

4.3 EIS Measurements on Thin-Film Microelectrodes 77

to the poor data quality in the respective frequency range. Most strikingly, the apparent chemical capacitances from the experimental impedance spectra are substantially smaller than the true chemical capacitances used to reproduce these spectra (Table 4.5), in accordance with the observations for Bi0.5Sr0.5FeO3-δ. The true chemical capacitances, on the other hand, in most cases even exceed the values extracted for bulk samples from TG measurements. The reason for these discrepancies is not yet clear. The true surface resistance of Bi0.8Sr0.2FeO3-δ was found to be by 15% higher than the apparent surface resistance or low-frequency resistance from experiment and might, therefore, partially be covered by the intermediate-frequency resistance. A strong deviation of the thin-film electrical conductivity of Bi0.8Sr0.2FeO3-δ determined in the course of the simulations from the value from electrical conductivity measurements was noticed (which could not be reduced significantly by decreasing the tip radius), but the electrical conductivity of BiSCF thin films on YSZ might be different from that on MgO due to differences in the film microstructure (chapter 4.2.1).

To summarize, all features observed in BiSCF microelectrode impedance spectra could be reproduced qualitatively and, with the exception of the high-frequency axis intercept, nearly quantitatively using the numerical model of Lynch et al..^[107]

This model does not take into account rate-limiting electrode-electrolyte interfacial processes. The good agreement between experimental and simulated spectra hence implies that these interfacial processes do not need to be considered in the interpretation of the experimental impedance spectra. The increased high-frequency axis intercept and the intermediate-high-frequency semicircle are most likely caused by high sheet resistance resulting from the exceptionally low electronic conductivity of BiSCF. Further experimental findings supporting this interpretation will be presented in the following chapters. The simulations revealed that the true chemical capacitance of an electrode material can not directly be extracted from experimental microelectrode impedance spectra with the equivalent circuit in Fig. 4.18 in the presence of significant sheet resistance. The apparent chemical capacitance deduced from such spectra is considerably smaller than the true chemical capacitance of the electrode material. The true surface resistance, on the other hand, is only slightly modified by sheet resistance for the materials under investigation.

From the simulations it also becomes evident that the equivalent circuit in Fig. 4.18 is actually not appropriate to evaluate BiSCF microelectrode impedance spectra. This equivalent circuit is based on a one-dimensional physical model.^[158]

The resistance and capacitance values extracted with it can strictly speaking not be correlated directly to physical properties of the BiSCF electrode. To account for sheet resistance effects, a two-dimensional model has to be applied, which can not simply be represented by an electrical circuit. The evaluation of impedance spectra as described above using the model of Lynch et al.^[107] is, however, time-consuming and not suited to be done routinely. Therefore, the equivalent circuit in Fig. 4.18

78 4 Results

was used, nevertheless, in this study for the evaluation of microelectrode impedance spectra on a more or less phenomenological basis. This approach seems to be justified, since the quantity of interest, the surface resistance of oxygen exchange, is in most cases still represented reasonably well by the low-frequency resistive element of the equivalent circuit.