How to check and correct the time variance of your system under EIS measurementsLatest updated: June 2, 2021
What is the problem?
For valid impedance measurements, the system under investigation should be linear, stable, causal, and stationary. EC-Lab®’s THD Quality Indicator can be used to test the linearity of the system. The THD Quality indicator, as well as the NSD (Non-Stationary Distorsion) and NSR (Noise to Signal Ratio), are also described in the BioLogic Learning center article “How to make reliable EIS measurements with your potentiostat or battery cycler”
Stationarity here means two different things: steady-state and time-invariance.
Steady-state is the state of a system that is not in a transient state. For example, an R/C circuit with a definite time constant is submitted to a potential or current step. Its response will vary over time until it reaches a steady state (Fig. 1).
Time variance is the property of a system whose parameters defining its transfer function change over time. For example, a corroding electrode whose polarization resistance changes over time, either because of corrosion or of passivation, is a time-variant system. A discharging battery also sees its various resistances (charge transfer, diffusion) evolve during a discharge (Fig. 2). Sometimes steady-state and time-invariance are difficult to distinguish.
The effects of transient state or time-variance on EIS measurements are generally visible at lower frequencies, because the changes are usually slower than the high frequencies used in EIS, say from 1 MHz to 10 Hz. When the potentiostat measurement time is long enough to capture the time variance, its effects will be seen. In the case of a change of polarization resistance of a corroding electrode, the effects can be seen at frequencies around or below 1 Hz. What happens is that the electrochemical impedance data at lower frequencies are deformed, as seen in Fig. 3.
In Fig. 3, the data shown below 32 mHz are due to the change of the polarization resistance of the system with time. As can be seen on the Non-Stationary Distorsion (NSD) factor shown in Fig. 4 (See “How to make reliable EIS measurements with your potentiostat or battery cycler” article for a definition of NSD), the impedance data seems to be affected by steady-state and time-variance effects from 1.1 Hz as it is the frequency at which the NSD increases. At 32 mHz the NSD reaches a value as small as 0.58 % and it can be seen in Fig. 3 that such a small value can be representative of the dramatic deformation of the impedance graph.
What is the solution?
One easy way to solve this issue is to use the analysis tool in the EC-Lab® potentiostat software called Z Inst, which implements a method discovered by Z. Stoynov and B. Savoya . The basic steps of this method are the following:
- Several impedance graphs are acquired sequentially on a system that changes with time (Fig. 5a).
- The impedance data are plotted as a function of time (Fig. 5b).
- The data at the same frequencies are connected and interpolated
- We obtain an impedance surface
- By choosing cross-sections of this tunnel we can have instantaneous impedance data (Figs. 5c and d).
The impedance graphs shown in Figs. 5c and d can be considered to have been corrected from the time variance and the change of polarization resistance of the corroding electrode. ZFit can be used to determine all the relevant parameters such as the double-layer capacitance Cdl, the charge transfer Rct, and the polarization Rp which is the limit at low frequencies. It should be noted that most of the impedance graphs have an inductive part at low frequencies, which shows that the reduction reaction of protons involves an adsorption step .
- Z. Stoynov, B. Savoya, J. Electroanal. Chem. 112 (1980) p. 157
- J.-P. Diard, P. Landaud, B. Le Gorrec, C. Montella, J. Electroanal. Chem. 255 (1988) p. 1