How to check and correct the time-variance of your system under EIS measurements?Latest updated: January 3, 2020
What is the problem?
For a valid impedance measurement, 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.
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 after reaching a steady-state (Fig. 1).
Time-variance is the property of a system whose parameters defining its transfer function change with 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 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 generally 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 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 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” topic for a definition of NSD), impedance data seem 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 dramatic deformation of the impedance graph.
What is the solution?
One easy way to solve this issue is to use a tool in EC-Lab® 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