I prefer to believe the result from EIS is more presentative to the electrochemical system. The ESR by EIS contains electrode resistance, electrolyte resistance and interfacial resistance combined in series, and thus is more reliable for an electrochemical device. As you mentioned, for the galvanostatic method the treatment is based on Ohm's law while most electrochemical systems are not resistors and their behaviors will deviate from Ohm's law.

For your 2I question, I believe your reasoning is correct. I haven't seen anyone ascribed the 2I to Faradaic processes.

+1 Tianyu.

EIS provides a lot more detail than just the Vdrop (or current interrupt) method, but only if your system is capable of making the EIS measurement. Many systems have trouble measuring EIS if the impedance is less than 1 mOhm (and some have issues even at 100 mOhms). ESR values can be in the uOhm region. At that level, you need to make sure that your EIS system is measuring the cell, and that the signal is not dominated by the cabling and system performance limits.

To your other question: current controlled charge and discharge doesn't care whether the current is Faradaic or not. The voltage profile is a better indicator of that. So long as the voltage is changing in a linear fashion with constant current, the current is non-Faradaic. If your voltage reaches plateaus where it doesn't change (much) while the current remains constant, then that is (more) likely the result of a Faradaic process.

Thank you Tianyu Liu and Jacob Ketter for sharing your thoughts/knowledge. Jacob, is this small impedance from micro systems? Any references?

Tianyu Liu , my guess is that at fast transient on charge discharge, the only response from the system is from ESR (capacitor works as wire, no time to response). So, maybe, that is why ohm's law fits. Actually problem is that I have seen so many papers considering Vdrop=ESR/Idischarge , which is a very silly mistake, right? We should apply current variation Vdrop=ESR/2I for charge discharge measurement.

Small impedance actually comes from large systems. The traditional setup for impedance is that it increases with distance but decreases with area. A 1 F double layer capacitor may have 0.01 Ohm resistance, while a 3000 F cap will have 0.00001 Ohm impedance. Micro[electrodes] tend to have high impedance (MOhm+).

So far as references, there are several applications notes including (https://www.gamry.com/application-notes/EIS/testing-electrochemical-capacitors-part-3-electrochemical-impedance-spectroscopy/) and I gave a presentation at ISE last year on making accurate EIS measurements of low Z [supercaps] that addressed several issues that can arise and limit your measurements when the system impedance is less than 1 mOhm. I don't have a link for that but I can send you the presentation.

I believe Vdrop=ESR/2I is the correct form. I clearly remember that I spent about a day figuring out why there is 2 in the equation. The papers published on electrochemical applications occasionally contain false information and unfortunately, they have been published and the reviewers failed to mention it to the authors. I try my best to scrutinize every paper I review to prevent releasing false information for readers.

according to supercapacitor manufacturer, ESR (DC) is measured from GCD as ESR= V/I, (no 2I factor involved). ESR (DC) contains all the resistances involved in the system and it can be used to determine the power density. The discharge current is applied generally as 10mA/F with recording time kept as 10ms or 1s.

ESR (AC) is measured at 1000Hz freq from the nyquist plot.

These are the standard measurement method used by manufacturer.

Please check this paper for ESR estimation from charge-discharge curve.

https://doi.org/10.1002/aenm.201401401

If there is a rest time in between the charging and discharging cycles, then ESR can be calculated as Vdrop/Idischarge , considering that the initial

response is only due to the voltage drop.

However, if you are doing continuous symmetrical constant current charge-discharge, we have to take care of the voltage drop during charging as well as discharging cycles, hence ESR = Vdrop/(2*I)

Regarding the accuracy of ESR measurement based on EIS:

I think the electrode resistance and sometimes the charge transfer resistances of many systems varies with frequency. This is reflected in the varying Real part of the EIS characteristics (Nyquist Plot). As the real part is considered as the ESR , it is difficult to choose the frequency at which we can calculate the nominal value of ESR. However, many papers have taken the low frequency component.

IEC 62391 says 30 min REST period between constant current charge and discharge and the formula Vdrop/Idischarge is used by the manufacturers.

After all in symmetric system you should use 2I factor.

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