Dear Ibrahim,

just few points that might be helpful:

1. The way to solve your problem, still  using your various quasi-automatic fits, is to increase your range of measured frequencies. In that way you will capture the "second semicircle".

2. If ad 1. can not be performed (for example you do not have the instrument), I would suggest to represent your data in a proper form, where the various equivalent circuit elements can be often very well identified already in the raw data. The representation in question is the log/log plot of real and imaginary part of the impedance versus frequency and/or real and imaginary part of the corresponding complex capacitance. You can then  try manual fits (very often better than some automatic fits!) to various R,C,L models.

3. Nyquist type plots are rather unfortunate historical legacy that still prevails to this date. They tell you really nothing more than what you already see in the raw data. They completely distort what is the most important independent variable, the frequency of the measurement !! it is my guess that that might be the partial reason for you not being successful with your fitting. To fit the data you showed above would normally take some 20-30 min of manual fit analysis to the complex impedance/capacitance.

4. And lastly word of caution. The circuit terms like constant phase element, Warburg impedance and others are purely phenomenological fitting parameters that tell you very little about real physics/chemistry going on in your system. Is then better to describe the response in the more general way, for example using Nagami-Havrilliak expression for "universal response" capacitance, often seen in condensed phase.

5. And an extra comment. If I would be to fit an R,C,L equivalent circuit to your data, it will consist of two parallel R,C, terms in series. The high frequency R,C term consists of the geometrical capacitance of the electrolyte and the corresponding dc resistance of the electrolyte. This term is called a bulk term. The metal electrode- electrolyte interface is then represented by the second R,C term (lower frequencies). Here the capacitance is more or less the Helmholtz double layer capacitance and the resistance is the dc current through the interface. Parallel to these two elements I will add Nagami-Havrilliak  universal capacitor. This term might already capture possible redox reactions at the interface.

I hope this will help you on the way

With bets regards

Petr

Dear Ibrahim,

just few points that might be helpful:

1. The way to solve your problem, still  using your various quasi-automatic fits, is to increase your range of measured frequencies. In that way you will capture the "second semicircle".

2. If ad 1. can not be performed (for example you do not have the instrument), I would suggest to represent your data in a proper form, where the various equivalent circuit elements can be often very well identified already in the raw data. The representation in question is the log/log plot of real and imaginary part of the impedance versus frequency and/or real and imaginary part of the corresponding complex capacitance. You can then  try manual fits (very often better than some automatic fits!) to various R,C,L models.

3. Nyquist type plots are rather unfortunate historical legacy that still prevails to this date. They tell you really nothing more than what you already see in the raw data. They completely distort what is the most important independent variable, the frequency of the measurement !! it is my guess that that might be the partial reason for you not being successful with your fitting. To fit the data you showed above would normally take some 20-30 min of manual fit analysis to the complex impedance/capacitance.

4. And lastly word of caution. The circuit terms like constant phase element, Warburg impedance and others are purely phenomenological fitting parameters that tell you very little about real physics/chemistry going on in your system. Is then better to describe the response in the more general way, for example using Nagami-Havrilliak expression for "universal response" capacitance, often seen in condensed phase.

5. And an extra comment. If I would be to fit an R,C,L equivalent circuit to your data, it will consist of two parallel R,C, terms in series. The high frequency R,C term consists of the geometrical capacitance of the electrolyte and the corresponding dc resistance of the electrolyte. This term is called a bulk term. The metal electrode- electrolyte interface is then represented by the second R,C term (lower frequencies). Here the capacitance is more or less the Helmholtz double layer capacitance and the resistance is the dc current through the interface. Parallel to these two elements I will add Nagami-Havrilliak  universal capacitor. This term might already capture possible redox reactions at the interface.

I hope this will help you on the way

With bets regards

Petr

Dear Ibrahim,

Since you perform EIS simulations using commercial softwares I believe that it would be useful to mention a few points regarding the working principles of the "black box" fitting tools embedded in such fitting softwares.

The general idea of a least square problem is rather simple. The main goal is to fit a parameterized function to a set of measured data points by minimizing the sum of squares of the errors between the data points and the function. The parameterized function in your case is derived through the predefined equivalent circuit, EQ (the one you input prior to each simulation experiment). During the fitting procedure the algorithm (usually a Levenberg-Marquardt type method) changes function parameters values in a specified range and always starting from predefined initial values, until a minimum error is attained (chi-square value in EIS experiments). One of the key points in the whole procedure is the choice of the initial values. In order to understand the importance of the initial values consider the following simplified example: let us assume a function with two minimums, at (x1,y1) and (x2,y2) where y2

Dear Petr and Anthanois,

Thank you very much for your valuable input. Your comments are very informative and helpful indeed! J

Please note I cannot go to lower frequencies. Following your comments, I tried 2 different approaches:

1st approach: To construct 2 parallel RC elements in series or parallel, as advised. I tried a circuit with 2 parallel RCPE elements in series, and I couldn’t get a fit, but I got a good fit using a 2 parallel RCPE elements, as in the figure (SEE FIG.1). Also, I added a resistor at the beginning of the circuit (Rs), as it wouldn’t fit otherwise. Furthermore, my understanding is that since my electrolyte is unstirred, and I go to low frequencies (down to 0.01 Hz, I should expect a Warburg element's behaviour). But, whenever I try to add a Warburg element to this system, it never fits, and this circuit (FIG. 1) was the best that I could get. Please note that I couldn’t figure how to add the Nagami-Havrilliak  universal capacitor to my circuit so just omitted it, for the time being.

Analysis: The fit follows experimental data, and seems good. So, this means that in my system the geometric capacitance of the electrolyte is derived from the 32.2mMho and 0.0886mMho, and the solution resistance is about 9ohms. The double layer capacitance is derived from 125mMho, and the Rct is 196 ohms.

Question1: Is my analysis correct please?

Question 2: Is my circuit correct please?

2nd approach: In literature, they always use a randles circuit for the same experimental setup, similar experimental results trend, and same materials, with the components: (R, CPE, W). Please SEE FIG. 3 (20ug pristine). Also, my understanding is that if the solution is unstirred, then we expect a Warburg term. Please note, I care to obtain 2 values only, Cdl and Rct, to study electrochemicaly active surface area of my electrode. So, I used this circuit model, and used Zfit with initial guesses. It was much easier and faster than NOVA. I didn’t fix any values, but the program just needed some initial values to make the software guess easier. I got a fairly good fit. (PLEASE SEE FIG. 2)

Analysis: Fit followed experimental results well. So, from these results (FIG.2) my solution resistance is also around 9 ohms. My Rct is 14.47 ohms. My capacitance Cdl is derived from the CPE values, to give values in uF. Here Cdl should be equal to 9.939 uF.

I think I trust the 2nd approach more than the first.

Question3: Is my analysis correct please for 2nd approach?

Question 4: Can I used this circuit if I aim to find Cdl and Rct only for my electrode please?

I have also attached the Z file of my experimental results. I really appreciate your guidance. Thank you very much for your help.

Ibrahim

Dear Ibrahim,

Regarding your 1st approach: The two randles modified circuit may seem to give a reasonable fit to your experimental data but as I implied to my previous post a good fit doesn't always mean that the chosen EQ is the proper one for modelling the system under study. EQs are used for modelling physicochemical phenomena. Are you able to extract a physical meaning through the chosen EQ? If not then you probably need to try a different one even if fitting seems to be reasonable from a mathematical point of view. To the point: What do these two time constants represent? i.e. in a porous electrode the first time constant would be related to the porosity of the material while the second one to the reaction under study. What's the case in your system? If you ask for my opinion I think that this circuit is not a suitable model for this particular system you study.

Regarding your 2nd approach: It seems that both mathematically and physicochemically (at least with a quick glance since I have little to no experience with RFBs) the chosen EQ fits the experimental data. In my opinion your arguments in the "2nd approach" are accurate and reasonable. As for your last question even if you want to determine Cdl and Rct only, you have to model the AC response of the system under study in total (most of the times). A different approach could lead to erroneous results/conclusions. Finally, I would recommend to examine each circuit element both independently and towards others while trying to find the physical meaning behind it. It is there for a reason right? If not then why use it? 

Cheers

PS1: Nova would probably fit the same EQ too (since the AC response of your system is quite clear), but it's more sensitive than ZView to the initial values so you might want to try a bit more.

PS2: In my previous post I referred to two time constants since (and I believe this was the case with Petr's argument too) the attached EIS spectra displayed a high frequency arc along with the medium/low frequency one. The last spectrum you attach doesn't seem to display a similar trend (I am just saying..it's not that I am trying to prove sth).   

Dear Ibrahim,

if you want to stick with the standard Randels/Warburg/CPE type fitting, I think Athanasios said it all and I have nothing to add to that, apart from other things also, because I do not use these R,C,L "building blocks" (elements). As Athanasios points out, a good fit does not always mean "good physics/electrochemistry".

 I have still some difficulties in identifying your equivalent R;C circuit with the standard Randels circuit (p.333 of Mark Ozarem book (Electrochemical Impedance Spectroscopy), but I might have missed something when interpreting your circuit symbols.

At present I can not do more for you, unless you send me some representative data with  information about electrode-sample geometry and a guess for dielectric constant of your electrolyte.

I tried to attach a figure or tywo to this answer, but I have not been successful. How one does this ?

With best regards

Petr

Hi Petr,

Thank you very much for your answers. Issue has been sorted. I have now got the hang of EIS fitting.

Very much appreciated.

You are welcome !

Best regards

Petr