The reason is that the equivalent circuit isn't actually suitable and the parallel resistor element shouldn't be there. The plots provided are typical of just a resistance and CPE in series, which fits well. However, a second resistance in parallel with the CPE means that at lower frequency the system should tend towards the real (x-) axis again and show the well-known depressed semicircle shape. This behaviour cannot be seen, and the program can't fit to it - so it ends up regressing to give a resistor with an absurdly high value with a huge error - it's the program's way of saying the element shouldn't be there. You could extend the range of the measurement to include lower frequencies but at the very least the R(RQ) equivalent circuit doesn't fit this data in this range.

I agree with Mathew that the biggest problem is the limited (at both ends !!) frequency range measured. In your case, the /Z/ curve at low frequencies does not seem to start to saturate and therefore the error of the fitting is large.

I have though a couple of points (warnings) more.

First of all, although I do not know what system you are actually measuring, the general response that is always present  is that of a bulk of the material and the response due to material-electrode interface (I understand you call  it barrier). This then can be modelled in the R,C,L network approximation, by TWO parallel R,C elements in series, one due to bulk (dc bulk conduction and the bulk polarisation) and one due to material-electrode interface. So, in your assumed model circuit, there is a capacitor (in parallel with the first resistor) missing. This capacitor is in most cases just the geometrical capacitance of the interface region, while the resistor describes the current dc limit (frequency w->0). In many cases it can be calculated analytically (see US patent 5 627 479, May 6, 1997).

Secondly, it is not clear at all, whether the "universal response part" (the Constant Phase Element) originates from the bulk or whether it comes from the interface region. It is our experience (liquids, ion conductors, semiconductors) that in many cases, the universal response is due to the polarisation processes within the material-electrode interface region and has nothing to do with the bulk response. The only way to resolve this problem is to establish which part of the response is due to bulk (doubling the thickness of your sample must increase the bulk resistance by a factor of two, while decreasing the bulk capacitance by a factor of two). When that is established , I would then try to fit introducing your CPE first to the bulk element and then to the interface element. The best fit then wins !

My last comment is that one must be rather sceptical of the various "ready to use" equivalent R,C,L simple circuits. The good fit is not equivalent to the "good" physics behind it. There are number of fallacies both in the Electro-chemical Impedance Spectroscopy that seem to prevail to this day. The same can be said about some of the models behind universal response, inclusive CPE.

I hope this can help on the way

With best regards

Petr

I appreciate your responses, thank you so much.

After I used your suggested circuit I obtain more fitted results but without physical meaning for my application.

My system is a Titanium alloy for implants and I tested it in electrolyte (Ringer's solution) and I expect a passive layer is formed on the virgin surface of the alloy and I want to find the barrier resistance of this passive layer.

Many literature in my field used this circuit R(RQ) or R(RQ(RQ)) and consider the second resistor as a barrier layer resistance.

If I use you suggestion RQ only, so What is the physical meaning of this circuit related to my system? 

What is the meaning of results without good explanation. I hope your help. 

In the RQ circuit the R is essentially the sum of the ionic resistance in the electrolyte and electronic resistances in the electrodes, wires and so on - I think it probably includes your barrier resistance as well. Based on the numbers it's reasonable to interpret Q as the (non-ideal) double layer capacitance. Presumably the oxide layer is extremely thin - this probably means that the time constant for the RQ unit you're actually looking for is very high.

If there is already similar literature on the measurements you are looking to make, hopefully they have included the Bode plot or some other indication of frequencies. In the Nyquist plot the RQ unit draws a (depressed) semicircle - in the Bode plot you see a peak in the imaginary part (or the phase, depending on how it's plotted). The frequency at that peak (the top of the semicircle is the relaxation frequency. This relaxation frequency *might* be higher than the maximum frequency of 100 kHz in your data. In which case you would need to scan to higher frequencies to resolve the semicircle - this may or may not be possible depending on your equipment.

I compare the values of the resistance R from RQ and R(QR) circuits but there is now difference, also both circuits are have the same values for constant time (n) and CPE.

I attached the bode plots which very close to the literature, and the fitting for both circuits are very good.

What can I do to find the resistance of the passive layer now?

I appreciate your helps so much.

As I said there's no evidence of a parallel RQ unit in the data so it's hard to get any information about the passive layer from it. As a general rule double layer capacitances are of the order of 20-40 uF/cm^2 so if your electrodes are of the order of 1 cm^2 then your Q value would seem to indicate it is the electrode/electrolyte interface that you see (the units are not exactly right because of the phase angle but for rough estimation it is near enough). I'm not that familiar experimentally with this type of measurement but I'm guessing that the capacitance values for the bulk/passive layer interface you're looking for are a lot smaller, and the semicircle you want to see is actually at higher frequency than your measurement makes. If that's true then the R in your existing fit is really the sum of the ionic resistance of the electrolyte, the electronic resistances in the electrodes and wires, and the barrier resistance you're looking for. That would be my interpretation of it at least but perhaps someone with more familiarity with using impedance to measure passive layers might be able to give more insight...

Thank you so much Prof Matthew for your valuable contributions and really I got a lot of information through this discussion, Thank you so much.

Any specialist here in the impedance of titanium alloy (passive layer on metal and its resistance) could give me his response please? 

Dear Yasser,

let me re-state again my comment concerning your Ringer solution - Titanium electrode(s) impedance measurement:

1. It is a rigorous result of the numerical ("first principles") calculation/simulation of the electrical response in solids and liquids that the systems like yours can be approximated by two parallel R,C elements in series in what one could call "first approximation" (please consult the above given reference to US patent) .

The first term describes the electrolyte (Ringer solution) response and from the Maxwell equations it is a trivial matter to show that this response consists of two physical processes in parallel - the dc conduction (your Rs) and the bulk polarisation (Ringer solution geometrical capacitance - this term is missing in your equivalent R,C circuit).

The second term describes the Ringer solution - Titanium interface. Here again, your Rb describes the dc conduction across this interface, while Cb (missing element in your equivalent R,C circuit) is the geometrical capacitance (with Ringer solution dielectric constant !!!) of the interface region.

2. The response of your system is more complicated and it seems that the best fit is abtained when one assumes an "universal response" (CPE) capacitance (again, consult Andrew Jonscher books on this topic) in parallel with Rb and Cb. This term approximates the slow Non-Debye polarisation process that takes place ONLY in the interface region and contributes to the dielectric "constant" here (dielectric response function would be a better term).

3. In general, the electrolyte - metal electrode interface consist of two, spatially separated regions, one that the electrochemists call double layer (immobile monolayer of polarised water molecule plus corresponding + or - ion monolayer and the second that the electrochemists call diffusion layer - sharply(exponentially) increasing/decreasing concentration of the respective ions in the Ringer solution - your Rb,Cb, CPE term. This then (and still in general) would add an extra series term to your equivalent R,C circuit. However, it is our experience that as far as the measurements are concerned, this extra term (double layer term) is not clearly frequency separated from the Rb,Cb and CPE part. Therefore one is forced to simulate and see what does and what does not have an effect on the fitted results.

I therefore disagree with Mathew that the terms are not there, the question is rather whether one should not extend the low frequency limit and in so doing could get hold of the resistance you are apparently after. The same argument goes for the high frequency regime where you neither got hold of the entire response from the term Rs and Cs.

Could I also suggest that you visualise (display) your data not as Niquist plot (real versus imaginary part of the measured impedance). This type of data presentation tells you almost absolutely nothing about the physics behind , although the same information is there. Just imagine that one would do the same thing in the optical spectroscopy !  The x-axis should be always the log of frequency. Plotting the real and imaginary parts of the measured complex impedance versus frequency (log-log plots) and real and imaginary part of the measured complex capacitance (remember that it is trivial transform that brings your data from impedance "Z" to admittance "Y" to capacitance "C") tells you much more !!. In plots like these you can directly read the values of your R,C circuit components.

Good luck with your work

with greetings

Petr

Thank you so much Prof. Petr for your interesting and valuable explanations. 

Actually, your issues seem to me to be better considered in the context of DC measurements rather than AC.  I did a lot of work during my career in the coatings industry on trying to utilize electrochemical measurements to assess the quality of coatings vs. their subsequent corrosion resistance.  Your passive layer can be seen in a similar light. 

To sum up over a decade of work, we found DC techniques to be considerably more informative regarding the passivation behavior of various metals in various electrolytes.  I suggest reviewing the following reference for possible application:

"Using DC electrochemical techniques to assess the relative corrosiveness of water-based coatings and their ingredients," Floyd, Tatti, and Provder; J. coatings Technology:  Research; vol 4, no. 2, pp. 111-129; 2007. 

Although not contained in this publication, we additionally found that we could use the same DC technique for zinc, aluminum, magnesium, and numerous current commercial alloys of each.  We have not tested titanium, but believe that the same technique would work, because of your interest in the passivation behavior of titanium. 

In many cases today, the same instrument can be used to obtain both AC and DC results, so you could conceivably run a few samples to see if the DC technique has any value for your interests. 

I realize that this means additional work, but from our experiences with AC approaches, it might be a more fruitful approach in the long run.

Good luck in your quest.

Lou Floyd

Thank you so much Prof Lou Floyd, and I will deeply analyzed your paper and see what can I do.

Dear Yasser,

Repeat the experiments with widening range of frequencies as corresponding to selected the suitable an equivalent circuit of EIS.

Regards,