Hello,

a RC element will produce a Dirac pulse in the DRT (ideal case), whereas an RQ element (resistor and constant phase element in parallel) will produce a broadened peak. The pattern of a Finite Length Warburg element is one main peak plus minor peaks towards higher frequencies. This is well portrayed in the paper by Leonide (see also the dissertation of Leonide by following the link).

For batteries, the Finite Space Warburg element is often encountered. As the imaginary part turns towards infinity for low frequencies, it is difficult to calculate a DRT. For a deeper analysis, there are other methods that have been developed recently, as introduced in the paper by Schönleber.

Best!

http://www.ksp.kit.edu/download/1000019173

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Thank you for the nice explanation.....

Dear Raam,

I do not really know what kind of relaxation time distribution you are talking about, but there is only one relaxation time both in "Warburg" impedance

Z(w)~1/sqrt(i*w*tau),

where tau is related to diffusion constant and

in parallel RC term(model for dc conduction and fast polarisation, both processes working in parallel), where the impedance is of the form

Z=R/(1+i*w*tau),

where tau=R*C (called conductivity relaxation time).

If you meant series RC term (model for "slow" polarisation, taking place bellow optical and infrared frequencies) , then the corresponding impedance is

Z=R(1+1/i*w*tau)

and here tau is characteristic dielectric relaxation time tau=R*C. But be aware, R here IS NOT ohmic resistance of the parallel RC term, but it is given by R=tau/C.

A word of warning though. Warburg impedance is La Place transform of the real space-time solution of the diffusion equation (the error function) for diffusing species. This might be a good approximation for electrically neutral species, but it is WRONG when electrically charged species are considered. Here it is not only the concentration gradient that is the "driving force" for the current, but equally important is the "drift" part of the current (j=el.conductivity*Elec.field). This is completely absent in Warburg type analysis and therefore Warburg impedance does not correspond to real situation in electrochemical cells.  

The monograph by Mark Orazem and Bernard Tribollet might be very useful if you decide to keep on this, in my view somewhat unfortunate, track.

With best regards

Petr

Thank you sir the concise explanation...

Thank you Petr for the qualified explanation about the nature of the Warburg element. I just like to add a recent paper by Boukamp where the DRT spectrum of a Finite Length Warburg element is derived by different methods.

Best,

Dino

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Dear Dino (and Raam),

thank you very much for the reference, I will read it with interest and if I come up with something that might be of interest to our discussion, I will let you both know.

With best regards

Petr 

Dear all,

just to add to the confusion, I just gave a presentation at the 12th ISSFIT in Kaunas (Lithuania), which hopefully will appear in Solid State Ionics (it is still in  the review process). I compared three methods of calculating the D(F)RT, my FT-transform method (ElChimActa), the Tikhonov regularization (using the nice Matlab program by the Ciucci group) and a multi LR(RQ)^n fit, which can exactly be transformed to the tau-domain. One of the test impedances was the Gerischer (Z0/sqrt(1+jw.tau0)). It turns out that there is an exact representation in the tau-domain ( G(tau) =(1/pi) sqrt(tau/(tau0-tau)) ). Neither of the methods was able to reproduce the exact DFRT, the FT and the Tikhonov were close, but with too broad peaks, whereas the multi-(RQ) fit gave too many peaks. But all gave a reasonable reconstruction of the original impedance.

So, with a FLW you may get several peaks in the FT, Tikhonov or multi-(RQ) fit transform, it is not necessary the most simple DFRT (remember, the transform method is an ill-posed inverse problem).

Best regards,

Bernard A. Boukamp

Thank you sir for your explanation

Dear Raam, Dino and Bernard,

first of all thank you Dino and Bernard for bringing to my attention(Dino) and sending me the full text of the article on Distribution Function of Relaxation Times (Bernard).

I have read the article (only quickly though !!) and my comments are more of a general nature, relating first of all to the more basic question of the existence of dielectric relaxation time distribution as an alternative to other possible models of the "universal relaxational response" in materials :

1. Bernard's article is very fine and brings up both the positive and negative aspects of the procedures to obtain the relaxation time distributions from the raw electrical impedance data. I do not understand though the equation(1). As long as there is a term 1/(1+i*w'tau) within the integral, it describes admittance and not the impedance (see also Jonscher's book (Dielectric relaxation in solids, Chelsea Dielectric Press, 1983), p.296).

2. One has to be very careful when considering DFRT . It might be that in some rather complex systems (complex electrode-sample interfaces for example), there is indeed a relaxation process where the characteristic relaxation time is distributed and where therefore one measures universal relaxation type response. But in general there is no evidence for this and the non-Debye response is a consequence of some other (and very general !!) complex polarisation process.

3. Raw electrical impedance data reflect the contribution to the response from both mobile electrical charges and from bound electrical charges. Both of these charges contribute to the observed polarisation of the system under test  and have to be separated in order to be able to start discussion of there being real dielectric (bound charges!)  relaxation time distribution . 

4. If you still want to fit Warburg element (Z(w)~1/sqrt(i*w*tau)) to your data, I would start with some simple (manual) fitting. First establish what is bulk response and what is interface response. Without Warburg diffusional contribution to the response, the equivalent R, C network consists of two parallel R,C terms in series. High frequency bulk response and low frequency interface response. The R terms represent dc resistances of the two respective regions while the C terms are the respective regions geometrical capacitances. The seconfd step then is to add Warburg element either to the bulk R,C element (in parallell !) or to the interface R,C element (in parallel) or to both. Once that is done and you get a good fit to your data, the really interesting question comes up , namely what it all means. Here I mean adding Warburg element to your equivalent R,C network. 

5. Problem with Warburg type analysis (diffusional motion of electrically charged particles) is that it in principle, it is non-physical. Equally important to the particle flow caused by the particle concentration gradient (this is Warburg analysis) is the particle drift flow caused by the applied electrical field !! You can not neglect one and use only the other. In fact , if the motion of electrically charged particles would be governed by Fick's diffusional laws only, you would never measure any electrical current when applying electrical field (drift term missing) ! This though does not mean that there is no diffusion, it  just goes hand in hand with drift.

I hope this will help you on the way.

With best regards

Petr

Just a quick response to section 1). What the integral portrays is an infinite series of parallel RC combinations as a model to construct the DFRT. In this sense it still is an impedance representation (the dimension of the right hand section of eq. 1 is 'ohm').

I will carefully read the other comments.

Best regards,

Bernard 

Dear Bernard,

but that is then another matter, if it is not a sum of  Debye-like processes, differing in the characteristic relaxation time values. I was not aware of that. Now it fits also in my world. Thank you for clarification. 

The physical interpretation of the topology I feel is still a problem, but the network fits and that is the important thing

Petr 

Dear Petr and Bernard,

Thanks for the fruitful discussion. There is just one more thing for me to add. There is a recent paper showing that any impedance (being linear, stable, causal and finite) can be broken down to a series of RC elements.

That confirms the validity of the mathematics but indeed, as Petr put it, it cannot make definite statements towards a physical interpretation. Although, if you don't know the physics behind your system, an analysis by the DRT can give good hints about the behavior of your system. It is possible to resolve and separate different processes quite effectively with the DRT. By this you can identify if there are diffusion limitations or not, for example.

Best,

Dino

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Dear Dino (and Bernard and Raam),

thanks again for the reference, I will look at it also with interest, because if the proof is rigorous, it is an important aspect of R,C,L modelling of the EIS data. My experience is that R,C,L models I use are quite unique, but I can not prove it. 

A parallel R,C term is correct. R is the resistive response and C is the capacitive response. Both are the integral part of the 3. Maxwell equation:

rot B= j+dD/dt, where j=n*e*mobility*E and D=eps*E. The two contributions to the total current are in-phase (j) and 90 degrees out-of phase(D) with the field. In spatially inhomogeneous samples (bulk and interface regions for example), there would be two paralellel R,C terms as I suggested to Raam. But many parallel R,C terms seem to be difficult to relate to a simple physical situation. But I might be wrong and therefore I will study your reference.

A better way to see the response clearly is to present the data in log-log plot as the real and imaginary part of the measured impedance. The R,C terms are often clearly apparent there.

With best regards

Petr