The 45 degree line, also known as the Warburg element in the equivalent circuit, is an element that represents a diffusion though a semi-infinite medium. Deviation from the 45 degree angle are a result of deviation from a semi-infinite diffusion condition. While I haven't looked into this in much detail, if I had to guess, I would say that higher angles are a result of something that adds additional capacitive behavior, such as extra porosity, which can charge, like a capacitor, but won't be part of the diffusion component.

I'm not sure that you can determine suitability of materials for LIBs based on this information alone.

Hi, I fully agree with the above answers.

If you want to know more details about the Warburg element in batteries, have a look at the publication I posted here. There are some quite descriptive figures plus a theoretical derivation of the battery impedance, starting from scratch.

Article A Consistent Derivation of the Impedance of a Lithium-Ion Ba...

In my opinion, there is a need to understand that the straight line we usullay observe in a Nyquist plot is also a semi circle with semi-infintie values of Zim and Zre. So think of it as a line associated to a very large semi circle. Now this kind of large semi circle at these lower frequencies CAN BE  associated to ionic diffusion inside electrode material which in circuit modeling is represented mainly by CPE, constant phase element. Warburg Element is just a special case of CPE where the phase angle is constant.  In a crude way, if an ionic specie is finding it dificult to diffuse inside the electrode material you might observe an even bigger semi circle which corresponds to higher anlges (In your case 65) and vice versa. Calling a material "best" based on impedance would be audacious, however, we can argue that lower agnles might be related to "ease in diffusion".  Ease in diffusion can be related to mircostructuring/porosity (As commented before) or new phase formations allowing fast lithium diffusion. Superficial or Pseudo-scientific answer would be "Lower angle is good higher angle is bad" 

Dear Firoz Khan,

the angle of the capacitor's Zc (impedance) is 90o, and the (EIS) Voltage is (always) fully compensated by capacitor's E-field, in contrast to the WE[1] of a Li-battery. The (AC over-) Voltage, now, is, only half, compensated by the pseudo-capacitor's E-field, due to the Li+ "over-concentration" in the WE. Half current is due to the e--current[4], e.g. the ZWE,real part. So, we have (always[2,3]) 45o for ZWE (Li+-diffusion) in the low frequency region.

1.   The Working Electrode (WE) in a Li-battery is a mixed[2] carrier conductor (e- and Li+) .

2.   Apart some limiting cases in the State of Charge (SOC) e.g. 0%, 100%.

3.   The 45 degrees in the low freq. ZWE, is a good indication for the State of Health (SOH) of a Li-battery.

4.   This e--current has a very different behavior from that due to a simple "R;C", a predication case from Umair Gulzar. It is not just a simple, a lumped R and C, resulting in a single (very high, RC) time constant. The ZWE angle is, always, near 45 degrees, not only at the freq. of -ZWE,imag,maximum.

I will agree with the Dr. Ioannis Samaras. In addition, The Warburg diffusion element (ZW) is a constant phase element (CPE), with a constant phase of 45° (phase independent of frequency).  Herewith I have attached a file and it may help to understand more about the constant phase of 45°.

Dear all thanks for your suggestions. I have compare the capacity of modified with graphene Li4Ti5O12 (LTO) with pure LTO using half cell. The capacity of modified LTO are 162 mAh/g and 125 mAh/g of modified  and pure LTO, respectively. I have measured EIS at initial stage (1) then cyclic performance then again EIS (2) then CV measured then EIS (3). The EIS of both the samples at above 3 stages are attached. The slope of the plot Zre vs w^-1/2 extracted form EIS (3) are 65 and 13. Is it correct? What is the reason behind these changes? Could you please explain why this happening?

Dear Firoz Khan,

I might be able to give you some advice/suggestion, but I need more infomation. The data you present are unlegible and should be presented as a double log plot of real and imaginary part of the response (Z or Y or C) versus frequency and not in a form of a "Nyquist plot" (quite unfortunate historical way to look at the data). Just like other electromagnetic spectroscopies (UV, VIS, IF, etc.,..).

I would also need a schematic drawing of your system (battery) with spatial dimensions.

Right now, a couple of general comments :

1. Warburg analysis is , in principle, un-physical. The problem is that it considers a simple diffusion process. But this is never the case if the diffusing species are electrically charged. Then the flux of given species is given by both the diffusion and the drift terms (gradient in the electrochemical potential). So, I would be very careful to infer too much from the the observation that real and imaginary part of the response have a constant phase angle of some 45 degrees. It might be though that it can indicate that the drift (movement of the charged species under the force field - electrical field ) might be small compared with the movement, caused by the  concentration gradient.

2. As has been already pointed out, the so called Warburg Element (WE) is just a special case of the general class  of electrical response, known as universal response , phenomenologicaly described by Nagami-Havrilliak function. It also goes under the name of Jonscher universal capacitor.

3. Your use of the electrical impedance spectroscopy in investigating battery function is very wise and in my yes quite essential. 

And a couple of references that might be useful :

Electrical Impedance Spectroscopy by M.E. Ozarem and B.Tribollet

Dielectric relaxation in Solids by A.K. Jonscher

With best regards

Petr

Dear Dr Petr,

Thanks for your suggestion, please find the double log plot for EIS along with UV-Vis absorbance data. Could you please explain the results?

Thanks again 

Dear Firoz,

The EIS data I would like you to send me should be log of the real part of impedance versus log of the measurement frequency and log of the imaginary part of the impedance versus log of measurement frequency. That was what I meant by the log-log plot. 

Could you also re-plot for me your UV-VIS absorbance data as the log of absorbance versus frequency rather than the wavelength ?

I will then try to find out whether I can come up with something more than what has been already said here.

With best regards

Petr 

Dear Dr. Petr,

Thanks for your cooperation. The required plot are attached.

Dear Firoz,

thank you for the plots. I will think about them, but right now I have the following suggestion as the next step.

Taking the first plot (log of Zreal and Zimag) versus log frequency), one can see that it is in the frequency region between 10^2 and 10^5 Hz that the slope is 1/2 ("Warburg solution"). The question now is , is this bulk or interface phenomenon ? In order to answer this question, you have to identify the bulk and the interface part of your response. Change the geometry of your cell (decrease the distance between the electrodes by a factor 2). Then the bulk resistance(the knee at ~10^2 Hz) should be halved and the bulk capacitance should increase by a factor of two. If this will not be the case, then even the response beyond 10^Hz is not the bulk response and any type of bulk modelling , like "Warburg" element is incorrect.

I would be interested in what you will measure and would like to see the results in the form of log(Zreal, ,Zimag) versus log of frequency, but also its transform to data presentation in a form og log(Creal,Cimag) versus log of frequency (C here means capacitance).

With best regards

Petr

Article Puzzles and confusions in supercapacitor and battery: Theory...

Please read this article...

I have same question about it but little bit different. I got different angle between Z real and Zimg for CNT tubes. What explanation behind of this? After fitting, I got 45 degree angle between Z real and Zimg. should I use raw data and fitted data or only raw data for paper?

Reflecting to the original question if the topic is still active...

The transfer function of the Warburg-element and so, its square root exponent comes from the solution of Fick II. law for 1D planar diffusion. The derivation of the final and well-known Warburg-impedance has been nicely and clearly written in Barsoukov's book from page 75 (Eq. 119) to 78 (Eq 136.).

Ref.: Evgenij Barsoukov, J. Ross Macdonald, Impedance Spectroscopy Theory, Experiment, and Applications, Second Edition, Wiley, 2005

Dear all,

there seems to be a fundamental misunderstanding as far as the so called "Warburg" impedance response is concerned. It is physically incorrect.

Why ? Because , the essential condition (E.Barsoukov and J.R. MacDonald in "Impedance Spectroscopy", p.54 ) :

"...In the absence of an electrical field..."

that leads to Fick's first law is never satisfied for the systems with real and non-neutral (blocking, partially blocking, accumulating,..) contacts, where there is finite and space-time changing electrical charge density. This means that there will always be finite electrical field and the flux of electrically active particles will always consist of TWO current contributions, the DRIFT and DIFFUSION (names from Semiconductor Physics). In general , this flux is given by the gradient in the electrocchemical potential (which is the same thing) !!

So, you can not solve the problem of the response within the interface regions by considering only one current density contribution. The pure phenomenology, used in electrochemistry to explain some EIS (Electrical Impedance Spectroscopy) response curves through various combinations of RCL elements, inclusive Warburg element , although apparently quite successful , does mean that it is correct.

The fact that the impedance curves have sometimes slope sqrt[1/((i*w)] might be though indicative of the possibility that an important part of the process is the diffusion of neutral species ("the bottle-neck" of electrochemical reactions at the interfaces). Otherwise it it is always a balance between the drift and diffusion fluxes (they are opposite to each other and the reason for thermodynamical equilibrium also, when there is no electrical charge neutrality in the system under consideration).

With best regards

Petr

Dear Petr Viscor ,

Bence Csomos (1-day ago, implicitly) referred, only, to the specific form of the "infinite" (semi-infinite thickness[1]) Warburg impedance element, the simple case of (Zr, -Zi) 45o, e.g. 'slope one' in (low freq. part of) Nyquist plot(s).

So, if we consider any additional 'driving forces' (say from: DRIFT, interfaces, "finite" thickness[2], etc.) then we shall include another, different, and a more complicated modeling case(s).

1. Diffusion in 'Basics of Electrochemical Impedance Spectroscopy' https://www.gamry.com/assets/Application-Notes/Basics-of-EIS.pdf

2. If the diffusion layer is bounded then the "finite" Warburg impedance element[1] might be, sometimes, helpful, also.

Dear Ioannis,

I disagree here :

1. As long as there exist, within the sample to be tested, spatial areas of non-zero electrical charge density, you must not neglect the effect of the local electrical field on the transport of the mobile charges. It is simply wrong to postulate that there is only diffusion. Or to put it differently, the flux of the electrical charges is determined by the gradient of the electrochemical potential. Full stop !

2. My criticism of various pseudo-RCL elements and their combinations is that they are in majority of cases totally random and are not based on a proper analysis, using the appropriate set of governing equations.

Warburg model is simply wrong in principle if one deals with electrically charged particles in systems with non-zero internal electrical fields present.

With best regards

Petr

P.S.: Even the proponents of the Warburg impedance acknowledge that it can be fitted only in a rather limited frequency range.

The gradient of a (finite) electrical potential, in an homogeneous medium having a (semi-) infinite thickness[1], is infinitesimally small, zero for common practice.

1. Thickness over diffusion length[2] of a mobile particle.

2. Diffusion length is the average distance traveled by a single mobile particle.

Dear Ioannis,

No, it is not. A semi-infinite system has a boundary at x=0 and therefore it is not homogeneous.

Your statement is correct only if the system is truly infinite. It is only then that any property of the system is the same wherever you are.

ANY interface (different materials or a surface) means charge transfer, means finite, non-zero electrical charge density, means finite voltage, means finite electrical field, means finite drift and diffusion current.

Also to your earlier comments:

"... to the specific form of the "infinite" (semi-infinite thickness[1]) Warburg impedance element, ..."

Yes, and in this semi-infinite thickness, there are TWO contributions to the particle flux given by the gradient of the electrochemical potential - drift flux proportional to local electrical field and diffusion flux, proportional to the particle concentration gradient. That is all I am saying. Fick's laws are not and can never be the whole story for mobile, electrically charged species..

With best regards

Petr