Hello !
I am trying to figure out how Dli (Lithium diffusion coefficient) is calculated. A method is from EIS measurement. Specifically, from the relationship between Warbung impedance and the inverse square root of angular frequency σ value can be extracted.
The equation is: D=0.5* (R*T)^2 / (A*n^2*F^2*c*σ)^2, where:
R is the gas constant (8.314 kg m^2 s^-2 K^-1 mol^-1)
T is the absolute temperature (298.15 K)
A is the working area of cathode (m^2)
n is the electronic transport ratio during redox process (1)
F is the Faraday constant (96485.33 A s mol^-1)
c is the molar density of li-ion in an electrode (1 mol/L) because I have 1M LiPF6 electrolyte solution
σ is the slope on the diffusion state of EIS measurement
Could you please inform me if the above values and this method are correct ??
Also, I am not sure if there is a need for converting the units because D is referred as cm^2 s^-1
Comments please !
Thank you very much for your time !
Dear Ioannis Gkionis,
in general, the method and equation you are using are correct. However, you need to remember that the equation is fulfilled only in a narrow frequency range, where the contribution from mass transport is equal to real and imaginary part of the impedance, thus giving EIS spectrum inclined at 45 degrees to Z' (or Z'') axis.
In some cases, that range can be easily seen on the impedance spectrum. Although there are also instances, in which the time constants of the processes present in the electrode are not separated enough from each other, giving a more complicated spectra in which diffusion part of the impedance can not be easily distinguished from other processes.
In that case, what you could do is to find the frequency region in which you should observe a response from the lithium ion diffusion, take a few points from there which are at a 45 degree slope and use those to create a Z' = const + σ*(omega)^-0.5 function and get the Warburg coefficient from it. But you must remember, that this will induce errors (quite large in some cases) in your calculations as you are not using the impedance response from the cell purely originating from diffusion of lithium ions.
Best regards,
MK.
Dear Hugh J Byrne
Actually, this graph is from SCIENTIFIC REPORTS. Article LiNi0.8Co0.15Al0.05O2: Enhanced Electrochemical Performance ...
In any case the topic of my question is the calculation method of lithium diffusion coefficient.
I am not so familiar with this field, but to me the data as plotted is not linear, it does not go through the origin, and I think you are correct, that the units of the equation do not balance.
You probably should check further literature.
Dear Ioannis Gkionis,
in general, the method and equation you are using are correct. However, you need to remember that the equation is fulfilled only in a narrow frequency range, where the contribution from mass transport is equal to real and imaginary part of the impedance, thus giving EIS spectrum inclined at 45 degrees to Z' (or Z'') axis.
In some cases, that range can be easily seen on the impedance spectrum. Although there are also instances, in which the time constants of the processes present in the electrode are not separated enough from each other, giving a more complicated spectra in which diffusion part of the impedance can not be easily distinguished from other processes.
In that case, what you could do is to find the frequency region in which you should observe a response from the lithium ion diffusion, take a few points from there which are at a 45 degree slope and use those to create a Z' = const + σ*(omega)^-0.5 function and get the Warburg coefficient from it. But you must remember, that this will induce errors (quite large in some cases) in your calculations as you are not using the impedance response from the cell purely originating from diffusion of lithium ions.
Best regards,
MK.
Dear Dr. Michal Krajewski,
Thank you very much for your analysis. Actually, I like this equation and I think that you can find the diffusion coefficient easily, only if you have a clear EIS measurement. However, I have a problem with the units ( I can't make the equivalences to get cm^2 s^-1) and I am not sure about n parameter (electronic transport ratio during redox process ). I think that "n" is a ratio, so there are no units regarding this parameter. For example, if I have a half cell (silicon and lithium metal) the value of n should be about 4 for fully lithiated state ?
If you have another idea or thoughts regarding my question feel free to answer me.
Dear Ioannis Gkionis,
when it comes to the units in the equation, they should be used like:
A (cm^2), F (C mol^-1), R (J(mol K)^-1), T (K), C (mol cm^-3) and σ (Ω s^-0.5). n is dimensionless. This should give you the diffusion coefficient in cm^2 s^-1.
And remember, that the concentration value is not the one you have in your electrolyte but, as you are examining the diffusion processes in the material grains, you should use the lithium concentration in the silicon particles.
Best regards,
MK.
Dear Michal Krajewski ,
Thank you very much for your kind response !
Best regards,
I.G
Dear Sara Costa,
the most accurate information about lithium concentration in active material particles would come from structural data at given potential. But as not everyone is equipped with in-situ instruments for simultaneous structural and electrochemical measurements, some other approaches could be made.
In the most simple way, for one-phase insertion/extraction mechanism, you could acquire the amount of lithium inserted/extracted from the charge passed through your cell while charging/discharging it to a given potential. Then, assuming no volume changes during insertion/extraction (which is mostly not true, especially for alloying and conversion materials) and using molar volume of lithium host, you can calculate lithium concentration in the active material at given potential.
This approach, however, gets more complicated for two-phase reaction compounds and conversion and alloying materials, as it involves multiple phases with different molar volumes. You could then try to find molar volume for the phase, which is currently active at given potential and calculate the lithium concentrations from it.
If you are researching the same compound family and doing impedance measurements at the same potential (which mostly mean the same lithium concentrations in active particles) you can easily compare diffusion coefficients calculated that way. The problem arise when you try to compare them to different family of compounds as then, even more factors come to place - not only the difficulty in determination of lithium concentration.
Best regards,
MK.
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