# During a Li-ion cell charge or discharge at 1C rate, are the electrode interfaces at thermodynamic equilibrium?

Is the cell voltage (at a modest charging rate) mostly determined by thermodynamics on the electrode surfaces and resistances of electrolyte and electrode particles, or there is a non-negligible kinetic component to the cell voltage (governed by the Butler-Volmer equation) on the electrode surfaces?

In equivalent-circuit models of cells, voltage (either of the full cell, or of a half-cell) is usually modelled as consisting of three components: open-circuit voltage, diffusion voltage, and hysteresis component.

Open-circuit voltage is determined by the thermodynamics of the electrode process at certain concentrations (stoichiometry) of the electrode (anode or cathode) and the electrolyte. The presence of a solid-electrolyte interface and some lattice effects don't allow to describe it with a single Nernst equation, but it's still thermodynamic in nature.

Diffusion voltage component arises from ion over- or under-concentration at the electrode surface. It's a reflection of the fact that stoichiometry at the surface of an electrode particle is not the average stoichiometry of the whole particle (which is used in the open-circuit voltage relationship).

The remaining hysteresis component is due to the polarisation of the electrode particle surfaces.

So, it looks like equivalent-circuit models disregard any kinetic contributions to the voltage. Does this mean that it's in fact negligible because the kinetics of Lithium-ion (de)intercalation at the electrode surfaces are so facile that a cell can be thought to always be at the thermodynamic equilibrium (when charged or discharged at 1C)? Does this differ appreciably for Graphite anode and typical cathode materials (NMC, LTO, LFP)?

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Please, have a look at this thesis, from P. 12 to the end.

The bottom line is that the Buttler-Volmer equation, which is in fact equation (13) derived in P. 12 to 16 (called by the author *the electrochemical diffusion model*), is said to be *the most accurate battery model*. So, it is certainly not disregarded, but on the contrary, IT IS CONSIDERED TO BE THE BEST BATTERY MODEL (despite its differential equation form).

Now, as always, you have a differential equation that is hardly tractable, so, you have to introduce simpler models that are more practical, albeit less accurate.

In this line, there are several equivalent circuit models of cells, that are applicable in more or less general conditions, depending upon the desired accuracy and the exterior conditions.

I don't know whether someone has derived the equivalent circuit models, usually first found by reckoning with more basic principles, as an approximation process of the Buttler-Volmer equation, but that's very probable to the best of my knowledge of the current state of academic research. Usually, the derivation of such models involves completing the equations with other principles depending upon the material into consideration, akin the equations of Electromagnetics in mater, where Maxwell equations in vacuum have to be completed by other principles like "Constitutive relations".

It is probably just a question of effort to find the relevant article that describes this derivation.

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