# How to adequately describe full range intercalation – a two-sided approach

**Researchers from the Max Planck Institute for Solid State Research (MPI-FKF Stuttgart) could show how Lithium- or Sodium intercalation can be thermodynamically described. Surprisingly, and in spite of the enormous significance and popularity of this phenomena enabling modern batteries, a satisfactory thermodynamic description of the charge-discharge curves has not yet been given. To demonstrate the procedure, the authors chose the most challenging case, namely nanocrystalline Li _{x}FePO_{4} as master example as here the entire range (from x = 0 to x = 1) is experimentally accessible.**

There is no need to stress the significance of batteries using intercalation of Lithium. These systems are changing our daily life noticeably – just think of electromobility. So the conferring of the 2019 Nobel prize to the inventors did not come as a surprise. Notwithstanding the enormous popularity of these electrochemical systems, a satisfactory thermodynamic description of the charge-discharge curve (i. e. of the equilibrium voltage as a function of the Lithium content in the solid) has not yet been provided.

The present paper shows how to do this. It used the most challenging system, namely nanocrystalline FePO_{4}. It was chosen less because FePO_{4} is the most promising next-generation storage electrode material, rather because here the entire storage range from 0 to 100% is available (i. e. Li* _{x}*FePO

_{4}from

*x*= 0 to 1).

The following steps are shown to lead to the successful treatment:

1) Point defect chemistry has to be applied rather than the usual lattice gas statistics.

2) The problem is tackled from the two endmember sites (FePO_{4} and LiFePO_{4}). Adding Li to FePO_{4} means that Li^{+} occupies interstitial sites, and excess electrons are injected into the conduction band. Removing Li from LiFePO_{4} means creation of vacancies and electron holes in the valence band. One should note that when going from one endmember to the other, the terms regular ion, interstitial site, vacancy on one hand and conduction band and valence band, on the other, undergo a “renormalization”.

3) The middle range is most difficult to treat as it is the range of interactions among the ions, the electrons and between ions and electrons. We give simple recipes of how to proceed.

4) For the pure thermodynamic reproduction of the curve, we play a “trick”. We interpolate between the two sides in a thermodynamic way using the theorem of phase stability which demands that the (thermal, mechanical, but especially the) chemical capacitance – if we may use our own nomenclature – be positive definite.

5) We show that the systematic experimental results of Wagemaker *et al*. can be beautifully described. If we use the interpolation approach. Then it suffices to consider ideal defect statistics plus modification by electronic saturation effects.

Again it is to be mentioned that the chosen electrode is not a special case but provides a general example. Should the storage be restricted to the environment of a given phase, a one-sided approach suffices. In any case, point defect chemistry is the approach to be taken.