Thermodynamic derivation of a Butler-Volmer model for intercalation in Li-ion batteries

We present an exclusively thermodynamics based derivation of a Butler-Volmer expression for the intercalation exchange current in Li ion insertion batteries. In this first paper we restrict our investigations to the actual intercalation step without taking into account the desolvation of the Li ions in the electrolyte. The derivation is based on a generalized form of the law of mass action for non ideal systems (electrolyte and active particles). To obtain the Butler-Volmer expression in terms of overpotentials, it is necessary to approximate the activity coefficient of an assumed transition state as function of the activity coefficients of electrolyte and active particles. Specific considerations of surface states are not necessary, since intercalation is considered as a transition between two different chemical environments without surface reactions. Differences to other forms of the Butler-Volmer used in the literature [1], [2] are discussed. It is especially shown, that our derivation leads to an amplitude of the exchange current which is free of singular terms which may lead to quantitative and qualitative problems in the simulation of overpotentials. This is demonstrated for the overpotential between electrolyte and active particles for a half cell configuration.