Thermodynamic derivation of a Butler–Volmer model forintercalation in Li-ion batteries

Abstract

We present an exclusively thermodynamics based derivation of a Butler–Volmer expression for the inter-calation exchange current in Li ion insertion batteries. In this first paper we restrict our investigationsto the actual intercalation step without taking into account the desolvation of the Li ions in the elec-trolyte. 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 isnecessary to approximate the activity coefficient of an assumed transition state as function of the activitycoefficients 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 withoutsurface reactions. Differences to other forms of the Butler–Volmer used in the literature [1,2] are dis-cussed. It is especially shown, that our derivation leads to an amplitude of the exchange current whichis free of singular terms which may lead to quantitative and qualitative problems in the simulation ofoverpotentials. This is demonstrated for the overpotential between electrolyte and active particles for ahalf cell configuration.