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

Abstract

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.

Introduction

Every continuum theory of Li ion insertion cells is based on models of Li ion transport in electrolyte and active particles on one side and on chemical kinetics at the interfaces between active particles and electrolyte on the other side. Most of the commonly used models concentrate on homogenized theories, which do not resolve single particles, but treat the electrodes as effective porous media [1], [2], [3], [4], [5], [6]. Therefore porous electrode theories are necessarily restricted to spatial scales much larger than the size of active particles, which range from the nanometer to the micrometer scale. Since electrodes often have a thickness of about 100 μm, the validity of this models is sometimes at least problematic. Only a few authors deal with microscopic models, which resolve the active particle scale [7], [8], [9], [10], [11], [12]. But also these models are restricted to scales above about 3 nm due to their continuum nature. Once the microscopic models are consistently formulated the homogenized porous electrode theory can be derived by e.g. volume averaging techniques [13], [14], [11]. Both type of models have to deal with the conceptual problem that the spatial scale of chemical kinetics is the scale of the molecules. On the scale of the transport models, reaction terms appear either as averaged source terms in the balance equations for mass, momentum, energy and entropy [15] or as boundary and interface conditions for the resulting partial differential equations. It is in both cases important to formulate them for exactly the same quantities, which are used in the transport equations. This goal is achieved best, if the missing expressions for chemical reactions are derived within the same theoretical frame and on the same spatial scale as the theory of transport. Pure thermodynamical theories of transport and chemical reactions in Li ion batteries are being developed in recent years [16], [17], [18], [19], [20], [11], [5], [6], dealing foremost with the problem of phase transitions in Li ion batteries or porous electrode theory. In this paper, we concentrate on a derivation of the expression for the exchange current due to ion intercalation at the interface between electrolyte and one active particle. Only concepts from nonequilibrium thermodynamics are used. They are applied to the assumption that ion intercalation can be simply understood as a transition between two different chemical environments where the actual ionic nature of e.g. the Li⁺ is preserved. There are strong experimental indications that this pictures is valid for graphitic materials in most of the cases [21], [22], [23]. We assume this picture also to hold for the positive electrodes. It is therefore not necessary to formulate interface reactions, in which Li⁺ is reduced to metallic Li. This reaction at the surface would rather be interpreted as the initiation of Li plating.

In Section 2 the underlying assumptions and the derivation of the intercalation model are presented. In Section 3 the new model is discussed and compared with the one proposed in [2]. Arguments on the thermodynamic consistency of the two different models are presented. In Section 4 the model is applied to a simple half cell configuration and the results for the overpotentials are compared with results of the Butler–Volmer expression used in [2].