(Invited) Multi-Site, Multi-Reaction Model for the Simulating Intercalation and Alloy Electrodes
Authors: Verbrugge Mark , Garrick Taylor R. 
Published: September 2023

We overview enabling research for the characterization and design of cells using intercalation and alloy electrodes. Specifically, we have recently developed and implemented a multi-site, multi-reaction (MSMR) model. We shall demonstrate the approach with various chemistries of interest today, and the creation of reduced order models to simply calculations.
We first discuss some of the similarities and differences between the MSMR model and the original Doyle-Fuller-Newman model [1, 2], which we shall refer to as the DFN model, first published in 1993. This model represents an intercalation electrode using porous electrode theory, but it also includes a detailed description of transport of Li ions in the electrolyte, transport of lithium in the solid phase, and the kinetics of the intercalation reaction as a Butler-Volmer equation. The MSMR model is based on the same porous electrode theory used in the original DFN model, and it uses the same treatment of transport in the electrolyte. However, the MSMR model introduces significant changes in the representation of both transport in the solid phase and the kinetics of the intercalation reaction. The main reason for the changes in the MSMR model is because it hypothesizes that intercalated lithium consists of not just one species, but multiple species. There is significant evidence to support this point of view, both from cyclic voltammetry experiments as well as X-ray diffraction analyses [4].
The DFN model, as originally described and often used today, represents solid phase diffusion as a linear equation with a constant diffusion coefficient. This representation is known to be highly inaccurate, for example, in the case of graphite, where measurements have shown that the diffusion coefficient can vary by several orders of magnitude depending on the mole fraction of intercalated lithium [3,4]. More sophisticated treatments of solid phase diffusion use the chemical potential as the driving force for diffusion and because the chemical potential can be written in terms of the OCV, this explains how the Open Circuit Voltage now plays a role in the diffusion equation. Since the MSMR model uses multiple species, the solid phase diffusion problem should also involve multiple species, but the entire system of multi-species diffusion equations simplifies to a single equation, if one assumes that the various species of lithium are locally equilibrated with each other within the active material particles. The details of this analysis are given in [6].
The DFN model uses a Butler-Volmer equation to represent the kinetics of the intercalation reaction, whereas the MSMR model uses a Butler-Volmer equation for each species; the intercalation reaction rate thus becomes the sum of these multiple Butler-Volmer equations, which are formulated using activities instead of concentrations, as is done in the original DFN model.
Finally, the MSMR model proposes a simple empirical relationship, based on a generalized Nernst equation, to describe the relationship between the mole fraction of each intercalated species and the OCV. This formulation provides a systematic way to derive analytical formulas to represent the OCV relationship x(U), where x is the total mole fraction of intercalated species and U is the corresponding OCV. Several examples [5,6] show that this formalism suffices to provide accurate fits of x(U) to experimental data for many different electrode materials in common use, and it also provides analytical expressions for the derivative dx/dU, which shows up in the expressions for the solid phase diffusion equation.
In summary, the MSMR model attempts to give a more accurate representation of solid phase diffusion and a more complete representation of intercalation interfacial kinetics, and, in this context, our hope is that the MSMR model provides a useful complement to the original DFN model.

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