# Basic and extendable framework for effective charge transport in electrochemical systems

## TL;DR

This paper introduces a simple, extendable mathematical framework for modeling charge transport in lithium batteries, capturing key processes and enabling efficient numerical simulations for system optimization.

## Contribution

It presents a basic, extendible model for lithium battery transport processes, including a novel Poisson equation and composite cathode equations, facilitating analysis and simulation.

## Key findings

- Complete description of mixed transport processes in batteries
- Upscaled equations for composite cathodes considering geometry and materials
- Numerical solutions without complex multiscale discretization

## Abstract

We consider basic and easily extendible transport formulations for lithium batteries consisting of an anode (Li-foil), a separator (polymer electrolyte), and a composite cathode (composed of electrolyte and intercalation particles). Our mathematical investigations show the following novel features: (i) \emph{complete and very basic description of mixed transport processes} relying on a neutral, binary symmetric electrolyte resulting in a non-standard Poisson equation for the electric potential together with interstitial diffusion approximated by classical diffusion; (ii) \emph{ upscaled and basic composite cathode equations allowing to take geometric and material features of electrodes into account}; (iii) \emph{the derived effective macroscopic model can be numerically solved with well-known numerical strategies for homogeneous domains} and hence does not require to solve a high-dimensional numerical problem or to depend on a computationally involved multiscale discretisation strategies where highly heterogeneous and realistic, nonlinear, and reactive boundary conditions are still unexplored. We believe that the here proposed basic and easily extendible formulations will serve as a basic and simple setup towards a systematic theoretical and experimental understanding of complex electrochemical systems and their optimization, e.g. Li-batteries.