# Multiphase Porous Electrode Theory

## TL;DR

This paper introduces a comprehensive multiphase porous electrode theory framework for batteries, incorporating phase separation, advanced reaction kinetics, and thermodynamics, implemented in an open-source software package called MPET.

## Contribution

It extends traditional porous electrode models to include multiphase materials and complex reaction kinetics using electrochemical thermodynamics, enabling more accurate battery simulations.

## Key findings

- Demonstrates the software's ability to model phase-separating materials.
- Shows integration of Marcus-Hush-Chidsey kinetics for electron transfer.
- Provides example simulations illustrating the model's capabilities.

## Abstract

Porous electrode theory, pioneered by John Newman and collaborators, provides a useful macroscopic description of battery cycling behavior, rooted in microscopic physical models rather than empirical circuit approximations. The theory relies on a separation of length scales to describe transport in the electrode coupled to intercalation within small active material particles. Typically, the active materials are described as solid solution particles with transport and surface reactions driven by concentration fields, and the thermodynamics are incorporated through fitting of the open circuit potential. This approach has fundamental limitations, however, and does not apply to phase-separating materials, for which the voltage is an emergent property of inhomogeneous concentration profiles, even in equilibrium. Here, we present a general theoretical framework for "multiphase porous electrode theory" implemented in an open-source software package called "MPET", based on electrochemical nonequilibrium thermodynamics. Cahn-Hilliard-type phase field models are used to describe the solid active materials with suitably generalized models of interfacial reaction kinetics. Classical concentrated solution theory is implemented for the electrolyte phase, and Newman's porous electrode theory is recovered in the limit of solid-solution active materials with Butler-Volmer kinetics. More general, quantum-mechanical models of Faradaic reactions are also included, such as Marcus-Hush-Chidsey kinetics for electron transfer at metal electrodes, extended for concentrated solutions. The full equations and numerical algorithms are described, and a variety of example calculations are presented to illustrate the novel features of the software compared to existing battery models.