## Abstract

An elementary 1-dimensional model is developed for a solid state lithium-ion battery having a single ion conductor electrolyte, a lithium metal negative electrode and a composite positive electrode. The battery topology is assumed to be of the layered variety, thereby justifying the 1-dimensional formulation. The governing equations for the electrochemical kinetics at the interface between the negative electrode and the electrolyte separator are stated, as are those for ion transport in the electrolyte. The positive electrode is assumed to be a particulate composite of storage material within a matrix of electrolyte. A mixture theory is developed for the positive electrode encompassing ion transport in the electrolyte matrix and storage and unstorage of lithium in the active material subject to electrochemical kinetics at the perimeter of the storage particles. Many simplifying assumptions are made with the advantage of them leading to closed-form or semi-closed-form solutions, including linearization of the equations governing the redox kinetics at interfaces in the battery between electrolyte and active material. An approximation is given for the concentration of lithium in the positive electrode of the battery during discharge, with the details depending on a length scale parameter that depends on the competition between the rate of lithium insertion into/extraction from the storage particles and the rate at which lithium ions are transported in the electrolyte. When the conductivity of the electrolyte is high and the redox reactions are relatively sluggish, this length scale parameter is comparable to the thickness of the positive electrode or larger than it. In that case lithium insertion into/extraction from storage particles occurs everywhere within an active zone of the positive electrode, but with the rates least at the current collector of the positive electrode. If the conductivity of the electrolyte is poor and the redox reactions rapid, the length scale for the solution is small compared to the thickness of the positive electrode and insertion into/extraction from storage particles occurs only in a narrow slice of the positive electrode. This slice moved along the positive electrode and separates a region of it that is completely filled/empty from a region of it that has not yet gained or lost any of its lithium. In all cases there will usually be a region of the positive electrode near the separator that is completely filled during discharge and completely empty during battery charging. Our results also give outcomes from which the internal resistance of the battery can be estimated.

Original language | English |
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Pages (from-to) | 207-221 |

Number of pages | 15 |

Journal | Journal of the Mechanics and Physics of Solids |

Volume | 123 |

Early online date | 10 Oct 2018 |

DOIs | |

Publication status | Published - Feb 2019 |

### Bibliographical note

As noted above, this contribution is dedicated to Norman Fleck on the occasion of his 60th birthday. I have spent many happy hours in collaboration with Norman, and it has been highly productive, educational, rewarding and enjoyable to do so. I look forward to further interactions with him on the subject of this paper, lithium-ion batteries.This work was funded by the University of California, Santa Barbara and by the University of Aberdeen.

## Keywords

- Composite positive electrode
- Linearized model
- Lithium-ion batteries
- Metal negative electrode
- Single ion conducting electrolyte