The evolving surface morphochemical reaction-diffusion system for battery modeling

It is well known that phase formation by electrodeposition yields films of poorly controllable morphology. This typically leads to a range of technological issues in many fields of electrochemical technology. Presently, a particularly relevant case is that of high-energy density next-generation batteries with metal anodes, that cannot yet reach practical cyclability targets, owing to uncontrolled elelctrode shape evolution. In this scenario, mathematical modelling is a key tool to lay the knowledge-base for materials-science advancements liable to lead to concretely stable battery material architectures. In this work, we introduce the Evolving Surface DIB (ESDIB) model, a reaction-diffusion system posed on a dynamically evolving electrode surface. Unlike previous fixed-surface formulations, the ESDIB model couples surface evolution to the local concentration of electrochemical species, allowing the geometry of the electrode itself to adapt in response to deposition. To handle the challenges related to the coupling between surface motion and species transport, we numerically solve the system by proposing an extension of the Lumped Evolving Surface Finite Element Method (LESFEM) for spatial discretisation, combined with an IMEX Euler scheme for time integration. The model is validated through six numerical experiments, each compared with laboratory images of electrodeposition. Results demonstrate that the ESDIB framework accurately captures branching and dendritic growth, providing a predictive and physically consistent tool for studying metal deposition phenomena in energy storage devices.