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The evolving surface morphochemical reaction-diffusion system for battery modeling

Benedetto Bozzini, Massimo Frittelli, Anotida Madzvamuse, Ivonne Sgura

TL;DR

This work addresses the challenge of predicting morphology evolution during metal electrodeposition in batteries. It introduces the Evolving Surface DIB (ESDIB) model, a morphochemical reaction-diffusion system posed on an evolving electrode surface, coupling surface growth to local species concentrations via a surface-velocity law. The authors develop a spatial discretisation using Lumped Evolving Surface Finite Elements (LESFEM) and a time integrator based on IMEX Euler to handle the coupling between surface motion and diffusion, and validate the approach with six numerical experiments compared to laboratory SEM images. The results demonstrate the model's ability to reproduce branching and dendritic patterns and their evolution, providing a predictive framework for electrodeposition phenomena in energy storage devices; limitations include handling surface self-intersections, with future work aimed at topology changes and phase-field formulations.

Abstract

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.

The evolving surface morphochemical reaction-diffusion system for battery modeling

TL;DR

This work addresses the challenge of predicting morphology evolution during metal electrodeposition in batteries. It introduces the Evolving Surface DIB (ESDIB) model, a morphochemical reaction-diffusion system posed on an evolving electrode surface, coupling surface growth to local species concentrations via a surface-velocity law. The authors develop a spatial discretisation using Lumped Evolving Surface Finite Elements (LESFEM) and a time integrator based on IMEX Euler to handle the coupling between surface motion and diffusion, and validate the approach with six numerical experiments compared to laboratory SEM images. The results demonstrate the model's ability to reproduce branching and dendritic patterns and their evolution, providing a predictive framework for electrodeposition phenomena in energy storage devices; limitations include handling surface self-intersections, with future work aimed at topology changes and phase-field formulations.

Abstract

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.

Paper Structure

This paper contains 19 sections, 4 theorems, 47 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. The DIB model and the derivation of an Evolving Surface DIB counterpart
  3. The original DIB model and its interpretation
  4. The SDIB model on a stationary surface
  5. Evolving surface reaction-diffusion systems: background
  6. Introducing the ESDIB model
  7. Spatial discretisation
  8. Weak formulation
  9. Lumped Evolving Surface Finite Elements
  10. Time discretisation
  11. Numerical examples and experimental comparisons
  12. Experimental cases
  13. Example 1: square, $B=30$, $C=3$, holes
  14. Example 2: square, $B=66$, $C=3$, labyrinth
  15. Example 3: square, $B=62$, $C=5$, spots
  16. ...and 4 more sections

Key Result

Theorem 1

If $\mathbf{g}:\mathcal{G} \rightarrow\mathbb{R}^3$ is a sufficiently smooth vector field tangent to $\Gamma(t)$ at all times, it holds that where $\bm{\mu}:\partial \Gamma(t) \rightarrow\mathbb{R}^3$ is the unit outward conormal vector field on $\partial \Gamma(t)$, see dziuk2013finite.

Figures (14)

  • Figure 1: How the evolution of the electrode shape is modeled (a) in the DIB model \ref{['dib_model']}, (b) in the SDIB model \ref{['sdib_model']} and (c) in the ESDIB model \ref{['model_evolving']}.
  • Figure 2: Pictorial comparison of surface evolution of type (i) and (ii). Color darkness indicates the absolute value of a given species. For type (i) evolution (left), a local increase in surface area generates a dilution effect which lowers concentration locally, in absolute value. For type (ii) evolution (right), there are no dilution effects, as the values of a given species are transported along the material trajectories.
  • Figure 3: Example 1. (a) $\eta$ component of the numerical solution of the ESDIB model at various times. (b) SEM micrograph of a zinc foil electrode charged in pure 6M KOH solution at 5 mA cm$^{-2}$ for 3 hours. Blue and red arrows indicate low and high current density regions, respectively.
  • Figure 4: Example 1. Left: $\eta$ component of the DIB model at the final time. Middle: time increment $\|\eta(t_{i+1}-t_i)\|_2$ of the $\eta$ component of the DIB and ESDIB models. Right: area of the evolving surface $\Gamma(t)$ in the ESDIB model.
  • Figure 5: Example 2. (a) $\eta$ component of the numerical solution of the ESDIB model at various times. (b) SEM micrograph of a zinc foil electrode charged in a 6M KOH solution containing 100 ppm cetyl-trimethyl ammonium chloride additive, at 20 mA cm$^{-2}$ for 30 min.
  • ...and 9 more figures

Theorems & Definitions (8)

  • Theorem 1: Integration by parts on evolving surfaces, dziuk2013finite
  • Theorem 2: Reynolds transport theorem, dziuk2013finite
  • Theorem 3: Green's formula on surfaces, dziuk2013finite
  • Remark 1: Interpretation of the balance laws \ref{['balance_law_eta']}-\ref{['balance_law_theta']}
  • Remark 2: Interpretation of the modified balance law \ref{['balance_law_eta_type2']}
  • Remark 3
  • Lemma 1: Transport property of the Lagrangian basis functions, dziuk2007finite
  • Remark 4: Discretisation of usual evolving RDSs