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Turing patterns in a 3D morpho-chemical bulk-surface reaction-diffusion system for battery modeling

Massimo Frittelli, Ivonne Sgura, Benedetto Bozzini

TL;DR

This work addresses how electrolyte diffusion and bulk-surface coupling influence pattern formation during electrodeposition. It introduces a 3D bulk-surface DIB (BS-DIB) model on a cube and solves it with a Bulk-Surface Virtual Element Method (BS-VEM) on a graded mesh, comparing to the original 2D DIB. Key findings show that bulk-surface coupling expands the Turing region and often changes pattern morphology relative to the 2D model, with numerical experiments demonstrating various morphologies such as holes, stripes, and worms and good agreement with MO-FEM where applicable. The combination of a flexible, efficient BS-VEM solver and the extended BS-DIB model provides a practical tool for battery morphology studies and electrolyte-electrode coupling in 3D.

Abstract

In this paper we introduce a bulk-surface reaction-diffusion (BSRD) model in three space dimensions that extends the DIB morphochemical model to account for the electrolyte contribution in the application, in order to study structure formation during discharge-charge processes in batteries. Here we propose to approximate the model by the Bulk-Surface Virtual Element Method on a tailor-made mesh that proves to be competitive with fast bespoke methods for PDEs on Cartesian grids. We present a selection of numerical simulations that accurately match the classical morphologies found in experiments. Finally, we compare the Turing patterns obtained by the coupled 3D BS-DIB model with those obtained with the original 2D version.

Turing patterns in a 3D morpho-chemical bulk-surface reaction-diffusion system for battery modeling

TL;DR

This work addresses how electrolyte diffusion and bulk-surface coupling influence pattern formation during electrodeposition. It introduces a 3D bulk-surface DIB (BS-DIB) model on a cube and solves it with a Bulk-Surface Virtual Element Method (BS-VEM) on a graded mesh, comparing to the original 2D DIB. Key findings show that bulk-surface coupling expands the Turing region and often changes pattern morphology relative to the 2D model, with numerical experiments demonstrating various morphologies such as holes, stripes, and worms and good agreement with MO-FEM where applicable. The combination of a flexible, efficient BS-VEM solver and the extended BS-DIB model provides a practical tool for battery morphology studies and electrolyte-electrode coupling in 3D.

Abstract

In this paper we introduce a bulk-surface reaction-diffusion (BSRD) model in three space dimensions that extends the DIB morphochemical model to account for the electrolyte contribution in the application, in order to study structure formation during discharge-charge processes in batteries. Here we propose to approximate the model by the Bulk-Surface Virtual Element Method on a tailor-made mesh that proves to be competitive with fast bespoke methods for PDEs on Cartesian grids. We present a selection of numerical simulations that accurately match the classical morphologies found in experiments. Finally, we compare the Turing patterns obtained by the coupled 3D BS-DIB model with those obtained with the original 2D version.
Paper Structure (11 sections, 1 theorem, 44 equations, 11 figures, 1 table)

This paper contains 11 sections, 1 theorem, 44 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. The bulk-surface DIB model on the cube
  3. Stability in absence of diffusion
  4. The Bulk Surface Virtual Element Method for the BS-DIB model
  5. Step 1: Rewriting the model with homogeneous boundary conditions
  6. Step 2: Weak formulation
  7. Step 3: Spatially discrete formulation
  8. Step 4: Fully discrete formulation in vector form
  9. Bespoke BS-VEM mesh for the BS-DIB model
  10. Numerical experiments
  11. Conclusions

Key Result

Theorem 1

If $\gamma = 0$, the equilibrium equilibrium is stable in the absence of diffusion if and only if In addition, if $A_1$, $k_2$, $k_3$, $\alpha$ are as in parameter_values, the condition stability_equilibrium_no_diffusion specializes to Furthermore, if $b_0 = q_0 = 1$, the condition stability_equilibrium_no_diffusion_specialized further reduces to

Figures (11)

  • Figure 1: Domain geometry and BCs for the bulk variables $b$ and $q$.
  • Figure 2: Graded polyhedral mesh used in the BS-VEM approximation of the model \ref{['model']}.
  • Figure 3: Simulation obtained with the MO-FEM approach. Bifurcation and coupling parameters as in Table \ref{['tab:experiments_recap']} (Experiment D3). The BS-DIB model \ref{['model']} shows a reversed spots pattern, while the DIB model \ref{['model2d']} exhibits a labyrinth pattern. In the BS-DIB model \ref{['model']}, the bulk components $(b,q)$ exhibit a spatial pattern only in a neighborhood of the surface $\Gamma$. This suggests the usage of a graded mesh that is highly refined close to $\Gamma$ and much coarser away from $\Gamma$.
  • Figure 4: Simulation T1. Bifurcation and coupling parameters as in Table \ref{['tab:experiments_recap']}. The BSDIB model \ref{['model']} shows a worm pattern, while the DIB model \ref{['model2d']} reaches the homogeneous steady state.
  • Figure 5: Simulation T2. Bifurcation and coupling parameters as in Table \ref{['tab:experiments_recap']}. The BSDIB model \ref{['model']} shows a slow-to-stabilize stripe pattern, while the DIB model \ref{['model2d']} reaches the homogeneous steady state.
  • ...and 6 more figures

Theorems & Definitions (1)

  • Theorem 1