Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence of solution to a system of PDEs modeling the crystal growth inside lithium batteries

Omar Lakkis, Alexandros Skouras, Vanessa Styles

TL;DR

The paper analyzes a three-field PDE model for lithium electrodeposition featuring a nonlinear anisotropic phase-field equation coupled to Nernst–Planck and Poisson equations. It establishes the existence of weak solutions via a time-discrete Rothe method, proves stability estimates, and demonstrates convergence to a continuous weak solution. It also provides a finite element discretization and numerical simulations that reveal dendritic growth patterns, showing how anisotropy and noise influence morphology. The work contributes a rigorous mathematical foundation for anisotropic phase-field models in lithium batteries and validates the model's qualitative dendritic behavior with simulations.

Abstract

We study a model for lithium (Li) electrodeposition on Li-metal electrodes that leads to dendritic pattern formation. The model comprises of a system of three coupled PDEs, taking the form of an Allen--Cahn equation, a Nernst--Planck equation and a Poisson equation. We prove existence of a weak solution and stability results for this system and present numerical simulations resulting from a finite element approximation of the system, which illustrate the dendritic nature of solutions to the model.

Existence of solution to a system of PDEs modeling the crystal growth inside lithium batteries

TL;DR

The paper analyzes a three-field PDE model for lithium electrodeposition featuring a nonlinear anisotropic phase-field equation coupled to Nernst–Planck and Poisson equations. It establishes the existence of weak solutions via a time-discrete Rothe method, proves stability estimates, and demonstrates convergence to a continuous weak solution. It also provides a finite element discretization and numerical simulations that reveal dendritic growth patterns, showing how anisotropy and noise influence morphology. The work contributes a rigorous mathematical foundation for anisotropic phase-field models in lithium batteries and validates the model's qualitative dendritic behavior with simulations.

Abstract

We study a model for lithium (Li) electrodeposition on Li-metal electrodes that leads to dendritic pattern formation. The model comprises of a system of three coupled PDEs, taking the form of an Allen--Cahn equation, a Nernst--Planck equation and a Poisson equation. We prove existence of a weak solution and stability results for this system and present numerical simulations resulting from a finite element approximation of the system, which illustrate the dendritic nature of solutions to the model.
Paper Structure (11 sections, 14 theorems, 121 equations, 4 figures, 1 table)

This paper contains 11 sections, 14 theorems, 121 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Weak formulation of the model
  3. The model
  4. The weak formulation
  5. Anisotropic diffusion tensor
  6. Time discretization
  7. Weak convergence of the limits
  8. Proof of Theorem \ref{['thmexistence']}
  9. Numerical Simulations
  10. Finite element approximation
  11. Numerical examples

Key Result

Lemma 1

Recalling anisotropy_function, for $\delta < 1/15$ , the functional is strictly convex in $\vec{p}$ for all $\vec{p}\in (\operatorname L\xspace_{2} (\Omega))^d$. It follows that the anisotropic Dirichlet energy $J_a(w)$ defined in (anisotropicdirichletenergy) is strictly convex in $w\in\operatorname H\xspace^{1}(\Omega)$.

Figures (4)

  • Figure 1: Dendritic formation with $\delta = 0.05$ and $\mu = 2$: $u_h^k$ (top row), $c_h^k$ (middle row) and $\phi_h^k$ (bottom row) displayed at $t_k = 0.061$ (left) $t_k = 0.244$ (middle) and $t_k = 0.427$ (right)
  • Figure 2: Comparison of $u_h^k$ at $t_k=0.366$ with $\mu = 2$ and different values for the anisotropy strength $\delta$, introduced in \ref{['anisotropy_function']}.
  • Figure 3: Comparison of $u_h^k$ at $t_k=0.122$ with $\delta = 0.05$ for different values of $\mu$.
  • Figure 4: Comparison of $u_h^k$ at $t_k=0.305$ with $\delta = 0.05$ and $\mu = 2$, for solutions to the model with \ref{['fullydiscretisedorderparameter']} replaced by \ref{['acnoise']} for different values of the noise amplitude $\beta$.

Theorems & Definitions (32)

  • Remark 1: cut-off of constitutive functions
  • Lemma 1: convexity of the anisotropic Dirichlet energy
  • proof
  • Remark 2: monotonicity
  • Remark 3: ellipticity
  • Theorem 2.4: existence of solutions
  • Remark 4: Right-hand side of \ref{['discretisedorderparameter']}
  • Theorem 3.1: existence and uniqueness of a time-discrete solution
  • Lemma 2: stability of the time-discrete order parameter
  • proof
  • ...and 22 more