Topic E — Diffusion-Induced Stress in Battery Electrode Particles

lithium diffusion + elasticity (the thermal-stress analogy) · spherical and ellipsoidal particles · fracture risk · tier: standard

Motivation

Lithium entering an electrode particle makes it swell; a non-uniform concentration therefore stresses the particle, and during fast charging the surface can crack. The stress depends on the charging rate, the particle size and its shape. Mathematically this is the thermal-stress problem of Week 3 with concentration in place of temperature — with closed-form solutions for a sphere that make it an ideal verification case — and it opens the door to non-spherical particles, concentration-dependent diffusivity and fracture.

Related fields: lithium-ion and post-lithium batteries: electrolytes, electrode materials, next-generation battery devices. Course weeks: 2 (axisymmetry), 3 (thermal stress, one-way coupling), 5 (phase-field fracture, stretch goal).

Model

Particle of radius R (sphere: 1D radial or 2D axisymmetric; prolate/oblate spheroid: 2D axisymmetric), dimensionless concentration \tilde c=c/c_{\max}. \partial_tc=\nabla\cdot(D(c)\nabla c),\qquad -D\partial_nc=-\frac{i}{F}\ \text{(galvanostatic charging)}, \nabla\cdot\boldsymbol\sigma=\mathbf 0,\qquad \boldsymbol\sigma=\lambda\operatorname{tr}\boldsymbol\varepsilon\,\mathbf I+2\mu\boldsymbol\varepsilon-(3\lambda+2\mu)\frac{\Omega}{3}(c-c_0)\mathbf I, (\Omega = partial molar volume; the “thermal expansion” is \Omega c/3), traction-free surface, quasi-static (diffusion is slow compared with elastic waves). One-way coupling first; the stress-enhanced diffusion D\nabla c\to D(\nabla c-\frac{\Omega c}{RT}\nabla\sigma_h) as a stretch goal.

Design outputs: the maximum tensile hoop stress at the surface and the maximum radial stress at the centre during charging, against the C-rate and the particle radius; the effect of the aspect ratio.

Suggested parameters

Order-of-magnitude values for a LiMn₂O₄-type cathode: R = 1–10 µm, D\sim10^{-14} m²/s, E\sim10 GPa, \nu=0.3, \Omega\sim3.5\times10^{-6} m³/mol, c_{\max}\sim2.3\times10^4 mol/m³ — take the exact set from Cheng & Verbrugge (2009) and quote it. Current density from 1C to 10C. Dimensionless: \tau=Dt/R^2, flux \frac{iR}{FDc_{\max}}, stress scale \frac{E\Omega c_{\max}}{3(1-\nu)}.

Required tasks

  1. Diffusion in a sphere (1D radial and 2D axisymmetric), galvanostatic, until the surface reaches c_{\max}.
  2. Stresses from the concentration at several times; maximum hoop and radial stress against \tau.
  3. C-rate and size study in dimensionless form.
  4. Spheroids with aspect ratio 0.5–2: where is the maximum stress?

Verification specific to this topic

  • Diffusion: a sphere with constant surface flux has a series solution (Crank, The Mathematics of Diffusion, Ch. 6); compare c(r,t).
  • Stresses (thermal analogy): for any radial c(r), \sigma_r=\frac{2\Omega E}{3(1-\nu)}\Big[\frac1{R^3}\int_0^Rc\,r^2dr-\frac1{r^3}\int_0^rc\,r^2dr\Big], \sigma_\theta=\frac{\Omega E}{3(1-\nu)}\Big[\frac2{R^3}\int_0^Rc\,r^2dr+\frac1{r^3}\int_0^rc\,r^2dr-c\Big] — evaluate the integrals of your FE concentration and compare with the FE stress (a check that separates the mechanics from the diffusion error).
  • Uniform concentration gives zero stress (free swelling) — the “stress-free field” test of Week 3.
  • MMS for the transient diffusion (time and space rates) and for the axisymmetric elasticity.

Stretch goals

  1. Concentration-dependent D(c) and two-way stress-enhanced diffusion (Newton on the coupled problem).
  2. Phase-field fracture (Week 5) driven by the diffusion-induced stress: critical C-rate for crack initiation.
  3. Core–shell particle (two materials, an interface).

Milestones

M1 1D radial diffusion with the series · M2 stresses with the integral check · M3 2D axisymmetric sphere = 1D · M4 C-rate/size study · M5 spheroids.

Pitfalls

  • The axisymmetric axis: no boundary condition on r=0, and the r-weight in every integral; the hoop strain u_r/r (Week 2).
  • The surface stress converges slowly with a steep concentration gradient at high C-rates: grade the mesh towards the surface.
  • Rigid-body modes of a free particle: fix the minimum (one point in 2D axisymmetric: u_z at one node) and check that the stresses do not change.

References

  • Y.-T. Cheng, M. W. Verbrugge, “Diffusion-induced stress, interfacial charge transfer, and criteria for avoiding crack initiation of electrode particles”, J. Electrochem. Soc. 157 (2010) A508–A516; and J. Power Sources 190 (2009) 453–460.
  • J. Christensen, J. Newman, “Stress generation and fracture in lithium insertion materials”, J. Solid State Electrochem. 10 (2006) 293–319.
  • J. Crank, The Mathematics of Diffusion, 2nd ed., Oxford 1975.