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An Analytical Model of Alkali Metal Dendrite Growth in Ceramic Solid Electrolytes based on Griffith's Theory

Ansgar Lowack

Abstract

In solid-state batteries, ceramic solid electrolytes are penetrated by dendrites when plating above a critical current density $J_\mathrm{crit}$. A dendrite will propagate by metal deposition at a pre-existing dendrite tip if the mechanical energy required to crack the ceramic open is less than the electrical energy (Joule heating) wasted by forcing the current to detour around the dendrite to the flat electrode surface. Based on this principle of minimal power dissipation, a dependence of $J_\mathrm{crit}\propto c_\mathrm{max}^{3/2}$ is derived. $c_\mathrm{max}$ is the length of the longest preexisting, sufficiently thin interfacial defect. Consequentially, scattering of $J_\mathrm{crit}$ between samples must follow a Weibull-distribution, similar to the tensile strength of ceramic components.

An Analytical Model of Alkali Metal Dendrite Growth in Ceramic Solid Electrolytes based on Griffith's Theory

Abstract

In solid-state batteries, ceramic solid electrolytes are penetrated by dendrites when plating above a critical current density . A dendrite will propagate by metal deposition at a pre-existing dendrite tip if the mechanical energy required to crack the ceramic open is less than the electrical energy (Joule heating) wasted by forcing the current to detour around the dendrite to the flat electrode surface. Based on this principle of minimal power dissipation, a dependence of is derived. is the length of the longest preexisting, sufficiently thin interfacial defect. Consequentially, scattering of between samples must follow a Weibull-distribution, similar to the tensile strength of ceramic components.
Paper Structure (19 sections, 84 equations, 4 figures)

This paper contains 19 sections, 84 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Discussing the model
  3. Geometrical assumptions
  4. Transport
  5. Dissipation
  6. Interfacial pressure field
  7. Onsager's principle of minimal dissipation
  8. Emergence of a local critical current density
  9. Dissipation estimation
  10. Global critical current density
  11. Weibull-statistics of critical current density scattering
  12. Numerical example and plausibility discussion
  13. Conclusion
  14. Supplementary derivations
  15. Gradient and normal derivative of $\Phi_c$
  16. ...and 4 more sections

Figures (4)

  • Figure 1: Schematic cross section of a SSB with unspecified counter electrode, indicating relevant variables.
  • Figure 2: Schematic depiction of the alkali metal volume injection in a crack-like interfacial defect via metal creep (left) and via crack growth (right).
  • Figure 3: Schematic visualization of the current field $\mathbf j_c(\mathbf x)$ around a defect at $\mathbf x_c$ for no current localization and the maximum possible current localization.
  • Figure 4: Approximation of defect as an ellipsoid using the method of mirror charge. The defect surface is $\Gamma_c$. Hence, the surface of the full ellipsoid is $\Gamma_c\cup\Gamma'_c$.