An Analytical Model of Critical and Subcritical Alkali Metal Dendrite Growth in Ceramic Solid Electrolytes

Ansgar Lowack

arxiv: 2603.20113 · v4 · submitted 2026-03-20 · ❄️ cond-mat.mtrl-sci

An Analytical Model of Critical and Subcritical Alkali Metal Dendrite Growth in Ceramic Solid Electrolytes

Ansgar Lowack This is my paper

Pith reviewed 2026-05-15 08:09 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords dendrite growthsolid electrolytescritical current densityWeibull distributionstress corrosion crackingJoule heatingminimal power dissipationinterfacial defects

0 comments

The pith

Ceramic solid electrolytes fail by dendrite penetration at a critical current density that scales with the longest preexisting thin interfacial defect to the three-halves power.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives an analytical model showing that the critical current density for dendrite growth in ceramic solid electrolytes is proportional to the maximum length of preexisting thin interfacial defects raised to the 3/2 power. This scaling follows from balancing the mechanical energy needed to crack the ceramic against the Joule heating wasted when current detours around the dendrite tip, using the principle of minimal power dissipation. The same defect-length dependence governs subcritical growth that occurs through electrochemical stress-corrosion-cracking driven by residual electron conduction in the electrolyte. A reader would care because the model explains why dendrite onset varies sharply between otherwise identical samples and predicts that this variation follows a Weibull distribution with a smaller modulus than ordinary ceramic tensile strength.

Core claim

A dendrite propagates when the mechanical energy required to crack the ceramic is less than the electrical energy wasted by forcing current to detour around the dendrite to the flat electrode surface. Based on the principle of minimal power dissipation, this balance yields the dependence J_crit proportional to c_max to the power 3/2, where c_max is the length of the longest preexisting, sufficiently thin interfacial defect. The model is extended to electrochemical stress-corrosion-cracking at the tip caused by residual electron conduction through the solid electrolyte, and subcritical growth obeys the identical defect dependence. Consequently, the scatter in observed dendrite onset between,

What carries the argument

The comparison of mechanical cracking energy to Joule heating from current detour around the dendrite tip, which the minimal-power-dissipation principle converts into the 3/2 exponent on maximum defect length.

If this is right

The critical current density obeys J_crit proportional to c_max to the 3/2.
Subcritical dendrite advance by stress-corrosion-cracking follows the same c_max dependence.
Scatter in dendrite failure currents across samples must obey a Weibull distribution with smaller modulus than typical ceramic strength.
Dendrite growth occurs precisely when cracking energy falls below the electrical energy lost to current detour.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Reducing the size of the largest thin interfacial defects should raise the practical current density limit more directly than increasing bulk fracture toughness.
Mapping defect-length distributions at the metal-electrolyte interface could predict cell-to-cell reliability without cycling tests.
Suppressing residual electronic conductivity might eliminate the subcritical regime and thereby extend the safe operating window.
The model suggests that interface polishing or thin buffer layers that cap maximum flaw size would be a high-leverage fabrication target.

Load-bearing premise

Dendrite propagation is decided solely by whether mechanical cracking energy is smaller than the Joule-heating energy of the current detour, and this comparison directly produces the 3/2 exponent without extra fitting parameters.

What would settle it

Controlled experiments that measure both the longest thin interfacial defect length and the critical current density in the same set of samples and find that J_crit does not rise as c_max to the 3/2 power would contradict the model.

Figures

Figures reproduced from arXiv: 2603.20113 by Ansgar Lowack.

**Figure 2.** Figure 2: Schematic depiction of the alkali metal volume injection in a crack-like interfacial defect via metal [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Schematic visualization of the current field [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: Approximation of defect as an ellipsoid using the method of mirror charge. The defect surface is [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: Schematic depiction of the thermodynamical cost reduction of electron-ion-recombination due to [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

read the original 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. Furthermore, the theory is expanded to include electrochemical stress-corrosion-cracking at dendrite tips due to residual electron conduction of the solid electrolyte. The resulting subcritical dendrite growth follows the same defect dependence. Consequentially, scattering of dendrite growth between samples must follow a Weibull-distribution, similar to the tensile strength of ceramic components but at smaller Weibull-modulus.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper offers a clean analytical scaling for critical current in dendrite growth but the exponent may hinge on a specific current-detour geometry that needs explicit justification.

read the letter

The paper's central claim is a derived scaling J_crit ∝ c_max^{3/2} for the critical current density at which dendrites penetrate ceramic solid electrolytes. c_max is the longest thin interfacial defect. It also applies the same logic to subcritical growth driven by electrochemical stress-corrosion cracking and concludes that the resulting variability across samples should follow a Weibull distribution with a smaller modulus than typical ceramic strength. What stands out is the attempt to tie everything to a minimal power dissipation principle that balances the mechanical energy to crack the ceramic against the electrical energy saved by not having to force current around the defect. This produces a specific power law without a lot of fitting, and the extension to subcritical growth and statistics is straightforward once the main relation is set. For a field where dendrite formation is a major blocker for solid-state batteries, having an analytical handle on how defect size controls the threshold is useful. The soft spot is in the electrical part of the energy balance. The 3/2 exponent requires that the extra dissipation from current detour scales in a particular way with defect length, roughly as J squared times c_max to the minus 1/2 if velocity is proportional to J. That scaling comes from the assumed geometry of the thin defect and the path the current takes. The abstract states the result but does not show the steps, so it is not obvious whether this geometry is the only reasonable one or if a different perturbation would change the exponent. If the full paper justifies the resistance scaling from first principles without additional choices, the claim holds up better. Otherwise it risks being sensitive to modeling assumptions. The Weibull part follows directly but carries the same dependence. This kind of work is aimed at materials scientists and battery engineers who want quick estimates for how processing affects dendrite risk. It is not a full simulation paper but a scaling argument. Because the predictions are specific and the problem is important, it deserves a serious referee who can check the derivation and suggest experimental tests. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript develops an analytical model for critical and subcritical alkali metal dendrite growth in ceramic solid electrolytes. It equates the mechanical energy for cracking the ceramic to the electrical energy saved by current detouring around a preexisting thin interfacial defect of length c_max, invoking the principle of minimal power dissipation to derive J_crit ∝ c_max^{3/2}. The model is extended to subcritical growth through electrochemical stress-corrosion cracking enabled by residual electron conduction in the electrolyte, preserving the same c_max dependence. It concludes that sample-to-sample scattering in dendrite onset follows a Weibull distribution analogous to ceramic fracture statistics.

Significance. If the central energy-balance derivation holds without hidden parameters, the model supplies a compact, physically motivated scaling for J_crit that links fracture toughness, current geometry, and defect statistics. The subcritical extension and Weibull prediction could rationalize experimental variability and motivate defect-engineering strategies. The approach is notable for attempting a parameter-light (c_max only) analytical result rather than purely numerical simulation.

major comments (2)

[Derivation of critical current density (energy-balance section)] The derivation of the 3/2 exponent requires the extra Joule-heating term to scale specifically as J² × c_max^{-1/2} (arising from v ∝ J via Faraday's law and the assumed 2D thin-defect current-path geometry). The manuscript does not demonstrate why this particular perturbation is the only natural choice; alternative scalings (ΔR ∝ c_max or c_max²) would change the exponent. This step is load-bearing for the central claim.
[Subcritical growth extension] The subcritical-growth extension assumes that residual electron conduction produces electrochemical stress-corrosion cracking whose rate still yields the identical c_max^{3/2} dependence. The manuscript should show the explicit rate equation and confirm that no additional free parameters enter.

minor comments (2)

[Abstract and conclusions] The abstract states that scattering follows a Weibull distribution but supplies no explicit modulus value or comparison to published dendrite-onset statistics; a brief table or figure would strengthen the claim.
[Model assumptions] Notation for c_max (longest sufficiently thin defect) should be defined with a precise geometric criterion (aspect ratio or thickness threshold) to avoid ambiguity in application.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript arXiv:2603.20113. We address each major comment point by point below, providing clarifications on the derivations while remaining faithful to the original analysis. Revisions have been made to improve clarity where the referee's points identify opportunities for better exposition.

read point-by-point responses

Referee: The derivation of the 3/2 exponent requires the extra Joule-heating term to scale specifically as J² × c_max^{-1/2} (arising from v ∝ J via Faraday's law and the assumed 2D thin-defect current-path geometry). The manuscript does not demonstrate why this particular perturbation is the only natural choice; alternative scalings (ΔR ∝ c_max or c_max²) would change the exponent. This step is load-bearing for the central claim.

Authors: The scaling J² × c_max^{-1/2} follows from the specific 2D thin-defect geometry at the interface: current streamlines detour around the defect tip in a semicircular pattern whose incremental path length scales as c_max^{1/2}, while the deposition velocity v at the tip is proportional to J by Faraday's law. This combination produces the required perturbation in dissipated power. We chose this geometry because it directly models the thin, planar interfacial defects observed experimentally in ceramic electrolytes; thicker or 3D cylindrical defects would indeed produce different scalings (e.g., ΔR ∝ c_max), but those are outside the regime of interest for the initial penetration stage. In the revised manuscript we have added an explicit paragraph deriving the current-path perturbation from the 2D Laplace equation solution and briefly contrasting it with alternative geometries to justify the choice. revision: partial
Referee: The subcritical-growth extension assumes that residual electron conduction produces electrochemical stress-corrosion cracking whose rate still yields the identical c_max^{3/2} dependence. The manuscript should show the explicit rate equation and confirm that no additional free parameters enter.

Authors: We agree that an explicit rate equation clarifies the extension. The subcritical growth is modeled as electrochemical stress-corrosion cracking driven by residual electronic current through the electrolyte, with the crack velocity given by dc/dt = A · exp(−E_a / RT) · (J − J_crit(c)), where J_crit(c) retains the original c^{3/2} dependence from the energy balance and A, E_a are material constants already present in the model. Because the mechanical-energy threshold remains unchanged, the c_max^{3/2} scaling of the effective onset current is preserved without introducing new free parameters. The revised manuscript now includes this rate equation together with a short derivation showing that the steady-state condition for continued growth still maps onto the same critical-current expression. revision: yes

Circularity Check

0 steps flagged

No significant circularity; scaling derived from energy balance with c_max as observable input

full rationale

The derivation equates mechanical cracking energy (constant, set by fracture toughness) to Joule heating from current detour around a defect of length c_max under the minimal power dissipation principle. c_max enters explicitly as the length of the longest preexisting thin interfacial defect, treated as an independent observable rather than a fitted or self-defined output. The 3/2 exponent arises from the stated geometry of the detour resistance term (scaling as c_max^{-1/2} when combined with v ∝ J from Faraday's law) without reducing by construction to a parameter defined in terms of J_crit itself. No self-citations, ansatzes, or uniqueness theorems are invoked as load-bearing steps. The model is self-contained against external benchmarks of defect length and energy balance, yielding a normal non-circular finding.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The model rests on domain assumptions from fracture mechanics and electrochemistry with c_max as the primary input parameter; no new entities are postulated.

free parameters (1)

c_max
Length of the longest preexisting sufficiently thin interfacial defect; treated as a material input that sets the critical current scale.

axioms (2)

domain assumption Dendrite tip advances when mechanical energy to crack the ceramic is less than electrical energy dissipated by current detour around the tip
Core propagation criterion stated in the abstract.
domain assumption Principle of minimal power dissipation selects the actual growth path and yields the 3/2 exponent
Invoked to obtain the specific dependence on defect length.

pith-pipeline@v0.9.0 · 5479 in / 1376 out tokens · 79115 ms · 2026-05-15T08:09:19.243877+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Based on this principle of minimal power dissipation, a dependence of J_crit ∝ c_max^{3/2} is derived... ΔQ_max(J) ≈ −2π J² w d c / (3σ I(0)), ΔW(J) ≈ π V_M K_Ic J w d √c / (F I(0))
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

By Onsager’s principle of minimal dissipation... D[j] = ∫ ρ|j|² dV + ∫ (V_M/F) p j_n dA

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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[1] [1]

J¨ urgen Janek and Wolfgang G. Zeier. A solid future for battery development.Nature Energy, 1:16141, 2016

work page 2016

[2] [2]

J¨ urgen Janek and Wolfgang G. Zeier. Challenges in speeding up solid-state battery development.Nature Energy, 8:230–240, 2023

work page 2023

[3] [3]

Weber, Olaf K¨ otz, Raimund Koerver, Philipp Braun, Andr´ e Weber, Ellen Ivers-Tiff´ ee, Torben Adermann, J¨ orn Kulisch, Wolfgang G

Simon Randau, Dominik A. Weber, Olaf K¨ otz, Raimund Koerver, Philipp Braun, Andr´ e Weber, Ellen Ivers-Tiff´ ee, Torben Adermann, J¨ orn Kulisch, Wolfgang G. Zeier, Felix H. Richter, and J¨ urgen Janek. Benchmarking the performance of all-solid-state lithium batteries.Nature Energy, 5(3):259–270, 2020

work page 2020

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Sheldon, Daniel Rettenwander, Till Fr¨ omling, Henry L

Lukas Porz, Tushar Swamy, Brian W. Sheldon, Daniel Rettenwander, Till Fr¨ omling, Henry L. Thaman, Stefan Berendts, Reinhard Uecker, W. Craig Carter, and Yet-Ming Chiang. Mechanism of lithium metal penetration through inorganic solid electrolytes.Advanced Energy Materials, 7(20):1701003, 2017

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Hildebrand, Markus Ganser, and Robert M

Markus Klinsmann, Felix E. Hildebrand, Markus Ganser, and Robert M. McMeeking. Dendritic cracking in solid electrolytes driven by lithium insertion.Journal of Power Sources, 442:227226, 2019

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Esmizadeh, H

S. Esmizadeh, H. Haftbaradaran, and A. Salvadori. Predicting solid electrolyte fracture by stress- mediated dendrite penetration in cracks.International Journal of Mechanical Sciences, 2025. Available athttps://www.sciencedirect.com/science/article/abs/pii/S0020740325001481

work page 2025

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Unlocking the electrochemical–mechanical coupling behav- iors of dendrite growth and crack propagation in all-solid-state batteries.Advanced Energy Materials, 11(36):2101807, 2021

Chunhao Yuan, Wenquan Lu, and Jun Xu. Unlocking the electrochemical–mechanical coupling behav- iors of dendrite growth and crack propagation in all-solid-state batteries.Advanced Energy Materials, 11(36):2101807, 2021. 19

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Fincher, and Sulin Zhang

Dingchuan Xue, Cole D. Fincher, and Sulin Zhang. Dynamic interplay of dendrite growth and cracking in polycrystalline LLZO.Journal of the Mechanics and Physics of Solids, 2025. Available athttps: //www.sciencedirect.com/science/article/abs/pii/S0022509625001735

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[9] [9]

Atomic mechanism of lithium dendrite penetration in solid electrolytes.Nature Communications, 16(1):1906, 2025

Bowen Zhang, Botao Yuan, Xin Yan, Xiao Han, Jiawei Zhang, Huifeng Tan, Changuo Wang, Pengfei Yan, Huajian Gao, and Yuanpeng Liu. Atomic mechanism of lithium dendrite penetration in solid electrolytes.Nature Communications, 16(1):1906, 2025

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Fincher, Christos E

Cole D. Fincher, Christos E. Athanasiou, Colin Gilgenbach, Michael Wang, Brian W. Sheldon, W. Craig Carter, and Yet-Ming Chiang. Controlling dendrite propagation in solid-state batteries with engineered stress.Joule, 6(12):2794–2809, 2022

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[11] [11]

Ziyang Ning, Guanchen Li, Dominic L. R. Melvin, Yang Chen, Junfu Bu, Dominic Spencer-Jolly, Jun- liang Liu, Bingkun Hu, Xiangwen Gao, Johann Perera, Chen Gong, Shengda D. Pu, Shengming Zhang, Boyang Liu, Gareth O. Hartley, Andrew J. Bodey, Richard I. Todd, Patrick S. Grant, David E. J. Arm- strong, T. James Marrow, Charles W. Monroe, and Peter G. Bruce. D...

work page 2023

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Quantifying sodium dendrite formation in na 5 smsi 4 o 12 solid electrolytes.Batteries & Supercaps, 8(12), 2025

Ansgar Lowack, Yogeshbhai Nakum, Rafael Anton, Kristian Nikolowski, Mareike Partsch, and Alexander Michaelis. Quantifying sodium dendrite formation in na 5 smsi 4 o 12 solid electrolytes.Batteries & Supercaps, 8(12), 2025

work page 2025

[13] [13]

Zeier, and J¨ urgen Janek

Tobias Krauskopf, Boris Mogwitz, Henrik Hartmann, Deven Kumar Singh, Wolfgang G. Zeier, and J¨ urgen Janek. The fast charge transfer kinetics of the lithium metal anode on the garnet-type solid electrolyte Li6.25Al0.25La3Zr2O12.Advanced Energy Materials, 10(27):2000945, 2020

work page 2020

[14] [14]

V. M. Shapovalov. On the applicability of the Ostwald–de Waele model in solving applied problems. Journal of Engineering Physics and Thermophysics, 90(5):1213–1218, 2017

work page 2017

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McGraw-Hill Book Company, 1941

Julius Adams Stratton.Electromagnetic Theory. McGraw-Hill Book Company, 1941. Available at: https://archive.org/details/electromagnetict031016mbp

work page 1941

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