Lithium Dendrites Are Batteries' Dandelions

Ceramic solid electrolytes are supposed to be the boring, safe part of a next-generation battery. Swap a flammable liquid for a dense, rigid, ion-conducting ceramic and a lithium or sodium metal electrode should no longer be able to grow the metallic filaments (dendrites) that puncture liquid-electrolyte separators and short a cell ^[1,2,3]. It is intuitive to believe a slab of solid ceramic is a mechanical wall. Experiments across a wide range of solid electrolyte chemistries have shown this intuition is simply wrong: dendrites penetrate ceramics too, just at a measurable critical current density $J_\text{crit}$ above which the cell reliably fails ^[1]. A March 2026 preprint by Ansgar Lowack derives, for the first time, a closed-form analytical expression for $J_\text{crit}$ from first principles, and in doing so accidentally wrote down the same physics that lets a dandelion break a driveway ^[1].

A pressure with two ways out

The starting picture is a ceramic separator riddled with microscopic interfacial defects, thin grain-boundary cracks between the alkali metal electrode and the ceramic, already filled with metal that crept in during cell assembly ^[1]. When current flows and metal ions plate out at the tip of one of these defects, that new volume of metal has to go somewhere. The paper works out that there are two ways to accommodate it.

The first way is creep: the soft alkali metal (lithium or sodium is ductile even at room temperature) simply deforms viscoplastically to make room, and the pressure inside the defect stays low ^[1]. The second way is crack propagation: if the defect is too thin and confined for creep to keep up, pressure builds until it reaches the Griffith threshold $p_\text{Griffith} = K_{Ic}/\sqrt{c}$, at which point it becomes thermodynamically cheaper for the crack to simply extend into the ceramic and make its own room ^[1]. $K_{Ic}$ is the fracture toughness of the ceramic (weakened, notably, by the metal already wetting the crack surface, equation 13 in the paper works out the reduction from the surface energies and the work of adhesion between metal and ceramic) ^[1].

I kept thinking about dandelions and asphalt while reading this section. A root tip pushing into a hairline crack in pavement does the exact same two things in the exact same order: it first swells by turgor pressure (measured at up to roughly 0.6 MPa, several times a car tire) deforming into whatever space the crack already offers, and only once that hydrostatic pressure exceeds what the asphalt can resist does the crack actually widen and propagate ^[4,5]. Nobody would call turgor pressure "Joule heating," but structurally it is the same fork: a confined volume increase either gets absorbed elastically/plastically, or it pays the fracture energy of the surrounding solid and the defect grows. Root biologists and fracture mechanics engineers arrived at the same two-branch decision independently, from opposite ends of materials science.

Minimizing dissipation

The reason the paper can turn this picture into an equation is Onsager's variational principle: of all the current distributions mathematically compatible with the boundary conditions, the one nature actually realizes is the one that minimizes total irreversible power dissipation ^[1]. This is the textbook statement "current takes the path of least resistance" but there are now two dissipation channels competing for the minimum.

Routing current uniformly straight through the ceramic, ignoring the metal-filled defect entirely, costs ordinary bulk Joule dissipation $\dot Q_\text{bulk}=\int_\Omega \rho\,|j|^2\,dV$ ^[1]. But the metal-filled defect is a good conductor sitting inside a much more resistive ceramic, so the current would electrically "prefer" to funnel toward the defect tip and use it as a shortcut. That funneling reduces Joule dissipation but it forces metal deposition (and hence crack propagation, if creep can't keep up) exactly at the tip, which costs mechanical dissipation $\dot W \propto p_\text{Griffith}\cdot I_c$ ^[1]. A dendrite only grows once the Joule-dissipation savings from current localization outweigh the mechanical cost of cracking the ceramic to get there. $\Delta\dot Q_\text{max}+\Delta\dot W=0$ is that break-even point ^[1].

To actually solve this, Lowack approximates each interfacial defect as one half of a flat ellipsoid, which turns an otherwise unsolvable Laplace-equation problem into a classic electrostatics exercise (image charges, worked out in Stratton and in Landau & Lifshitz) with a known analytical potential field ^[1,6,7]. Plugging that potential back into the dissipation functional and solving the break-even condition yields the paper's central result:

where $\sigma$ is ionic conductivity, $V_M$ the molar volume of the alkali metal, $F$ the Faraday constant, $Y\sim 3/2$ a geometry constant, and $c_\text{max}$ the length of the single longest, sufficiently thin interfacial defect ^[1].

The weakest link

That $c_\text{max}^{-3/2}$ dependence means $J_\text{crit}$ is set entirely by the single worst defect. Assuming defect lengths follow a Pareto distribution across the interface (a small shape parameter $k$ meaning some defects are much longer than average), the largest defect in any given sample follows extreme-value statistics, and propagating that through $J_\text{crit}\propto c_\text{max}^{-3/2}$ produces a Weibull distribution for the critical current density across nominally identical cells ^[1]. This is precisely the weakest-link argument Weibull used in 1939 to describe the scattering of tensile strength between ceramic samples, and the paper shows the electrochemical Weibull modulus should come out to roughly a third of the mechanical one for the same material ^[1].

This is the second place the asphalt analogy held up better than I expected. Forestry and pavement-engineering literature on tree roots reports the same weakest-link behavior and the same feedback loop of significantly more root growth found under pre-existing cracks than under intact pavement, because a crack is where moisture and aeration concentrate, so a crack promotes more root growth, which enlarges the crack, which attracts still more root growth ^[8,9]. A ceramic dendrite doesn't get attracted by moisture, but the logic is the same.

Slow failure

The paper doesn't stop at $J_\text{crit}$. Below the critical current, ceramic solid electrolytes still conduct a small residual electron current (quantified by the electron transference number $t_e$), and that residual conduction lets alkali ions and electrons recombine directly at a defect tip without needing the full Griffith pressure (quantified by electrochemical stress-corrosion cracking) ^[1]. This gives a second threshold, $J_\text{subcrit}\approx Y\,\sigma\, K_{Isc}\, /(F\, c_\text{max}^{3/2})$, below $J_\text{crit}$, above which a dendrite grows slowly rather than catastrophically, at a failure time $\tau_\text{fail}\propto 1/(t_e\, J\, c_0)$ that the paper estimates at anywhere from tens of minutes to roughly ten hours for realistic garnet parameters ^[1]. Suppressing $t_e$ (making the ceramic a worse electron conductor) is therefore a lever completely independent of increasing fracture toughness or ionic conductivity, and the paper is explicit that it can shut down subcritical growth even when the classic critical-current picture would predict failure ^[1].

Root growth into pavement has an analogous slow regime: pavement studies describe damage accumulating over years of gradual root thickening long before any sudden failure, governed by the same tension-on-the-surface mechanic (concrete and asphalt are strong in compression but weak in tension) that governs Griffith crack growth in the ceramic case ^[9,10]. Slow, sub-threshold damage accumulation followed by sudden failure once a critical length is reached is apparently just what happens whenever something tries to wedge open a brittle, weakly conductive/porous barrier, batteries and driveways included.

Battery engineering

Numerically, the model is honest about where it still falls short: plugging in FESEM-measured defect sizes ($c_\text{max}\sim30$ µm, width $w\sim50$ nm) and typical Na₅SmSi₄O₁₂ material constants predicts $J_\text{crit}\sim50$ mA/cm², one to two orders of magnitude above the experimentally observed range of roughly 1 mA/cm² ^[1,11]. The paper attributes the gap to treating $K_{Ic}$ as a fixed bulk material constant, when atomistic and molecular-dynamics work suggests fracture toughness likely drops sharply right at a growing dendrite tip, for reasons still under active debate ^[1]. Even with that caveat, the design guidance is to suppress long, thin grain-boundary defects (since $J_\text{crit}$ only cares about the single worst one, not the average), push ionic conductivity and fracture toughness up, and push electron transference number down to kill the subcritical pathway. There's also a genuinely counterintuitive corollary. Since the model treats the growing dendrite as always metal-filled, a stronger interfacial adhesion between electrode and ceramic actually lowers the effective fracture toughness $K_{Ic}$ (via the surface-energy/work-of-adhesion relation), creating a real design tension with other interface engineering goals like suppressing void formation on discharge ^[1,12].

The next time your battery short-circuits, I hope you are thinking about dandelions.