Active fluids are a strange class of matter. Their microscopic constituents — motor proteins dragging filaments, bacteria propelling themselves through liquid — continuously convert chemical energy into motion, keeping the system permanently out of equilibrium. At low driving, the flows are weak and disorganized. Crank up the activity, and the fluid erupts into a self-sustained storm of vortices and jets known as active turbulence. The problem, the authors argue, is that unlike ordinary inertial turbulence, where well-established control parameters and asymptotic regimes exist, active turbulence has never had a clear dividing line separating a weakly disordered flow from a genuinely collective, system-scale state. Equilibrium statistical physics classifies phases through free energies, symmetry breaking and universal scaling, all rooted in detailed balance — tools that driven systems, lacking an underlying free energy, do not obviously inherit.

The team’s solution borrows from one of the most elegant developments in modern mathematical physics: Schramm–Loewner evolution, or SLE. SLE describes a one-parameter family of random, conformally invariant curves, labeled by a diffusivity κ, that arise as the scaling limits of interfaces in critical two-dimensional models. Specific values of κ correspond to famous systems — loop-erased random walks at κ = 2, Ising model interfaces at κ = 3, and critical percolation cluster boundaries at κ = 6. Because the parameter fully determines the curve statistics, measuring κ in a complex system provides a stringent test of whether that system shares the universality class of a known critical model. Demonstrating SLE behavior also implies conformal invariance — invariance under translations, rotations, rescalings and angle-preserving transformations — an exceptionally restrictive symmetry that tightly constrains the geometry of fluctuations across all scales.

Previous work had hinted that living matter might possess such hidden symmetry. In epithelial cell layers and bacterial colonies, the nodal lines of the vorticity field — the zero-vorticity contours separating regions of clockwise and counterclockwise rotation — had been found to follow the statistics of critical percolation, corresponding to SLE with κ = 6. But it remained unclear whether these statistics reflected a robust transition controlled by activity or were incidental features of particular biological systems. The new study settles the question by combining two independent experimental platforms with large-scale simulations. The first experiment used the canonical microtubule–kinesin active nematic, where activity is tuned through ATP concentration; the second used dense suspensions of swimming Bacillus subtilis, where oxygen availability controls self-propulsion.

The results were strikingly consistent. In the microtubule–kinesin system, at ATP concentrations above roughly 18 micromolar, the extracted vorticity nodal lines obeyed the analytical left-passage probability prediction for κ = 6, and the Loewner driving function behaved as a Brownian process with variance growing linearly in time at exactly the rate expected for SLE6. Below about 8 micromolar ATP, both signatures vanished. In the bacterial suspensions, high-oxygen conditions produced the same critical percolation statistics, while oxygen depletion — with the transition estimated to fall between 0.07 and 0.125 millimolar dissolved oxygen — destroyed them. Crucially, the velocity and vorticity fields look visually indistinguishable on either side of the threshold; the difference is not in how the flow appears, but in the symmetry class of its geometry.

Simulations revealed why. In the hydrodynamic active nematic model, where a nematic order parameter couples to flow through an active stress proportional to the activity parameter ζ, the same SLE6 signatures emerge only above a critical activity ζc. Below that threshold, the left-passage probability deviates from the analytical form and the driving function loses its Brownian character. Within the SLE framework, this breakdown corresponds directly to the loss of conformal invariance. The activity-controlled crossover is therefore a genuine symmetry-breaking transition — but of an unusual kind, in which the symmetry gained or lost is conformal, imposing exceptionally strong constraints on the flow geometry across scales. Even more remarkably, when the team replaced explicit active stress with stochastic forcing in a fluctuating nematohydrodynamic model, the conformally invariant state still appeared whenever detailed balance was violated, and disappeared when the fluctuations were constrained to satisfy the fluctuation–dissipation relation. Conformal symmetry, it turns out, is a generic consequence of non-equilibrium driving, not a property of any particular microscopic mechanism.

The transition is not merely geometric. Vortices are carriers of stress and momentum, so the team went on to ask how the vortex population reorganizes at the same threshold. Detecting vortex cores from the winding of the velocity field and connecting nearby vortices into geometric graphs, they found that below the transition, a spanning cluster forms only at connection distances far larger than the natural vortex spacing, while above ζc the percolation threshold snaps to roughly one characteristic length. More tellingly, a rigidity percolation analysis using the pebble game algorithm showed that precisely at ζc, a giant mechanically rigid cluster spanning the entire system emerges — the flow switches from a floppy collection of independent vortices to a self-supporting mechanical network. Both experimental systems showed the same trend, with high-activity states exhibiting larger geometric clusters and rigid clusters than their low-activity counterparts.

A third, fully independent diagnostic came from topology. Using persistent homology — a method that tracks how connected components and loops in the vorticity landscape are born and die as the field is systematically ‘flooded’ — the researchers converted persistence diagrams into persistence images and applied principal component analysis. The low-activity and high-activity regimes separated cleanly along the dominant principal component, with the transition occurring at the same critical activity identified by the SLE diagnostics and the vortex network analyses. Below the threshold, topological features had similar lifetimes, reflecting smooth, weakly structured flows; above it, the lifetime distributions broadened into long tails characteristic of heterogeneous, intermittent fields. Three independent lenses — nodal line geometry, vortex mechanics and field topology — all converge on the same critical point, establishing the transition as a robust, system-wide reorganization.

The implications reach well beyond soft matter laboratories. Because changes in flow organization in biological systems are tied to transport, mixing and force transmission, the identified threshold marks the point at which an active flow becomes coordinated across the entire system, both geometrically and mechanically. The work also connects active turbulence to a growing family of non-equilibrium systems in which conformal invariance has recently been spotted, including driven amorphous solids, water-wave turbulence and liquid-crystal interfaces, and it dovetails with theoretical efforts linking conformal symmetry, topology and non-equilibrium field theory. Open questions remain — most strikingly why the critical percolation of nodal lines and the rigidity percolation of vortex centres, distinct universality classes in equilibrium settings, coincide here. But the message is clear: fully developed active turbulence is not merely a disordered regime. It is a non-equilibrium critical state, and the boundary of that state can now be measured, in a microscope, with nothing more than the zeros of a vorticity field.

Subject of Research: The onset of fully developed active turbulence as an activity-driven non-equilibrium phase transition with critical percolation and rigidity statistics

Article Title: Fully developed active turbulence defined through a non-equilibrium phase transition

Article References: Bonn, L., Ma, T., Bantysh, O., Feng, W., Pedersen, M. C., Jing, G., Ignés-Mullol, J., Sagués, F., Araujo, N. A. M., & Doostmohammadi, A. (2026). Fully developed active turbulence defined through a non-equilibrium phase transition. Nature Physics. https://doi.org/10.1038/s41567-026-03408-y

Image Credits: AI Generated

DOI: 10.1038/s41567-026-03408-y

Keywords: active matter, active turbulence, phase transition, percolation, Schramm-Loewner evolution, conformal invariance, active nematics, bacterial suspensions, rigidity percolation, persistent homology, non-equilibrium physics, vorticity

The stakes could hardly be higher. Lithium metal is the dream anode, packing a theoretical specific capacity of 3860 mAh per gram with a remarkably low mass density of 0.534 g per cubic centimeter and an aggressive reducing potential of −3.04 volts against the standard hydrogen electrode. Paired with nonflammable inorganic solid electrolytes such as garnet-type LLZO, it underpins not only next-generation solid-state cells but also exotic lithium–sulfur and lithium–air chemistries. The problem is that a solid–solid contact is never truly conformal. Unlike the liquid electrolytes that wet every crevice of a conventional electrode, a rigid ceramic pressed against soft lithium leaves microscopic gaps, and those imperfections amplify into spatially heterogeneous deposition and dissolution. Years of experiments, from pressure-dependent stripping tests on Li–LLZO cells to operando synchrotron X-ray tomography, have converged on a single conclusion: contact loss during stripping, not interphase growth, is the primary driver of current constriction and cell failure.

What has been missing is a mechanistic account of how an individual void actually behaves once it forms. Does it simply grow monotonically under stripping, as many models assumed? Or can it shrink, migrate, or even detach from the interface entirely? The research team behind the new work, using a three-dimensional kinetic Monte Carlo framework, simulated a 50-by-50-by-50 cubic lattice representing a 17.5-nanometer cube of lithium against LLZO, and tracked a single hemispherical void through the competing atomic events that shape it. Four mechanisms were woven into the model: bulk migration of lithium vacancies away from the interface, surface diffusion of lithium atoms across the evolving void boundary, electro-dissolution at active contact sites governed by Butler–Volmer kinetics, and transport of lithium ions into the electrolyte bulk. Each event carried an Arrhenius rate constant calibrated to literature values, with activation barriers of 0.3 eV for terrace diffusion, 0.15 eV for step diffusion, and 0.5 eV for interlayer diffusion.

The simulations revealed a strikingly choreographed two-act drama. In the first act, current focusing concentrates the electric field at the sharp circular edge of the hemispherical void, where the electrolyte’s potential distribution distorts around the non-conducting cavity. The enhancement decays with a 1/r-cubed dependence, so dissolution attacks the void rim preferentially, and the void spreads laterally along the interface while its depth barely changes. In the second act, surface diffusion takes over: vacancies redistribute from the interface down along the void surface, the void’s footprint retracts slightly, and its depth increases substantially. The team quantified this morphological ballet with geometric descriptors, including a circularity index that dropped as the footprint elongated and roughened, then recovered as surface diffusion smoothed the shape back toward symmetry, and a vertical shape factor that stayed flat during lateral growth before climbing sharply during the deepening phase.

The most consequential discovery concerns the three distinct modes of surface diffusion, classified through a terrace-step-kink framework that distinguishes atomic hops by their coordination environment. Terrace diffusion, in which adatoms glide across flat crystalline planes, and step diffusion, in which they travel along step edges, both act to keep voids compact and anchored at the interface. Interlayer diffusion, by contrast, shuttles atoms between adjacent atomic layers and directs vacancies into the depth of the electrode. When interlayer diffusion dominates, voids detach from the interface and drift into the lithium bulk, which paradoxically allows the interface to heal as fresh contact is restored. Phase maps sweeping the activation barriers from 0.05 to 0.5 eV showed that suppressing interlayer diffusion while keeping terrace diffusion active produces flat, pancake-shaped voids, whereas strong step diffusion inflates voids in both lateral and depth directions, and combined dominance of interlayer and bulk diffusion can dissolve a void entirely into the electrode.

To unify these observations, the researchers introduced an electrochemical Damköhler number, the ratio of the reaction rate to the bulk vacancy diffusion rate, which compares how fast vacancies are generated at the interface with how fast they can be swept away into the anode interior. When this number is far below unity, vacancy removal outpaces generation, vacancies disperse, and voids shrink or vanish altogether, defining a no-void regime. When it is far above unity, vacancies pile up around the void perimeter, and the resulting morphology is dictated by the surface diffusion ratios. Strong terrace diffusion yields compact but irregular voids; strong step diffusion carves the deepest attached voids; strong interlayer diffusion promotes detachment at moderate Damköhler numbers and complete dissolution when bulk diffusion is also strong. Notably, the analysis showed that bulk diffusion and interlayer diffusion are two independent routes by which a void can effectively disappear into the electrode, a finding with direct implications for how operating conditions might be tuned to favor healing over degradation.

Temperature emerged as a double-edged lever. Sweeping the operating temperature from −10 to 80 degrees Celsius, a range that spans subzero winter starts to sustained elevated-temperature operation in real cells, the model showed that higher temperatures accelerate both surface diffusion and bulk vacancy transport through Arrhenius kinetics, effectively enabling a transport-driven healing mechanism that suppresses void growth. But the authors caution that this benefit is chemistry-dependent. For garnet oxides like LLZO, whose interface with lithium is kinetically passivated, the morphological advantage of warmth is captured directly by the model. Sulfide electrolytes, however, can undergo interfacial reduction and form interphases that grow faster, and sometimes more dangerously, at elevated temperature, so chemical degradation may offset the healing benefit. The optimal operating temperature therefore depends on the balance between morphological recovery and chemical stability, and it will vary with electrolyte chemistry in ways the current framework deliberately leaves for future work.

Perhaps the most dramatic finding is the percolation transition. As stripping proceeds, thousands of single-site vacancy clusters nucleate randomly at the interface and along void edges, where current focusing makes dissolution most aggressive. These point defects act as nuclei that grow through aggregation and coalescence. The team tracked cluster populations across four stages of stripping and watched the original 6-cubic-nanometer void swell to 9 cubic nanometers as neighbors merged into it. Two coupled mechanisms drive the merging: current focusing is strongest in the narrow lithium ligaments separating adjacent voids, so those bridges dissolve preferentially, and curvature-mediated surface diffusion then transports atoms away from the high-curvature necks, lowering surface energy and consolidating each merged pair into a single compact void. At a critical void fraction of 0.0256, the void network suddenly connects across the simulation domain, snapping from an isolated-defect state to a percolated structure composed entirely of one giant cluster.

The electrochemical consequences of that geometric transition are severe. Using a flux-tube formulation, the researchers estimated the interfacial constriction resistance directly from the simulated contact morphology and found that it climbs by roughly two orders of magnitude across the percolation window, peaking at the minimum contact fraction before partially recovering as void detachment restores contact. The percolation threshold itself is tunable: strong terrace diffusion keeps voids as isolated compact clusters and prevents percolation entirely, while strong interlayer diffusion detaches voids before they can link laterally. A connected, contact-destroying network emerges only when both modes are sufficiently weak, and within that regime, step diffusion acts as the controlling mechanism, with strong step diffusion delaying the onset of percolation by carving deeper, more isolated voids instead of shallow lateral ones.

The model’s predictions align qualitatively with a decade of experimental evidence: the lateral-then-deep growth sequence matches operando dilatometry observations, the Damköhler threshold echoes the critical stripping current above which voids accumulate, and the partial healing under enhanced surface diffusion mirrors the recovery of dewetted interfaces after thermal treatment. The framework deliberately excludes mechanical effects such as stack pressure and lithium creep, which would supply material toward the interface and slow void growth, meaning the present results represent a conservative, pressure-free bound on degradation. Even so, the study delivers something solid-state battery engineers have lacked: a theoretical baseline identifying which surface diffusion modes dictate void morphology, and therefore a design compass for selecting electrolyte chemistries, operating temperatures, and current densities that keep lithium’s molecular potholes from connecting into a highway to failure.

Subject of Research: Void formation and percolation dynamics at the lithium metal–solid electrolyte interface during stripping in solid-state batteries

Article Title: Mechanistic Insights Into Instability From Single Void Dynamics at the Lithium Metal‐Solid Electrolyte Interface

Article References: Banerjee, S., Vishnugopi, B. S., Lin, G., & Mukherjee, P. P. (2026). Mechanistic Insights Into Instability From Single Void Dynamics at the Lithium Metal‐Solid Electrolyte Interface. Advanced Science, Article e78028. https://doi.org/10.1002/advs.78028

Image Credits: AI Generated

DOI: 10.1002/advs.78028

Keywords: solid-state batteries, lithium metal anode, solid electrolyte, void formation, kinetic Monte Carlo, surface diffusion, vacancy transport, percolation, LLZO, electrochemical stripping, interfacial stability, dendrite formation