Fig. 1: Emergence of chemomechanical microparticle self-oscillation.

a Schematic of a self-limited system of a single particle resting still at the air–liquid interface of a H₂O₂ drop. The particle is composed of a catalytic patch of Pt (yellow) underneath a polymeric disc (blue). The O₂ formation slows down asymptotically over time as the gas bubble restricts the available catalytic surface area. b A 2-particle system, in contrast, exhibits an emergent and self-sustained beating behaviour as the bubble merger restores the previously hindered reactivity, thus disrupting the equilibrium state. c, d Micrograph sequence (c) and tracked particle coordinates (d) of a 1-particle system that remains still for an extended period of time. e, f Micrograph sequence (e) and tracked coordinates (f) of a 2-particle system with emergent beating. The breathing radius, r(t), is the distance from the collective's centroid to each particle, averaged over all particles. g The long-term breathing radius trajectory of the same system as in e and f demonstrates the robustness of the beating behaviour. The shaded portion is magnified in the right panel, where the mechanistic model simulations (black, Supplementary Note 1) are shown to match the experimental curve (blue). h The phase portraits of 4 independent 2-particle experiments demonstrate reproducible limit cycles with closed-loop orbits, confirming the periodicity of collective beating. Note that to calculate the phase portraits the system's bubble-driven discontinuities were processed through a standard finite-impulse response filter (see Methods). All phase portraits share the same axes. i The recurrence histograms of the same 4 experiments all display a narrow peak centred at a period of 3.2 s, consistent with visual evidence in e. All histograms share the same axes. j The beating frequency can be tuned with the concentration of H₂O₂. The dependence predicted by the mechanistic simulations, on the basis of a Langmuir–Hinshelwood kinetics (black curve), matches the experimental measurements (blue markers). Scale bars, 500 μm.