Emergent microrobotic oscillators via asymmetry-induced order

Abstract

Spontaneous oscillations on the order of several hertz are the drivers of many crucial processes in nature. From bacterial swimming to mammal gaits, converting static energy inputs into slowly oscillating power is key to the autonomy of organisms across scales. However, the fabrication of slow micrometre-scale oscillators remains a major roadblock towards fully-autonomous microrobots. Here, we study a low-frequency oscillator that emerges from a collective of active microparticles at the air-liquid interface of a hydrogen peroxide drop. Their interactions transduce ambient chemical energy into periodic mechanical motion and on-board electrical currents. Surprisingly, these oscillations persist at larger ensemble sizes only when a particle with modified reactivity is added to intentionally break permutation symmetry. We explain such emergent order through the discovery of a thermodynamic mechanism for asymmetry-induced order. The on-board power harvested from the stabilised oscillations enables the use of electronic components, which we demonstrate by cyclically and synchronously driving a microrobotic arm. This work highlights a new strategy for achieving low-frequency oscillations at the microscale, paving the way for future microrobotic autonomy.

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.

Fig. 2. Observations of emergent order via symmetry-breaking.

a Schematic of interarrival times in a system of beating microparticles, defined as the time that transpires between two consecutive bubble collapses. The interarrival time distribution should be tight (i.e., a single peak) in a perfectly periodic system, and broad in an aperiodic system. b (top to bottom) Interarrival time distributions and optical micrographs for homogeneous systems of N = 2, 3, 5, and 8 identical particles. As N increases, the collective system periodicity gradually decays and transitions to an exponential interarrival distribution at N = 8 (bottom, black curve). Scale bar, 500 μm. c Indeed, we observe that the breathing radius of a homogeneous N = 8 system is not periodic. d Asymmetry-induced order across N predicted by Rattling Theory. A quantification of collective disorder, the system's Rattling $\mathcal{R}$ is predicted to be lower (i.e. more orderly) if the relative burst intensity of one particle is increased beyond or decreased below 1x, which signifies homogeneity. This is experimentally realised by modulating the Pt patch size on a “designated leader” (DL) particle relative to the others. The curves are offset to make all $\mathcal{R}=0$ at 1x intensity to highlight the effect of system heterogeneity on Rattling. See Supplementary Note 2 for a detailed discussion of the analytical model. e Same as (b), but for heterogeneous systems of equal particle numbers, where the DL broke the permutation symmetry. In contrast to the homogeneous systems (b), they remain robustly periodic across N. It is important to recognise that the polymeric disc size of a DL is unchanged. Scale bar, 500 μm. f, Breathing radius for an 8-particle DL system (i.e., N = 7 + 1DL), which reliably beats periodically. The period of 14.2s extracted from r(t) coincides with the most probable interarrival time in e (bottom).

Fig. 3. Designated leaders induce periodic limit cycles.

a, b Features of DL beating explained with schematic (a) and micrograph sequence (b) of a 2-particle heterogeneous system. The leader particle is able to grow a large bubble promptly and subsume the smaller bubbles of neighbouring particles across several rounds of bubble coalescence. Scale bars, 1 mm. c, d, Phase portraits of homogeneous (c) and heterogeneous (d) systems of N = 2, 3, 6, and 8. Only the latter is able to maintain the closed-loop orbits at high particle counts. e Schematic of recurrence time calculation. The recurrence time is the time it takes to return from a given system configuration to the neighbourhood of said configuration (see “Methods”). f Recurrence histogram compiling all of the recurrence times observed across experiments of the 2-particle heterogeneous system (N = 1 + 1DL). g Recurrence entropy as a function of N for both homogeneous (yellow) and heterogeneous/DL (blue) systems. Low recurrence entropy is a quantitative indicator of periodic behaviour. The homogeneous system's recurrence entropy trends upward, suggesting a decay in periodicity, while the DL system's entropy remains low in accordance with its observed periodicity even at high N.

Fig. 4. Self-organised oscillation powers a microrobotic arm.

a Schematics of the generation of an oscillatory electrical current from chemomechanical beating. The pair of metals (Pt–Ru or Pt–Au) patterned on a polymer base constitute the electrodes of a H₂O₂ fuel cell, which serves as an on-board voltage source. The periodic bubble growth and collapse in a beating system separately modulates the electrical resistance between the electrodes, leading to an oscillatory current. b Optical micrograph of a typical Pt-Ru fuel cell particle. The entire surface, less the electrode area, is passivated with a thin layer of insulating SU-8 polymer (shaded). The metallic leads on the left are not necessary for device operation and are included to facilitate measurement. Scale bar, 100 μm. c Short-circuit current density as a function of H₂O₂ concentration for a Pt-Ru device. d, e Cyclic motion of a microrobotic actuator driven by the oscillatory current. The schematics and micrographs in d show the extended and contracted states of the actuator respectively under the ON and OFF current conditions, as modulated by the bubble size. The current measurement over time and the actuator length change (e) closely match, confirming that the cyclic actuation is driven by the oscillatory current, which itself is emergent from the particle beating. Scale bar, 2 μm.