Relationship between the size of a camphor-driven rotor and its angular velocity

Mathematical modeling for the synchronization of two interacting active rotors

We investigate the synchronization of active rotors. A rotor is composed of a free-rotating arm with a particle that releases a surface-active chemical compound. It exhibits self-rotation due to the surface tension gradient originating from the concentration field of the surface-active compound released from the rotor. In a system with two active rotors, they should interact through the concentration field. Thus, the interaction between them does not depend only on the instantaneous positions, but also on the dynamics of the concentration field. By numerical simulations, we show that in-phase and antiphase synchronizations occur depending on the distance between the two rotors. The stability of the synchronization mode is analyzed based on phase reduction theorem through the calculation of the concentration field in the co-rotating frame with the active rotor. We also confirm that the numerical results meet the prediction by theoretical analyses.

New types of complex motion of a simple camphor boat

We discuss the motion of a rectangular camphor boat, considering the position of a camphor pill in relation to the boat's stern as the control parameter. The boat moves because the pill releases surface active molecules that decrease the surface tension and support the motion. We introduce a new experimental system in which the boat rotates on a long arm around the axis located at the centre of a Petri dish; thus, the motion is restricted to a circle and can be studied under stationary conditions for a long time. The experiments confirmed two previously reported modes of motion: continuous motion when the pill was located at the boat edge and pulsating (intermittent) motion if it was close to the boat centre (Suematsu et al., J. Phys. Chem. C, 2010, 114(21), 9876-9882). For intermediate pill locations, we observed a new, unreported type of motion characterised by oscillating speed (i.e. oscillating motion). Different modes of motion can be observed for the same pill location. The experimental results are qualitatively confirmed using a simple reaction-diffusion model of the boat evolution used in the above-mentioned paper.

High-performance Marangoni hydrogel rotors with asymmetric porosity and drag reduction profile

Miniaturized rotors based on Marangoni effect have attracted great attentions due to their promising applications in propulsion and power generation. Despite intensive studies, the development of Marangoni rotors with high rotation output and fuel economy remains challenging. To address this challenge, we introduce an asymmetric porosity strategy to fabricate Marangoni rotor composed of thermoresponsive hydrogel and low surface tension anesthetic metabolite. Combining enhanced Marangoni propulsion of asymmetric porosity with drag reduction of well-designed profile, our rotor precedes previous studies in rotation output (~15 times) and fuel economy (~34% higher). Utilizing thermoresponsive hydrogel, the rotor realizes rapid refueling within 33 s. Moreover, iron-powder dopant further imparts the rotors with individual-specific locomotion in group under magnetic stimuli. Significantly, diverse functionalities including kinetic energy transmission, mini-generator and environmental remediation are demonstrated, which open new perspectives for designing miniaturized rotating machineries and inspire researchers in robotics, energy, and environment.

Pairing-induced motion of source and inert particles driven by surface tension

We experimentally and theoretically investigate systems with a pair of source and inert particles that interact through a concentration field. The experimental system comprises a camphor disk as the source particle and a metal washer as the inert particle. Both are floated on an aqueous solution of glycerol at various concentrations, where the glycerol modifies the viscosity of the aqueous phase. The particles form a pair owing to the attractive lateral capillary force. As the camphor disk spreads surface-active molecules at the aqueous surface, the camphor disk and metal washer move together, driven by the surface tension gradient. The washer is situated in the front of the camphor disk, keeping the distance constant during their motion, which we call a pairing-induced motion. The pairing-induced motion exhibited a transition between circular and straight motions as the glycerol concentration in the aqueous phase changed. Numerical calculations using a model that considers forces caused by the surface tension gradient and lateral capillary interaction reproduced the observed transition in the pairing-induced motion. Moreover, this transition agrees with the result of the linear stability analysis on the reduced dynamical system obtained by the expansion with respect to the particle velocity. Our results reveal that the effect of the particle velocity cannot be overlooked to describe the interaction through the concentration field.

Revealing the deterministic components in active avalanche-like dynamics

Avalanche dynamics in an ensemble of self-propelled camphor boats are studied. The self-propelled agents are camphor infused circular paper disks moving on the surface of water. The ensemble exhibits bursts of activity in the autonomous state triggered by stochastic fluctuations. This type of dynamics has been previously reported in a slightly different system (J. Phys. Soc. Jpn., 2015, 84, 034802). Fourier analysis of the autonomous ensemble's average speed reveals a unimodal spectrum, indicating the presence of a preferred time scale in the dynamics. We therefor, entrain such an ensemble by external forcing by using periodic air perturbations on the surface of the water. This forcing is able to replace the stochastic fluctuations which trigger a burst in the autonomous ensemble, thus entraining the system. Upon varying the periodic forcing frequency, an optimal frequency is revealed at which the quality of entrainment of the ensemble by the forcing is augmented. This optimal frequency is found to be in the vicinity of the Fourier spectrum peak of the autonomous ensemble's average speed. This indicates the existence of an underlying deterministic component in the apparent aperiodic bursts of motion of the autonomous ensemble of active particles. A qualitative reasoning for the observed phenomenon is presented.

Surfactant-loaded capsules as Marangoni microswimmers at the air-water interface: Symmetry breaking and spontaneous propulsion by surfactant diffusion and advection

We present a realization of a fast interfacial Marangoni microswimmer by a half-spherical alginate capsule at the air-water interface, which diffusively releases water-soluble spreading molecules (weak surfactants such as polyethylene glycol (PEG)), which act as "fuel" by modulating the air-water interfacial tension. For a number of different fuels, we can observe symmetry breaking and spontaneous propulsion although the alginate particle and emission are isotropic. The propulsion mechanism is similar to soap or camphor boats, which are, however, typically asymmetric in shape or emission to select a swimming direction. We develop a theory of Marangoni boat propulsion starting from low Reynolds numbers by analyzing the coupled problems of surfactant diffusion and advection and fluid flow, which includes surfactant-induced fluid Marangoni flow, and surfactant adsorption at the air-water interface; we also include a possible evaporation of surfactant. The swimming velocity is determined by the balance of drag and Marangoni forces. We show that spontaneous symmetry breaking resulting in propulsion is possible above a critical dimensionless surfactant emission rate (Peclet number). We derive the relation between Peclet number and swimming speed and generalize to higher Reynolds numbers utilizing the concept of the Nusselt number. The theory explains the observed swimming speeds for PEG-alginate capsules, and we unravel the differences to other Marangoni boat systems based on camphor, which are mainly caused by surfactant evaporation from the liquid-air interface. The capsule Marangoni microswimmers also exhibit surfactant-mediated repulsive interactions with walls, which can be qualitatively explained by surfactant accumulation at the wall.

Imperfect bifurcation in the rotation of a propeller-shaped camphor rotor

We investigated the bifurcation structure on the self-propelled motion of a camphor rotor at a water surface. The center of the camphor rotor was fixed by the axis, and it showed rotational motion around it. Due to the chiral asymmetry of its shape, the absolute values of the angular velocities in clockwise and counterclockwise directions were different. This asymmetry in the angular velocities implies an imperfect bifurcation. From the numerical simulation results, we discuss the condition for the occurrence of the imperfect bifurcation.

Chimeralike states in a minimal network of active camphor ribbons

A chimeralike state is the spatiotemporal pattern in an ensemble of homogeneous coupled oscillators, described as the emergence of coexisting coherent (synchronized) and incoherent (unsynchronized) groups. We demonstrate the existence of these states in three active camphor ribbons, which are camphor infused rectangular pieces of paper. These pinned ribbons rotate on the surface of the water due to Marangoni effect driven forces generated by the surface tension gradients. The ribbons are coupled via a camphor layer on the surface of the water. In the minimal network of globally coupled camphor ribbons, chimeralike states are characterized by the coexistence of two synchronized and one unsynchronized ribbons. We present a numerical model, simulating the coupling between ribbons as repulsive Yukawa interactions, which was able to reproduce these experimentally observed states.

Periodic oscillations in a string of camphor infused disks

The rhythmic beating motion of autonomously motile filaments has many practical applications. Here, we present an experimental study on a filament made of camphor infused paper disks, stitched together adjacent to each other using nylon thread. The filament displays spontaneous translatory motion when it is placed on the surface of water due to the surface tension gradients created by camphor molecules on the water surface. When this filament is clamped on one end, we obtain regular oscillatory motion instead of translation. The filament shows qualitatively different dynamics at different activity levels, which is controlled by the amount of camphor infused into the paper disks. For a better physical understanding of the filament dynamics, we develop a minimal numerical model involving a semi-flexible filament made of active polar disks, where the polarity is coupled to the instantaneous velocity of the particle. This model qualitatively reproduces different oscillatory modes of the filament. Moreover, our model reveals a rich dynamical state diagram of the system, as a function of filament activity and the coupling strength.

First passage of an active particle in the presence of passive crowders

We experimentally study the stochastic transport of a self-propelled camphor boat, driven by Marangoni forces, through a crowd of passive paper discs floating on water. We analyze the statistics of the first passage times of the active particle to travel from the center of a circular container to its boundary. While the mean times rise monotonically as a function of the covered area fraction φ of the passive paper discs, their fluctuations show a non-monotonic behavior - being higher at low and high value of φ compared to intermediate values. The reason is traced to an interplay of two distinct sources of fluctuations - one intrinsic to the dynamics, while the other due to the crowding.

Rotational synchronization of camphor ribbons in different geometries

We present experiments on multiple pinned self-propelled camphor ribbons, which is a rectangular piece of paper with camphor infused in its matrix. Experiments were performed on three, four, and five ribbons placed in linear and polygonal geometries. The pinned ribbons rotate on the surface of water, due to the surface tension gradient introduced by the camphor layer in the neighborhood of the ribbon. This camphor layer leads to a chemical coupling between the ribbons. In different geometries, the ribbons have been observed to rotationally synchronize in all the possible configurations. A numerical model, emulating the interactions between the ribbons as Yukawa interaction was studied, which was qualitatively able to reproduce the experimental findings.

On a simple model that explains inversion of a self-propelled rotor under periodic stop-and-release-operations

We propose a simple mathematical model that describes the time evolution of a self-propelled object on a liquid surface using variables such as object location, surface concentration of active molecules, and hydrodynamic surface flow. The model is applied to simulate the time evolution of a rotor composed of a polygonal plate with camphor pills at its corners. We have qualitatively reproduced results of experiments, in which the inversion of rotational direction under periodic stop-and-release-operations was investigated. The model correctly describes the probability of the inversion as a function of the duration of the phase when the rotor is stopped. Moreover, the model allows to introduce the rotor asymmetry unavoidable in real experiments and study its influence on the studied phenomenon. Our numerical simulations have revealed that the probability of the inversion of rotational direction is determined by the competition among the transport of the camphor molecules by the flow, the intrinsic asymmetry of the rotor, and the noise amplitude.

A hybrid camphor-camphene wax material for studies on self-propelled motion

A new material that combines the self-propelling properties of camphor with the malleability of camphene is reported. It has wax-like mechanical properties at room temperature and can be formed into required shapes. The speed of the self-propelled objects and the trajectory depend on the shape and camphor-camphene weight ratio.

Rotational motion of a camphor disk in a circular region

In a two-dimensional axisymmetric system, the system symmetry allows rotational or oscillatory motion as stable stationary motion for a symmetric self-propelled particle. In the present paper, we studied the motion of a camphor disk confined in a two-dimensional circular region. By reducing the mathematical model describing the dynamics of the motion of a camphor disk and the concentration field of camphor molecules on a water surface, we analyzed the reduced equations around a bifurcation point where the rest state at the center of the system becomes unstable. As a result, we found that rotational motion is stably realized through the double-Hopf bifurcation from the rest state. The theoretical results were confirmed by numerical calculation and corresponded well to the experimental results.

Rotational synchronization of camphor ribbons

Experiments on interacting pinned self-propelled rotators are presented. The rotators are made from paper with camphor infused in its matrix. The ribbons rotate due to Marangoni effect driven forces arising by virtue of surface tension gradients. Two such self-rotating camphor ribbons are observed to experience a repulsive coupling via the camphor layer in the common water medium. Lag synchronization in both corotating (same sense of rotation) and counterrotating (opposite sense of rotation) ribbons is reported for the experiments. This synchronization is found to be dependent on the pivot to pivot distance l. For distances less than the span of both the ribbons, l_{c}, the rotators successfully synchronize. Furthermore, it is experimentally perceived that synchronization in the counterrotating ribbons is more robust than in the corotating ribbons. We rationalize the mechanism of this synchronization via a theoretical model involving a Yukawa type interaction which is analyzed numerically.

Bifurcation in the angular velocity of a circular disk propelled by symmetrically distributed camphor pills

We studied rotation of a disk propelled by a number of camphor pills symmetrically distributed at its edge. The disk was put on a water surface so that it could rotate around a vertical axis located at the disk center. In such a system, the driving torque originates from surface tension difference resulting from inhomogeneous surface concentration of camphor molecules released from the pills. Here, we investigated the dependence of the stationary angular velocity on the disk radius and on the number of pills. The work extends our previous study on a linear rotor propelled by two camphor pills [Y. Koyano et al., Phys. Rev. E 96, 012609 (2017)]. It was observed that the angular velocity dropped to zero after a critical number of pills was exceeded. Such behavior was confirmed by a numerical model of time evolution of the rotor. The model predicts that, for a fixed friction coefficient, the speed of pills can be accurately represented by a function of the linear number density of pills. We also present bifurcation analysis of the conditions at which the transition between a standing and a rotating disk appears.