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.

Complexity and bifurcations in the motion of a self-propelled rectangle confined in a circular water chamber.

We consider the motion of a self-propelled object of rectangular shape inside a circular water chamber. The mathematical model of self-motion includes equations for the orientation and location of the rectangle and reaction-diffusion equation with an effective diffusion coefficient for the time evolution of the surface concentration of active molecules. Numerical simulations of motion were performed for different values of the ratio between the supply rate S and the evaporation rate a of active molecules. Treating S0 = S/a as a control parameter, we found the critical behavior in variables characterizing the trajectory and identified different types of motion. If the value of S0 is small, the rectangle rests at the chamber center. For larger S0, a reciprocal motion during which the rectangle passes through the center is observed. At yet higher supply rates, the star-polygonal motion appears, and the trajectory remains at a distance from the chamber center. In the experiments with a rectangle made of camphor-camphene-polypropylene plastic moving in a Petri dish, we observed the transition from the star-polygonal motion to the reciprocal motion in time. This transition can be understood on the basis of the developed model if we assume that the supply rate decreases in time.

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.

Start of Micrometer-Sized Oil Droplet Motion through Generation of Surfactants.

Self-propelled motion of micrometer-sized oil droplets in surfactant solution has drawn much attention as an example of nonlinear life-like dynamics under far-from-equilibrium conditions. The driving force of this motion is thought to be induced by Marangoni convection based on heterogeneity in the interfacial tension at the droplet surface. Here, to clarify the required conditions for the self-propelled motion of oil droplets, we have constructed a chemical system, where oil droplet motion is induced by the production of 1,2,3-triazole-containing surfactants through the Cu-catalyzed azide-alkyne cycloaddition reaction. From the results of the visualization and analysis of flow fields around the droplet, the motion of the droplets could be attributed to the formation of flow fields, which achieved sufficient strength caused by the in situ production of surfactants at the droplet surface.

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.

Unidirectional motion of a camphor disk on water forced by interactions between surface camphor concentration and dynamically changing boundaries.

We study the motion of a camphor disk on the water surface in a system with flexible boundaries. The boundaries can be dynamically modified by non-uniform surface tension resulting from the nonhomogeneous surface concentration of the camphor molecules dissipated by the disk. We investigate the geometry of the boundaries that forces unidirectional motion of the disk. The studied system can be regarded as a signal diode if the presence or absence of a camphor disk at a specific point is interpreted as the binary TRUE and FALSE variables. The diode can be incorporated into more complex devices, like a ring that imposes unidirectional rotation of camphor disks.

General criteria for determining rotation or oscillation in a two-dimensional axisymmetric system.

A self-propelled particle in a two-dimensional axisymmetric system, such as a particle in a central force field or confined in a circular region, may show rotational or oscillatory motion. These motions do not require asymmetry of the particle or the boundary, but arise through spontaneous symmetry breaking. We propose a generic model for a self-propelled particle in a two-dimensional axisymmetric system. A weakly nonlinear analysis establishes criteria for determining rotational or oscillatory motion.

Activity-induced diffusion recovery in crowded colloidal suspensions.

We show that the forces generated by active enzyme molecules are strong enough to influence the dynamics of their surroundings under artificial crowded environments. We measured the behavior of polymer microparticles in a quasi-two-dimensional system under aqueous environment, at various area fraction values of particles. In the presence of enzymatic activity, not only was the diffusion of the suspended particles enhanced at shorter time-scales, but the system also showed a transition from subdiffusive to diffusive dynamics at longer time-scale limits. Similar observations were also recorded with enzyme-functionalized microparticles. Brownian dynamics simulations have been performed to support the experimental observations.

Erratum: Pairing-induced motion of source and inert particles driven by surface tension [Phys. Rev. E 106, 024604 (2022)].

This corrects the article DOI: 10.1103/PhysRevE.106.024604.

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.

Anomalous diffusion and transport by a reciprocal convective flow.

Under low-Reynolds-number conditions, dynamics of convection and diffusion are usually considered separately because their dominant spatial and temporal scales are different, but cooperative effects of convection and diffusion can cause diffusion enhancement [Koyano et al., Phys. Rev. E 102, 033109 (2020)2470-004510.1103/PhysRevE.102.033109]. In this paper, such cooperative effects are investigated in detail. Numerical simulations based on the convection-diffusion equation revealed that anisotropic diffusion and net shift as well as diffusion enhancement occur under a reciprocal flow. Such anomalous diffusion and transport are theoretically derived by the analyses of the Langevin dynamics.

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.

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.

Diffusion enhancement in a levitated droplet via oscillatory deformation.

Recent experimental results indicate that mixing is enhanced by a reciprocal flow induced inside a levitated droplet with an oscillatory deformation [T. Watanabe et al., Sci. Rep. 8, 10221 (2018)2045-232210.1038/s41598-018-28451-5]. Generally, reciprocal flow cannot convect the solutes in time average, and agitation cannot take place. In the present paper, we focus on the diffusion process coupled with the reciprocal flow. We theoretically derive that the diffusion process can be enhanced by the reciprocal flow, and the results are confirmed via numerical calculation of the over-damped Langevin equation with a reciprocal flow.

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.

Publisher's Note: Oscillatory motion of a camphor grain in a one-dimensional finite region [Phys. Rev. E 94, 042215 (2016)].

This corrects the article DOI: 10.1103/PhysRevE.94.042215.

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

We consider a rotor made of two camphor disks glued below the ends of a plastic stripe. The disks are floating on a water surface and the plastic stripe does not touch the surface. The system can rotate around a vertical axis located at the center of the stripe. The disks dissipate camphor molecules. The driving momentum comes from the nonuniformity of surface tension resulting from inhomogeneous surface concentration of camphor molecules around the disks. We investigate the stationary angular velocity as a function of rotor radius ℓ. For large ℓ the angular velocity decreases for increasing ℓ. At a specific value of ℓ the angular velocity reaches its maximum and, for short ℓ it rapidly decreases. Such behavior is confirmed by a simple numerical model. The model also predicts that there is a critical rotor size below which it does not rotate. Within the introduced model we analyze the type of this bifurcation.

Oscillatory motion of a camphor grain in a one-dimensional finite region.

The motion of a self-propelled particle is affected by its surroundings, such as boundaries or external fields. In this paper, we investigated the bifurcation of the motion of a camphor grain, as a simple actual self-propelled system, confined in a one-dimensional finite region. A camphor grain exhibits oscillatory motion or remains at rest around the center position in a one-dimensional finite water channel, depending on the length of the water channel and the resistance coefficient. A mathematical model including the boundary effect is analytically reduced to an ordinary differential equation. Linear stability analysis reveals that the Hopf bifurcation occurs, reflecting the symmetry of the system.

Hydrodynamic collective effects of active proteins in biological membranes.

Lipid bilayers forming biological membranes are known to behave as viscous two-dimensional fluids on submicrometer scales; usually they contain a large number of active protein inclusions. Recently, it was shown [A. S. Mikhailov and R. Kapral, Proc. Natl. Acad. Sci. USA 112, E3639 (2015)PNASA60027-842410.1073/pnas.1506825112] that such active proteins should induce nonthermal fluctuating lipid flows leading to diffusion enhancement and chemotaxislike drift for passive inclusions in biomembranes. Here, a detailed analytical and numerical investigation of such effects is performed. The attention is focused on the situations when proteins are concentrated within lipid rafts. We demonstrate that passive particles tend to become attracted by active rafts and are accumulated inside them.

Deformable Self-Propelled Micro-Object Comprising Underwater Oil Droplets.

The self-propelled motion with deformation of micrometer-sized soft matter in water has potential application not only for underwater carriers or probes in very narrow spaces but also for understanding cell locomotion in terms of non-equilibrium physics. As far as we know, there have been no reports about micrometer-sized self-propelled soft matter mimicking amoeboid motion underwater. Here, we report an artificial molecular system of underwater oil droplets exhibiting self-propelled motion with deformation as an initial experimental model. We describe the heterogeneity in a deformable self-propelled oil droplet system in aqueous and oil phases and at their interface based on the behavior and interaction of surfactant and oil molecules. The current results have great importance for scientific frontiers such as developing deformable micro-swimmers and exploring the emergence of self-locomotion of oil droplet-type protocells.