Camphor Boats Research Articles

Quantitative Analysis of the Complex Time Evolution of a Camphor Boat

The motion of a camphor boat on the water’s surface is a long-studied example of the direct transformation of chemical energy into a mechanical one. Recent experimental papers have reported a complex character of boat motion depending on the location of the camphor source. If the source is close to the stern, the boat moves at a constant speed. When it is shifted towards the boat center, oscillations of speed are observed. When the source is close to the boat center, pulses of speed followed by oscillations appear. Here, we focus on numerical simulations of camphor boat motion. We discuss approximations that allow us to reduce the numerical complexity of the problem and formulate a model in which the equation for boat velocity is coupled with a one-dimensional reaction–diffusion equation for camphor surface concentration. We scanned the phase space of model parameters and found the values that give qualitative agreement with the experiments. The model predicts all types of boat motion (continuous, oscillating, and pulsating) observed in experiments. Moreover, the model with selected parameter values shows that for specific locations of the camphor source, a spike in speed is followed by transient oscillations, which are an inherent part of speed relaxation.

Time-Resolved Measurements of Interfacial Tension and Flow Speed of the Inclined Water Surface around a Self-propelled Camphor Boat by the Quasi-elastic Laser Scattering Method.

An inclined liquid surface, such as a meniscus, plays an important role in advection and transport phenomena at a liquid's surface. However, there is no time-resolved measurement method for the interfacial tension of an inclined liquid-air interface. Here, a noninvasive method for simultaneous measurements of the interfacial tension and surface flow speed for an inclined water surface is described. This is an upgrade of the quasi-elastic laser scattering method with a closed-loop control system that introduces the dynamically tracked scattered and referential light into the detector. For the evaluation of the tilt compensation by dynamic tracking, the relationship between the apparent interfacial tension and surface inclination was examined for a water meniscus at 0-5° inclinations. It was also demonstrated that simultaneous measurements of the interfacial tension and surface flow speed around a self-propelled camphor boat on a pure water surface inclined by >3° at the back end of the boat are difficult to conduct accurately without dynamic tracking. Both the interfacial tension difference and the backward flow speed increased as the boat speed increased to 0.1 m/s; that had not been evaluated to date because of the high velocity of the boat and the surface inclination of the water around it. The direct experimental evaluation of the interfacial tension and the flow speed supported the model that the driving force of the camphor boat is the interfacial tension difference and the resistance force proportional to the boat velocity reduces its acceleration.

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.

Center manifold theory for the 1-dimensional collective motions of camphor disks with delta functions in the <inline-formula><tex-math id="M1">$ L^2 $</tex-math></inline-formula>-framework

Camphor boat and disk are types of interfacial self-propellers that rely on the modification of forces present at the air-liquid interface. In a one-dimensional circuit, self-organized congestion and clustering phenomena are observed experimentally and numerically [11,43]. Although the center manifold theories proposed in [5,7] are useful in the analysis of collective motions, the requirement in the reduction process is not fulfilled because the linearized operator involves Dirac delta functions. To overcome such mathematical difficulties, a theory in the $ (H^1)^* $-framework was developed in [17]. However, the reduced equation does not include any nonlinearity. Thus, information on the collective motions could not be obtained. The objectives of the present study are to propose a new approach to establish the existence and uniqueness of a center manifold in a system with delta functions in the $ L^2 $-framework and to derive a reduced system with suitable nonlinearity from the original model in [33].

Surfactant-loaded capsules as Marangoni microswimmers at the air\u2013water 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.Graphic

From diffusive mass transfer in Stokes flow to low Reynolds number Marangoni boats

We present a theory for the self-propulsion of symmetric, half-spherical Marangoni boats (soap or camphor boats) at low Reynolds numbers. Propulsion is generated by release (diffusive emission or dissolution) of water-soluble surfactant molecules, which modulate the air–water interfacial tension. Propulsion either requires asymmetric release or spontaneous symmetry breaking by coupling to advection for a perfectly symmetrical swimmer. We study the diffusion–advection problem for a sphere in Stokes flow analytically and numerically both for constant concentration and constant flux boundary conditions. We derive novel results for concentration profiles under constant flux boundary conditions and for the Nusselt number (the dimensionless ratio of total emitted flux and diffusive flux). Based on these results, we analyze the Marangoni boat for small Marangoni propulsion (low Peclet number) and show that two swimming regimes exist, a diffusive regime at low velocities and an advection-dominated regime at high swimmer velocities. We describe both the limit of large Marangoni propulsion (high Peclet number) and the effects from evaporation by approximative analytical theories. The swimming velocity is determined by force balance, and we obtain a general expression for the Marangoni forces, which comprises both direct Marangoni forces from the surface tension gradient along the air–water–swimmer contact line and Marangoni flow forces. We unravel whether the Marangoni flow contribution is exerting a forward or backward force during propulsion. Our main result is the relation between Peclet number and swimming velocity. Spontaneous symmetry breaking and, thus, swimming occur for a perfectly symmetrical swimmer above a critical Peclet number, which becomes small for large system sizes. We find a supercritical swimming bifurcation for a symmetric swimmer and an avoided bifurcation in the presence of an asymmetry.Graphic abstract

Spiral Thermal Waves Generated by Self-Propelled Camphor Boats

Spiral thermal surface waves arising from self-propulsion of the camphor-driven objects are reported. Spiral thermal waves were registered for dissolution and evaporation-guided self-propulsion. Soluto-capillarity is accompanied by thermo-capillarity under self-propulsion of camphor boats. The jump in the surface tension due to the soluto-capillarity is much larger than that due to the thermo-capillarity. The spiral patterns inherent for the surface thermal waves are imposed by the self-rotational motion of camphor grains. The observed thermal effect is related to the adsorption of camphor molecules at the water/vapor interface. The observed spirals are shaped as Archimedean ones.

Steady state dynamic dependence between local mobility and non-affine fluctuations in two-dimensional aggregates

Motivated by qualitative experimental observations in collective behavior of self-propelled camphor particles at air-water interfaces, we study a generic aggregate forming system in two dimensions using canonical ensemble constant temperature molecular dynamics simulation. The aggregates form due to the competition between short-range attraction and long-range repulsion of pair-wise interactions as a generic proxy for the specific case of short-range capillary attraction competing with long-range Marangoni-assisted repulsion in camphor boat systems. Choosing the appropriate set of interaction parameters, we focus on characterising the local dynamics in two specific limiting morphologies, viz. compact and string-like aggregates. We focus on the temporal evolution of the mobility of an individual particle and the dynamic change in its nearest neighbourhood, measured in terms of the Debye–Waller factor () and the non-affine parameter (), respectively (both defined in the text), and their interrelation over several lengths of observation time . The distribution for both measures are found to follow the relation: for the measured quantity x. The exponent is equal to two and one respectively, for the compact and string-like morphologies following the respective ideal fractal dimension of these aggregates. A functional dependence between these two observables is determined from a detailed statistical analysis of their joint and conditional distributions. The results obtained can readily be used and verified by experiments on aggregate forming systems more generic than the specific camphor boat system that motivated us, such as globular proteins, nanoparticle self-assembly etc. Further, the insights gained from this study might be useful to understand the evolution of collective dynamics in diverse glass-forming systems.

Center Manifold Theory for the Motions of Camphor Boats with Delta Function

Various collective motions of camphor boats have been studied. Camphor boats are self-driven particles that interact with each other through the change of surface tension on water surface by camphor molecules. Consequently, even in a one-dimensional circuit, a congested state or jamming can be observed (Suematsu et al. in Phys Rev E 81:056210, 2010). In this phenomenon, the unidirectional motion of each particle is considered a traveling wave, and the concentration of camphor molecules forms a pulse shape. Hence, each pair of particles interacts with each other like a pulse–pulse interaction. Thus, we expect that the center manifold theories proposed in Ei (J Dyn Differ Equ 14:85–137, 2002) and Ei et al. (Physica D 165:176–198 2002) are applicable for the analysis of the collective motion of camphor boats. However, spatial discontinuity in our model, in particular the existence of Dirac delta functions in a linearized operator, does not fulfill the requirement in the reduction process because the authors developed their theories in $$L^2$$ -framework and for smooth nonlinearity in Ei (2002) and Ei et al. (2002). In this article, we extend the results obtained in Ei (2002) and Ei et al. (2002) and establish a new center manifold approach in $$(H^1)^*$$ -framework. Finally, we succeed to rigorously reduce a mathematical model (Nagayama et al. in Physica D: Nonlinear Phenom 194:151–165, 2004) coupled with a Newtonian equation and a reaction–diffusion equation including delta functions to an ordinary differential system that represents the motions of camphor boats.

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.

Reduced model of a reaction-diffusion system for the collective motion of camphor boats.

The unidirectional motion of a camphor boat along an annular water channel is observable. When camphor boats are placed in a water channel, both homogeneous and inhomogeneous states occur as collective motions, depending on the number of boats. The inhomogeneous state is a type of congestion, that is, the velocities of the boats change with temporal oscillation, and the shock wave appears along the line of travel of the boats. The unidirectional motion of a single camphor boat and the homogeneous state can be represented by traveling wave solutions in a mathematical model. Because the experimental results described here are thought of as a type of bifurcation phenomenon, the destabilization of traveling wave solutions may indicate the emergence of congestion. We previously attempted to study a linearized eigenvalue problem associated with a traveling wave solution. However, the problem is too difficult to analyze rigorously, even for just two camphor boats. Therefore we developed a center manifold theory and derived a reduced model in our previous work. In the present paper, we study the reduced model and show that the original model and our reduced model qualitatively exhibit the same properties by applying numerical techniques. Moreover, we demonstrate that the numerical results obtained in our models for camphor boats are quite similar to those in a car-following model, the OV model, but there are some different features between our reduced model and a typical OV model.

Construction of a chemical motor-movable frame assembly based on camphor grains using water-floating 3D-printed models

Using water-floating 3D-printed models, we investigated the dynamics of camphor boats, which are composed of a one-holed board and a camphor grain, under constraint conditions where a rod hooks the camphor boat on the water surface. When a camphor boat was connected to a water-floating disk or another camphor boat through a rod, we found that regular dynamics of the camphor boat appear and depend on the mobility restriction of the disk, the rod length, and the position of a slit of the camphor boat. We performed numerical simulations to reproduce the observed motions with theoretical models including constraint conditions.

Synthetic Chemotaxis and Collective Behavior in Active Matter.

The ability to navigate in chemical gradients, called chemotaxis, is crucial for the survival of microorganisms. It allows them to find food and to escape from toxins. Many microorganisms can produce the chemicals to which they respond themselves and use chemotaxis for signaling, which can be seen as a basic form of communication, allowing ensembles of microorganisms to coordinate their behavior, for example, during embryogenesis, biofilm formation, or cellular aggregation. For example, Dictyostelium cells use signaling as a survival strategy: when starving, they produce certain chemicals toward which other cells show taxis. This leads to aggregation of the cells resulting in a multicellular aggregate that can sustain long starvation periods. Remarkably, the past decade has led to the development of synthetic microswimmers, which can self-propel through a solvent, analogously to bacteria and other microorganisms. The mechanism underlying the self-propulsion of synthetic microswimmers like camphor boats, droplet swimmers, and in particular autophoretic Janus colloids involves the production of certain chemicals. As we will discuss in this Account, the same chemicals (phoretic fields) involved in the self-propulsion of a (Janus) microswimmer also act on other ones and bias their swimming direction toward (or away from) the producing microswimmer. Synthetic microswimmers therefore provide a synthetic analogue to motile microorganisms interacting by taxis toward (or away from) self-produced chemical fields. In this Account, we review recent progress in the theoretical description of synthetic chemotaxis mainly based on simulations and field theoretical descriptions. We will begin with single motile particles leaving chemical trails behind with which they interact themselves, leading to effects like self-trapping or self-avoidance. Besides these self-interactions, in ensembles of synthetic motile particles each particle also responds to the chemicals produced by other particles, inducing chemical (or phoretic) cross-interactions. When these interactions are attractive, they commonly lead to clusters, even at low particle density. These clusters may either proceed toward macrophase separation, resembling Dictyostelium aggregation, or, as shown very recently, lead to dynamic clusters of self-limited size (dynamic clustering) as seen in experiments in autophoretic Janus colloids. Besides the classical case where chemical interactions are attractive, this Account discusses, as its main focus, repulsive chemical interactions, which can create a new and less known avenue to pattern formation in active systems leading to a variety of pattern, including clusters which are surrounded by shells of chemicals, traveling waves and more complex continuously reshaping patterns. In all these cases "synthetic signalling" can crucially determine the collective behavior of synthetic microswimmer ensembles and can be used as a design principle to create patterns in motile active particles.

Power law observed in the motion of an asymmetric camphor boat under viscous conditions.

We investigated the velocity of an asymmetric camphor boat moving on aqueous solutions with glycerol. The viscosity was controlled by using several concentrations of glycerol into the solution. The velocity decreased with an increase in the glycerol concentration. We proposed a phenomenological model, and we showed that the velocity decreased with an increase in the viscosity according to power law. Our experimental result agreed with the one obtained from our model. These results suggest that a decay length of the camphor concentration at the front side of the boat is sufficiently shorter than that of the rear side.

Period of Oscillatory Motion of a Camphor Boat Determined by the Dissolution and Diffusion of Camphor Molecules.

Here, we investigated the oscillatory motion of a camphor boat on water to clarify how the dynamics of camphor concentration profile determines the period of oscillation. The boat, which was made of a plastic plate and a camphor disk, was glued below the plate at a distance from the edge. The dependence of oscillation period on temperature and viscosity of the water phase was measured in experiments. We reproduced the experimental results by calculating the period of oscillatory motion by considering the experimental values of physicochemical parameters describing the time evolution of camphor concentration profile and the friction acting on a boat, such as diffusion and dissolution rates of camphor, viscosity of the water phase, and the threshold concentration of camphor necessary to accelerate the boat from the resting state. The increase in the period of oscillatory motion at low temperatures was explained by the reduced dissolution rate of camphor into the water phase.

Motion modes of two self-propelled camphor boats on the surface of a surfactant-containing solution

HypothesisThe existence of a surface tension difference and a surface flow around self-propelled objects, such as camphor boats, has been confirmed by many studies. However, the interactions in the collective motion of several camphor boats have not been explicitly discussed. Here, a model system of two camphor boats was investigated for the first time from the viewpoint of surface tension and surface flow. ExperimentsThe behavior of one fixed boat and one movable boat on a sodium dodecylsulfate aqueous solution in an oval track has been studied as a model system of two-body interactions. The surface tension around the movable boat was measured using the non-invasive, double-beam quasi-elastic laser scattering method. The Marangoni surface flow was quantitatively estimated using a mock boat in order to evaluate the net repulsive interaction between the two boats. FindingsWe succeeded in controlling the motion of the two camphor boats by varying the boat characteristics and the surfactant concentration. The balance/unbalance between the surface tension distribution and the surface flow around the boats was found to drive two different behavior modes: stationary and oscillatory. These results allow us to gain further insight into the dynamics of the interactions in the collective behavior of autonomous inanimate objects.

Acceleration or deceleration of self-motion by the Marangoni effect

We investigated the water-depth dependence of the self-motion of a camphor disk and camphor boat. With increasing water depth, the speed of motion of the camphor disk increased, but that of the camphor boat decreased in an annular one-dimensional system. We discussed the difference in the water-depth dependence of the speed of the camphor objects in relation to Marangoni flow. We concluded that Marangoni flow, which became stronger with increasing the water depth, positively and negatively affected the speed of the disk and boat, respectively.

Characteristic oscillatory motion of a camphor boat sensitive to physicochemical environment.

A self-propelled camphor boat on water was investigated from the viewpoint of characteristic features of motion and mode-bifurcation depending on the diffusion length of camphor molecules. When a camphor disk was connected to the bottom of a larger plastic plate and then was placed on water, either oscillatory motion (repetition between rest and motion) or continuous motion was observed. In this paper, we report the novel features of this motion and mode-bifurcation as a function of the diffusion length of camphor molecules, e.g., multiple accelerations during oscillation, period-2 or irregular oscillatory motion, and reciprocating oscillation. These characteristic motion and mode-bifurcation are discussed in relation to the diffusion length of camphor molecules under the camphor boat and the development of camphor molecules from the camphor boat on water.

Reduced model from a reaction-diffusion system of collective motion of camphor boats

Various motions of camphor boats in the water channel exhibit both a homogeneous and an inhomogeneous state, depending on the number of boats, when unidirectional motion along an annular water channel can be observed even with only one camphor boat. In a theoretical research, the unidirectional motion is represented by a traveling wave solution in a model. Since the experimental results described above are thought of as a kind of bifurcation phenomena, we would like to investigate a linearized eigenvalue problem in order to prove the destabilization of a traveling wave solution. However, the eigenvalue problem is too difficult to analyze even if the number of camphor boats is 2. Hence we need to make a reduction on the model. In the present paper, we apply the center manifold theory and reduce the model to an ordinary differential system.

Quasi-elastic laser scattering for measuring inhomogeneous interfacial tension in non-equilibrium phenomena with convective flows.

An inhomogeneous distribution of interfacial tension can induce different types of non-equilibrium spontaneous motion at the interface by convective flow, or by the solutal Marangoni effect. Several applications of the quasi-elastic laser scattering (QELS) method used to study these effects are presented here. The relationship between the interfacial tension and the non-equilibrium phenomena has been verified experimentally for each application. In a water/nitrobenzene oscillatory system with continuous surfactant addition to the interface, the local adsorption of surfactants at the interface has been demonstrated and shown to be strongly related to the presence of electrolytes. In a donor/membrane/acceptor system, the dual-beam QELS method shows that surfactant adsorption at the membrane/acceptor interface is responsible for oscillations in the electric potential. The differences in the adsorption/desorption behavior of metal complex catalysts between air/liquid and liquid/liquid interfaces were considered in the propagating chemical waves of the Belousov-Zhabotinsky reaction. We successfully measured the distribution of interfacial tension around a self-propelled camphor boat and an alcohol droplet floating on an aqueous phase, and compared the mechanisms of their motion.