Synchronized Sailing of Two Camphor Boats in Polygonal Chambers

Satoshi Nakata,*,† Yukie Doi,† and Hiroyuki Kitahata ‡

Department of Chemistry, Nara UniVersity of Education, Takabatake-cho, Nara 630-8528, Japan, andDepartment of Physics, Graduate School of Science, Kyoto UniVersity, Kyoto 606-8502, Japan

ReceiVed: May 6, 2004; In Final Form: October 10, 2004

The synchronized self-motion of two camphor boats on polygonal water chambers was investigated. The twoboats synchronously moved depending on the number of corners in the polygon by changing the distancebetween the two boats through the corners. We regard the self-motion of a camphor boat as an oscillator; i.e.,one cycle on the polygonal chamber corresponds to 2π. Phase-locked synchronization at a phase differenceof 2π/3, which corresponds to the length of one side of the chamber, was observed with a triangular chamber.Two types of synchronized motion at phase differences ofπ/2 andπ, which correspond to the length of oneand two sides of the chamber, respectively, were observed with a square chamber. These characteristic featuresof synchronized self-motion were qualitatively reproduced by a numerical calculation that regarded the surfacetension as the driving force and the number of corners in the chamber as a velocity-regulating mechanism.We believe that the present system may be a simple model of synchronization which depends on the geometryof the system.

1. Introduction

When two or more nonlinear oscillators are coupled together,the phases can be locked at a constant value. Such a phenom-enon is called “synchronization”, which is one of the mostinteresting phenomena observed in nonlinear systems.1,2 Thesynchronization in living organisms can be widely observed,for example, the beating heart, circadian rhythm, the organizedrhythm in the flashing of swarms of fireflies, and so on.3 Therehave also been many experimental and theoretical studies onthe synchronization of physicochemical coupled oscillators. Forexample, the synchronized chemical wave among small-beads4

or stirred containers5 was reported by using the Belousov-Zhabotinsky reaction. Hudson et al. reported the synchronizationof the electrochemical oscillations on electrode arrays.6,7 Thecoupling of saltwater oscillators, which was constructed by twoor three cups with saltwater and a larger vessel with pure waterconnected through a small orifice on the bottom of the individualcup, exhibited various natures of synchronization.8-11

To describe the nature of a single oscillator, the phasedescription is generally available as

whereθ is the phase of oscillator. If the oscillator is symmetricon the phase,f(θ) is constant; i.e., the angular velocity isconstant. Otherwise, the angular velocity of oscillator is afunction of the phase.

On the other hand, autonomous motors have been studied tocreate artificial chemomechanical transducers that mimic bio-logical and molecular motors.12 For example, the self-motionof a liquid droplet on a solid surface or a liquid surface isinduced by differences in surface tension,13-27 and a gel

autonomously changes its shape according to the surroundingconditions.28-30 With regard to the interaction between twoautonomous motors, they can synchronize with each otherbecause of potent nonlinearity.

Recently, we used a camphor boat at an air/water interfaceas a simple model of an autonomous motor. The camphor boatis driven by the heterogeneous distribution of the molecular layerof camphor developed from the solid fragment.31-39 Weinvestigated systems with camphor that show various types ofself-motion, e.g., unidirectional motion,32 characteristic motiondepending on the shape of the water chamber,33 mode-switchingbetween different types of motion,34 and the characteristicmotion of camphor derivatives.35,36

We also previously discussed the synchronized sailing of twoor three camphor boats on a circular chamber.37-39 Here, weregarded the self-motion of a camphor boat as an oscillator,i.e., one cycle on the circular chamber corresponds to 2π, anddiscussed the nature of self-motions of two boats on the samechamber as the coupled oscillators. In this system, the phasedifference between two boats was locked or oscillated dependingon the temperature of aqueous phase and the inherent velocityof the boats. It is noted that there is no specific position on thecircular chamber to artificially change the velocity of thecamphor boat because of its completely symmetric shape, i.e.,f(θ) is constant in eq 1.

In the present study, we investigated that two camphor boatssynchronously moved while regulating the individual velocitiesat the corners of polygonal water chambers. If the camphor boatis decelerated at the corner,f(θ) in eq 1 depends on the phase,i.e., it is possible to create various nature of synchronizationbetween two oscillators regulated depending on the phase. Thetwo camphor boats exhibited synchronized sailing that dependedon the shape of the water chamber. These characteristic featuresof synchronized self-motion were qualitatively reproduced bya numerical calculation based on a Newtonian equation and thenumber of corners of the chamber. This system may be a simple

* To whom correspondence should be addressed. Tel and fax:+81-742-27-9191. E-mail: nakatas@nara-edu.ac.jp.

† Nara University of Education.‡ Kyoto University.

dθdt

) f(θ) (1)

1798 J. Phys. Chem. B2005,109,1798-1802

10.1021/jp0480605 CCC: $30.25 © 2005 American Chemical SocietyPublished on Web 01/15/2005

model of synchronization between two oscillators regulateddepending on the phase.

2. Experimental Section

Camphor was obtained from Wako Chemicals (Kyoto, Japan).Water used in the chamber was first distilled and then purifiedwith a Millipore Milli-Q filtering system. To obtain similarcamphor boats, a camphor disk (diameter, 3 mm; thickness, 1mm) was prepared with a pellet die set for FTIR, and a boatwas drawn using computer software and printed on a polyestersheet (thickness: 0.1 mm) with a laser printer. The camphordisk was stuck to the stern of the boat with an adhesive.38 Thetwo camphor boats were floated in the same direction on thewater surface in a polygonal chamber made of Teflon (widthof the route: 5 mm, thickness: 2 mm, length of a side: 50 mm(triangular), 40 mm (square)). The temperature of the waterchamber was adjusted to 293( 1 K with a thermoplate (TP-80, AS ONE Co. Ltd., Japan). The movement of the camphordisk was monitored with a digital video camera (SONY DCR-VX700, minimum time-resolution: 1/30 s) and then analyzedby an image-processing system (Himawari, Library Inc., Japan).

To quantitatively evaluate the features of the synchronization,we used the phase description. Figure 1 shows a schematicrepresentation of the (a) triangular and (b) square chambers toanalyze synchronized motion in terms of a phase description;i.e., the phase of the camphor boat in the chamber,θ, is definedas 2πL/nlc (n, number of corners in the polygon;L, distancealong the water chamber from a reference point in the chamberin a clockwise direction as denoted in Figure 1a;lc, the lengthof one side of the polygonal chamber).

3. Results

Figure 2 shows snapshots of the synchronized self-motionbetween two equivalent camphor boats in (a) a triangularchamber and (b) a square chamber. The individual boatsdecelerated when they passed through a corner and acceleratedafterward.

For the triangular chamber, the two camphor boats movedwhile maintaining the distance between them at∆θ ) 2π/3 (L) lc) for ca. 20 cycles (∆θ: the difference in phase correspond-ing to the distance between two camphor boats). This synchro-nized motion was independent of the initial locations of the twocamphor boats.

For the square chamber, the two camphor boats moved whilemaintaining∆θ ∼ π (L ∼ 2lc) (mode I) for ca. 15 cycles (Figure1b-I) when∆θ ) 3π/4 to π (L ) 1.5lc to 2lc) was the initiallocation andVR was nearly equal to but slightly higher thanVâ(V: inherent velocity of a single camphor boat). When∆θ )π/4 to π/2 (L ) 0.5lc to lc) was the initial condition andVR wasnearly equal to or slightly lower thanVâ, they moved while

maintaining∆θ ∼ π/2 (L ∼ lc) (mode II) for ca. 20 cycles(Figure 1b-II). As for∆θ ) π/2 to 3π/4 (L ) lc to 1.5lc) as theinitial condition, either mode I or II is selected randomly dueto the experimental fluctuation. The experiments for∆θ )0 toπ/4 (L ) 0 to 0.5lc) as the initial condition could not beperformed because of the finite size of a camphor boat. Theduration time of synchronization was limited because theimpurity in the camphor disk and the remained camphor layerlowered the surface tension with time.

Figure 3 shows (1) aθR versusθâ curve and (2) aθR versus∆θ () θR - θâ) curve for the (a) triangular and (b) squarechambers. When the preceding boatR passed a corner in thetriangular chamber, the following boatâ approached boatR,and then∆θ became 2π/3, as seen in Figure 3a. BoatR then

Figure 1. Schematic representation of the (a) triangular and (b) squarechambers to analyze synchronized motion in the polar coordinates alongthe polygonal route. The phase of the camphor boat in the chamber,θ,is defined as 2πL/nlc.

Figure 2. Snapshots of the self-motion of two camphor boats (R, â)in (a) triangular and (b) square chambers at (a)t ) 36-41 s, (b-I)t )9-13 s, and (b-II)t ) 58-62 s after the boats were floated on thewater surface with a time interval of (a) 5/3 and (b) 4/3 s (top view).

Figure 3. Experimental results regarding the (1)θR vs θâ curve and(2) θR vs ∆θ () θR - θâ) curve for the (a) triangular and (b) squarechambers. The data in (a), I in (b), and II in (b), correspond to thosein Figure 1a, b-I, and b-II, respectively. Dotted lines denote theindividual corners of the polygonal chamber, and their thicknessescorrespond to the length of the corners.

Synchronized Sailing of Camphor Boats J. Phys. Chem. B, Vol. 109, No. 5, 20051799

accelerated and backed away from boatâ, i.e.,∆θ > 2π/3 whenthe boatâ passed a corner.

In the square chamber, modes I and II corresponded to phasedifferences ofπ andπ/2, respectively. When boatR passed acorner in the square chamber, boatâ approached boatR at ∆θ) π andπ/2, respectively, in modes I and II, as seen in Figure3b. BoatR then accelerated and backed away from boatâ, i.e.,∆θ > π in mode I and∆θ > π/2 in mode II, when boatâpassed a corner. The amplitude of∆θ in mode I was lowerthan that in mode II.

When the inherent velocity of one boat (â) was somewhatfaster than that of the other boat (R), mode switching wasobserved in the square chamber. Figure 4 shows the timevariation of (a)θR and θâ, and (b)∆θ upon switching frommode I as the initial condition to mode II.∆θ gradually deviatedfrom π with time, mode I switched to mode II for ca. 10 s (ortwo cycles), and∆θ approachedπ/2 for a few seconds. Theamplitude of∆θ was enhanced in the vicinity of the switching.The system maintained mode II for ca. 20 cycles, and finally itreached coordinated motion, i.e., the two boats moved togetherat ∆θ ∼ 0 (L ∼ 0).

4. Discussion

Figures 2 and 3 suggest that the distance between the twoboats is regulated as an integral multiple oflc at the corner evenif either VR is different fromVâ, or boatR moves away fromboatâ on the linear route. The regulation at the corner is dueto the velocity and distance between the two boats. Thus, acamphor layer accumulates around the corner because thepreceding camphor boat has physically settled there, and themotion of the following boat is decelerated by this layer.

To theoretically understand the mechanism of the synchro-nized sailing of two autonomous motors, we introduce amathematical model for a camphor boat in a closed route andsolve it numerically. Based on the experimental results andprevious papers,37-39 the motion of a camphor boat in apolygonal cell may be considered as a one-dimensional motion.Therefore, we consider a one-dimensional model with a periodicboundary condition. As in the experimental results, the phaseof the camphor boat on a closed route,θ, can be written as

We describe the motion of a single camphor boat by theNewtonian motion equation as

wherem is the mass of the boat,k is a constant for viscousresistance, andf(θ) is the driving force due to the difference inthe surface tension between the bow and stern of the boataccording to the camphor layer that develops from the camphorgrain and is approximately expressed as

Equation 4 suggests that the maximum driving force,f0 + f1,is achieved when the camphor boat moves on a linear chamberwith an infinite length. The sinusoidal part (or the second termon the right side of eq 4) is the physical effect of the polygonalcell, i.e., the velocity of the camphor boat is reduced as it passesaround a corner. Figure 5 shows (a) experimental and (b)numerical results on the velocity of the single camphor boatfor (1) triangular and (2) square chambers. The time variationsof the velocity for the experimental results were similar to those

Figure 4. Experimental results regarding the time variation of (a)θRandθâ, and (b)∆θ upon switching from mode I as the initial conditionto mode II for the square chamber.

θ ) 2πnlc

L (2)

Figure 5. Time variation of the inherent velocity of a single boat on(1) triangular and (2) square chamber in the (a) the experiments and(b) numerical simulation. Parameters used in (b) arem ) 0.01, k )10.0, f0 ) 1.0, andf1 ) 0.9.

md2θdt2

) -kdθdt

+ f(θ) (3)

f(θ) ) f0 + f1 cosnθ, f0 > f1 (4)

1800 J. Phys. Chem. B, Vol. 109, No. 5, 2005 Nakata et al.

for the numerical ones, and therefore we adapted eq 4 as asimple approximation to reproduce the nature of self-motion.

As for the motion of the camphor boatsR andâ, we introducean interaction term as a function,g, as follows

where

g(∆θ,∆ω) is expressed by

wherea and b are positive constants,∆θ is the difference inphase, and∆ω is the difference in angular velocity. Equation 8represents the repulsive force from the preceding boat. Thecamphor layer develops from preceding boat and decreases thedriving force of the following boat. As the two boats are nearer,the repulsive force becomes stronger. Here, we introduce thetermb∆ω to consider the effect of the relevant velocity betweentwo boats, i.e., the repulsive force for the following boatbecomes stronger when the following boat approaches to thepreceding one (∆ω < 0), and vice versa. This effect may bereasonable if the camphor layer on the water surface hasviscoelasticity on the surface pressure.

Using eqs 5-8, numerical calculations were performed.Figure 6 shows the numerical results for the (a) triangular and(b) square chambers. For the triangular chamber, the twocamphor boats exhibit phase-locking synchronization at around2π/3. This synchronized motion is exhibited even if theindividual inherent velocities are slightly different. When the

preceding boat approaches to the corners,∆θ decreases to 2π/3. In contrast,∆θ increases after the two boats pass throughthe corners. For the square chamber, either mode I or mode IIis selected depending on the initial phases of the two boats whenthe inherent velocity of one boat is similar to that of the other.When the inherent velocity of one boat is slightly different fromthat of the other, mode II is more stable than mode I. When theinitial state is mode I, it switches suddenly to mode II, as shownin Figure 7. Thus, these numerical results in Figures 6 and 7qualitatively correspond to the experimental results shown inFigures 3 and 4.

When the single camphor boat was floated on the triangularand square chambers in the experiment, the average velocities(and angular velocities) of the single camphor boat were 55mm s-1 (2.3 rad s-1) and 50 mm s-1 (2.0 rad s-1), respectively.On the other hand, when the two camphor boats were floatedon these chambers, the average velocities (and angular veloci-ties) of two camphor boats were decreased to 33 mm s-1 (1.4rad s-1) for the triangular chamber, 32 mm s-1 (1.3 rad s-1) inmode I for the square chamber, and 35 mm s-1 (1.4 rad s-1) inmode II for the square chamber. The reason such an additionof another boat decreases in the velocity of the single boat isthat the following camphor boats are decelerated by the camphorlayer developed from the preceding boats.

On the contrary, in the numerical calculation, the averageangular velocities of the single boat are 0.0436 and 0.0435 forthe triangular and rectangular chambers, respectively. When twoboats are coupled together, the average angular velocities are0.0426 for the triangular chamber, 0.0361 in mode I for thesquare chamber, and 0.0419 in mode II for the square chamber.Thus, the experimental results of the angular velocity arequalitatively reproduced by the numerical simulation.

As for the size effect of the chamber in the experiments, thecharacteristic synchronization depending on the length of thecorner was not observed with a larger chamber thanlc ) 60mm for the triangular chamber. The two boats proceeds in oneside and the phase is approximately locked according to thedifference in inherent velocities. When using a smaller chamber

Figure 6. Numerical results regarding the (1)θR vs θâ curve and (2)θR vs∆θ () θR - θâ) curve for the (a) triangular and (b) square routes.The parameters aremR ) mâ ) 0.01,kR ) kâ ) 10.0,f0R ) f0â ) 1.0,f1R ) f1â ) 0.9, a ) 30.0, andb ) 0.3. The individual curves weredrawn when the synchronized motions were complete (or the trajectoriesof the curves were almost closed). The initial phases are (a)θR)2.0,θâ)0.0; mode I in (b)θR ) 2.0,θâ ) 0.0; and mode II in (b)θR ) 3.0,θâ ) 0.0.

mR

d2θR

dt2) -kR

dθR

dt+ fR(θR) + g(θâ - θR,

dθâ

dt-

dθR

dt ) (5)

mâ

d2θâ

dt2) -kâ

dθâ

dt+ fâ(θâ) + g(θR - θâ,

dθR

dt-

dθâ

dt ) (6)

fi(θ) ) f0i + f1i cosnθ, for i ) R or â (7)

g(∆θ,∆ω) ) - 1

∆θ2(a - b∆ω) (8)

Figure 7. Numerical results regarding the time variation of (a)θR andθâ, and (b)∆θ upon switching from mode I as the initial condition tomode II for the square chamber. The gray and black lines in (a)correspond toθR andθâ, respectively. The parameters are the same asthose in Figure 6 except forf1â )1.003. The initial phases areθR )3.0, θâ ) 0.0.

Synchronized Sailing of Camphor Boats J. Phys. Chem. B, Vol. 109, No. 5, 20051801

than lc ) 40 mm for the triangular chamber, the interactionbetween two boats becomes stronger, and they tend to lock ata phase difference ofπ (data not shown). In the numericalmodel, the size effect can be considered as a change in theparametersa andb. When the chamber is small, the parametersa andb become larger, and vice versa. By changing parameters,we can reproduce the tendency on the size effect.

5. Conclusion

The characteristic synchronized sailing of two camphor boatswas exhibited on polygonal chambers, where the corners of thechambers contribute to regulating the velocity of the boats. Forthe triangular chamber, the two boats proceeded at an intervalof the length of one side. For the square chamber, theyproceeded at an interval of the length of either one or two sidesaccording to their initial positions. This synchronized sailingwas discussed using a phenomenological model and a numericalcalculation reproduced the experimental trends in a qualitativemanner. The present results suggest that autonomous motorsmay proceed by adapting themselves to external conditions. Webelieve that the present experimental system may be a simplemodel for creating various nature of synchronization whichdepends on the geometry of the system.

Acknowledgment. We thank Professor Masaharu Nagayama(Kanazawa University, Japan) for his helpful discussionsregarding the mechanism and Mr. Shin-ichi Hiromatsu (NaraUniversity of Education, Japan) for his technical assistance. Thisstudy was supported by a Grant-in-Aid for Scientific Researchfrom the Ministry of Education, Culture, Sports, Science andTechnology of Japan to S.N.

References and Notes

(1) Pikovsky, A.; Rosenblum, M.; Kurths, J.Synchronization, AUniVersal Concept on Nonlinear Sciences; Cambridge University Press:Cambridge, 2001.

(2) Kuramoto, Y. Chemical Oscillations, WaVes, and Turbulence;Springer: New York, 1984.

(3) Winfree, A. T.The Geometry of Biological Time; Springer: NewYork, 1980.

(4) Nishiyama, N.; Eto, K.J. Chem. Phys.1994, 100, 6977.(5) Yoshimoto, M.; Yoshikawa, K.; Mori, Y.Phys. ReV. E 1993, 47,

864.

(6) Kiss, I. Z.; Wang, W.; Hudson, J. L.J. Phys. Chem. B1999, 103,11433.

(7) Zhai, Y.; Kiss, I. Z.; Hudson, J. L.Ind. Eng. Chem. Res.2004, 43,315.

(8) Nakata, S.; Miyata, T.; Ojima, N.; Yoshikawa, K.Physica D1998,115, 313.

(9) Miyakawa, K.; Yamada, K.Physica D2001, 151, 217.(10) Yoshikawa, K.; Oyama, N.; Shoji, M.; Nakata, S.Am. J. Phys.

1991, 59, 137.(11) Das, A. K.; Srivastava, R. C.J. Chem. Faraday Trans.1993, 89,

905.(12) Schro¨dinger, E. What is Life?; Cambridge University Press:

Cambridge, 1944.(13) Linde, H.; Schwartz, P.; Wilke, H. InDynamics and Instability of

Fluid Interfaces; Sørensen, T. S., Ed.; Springer-Verlag: Berlin, 1979.(14) Levich, V. G. InPhysicochemical Hydrodynamics; Spalding, D.

B., Ed.; Advance Publication: London, 1977.(15) Brochard, F.Langmuir1989, 5, 432.(16) Kitahata, H.; Aihara, R.; Magome, N.; Yoshikawa, K.J. Chem.

Phys.2002, 116, 5666.(17) Yamaguchi, T.; Shinbo, T.Chem. Lett.1989, 935.(18) Chaudhury, M. K.; Whitesides, G. M.Science1992, 256, 1539.(19) Magome, N.; Yoshikawa, K.J. Phys. Chem.1996, 100, 19102.(20) Scriven, L. E.; Sternling, C. V.Nature1960, 187, 186.(21) Stoilov, Yu. Yu.Langmuir1998, 14, 5685.(22) Bain, C.; Burnett-Hall, G.; Montgomerie, R.Nature1994, 372, 414.(23) dos Santos, F. D.; Ondarc¸uhu, T.Phys. ReV. Lett.1995, 75, 2972.(24) Nakata, S.; Komoto, H.; Hayashi, K.; Menzinger, M.J. Phys. Chem.

B 2000, 104, 3589.(25) de Gennes, P. G.Physica A1998, 249, 1998.(26) Kovalchuk, N. M.; Vollhardt, D.J. Phys. Chem. B2001, 105, 41.(27) Kovalchuk, N. M.; Vollhardt, D.J. Phys. Chem. B2000, 104, 7987.(28) Yoshida, R.; Takahashi, T.; Yamaguchi, T.; Ichijo, H.J. Am. Chem.

Soc.1996, 118, 5134.(29) Sakai, T.; Yoshida, R.Langmuir2004, 20, 1036.(30) Sasaki, S.; Koga, S.; Yoshida, R.; Yamaguchi, T.Langmuir2003,

19, 5595.(31) Rayleigh, L.Proc. R. Soc. London1890, 47, 364.(32) Nakata, S.; Iguchi, Y.; Ose, S.; Kuboyama, M.; Ishii, T.; Yoshikawa,

K. Langmuir1997, 13, 4454.(33) Hayashima, Y.; Nagayama, M.; Nakata, S.J. Phys. Chem. B2001,

105, 5353.(34) Nakata, S.; Hiromatsu, S.Colloids Surf. A2003, 224, 157.(35) Hayashima, Y.; Nagayama, M.; Doi, Y.; Nakata, S.; Kimura, M.;

Iida, M. Phys. Chem. Chem. Phys.2002, 4, 1386.(36) Nakata, S.; Doi, Y.; Hayashima, Y.J. Phys. Chem. B2002, 106,

11681.(37) Nakata, S.; Kohira, M. I.; Hayashima, Y.Chem. Phys. Lett.2000,

322, 419.(38) Kohira, M. I.; Hayashima, Y.; Nagayama, M.; Nakata, S.Langmuir

2001, 17, 7124.(39) Nakata, S.; Doi, Y.AdV. Compl. Sys.2003, 6, 127.

1802 J. Phys. Chem. B, Vol. 109, No. 5, 2005 Nakata et al.