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J. Phys. Chem. B 2005, 109, 1798-1802
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, and Department 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 two boats synchronously moved depending on the number of corners in the polygon by changing the distance between 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 difference of 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 of π/2 and π, which correspond to the length of one and two sides of the chamber, respectively, were observed with a square chamber. These characteristic features of synchronized self-motion were qualitatively reproduced by a numerical calculation that regarded the surface tension 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 geometry of the system.
1. Introduction When two or more nonlinear oscillators are coupled together, the phases can be locked at a constant value. Such a phenomenon is called “synchronization”, which is one of the most interesting phenomena observed in nonlinear systems.1,2 The synchronization in living organisms can be widely observed, for example, the beating heart, circadian rhythm, the organized rhythm in the flashing of swarms of fireflies, and so on.3 There have also been many experimental and theoretical studies on the synchronization of physicochemical coupled oscillators. For example, the synchronized chemical wave among small-beads4 or stirred containers5 was reported by using the BelousovZhabotinsky reaction. Hudson et al. reported the synchronization of the electrochemical oscillations on electrode arrays.6,7 The coupling of saltwater oscillators, which was constructed by two or three cups with saltwater and a larger vessel with pure water connected through a small orifice on the bottom of the individual cup, exhibited various natures of synchronization.8-11 To describe the nature of a single oscillator, the phase description is generally available as
dθ ) f(θ) dt
(1)
where θ is the phase of oscillator. If the oscillator is symmetric on the phase, f(θ) is constant; i.e., the angular velocity is constant. Otherwise, the angular velocity of oscillator is a function of the phase. On the other hand, autonomous motors have been studied to create artificial chemomechanical transducers that mimic biological and molecular motors.12 For example, the self-motion of a liquid droplet on a solid surface or a liquid surface is induced by differences in surface tension,13-27 and a gel * To whom correspondence should be addressed. Tel and fax: +81742-27-9191. E-mail:
[email protected]. † Nara University of Education. ‡ Kyoto University.
autonomously changes its shape according to the surrounding conditions.28-30 With regard to the interaction between two autonomous motors, they can synchronize with each other because of potent nonlinearity. Recently, we used a camphor boat at an air/water interface as a simple model of an autonomous motor. The camphor boat is driven by the heterogeneous distribution of the molecular layer of camphor developed from the solid fragment.31-39 We investigated systems with camphor that show various types of self-motion, e.g., unidirectional motion,32 characteristic motion depending on the shape of the water chamber,33 mode-switching between different types of motion,34 and the characteristic motion of camphor derivatives.35,36 We also previously discussed the synchronized sailing of two or three camphor boats on a circular chamber.37-39 Here, we regarded the self-motion of a camphor boat as an oscillator, i.e., one cycle on the circular chamber corresponds to 2π, and discussed the nature of self-motions of two boats on the same chamber as the coupled oscillators. In this system, the phase difference between two boats was locked or oscillated depending on the temperature of aqueous phase and the inherent velocity of the boats. It is noted that there is no specific position on the circular chamber to artificially change the velocity of the camphor 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 boats synchronously moved while regulating the individual velocities at the corners of polygonal water chambers. If the camphor boat is decelerated at the corner, f(θ) in eq 1 depends on the phase, i.e., it is possible to create various nature of synchronization between two oscillators regulated depending on the phase. The two camphor boats exhibited synchronized sailing that depended on the shape of the water chamber. These characteristic features of synchronized self-motion were qualitatively reproduced by a numerical calculation based on a Newtonian equation and the number of corners of the chamber. This system may be a simple
10.1021/jp0480605 CCC: $30.25 © 2005 American Chemical Society Published on Web 01/15/2005
Synchronized Sailing of Camphor Boats
J. Phys. Chem. B, Vol. 109, No. 5, 2005 1799
Figure 1. Schematic representation of the (a) triangular and (b) square chambers to analyze synchronized motion in the polar coordinates along the polygonal route. The phase of the camphor boat in the chamber, θ, is defined as 2πL/nlc.
model of synchronization between two oscillators regulated depending on the phase. 2. Experimental Section Camphor was obtained from Wako Chemicals (Kyoto, Japan). Water used in the chamber was first distilled and then purified with a Millipore Milli-Q filtering system. To obtain similar camphor boats, a camphor disk (diameter, 3 mm; thickness, 1 mm) was prepared with a pellet die set for FTIR, and a boat was drawn using computer software and printed on a polyester sheet (thickness: 0.1 mm) with a laser printer. The camphor disk was stuck to the stern of the boat with an adhesive.38 The two camphor boats were floated in the same direction on the water surface in a polygonal chamber made of Teflon (width of the route: 5 mm, thickness: 2 mm, length of a side: 50 mm (triangular), 40 mm (square)). The temperature of the water chamber was adjusted to 293 ( 1 K with a thermoplate (TP80, AS ONE Co. Ltd., Japan). The movement of the camphor disk was monitored with a digital video camera (SONY DCRVX700, minimum time-resolution: 1/30 s) and then analyzed by 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 schematic representation of the (a) triangular and (b) square chambers to analyze synchronized motion in terms of a phase description; i.e., the phase of the camphor boat in the chamber, θ, is defined as 2πL/nlc (n, number of corners in the polygon; L, distance along the water chamber from a reference point in the chamber in a clockwise direction as denoted in Figure 1a; lc, the length of one side of the polygonal chamber). 3. Results Figure 2 shows snapshots of the synchronized self-motion between two equivalent camphor boats in (a) a triangular chamber and (b) a square chamber. The individual boats decelerated when they passed through a corner and accelerated afterward. For the triangular chamber, the two camphor boats moved while maintaining the distance between them at ∆θ ) 2π/3 (L ) lc) for ca. 20 cycles (∆θ: the difference in phase corresponding to the distance between two camphor boats). This synchronized motion was independent of the initial locations of the two camphor boats. For the square chamber, the two camphor boats moved while maintaining ∆θ ∼ π (L ∼ 2lc) (mode I) for ca. 15 cycles (Figure 1b-I) when ∆θ ) 3π/4 to π (L ) 1.5lc to 2lc) was the initial location and VR was nearly equal to but slightly higher than Vβ (V: inherent velocity of a single camphor boat). When ∆θ ) π/4 to π/2 (L ) 0.5lc to lc) was the initial condition and VR was nearly equal to or slightly lower than Vβ, they moved while
Figure 2. Snapshots of the self-motion of two camphor boats (R, β) 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 the water 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 θβ curve and (2) θR vs ∆θ () θR - θβ) curve for the (a) triangular and (b) square chambers. The data in (a), I in (b), and II in (b), correspond to those in Figure 1a, b-I, and b-II, respectively. Dotted lines denote the individual corners of the polygonal chamber, and their thicknesses correspond to the length of the corners.
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 the initial condition, either mode I or II is selected randomly due to the experimental fluctuation. The experiments for ∆θ )0 to π/4 (L ) 0 to 0.5lc) as the initial condition could not be performed because of the finite size of a camphor boat. The duration time of synchronization was limited because the impurity in the camphor disk and the remained camphor layer lowered the surface tension with time. Figure 3 shows (1) a θR versus θβ curve and (2) a θR versus ∆θ () θR - θβ) curve for the (a) triangular and (b) square chambers. When the preceding boat R passed a corner in the triangular chamber, the following boat β approached boat R, and then ∆θ became 2π/3, as seen in Figure 3a. Boat R then
1800 J. Phys. Chem. B, Vol. 109, No. 5, 2005
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Figure 4. Experimental results regarding the time variation of (a) θR and θβ, and (b) ∆θ upon switching from mode I as the initial condition to mode II for the square chamber.
accelerated and backed away from boat β, i.e., ∆θ > 2π/3 when the boat β passed a corner. In the square chamber, modes I and II corresponded to phase differences of π and π/2, respectively. When boat R passed a corner in the square chamber, boat β approached boat R at ∆θ ) π and π/2, respectively, in modes I and II, as seen in Figure 3b. Boat R then accelerated and backed away from boat β, i.e., ∆θ > π in mode I and ∆θ > π/2 in mode II, when boat β passed a corner. The amplitude of ∆θ in mode I was lower than that in mode II. When the inherent velocity of one boat (β) was somewhat faster than that of the other boat (R), mode switching was observed in the square chamber. Figure 4 shows the time variation of (a) θR and θβ, and (b) ∆θ upon switching from mode I as the initial condition to mode II. ∆θ gradually deviated from π with time, mode I switched to mode II for ca. 10 s (or two cycles), and ∆θ approached π/2 for a few seconds. The amplitude of ∆θ was enhanced in the vicinity of the switching. The system maintained mode II for ca. 20 cycles, and finally it reached coordinated motion, i.e., the two boats moved together at ∆θ ∼ 0 (L ∼ 0).
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) are m ) 0.01, k ) 10.0, f0 ) 1.0, and f1 ) 0.9.
4. Discussion
We describe the motion of a single camphor boat by the Newtonian motion equation as
Figures 2 and 3 suggest that the distance between the two boats is regulated as an integral multiple of lc at the corner even if either VR is different from Vβ, or boat R moves away from boat β on the linear route. The regulation at the corner is due to the velocity and distance between the two boats. Thus, a camphor layer accumulates around the corner because the preceding camphor boat has physically settled there, and the motion of the following boat is decelerated by this layer. To theoretically understand the mechanism of the synchronized sailing of two autonomous motors, we introduce a mathematical model for a camphor boat in a closed route and solve it numerically. Based on the experimental results and previous papers,37-39 the motion of a camphor boat in a polygonal cell may be considered as a one-dimensional motion. Therefore, we consider a one-dimensional model with a periodic boundary condition. As in the experimental results, the phase of the camphor boat on a closed route, θ, can be written as
θ)
2π L nlc
(2)
d2θ dθ ) -k + f(θ) dt dt2
m
(3)
where m is the mass of the boat, k is a constant for viscous resistance, and f(θ) is the driving force due to the difference in the surface tension between the bow and stern of the boat according to the camphor layer that develops from the camphor grain and is approximately expressed as
f(θ) ) f0 + f1 cos nθ, f0 > f1
(4)
Equation 4 suggests that the maximum driving force, f0 + f1, is achieved when the camphor boat moves on a linear chamber with an infinite length. The sinusoidal part (or the second term on the right side of eq 4) is the physical effect of the polygonal cell, i.e., the velocity of the camphor boat is reduced as it passes around a corner. Figure 5 shows (a) experimental and (b) numerical results on the velocity of the single camphor boat for (1) triangular and (2) square chambers. The time variations of the velocity for the experimental results were similar to those
Synchronized Sailing of Camphor Boats
J. Phys. Chem. B, Vol. 109, No. 5, 2005 1801
Figure 6. Numerical results regarding the (1) θR vs θβ curve and (2) θR vs ∆θ () θR - θβ) curve for the (a) triangular and (b) square routes. The parameters are mR ) mβ ) 0.01, kR ) kβ ) 10.0, f0R ) f0β ) 1.0, f1R ) f1β ) 0.9, a ) 30.0, and b ) 0.3. The individual curves were drawn when the synchronized motions were complete (or the trajectories of the curves were almost closed). The initial phases are (a) θR)2.0, θβ)0.0; mode I in (b) θR ) 2.0, θβ ) 0.0; and mode II in (b) θR ) 3.0, θβ ) 0.0.
for the numerical ones, and therefore we adapted eq 4 as a simple approximation to reproduce the nature of self-motion. As for the motion of the camphor boats R and β, we introduce an interaction term as a function, g, as follows
mR
mβ
d2θR dt
2
) -kR
( (
) )
dθR dθβ dθR + fR(θR) + g θβ - θR, dt dt dt
d2θβ
dθR dθβ dθβ + fβ(θβ) + g θR - θβ, ) -kβ dt dt dt dt 2
(5)
(6)
where
fi(θ) ) f0i + f1i cos nθ, for i ) R or β
(7)
g(∆θ,∆ω) is expressed by
g(∆θ,∆ω) ) -
1 (a - b∆ω) ∆θ2
(8)
where a and b are positive constants, ∆θ is the difference in phase, and ∆ω is the difference in angular velocity. Equation 8 represents the repulsive force from the preceding boat. The camphor layer develops from preceding boat and decreases the driving force of the following boat. As the two boats are nearer, the repulsive force becomes stronger. Here, we introduce the term b∆ω to consider the effect of the relevant velocity between two boats, i.e., the repulsive force for the following boat becomes stronger when the following boat approaches to the preceding one (∆ω < 0), and vice versa. This effect may be reasonable if the camphor layer on the water surface has viscoelasticity 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 two camphor boats exhibit phase-locking synchronization at around 2π/3. This synchronized motion is exhibited even if the individual inherent velocities are slightly different. When the
Figure 7. Numerical results regarding the time variation of (a) θR and θβ, and (b) ∆θ upon switching from mode I as the initial condition to mode II for the square chamber. The gray and black lines in (a) correspond to θR and θβ, respectively. The parameters are the same as those in Figure 6 except for f1β )1.003. The initial phases are θR ) 3.0, θβ ) 0.0.
preceding boat approaches to the corners, ∆θ decreases to 2π/ 3. In contrast, ∆θ increases after the two boats pass through the corners. For the square chamber, either mode I or mode II is selected depending on the initial phases of the two boats when the inherent velocity of one boat is similar to that of the other. When the inherent velocity of one boat is slightly different from that of the other, mode II is more stable than mode I. When the initial state is mode I, it switches suddenly to mode II, as shown in Figure 7. Thus, these numerical results in Figures 6 and 7 qualitatively correspond to the experimental results shown in Figures 3 and 4. When the single camphor boat was floated on the triangular and square chambers in the experiment, the average velocities (and angular velocities) of the single camphor boat were 55 mm 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 floated on these chambers, the average velocities (and angular velocities) of two camphor boats were decreased to 33 mm s-1 (1.4 rad s-1) for the triangular chamber, 32 mm s-1 (1.3 rad s-1) in mode I for the square chamber, and 35 mm s-1 (1.4 rad s-1) in mode II for the square chamber. The reason such an addition of another boat decreases in the velocity of the single boat is that the following camphor boats are decelerated by the camphor layer developed from the preceding boats. On the contrary, in the numerical calculation, the average angular velocities of the single boat are 0.0436 and 0.0435 for the triangular and rectangular chambers, respectively. When two boats are coupled together, the average angular velocities are 0.0426 for the triangular chamber, 0.0361 in mode I for the square chamber, and 0.0419 in mode II for the square chamber. Thus, the experimental results of the angular velocity are qualitatively reproduced by the numerical simulation. As for the size effect of the chamber in the experiments, the characteristic synchronization depending on the length of the corner was not observed with a larger chamber than lc ) 60 mm for the triangular chamber. The two boats proceeds in one side and the phase is approximately locked according to the difference in inherent velocities. When using a smaller chamber
1802 J. Phys. Chem. B, Vol. 109, No. 5, 2005 than lc ) 40 mm for the triangular chamber, the interaction between two boats becomes stronger, and they tend to lock at a phase difference of π (data not shown). In the numerical model, the size effect can be considered as a change in the parameters a and b. When the chamber is small, the parameters a and b 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 boats was exhibited on polygonal chambers, where the corners of the chambers contribute to regulating the velocity of the boats. For the triangular chamber, the two boats proceeded at an interval of the length of one side. For the square chamber, they proceeded at an interval of the length of either one or two sides according to their initial positions. This synchronized sailing was discussed using a phenomenological model and a numerical calculation reproduced the experimental trends in a qualitative manner. The present results suggest that autonomous motors may proceed by adapting themselves to external conditions. We believe that the present experimental system may be a simple model for creating various nature of synchronization which depends on the geometry of the system. Acknowledgment. We thank Professor Masaharu Nagayama (Kanazawa University, Japan) for his helpful discussions regarding the mechanism and Mr. Shin-ichi Hiromatsu (Nara University of Education, Japan) for his technical assistance. This study was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan to S.N. References and Notes (1) Pikovsky, A.; Rosenblum, M.; Kurths, J. Synchronization, A UniVersal 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: New York, 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.
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