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Langmuir 1997, 13, 4454-4458
Self-Rotation of a Camphor Scraping on Water: New Insight into the Old Problem Satoshi Nakata,*,† Yasutaka Iguchi,† Sachie Ose,† Makiko Kuboyama,† Toshio Ishii,‡ and Kenichi Yoshikawa§ Department of Chemistry, Nara University of Education, Takabatake-cho, Nara 630, Japan, School of Dental Medicine, Tsurumi University, Tsurumi, Yokohama 230, Japan, and Graduate School of Human Informatics, Nagoya University, Nagoya 464-01, Japan Received February 24, 1997. In Final Form: May 7, 1997X
The translational and rotational motion of a camphor scraping at an air/water interface was investigated. The characteristics of the rotation depended on the temperature, surface tension, and the chemical structure of the camphor derivative. The direction of rotation, either clockwise or counterclockwise, is determined by asymmetry in the shape of the solid camphor scraping. The essential features of the motion of a camphor scraping were reproduced by a computer simulation. The driving force on this motion is believed to be the spatial heterogeneity of the camphor layer at the air-water interface.
Introduction There are several types of motor organs, or organelles, in living organisms. All of them work through the dissipation of chemical energy under almost isothermal conditions. On the other hand, modern industrial technology relies on heat engines, such as the gasoline engine, as a main source for power. It is well-known that heat engines cannot violate the second law of thermodynamics. The Carnot cycle tells us that the coefficient of energy transduction, η, should be less than (Th - Tl)/Th, where Th and Tl are the absolute temperatures of the heat reservoirs at the high and low temperatures, respectively. Thus, it is clear that biological motors operate through some mechanism other than the Carnot cycle. The development of such a chemical engine, which would work under isothermal conditions, is an important topic in modern science.1 More than a century ago, the self-motion of small camphor scrapings floating on water was explained by Van der Mensbrugghe as being due to the diminished surface tension of water.2 Subsequently, Rayleigh studied the retarding effect of contaminating oily substances on the self-motion of a camphor scraping.2 Since then, there have been several experimental trials to explain these observations. In our experiments on the self-motion of camphor, we have noticed that some scrapings undergo self-rotational motion. The present study was performed with the end goal of being able to control the direction, clockwise or counterclockwise, of this rotational motion, since experiments on such motion imply the realization of chemo-mechanical energy transduction under isothermal conditions.3-13 In a related experiment, we recently * Author to whom correspondence should be address. Tel.: +81742-27-9191. Fax: +81-742-27-9190. E-mail: nakatas@nara-edu. ac.jp. † Nara University of Education. ‡ Tsurumi University. § Nagoya University. X Abstract published in Advance ACS Abstracts, July 15, 1997. (1) Astumian, R. D.; Bier, M. Phys. Rev. Lett. 1994, 72, 1776. (2) Rayleigh, L. Proc. R. Soc. London 1890, 47, 364. (3) Dupeyrat, M.; Nakache, E. Bioelectrochem. Bioenerg. 1978, 5, 134. (4) Kai, S.; Ooishi, E.; Imasaki, M. J. Phys. Soc. Jpn. 1985, 54, 1274. (5) Yamaguchi, T.; Shinbo, T. Chem. Lett. 1989, 935. (6) Barton, K. D.; Subramanian, R. S. J. Colloid Interface Sci. 1989, 133, 211. (7) Chaudhury, M. K.; Whitsides, G. M. Science 1992, 256, 1539.
S0743-7463(97)00196-0 CCC: $14.00
examined on spontaneous vectorial movement in an oilwater system by the introduction of asymmetry to the shape of the rotor.9 Experimental Section (+)-Camphor and its derivative (borneol) were obtained from Tokyo Kasei Kogyo (Tokyo, Japan). When necessary, fragments of camphor were produced in asymmetric shapes. Water was first distilled and then purified with a Millipore Milli-Q filtering system (pH of the obtained water: 6.3). A camphor scraping (diameter, ca. 0.7 mm) was dropped onto pure water (50 mL) in a petri dish (diameter, 86 mm). The movement of camphor was monitored with a digital video camera (Sony DCR-VX700) and recorded on video tape. The two-dimensional position of the camphor and the center of gravity for the rotation were measured using a digitizer. The minimum time resolution was 1/30 s. The surface tension at the air-liquid interface was measured by the standard Wilhelmy method.14
Results Revolution and Translation of a Camphor Scraping. Figure 1 shows (a) the time trace of the circular orbital motion (revolution) together with the rotation around its centroid and translational motion of a camphor scraping with a time interval of 1/30 s (top view) and (b) snapshots of the orbital motion of a camphor scraping with a time interval of 1/15 s (top view). The camphor scraping keeps the circular orbital motion together with the rotation around its centroid for several tens of minutes with intermittent translational motion, as seen in Figure 1(a). The direction of rotation will sometimes spontaneously reverse. The period of scraping’s orbital motion is equal to that of its rotation about its centroid, as seen in Figure 1b. The frequency, radius of orbital motion, and the characteristics of camphor motion depend on the shape and size of the camphor scraping. (8) Sackmann, E. In Temporal Order; Rensing, L., Jaeger, N. I., Eds.; Springer-Verlag: Berlin, 1985; pp 153-162. (9) Yoshikawa, K.; Magome, N. Bull. Chem. Soc. Jpn. 1993, 66, 3352. (10) Magome, N.; Yoshikawa, K. J. Phys. Chem. 1996, 100, 19102. (11) Nakata, S.; Yoshikawa, K. Trends Chem. Phys. 1996, 4, 23. (12) Nakata, S.; Ukitsu, M.; Yoshikawa, K. In Proceedings of the International Conference on Dynamical Systems and Chaos; Aoki, N., Shiraiwa, K., Takahashi, Y., Eds.; World Scientific: Hong Kong, 1994; pp 339-342. (13) Yoshikawa, K.; Kusumi, T.; Ukitsu, M.; Nakata, S. Chem. Phys. Lett. 1993, 211, 211. (14) Adamson, A. W. Physical Chemistry of Surfaces, Interscience, New York (1960).
© 1997 American Chemical Society
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Figure 2. Logarithm of the frequency of revolution, ln f (Hz), versus temperature (expressed as T-1 × 1000 (K-1)) for (+)camphor and borneol.
Figure 1. (a) Time trace of the motion of a camphor scraping (top view). (b) Snapshots of the revolution of a camphor scraping with a time interval of 2/15 s (top view). (c) Time variation of the angular velocity of rotation for a camphor scraping on pure water.
Figure 1c shows the time-dependent change of the angular velocity in the rotation for the experiment given in Figure 1a, indicating that the angular velocity remains almost constant before and after the translational episode. In addition, Figure 1c shows that the angular velocity is minimized during the translation. Dependence of the Frequency of Revolution on the Temperature and the Surface Tension. Graphs of the logarithm of the frequency of revolution, ln f (Hz), versus temperature (shown as T-1 × 1000 (K-1)) for (+)camphor and borneol are shown in Figure 2. Here, the frequency of revolution was measured with a radius of revolution of ca. 5 mm. For both camphor and borneol, the rotational frequency increased with an increase in temperature, but the dependence of the rotational frequency of borneol on temperature is less than that of camphor. From the slopes of the lines in Figure 2, the activation energies of the rotation for camphor and borneol were evaluated to be 28 and 17 kJ mol-1, respectively.13,15 Figure 3 shows a graph of revolutional frequency versus surface tension at the air-water interface, where the surface tension was changed with the addition of sodium dodecyl sulfate (SDS) to the aqueous solution at different concentrations. No movement of the camphor scraping was observed for an interfacial tension below 50 mN m-1 or at an SDS concentration above 1.0 mM. Unidirectional Revolution of a Camphor Scraping with an Asymmetric Shape. Figure 4 shows a schematic representation of camphor fragments with asymmetric shapes and snapshots of their motions. We have made small fragments with the shapes shown in Figure 4 using a plastic plate (thickness, 1.0 mm). It is noted that the shapes in Figure 4a and 4b are mirror images of each other when these scrapings are placed on the air(15) Nakata, S.; Ukitsu, M.; Morimoto, M.; Yoshikawa, K. Mater. Sci. Eng. C 1995, 3, 191.
Figure 3. Dependence of the frequency of revolution, f (Hz), on the surface tension, γ (mN m-1), at the air-water interface.
water interface. The direction of rotation is almost always clockwise for the camphor scraping with the shape shown in Figure 4a, but is almost always counterclockwise for the asymmetric scraping as in Figure 4b. For these asymmetric shapes, rotational motion was predominant, with very little translational motion. On the other hand, when the floating camphor scraping has no chiral symmetry at the interface, as shown in Figure 4c, the translational motion becomes almost predominant without the occurrence of the revolution. Discussion Mechanism of Camphor Revolution. Camphor and its derivative exhibit characteristic motions under various experimental conditions. The frequency of rotation depends on the temperature, surface tension, and chemical structure of the camphor derivative. The direction of rotation is determined by the shape of the camphor scraping. We can discuss the mechanism of camphor motion on the basis of these results. Camphor motion may be induced by the following mechanisms (1-4), as shown in Figure 5a:2 (1) The camphor scraping dissolves into the outer environment by forming a thin layer at the interface around the scraping, since the hydrophobic camphor molecules have weak surface activity. (2) A gradient in the surface tension is induced, due to the spatial inhomogeneity in the surface density of the camphor layer. (3) The camphor scraping accelerates due to the spatial gradient of the surface tension. (4) Since the camphor molecules in a thin film can easily sublimate into the bulk
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Figure 4. Schematic representation of camphor fragments and snapshots of their motions with a time interval of 0.2 s (top view). (a) clockwise rotation, (b) counterclockwise rotation, and (c) translation. When the camphor fragments in (a) and (b) are placed at an air-water interface, they are mirror images of each other.
air phase, the spatial inhomogeneity of the camphor layer is maintained. The temperature dependence of the frequency of rotation is attributed to enhanced spatial inhomogeneity due to the increase in the rate of sublimation from the interface (Figure 2). The weaker temperature dependence of borneol may be due to the presence of a hydrophilic hydroxyl group in borneol. The borneol molecules will tend to stay at the interface even at elevated temperatures due to the attractive interaction between the hydroxyl group and water molecules, and this reduction in the sublimation of borneol may weaken the driving force on the motion of the scraping. The nearly constant angular velocity suggests that the driving force of this revolution is constant, i.e., the spatial inhomogeneity of the camphor layer is maintained by the scraping (Figure 1). The finding that the camphor scraping did not move a concentration of SDS higher than 1 mM in aqueous solution suggests that the driving force behind the movement of the camphor scraping is lower than the
surface pressure of the SDS molecular layer (Figure 3). Thus, the driving force is equal to the threshold value of the surface pressure, 15 mN m-1, (the surface tension of 1.0 mM SDS aqueous solution is 57 mN m-1) for camphor motion. The nature of diffusion of the camphor layer at the water surface could be observed by adding CaSO4 powder, with which the diameter of diffusion around the camphor scraping was found to be ca. 15 mm at 298 K on pure water. When a petri dish with a diameter of 15 mm was used for this experiment, neither rotation nor translation were observed. Therefore, diffusion of the camphor scraping plays an important role, and the spatial inhomogeneity of the surface tension around the camphor scraping due to the spatial distribution of the camphor layer is expected to be the driving force on camphor motion.4-10 Unidirectional rotation is attributable to the nonuniformity of surface tension around the camphor scraping, as indicated in Figure 5b. The density gradient of a camphor layer at a concave point (point B in Figure 5 (b-1)) is expected to be higher than that at a convex point
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(point C in Figure 5 (b-1)) as indicated in the model in Figure 5 (b-2), even at the same distance from the edge of the camphor scraping. Thus, the asymmetric density gradient of a camphor layer due to the shape of a camphor fragment may determine the direction of rotation. Although the switching between the orbital motion and the translation, and the reversal of the rotational direction are partly attributable to the gradual change in the shape of the fragment through its dissolution, we have failed to find any clear correspondence between the morphological change of the fragment and the switching of the motion. In other words, the morphology is kept almost unchanged during the period of several minutes. In addition to the effect of the gradient in the surface tension, a solutocapillary effect (or soluto-Marangoni effect), which involves surface convection from low to high surface tension, should be taken into account as the important factor on the camphor motion.16 In other words, the effect of the surface convection, owing to the gradient of the surface tension, is induced by no-slip between the peripheral edge of the particle lying in the interface and the interfacial liquid, and bulk motions will be induced by no-slip between the surface of the particle at the airwater interface and the liquid below it. Actually, we have noticed the occurrence of the convective motion of the fluid induced by the capillary effect in our experiment. As will be discussed in the next section, the irregularity of the orbital motion will be discussed, attributed, as a dominant cause, to the characteristics included in the nonlinear property of the fluid motion. Computer Simulation of Camphor Motion. Referring to the scheme of movement shown in Figure 5a and the above discussion on the mechanism of camphor motion, revolution and translation can be described by the following differential equations. The equation for rotation is
I dω(t)/dt ) N - ηω(t) + aξI(t)
Figure 5. (a) Schematic diagram of the mechanism of the selfmotion of a camphor scraping at an air-water interface (side view). The spatial gradient of the surface tension around the scraping is the driving force. (b-1) Schematic representation of a camphor fragment and its observed motion (top view). A-D are marked to explain the deterministic rotation in (b-2). (b-2) Proposed model of the spatial gradient of the camphor layer around a camphor scraping. A-D correspond to the points in (b-1).
(1)
and those for translation are
m dvx(t)/dt ) f cos θ(t) - ηvx(t) + bξx(t)
(2)
m dvy(t)/dt ) f sin θ(t) - ηvy(t) + bξy(t)
(3)
where I denotes the moment of inertia, ω the angular velocity, N torque, caused by a change in the symmetry of the camphor scraping, η the surface viscosity, m the weight of the camphor scraping, vx the velocity of translation in the x-axis, vy the velocity of translation in the y-axis, f the driving force for translational movement of the camphor scraping, θ the angular displacement of the rotation (dθ/dt ) ω), and a and b coefficients. In the above system of the equations, the first and second terms imply the acceleration and viscous damping of the scraping motion where the viscous term is interpreted as proportional to the velocity. This assumption implies that the origin of the viscous damping is attributed to the differential flow of the fluid near the scraping surface because of the non-slip condition at the solid-liquid interface.17 For fluctuation, ξ(t), which partly corresponds to the time variation in the shape of the camphor scraping and also to some hydrodynamic effects,
ξ(t) ) [white noise] + pδ(t ) tp)
Figure 6. Computer simulation of camphor revolution based on eqs 1-4. (a) N ) 1.0, η ) 0.5, a ) 0, b ) 0, p ) 0, x(0) ) 0, y(0) ) 0, θ(0) ) 0, vx(0) ) 0, vy(0) ) 0, and ω(0) ) 0. (b) N ) -1.0, η ) 1.0, a ) -0.3, b ) 0.3, p ) 100, tp ) 21.5, x(0) ) 0.70, y(0) ) -2.06, and θ(0) ) -0.05 (rad). m ) 1.0, I ) 1.0, and f ) 1.0. Without fluctuation, the x-y curve exhibits a revolution with a finite amplitude, as seen in the inset in (a). The filled arrow is the direction of self-motion, and the empty arrow in (a) is the vectorial direction of the torque N.
σ ) 1,18 and p is stimulation which is caused by a sudden change in the shape of the camphor. In relation to the framework in eq 4, it has been found that, owing to chemoMarangoni effect, intermittent pulsative change in the interfacial tension is generated at the oil/water interface.19,20 Such a pulsation has been interpreted as a kind of chaotic phenomena due to the nonlinear characteristics of the fluid near the interface. Numerical results for eqs 1-4 are shown in Figure 6. Without fluctuation, the x-y curve exhibits revolution with a finite amplitude, as shown in the inset in Figure 6a. Here, the filled arrow is the direction of camphor motion, and the empty arrow is the vectorial direction of N. The frequency of revolution is equal to that of rotation
(4)
where δ(t) is the delta function, white noise is Gaussian white noise with zero mean and a standard deviation of (16) Levich, V. G. In Physicochemical Hydrodynamics; Spalding, D. B., Ed.; Advance Publication: London, 1977.
(17) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics, 2nd ed.; Pergamon Press: London, 1987; Vol. 6 (of Course of Theoretical Physics). (18) Yoshimoto, M.; Nakaiwa, M.; Akiya, T.; Ohmori, T.; Yamaguchi, T. Physica D 1995, 84, 310. (19) Yoshikawa, K.; Shoji, M.; Nakata, S.; Maeda, S.; Kawakami, H. Langmuir 1988, 4, 759. (20) Yoshikawa, K.; Makino, M. Chem. Phys. Lett. 1989, 160, 623.
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(Figures 6a and 1b). The resistive surface viscosity for rotation, η, changes the radius of revolution. The direction of revolution is determined by the sign of N despite the initial angular velocity. The direction of torque may be related to the nonuniformity of the surface tension around the camphor scraping caused by the asymmetric shape, as indicated in Figure 4. The closed cycle fluctuates with noise and moves to another location due to stimulation (Figures 6b and 1a). Although the torque depends on the velocities of revolution and translation, we assume a constant torque for simplification in the present simulation. The frequency of rotation increases linearly with torque N. The N-dependent frequency may correspond to the surface-tension-dependent frequency in Figure 3, since the torque of the camphor scraping may be related to a spatial gradient in its surface free energy induced by chemical nonequilibrium at the air-water interface. New Insight into the Development of ChemoMechanical Energy Transduction under Isothermal Condition. According to the Curie-Prigogine theorem,21-23 vector processes cannot couple with scalar variables, such as an ideal chemical reaction, in a “linear” system under isotropic conditions. This implies that chemo-mechanical
Nakata et al.
coupling becomes possible with a violation of either the linear or isotropic conditions. Thus, it may be natural to expect that it is expected that effective chemo-mechanical coupling can be realized in the absence of both linear and isotropic conditions. In this study, we have demonstrated that the generation of macroscopic “deterministic” vectorial movement is generated under asymmetric conditions for a nonequilibrium system which is driven by a gradient in the chemical potential. Acknowledgment. The authors would like to thank Dr. N. Magome for his technical assistance. The present work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture. LA970196P (21) Katchalsky, A.; Curie, P. F. Nonequilibrium Thermodynamics in Biophysics; Harvard University Press: Cambridge, 1965. (22) Prigogine, I. Introduction to the Thermodynamics of Irreversible Process, 2nd ed.; John Wiley & Sons: New York, 1961. (23) Boccara, N. Symmetries and Broken Symmetries in Condensed Matter Physics; IDSET: Paris, 1981.
