Selection of the Rotation Direction for a Camphor Disk Resulting from

Aug 8, 2016 - ∥Graduate School of Advanced Mathematical Sciences and ⊥Meiji Institute of Advanced Study of Mathematical Sciences, Meiji University...
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Selection of the Rotation Direction for a Camphor Disk Resulting from Chiral Asymmetry of a Water Chamber Satoshi Nakata,*,† Hiroya Yamamoto,† Yuki Koyano,‡ Osamu Yamanaka,† Yutaka Sumino,§ Nobuhiko J. Suematsu,∥,⊥ Hiroyuki Kitahata,‡ Paulina Skrobanska,# and Jerzy Gorecki# †

Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima 739-8526, Japan Department of Physics, Graduate School of Science, Chiba University, 1-33 Inage-ku, Chiba 263-8522, Japan § Department of Applied Physics, Faculty of Science, Tokyo University of Science, Tokyo 125-8585, Japan ∥ Graduate School of Advanced Mathematical Sciences and ⊥Meiji Institute of Advanced Study of Mathematical Sciences, Meiji University, 4-21-1 Nakano, Tokyo 164-8525, Japan # Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland ‡

S Supporting Information *

ABSTRACT: Self-motion of a camphor disk rotating inside a water chamber composed of two half-disks was investigated. The half-disks were joined along their diameter segments, and the distance between their midpoints (ds) was considered as the control parameter. Various types of camphor disk motions were observed depending on ds. When ds = 0, the chamber had a circular shape, so it was symmetric. A camphor disk showed either a clockwise (CW) or counterclockwise (CCW) rotation with the direction determined by its initial state. The symmetry of the chamber was broken for ds > 0. For moderate distances between the midpoints, a unidirectional orbital motion of the disk was observed. The preferred rotation direction was determined by the shape of the chamber, and it did not depend on the initial rotation direction. For yet larger ds, the unidirectional circular motion was no longer observed and the trajectory became irregular. A mathematical model coupling the camphor disk motion with the dynamics of the developed camphor molecular layer on water was constructed, and the numerical results were compared with the experimental results. The selection of motion type can be explained by considering the influence of camphor concentration on the disk trajectory through the surface tension gradient.

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and the shape of the water chamber.1,19,20 However, as far as we know, there were no attempts to systematically investigate the effect of boundary shape. In this article, we describe the effect of the anisotropy in the boundary shape on the self-propelled motion of a camphor disk. We designed a water chamber characterized by a constant surface and the boundary shape that could be continuously changed. The water chamber was composed of two half-disks joined together along their diameter segments (cf. Figure 1). One of the half-disks could be shifted with respect to the other; thus, the shape of the chamber boundary was changed from rotationally symmetric to asymmetric. Using such a water chamber, we systematically studied the characteristics of selfmotion of a camphor disk depending on the boundary shape. A mathematical model1 commonly used to describe self-motion of a camphor disk was applied to calculate the disk trajectory in the chamber. The numerical results successfully reproduced the

INTRODUCTION Many examples of self-propelled objects responsive to the external environment, such as spatial gradients of chemical concentrations, light irradiation, and electromagnetic fields, have been investigated.1−7 It is known that the motion of selfpropelled objects can be strongly influenced by the shape of environment boundary. Therefore, investigation on the coupling between the motion and boundary shape is important for systematic control of self-propelled motion.8−12 Camphor particles on water have been investigated as an example of a self-propelled object.1,13−21 The driving force of a camphor particle is the difference in the surface tension around it. The difference appears due to spatial inhomogeneities of the camphor molecular layer around the solid camphor when the developed camphor molecules are dissolved into the water phase or sublimated into the air phase. A long time of motion with a constant speed and an easy control of experiment are the advantages of the camphor system. As often observed for other nonequilibrium systems, the self-motion of a camphor fragment can be strongly influenced by the shape of boundaries in the system, including the shape of the camphor fragment itself17,18 © 2016 American Chemical Society

Received: May 30, 2016 Revised: August 7, 2016 Published: August 8, 2016 9166

DOI: 10.1021/acs.jpcb.6b05427 J. Phys. Chem. B 2016, 120, 9166−9172

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Ltd., Japan; accuracy: 2 μm) to control its shift along the connecting diagonal segment. The camphor disk was placed at a water surface inside the chamber at t = 0 s, and at the beginning, it showed either a clockwise (CW) or counterclockwise (CCW) rotation. At t = 100 s, the mobile sheet was shifted by the distance, ds, required to make the chamber asymmetric. The shift was controlled by the stepping motor as illustrated in Figure 1, and it was completed within 0.5 s. The experiments were performed at room temperature (298 ± 2 K). Our preliminary experiments showed that in a chamber made of half-disks with a small radius the direction of circular motion rarely changed and was not affected much by the anisotropy of the boundary shape. On the other hand, in large chambers, the direction of motion was easily perturbed. Therefore, we selected the radius r of the water chamber to be 15 mm. For such a radius, the camphor disk rotation in a circular chamber is stable, and the disk trajectory can be easily perturbed if asymmetry is introduced.

Figure 1. Schematic illustration of the experimental system for studying the self-motion of a camphor disk in a water chamber made of two half-disk holes. The radius, r, of each disk was 15 mm. The asymmetry of the water chamber shape was introduced by sliding the mobile sheet connected to the stepping motor by distance ds at t = 100 s. The case ds = 0 corresponds to a circular boundary. A black mark with a diameter of 2 mm was attached to the top surface of the camphor disk for better visualization.

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RESULTS In the case of a circular boundary (ds = 0), the camphor disk continued unidirectional rotation either in CW or CCW direction, as shown in Figure 2a,b. The rotation direction

qualitative features of motion and the selection of circular motion direction observed in our experiments. The study on self-propelled motion in the considered geometry was also motivated by our interest in unconventional computation. Unconventional computing22,23 is a field of research dedicated to chemistry-, physics-, and biology-inspired computational strategies, structures, and substrates. In contrast to the conventional computer technology based on semiconductors and the von Neumann concept of computer architecture,24 unconventional computing interprets the natural time evolution of a medium as a series of information processing operations. For example, in the theoretical concept of billiard-ball computing,25 information is coded in the presence (or absence) of an object in a selected region of space at a given time. It has been found that geometrical constraints play an important role in unconventional information processing.25,26 For example, in reaction-diffusion computing,27,28 it has been observed that a junction between excitable media formed by triangular and rectangular ends can transmit pulses of excitation in one direction only, so it makes a signal diode. Here, we verify whether a water chamber with broken symmetry can force unidirectional motion of a camphor disk. A system of two shifted half-disks is a simple experimental realization of such a channel.

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Figure 2. Unidirectional rotation (motion I) in (a) CW and (b) CCW directions observed for ds = 0. (1) Superimposed images of snapshots of a camphor disk (top view; time interval: 0.1 s) and (2) trajectories of the camphor motion (t = 60.0−70.0 s).

EXPERIMENTS Camphor was purchased from Wako Pure Chemical Industries, Ltd., Japan, and used without further purification. Water was prepared using a water purifying system (WG23; Yamato Scientific, Japan). A camphor disk (diameter: 3 mm, thickness: 1 mm) was prepared in the same manner as described in our previous studies.1 A plastic, black circular dot (diameter: 2 mm) was glued on the camphor disk. Water (120 mL) was poured into a rectangular plastic vessel (154 mm × 154 mm × 17 mm), and the water level was 5 mm. Two rectangular plastic sheets (35 mm × 100 mm, thickness: 0.1 mm) with half-disk cuts (radius: r = 15 mm) were placed on water, such that both halfdisk cuts composed a circular water area. The position of one of the sheets was fixed. The other (the mobile sheet) was connected to a stepping motor (PM40B-100X, COMS Co.

depended on the initial state. The experiments confirmed the stability of unidirectional rotation for at least 200 s if the mobile sheet was not shifted from ds = 0. In the case in which ds > 0 was changed at t = 100 s, we found three characteristic types of camphor motions (motions I, II, and III) depending on the value of ds. When the mobile sheet was shifted by ds = 2 mm at t = 100 s, the direction of circular motion was maintained after the change in ds. We defined such a behavior as motion I. For such a small ds, the anisotropy in the boundary shape did not affect the rotation direction. In the case of shift distance of ds = 5 mm, CCW circular motion was eventually selected irrespective of the initial 9167

DOI: 10.1021/acs.jpcb.6b05427 J. Phys. Chem. B 2016, 120, 9166−9172

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Figure 3. Time evolution of disk orbit observed for ds = 5 mm (motion II) starting from the initial (a) CW and (b) CCW rotations. (1) Superimposed images of snapshots of a camphor disk (top view; time interval: 0.1 s) and (2) trajectories of the camphor motion (t = 60.0−70.0 s for (a-2i), 126.6−128.3 s for (a-2ii), 189.8−199.8 s for (a-2iii), 60.0−70.0 s for (b-2i), 121.2−121.9 s for b-2ii), and 172.7−182.7 s for (b-2iii)).

Figure 4. Irregular motion (motion III) observed for ds = 15 mm starting from the initial (a) CW and (b) CCW rotations. (1) Superimposed images of snapshots of a camphor disk (top view; time interval: 0.1 s) and (2) trajectories of the camphor motion (t = 87.0−97.0 s for (a-2i), 166.6−176.6 s for (a-2iii), 80.0−90.0 s for (b-2i), and 181.7−191.7 s for (b-2iii)).

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stable unidirectional rotation, whose direction was either CW or CCW. The probabilities of CW and CCW rotations were equal for ds = 0. The probability of CCW circular motion increased with ds, but the distribution of curvature for small ds retains its bimodal character with a minimum at κ = 0. For motion II (ds = 3−10 mm), the camphor disk always rotated in the CCW direction. Such a selection of rotation direction is represented by a single peak at positive κ, as seen in Figure 5. When ds was within the regime of motion III (>10 mm), the distribution of κ was almost symmetric again at about the axis at κ = 0 but with a single broad peak.

rotation direction, as shown in Figure 3. The transition between CW and CCW orbits occurred within 30 s after the shift was executed (for t = 127 s in Figure 3a). Therefore, the rotation direction was determined by the anisotropy of the boundary shape. We denoted such a behavior motion II. Further increase in ds changed the characteristics of the camphor motion. The trajectory combined the parts in which the camphor disk moved within one of the half-disks with transitions between them. An example of such a trajectory observed for ds = 15 mm is shown in Figure 4. We defined such a behavior as motion III. It comes out that the quantitative features of the camphor motion can be found in the curvature of the trajectory of motion, κ. The method applied to evaluate κ from the experimental data is described in the Supporting Information (SI). Figure 5 shows the probability distribution of κ measured for t between 150 and 200 s, which was obtained from four independent experiments for each value of ds. In the case of motion I (data for ds = 0−2 mm), the camphor disk exhibited

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NUMERICAL SIMULATIONS OF CAMPHOR DISK MOTION We describe the time evolution of a camphor disk in a water chamber considering the time dependence of camphor concentration at water surface, c(t, x, y), and the center position of the camphor disk, rc(t).1,18,19 The time development of c is represented as ∂c = D∇2 c − ac + fθ(ρ − |r − rc|) ∂t

(1)

where D is the diffusion coefficient, a is the sublimation rate of camphor molecules into air, f is the dissolution rate of camphor molecules onto the water surface, and ρ is the radius of the camphor disk. Here, θ(x) is a step function defined as ⎧1, (x ≥ 0) θ (x ) = ⎨ ⎩ 0, (x < 0) ⎪ ⎪

(2)

The position of the camphor disk is described by the following Newton’s equation of motion m

d2rc dt

2

= −η

drc + dt

∫0

2π

γ(c(rc + ρe(θ )))e(θ )ρ dθ + ξ(t ) (3)

where m is the mass of the camphor disk, η is the friction coefficient of the camphor disk on water, e(θ) is a unit vector in the direction of θ, that is, e(θ) = (cos θ, sin θ), and γ(c) is the function that describes the relation between the surface tension, γ, and camphor concentration, c. ξ(t) represents the Gaussian white noise exerted on the camphor disk, which is defined as ⟨ξ(t )⟩ = 0

(4)

⟨ξ(t ) ·ξ(t ′)⟩ = 2Ξδ(t − t ′)

(5)

where Ξ is the noise intensity. As the simplest approximation, we assume that γ(c) = γ0 − kc

(6)

reflecting that the camphor molecules decrease the surface tension. Here, γ0 is the surface tension of pure water and k is a positive constant. To simplify the description, we introduced scaled variables of length, time, and concentration as follows: r = r ̃ D/a , t = t/̃ a, and c = fc̃. Such a scaling defines the unit of length and unit of time with quantities that are independent of the chamber geometry: diffusion of camphor and its rate of sublimation from water surface. Then, equations 1 and 3−5 are rewritten as ∂c ̃ = ∇2 c ̃ − c ̃ + θ(ρ ̃ − |r ̃ − rc̃ |) ∂t ̃

Figure 5. Probability distribution of curvature κ obtained from experimental trajectories of camphor disks for various ds values. 9169

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Figure 6. Numerical results of the motion of a camphor disk in the chamber with a shape of two combined half-disks. The trajectories of the camphor disk in the chamber with (a) d = 0, (b) 2, and (c) 6 are shown. The initial conditions were set to be (1) (x, y) = (0, 1.5), (dx/dt, dy/dt) = (0.1, 0) and (2) (x, y) = (0, 1.5), (dx/dt, dy/dt) = (−0.1, 0). Here, we adopted the Cartesian coordinates, where the center point between the midpoints of two half-disks corresponds to the origin and the diameter segments of half-disks are on the x axis.

d2rc̃ dr ̃ = −η ̃ c − k ̃ 2̃ dt ̃ dt

∫0

2π

time. The results are shown in Figure 7. When d was sufficiently small, that is, d < 1, the distribution of curvature had two peaks

c (̃ rc̃ + ρ ̃e(θ ))e(θ ) dθ + ξ (̃ t ) (8)

̃ = where ρ ̃ = ρ / D/a , η̃ = η/(ma), k ̃ = kfρ /(m Da3 ), ξ(t) ξ(t)a3/2/(mD1/2), and the noise term is transformed as ⟨ξ (̃ t )⟩ = 0

(9)

⟨ξ (̃ t ) ·ξ (̃ t ′)⟩ = 2Ξ̃δ(t − t ′)

(10)

where Ξ̃ = Ξa /(m D). We adopted the Neumann boundary condition for eq 7 at the periphery of the water surface. The numerical calculation was performed using the Euler method for eqs 7 and 8 with a time step of Δt = 0.0005 and spatial unit of Δx = 0.05. For introducing the arbitrary boundary shape, we first determined whether each point is inside or outside of the region, and then we set the diffusion flow across the boundary to be zero. The parameters used in the calculations are ρ̃ = 0.6, η̃ = 450, k̃ = 5000, and Ξ̃ = 62.5. The radius of the half-disk was set to be 6 and the radius of camphor disk was 0.6. Therefore, the ratio between the chamber radius and the camphor disk radius was the same as that in experiments. We calculated the time series of the trajectory of the camphor disk together with the surface concentration of camphor. The trajectories for d = 0, 2, and 6 with two different initial conditions are shown in Figure 6. When d = 0, rotations in both CW and CCW directions were seen with almost equal probability, depending on the initial condition and noise, that is, motion I, as shown in Figure 6a. When d = 2, the unidirectional circular motion, depending on the shape of the chamber (motion II), was observed, as shown in Figure 6b. In this case, for every initial condition, the camphor disk finally rotated in the CCW direction. When d = 6, no regular orbits were observed, as shown in Figure 6c. To systematically investigate the relationship between motion type and shift, we varied d from 0 to 6, with a step of 0.5, and evaluated the curvature of trajectory, κ, using the same method that was applied to experimental results (see SI). Numerical calculations were performed with 80 different initial conditions and noise series for each d, and the distribution of κ was calculated for each d, using the trajectory after sufficiently long 4

2

Figure 7. Probability distribution of the curvature of trajectories obtained in numerical simulations. Eighty trajectories of 1000 < t < 2000 with different initial conditions and noise series were used for each d.

with positive and negative values, which corresponded to rotations in CCW and CW directions, respectively. At 1 < d < 4, the distribution had only one peak at a positive value of κ. This means that the camphor disk always exhibited CCW circular motion regardless of the initial condition. When d was greater than 4, the distribution was broad around κ = 0, that is, the camphor disk exhibited irregular motion. 9170

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DISCUSSION

CONCLUSIONS We investigated the influence of the anisotropic shape of a water chamber on the characteristics of the self-motion of a camphor disk. The considered system was composed of two half-disks, and the distance between their midpoints was the control parameter. On the basis of the probability distribution of the curvature of camphor disk trajectory, we were able to identify three types of camphor motions. For a small distance between the midpoints of the half-disks, a unidirectional rotation determined by the initial condition was observed (motion I). For a moderate distance, the direction of circular motion was determined by the boundary condition (motion II). If the distance between the midpoints was larger than 2/3 of the half-disk radius, an irregular motion was dominant (motion III). Three types of motions were reproduced in numerical simulations based on the mathematical model considering the dynamics of camphor concentration. The selection of motion can be explained on the basis of the effect of boundary shape on the trajectory through the surface tension gradient. Our results indicate that the features and the direction of camphor motion can be controlled by the boundary shape. We believe that our results bring an interesting example on the response of a selfpropelled object to the constraints defined by the environment.

On the basis of the results of experiments and numerical simulations, we can qualitatively explain which effects are responsible for a given type of camphor disk motion. At the beginning, let us consider the threshold between motions I and II. In a circular region, the camphor disk rotates along the periphery, but the rotation radius is slightly smaller than the inner radius of the circular region, as in Figure 2. Numerical calculation suggests that the radius of the circular trajectory of a camphor disk is less than the radius of the circular boundary by the diffusion length, which is determined by the diffusion constant and sublimation rate of the camphor molecules on water.21 It can be explained by the fact that camphor concentration near the periphery of the circular field is high and the camphor disk cannot approach the periphery. Even if there is anisotropy at the boundary, the region with a higher camphor concentration around the periphery is maintained for small values of ds or d. Thus, the camphor disk does not sense the real boundary shape, and it can rotate in either direction depending on the initial condition, as seen in motion I. However, when the anisotropy is greater, the motion of the camphor disk is more affected by the boundary shape. Actually, numerical results suggest that motion I is changed to motion II at around d = 1, that is, when d is comparable to the diffusion length. In our system, the boundary shape breaks the chiral symmetry, and as a result, one rotation direction of a camphor disk is selected as seen in motion II. To describe the transition between motions II and III, we have to consider the area outside of the trajectory. During camphor motion, the camphor molecules are dissolved into the water phase. Although almost all camphor molecules sublimate to the air, some camphor molecules remain, and the concentration of camphor increases around the trajectory. Therefore, the camphor concentration in the area outside of the trajectory does not increase so much. As ds increases, the ratio of the area outside of the trajectory becomes greater, and the surface tension gradient induced by such a concentration difference disturbs the regular trajectory seen in motion II. This can be considered to be the cause of the irregular motion, which was defined as motion III. A camphor particle driven by the surface concentration of camphor molecules on water is a typical example of a selfpropelled motion driven by the surrounding field. The arguments presented above indicate that the disk motion is a complex function of system geometry and disk size. The results presented in Figures 5 and 7 illustrate that all types of motions can be observed for a single ratio between the surfaces of the disk and chamber. In our opinion, it is unlikely that there is a simple scaling between the geometrical parameters and the motion type. For example, the characteristics of motion can change if all geometrical parameters are rescaled by a constant factor. The stationary density profile obtained from eq 1 with a fixed rc indicates that the radius of the area with a higher camphor concentration is a slowly increasing function of ρ. Therefore, it changes slower than the system size. As a result, the transition between motions I and II is expected if the size of a chamber with small asymmetry and the disk diameter are uniformly rescaled. If the boundary is gradually changed, hysteresis phenomena or bistability may be seen in the process of the change in the direction of circular motion. The extended study on this problem is planned.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.6b05427. Measurement of the curvature from the trajectory (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel/Fax: +81-82-4247409. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful to Prof. B. Nowakowski for valuable comments. This work was supported by JSPS Bilateral Joint Research Program between Japan and the Polish Academy of Sciences and JSPS KAKENHI Grant Numbers 25410094, 25103008, 15K05199, and 16K13866. This work was also performed under the Cooperative Research of “Network Joint Research Center for Materials and Devices” with Hokkaido University (Nos. 20163002 and 20163003).

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