Period of Oscillatory Motion of a Camphor Boat Determined by the

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Period of Oscillatory Motion of a Camphor Boat Determined by the Dissolution and Diffusion of Camphor Molecules Ryoichi Tenno, You Gunjima, Miyu Yoshii, Hiroyuki Kitahata, Jerzy Gorecki, Nobuhiko Jessis Suematsu, and Satoshi Nakata J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b11903 • Publication Date (Web): 06 Feb 2018 Downloaded from http://pubs.acs.org on February 9, 2018

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The Journal of Physical Chemistry

Period of oscillatory motion of a camphor boat determined by the dissolution and diffusion of camphor molecules

Ryoichi Tenno,1 You Gunjima,1 Miyu Yoshii,1 Hiroyuki Kitahata,2 Jerzy Gorecki,3 Nobuhiko J. Suematsu,4,5 and Satoshi Nakata1,*

1

Graduate School of Science, Hiroshima University, Kagamiyama 1-3-1, Higashi-Hiroshima 739-8526, Japan

2

Department of Physics, Chiba University, Yayoi-cho 1-33, Inage-ku, Chiba 263-8522, Japan

3

Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland

4

Meiji Institute for Advanced Study of Mathematical Sciences (MIMS), Meiji University, 4-21-1 Nakano, Nakano-ku,

Tokyo 164-8525, Japan 5

Graduate School of Advanced Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525,

Japan

*Corresponding contributor. E-mail: [email protected]

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Abstract 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 were 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 period of oscillatory motion at low

temperatures was explained by the reduced dissolution rate of camphor into the water phase.


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1. Introduction The ability to move is one of the basic features of living organisms. Therefore, investigation on different strategies of motion generated by physicochemical phenomena attracts attention of those researchers who wish to bridge chemistry with biology. For example, self-propelled objects have been studied as potential carriers to transport materials or themselves to a target location in a confined space, mimicking the behavior of bacteria.1-3 Most self-propelled objects exhibit monotonous or random motion, since the direction of motion is determined by the anisotropy of the motor shape or by the external field. On the other hand, biological motors such as bacteria can flexibly change the character of their motion while responding to the physicochemical environment, and their characteristic behavior, such as chemotaxis, is induced consequently.4, 5 Designing artificial self-propelled systems that mimic biological motors, can help us to understand how the variety and autonomy of self-propelled motion seen in nature emerge.6-21 The coupling of self-propelled motion with other nonlinear phenomena,6, 22, 23 e.g., spatio-temporal pattern formation, oscillation, bifurcation, synchronization is one of the available strategies to realize self-propelled motors with diversity and autonomy.6, 21,24

We have investigated simple self-propelled objects for which the difference in the surface tension on water is the driving force6, 25 and that exhibit a qualitative change in the character of their motion.6, 26-30 Here we focus our attention on a system in which the temperature can cause such qualitative changes. The motion of considered self-propelled objects can be described by Newton equation with forces and momenta including the surface tension that depends on the surface concentration of surface active molecules. The time evolution of surface concentration of active molecules can be obtained from reaction-diffusion equations including transport, dissolution, and evaporation. For example, it is well known that camphor molecules developed from a piece of solid camphor decrease the surface tension. As a result, a plastic plate with a camphor disk attached close to its end (a plastic boat) moves in the direction of the side without the camphor disk 31 because on this side the surface tension is higher.25, 32 3 ACS Paragon Plus Environment


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The character of camphor boat motion depends on the position of camphor disk with respect to the plate edge.26 If the position of camphor disk is close to an edge, an uninterrupted boat motion are observed. We reported that a camphor boat can exhibit oscillatory motion composed of short time intervals when a boat moves fast followed by long intervals when the boat stands still (cf. Fig 2 (d-1)). Following the terminology introduced in the reference 6, we call these types of motion continuous and oscillatory, respectively. In the continuous motion, the boat speed can drop to zero only if the boat collides with a wall of water container. The term oscillatory motion reflects repeatedly occurring transitions between stop and go. Such oscillatory motion occurs when a camphor disk is attached close to the center of plastic plate. The period of oscillatory motion is an increasing function of the distance between the edge of the plastic plate and the edge of the camphor disk.26-28 The analysis of oscillatory motion suggests that a stopped boat starts to accelerate after the camphor concentration at the side closed to the camphor disk exceeds the threshold value. Therefore, the period of oscillations is equal to the time, at which the required concentration appears. As far as we know, the quantitative analysis of oscillation period based on the model of time evolution of camphor concentration has not been reported yet. In this paper, we investigated the oscillatory motion of the camphor boat by changing the temperature and viscosity of the water phase to quantitatively clarify the relationship between the period of oscillatory motion and the camphor concentration at the edge of the plastic boat. We confirmed experimentally the existence of the threshold concentration needed to accelerate the resting boat and we found that it does not strongly depend on temperature nor on viscosity. On the basis of a model that included diffusion and dissolution of camphor molecules, we calculated the time needed to develop the threshold value of concentration and compared it with the period. We believe that the experimental and numerical results are important to design other self-propelled objects which exhibit complex characteristic features of motion depending on the diffusion of active surface molecules.


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2. Experiments Camphor and glycerol were purchased from Wako Chemicals (Kyoto, Japan) and Nacalai Tesque, Inc. (Kyoto, Japan), respectively. These chemicals were used without further purification. The water was purified by filtering it through active carbon, an ion-exchange resin, and Millipore Milli-Q filtering system (Merck Direct-Q 3UV, Germany). A camphor boat was composed of a camphor disk and a bended plastic plate, which was made using a 3D plotter (Roland, SRM-20, Japan) just as described in our previous papers.25-27 The length of plastic plate was L (= 10 mm). The camphor disk (diameter: 3 mm, thickness: 1 mm, mass: 5 mg) was attached to the plate as shown in Fig.1. The distance between the camphor disk center and the plate center p was 1.5 mm. The distance from the edge of the plastic plate to the closest periphery of the camphor disk d was 2 mm. In the following, we call d diffusion distance. A glass Petri dish (inner diameter: 125 mm) was used as a water chamber. The volume and depth of the water phase were 50 mL and 4 mm, respectively. In the experiments on camphor boat motion the temperature of the water phase was controlled by thermostat (AS ONE TP-80, Japan, precision of temperature: 1 K). The used thermostat did not allow for reliable temperature stabilization below 285 K. The motion of the camphor boat was recorded with a digital video camera (SONY HDRCX590V, Japan, maximum frame rate: 30 fps) and next analyzed on a computer using the imageanalysis software “ImageJ 1.41” (National Institutes of Health, USA). For each set of experimental conditions, the presented experimental data were obtained in least four separate experiments.

To

evaluate the rate of solution for solid camphor into the water phase, the camphor disk was immersed into the water phase at different temperatures and the mass of the disk was measured at different immersion time. In this experiment, the temperature was controlled using a thermostat (Yamato Scientific Co., Ltd., CLH300, Japan, precision of temperature stabilization: 1 K).



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Figure 1. Schematic illustration of the camphor boat used in this study. p (= 1.5 mm) is the distance between the center of a camphor disk and that of a plastic plate, L (= 10 mm) is the length of the plastic plate, and d (= 2 mm) is the diffusion distance between the edge of the plastic plate and the camphor disk.

3. Results

3-1. Period of oscillatory motion depending on the temperature of the water phase

The relationship between the average time between velocity spikes of oscillatory motion and the diffusion distance at a room temperature was previously published.25 In the experiment reported below the oscillatory motion was investigated at different temperatures in order to obtain the relationship between the period of oscillatory motion and the temperature shown in Figure 2. With the increase in temperature, the average time between spikes (the period) decreased and the transformation to continuous motion was observed at T ≥ 312 K. In addition, the period of oscillatory motion at T ≤ 288 K was much higher than that at T > 288 K.


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Figure 2. (a) The period of oscillatory motion (filled circle) of the camphor boat as a function of water temperature, T. Continuous motion (marked by empty circles), for which the period is set as zero was observed at T ≥ 312 K. (b) Superimposed images of a boat in the oscillatory and continuous motion types (time interval: 0.5 s) and (c) the time evolution of speed for the camphor boat on two typical motion of the camphor boat, i.e., (1) oscillatory motion at 298 K and (2) continuous motion at 313 K. (d) Long time behavior of the camphor boat speed in (1) oscillatory motion at 298 K and (2) continuous motion at 313 K. Black circles in Fig. 2b show the positions of a mark attached to the boat above the camphor disk. Numbers of corresponding to positions in (b-1) and (b-2) match the marks in (c-1) and (c2), respectively.

3-2. Rate of dissolution for solid camphor in the water phase at different temperatures



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The dissolution rate of solid camphor in the water phase is one of the parameters needed to calculate the period of oscillatory as a function of temperature. The amount of camphor dissolved from the solid disk per unit time, v (mol s-1) is expressed by Eq. 1.29, 30

v = − dMs(t)/dt = dMl(t)/dt = kd Ms(t),

(1)

where Ml(t) (mol) and Ms(t) (mol) are the amounts of camphor dissolved into the water phase from the disk and the amount of camphor remaining in the disk, and kd (s-1) is the dissolution rate of camphor. Equation 1 leads to the relationship:

ln (Ms(t) /Ms(0) ) = − kd t,

(2)

The values of v and kd were extracted from the experimental results using Eqs. 1 and 2. In experiments, we measured the mass of the camphor disk w(t) (kg) = mwMs(t) (mw (kg mol-1): molecular weight of camphor) at time t (s) for different temperatures (see Fig. S1). The initial mass of the camphor disk w(0) (kg) = mwMs(0) was 5×10-6 kg. The time dependence of ln (w(t)/w(0)) = ln (Ms(t)/Ms (0)) was almost proportional to time for the considered temperatures. The values of kd were obtained from the linear fit of ln (w(t)/w0)).

Considering that the dissolution of camphor is sufficiently slow, and thus we

approximate that the dissolution rate v does not change in time, and is equal to kdMs(0). Figure 3 shows v for different temperatures of the water phase. For the following analysis, v was approximated by a linear function of T.


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Figure 3. Amount of camphor supplied from the solid disk per unit time, v, depending on the temperature, T.

3-3. Period of oscillatory motion depending on the viscosity of the water phase

To study the influence of viscosity of the water phase on the period of oscillatory motion, we used glycerol aqueous solutions at different concentrations as the liquid phase. Figure 4 shows the period of oscillatory motion of the camphor boat for different viscosity of the water phase (µ) at 298 K. The observed period of oscillation increases with an increase in viscosity of the water phase.



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Figure 4. Period of oscillatory motion of the camphor boat as a function of the viscosity (µ) of the water phase at 298 K.

3-4. Determination of the threshold camphor concentration to accelerate the boat

Camphor molecules are developing from the edge of the boat and then the camphor boat moves slightly before the rapid acceleration.27

Although the detailed mechanism to explain this rapid

acceleration has not yet been uncovered, the existence of threshold concentration of camphor, Cth (mol L-1), can explain for the rapid acceleration from the resting state via very slow motion. The value of Cth was estimated by the following experiment. The plastic boat without a camphor disk was floating on the water phase. A droplet of camphor aqueous solution (volume: 10 µL) was placed on water close to the edge of the resting boat with a syringe and the maximum boat speed was measured (see Figure S3). Figure 5 shows the maximum speed of the plastic plate after the addition of water droplets with different concentrations of camphor at 298 K. When a pure water droplet was placed on the water surface, the maximum speed was 11 mm s-1 (see Fig.5). We confirmed that the boat speed at 11 mm s-1 is due to the deformation of water surface by a falling droplet. Therefore, the threshold value of the maximum speed between motion and no motion was determined as 11 mm s-1. We estimated Cth to be 0.8 mM which was


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obtained from an intersection point between the fitted line on the maximum speed vs. camphor concentration and the line on the maximum speed at 11 mm s-1 (see Fig.5).

Figure 5. The maximum speed of plastic boat after the addition of a camphor aqueous solution droplet at T =298 K. The dotted black line is the maximum speed of the plastic plate with the addition of a water droplet (11 mm s-1). Cth was obtained from the intersection point between the dotted horizontal line and another dotted line fitting the experimental results for the maximum boat speed for concentrations of camphor in the range of 1 ~ 3 mM. Such estimation gives Cth = 0.8 mM.

4. Discussion Based on the experimental results and related papers,6, 26, 28, 30, 32 we theoretically estimate the period of oscillatory motion as a function of the temperature and viscosity of the water phase. Here, we assume that camphor molecules diffuse from the camphor disk under the plastic plate when the plate stand still. The camphor boat accelerates when the concentration of camphor at the edge of the plastic plate reaches a threshold value. The time, tm, to reach the threshold value of concentration is regarded as the period of oscillatory motion.

We assume that at t =0 there is no camphor in the water phase. The concentration of camphor, C(x, t) (mol L-1), on the boat symmetry axis at the distance x from the disk and at time t is expressed by:25 11 ACS Paragon Plus Environment


10-3 v

C(x, t) =

S

10-3 v

=

S

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0 √4D exp 4D d t

-x2

1

D exp 4Dt − 2D Erfc -x2

t

x

x

√4Dt

,

(3)

where D (m2 s-1) is the diffusion coefficient of camphor, S is the cross sectional area for the diffusion of camphor at the bottom of the plastic plate (= 4 mm (width of the boat) × 4 mm (the depth of water phase) = 1.6×10-5 m2), and Erfc is the complementary error function defined as: Erfc(z) = π z e-t dt. √ 2

∞

2

(4)

The time evolution of camphor concentration at the edge of the boat, C(d, t), can be calculated using Eqs. 3 and 4 and the parameter values obtained from experiments.

At the beginning, let us introduce the empirical relationships describing the dependence of v and D on temperature. The relationship between v (mol s-1) and T (K) is approximated by the linear fit of the experimental data in Fig.3 in the range of temperatures 280 < T < 320 (K). v = (0.2925 T − 81.76) × 10-9.

(5)

For the diffusion coefficient of camphor, D (m2 s-1), depending on the temperature of the water phase, T (K), we adopt the approximate expression based on the linear fit of experimental results which were obtained by the other groups33 for the temperature range 280 < T < 320 (K): D = (0.0225 T – 5.8727) ×10-9 .

(6)

The viscosity of the water phase decreases with an increase in the temperature,34 and the diffusion coefficient is inversely proportional to the viscosity according to the Stokes-Einstein equation. The influence of temperature on D through the temperature dependent µ is included in Eq.6. 12 ACS Paragon Plus Environment

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Figure 4 suggests that the period of oscillatory motion depends on the viscosity of the water phase at a constant temperature at 298 K. We assume the relationship between D (m2 s-1) and µ (N s m-2) at 298 K in the form: D = 0.7407 ×10-12 / µ

(7)

that can be derived from the Stokes-Einstein equation (see SI on the deviation of eq.7).

The

approximation given by Eq.7 is valid if the viscosity of the water phase is 0.85 ×10-3 < µ < 1.25×10-3 (N s m-2).

The basic assumption of our analysis is that the camphor boat accelerates when C(d, t) reaches Cth. We experimentally estimated the value of Cth by changing the temperature or the viscosity of the water phase. We did not observe a significant dependence of Cth on temperature or viscosity (see Fig.S2). Therefore, Cth (mol L-1) was estimated as 0.8 mM in agreement with the experimental results in Fig 5. Let us introduce tm as the time when C(d, t) to reaches Cth. According to our assumption it is equal to the oscillation period. Figure 6 shows tm as a function of T calculated from Eqs. 3-7.

In this calculation,

we used the relationship between µ (N s m-2) and T (K):

µ = (7.27− 0.0212 T)×10-3

(8)

given in the reference 33 and valid in the temperature range is 280 < T < 320 (K). The temperature dependence of tm is similar to that for period of oscillation and tm around room temperature is close to the period at room temperature. The difference between numerical and experimental results, i.e., between tm and period observed at the other temperatures, may come from the initial state. Before the camphor boat is in contact with the water surface, especially at the higher temperatures, the camphor molecules have already diffused due to their fluctuation and sublimation. Among the other factors that 13 ACS Paragon Plus Environment


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can cause the difference, we can consider camphor molecules partly remained at the bottom of the boat for the individual oscillatory motion. In addition, we should consider the effect of advection flow at the edge of the plastic plate. Slight development of camphor molecules may induce the advection flow, and as a result, the camphor concentration at the edge of the plastic plate can be additionally increased especially at the higher temperature.

Here, we have only focused on reproducing the period of

oscillatory motion by numerical calculation including such physicochemical parameters as: the dissolution of camphor molecules depending on the temperature of the water phase, the temperature dependent diffusion of camphor molecules, and the viscosity of the water phase as well as the threshold concentration of camphor molecules.

In experiments, we observed a bifurcation between continuous

motion and oscillatory motion with the change in temperature as a bifurcation parameter (Figure 2a). It can be noted that the bifurcation point does not correspond to the temperature at which the period becomes zero. It has been already reported that the bifurcation occurs before the period becomes zero.26

Figure 6. Numerical results on tm (solid line) and experimental results of the period of oscillatory motion as a function of (a) temperature, T, and (b) viscosity, µ, at 298 K. The experimental results are shown as filled circles and the vertical lines represent the errors. The experimental results in (a) and (b) correspond to those in Figs. 2 and 4, respectively.


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We examined the period of oscillatory motion depending on the density of the water phase by using aqueous solutions with different concentrations of KBr. The period of oscillation for 1.4 g/mL KBr aqueous solution was similar to that for the pure water at 298 K, that is, the period of oscillatory motion was not sensitive to the density of the water phase.

5. Conclusion

The period of oscillatory motion of a camphor boat characteristically depends on the temperature and viscosity of the water phase. With an increase in temperature, the period of oscillatory motion decreased and oscillatory motion changed to continuous motion at T ≥ 312 K. We found experimentally and numerically that markedly high period of oscillatory motion at T ≤ 288 K can be associated with a lower dissolution rate of the camphor disk. The period of oscillatory motion was semi-quantitatively reproduced by numerical calculation including physicochemical parameters, i.e., the dissolution of camphor molecules depending on the temperature of the water phase, the diffusion of camphor molecules depending on the temperature and viscosity of the water phase, the threshold concentration of camphor molecules. In the future studies on complex motion of a camphor boat, we also plan to consider the period of oscillatory motion as a function the surface tension as well as the to include effect of advection flow at the edge of the plastic plate.

Acknowledgements This work was supported in part by JSPS KAKENHI through Grants No. JP17K05835, No. JP17KT0123, No. JP25103008, No. JP15K05199, and No. JP16H03949, Electric Technology Research



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Foundation of Chugoku, Satake Technical Foundation to SN, and the Cooperative Research of ‘‘Network Joint Research Center for Materials and Devices’’ (No. 20173006) to SN and HK.

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Table of Contents 463x405mm (150 x 150 DPI)

ACS Paragon Plus Environment

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Figure 1 :Schematic illustration of the camphor boat used in this study. p (= 1.5 mm) is the distance between the center of a camphor disk and that of a plastic plate, L (= 10 mm) is the length of the plastic plate, and d (= 2 mm) is the diffusion distance between the edge of the plastic plate and the camphor disk. 217x84mm (300 x 300 DPI)



Caption: Figure 2 : (a) The period of oscillatory motion (filled circle) of the camphor boat as a function of water temperature, T. Continuous motion (marked by empty circles), for which the period is set as zero was observed at T ≥ 312 K. (b) Superimposed images of a boat in the oscillatory and continuous motion types (time interval: 0.5 s) and (c) the time evolution of speed for the camphor boat on two typical motion of the camphor boat, i.e., (1) oscillatory motion at 298 K and (2) continuous motion at 313 K. (d) Long time behavior of the camphor boat speed in (1) oscillatory motion at 298 K and (2) continuous motion at 313 K. Black circles in Fig. 2b show the positions of a mark attached to the boat above the camphor disk. Numbers of corresponding to positions in (b-1) and (b-2) match the marks in (c-1) and (c-2), respectively. 190x270mm (300 x 300 DPI)


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Figure 3 : Amount of camphor supplied from the solid disk per unit time, v, depending on the temperature, T. 101x101mm (300 x 300 DPI)



Figure 4: Period of oscillatory motion of the camphor boat as a function of the viscosity (µ) of the water phase at 298 K. 151x102mm (300 x 300 DPI)


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Figure 5: The maximum speed of plastic boat after the addition of a camphor aqueous solution droplet at T =298 K. The dotted black line is the maximum speed of the plastic plate with the addition of a water droplet (11 mm s-1). Cth was obtained from the intersection point between the dotted horizontal line and another dotted line fitting the experimental results for the maximum boat speed for concentrations of camphor in the range of 1 ~ 3 mM. Such estimation gives Cth = 0.8 mM. 70x70mm (300 x 300 DPI)



Figure 6: Numerical results on tm (solid line) and experimental results of the period of oscillatory motion as a function of (a) temperature, T, and (b) viscosity, µ, at 298 K. The experimental results are shown as filled circles and the vertical lines represent the errors. The experimental results in (a) and (b) correspond to those in Figs. 2 and 4, respectively. 204x98mm (300 x 300 DPI)


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Period of Oscillatory Motion of a Camphor Boat Determined by the

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