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Self-Propelled Motion of a Coumarin Disk Characteristically Changed in Couple With Hydrolysis on an Aqueous Phase Satoshi Nakata, Yasutaka Irie, and Nobuhiko Jessis Suematsu J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b11534 • Publication Date (Web): 22 Apr 2019 Downloaded from http://pubs.acs.org on April 22, 2019

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The Journal of Physical Chemistry

Self-Propelled Ｍotion of a Ｃoumarin Disk Characteristically Changed in Couple with Hydrolysis on an Aqueous Phase

Satoshi Nakata,*,1 Yasutaka Irie,1 Nobuhiko J. Suematsu2,3

1Graduate

School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima

739-8526, Japan 2Meiji

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

Nakano, Nakano-ku, Tokyo 164-8525, Japan 3Graduate

School of Advanced Mathematical Sciences, Meiji University, 4-21-1 Nakano,

Nakano-ku, Tokyo 164-8525, Japan

*Corresponding author. Tel. & fax: +81-824-24-7409 E-mail: [email protected]

ACS Paragon Plus Environment

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ABSTRACT

In this study, a coumarin disk was examined as a simple self-propelled object under a chemical reaction. A coumarin disk placed on an aqueous phase containing Na3PO4 as a base exhibited continuous and oscillatory motion at lower and higher initial concentrations of Na3PO4, [Na3PO4]0, respectively. In addition, the period of the oscillation between rest and motion increased with increasing [Na3PO4]0. The mechanism of mode bifurcation between continuous and oscillatory motion and a change in the period of oscillation were discussed in terms to hydrolysis of coumarin and the surface tension of the aqueous solution as a driving force. A reduced mathematical model based on the reaction kinetics of coumarin around the air/aqueous interface, which adequately reproduced the experimental observation, was constructed. These results suggest that the characteristics of the self-propelled motion were determined by the kinetics of hydrolysis.


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INTRODUCTION Self-propelled objects have been widely investigated not only to understand characteristic biological motions, such as collective motion and chemotaxis, but also to artificially realize such characteristic biological motions in small spaces.1-13 Although several types of self-propelled objects have been artificially developed, the direction of motion is determined by the initial conditions or an external force.13-18 Consequently, the flexibility of motion of artificial selfpropelled objects is limited, and their autonomy is clearly lower than that of biological selfpropelled motors. The autonomy of self-propelled systems can be enhanced by introducing nonlinearity.1-3 Note that nonlinear phenomena19,

20

(e.g., oscillation, synchronization, bifurcation, and

spatiotemporal pattern formation) are important for maintaining living organisms. We have experimentally and numerically investigated the introduction of nonlinearity into self-propelled camphor or camphor derivative systems and created characteristic types of motion, i.e., oscillatory motion, synchronized motion, collective motion, motion with memory, and bifurcation of motion.2, 3, 21, 22

In this study, coumarin was used as an energy source for self-propelled motion driven by a difference in surface tension to induce characteristic features of self-motion resulting from hydrolysis of coumarin. We successfully obtained experimental results on mode bifurcation between continuous and oscillatory motion. The period of the oscillatory motion, which depended on the concentration of the base, Na3PO4, can be numerically reproduced according to the kinetics of coumarin hydrolysis.


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EXPERIMENTS The coumarin used to fabricate a self-propelled object and the Na3PO4 that served as a base were purchased from Nacalai Tesque, Inc. (Japan). The experiments were performed in an airconditioned room at 298 ± 2 K. The coumarin disk was prepared as follows (see Fig. S1). A hole (diameter: 3.0 mm, depth: 1.0 mm) was drilled in a Teflon plate (thickness: 1.5 mm) using a 3D plotter (Roland, monoFab, SRM-20, Japan) and filled with ~15 mg of coumarin grains. The coumarin particles (melting point: 344 K) in the hole were melted using a hot plate (As-one, CHS180, Japan) set at ~373 K. The system was then allowed to cool to room temperature. A solid object with a diameter of 3 mm was obtained. The excess coumarin around the top of the object was removed to make a disk with a constant thickness of 1 mm. Finally, the coumarin disk was released from the hole (final mass of the disk: 9 mg). A glass Petri dish (inner diameter: 145 mm, height: 18 mm) was used as an aqueous solution chamber. The volume and depth of the water phase were 250 mL and 15 mm, respectively. To reduce the effect of the boundary of the water chamber, we used a large water chamber in this study. When a smaller water chamber was used instead, the duration of motion was shorter. This indicates that the development of a coumarin molecular layer from the coumarin disk influences the speed of motion on the smaller water surface. The motion of the coumarin disk was monitored using a digital video camera (Sony, HDR-CX560, Japan, minimum time resolution: 1/30 s) and was analyzed by software (ImageJ, the National Institutes of Health, USA). At least four disks were used for each experimental condition.

RESULTS


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Figure 1 shows the time series of the speed of the coumarin disk on aqueous solutions composed of Na3PO4 [initial concentrations of Na3PO4 ([Na3PO4]0) = (a) 0, (b) 0.08, (c) 0.25, and (d) 0.50 M]. Continuous motion was observed when little or no Na3PO4 was present (Figs. 1a and 1b) and was maintained for at least 30 min. Oscillatory motion, i.e., repeated alternation of rest and motion, was observed at higher values of [Na3PO4]0 (Figs. 1c and 1d). In addition, the period of oscillation at 0.5 M was longer than that at 0.25 M. Oscillatory motion continued for at least 30 min at 0.25 and 0.50 M.


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Figure 1. Motion of the coumarin disk on aqueous solutions with different initial concentrations of Na3PO4 ([Na3PO4]0): (a) 0, (b) 0.08, (c) 0.25, and (d) 0.50 M. (1) Time series of speed of the coumarin disk and (2) top view of trajectory of the coumarin disk (see Movies S1, S2, S3, and S4). The color of trajectory indicates that the speed of the disk. The horizontal bars in (1) correspond to the data in (2).

The average speed of the continuous motion gradually decreased with increasing [Na3PO4]0 up to 0.21 M and then exhibited a sharp drop accompanied by mode bifurcation to oscillatory motion at [Na3PO4]0 = 0.25 M (see Fig. S2). As seen in Figs. 1c and 1d, the oscillatory motion consisted mostly of the resting state (v < 4 mm s−1), and rapid motion occurred for only a short time. This explains why the average speed decreased dramatically for oscillatory motion. We define the threshold value of the average speed marking the boundary between continuous and oscillatory motion as 4 mm s−1 in this study (see Fig. S2). Figure 2 shows the period of oscillation (Tp) between motion and rest as a function of [Na3PO4]0. TP was regarded as zero when continuous motion was observed. Continuous motion was observed at [Na3PO4]0 ≤ 0.21 M. In contrast, oscillatory motion was observed when [Na3PO4]0 ≥ 0.25 M, and Tp increased with increasing [Na3PO4]0.


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Figure 2. Period of oscillatory motion (Tp) between rest and motion as a function of [Na3PO4]0. The error bars are based on four examinations for each experimental condition.

The surface tension, i.e., the driving force for self-propelled motion, was measured while varying the concentrations of Na3PO4 and coumarin to clarify the relationship between the driving force of motion and coumarin hydrolysis. In this experiment, the surface tension was measured 1 h after the aqueous solution was prepared so the solution could reach the equilibrium condition for coumarin hydrolysis (see Scheme 1). For [Na3PO4]0 = 0 M, the surface tension decreased monotonically with increasing coumarin concentration and reached its minimum value at ~10 mM (Fig. 3a). The surface tension of the 10 mM coumarin aqueous solution increased with [Na3PO4]0 up to 0.02 M and converged to that on pure water (Fig. 3b). In contrast, the surface tension of the coumarin-free aqueous solution was independent of [Na3PO4]0. Thus, the difference in surface tension between the 0 and 10 mM coumarin solution decreased with increasing [Na3PO4]0 and was zero at [Na3PO4]0 ≥ 0.03 M.


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Scheme 1. Hydrolysis of coumarin and reverse reaction of the product (o-coumaric acid). k1 and k−1 are the rate constants for the forward and reverse reactions, respectively.23-25

Figure 3. Surface tension of aqueous solutions as a function of (a) concentration of coumarin at [Na3PO4]0 = 0 M and (b) [Na3PO4]0 at 0 (empty circles) and 10 mM (filled circles) coumarin. The surface tension was measured four times for each experimental condition.

DISCUSSION


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First, the driving force of the coumarin disk is the difference in surface tension around the disk, which appears because the coumarin molecular layer that develops from the disk decreases the surface tension. The mechanism of self-motion of the coumarin disk without hydrolysis is similar to that reported for a camphor disk.1-3 Although the direction of motion of the coumarin disk is not determined, the disk can move because the asymmetric floating state on water induces an inhomogeneous distribution of the coumarin molecular layer around the disk. This inhomogeneity originates from the enhance of fluctuation through a positive feedback process as is explained previously.26 At the initial stage, net force working to the disk is zero due to the symmetry. Here, once a perturbation is added to the disk, e.g., fluctuation of the disk position or distribution of camphor molecular layer, the symmetry is little bit broken. In this time, the net force originating from difference in surface tension works toward the direction of which the small initial perturbation is enhanced, i.e., positive feedback process. As the result, the disk accelerates and finally reaches the steady state. When unidirectional motion starts on an annular water channel, the direction of motion is maintained.26 Second, we consider that the bifurcation between continuous and oscillatory motion at approximately [Na3PO4]0 = 0.23 M (Fig. 2) is related to hydrolysis of coumarin, as indicated in Scheme 1.23-25 Coumarin reacts with OH− and produces o-coumaric acid, whose surface activity is lower than that of coumarin. In fact, the surface tension of the coumarin aqueous solution increases with [Na3PO4]0 under the equilibrium condition (Fig. 3b). OH− is produced by the buffering action between PO43− and HPO42− (PO43 + H2O ⇄ HPO42 + OH), and the concentration of OH−, [OH−], is described as a function of [Na3PO4]0. Therefore, both the hydrolysis and buffering action occur in an aqueous solution composed of coumarin and Na3PO4. Assuming that Scheme 1 always


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reaches equilibrium, [OH−] can be calculated. [The derivation of Eq. 1 is presented in Eqs. S1–S6 in the Supporting Information (SI).]

{

1

𝐾𝑤

(

[OH ― ] = 2 ― 𝐾𝑎3 + 10 ―7 ― 𝑤b +

𝐾𝑤

―7 + 𝑤b 𝐾𝑎3 ― 10

)

2

𝐾𝑤

}

+ 4𝐾𝑎3([Na3PO4]0 + 10 ―7 ― 𝑤b) , (1)

where 𝑤b is the bulk concentration of o-coumaric acid, which indicates the progress of coumarin hydrolysis. By considering the equilibrium conditions for both hydrolysis and the buffering action, the bulk concentration of coumarin (𝑢b) and 𝑤b can be numerically calculated for the aqueous solution of coumarin and Na3PO4. Here, the reaction rates for hydrolysis, k1 and k−1, are set to 0.385 M−1 s−1 and 1.32×10−4 s−1, respectively, which were experimentally estimated using the time series of the absorbance of the coumarin mixture and Na3PO4 (Figs. S4–S7). Eq. 2 holds under the equilibrium conditions of Scheme 1. 𝑘1𝑢b[OH ― ] = 𝑘 ―1𝑤b,

(2)

where [OH] is the equilibrium concentration described by Eq. 1. The relationship between wb and ub is described by Eq. 3. 𝑤b = 𝑢b0 ― 𝑢b,

(3)

where 𝑢b0 is the initial concentration of coumarin. The calculated result indicates that 𝑢b decreases with increasing [Na3PO4]0 at low values of [Na3PO4]0 up to 0.02 M and gradually approaches zero (Fig. 4). Using the relationship between the coumarin concentration and surface tension (Fig. 3a), the calculated concentration is converted to the surface tension (Fig. 4). The numerical results in Fig. 4 well reproduce the experimental observations, as shown in Fig. 3b.27, 28


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Figure 4. Bulk concentration of coumarin, ub, and surface tension of coumarin aqueous solution as a function of [Na3PO4]0 under the equilibrium condition. These results were obtained by numerical calculation using Mathematica. The initial concentration of coumarin was 0.010 M.

The oscillatory motion mechanism has been proposed on the basis of the chemical reaction of surface-active molecules at air/aqueous interfaces.1-3 Our observation can be explained by a similar mechanism. A coumarin disk can spontaneously move on water owing to spreading of coumarin molecules on water, which decreases the surface tension around the disk.1-3 However, at higher OH− concentrations, the coumarin disk cannot move during hydrolysis of coumarin because o-coumaric acid does not decrease the surface tension; this suggests that there is a critical value ([OH−]c) needed to obtain a driving force for the disk. For [OH−] > [OH−]c, the disk is in the resting state. However, the [OH−] value around the disk decreases locally with time as hydrolysis proceeds. When [OH−] becomes lower than [OH−]c, the disk regains its driving force and is rapidly accelerated. This is the phenomenological explanation of the oscillatory motion of the disk. This mechanism reveals that [OH−] is one of the bifurcation parameters.


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Mathematical analysis shows the necessary conditions for continuous motion of the disk. As indicated in Eq. 4, the established mathematical model for the motion of a camphor disk on a one-dimensional system3 can easily be modified for the motion of the coumarin disk by adding the chemical reaction term ( ― 𝑘1[OH ― ]𝑢), where the reverse hydrolysis reaction is neglected for simplicity because 𝑘1 >> 𝑘 ―1 (see Figs. S3–S6 and Eq. S16 in the SI). ∂𝑢 ∂𝑡

∂2𝑢

= 𝑆0𝛿(𝑥 ― 𝑥𝑐) +𝐷 ∂𝑡2 ― 𝑘d𝑢 ― 𝑘1[OH ― ]𝑢,

(4)

where u is the surface concentration of coumarin, [OH−] is the bulk concentration of OH−, D is the effective diffusion coefficient, kd is the rate constant for dissolution of coumarin into the bulk water phase, S0 is the supply rate of coumarin molecules on the water surface from the solid coumarin, 𝛿

(𝑥 ― 𝑥𝑐) is the delta function, x is the coordinate in the one-dimensional system, and xc is the location of the center of the coumarin disk. Here, the additional term ( ― 𝑘1[OH ― ]𝑢) is obtained by assuming that the surface concentrations are proportional to their bulk concentrations (see Eqs. S14–S16). In addition, ― 𝑘1[OH ― ]𝑢 can be considered a linear function of u by assuming that [OH−] is constant in time for continuous motion. This is a reasonable assumption because the [OH−] value around the disk is refreshed by the movement of the disk, and the surface area of the bulk water phase is quite large under actual examination. The force balance around the disk is described by a Newtonian equation (Eq. 5) in which the first term represents the frictional force, and the second term represents the driving force originating from the difference in surface tension: 𝑑𝑣

𝑚 𝑑𝑡 = ―𝜇𝑙𝑣 + 𝑙(𝛾(𝑥𝑐 + 𝜀) ― 𝛾(𝑥𝑐 ― 𝜀)),


12

(5)

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where m [kg] is the mass of the disk, v [m s−1] is the speed of motion,  [N s m−2] is the friction coefficient, l [m] is the length of the three phase line around the disk, and  [m] is the radius of the disk (1.5 mm). To determine the surface tension, the profile of u is required. u is dominated by the reaction–diffusion equation (Eq. 4). In addition, we suppose that the surface tension () decreases linearly with increasing u: 𝛾 = 𝛾0 ―𝛼𝑢(𝑥),

(6)

where  [N m−1] is the surface tension at x, 0 [N m−1] is the surface tension of water, and  [N m mol−1] is the activity of coumarin, which is defined as the slope of the function (u) (

[𝑑𝑢𝑑𝛾]𝑢 = 0 ≈

0.3 N m mol−1, as shown in Fig. 3a). To mathematically analyze the above equations, we first construct dimensionless equations and then consider the steady-state condition. We obtain the following dimensionless equations. 𝑑𝑉 𝑑𝑇

= ―𝑉 + Γ(𝑋𝑐 + 𝜖) ― Γ(𝑋𝑐 ― 𝜖),

𝑑𝑈

(7)

∂2𝑈

𝜏 𝑑𝑇 = ∂𝑋2 ―𝑈 + 𝛿(𝑋 ― 𝑋𝑐),

(8)

Γ = Γ0 ―𝛽𝑈.

(9)

They are all constructed using the dimensionless parameters defined as follows:

[

𝑡 = 𝑡0𝑇,𝑣 = 𝑣0𝑉,𝑥 = 𝑥0𝑋,𝑢 = 𝑢0𝑈,𝛾 = 𝜇𝑣0Γ,𝛾0 = 𝜇𝑣0Γ0,𝜀 = 𝑥0𝜖 𝑡0 =

𝑚 𝑙𝜇,𝑣0

𝐷

𝑠0

]

― = 4(𝑘𝑑 + 𝑘1[OH ])𝐷,𝑥0 = (𝑘𝑑 + 𝑘1[OH ― ]),𝑢0 = (𝑘 + 𝑘 [OH ― ])𝐷 . 𝑑 1

𝜏 = (𝑘

𝑑

𝛼𝑢0

𝑙𝜇 + 𝑘1[OH

―

])𝑚,𝛽 = 𝜇𝑣0 = 2(𝑘

𝛼𝑠0

𝑑

+ 𝑘1[OH ― ])𝐷𝜇


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Assuming that 𝜖 ≪ 1, the disk is approximated as a mass point. Considering the Taylor series around Xc, the surface tension can be written as Γ(𝑋𝑐 ± 𝜖) = Γ(𝑋𝑐) ±

(∂𝑋∂Γ)𝑋 ± 0𝜖 + ⋯ = Γ(𝑋𝑐) ± (𝑑𝑈𝑑Γ )𝑈(𝑋 )(∂𝑈 ∂𝑋)𝑋 ± 0𝜖 + ⋯. 𝑐

According to Eq. 9,

𝑐

𝑐

(𝑑𝑈𝑑Γ )𝑈(𝑋 ) = 𝛽. Thus, Eqs. 7 and 8 can be rewritten as follows: 𝑐

𝑑𝑉 𝑑𝑇

[( )

∂𝑈 ∂𝑋 𝑋 + 0 𝑐

= ―𝑉 + 𝛽

+

(∂𝑈 ∂𝑋)𝑋 ― 0]𝜖,

(10)

𝑐

∂2𝑈

∂𝑈

𝜏 ∂𝑇 = ∂𝑋2 ―𝑈 + 𝛿(𝑋 ― 𝑋𝑐),

(11)

where 𝛿(𝑋 ― 𝑋𝑐) is the delta function. Here, because 𝜏 ≪ 1, we consider the steady state of U. To obtain the steady state, Eq. 11 has to be rewritten under moving coordinates with speed V, where 𝑋 = 𝑋 ― ∂𝑈 ∂𝑋

∂𝑈

𝑣0𝑡0 𝑥0

𝑑𝑈

∂𝑈

∂𝑋

𝑉𝑇, and 𝜏 𝑑𝑇 = 𝜏 ∂𝑇 +𝜏∂𝑇

∂𝑈

= 𝜏 ∂𝑇 ―2𝑉∂𝑋. Therefore, Eq. 11 is replaced with the following equation. ∂𝑈

∂2𝑈

∂𝑈

𝜏 ∂𝑇 = ∂𝑋2 +2𝑉∂𝑋 ―𝑈 + 𝛿(𝑋), where we set 𝑋𝑐 = 0. Considering

∂𝑈 ∂𝑇

(12)

= 0, Eq. 12 becomes a second-order ordinary differential

equation. Therefore, we obtain the following function for the surface concentration U under the steady state [𝑈𝑆𝑆(𝑋)]. 𝑈𝑆𝑆(𝑋) = 2

1 𝑉2 + 1

exp( ― (𝑉 ± 𝑉2 + 1)𝑋),


14

(13)

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where the sign of the second term in the power of exponential function is positive for 𝑋 ≤ 0 and negative for the opposite. Next, we consider the time evolution of V. Using Eq. 13, we can obtain the driving force. Inserting Eq. 13 into Eq. 10, we obtain the following ordinary differential equation. 𝑑𝑉 𝑑𝑇

= ―𝑉 + 𝑟

𝑉

.

(14)

𝑉2 + 1

Here, r is a dimensionless bifurcation parameter and is defined as follows.

𝑟 = 𝛽𝜖 =

𝛼𝑠0𝜀 ― 𝜇𝐷 4(𝑘𝑑 + 𝑘1[OH ])𝐷

.

(15)

This corresponds to the ratio of the driving force against the frictional force. In fact, Eq. 14 indicates that the stationary state (V = 0) is stable for r < 1 but unstable for the opposite condition, where other stable fixed points (𝑉 ≠ 0) appear.2, 30 In the case of r > 1, the driving force overcomes the frictional force, and thus, the disk gets steady driving force. Here, we briefly consider how the disk gets the driving force. At the initial condition, V = 0 is considered and the net force is also 0 due to the symmetry. If a small fluctuation is added to the disk, the symmetry is broken and V has a small value. Here, we consider the case that V has small positive value as an example. In this case, the value of the first term in the right hand side of Eq. 14 (frictional term) becomes negative and the second term (driving force term) is positive. Due to r > 1, the right hand side of Eq. 14 becomes positive value, this means that the V increases with time and finally reaches the steady state where the driving force equal to the frictional force. On the other hand, in the case of r < 1, the frictional force is strong enough to restrict the small fluctuation, and namely, the right hand side of Eq. 14 becomes negative value. Therefore, V decreased in time. Thus, the disk cannot get steady driving force and backs to stationary state.26 Therefore, with an increase in [OH−], which


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corresponds to reducing r, the dynamics of the disk transits from continuous motion to stationary state when [OH−] = [OH−]c, in the case of r = 1. Next, we consider the oscillation between the motion and rest conditions using the bifurcation parameter, r. When [OH−]0 is higher than [OH−]c, r ≤ 1, and the disk is initially in the stationary state. In this case, the [OH−] value around the disk is no longer constant and gradually decreases with hydrolysis of coumarin, which is continuously supplied from the disk. When [OH−] falls below [OH−]c as hydrolysis proceeds, the disk fulfills the necessary conditions for beginning to move, i.e., r > 1. Here, we define the elapsed time tm as the time required for the concentration [OH] to become lower than [OH−]c through hydrolysis, assuming that the surface concentration profile of coumarin (u) and the speed of the disk (v) rapidly reach steady values. In the actual experiment, the acceleration time for one oscillatory motion is very short; that is, the resting time, which corresponds to tm, is similar to the period of the oscillation. Therefore, we can calculate the period of the oscillation between rest and motion using only the dynamics of [OH−] around the disk. The kinetics of the concentration of coumarin around the disk can be described in terms of (i) the supply of coumarin molecules, (ii) dissolution of coumarin into the water phase, (iii) hydrolysis of coumarin with OH− which is determined by phosphate dissociation, and (iv) the reverse hydrolysis reaction. Thus, the surface concentration of coumarin around a stationary disk (u) can be obtained using the following ordinary differential equation. 𝑑𝑢 𝑑𝑡

= 𝑆0 ― 𝑘d𝑢 ― 𝑘1𝑢[OH ― ] + 𝑘 ―1𝑤.

(16)


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We numerically calculated the dynamics of [OH−] using Eq. 16 and obtained the period of the oscillation using appropriate values for each physicochemical parameter. Here, we assumed that the dissociated phosphate rapidly reaches the equilibrium condition. This implies that Eq. 1 is valid for calculating [OH−]. The value of [OH−]c is set to 5.97 × 10−2 M, which was obtained using Eq. 1 and the bifurcation concentration at [Na3PO4]0 = 0.23 M (Fig. 2). In addition,  and D are set to 5.0 × 10−2 N s m−1 and 1.0 × 10−3 m2 s−1, respectively, which are the values estimated for self-propelled motion of a camphor disk or boat.27, 29 In Fig. 5, the calculated tm is plotted against [Na3PO4]0; the obtained values are in good agreement with our experimental results for Tp, where kd and S0 are used as fitting parameters. Note, however, that kd and S0 are not independent variables but must fulfill the following bifurcation condition: r = 1 at [OH−] = [OH−]c. Thus, when we set one parameter, the value of the other parameter is automatically determined. That is, there is only one fitting parameter. At [Na3PO4]0 < 0.23 M, [OH−] is lower than [OH−]c in the initial state, implying that the period is 0 s. The period increases monotonically with [Na3PO4]0 above the critical concentration (0.23 M), and its value is similar to the experimentally observed value. This agreement suggests that our assumption is suitable for understanding the oscillatory intermittent motion of a coumarin disk on an aqueous surface.


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Figure 5. Experimental results for Tp (filled circles) and numerical calculation of tm (empty circles) as functions of [Na3PO4]0. The data for Tp correspond to those in Fig. 2. S0 = 0.00108 M s−1, and kd = 0.001 s−1.

CONCLUSIONS We reported that a coumarin disk placed on a Na3PO4 aqueous solution exhibited bifurcation between continuous and oscillatory motion and that the period of the oscillation between rest and motion increased as a function of the initial concentration of Na3PO4. Coumarin decreased the surface tension of an aqueous solution of Na3PO4, but the surface tension of an aqueous solution of coumarin hydrolyzed with Na3PO4 was similar to that of pure water (~72 mN m–1), which suggests that coumarin is the driving force, rather than o-coumaric acid, which is the hydrolysis product. We previously used a mathematical model based on partial differential equations to obtain the period of the oscillation.31 In this study, we proposed a simple numerical method to understand the


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bifurcation of the movement behavior of the coumarin disk and to estimate the period of the oscillation. We used only an ordinary differential equation based on the chemical reaction of surface-active molecules. We should consider the effect of the water surface and the depth of the water chamber. When we used the smaller water chamber (diameter: 70 mm), motion was suppressed due to the development of the coumarin molecular layer to the smaller water surface area (see Figure S8). Marangoni flow should be considered to understand the effect of the depth of the water chamber since the driving force of motion is the difference in the surface tension.32 The effect of the depth on self-motion of a camphor disk was reported separately.33 Our approach successfully reproduces the experimentally observed dynamics of self-propelled motion. This agreement between the numerical and experimental results indicates the potential of our method. We believe that our simple approach will improves the understanding of the complex behavior of self-propelled objects.

ASSOCIATED CONTENT The Supporting Information is available free of charge on the ACS Publications website at DOI: xxxxx. Preparation of a coumarin disk; other analyses of motion for a coumarin disk; phosphate buffering action; evaluation of k1 and k–1; kinetics of surface coumarin concentration. Movie corresponding to Figure 1(a) (video rate: 10 times the actual speed) (Movie S1) (AVI) Movie corresponding to Figure 1(b) (video rate: 10 times the actual speed) (Movie S2) (AVI) Movie corresponding to Figure 1(c) (video rate: 10 times the actual speed) (Movie S3) (AVI)


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Movie corresponding to Figure 1(d) (video rate: 10 times the actual speed) (Movie S4) (AVI)

ACKNOWLEDGMENTS This work was supported in part by JSPS KAKENHI through Grants No. JP17K05835 and No. JP17KT0123, funding from the Electric Technology Research Foundation of Chugoku to SN, and funding from the Cooperative Research on “Network Joint Research Center for Materials and Devices” (No. 20183003) to SN and HK.

REFERENCES 1. Nakata, S.; Pimienta, V.; Lagzi, I.; Kitahata, H.; Suematsu, N. J., Eds. Self-Organized Motion: Physicochemical Design Based on Nonlinear Dynamics. The Royal Society of Chemistry, Cambridge, 2019. 2. Suematsu, N. J.; Nakata, S. Evolution of Self-Propelled Objects: From the Viewpoint of Nonlinear Science. Chem. Eur. J. 2018, 24, 6308–6324. 3. Nakata, S.; Nagayama, M.; Kitahata, H.; Suematsu, N. J.; Hasegawa, T. Physicochemical Design and Analysis of Self-Propelled Objects that are Characteristically Sensitive to Environments. Phys. Chem. Chem. Phys. 2015, 17, 10326–10338. 4. Tu, Y.; Peng, F.; Wilson, D. A. Motion Manipulation of Micro- and Nanomotors. Adv. Mater. 2017, 29, 1701970.


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14. Zarei, M.; Zarei, M. Self-Propelled Micro/Nanomotors for Sensing and Environmental Remediation. Small 2018, 14, 1800912. 15. Wang, H.; Pumera, M. Fabrication of Micro/Nanoscale Motors. Chem. Rev. 2015, 115, 8704−8735. 16. Villa, K.; Parmar, J.; Vilela, D.; Sánchez, S. Metal-Oxide-Based Microjets for the Simultaneous Removal of Organic Pollutants and Heavy Metals. ACS Appl. Mater. Interfaces 2018, 10, 20478−20486. 17. Han, K.; Shields, C. W.; Velev, O. D. Engineering of Self-Propelling Microbots and Microdevices Powered by Magnetic and Electric Fields. Adv. Funct. Mater. 2018, 28, 1705953. 18. Zhang, L.; Abbott, J. J.; Dong, L.; Kratochvil, B. E.; Bell, D.; Nelson, B. J. Artificial Bacterial Flagella: Fabrication and Magnetic Control. Appl. Phys. Lett. 2009, 94, 064107. 19. Suematsu, N. J.; Awazu, A.; Izumi, S.; Noda, S.; Nakata, S.; Nishimori, H. Localized Bioconvection of Euglena Caused by Phototaxis in the Lateral Direction. J. Phys. Soc. Jpn. 2011, 80, 064003. 20. Horikawa, K.; Ishimatsu, K.; Yoshimoto, E.; Kondo, S.; Takeda, H. Noise-Resistant and Synchronized Oscillation of the Segmentation Clock. Nature 2006, 441, 719−723. 21. Nakata, S.; Hata, M.; Ikura, Y. S.; Heisler, E.; Awazu, A.; Kitahata, H.; Nishimori, H. Motion with Memory of a Self-Propelled Object. J. Phys. Chem. C 2013, 117, 24490−24495.


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22. Nakata, S.; Kayahara, K.; Yamamoto, H.; Skrobanska, P.; Gorecki, J.; Awazu, A.; Nishimori, H.; Kitahata, H. Reciprocating Motion of a Self-Propelled Rotor Induced by Forced Halt and Release Operations. J. Phys. Chem. C 2018, 122, 3482−3487. 23. Mattoo, B. N. Spectrophotometric Study of the Hydrolysis of Coumarin and Dissociation of cis-Coumarinic Acid. Trans. Faraday Soc. 1957, 53, 760−766. 24. Maresca, A.; Temperini, C.; Vu, H.; Pham, N. B.; Poulsen, S.; Scozzafava, A.; Quinn, R. J.; Supuran, C. T. Non-Zinc Mediated Inhibition of Carbonic Anhydrases: Coumarins Are a New Class of Suicide Inhibitors. J. Am. Chem. Soc. 2009, 131, 3057–3062. 25. Sethna, S. M.; Shah, N. M. The Chemistry of Coumarins. Chem. Rev. 1945, 36, 1–62. 26. Nagayama, M.; Nakata, S.; Doi, Y.; Hayashima, Y. A Theoretical and Experimental Study on the Unidirectional Motion of a Camphor Disk. Physica D 2004, 194, 151–165. 27. Suematsu, N. J.; Sasaki, T.; Nakata, S.; Kitahata, H. Quantitative Estimation of the Parameters for Self-Motion Driven by Difference in Surface Tension. Langmuir 2014, 30, 8101–8108. 28. Ikura, Y. S.; Tenno, R.; Kitahata, H.; Suematsu, N. J.; Nakata, S. Suppression and Regeneration of Camphor-Driven Marangoni Flow with the Addition of Sodium Dodecyl Sulfate. J. Phys. Chem. B 2012, 116, 992−996. 29. Tenno, R.; Gunjima, Y.; Yoshii, M.; Kitahata, H.; Gorecki, J.; Suematsu, N. J.; Nakata, S. Period of Oscillatory Motion of a Camphor Boat Determined by the Dissolution and Diffusion of Camphor Molecules. J. Phys. Chem. B 2018, 122, 2610−2615.


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30. Koyano, Y.; Sakurai, T.; Kitahata, H. Oscillatory Motion of a Camphor Grain in a OneDimensional Finite Region. Phys. Rev. E 2016, 94, 042215. 31. Nakata, S.; Nagayama, M.; Kitahata, H.; Suematsu, N. J.; Hasegawa, T. Physicochemical Design and Analysis of Self-Propelled Objects that Are Characteristically Sensitive to Environments. Phys. Chem. Chem. Phys. 2015, 17, 10326−10338. 32. Scriven, L. E.; Sterling, C. V. Marangoni Effects. Nature 1960, 187, 186–188. 33. Matsuda, Y.; Suematsu, N. J.; Kitahata, H.; Ikura, Y. S.; Nakata, S. Acceleration or Deceleration of Self-Motion by the Marangoni Effect. Chem. Phys. Lett. 2016, 654, 92-96.


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TOC Graphic


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Scheme 1. Hydrolysis of coumarin and reverse reaction of the product (o-coumaric acid). k1 and k–1 represent the rate constants for the forward and reverse reactions, respectively. 156x45mm (150 x 150 DPI)


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Figure 1. Motion of the coumarin disk on aqueous solutions with different initial concentrations of Na3PO4 ([Na3PO4]0): (a) 0, (b) 0.08, (c) 0.25, and (d) 0.50 M. (1) Time series of speed of the coumarin disk and (2) top view of trajectory of the coumarin disk (see Movies S1, S2, S3, and S4). The color of trajectory indicates that the speed of the disk. The horizontal bars in (1) correspond to the data in (2).



Figure 2. Period of oscillation (Tp) as a function of [Na3PO4]0. 97x93mm (150 x 150 DPI)


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Figure 3. Surface tension of aqueous solutions as a function of (a) concentration of coumarin at [Na3PO4]0 = 0 M, and (b) [Na3PO4]0 at 0 (empty circle) and 10 mM (filled circle) coumarin. 200x97mm (150 x 150 DPI)



Figure 4. Bulk concentration of coumarin, ub, and surface tension of coumarin aqueous solution as a function of [Na3PO4]0 under the equilibrium condition. These results were obtained by numerical calculation using Mathematica. The initial concentration of coumarin was 0.010 M. 98x82mm (150 x 150 DPI)


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Figure 5. Experimental results for Tp (filled circles) and numerical calculation of tm (empty circles) as functions of [Na3PO4]0. The data for Tp correspond to those in Fig. 2. S0 = 0.00108 M s−1, and kd = 0.001 s−1. 97x91mm (150 x 150 DPI)



Table of content: During the hydrolysis of coumarin with hydroxide ion, a coumarin disk cannot move since the production o-coumaric acid does not decrease the surface tension. When the hydrolysis is finished, the coumarin disk can move since coumarin decreases the surface tension, i.e., the driving force can be obtained.


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Self-Propelled Motion of a Coumarin Disk ... - ACS Publications

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