Analysis of convection flow of a self-propelled alcohol droplet in an exoskeleton frame

Abstract

This study aims to analyze the convection flow of a self-propelled 1-pentanol droplet. The droplets move spontaneously when 1-pentanol droplets are dropped into an aqueous 1-pentanol solution. This self-propulsion is due to the interfacial tension gradient caused by the concentration differences. The shape of the droplet is closely related to its behavior because the shape of the droplet changes the interfacial tension gradient. In this study, an exoskeleton is used to fix the droplet shape. In our preliminary experiments, we observed Marangoni convection in droplets dropped in exoskeleton frames with boomerang and round holes. The results showed that a large difference in surface tension was necessary to control the self-propulsion of the 1-pentanol droplets. Herein, we prepared two exoskeletons with different holes, an elongated symmetrical elliptical shape, and an asymmetrical shape to fix the shape of the droplet. The droplets were then dropped into each exoskeleton, and the droplet behavior, Marangoni convection inside the droplet, and convection in the aqueous phase were analyzed. We found that the direction of the self-propulsion of the droplet was determined by these exoskeletons, particularly in the case of the asymmetrical exoskeleton, and the direction of self-propulsion was fixed in one direction. Marangoni convection was observed in the droplet from the direction of lower surface tension to that of higher surface tension. In the aqueous phase, two convections were generated from the aqueous phase to the droplet because of the diffusion of 1-pentanol. In particular, when an asymmetrical exoskeleton was used, two convections of different sizes and velocities were observed in the aqueous phase. Based on these experimental results, the relationship between droplet behavior and convection is discussed.

Introduction

The mechanisms, by which living and nonliving materials convert energy into motion through a unique mechanism, have drawn much attention in the studies of soft robotics [1,2,3,4,5,6]. Their complex structures and structural flexibilities have been explored as synthetic active materials, and they have biological characteristics that possess dynamic properties along with chemical reactivity. For example, biological cells [7,8,9], microorganisms [8, 9], camphor [10,11,12], and self-propelled droplets [13,14,15,16,17,18,19,20,21,22,23,24] have mechanisms for converting chemical energy into kinetic energy. Droplets have been studied in many research fields, including physical chemistry, chemical engineering and fluid mechanics [25, 26]. Within those fields, the combining chemical robotics with biological mechanisms could lead to the creation of robots specialized for living things [27]. In particular, self-propelled droplets have been reported to exhibit lifelike properties and can be applied in computing, such as solving mazes and operations in logic gates [28, 29].

The self-propulsion of alcohol droplets is caused by a simple factor: the difference in concentration at the interface between a droplet and aqueous solution [30]. Using the Marangoni effect, Nagai et al. interpreted the relationship between 1-pentanol droplet size and droplet behavior in terms of internal instability. Nagai et al. proposed that the droplet introduced into the aqueous phase deforms into a boomerang shape and self-propels in a convex direction when a droplet volume between 0.1 and 200 µL was dropped. The difference in curvature between the front and back of the boomerang-shaped droplet causes a difference in the diffusion velocity of 1-pentanol, resulting in a concentration gradient. This concentration gradient becomes a difference in interfacial tension and generates convection inside the droplet, as shown in Fig. 1 [27, 31]. This internal convection is called Marangoni convection [32, 33], which generates self-propulsion of the droplet.

Fig. 1

Marangoni convection inside the droplet and direction of self-propulsion [30]

The concentration gradient in the aqueous phase generated by an exoskeleton frame to fix the shape of a droplet was investigated in our previous study [27] to control droplet behavior. We have previously reported that the behavior of anhydrous oleic acid droplets could be successfully controlled using an exoskeleton. Furthermore, this method was applied to a transporting robot that utilizes the self-propulsion of a droplet.

This study focuses on the convection generated when 1-pentanol droplets are self-propelled in an aqueous solution. We examine the relationship between the droplet behavior and the convection generated when the droplet dropped into the exoskeleton. We first observe convection from two directions, from the top of the droplet and from the side of the aqueous phase, and then analyze the convection generated in the droplet and aqueous phase respectively.

Preparation of experiments

Materials

We mix 100 mL of pure water and 2.3 mL of 1-pentanol (FUJIFILM) and stir by a magnetic stirrer (AS ONE, RS-6AN) for 1 h. A droplet of 1-pentanol only is dropped into the aqueous solution.

Preparing the exoskeleton frame

The shapes of the exoskeleton frames are designed using an illustrator (Brother, CanvasWorkspace). An overhead projector (OHP, polyester) film of 0.10 mm thickness is cut using a cutting plotter (Brother, SDX85) and double bond with a bond (Konishi Co., Ltd., #10122).

Preliminary experiment

Exoskeleton frames

To determine the optimal shape for self-propulsion, two exoskeletons were prepared by cutting different internal shapes as shown in Fig. 2. The boomerang-shaped exoskeleton was adopted because it was a shape in which the direction of the anhydrous oleic acid droplets was controlled in our previous study [27]. The round exoskeleton had an equal surface tension at the circumference, and was prepared for comparison with a boomerang-shaped exoskeleton. The outer diameter of the two exoskeletons was 10 mm, and the thickness was 0.20 mm by bonding two OHP films of 0.10 mm thickness.

Fig. 2

Two exoskeletons used in a preliminary experiment

Experimental method

To investigate the relationship between Marangoni convection and self-propulsion, we observed Marangoni convection inside a 1-pentanol droplet three times each using particle image velocimetry (PIV). Internal convection was visualized by tracking the particles using the image analysis software, FlowExpert2D2C (Kato Optical Laboratory). For the stable observation of Marangoni convection inside a droplet, the droplet must remain a fixed location. An OHP film plate was used, as shown in Fig. 3a.

Fig. 3

Preliminary experimental system

First, acrylic particles (Lumisis marker, specific gravity 1.3) were mixed with 1-pentanol droplets. The measurements were performed using the system shown in Fig. 3 under the experimental conditions listed in Table 1. After injecting the aqueous solution (3.5 mL) into a Petri dish using a micropipette, we floated the OHP film shown in Fig. 3a on the aqueous phase to fix the position of the exoskeleton. Then, we floated an exoskeleton in the hole of the OHP film (Fig. 3a), and a 10 µL droplet mixed with acrylic particles was introduced inside the exoskeleton using an automatic micropipette (INTEGRA, VOYAGER). While illuminating the droplet with ultraviolet black light, the Marangoni convection inside the droplet was captured using a high-speed camera (Baumer, VCXU-13 M, 1280 × 1024 px, 100 fps) placed directly above the droplet. The movies were analyzed using FlowExpert2D2C software to track the particles in internal convection.

Table 1 Experimental conditions

Result and discussion

In the preliminary experiment, we analyzed the Marangoni convection inside a droplet that was dropped into two exoskeletons three times each. As shown in Fig. 4, Marangoni convection appeared in all the droplets dropped in the two exoskeletons. Figure 4b presents the moment when convections were generated.

Fig. 4

Observation of Marangoni convection in droplets dropped in two different exoskeletons

A pair of Marangoni convections was generated inside the boomerang-shaped droplet (Fig. 4a). Marangoni convection was generated from the concave to the convex region inside the boomerang-shaped anhydrous oleic acid droplet because of surface tension differences, which was discovered in our previous study [27]; however, the position of convection inside the 1-pentanol droplet changed in repeated experiments. This result implies that the difference in the surface tension between the convex and concave regions must be greater to maintain the position of the Marangoni convection inside the 1-pentanol droplet.

In a round-shaped droplet, Marangoni convection was also observed; however, the locations were not identical. Figure 4b shows one of the results when a round exoskeleton was used. A pair of Marangoni convections was generated initially as shown in Fig. 4b(i), then 11 s later, each convection was reversed to flow in the opposite direction as presented in Fig. 4b(ii). The reason is considered as follows. Once internal convection is generated (Fig. 4b(i)) due to the minute asymmetry of the surface tension, the liquid flow shown in a yellow arrow bifurcates in the two directions along the boundary of the droplet surface, and moves to the root of the arrow. These two flows increase the surface tension at the root, and the flows are suddenly switched in the opposite direction (Fig. 4b(ii)).

These results suggest that these two exoskeletons could not control the self-propulsion of 1-pentanol droplets because the differences in surface tension were too small owing to their shape. To control the self-propulsion, it is necessary to obtain a greater difference in the surface tension of the droplet.

Experiments and analysis

Exoskeleton frames

Based on the preliminary experiment results, we determined the necessity of preparing exoskeletons that cause greater differences in the surface tension inside a droplet. We fabricated exoskeletons with elongated symmetrical (Fig. 5a) and asymmetrical (Fig. 5b) holes. The directions of self-propulsion relative to the exoskeleton are indicated by the arrows in the figure. The outer diameter of both exoskeletons was 12 mm. Droplets of 6 µL were dropped inside the symmetrical exoskeleton, and 5 µL inside the asymmetrical exoskeleton by considering the volume of the holes. The two exoskeletons were made by using cutting plotter, and the qualities of two exoskeletons were the same. We conducted three experiments to investigate the behavior of self-propulsion associated with Marangoni convection inside the droplet and in the aqueous phase.

Fig. 5

Exoskeleton frames with symmetrical and asymmetrical holes

Experiment 1: observation of behavior of the droplet

Experimental method

In Experiment 1, we observed the behavior of a droplet dropped into an exoskeleton five times for each exoskeleton. Under the experimental conditions listed in Table 1, the experiments were conducted in a system consisting of a Petri dish and a single-lens reflex camera (Canon, EOS 80D, 6000 × 4000 px, 29.97 fps), as shown in Fig. 6. After filling 7.0 mL of aqueous solution in the Petri dish, a 1-pentanol droplet was introduced into the exoskeleton, which floated on the aqueous phase using an automatic micropipette. These experiments were performed on an experimental desk placed horizontally. The self-propulsion behavior was captured using a camera. The movies were analyzed using Kinovea software to measure the behavior and velocity of the droplet.

Fig. 6

Experimental system in Experiment 1

Result and discussion

The behaviors of the 1-pentanol droplets in the symmetrical and asymmetrical exoskeletons are shown in Fig. 7. Figure 7 presents the moment when convections were generated.

Fig. 7

Behavior of a droplet

In the case of a symmetrical exoskeleton, we observed the droplet self-propelling in the lateral direction a few seconds after the droplet was dropped into the exoskeleton (Fig. 7a(i)). It was not determined whether the droplet self-propelled to the rightward or to the leftward direction. After it moved laterally, the droplet moved along the Petri dish wall. (Fig. 7a(ii)) Similar behavior was observed when an asymmetrical exoskeleton was used. However, the self-propelling direction of the droplet was always leftward (Fig. 7b(i)).

Figure 8 shows the time evolution of the velocity of the self-propelled droplets. The velocity reached a maximum of approximately 30 s for the symmetrical exoskeleton (Fig. 8a) and 35 s for the asymmetrical exoskeleton (Fig. 8b). We found that the velocity was maximum in both the symmetrical and asymmetrical exoskeletons when the droplet self-propelled in the lateral or leftward direction.

Fig. 8

Time evolution of the self-propelled velocity of the droplet

We calculated the average and standard deviation of the velocity of self-propulsion and the time required for the droplet to drop into the exoskeleton for the self-propulsion. In the case of the symmetrical exoskeleton, the average velocity of self-propulsion in the lateral direction was 10.4 mm s⁻¹, and the standard deviation was 6.81 mm s⁻¹. The average time from dropping to self-propulsion was 10.5 s, and the standard deviation was 6.16 s. When the asymmetrical exoskeleton was used, the average velocity of self-propulsion in the leftward direction was 3.14 mm s⁻¹, and the standard deviation was 0.87 mm s⁻¹. The average time from dropping to self-propulsion was 20.0 s, and the standard deviation was 8.07 s.

Experiment 2: observation of convection inside the droplet

Experimental method

In Experiment 2, we observed Marangoni convection inside a 1-pentanol droplet using PIV. The measurements were performed using the same system used in the preliminary experiment, as shown in Fig. 3 and Table 1.

Result and discussion

The Marangoni convection inside the 1-pentanol droplets dropped into the symmetrical and asymmetrical exoskeletons is shown in Fig. 9. Figure 9 shows the Marangoni convection observed in a droplet. In each figure (a) and (b), the picture on the left presents the observed convection, and the right schematic figure shows the definition against the exoskeleton. The experiment on the left and the direction relative to the exoskeleton on the right. When a symmetrical exoskeleton was used, strong convection inside the droplet was observed at the edges of the ellipse. (Fig. 9a) However, when an asymmetrical exoskeleton was used, convection was generated in the leftward direction relative to the exoskeleton. (Fig. 9b).

Fig. 9

Marangoni convection inside a droplet

The difference in the interfacial tension gradient, which depends on the droplet shape, causes the Marangoni convection inside the droplet. The force strength per unit length is $\gamma$, which is proportional to the concentration gradient of pentanol. In the case that a symmetrical exoskeleton is used, since the convection is generated from the center to the edge of the droplet, the force ${\gamma }_{\text{S}}$ generated in the droplet is derived as

$$\begin{array}{c}{\gamma }_{\text{S}}=a\left[\left|\left({\left.\frac{dc}{dr}\right|}_{\text{SE}}\right)\right|-\left|\left({\left.\frac{dc}{dr}\right|}_{\text{SC}}\right)\right|\right],\end{array}$$

(1)

where $a$ is a constant, $c$ is the 1-pentanol aqueous solution concentration, $r$ is the relative position, ${\left.dc/dr\right|}_{\text{SE}}$ is the concentration gradient at the edge, and ${\left.dc/dr\right|}_{\text{SC}}$ is the concentration gradient at the center. When an asymmetrical exoskeleton is used, since the convection is generated from the right edge to the left edge of the droplet, the force ${\gamma }_{\text{A}}$ generated in the droplet is given by

$$\begin{array}{c}{\gamma }_{\text{A}}=a\left[\left|\left({\left.\frac{dc}{dr}\right|}_{\text{AL}}\right)\right|-\left|\left({\left.\frac{dc}{dr}\right|}_{AR}\right)\right|\right],\end{array}$$

(2)

where ${\left.dc/dr\right|}_{\text{AL}}$ is the concentration gradient at the left edge, and ${\left.dc/dr\right|}_{\text{AR}}$ is the concentration gradient at the right edge.

The necessary condition to generate convection inside the droplet is that ${\gamma }_{\text{S}}$ and ${\gamma }_{\text{A}}$ have to be positive. Thus, for a symmetrical exoskeleton

$$\begin{array}{c}\left|\left({\left.\frac{dc}{dr}\right|}_{\text{SE}}\right)\right|>\left|\left({\left.\frac{dc}{dr}\right|}_{\text{SC}}\right)\right|,\end{array}$$

(3)

and in the case of the asymmetrical exoskeleton,

$$\begin{array}{c}\left|\left({\left.\frac{dc}{dr}\right|}_{\text{AL}}\right)\right|>\left|\left({\left.\frac{dc}{dr}\right|}_{\text{AR}}\right)\right|.\end{array}$$

(4)

When Eqs. (3) and (4) are satisfied, Marangoni convection is generated inside the droplet. Figure 10 shows the relationship between the concentration gradient and the Marangoni convection.

Fig. 10

Relationship between concentration gradient and Marangoni convection

Based on the results of the experiment with the symmetrical exoskeleton, we analyzed the velocity of Marangoni convection and self-propulsion. As shown in Fig. 11, the synchronization of the velocity of the Marangoni convection and self-propulsion was confirmed. We also found a positive correlation between the velocity of Marangoni convection and self-propulsion, as shown in Fig. 12.

Fig. 11

Time evolution of the velocity of Marangoni convection and self-propulsion

Fig. 12

Relationship between the velocity of Marangoni convection and self-propulsion

Experiment 3: observation of convection inside the aqueous phase

Experimental method

In Experiment 3, we observed Marangoni convection inside the aqueous phase using the PIV method. In this experiment, the glass experimental equipment shown in Fig. 13 was used instead of a Petri dish.

Fig. 13

Experimental equipment used in Experiment 3

First, the acrylic particles were mixed with a 1-pentanol droplet and a 1-pentanol solution. The measurements were performed using the system shown in Fig. 14 under the experimental conditions listed in Table 1. After injecting 7.0 mL of an aqueous solution mixed with acrylic particles into the glass experimental equipment shown in Fig. 13, a droplet mixed with acrylic particles was introduced inside the exoskeleton using an automatic micropipette that floated on the aqueous phase. While illuminating the aqueous phase with ultraviolet black light, the Marangoni convection inside the aqueous phase was captured using a high-speed camera. The movies were analyzed using FlowExpert2D2C software to visualize the internal convection.

Fig. 14

Experimental system in Experiment 3

Results and discussion

The convection observed inside the aqueous phase when 1-pentanol droplets were dropped inside the symmetrical and asymmetrical exoskeletons is shown in Fig. 15. In both the symmetrical and asymmetrical exoskeletons, two convections were observed on the left and right sides of the exoskeleton frames. For the asymmetrical exoskeleton, the convection generated on the right side was greater than that on the left side (Fig. 16b).

Fig. 15

Marangoni convection inside the aqueous phase

Fig. 16

Time evolution of the velocity of the convection on the right and left sides

Figure 16 shows the time evolution of the convection velocity on the right and left sides. In both exoskeletons, there was a time when convection velocity of one was higher than that of the other. When an asymmetrical exoskeleton was used, the convection velocity on the right side was higher than that on the left side (Fig. 16b), indicating that the differences in velocity were related to self-propulsion. The droplet self-propels because the convection on the faster side pushes it.

Table 2 compares the velocities of each convection at 24 s. We found that the velocities of the convection differed between cases when the symmetrical and asymmetrical exoskeletons were used. For the asymmetrical exoskeleton, the velocities were also different between the convections appearing on the left and right sides.

Table 2 Velocity of each convection at 24 s

Here, we discuss the cause of convection and the differences in the velocity of each type of convection when symmetrical and asymmetrical exoskeletons are used.

First, we consider the cause of convection. Convection inside the aqueous phase occur because the 1-pentanol droplets diffuse into the aqueous solution. Figure 17 shows convection in the aqueous phase. Diffusion causes convection away from the droplet in the aqueous phase. 1-pentanol is lighter than water (specific gravity = 0.82); therefore, convection is generated in the upper part of the aqueous phase. This causes convection inside the aqueous phase to return to the droplet. In the case of an asymmetrical exoskeleton, the right side of the droplet, which has a higher surface area in contact with the aqueous phase, diffuses more than the left side of the droplet. Therefore, convection on the right side is larger than that on the left side (Fig. 17b).

Fig. 17

Relationship between diffusion and convection generation

We further discuss the differences in the velocity of each convection when the symmetrical and asymmetrical exoskeletons are used.

From Table 2, the relationship between the velocities of each convection is

$$\begin{array}{c}{v}_{\text{S}}>{v}_{\text{AR}}>{v}_{\text{AL}},\end{array}$$

(5)

where ${v}_{\text{S}}$ is the convection velocity with a symmetrical exoskeleton, ${v}_{\text{AR}}$ is the velocity on the right side of the convection with a symmetrical exoskeleton, and ${v}_{\text{AL}}$ is the velocity on the left side of the convection with an asymmetrical exoskeleton. As shown in Fig. 17, because convection is caused by the diffusion of 1-pentanol droplets into the aqueous phase, the following relationship holds for the diffusion capacity per unit area and time, J:

$$\begin{array}{c}{J}_{\text{S}}>{J}_{\text{AR}}>{J}_{\text{AL}}.\end{array}$$

(6)

According to Fick’s first law of diffusion, diffusion capacity is proportional to the concentration gradient.

$$\begin{array}{c}J=-D\frac{dc}{dr},\end{array}$$

(7)

where $D$ denotes the diffusion coefficient. Thus, the relationship for the absolute value of the concentration gradient $|dc/dr|$ is:

$$\begin{array}{c}\left|\left({\left.\frac{dc}{dr}\right|}_{\text{S}}\right)\right|<\left|\left({\left.\frac{dc}{dr}\right|}_{\text{AR}}\right)\right|<\left|\left({\left.\frac{dc}{dr}\right|}_{\text{AL}}\right)\right|.\end{array}$$

(8)

This concentration gradient is caused by the droplet shape, and the relationship between the concentration gradients when an asymmetrical exoskeleton is used is consistent with the necessary conditions required for convection to be generated inside the droplet, as shown in Eq. (4).

Total analysis

The results of Experiment 1 show that the droplets self-propelled laterally when a symmetrical exoskeleton was used. The asymmetrical exoskeleton allowed the droplet to self-propel in the leftward direction.

These two convections are related to the self-propulsion of 1-pentanol droplets. The Marangoni convection was first generated locally inside the droplet because of the difference in surface tension. Convection occurred in the aqueous phase because of the diffusion of 1-pentanol. These convections were caused by the asymmetry of the exoskeleton.

Based on these results, we examine the equation of motion for the behavior of a droplet with an exoskeleton [34]. Three forces are related to the self-propulsion of a droplet: the driving force of the Marangoni convection inside the droplet, the force of convection in the aqueous phase pushing the droplet, and the inertial force. The equation of motion is derived as follows:

$$\begin{array}{c}{m}_{d}\frac{{d}^{2}{{\varvec{r}}}_{{\varvec{d}}}}{d{t}^{2}}={F}_{1}+{F}_{2}-\zeta \frac{d{{\varvec{r}}}_{{\varvec{d}}}}{dt},\end{array}$$

(9)

where ${m}_{d}$ is the mass of a droplet, ${{\varvec{r}}}_{{\varvec{d}}}$ is its position, and $\zeta$ is the friction coefficient. ${F}_{1}$ is the driving force of Marangoni convection inside the droplet, which coincides with ${\gamma }_{\text{S}}$ or ${\gamma }_{\text{A}}$ depending on the exoskeleton. ${F}_{2}$ is the force of convection in the aqueous phase pushing the droplet, which is caused by the interfacial tension field $\Gamma \left({\varvec{r}}, t\right)$. We assume

$$\begin{array}{c}\Gamma \left({\varvec{r}}, t\right)={\Gamma }_{0}-{\Gamma }_{p}\frac{c\left({\varvec{r}}, t\right)}{{c}_{p}},\end{array}$$

(10)

where ${\Gamma }_{0}$ is the interfacial tension of water, ${\Gamma }_{p}$ is the interfacial drops at saturation, and ${c}_{p}$ is the saturation concentration. Thus, ${F}_{2}$ is given by

$$\begin{array}{c}{F}_{2}={\int }_{{C}_{d}}\Gamma \left({\varvec{r}}, t\right)\frac{{\varvec{r}}-{{\varvec{r}}}_{{\varvec{d}}}}{\left|{\varvec{r}}-{{\varvec{r}}}_{{\varvec{d}}}\right|} dl,\end{array}$$

(11)

where integration is conducted along the perimeter of the droplet ${C}_{d}$. When the sum of these three forces is positive, the droplet is self-propelled.

Conclusions

In this study, we observed the self-propulsive behavior of 1-pentanol droplets dropped into exoskeletons, and analyzed the relationship between self-propulsion and internal and external convection. We prepared two types of exoskeletons: an elongated symmetrical and an asymmetrical exoskeleton.

When a droplet was dropped into the symmetrical exoskeleton, it self-propelled in the direction of the extended major axis of the ellipse. By contrast, when a droplet dropped into the asymmetrical exoskeleton, it self-propelled in a direction identical from the wider-hole area to the narrow-hole area.

The convection inside the droplets and in the outer aqueous phase was studied to determine the cause of self-propulsion. Inside the droplets, Marangoni convection was generated from lower to higher surface tensions because of the shape of the droplet. Two convections were generated in the aqueous phase because of diffusion of the droplet. We found that a larger convection pushed the droplet toward self-propulsion.

We will further investigate the optimal conditions for manipulating and controlling self-propelled droplets in future studies. The advantage of the present method is that self-propulsion is autonomously achieved using the chemical energy generated by the chemical reaction, and no external energy sources, such as electricity, are required. We believe that clarifying and controlling the mechanism of self-propelled droplets will enable their application in driving and controlling chemical-based actuators without supplying external energy.