AN OPTIMAL CURVE FOR FASTEST TRANSPROTATION OF LIQUID DROPS ON A SUPERHYDROPHOBIC SURFACE

Kwangseok Seo, Minyoung Kim, Do Hyun Kim Department of Chemical and Biomolecular Engineering, Korea Advanced Institute of Science and Technology, Daejeon, 305-701, South Korea [email protected]

ABSTRACT

A trajectory that will carry a particle from one place to another in the shortest time is called the brachistochrone curve. In 17th century, Johann Bernoulli posed the brachistochrone problem for the first time and discovered that the trajectory on a curve without friction is a cycloid. In most brachistochrone problems, a movable solid particle, especially a sphere, has been dealt with assuming its free movement. Generally, the transport of small liquid drops on a solid surface is not a simple process, because the nature of the contact between the two phases is complicated. However, liquid drops can easily roll off on a superhydrophobic surface. Thus, the drops can also be treated as an easily movable particle on the surface. In this study, the brachistochrone curves are discussed with water drops on a superhydrophobic surface. We expect that this study will be helpful for applications in droplet transport technology and lab-on-a-chip systems where discrete droplets are manipulated.

INTRODUCTION

A brachistochrone curve is a trajectory that will carry a particle from one place to another in the shortest time. The trajectory on a curve without friction was known to be a cycloid (1). In most brachistochrone problems, liquid drops have not been dealt with, because the nature of the contact between the two phases, a solid surface and a liquid drop, causes complicated wetting problems.

A superhydrophobic surface has properties of extreme water repellency and low contact angle hysteresis. These properties originate from having an immensely small area of the solid surface actually coming into contact with water (2). A water droplet can easily roll off on the surface with a slight slope without any liquid film left behind. Thus, the drops can also be treated as an easily movable particle on the surface.

There have been prolific researches on the dynamic behaviors of liquid drops on a superhydrophobic surface (3-5). Usually, a small water droplet of a few millimeters shows accelerated motion on a superhydrophobic surface shorter than 1 m (5). Thus, it is meaningful to verify whether the water drop on a cycloid path can also run down faster than on any other paths like a line path. Here, to the best of our knowledge, brachistochrone problem of a water droplet on superhydrophobic surfaces is studied for the first time.

THEORETICAL REVIEW

The classical brachistochrone problem is to find the shape of the path so that a solid 4-35 particle travels between prescribed points or distance in the least time under a uniform gravitational field.

For simplicity, it is assumed that the original point coincides with the origin of coordinates so that and that the initial velocity of the particle is zero. When the particle move along the curve, the infinitesimal travel time is as follows

Thus, the total time taken to travel a whole path is given by the integral.

When there is no friction (pure acceleration), the speed of the particle due to gravity is as follows

where the acceleration of gravity g is 9.8 m s-2.

The total time is rewritten as

We can find function y that makes T have an extreme value by applying the Euler equation

where

The equation can be rearranged as,

Finally, the problem becomes the following non-linear differential equation,

where

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The general solution of the corresponding Euler equation consists of a family of cycloids.

Since the curve must pass through the origin, k3 is zero. To determine k2, we use the second condition from the boundary condition at the other end.

Figure 1. (a) In a fixed end point, the end point of the curve is initially fixed. (b) In a variable end point, the end point of the curve are free to move along a given vertical line .

We consider two types of boundary conditions, a fixed end point and a variable end point. Figure 1(a) shows the fixed end point problem. In this case, the value of the arbitrary constants k2 is determined by the condition . Figure 1(b) shows the variable end point problem. The end point of the admissible curve are free to move along a given vertical line . To determine the values of the arbitrary constants, we use a natural boundary condition, (1). In this case, it is possible to obtain a curve that make the particle transport to a place at a horizontal distance away from the origin in the least time.

EXPERIMENTAL

1. Fabrication of a superhydrophobic ramp

A superhydrophobic ramp was basically made of polydimethylsiloxane (PDMS). The front view of a ramp is presented in Fig. 2. Curvature was determined assuming the surface is a part of a pipe of a radius of 3 cm, to hold water drops on the ramp. PDMS was poured over the mold with the dimension of 2.5 cm in width, 0.3 cm in height, and 100 cm in length. Next, the pipe was placed on the mold filled with PDMS and solidified for three days at room temperature (20 ˚C). Finally, the solidified PDMS was coated several times with a commercial superhydrophobic spray (NeverWet, Rust-Oleum) to make a superhydrophobic surface over a large area without defects.

Figure 2. Scheme for the fabrication of a superhydrophobic ramp. 4-37

2. Water drop behavior on a linear path according to different slope angles.

In order to investigate the behavior of a water drop with respect to slope angles, droplets of 15 μL were rolled off at various slope angles from 5° to 75°. To avoid unwanted initial velocity of a drop, a pipette was not used to dispense drops. Instead, a superhydrophobic blade was used to release the drops, minimizing possible effects of an initial velocity that can affect the droplet motion. The positions of drops were recorded with a high speed camera (Casio® EX-FC100). The accelerations of moving drops were measured using their positions in the captured images. The droplets became elongated at the terminal stage of the travel. Therefore, the center of the droplet was used for the position and the position was measured by repeated runs at least 5 times for each inclined angle.

3. Comparison of drop behaviors on line and cycloid paths.

In order to create cycloid path, the styrofoam was cut along the path line and the flexible PDMS ramp with the superhydrophobic surface was attached on the tailored styrofoam. The time taken for the traveling of a drop was measured at least 7 times. Using superhydrophobic blades, both droplets on linear and cycloid path could be simultaneously released.

RESULTS AND DISCUSSION

Figure 3(a) shows a scanning electron microscope (SEM) image of the superhydrophobic coating. The coating consists of very small spherical particles of tens of nanometer in size. These nanoparitcles had coagulated and formed irregular agglomerates. Thus, the surface was rough enough to create a superhydrophobic surface. The static contact angle of a water drop on a superhydrophobic surface was about 163°, as shown in Fig. 3(b). The contact angle hysteresis was about 3° (advancing angle : 165°, receding angle 162°). The roll-off angles of a water drop (10 μL) were less than 2°.

Figure 3. (a) SEM images of the superhydrophobic coating, and (b) Static contact angle on the superhydrophobic coating

Figure 4 shows the position versus time relationship when a 15uL water droplet is dropped at different slope angles. Over approximate distance of 80 cm, the water droplet appears to show accelerated motion. The position-time profiles of droplet motion obtained at different slope angles are superimposed in Fig. 5. It can be observed that the droplet speed gradually increases with the inclined angle.

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Figure 6 shows the tendency of the observed droplet accelerations depending on the slope angles. It can be shown that the acceleration varies linearly with the degree of inclination. By data fitting, the observed acceleration was calculated and a value of about 7.696 m s-2 was obtained. This is lower compared to the value for the gravitational acceleration of 9.8 m s-2 and there are two possible reasons for this discrepancy. First, the inertial moment for the case of a rolling drop was neglected. On a superhydrophobic surface, the water droplet initially moves in a sliding manner and not by rolling motion (4). Thus, it can be considered that the acceleration during the initial motion is not affected by inertial moment. However, as the velocity gradually increases, the droplet moves both by rolling and sliding motion (5). Therefore, the effects due to inertial moment at the later stage could explain the lower acceleration actually observed.

The other factor that was not taken into account is air resistance and viscous dissipation. We have used a superhydrophobic surface 80 cm long. Although plain accelerated motion is observed when the initial droplet speed is small, acceleration is not maintained due to air resistance and viscous dissipation between the droplet and the surface, so that the droplet speed approaches a certain terminal velocity. Due to these reasons, the actual observed acceleration was measured to be lower than the gravitational acceleration.

Regardless of the actual physical phenomena occurring, the observed acceleration was almost proportional to the inclination. Therefore, the optimum path for the brachistochrone problem of a water drop can be expected to be a cycloid as previously derived.

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Figure 4. Motion of the water drops at 9 different slope angles was captured and the position of the center of the droplet is reported as a function of time.

Figure 5. The superimposed position vs time profiles of water drops obtained at different slope angles.

To compare the speed of a drop on a cycloid path and a line path, two types of the paths were prepared. Figure 7(a) shows the cycloid curve which a drop can move along from the origin to the fixed end point (50, 20) in the least time. This curve will henceforth be called as cycloid path A. Figure 7(b) shows the cycloid curve which a particle can move along to a place at a distance from the origin in the least time. The end point, (50, 31.83), could be obtained from the natural boundary condition. This curve will be called as cycloid path B. For comparison, straight line paths connecting the origin and the end point of each cycloid path were prepared. These line paths will be called as line path A and line path B, respectively.

Figure 6. The tendency of the apparent droplet acceleration with respect to the slope angle.

The travelling times taken when the water drop move along a cycloid path can be calculated as,

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The above equation can be rearranged as

Likewise, considering a line path is expressed by the linear equation, , thus, , the travel times on a linear path can be calculated as,

Integration of the above equation gives rise to

Therefore, a ratio of the travel times of the line path and the cycloid path is given as follows,

It can be noted that the ratio of the travel times does not depend on the acceleration of gravity.

Figure 7. (a) The cycloid curve (cycloid path A) where a water drop can move down from the origin to the point (50, 20) in the least time and the straight line path (line path A), and (b) The cycloid curve (cycloid path B) where a water drop can move down to a place away from the origin in the least time and the straight line path (line path B). 4-41

Table 1. The travel times of the line path A and the cycloid path A and the ratio of the times. Theoretical value (ms) Measured value (ms) Trevel time on line path A 544* 610 ± 10 Trevel time on cycloid path A 409* 522 ± 25 Ratio of TL /TC 1.33 1.17 * The acceleration of gravity of 9.8 m s-2 was used for calculation.

Table 2. The travel times of the line path B and the cycloid path B and the ratio of the times. Theoretical value (ms)* Measured value (ms) Trevel time on line path B 475* 498±5 Trevel time on cycloid path B 400* 443±8 Ratio of TL /TC 1.19 1.12 * The acceleration of gravity of 9.8 m s-2 was used for calculation.

As shown in Table 1 and Table 2, water drops on cycloid paths have shorter travel times than on line paths. Also, it was verified that the cycloid path B among four paths makes water drops transport to a place horizontally away 50 cm from the origin in the least time.

In the case of line path A and cycloid path A, the ratio of the theoretical travel time was 1.33. However, the ratio of the measured travel time was 1.17. In the case of line path B and cycloid path B, the ratios of the theoretical travel time and the measured travel time were 1.19 and 1.12, respectively. It means that the difference between the measured and theoretical travel times is longer for the cycloid paths than the line paths.

It can be explained as follows. Because water drops on the cycloid path are more quickly accelerated than on the line paths, the drops will experience strong viscous dissipation and air resistance from the early stage of the travel. So, it might cause a difference with the theoretical values of which the friction factors were not considered during calculation.

The greater the normal force of a water drop acting on the bottom, the larger viscous dissipation near the surface will be induced. Unlike the line paths, when a water drop moves along the cycloid paths, the viscous dissipation is greatly increased due to centrifugal force. Thus, an additional energy loss occurs in a cycloid path, compared to a line path. The greater the curvature, the larger the energy loss will be induced. It should be noted that the difference between the measured and theoretical travel times in cycloid path A is larger than in the cycloid path B.

CONCLUSION 4-42

A water droplet of 15 μL on the superhydrophobic surface showed accelerated motion within the travel length shorter than 1 m. The acceleration was observed to be nearly proportional to the degree of inclination. The trajectory of a solid particle on a curve without friction traveling in the least time is known to be a cycloid. For the first time, we experimentally demonstrated that when a water drop on a superhydrophobic surface move along a cycloid path, a water drop travels between prescribed points or distance in the least time. We expect that this study will be helpful in designing the shapes of solar cell panels for quickly eliminating rain drops and for applications in droplet transport technology

REFERENCES

1. Gelfand I N, Fomin S V: Calculus of Variations. New York, Dover, 2000. 2. Seo K, Kim M, Kim D H: ‘Candle-based process for creating a stable superhydrophobic surface’. Carbon 2014 68 583-96. 3. Hao P, Lv C, Yao Z, He F: ‘Sliding behavior of water droplet on superhydrophobic surface’. Europhysics Letter 2010 90 (6) 66003. 4. Sakai M, Song J H, Yoshida N, Suzuki S, Kameshima Y, Nakajima A: ‘Direct observation of internal fluidity in a water droplet during sliding on hydrophobic surfaces’. Langmuir 2006 22 (11) 4906-4909. 5. Faraday Discuss., 2010, 146, 19–33 Mathilde Reyssat M, Richard D, Clanet C, Quéré D: ‘Dynamical superhydrophobicity’. Faraday Discussions 2010 146 19-33.

ACKNOWLEDGEMENTS

This research was supported by Basic Science Research Program through a National Research Foundation of Korea (NRF) grant funded by the Ministry of Education, Science, and Technology (2012R1A2A2A01047371).

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