Article pubs.acs.org/JPCB
Mode Pattern of Internal Flow in a Water Droplet on a Vibrating Hydrophobic Surface Hun Kim and Hee-Chang Lim* School of Mechanical Engineering, Pusan National University, San 30, Jangjeon-Dong, Geumjeong-Gu, Busan 609-735, South Korea S Supporting Information *
ABSTRACT: The objective of this study is to understand the mode pattern of the internal flow in a water droplet placed on a hydrophobic surface that periodically and vertically vibrates. As a result, a water droplet on a vibrating hydrophobic surface has a typical shape that depends on each resonance mode, and, additionally, we observed a diversified lobe size and internal flows in the water droplet. The size of each lobe at the resonance frequency was relatively greater than that at the neighboring frequencies, and the internal flow of the nth order mode was also observed in the flow visualization. In general, large symmetrical flow streams were generated along the vertical axis in each mode, with a large circulating movement from the bottom to the top, and then to the triple contact line along the droplet surface. In contrast, modes 2 and 4 generated a Y-shaped flow pattern, in which the flow moved to the node point in the lower part of the droplet, but modes 6 and 8 had similar patterns, with only a little difference. In addition, as a result of the PIV measurement, while the flow velocity of mode 4 was faster than that of model 2, those of modes 6 and 8 were almost similar.
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INTRODUCTION There has been much research interest in water drop behavior, particularly for its oscillation and evaporation, which have been demonstrated by many researchers because it is an interesting issue in terms of its basic properties and application in chemical and mechanical engineering. Early research on the vibration of a water droplet when it is surrounded by a different fluid began with studies by Kelvin1 and Rayleigh2 on water droplet behavior caused by resonance. This research was undertaken on a free water droplet, in which the viscosity could be neglected and where the amplitude is marginal. Later, Strani and Sabetta3 conducted an analysis of a nonviscous water droplet in contact with a solid surface, showing that its vibration has marginal amplitude. Subsequently, Daniel, Chaudhury, and Brunet conducted research4−10 on the motion of a water droplet that moves in only one direction on sloped surfaces and the vibration of a water droplet that moves in a vertical or horizontal direction as well as the change in its motion features. Research11−13 using the electro-wetting phenomenon, in which electric energy is applied, is being actively carried out. In particular, electro wetting is applied and used on micro systems, such as lab-on-a-chip, electro display, and liquid lens systems, due to its fast response and low power consumption. The water drop vibration generated by a periodic force is able to overcome micro-/nanoscale problems, such as the contact line pinning, chemical synthesis in solution, and the self-aggregation placed on a solid surface, and these advantages enable water droplets driven by forced vibration to be applied in the industrial sites as well. Applications of water droplet vibration include humidifiers (an air supply/exhaust system); heating, ventilation, and air © 2015 American Chemical Society
conditioning (HVAC); and diversified applications for industrial sites to help achieve the maximum heat transfer efficiency. For this reason, research14−16 on not only the water droplet hung on the upper part of the wall, which is attached to the lateral wall,8,17 but also the measurement18,19 of its surface tension and contact angle is being actively carried out. Furthermore, studies have performed not only experiments on water droplet behavior20 under simple resonance but also complex water droplet experiments, such as an experiment involving the evaporation of a water droplet hung in acoustic resonance and an experiment on the behavior of a water droplet placed on a heated plane plate. Generally, when a water droplet is evaporated or heated, a thermocapillary flow (Marangoni flow) occurs, and this flow significantly influences the sediment pattern.21,22 Research on the application of this phenomenon to various mechanical industries, such as porous thin film patterning, spray painting, and thin film deposition, is in progress. However, most of the research on internal flow was based on the Marangoni flow23−25 using natural evaporation and heating, and research on the internal flow of a vibrating water droplet was attempted using only numerical analysis.26 Such research was not performed in much detail, as evident from the scarce experimental results available at present. Therefore, the objective of this study was to confirm whether the actual flow exists inside the water droplet by applying a resonance to a water droplet placed on a hydrophobic surface. Received: March 28, 2015 Revised: May 8, 2015 Published: May 18, 2015 6740
DOI: 10.1021/acs.jpcb.5b02975 J. Phys. Chem. B 2015, 119, 6740−6746
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(HIFAX32C20-8, HiWave). A function generator (function generator, 33522A, Agilent) was used to supply a sinusoidal wave with a range of frequencies and voltages to the system. In addition, the laser used for the experiment was a CW laser (Excel 532-2000) with a wavelength of 532 nm. The internal flow was observed by penetrating the dead center of the water droplet by making a thin laser sheet by using a cylindrical lens after expanding the light emitted from the laser by a laser beam expander. The thickness of the laser sheet was about 0.2 mm, and, for visualization, a polystyrene nile red fluorescent sized of about 2 m was used. Furthermore, by using two sets of FASTCAM SA3, a Photron camera, photographs of the vibration cycle of the top lobe of the water droplet (camera 1) and its SI wafer substrate (camera 2) were captured with a shutter speed of 2000−4000 frames/s, and then the velocity value was calculated by using the commercial software MATLAB, Insight 4G, and Tecplot. A camera lens, zoom lens (Canon Macro 100 mm), and supermacro lens (Canon MP-E 65 mm f/2.8) were used. In this study, the flow pattern inside the water droplet and its velocity field, which depends on four types of water droplet modes (i.e., 2, 4, 6, and 8), were identified, and to apply heat and the substance transfer of the water droplet being induced, thereby the following experiment process was performed. First, a water droplet of 5 μL was generated on the surface of the silicon wafer by using a micropipette. As the evaporation of the water droplet was taking place due to the difference of external temperature and humidity, the temperature and the humidity were maintained by using a chamber. After the voltage was fixed to 2 V (modes 2, 4) and 10 V (modes 6, 8) by using function generator (33522A, Agilent), it was matched with a resonance frequency value obtained through a theoretical formula. While adjusting the frequency, the resonance frequency of the actual water droplet (its vibrating shape) was analyzed by using a live mode of a super high speed camera. The vibration cycle then was compared by simultaneously taking photos of the top lobe vibration cycle of the water droplet and the vibration cycle of the substrate by using two sets of a high speed camera based on 2000−4000 frames/s. Finally, the images obtained by the repetitive measurements were processed to make the precise mode shape, the internal flow pattern inside water droplet, and the internal velocity.
This fundamental study is aimed at understanding the applicability of such water droplets in diversified fields, such as the process, handling, and heat transfer using a water droplet, and to identify the mode pattern and the features of the internal flow inside a water droplet. To achieve this objective, the flow inside a water droplet was visualized by applying a forced vibration to the surface of the hemispherical water droplet having hydrophobic features, and its velocity field was investigated.
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EXPERIMENTAL DEVICES AND METHOD Surface Treatment. In general, the larger the droplet’s contact angle becomes, the higher the driving force required to stimulate the internal velocity would be, whereas the higher the homogeneity and the lower the roughness of the surface are, the smaller is the contact angle hysteresis.27 Therefore, to observe the shape and the internal flow in detail, Teflon, a low surface energy material, should be spin-coated onto the surface. In this experiment, to fabricate a silicon wafer that is a hydrophilic surface that acts as a hydrophobic surface, a Teflon solvent of concentration 0.6 wt % was prepared by dissolving Teflon AF (601S2-100-6, Dupont) in a fluorocarbon solvent (FC-40, 3 M). First, to remove the impurities and dust on the surface of the silicon wafer, the Piranha cleaning process was performed in a clean room. Afterward, the silicon wafer surface was spin-coated for around 5 s at 500 rpm, and then for 30 s at 2000 rpm, respectively. To strengthen the adhesion of the Teflon thin film, it was baked for 60 min on a hot plate of 165 °C. As a result, the coating thickness of about 100 nm was produced. Deionized water was used as the fluid; the size of the water droplet was 5 μL, and the equilibrium contact angle of the water droplet was determined to be 115° ± 1° by using a contact angle analyzer (CAM 100, KSV) after measuring it a total of 5 times. Furthermore, to reduce the experiment error, the temperature and the humidity were maintained constant by using a chamber, and, at this time, the experimental conditions were 25 ± 1 °C and 45% ± 5%, respectively. Vibration and Visualization Experiment. To observe the characteristics of the vertical vibration and to visualize the internal flow of the water droplet, a PIV (particle image velocimetry) measurement was performed, as shown in Figure 1. To observe the internal flow pattern inside the water droplet that was placed vertically in a periodic forced vibration, the spin-coated silicon wafer, which was made earlier, was chemically bonded with the front surface of a speaker
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MATHEMATICAL MODEL Velocity on the Droplet Surface. The droplet is placed on a plate and oscillated while maintaining a hemispherical axissymmetric shape. The flow inside a droplet is assumed to be inviscid, incompressible, and the axis-symmetrical shape under a vertical free oscillation. The equation for nth order resonance frequency is based on the governing equations (i.e., equation of continuity and Laplace equation) in the spherical coordinate system (r,θ,φ)28 (for the diagram of the droplet in detail, see Figure 2). The equation of continuity can be expressed as ∂ρ + ∇·(ρ u) = 0 ∂t
(1)
where u is the velocity of the fluid element at position x(r,θ,φ) and time t. The Laplace equation for the velocity potential ϕ of the fluid element is given by
Figure 1. Schematic diagram of the experimental setup. 6741
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Oh et al.12) Therefore, the governing equation for this interfacial phenomenon can be expressed as follows: rs = R + εPn
(5)
To obtain the surface pressure on the droplet, we need another boundary condition, the kinematic boundary condition. Depending on the sign of the perturbation part of the droplet, that is, if the movement is toward the inside (r < R) or the outside (r > R) of the droplet, the velocity potential can be expressed as follows: ⎛ R2Ṙ ⎞ n ϕ = ϕ1 = ⎜ ⎟ + A nr Pn r ⎠ ⎝
⎛ R2Ṙ ⎞ ⎛ Pn ⎞ ϕ = ϕ2 = ⎜ ⎟ + Bn⎜ n + 1 ⎟ ⎝r ⎠ ⎝ r ⎠
Figure 2. Descriptive diagram of a droplet in a fluid medium on a wetted substrate.
⎛ R2Ṙ ⎞ ⎛ rn Ṙ ⎞ P ε ̇ + 2ε ⎟ ϕ = ϕ1 = ⎜ ⎟+ n − 1 n⎜ ⎝ r R⎠ ⎠ nR ⎝
(8)
⎛ R2Ṙ ⎞ ⎛ Rn + 2 Ṙ ⎞ P ε ̇ + 2ε ⎟ ϕ = ϕ2 = ⎜ ⎟+ n − 1 n⎜ ⎝ R⎠ ⎝ r ⎠ (n + 1)r
(9)
Using the Bernoulli integral, the pressures at the inner and outer surfaces can be obtained as
(3)
⎡⎛ ∂ϕ ⎞ 1 ⎤ p1 = f1 (t ) + ρ1⎢⎜ 1 ⎟ − |∇ϕ1|2 ⎥ 2 ⎣⎝ ∂t ⎠ ⎦
(10)
⎡⎛ ∂ϕ ⎞ 1 ⎤ p2 = f2 (t ) + ρ2 ⎢⎜ 2 ⎟ − |∇ϕ2|2 ⎥ 2 ⎣⎝ ∂t ⎠ ⎦
(11)
where f n(t) is an arbitrary function of time. Therefore, the pressure difference at the droplet surface can be expressed as
If there is no vaporization and condensation at the interface between the droplet and the surrounding, the surface velocity at point r in the droplet can be expressed as R2 =ϕ r2
(toward the outside)
where An and Bn are the unknown constants that can be obtained by substituting eq 5 into eqs 6 and 7. In the result, the velocity potential can be expressed as
where θ is the polar angle measured down from the north pole, and φ is the azimuthal angle, analogous to longitude in earth measuring coordinate. Therefore, ((∂ϕ)/(∂r)), ((∂ϕ)/(∂θ)), and ((∂ϕ)/(∂φ)) are the velocities in the radial, azimuthal, and polar directions, respectively. Because fluid incompressibility and axis-symmetry are assumed, the equations can be substantially simplified. The continuity equation becomes ∇·u = 0 and the ((∂ϕ)/(∂θ)) and ((∂ϕ)/(∂φ)) velocities vanish so that the Laplace equation only depends on the radial direction. Therefore, the Laplace equation in the spherical coordinate system can be expressed as
Ṙ
(6)
(7)
∂ϕ ⎞ 1 ∂ ⎛ ∂ϕ ⎞ 1 ∂ ⎛ ⎜sin θ ⎟ ∇·u = ∇ ϕ = 2 ⎜r 2 ⎟ + 2 ∂θ ⎠ r ∂r ⎝ ∂r ⎠ r sin θ ∂θ ⎝ ⎛ ∂ 2ϕ ⎞ 1 + 2 2 ⎜ 2⎟ r sin θ ⎝ ∂φ ⎠ (2) 2
1 ∂ ⎛ 2 ∂ϕ ⎞ ⎜r ⎟ = 0 r 2 ∂r ⎝ ∂r ⎠
(toward the center)
⎛ 3 ⎞ p2 − p1 = f2 (t ) − f1 (t ) + (ρ2 − ρ1)⎜RR̈ + Ṙ 2⎟ ⎝ 2 ⎠ ⎡ R̈ +⎢ (ρ (n + 1)(n + 2) − ρ2 n(n − 1))ε ⎣ n(n + 1) 1 R 3Ṙ + (ρ1(n + 1) + ρ2 n)ε ̇ + n(n + 1) n(n + 1) ⎤ (ρ1(n + 1) + ρ2 n)ε ⎥̈ Pn ⎦
(4)
where R and Ṙ are a radius and the velocity at the radius, respectively. Pressure on the Droplet Surface. When a droplet oscillates with a small amplitude, the surface distortion of the droplet can grow or diminish (r < R, r > R) as shown in Figure 3. In addition, Figure 3 delineates a descriptive diagram of a droplet shape assuming that the droplet has a unique shape placed under a mode oscillation, in particular, modes 2−8. (For the pattern study of droplet oscillation in electro-wetting, see
(12)
The Laplace pressure is the pressure difference between the inside and outside of a curved surface, and it can be determined from the Young−Laplace equation as below. ⎛1 1 ⎞ △p = pinside − poutside = p1 − p2 = σ ⎜ + ⎟ R2 ⎠ ⎝ R1 =
(n − 1)(n + 1) 2σ + σεPn R R2
(13)
where R1 and R2 are the radii of curvature and σ is surface tension. If the droplet remains stationary and the radii of curvature are same, the Laplace pressure becomes 2σ/R.
Figure 3. Shape characteristics of a droplet at modes 2−8. 6742
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particularly important for verification. However, approximately 10−21% of the discrepancy occurred between the theoretical and experimental values for various reasons. The discrepancy tends to occur due to an effect of the dynamic features and the surrounding environment of a water droplet, and the contact friction appearing between a solid surface and the droplet is caused by the motion status of the contact line, where the water droplet, the solid surface, and three gas phases are simultaneously in contact. Here, the resonance frequency value of the nth mode could be accurately found when the largest peak of the lobe amplitude was experimentally observed around the theoretical resonance frequency, as shown in Figure 4. In addition, the experimental resonance frequency value of
However, when the droplet oscillates, the perturbation term is added to 2σ/R as shown in eq 13. Theoretical Frequency of Resonance in an Oscillating Droplet. Equations 12 and 13 represent the pressure difference on the droplet surface. Combining both equations, we obtain the following expression, in terms of the perturbation (ε): ε̈ +
⎡⎛ 3Ṙ ε ̇ + ⎢⎜R̈ρ1(n + 1)(n + 2) − ρ2 n(n − 1) ⎣⎝ R ⎞⎤ (n − 1)(n + 2)σ ⎞ ⎛ + ⎟ ⎜Rρ1(n + 1) + ρ2 n⎟⎥ 2 ⎠ ⎝ ⎠⎦ R ·n(n + 1)ε = 0
(14)
It is a second-order linear homogeneous ordinary differential equation. The solution of harmonic function (ωn) can be arranged as follows: ⎡⎛ ωn2 = ⎢⎜R̈ρ1(n + 1)(n + 2) − ρ2 n(n − 1) ⎣⎝ +
⎞⎤ (n − 1)(n + 2)σ ⎞ ⎛ ⎟ ⎜Rρ1(n + 1) + ρ2 n⎟⎥ ·n(n + 1) 2 ⎠ ⎝ ⎠⎦ R (15)
=0
If ρ2 is zero, the resonance frequency of nth order becomes as follows: fn =
ωn = 2π
n(n − 1)(n + 2)σ 4π 2ρR3
Figure 4. Frequency-dependent oscillation magnitude of the lobe displacement.
(16)
In eq 16, when the radius of the droplet becomes bigger and the mode frequency smaller, the resonance frequency of the droplet becomes smaller.
the nth mode is a value obtained by measuring the area where the lobe amplitude is the largest in the peripheral frequency domain of the theoretical resonance frequency, as shown in Figure 4. The maximum change of the lobe size was measured to be 0.22 and 0.15 mm at the second and fourth resonance modes, respectively. However, for the sixth and eighth, the variation of the lobe size was tiny at the applied voltage of 2 V. Therefore, the applied voltage was further increased from 2 to 10 V. As a result, the maximum change of the lobe size was measured to be 0.06 and 0.047 mm at the sixth and eighth resonance modes, respectively. Although the supply voltage was increased to be about 5 times higher (i.e., 10 V), the change of the upper lobe size at the sixth and eighth resonance modes was still less than for the second and fourth resonance modes at 2 V. From this result, it could be noted that the lobe size of the water droplet in the resonance frequency was one of most crucial parameters to observe, and at the other frequency range, the lobe size was substantially reduced. Internal Flow Features by Each Mode of the Water Droplet. In this study, the experiment was performed under 2 V (modes 2, 4) and 10 V (modes 6, 8), and, by using flow visualization, the flow pattern change inside the water droplet placed on a vibrating hydrophobic surface was confirmed. However, the reason the input voltages were different from each other in modes 2, 4 and 6, 8 was that the shape corresponding to each mode could not be distinguished even when using a high-speed camera. The change in the lobe size of the water droplet corresponding to modes 6, 8 under a relatively low voltage of 2 V was marginal. Additionally, even though the water droplet detachment and its separation from the silicon wafer surface were not our focus, they were also
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EXPERIMENT RESULTS AND DISCUSSION Comparison of Theoretical and Experimental Resonance Frequency. The theoretical resonance frequency formula being used in this study was deduced by developing a study on the resonance vibration of noncompressive, nonviscous water droplets by the forced vibration of Rayleigh and the velocity potential of a water droplet surface in the amplitude area, where a free water droplet was low and used Lamb’s28 theoretical resonance frequency formula, which satisfies both the Laplace equation and the boundary condition equation. In formula 16, f n is a resonance frequency corresponding to the shape of the oscillation in the nth mode, σ is surface tension, R is radius of water droplet, and ρ is the density of water droplet. The surface tension and the density of the water being used in this experiment are 0.07197 N/m and 997 kg/m3, respectively. Table 1 presents a comparison of the theoretical and experimental values of the resonance frequencies of each mode; this information is Table 1. Comparison of Theoretical and Experimental Results for Resonance Frequency of a 5 μL Droplet mode no.
resonance freq [Hz] (theory)
resonance freq [Hz] (experiment)
difference [%]
2 4 6 8
96 288 526 804
85 226 469 635
14.4 21.5 10.8 21.0 6743
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and the phase at the first and second cycles are identical; therefore, the velocity field obtained from the PIV measurement is confirmed to be valid. In addition, when observing the sine curve showing a resonance cycle of substrate in Figure 6, it could be confirmed that even though the resonance displacement of the substrate was not proportional to the lobe size, the vibration cycle and the size of the substrate affected the flow velocity inside the water droplet. When observing Figure 7, that is, the flow pattern image inside the water droplet being obtained through exposure of DSLR for 2 s, in each mode, bisymmetrical flow was commonly represented together with a vortex based on the central axis of the water droplet flow, but it could be seen that as the mode was increased, diversified flow patterns were represented. As the water droplet ascended from down to up, the Y-shaped flow pattern flowing to node point was taking place in modes 2 and 4, and the big vortex was taking place in modes 6 and 8. Here, the reason for the flow pattern being represented bisymmetrically was that the number and the size of the lobe that determine the shape of the water droplet were identical on both sides of the central axis of the water droplet. In addition, it could also be assumed that if the point of generation of the lobe is toward one side, the flow pattern would not be symmetric. Figure 7a shows an image of mode 2 and its Y-shaped flow pattern that flows to node point where the lobe is duplicated after ascending to about 2/3 point from down to up based on the central axis of the water droplet. In addition, large symmetrical flow streams along the vertical axis were generated in each mode, and they had a large circulating movement going from bottom to top, and then to the triple contact line along the droplet surface. Figure 7b shows an image of mode 4, in which the vortex is generated on the basis of two additional lobe centers as the axis, and it shows the flow being a little longer than mode 2 and the Y-shaped flow pattern flowing to each lobe is represented. Also, when observing the vortex being generated in the lobe of modes 2 and 4, it could be realized that
taking place because the energy of the modes 2, 4 at 10 V was that strong. Therefore, to understand the internal flow features for each mode of the water droplet, which is the goal of this study, an experiment was performed as detailed above. Figure 5 shows the variation of the resonance frequency (i.e., modes 2, 4, 6, and 8) obtained by using a high-speed camera.
Figure 5. Shape oscillation patterns that are obtained by high-speed camera under 2 and 10 V at the mode numbers.
As the mode number increases, it was observed that the number of lobes also increases, but the size gets smaller gradually. In addition, to have a proper identification of the number of lobes, the node points were also highlighted by the arrows in the figure. Therefore, the higher mode numbers have more arrows and more lobes, as shown in the figure. As a result, it can be shown that the droplet shape under the mechanical vibration is almost identical to those with free vibration. Figure 6 shows a graph measuring the vertical displacement of the top lobe and the Si-wafer substrate in four resonance modes when a forced vibration was applied to the hydrophobic surface. The abscissa in Figure 6 shows the elapsed time and the ordinate vertical displacement of the droplet and the substrate, respectively. As the mode number increases, the oscillation period of the lobe gradually becomes faster, about 12.5, 5, 2.25, and 1.5 ms. It could be realized that in all of the modes, the displacement of the lobe and substrate, depending on time, shows a constant cycle. When observing the images from the high-speed camera, the vibrating substrate location
Figure 6. Oscillations of the droplet and substrate height (h) for 5 μL droplets at the resonance modes 2 (a), 4 (b), 6 (c), and 8 (d), respectively. 6744
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Table 2. Averaged Vertical Velocity at the Central Bottom Region of the Droplet in Modes 2, 4, 6, and 8 mode no.
velocity [mm/s]
error [%]
2 4 6 8
0.36 0.79 3.87 3.61
1.35 2.5 1.37 1.29
Figure 8. Distribution of the velocity vector of the internal flow field in the vibrating water droplet.
Figure 7. Visualized flow pattern inside a droplet on the vibrating hydrophobic surface. The exposure time is 2 s.
flow direction is the same, except for the different size and generation location only. Figure 7c and d shows the images of modes 6 and 8, and each big elliptical vortex is created at both sides based on the centerline of the water droplet, which is slightly different from the previous modes (2 and 4). These results highlight the fact that as the number of lobe increases, the size of lobe decreases accordingly, and hence these lobes become smaller. This means that the wobbly surface of a droplet at modes 2 and 4 gradually becomes smoother at modes 6 and 8, and we observed this fact several times, so that this phenomenon seems to generate a large circulating vortex inside the droplet rather than the Y-shape. The internal velocity value of the water droplet being obtained through the visualization experiment was calculated by using Insight 4G after confirming that the vibration cycle of the lobe and that of the substrate of the water droplet are constant in each mode, as shown in the previously mentioned Figure 6. However, due to the water droplet distortion by shape vibration, the expansion or the sagging of the particles is taking place, and so, by taking photos in a way of magnifying high-speed camera frame from 2000 to 4000, the least distorted image was selected. In addition, Table 2 lists the values obtained from the velocity field in modes 2, 4, 6, and 8 by designating a specific area (for the diagram of the droplet in detail, see Figure 9). To obtain the velocity values as accurately as possible, the least distorted image, in which the water droplet shape was closest to the original hemisphere, was selected in each mode, but the complete velocity field of the water droplet was not successfully acquired due to the shape distortion.
Figure 9. Designated specific area of a water droplet.
Figure 8 shows the distribution of the velocity vector of the internal flow field inside the vibrating water droplet. The velocity fields are mostly well depicted except in the node point regions, which are transient regions between nearby up−down moving surfaces (i.e., lobes) making complicated flow patterns. As the mode increases, interestingly, the flow patterns become gradually well visualized because of the lower magnitude of lobes. However, mode 2, as an exception, has a better flow pattern inside the droplet (Figure 8a), which seems to have only two nodes on the droplet surface and a simple countervortex circulation region inside the droplet. The video files of Figure 8 are attached as a reference (please see the movie clips given in the Supporting Information). In particular, the flow features by each mode of 2, 4, 6, and 8 of the diversified forms could be clearly confirmed in the Supporting Information.
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CONCLUSIONS In this study, the mode shape and the internal flow pattern for different modes (2, 4, 6, and 8) of a deionized water droplet placed on a hydrophobic surface were identified when a forced vibration was applied to the hydrophobic surface. The hydrophobic surface used in this study had a high contact angle of 115°, and we obtained the actual resonant frequency values for each mode and compared them to each other in terms of the size of the lobes. As a result, the actual value of the resonance frequency was lower than the theoretical value of the 6745
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(10) Hao, P.; Lv, C.; Zhang, X.; Yao, Z.; He, F. Chem. Eng. Sci. 2011, 66, 2118−2123. (11) Hong, F. J.; Jiang, D. D.; Cheng, P. J. Micromech. Microeng. 2012, 22, 1−9. (12) Oh, J. M.; Ko, S. H.; Kang, K. H. Langmuir 2008, 24, 8379− 8386. (13) McHale, G.; Elliott, S. J.; Newton, M. I.; Herbertson, D. L.; Esmer, K. Langmuir 2009, 25, 529−533. (14) Depaoli, D. W.; Feng, J. Q.; Basaran, O. A.; Scott, T. C. Phys. Fluids 1995, 7, 1181−1183. (15) Wilkes, E. D.; Basaran, O. A. Phys. Fluids 1997, 9, 1512−1528. (16) Kim, H. Y. Phys. Fluids 2004, 16, 474−477. (17) Brunet, P.; Eggers, J.; Deegan, D. R. Phys. Rev. Lett. 2007, 99, 144501−144504. (18) Matsumoto, T.; Fujii, H.; Ueda, T.; Kamai, M.; Nogi, K. Meas. Sci. Technol. 2005, 16, 432−437. (19) Yamakita, S.; Matsui, Y.; Shiokawa, S. Jpn. J. Appl. Phys. 1999, 38, 3127−3130. (20) Makino, K.; Michiyoshi, I. Int. J. Heat Mass Transfer 1984, 27, 781−791. (21) Wang, H.; Wang, Z.; Huang, L.; Mitra, A.; Yan, Y. Langmuir 2001, 17, 2572−2574. (22) Truskett, V. N.; Stebe, K. J. Langmuir 2003, 19, 8271−8279. (23) Scriven, L. E.; Sternling, C. V. Nature 1960, 187, 186−188. (24) Hu, H.; Larson, R. G. J. Phys. Chem. B 2006, 110, 7090−7094. (25) Xu, X.; Luo, J. Appl. Phys. Lett. 2007, 91, 12410. (26) Oh, J. M.; Legendre, D.; Mugele, F. Europhys. Lett. 2012, 98, 34003. (27) Soolaman, D. M.; Yu, H. Z. J. Phys. Chem. B 2012, 98, 34003. (28) Lamb, H. Hydrodynamics; Cambridge University Press: New York, 1932.
resonance frequency, and it was confirmed that as the mode number increased, the measurement uncertainty increased. In addition, the size of each lobe at the resonance frequency was relatively greater than that at the neighboring frequencies, and the internal flow of the nth order mode was observed through the flow visualization. Subsequently, a central bottom region of the droplet was chosen for the PIV measurements, and the velocity values were compared to each other. In all of the modes, a bisymmetrical flow pattern was generated about the central axis of the water droplet. In modes 2 and 4, a Y-shaped flow to the node point was observed, and in modes 6 and 8, a big elliptical vortex was confirmed. In addition, a small vortex was generated inside the lobe generated in each mode. Furthermore, it could be observed from the inner lobe that a small vortex is generated in each mode. In the case of the velocity field at the lower center of the water droplet, mode 4 exhibited higher velocity than mode 2, and the velocities of modes 6 and 8 were almost similar, but the velocity of mode 6 was approximately 7.2% higher than that of mode 8. Further research taking into account diversified variables affecting the water droplet pattern, such as forced vibration, temperature change, particle density inside water droplets, viscosity, and surface tension of water droplets, would be required for a better understanding of the behavior of water droplets.
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ASSOCIATED CONTENT
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AUTHOR INFORMATION
S Supporting Information *
Additional video files of Figure 8. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.5b02975. Corresponding Author
*Phone: +82 (0)51 5102302. Fax: +82 (0)51 5125236. E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy (no. 20114010203080). In addition, this research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2013005347).
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REFERENCES
(1) Kelvin, B. Mathematical and Physical Papers; Cambridge University Press: New York, 1882. (2) Rayleigh, B. Proc. R. Soc. London 1879, 29, 196−199. (3) Strani, M.; Sabetta, F. J. Fluid Mech. 1984, 141, 233−247. (4) Daniel, S.; Sircar, S.; Gliem, J.; Chaudhury, M. K. Langmuir 2004, 20, 4085−4098. (5) Daniel, S.; Chaudhury, M. K.; Gennes, P. G. D. Langmuir 2005, 21, 4240−4248. (6) Dong, L.; Chaudhury, A.; Chaudhury, M. K. Eur. Phys. J. E: Soft Matter Biol. Phys. 2006, 21, 231−242. (7) Noblin, X.; Buguin, A.; Brochard-Wyart, F. Eur. Phys. J.: Spec. Top. 2009, 166, 7−10. (8) Brunet, P.; Eggers, J.; Deegan, D. R. Eur. Phys. J.: Spec. Top. 2009, 166, 11−14. (9) Shin, Y.; Lim, H. Eur. Phys. J. E 2014, 37, 74−5. 6746
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Mode Pattern of Internal Flow in a Water Droplet on a Vibrating Hydrophobic Surface Hun Kim and Hee-Chang Lim* School of Mechanical Engineering, Pusan National University, San 30, Jangjeon-Dong, Geumjeong-Gu, Busan 609-735, South Korea S Supporting Information *
ABSTRACT: The objective of this study is to understand the mode pattern of the internal flow in a water droplet placed on a hydrophobic surface that periodically and vertically vibrates. As a result, a water droplet on a vibrating hydrophobic surface has a typical shape that depends on each resonance mode, and, additionally, we observed a diversified lobe size and internal flows in the water droplet. The size of each lobe at the resonance frequency was relatively greater than that at the neighboring frequencies, and the internal flow of the nth order mode was also observed in the flow visualization. In general, large symmetrical flow streams were generated along the vertical axis in each mode, with a large circulating movement from the bottom to the top, and then to the triple contact line along the droplet surface. In contrast, modes 2 and 4 generated a Y-shaped flow pattern, in which the flow moved to the node point in the lower part of the droplet, but modes 6 and 8 had similar patterns, with only a little difference. In addition, as a result of the PIV measurement, while the flow velocity of mode 4 was faster than that of model 2, those of modes 6 and 8 were almost similar.
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INTRODUCTION There has been much research interest in water drop behavior, particularly for its oscillation and evaporation, which have been demonstrated by many researchers because it is an interesting issue in terms of its basic properties and application in chemical and mechanical engineering. Early research on the vibration of a water droplet when it is surrounded by a different fluid began with studies by Kelvin1 and Rayleigh2 on water droplet behavior caused by resonance. This research was undertaken on a free water droplet, in which the viscosity could be neglected and where the amplitude is marginal. Later, Strani and Sabetta3 conducted an analysis of a nonviscous water droplet in contact with a solid surface, showing that its vibration has marginal amplitude. Subsequently, Daniel, Chaudhury, and Brunet conducted research4−10 on the motion of a water droplet that moves in only one direction on sloped surfaces and the vibration of a water droplet that moves in a vertical or horizontal direction as well as the change in its motion features. Research11−13 using the electro-wetting phenomenon, in which electric energy is applied, is being actively carried out. In particular, electro wetting is applied and used on micro systems, such as lab-on-a-chip, electro display, and liquid lens systems, due to its fast response and low power consumption. The water drop vibration generated by a periodic force is able to overcome micro-/nanoscale problems, such as the contact line pinning, chemical synthesis in solution, and the self-aggregation placed on a solid surface, and these advantages enable water droplets driven by forced vibration to be applied in the industrial sites as well. Applications of water droplet vibration include humidifiers (an air supply/exhaust system); heating, ventilation, and air © 2015 American Chemical Society
conditioning (HVAC); and diversified applications for industrial sites to help achieve the maximum heat transfer efficiency. For this reason, research14−16 on not only the water droplet hung on the upper part of the wall, which is attached to the lateral wall,8,17 but also the measurement18,19 of its surface tension and contact angle is being actively carried out. Furthermore, studies have performed not only experiments on water droplet behavior20 under simple resonance but also complex water droplet experiments, such as an experiment involving the evaporation of a water droplet hung in acoustic resonance and an experiment on the behavior of a water droplet placed on a heated plane plate. Generally, when a water droplet is evaporated or heated, a thermocapillary flow (Marangoni flow) occurs, and this flow significantly influences the sediment pattern.21,22 Research on the application of this phenomenon to various mechanical industries, such as porous thin film patterning, spray painting, and thin film deposition, is in progress. However, most of the research on internal flow was based on the Marangoni flow23−25 using natural evaporation and heating, and research on the internal flow of a vibrating water droplet was attempted using only numerical analysis.26 Such research was not performed in much detail, as evident from the scarce experimental results available at present. Therefore, the objective of this study was to confirm whether the actual flow exists inside the water droplet by applying a resonance to a water droplet placed on a hydrophobic surface. Received: March 28, 2015 Revised: May 8, 2015 Published: May 18, 2015 6740
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(HIFAX32C20-8, HiWave). A function generator (function generator, 33522A, Agilent) was used to supply a sinusoidal wave with a range of frequencies and voltages to the system. In addition, the laser used for the experiment was a CW laser (Excel 532-2000) with a wavelength of 532 nm. The internal flow was observed by penetrating the dead center of the water droplet by making a thin laser sheet by using a cylindrical lens after expanding the light emitted from the laser by a laser beam expander. The thickness of the laser sheet was about 0.2 mm, and, for visualization, a polystyrene nile red fluorescent sized of about 2 m was used. Furthermore, by using two sets of FASTCAM SA3, a Photron camera, photographs of the vibration cycle of the top lobe of the water droplet (camera 1) and its SI wafer substrate (camera 2) were captured with a shutter speed of 2000−4000 frames/s, and then the velocity value was calculated by using the commercial software MATLAB, Insight 4G, and Tecplot. A camera lens, zoom lens (Canon Macro 100 mm), and supermacro lens (Canon MP-E 65 mm f/2.8) were used. In this study, the flow pattern inside the water droplet and its velocity field, which depends on four types of water droplet modes (i.e., 2, 4, 6, and 8), were identified, and to apply heat and the substance transfer of the water droplet being induced, thereby the following experiment process was performed. First, a water droplet of 5 μL was generated on the surface of the silicon wafer by using a micropipette. As the evaporation of the water droplet was taking place due to the difference of external temperature and humidity, the temperature and the humidity were maintained by using a chamber. After the voltage was fixed to 2 V (modes 2, 4) and 10 V (modes 6, 8) by using function generator (33522A, Agilent), it was matched with a resonance frequency value obtained through a theoretical formula. While adjusting the frequency, the resonance frequency of the actual water droplet (its vibrating shape) was analyzed by using a live mode of a super high speed camera. The vibration cycle then was compared by simultaneously taking photos of the top lobe vibration cycle of the water droplet and the vibration cycle of the substrate by using two sets of a high speed camera based on 2000−4000 frames/s. Finally, the images obtained by the repetitive measurements were processed to make the precise mode shape, the internal flow pattern inside water droplet, and the internal velocity.
This fundamental study is aimed at understanding the applicability of such water droplets in diversified fields, such as the process, handling, and heat transfer using a water droplet, and to identify the mode pattern and the features of the internal flow inside a water droplet. To achieve this objective, the flow inside a water droplet was visualized by applying a forced vibration to the surface of the hemispherical water droplet having hydrophobic features, and its velocity field was investigated.
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EXPERIMENTAL DEVICES AND METHOD Surface Treatment. In general, the larger the droplet’s contact angle becomes, the higher the driving force required to stimulate the internal velocity would be, whereas the higher the homogeneity and the lower the roughness of the surface are, the smaller is the contact angle hysteresis.27 Therefore, to observe the shape and the internal flow in detail, Teflon, a low surface energy material, should be spin-coated onto the surface. In this experiment, to fabricate a silicon wafer that is a hydrophilic surface that acts as a hydrophobic surface, a Teflon solvent of concentration 0.6 wt % was prepared by dissolving Teflon AF (601S2-100-6, Dupont) in a fluorocarbon solvent (FC-40, 3 M). First, to remove the impurities and dust on the surface of the silicon wafer, the Piranha cleaning process was performed in a clean room. Afterward, the silicon wafer surface was spin-coated for around 5 s at 500 rpm, and then for 30 s at 2000 rpm, respectively. To strengthen the adhesion of the Teflon thin film, it was baked for 60 min on a hot plate of 165 °C. As a result, the coating thickness of about 100 nm was produced. Deionized water was used as the fluid; the size of the water droplet was 5 μL, and the equilibrium contact angle of the water droplet was determined to be 115° ± 1° by using a contact angle analyzer (CAM 100, KSV) after measuring it a total of 5 times. Furthermore, to reduce the experiment error, the temperature and the humidity were maintained constant by using a chamber, and, at this time, the experimental conditions were 25 ± 1 °C and 45% ± 5%, respectively. Vibration and Visualization Experiment. To observe the characteristics of the vertical vibration and to visualize the internal flow of the water droplet, a PIV (particle image velocimetry) measurement was performed, as shown in Figure 1. To observe the internal flow pattern inside the water droplet that was placed vertically in a periodic forced vibration, the spin-coated silicon wafer, which was made earlier, was chemically bonded with the front surface of a speaker
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MATHEMATICAL MODEL Velocity on the Droplet Surface. The droplet is placed on a plate and oscillated while maintaining a hemispherical axissymmetric shape. The flow inside a droplet is assumed to be inviscid, incompressible, and the axis-symmetrical shape under a vertical free oscillation. The equation for nth order resonance frequency is based on the governing equations (i.e., equation of continuity and Laplace equation) in the spherical coordinate system (r,θ,φ)28 (for the diagram of the droplet in detail, see Figure 2). The equation of continuity can be expressed as ∂ρ + ∇·(ρ u) = 0 ∂t
(1)
where u is the velocity of the fluid element at position x(r,θ,φ) and time t. The Laplace equation for the velocity potential ϕ of the fluid element is given by
Figure 1. Schematic diagram of the experimental setup. 6741
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Oh et al.12) Therefore, the governing equation for this interfacial phenomenon can be expressed as follows: rs = R + εPn
(5)
To obtain the surface pressure on the droplet, we need another boundary condition, the kinematic boundary condition. Depending on the sign of the perturbation part of the droplet, that is, if the movement is toward the inside (r < R) or the outside (r > R) of the droplet, the velocity potential can be expressed as follows: ⎛ R2Ṙ ⎞ n ϕ = ϕ1 = ⎜ ⎟ + A nr Pn r ⎠ ⎝
⎛ R2Ṙ ⎞ ⎛ Pn ⎞ ϕ = ϕ2 = ⎜ ⎟ + Bn⎜ n + 1 ⎟ ⎝r ⎠ ⎝ r ⎠
Figure 2. Descriptive diagram of a droplet in a fluid medium on a wetted substrate.
⎛ R2Ṙ ⎞ ⎛ rn Ṙ ⎞ P ε ̇ + 2ε ⎟ ϕ = ϕ1 = ⎜ ⎟+ n − 1 n⎜ ⎝ r R⎠ ⎠ nR ⎝
(8)
⎛ R2Ṙ ⎞ ⎛ Rn + 2 Ṙ ⎞ P ε ̇ + 2ε ⎟ ϕ = ϕ2 = ⎜ ⎟+ n − 1 n⎜ ⎝ R⎠ ⎝ r ⎠ (n + 1)r
(9)
Using the Bernoulli integral, the pressures at the inner and outer surfaces can be obtained as
(3)
⎡⎛ ∂ϕ ⎞ 1 ⎤ p1 = f1 (t ) + ρ1⎢⎜ 1 ⎟ − |∇ϕ1|2 ⎥ 2 ⎣⎝ ∂t ⎠ ⎦
(10)
⎡⎛ ∂ϕ ⎞ 1 ⎤ p2 = f2 (t ) + ρ2 ⎢⎜ 2 ⎟ − |∇ϕ2|2 ⎥ 2 ⎣⎝ ∂t ⎠ ⎦
(11)
where f n(t) is an arbitrary function of time. Therefore, the pressure difference at the droplet surface can be expressed as
If there is no vaporization and condensation at the interface between the droplet and the surrounding, the surface velocity at point r in the droplet can be expressed as R2 =ϕ r2
(toward the outside)
where An and Bn are the unknown constants that can be obtained by substituting eq 5 into eqs 6 and 7. In the result, the velocity potential can be expressed as
where θ is the polar angle measured down from the north pole, and φ is the azimuthal angle, analogous to longitude in earth measuring coordinate. Therefore, ((∂ϕ)/(∂r)), ((∂ϕ)/(∂θ)), and ((∂ϕ)/(∂φ)) are the velocities in the radial, azimuthal, and polar directions, respectively. Because fluid incompressibility and axis-symmetry are assumed, the equations can be substantially simplified. The continuity equation becomes ∇·u = 0 and the ((∂ϕ)/(∂θ)) and ((∂ϕ)/(∂φ)) velocities vanish so that the Laplace equation only depends on the radial direction. Therefore, the Laplace equation in the spherical coordinate system can be expressed as
Ṙ
(6)
(7)
∂ϕ ⎞ 1 ∂ ⎛ ∂ϕ ⎞ 1 ∂ ⎛ ⎜sin θ ⎟ ∇·u = ∇ ϕ = 2 ⎜r 2 ⎟ + 2 ∂θ ⎠ r ∂r ⎝ ∂r ⎠ r sin θ ∂θ ⎝ ⎛ ∂ 2ϕ ⎞ 1 + 2 2 ⎜ 2⎟ r sin θ ⎝ ∂φ ⎠ (2) 2
1 ∂ ⎛ 2 ∂ϕ ⎞ ⎜r ⎟ = 0 r 2 ∂r ⎝ ∂r ⎠
(toward the center)
⎛ 3 ⎞ p2 − p1 = f2 (t ) − f1 (t ) + (ρ2 − ρ1)⎜RR̈ + Ṙ 2⎟ ⎝ 2 ⎠ ⎡ R̈ +⎢ (ρ (n + 1)(n + 2) − ρ2 n(n − 1))ε ⎣ n(n + 1) 1 R 3Ṙ + (ρ1(n + 1) + ρ2 n)ε ̇ + n(n + 1) n(n + 1) ⎤ (ρ1(n + 1) + ρ2 n)ε ⎥̈ Pn ⎦
(4)
where R and Ṙ are a radius and the velocity at the radius, respectively. Pressure on the Droplet Surface. When a droplet oscillates with a small amplitude, the surface distortion of the droplet can grow or diminish (r < R, r > R) as shown in Figure 3. In addition, Figure 3 delineates a descriptive diagram of a droplet shape assuming that the droplet has a unique shape placed under a mode oscillation, in particular, modes 2−8. (For the pattern study of droplet oscillation in electro-wetting, see
(12)
The Laplace pressure is the pressure difference between the inside and outside of a curved surface, and it can be determined from the Young−Laplace equation as below. ⎛1 1 ⎞ △p = pinside − poutside = p1 − p2 = σ ⎜ + ⎟ R2 ⎠ ⎝ R1 =
(n − 1)(n + 1) 2σ + σεPn R R2
(13)
where R1 and R2 are the radii of curvature and σ is surface tension. If the droplet remains stationary and the radii of curvature are same, the Laplace pressure becomes 2σ/R.
Figure 3. Shape characteristics of a droplet at modes 2−8. 6742
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particularly important for verification. However, approximately 10−21% of the discrepancy occurred between the theoretical and experimental values for various reasons. The discrepancy tends to occur due to an effect of the dynamic features and the surrounding environment of a water droplet, and the contact friction appearing between a solid surface and the droplet is caused by the motion status of the contact line, where the water droplet, the solid surface, and three gas phases are simultaneously in contact. Here, the resonance frequency value of the nth mode could be accurately found when the largest peak of the lobe amplitude was experimentally observed around the theoretical resonance frequency, as shown in Figure 4. In addition, the experimental resonance frequency value of
However, when the droplet oscillates, the perturbation term is added to 2σ/R as shown in eq 13. Theoretical Frequency of Resonance in an Oscillating Droplet. Equations 12 and 13 represent the pressure difference on the droplet surface. Combining both equations, we obtain the following expression, in terms of the perturbation (ε): ε̈ +
⎡⎛ 3Ṙ ε ̇ + ⎢⎜R̈ρ1(n + 1)(n + 2) − ρ2 n(n − 1) ⎣⎝ R ⎞⎤ (n − 1)(n + 2)σ ⎞ ⎛ + ⎟ ⎜Rρ1(n + 1) + ρ2 n⎟⎥ 2 ⎠ ⎝ ⎠⎦ R ·n(n + 1)ε = 0
(14)
It is a second-order linear homogeneous ordinary differential equation. The solution of harmonic function (ωn) can be arranged as follows: ⎡⎛ ωn2 = ⎢⎜R̈ρ1(n + 1)(n + 2) − ρ2 n(n − 1) ⎣⎝ +
⎞⎤ (n − 1)(n + 2)σ ⎞ ⎛ ⎟ ⎜Rρ1(n + 1) + ρ2 n⎟⎥ ·n(n + 1) 2 ⎠ ⎝ ⎠⎦ R (15)
=0
If ρ2 is zero, the resonance frequency of nth order becomes as follows: fn =
ωn = 2π
n(n − 1)(n + 2)σ 4π 2ρR3
Figure 4. Frequency-dependent oscillation magnitude of the lobe displacement.
(16)
In eq 16, when the radius of the droplet becomes bigger and the mode frequency smaller, the resonance frequency of the droplet becomes smaller.
the nth mode is a value obtained by measuring the area where the lobe amplitude is the largest in the peripheral frequency domain of the theoretical resonance frequency, as shown in Figure 4. The maximum change of the lobe size was measured to be 0.22 and 0.15 mm at the second and fourth resonance modes, respectively. However, for the sixth and eighth, the variation of the lobe size was tiny at the applied voltage of 2 V. Therefore, the applied voltage was further increased from 2 to 10 V. As a result, the maximum change of the lobe size was measured to be 0.06 and 0.047 mm at the sixth and eighth resonance modes, respectively. Although the supply voltage was increased to be about 5 times higher (i.e., 10 V), the change of the upper lobe size at the sixth and eighth resonance modes was still less than for the second and fourth resonance modes at 2 V. From this result, it could be noted that the lobe size of the water droplet in the resonance frequency was one of most crucial parameters to observe, and at the other frequency range, the lobe size was substantially reduced. Internal Flow Features by Each Mode of the Water Droplet. In this study, the experiment was performed under 2 V (modes 2, 4) and 10 V (modes 6, 8), and, by using flow visualization, the flow pattern change inside the water droplet placed on a vibrating hydrophobic surface was confirmed. However, the reason the input voltages were different from each other in modes 2, 4 and 6, 8 was that the shape corresponding to each mode could not be distinguished even when using a high-speed camera. The change in the lobe size of the water droplet corresponding to modes 6, 8 under a relatively low voltage of 2 V was marginal. Additionally, even though the water droplet detachment and its separation from the silicon wafer surface were not our focus, they were also
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EXPERIMENT RESULTS AND DISCUSSION Comparison of Theoretical and Experimental Resonance Frequency. The theoretical resonance frequency formula being used in this study was deduced by developing a study on the resonance vibration of noncompressive, nonviscous water droplets by the forced vibration of Rayleigh and the velocity potential of a water droplet surface in the amplitude area, where a free water droplet was low and used Lamb’s28 theoretical resonance frequency formula, which satisfies both the Laplace equation and the boundary condition equation. In formula 16, f n is a resonance frequency corresponding to the shape of the oscillation in the nth mode, σ is surface tension, R is radius of water droplet, and ρ is the density of water droplet. The surface tension and the density of the water being used in this experiment are 0.07197 N/m and 997 kg/m3, respectively. Table 1 presents a comparison of the theoretical and experimental values of the resonance frequencies of each mode; this information is Table 1. Comparison of Theoretical and Experimental Results for Resonance Frequency of a 5 μL Droplet mode no.
resonance freq [Hz] (theory)
resonance freq [Hz] (experiment)
difference [%]
2 4 6 8
96 288 526 804
85 226 469 635
14.4 21.5 10.8 21.0 6743
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and the phase at the first and second cycles are identical; therefore, the velocity field obtained from the PIV measurement is confirmed to be valid. In addition, when observing the sine curve showing a resonance cycle of substrate in Figure 6, it could be confirmed that even though the resonance displacement of the substrate was not proportional to the lobe size, the vibration cycle and the size of the substrate affected the flow velocity inside the water droplet. When observing Figure 7, that is, the flow pattern image inside the water droplet being obtained through exposure of DSLR for 2 s, in each mode, bisymmetrical flow was commonly represented together with a vortex based on the central axis of the water droplet flow, but it could be seen that as the mode was increased, diversified flow patterns were represented. As the water droplet ascended from down to up, the Y-shaped flow pattern flowing to node point was taking place in modes 2 and 4, and the big vortex was taking place in modes 6 and 8. Here, the reason for the flow pattern being represented bisymmetrically was that the number and the size of the lobe that determine the shape of the water droplet were identical on both sides of the central axis of the water droplet. In addition, it could also be assumed that if the point of generation of the lobe is toward one side, the flow pattern would not be symmetric. Figure 7a shows an image of mode 2 and its Y-shaped flow pattern that flows to node point where the lobe is duplicated after ascending to about 2/3 point from down to up based on the central axis of the water droplet. In addition, large symmetrical flow streams along the vertical axis were generated in each mode, and they had a large circulating movement going from bottom to top, and then to the triple contact line along the droplet surface. Figure 7b shows an image of mode 4, in which the vortex is generated on the basis of two additional lobe centers as the axis, and it shows the flow being a little longer than mode 2 and the Y-shaped flow pattern flowing to each lobe is represented. Also, when observing the vortex being generated in the lobe of modes 2 and 4, it could be realized that
taking place because the energy of the modes 2, 4 at 10 V was that strong. Therefore, to understand the internal flow features for each mode of the water droplet, which is the goal of this study, an experiment was performed as detailed above. Figure 5 shows the variation of the resonance frequency (i.e., modes 2, 4, 6, and 8) obtained by using a high-speed camera.
Figure 5. Shape oscillation patterns that are obtained by high-speed camera under 2 and 10 V at the mode numbers.
As the mode number increases, it was observed that the number of lobes also increases, but the size gets smaller gradually. In addition, to have a proper identification of the number of lobes, the node points were also highlighted by the arrows in the figure. Therefore, the higher mode numbers have more arrows and more lobes, as shown in the figure. As a result, it can be shown that the droplet shape under the mechanical vibration is almost identical to those with free vibration. Figure 6 shows a graph measuring the vertical displacement of the top lobe and the Si-wafer substrate in four resonance modes when a forced vibration was applied to the hydrophobic surface. The abscissa in Figure 6 shows the elapsed time and the ordinate vertical displacement of the droplet and the substrate, respectively. As the mode number increases, the oscillation period of the lobe gradually becomes faster, about 12.5, 5, 2.25, and 1.5 ms. It could be realized that in all of the modes, the displacement of the lobe and substrate, depending on time, shows a constant cycle. When observing the images from the high-speed camera, the vibrating substrate location
Figure 6. Oscillations of the droplet and substrate height (h) for 5 μL droplets at the resonance modes 2 (a), 4 (b), 6 (c), and 8 (d), respectively. 6744
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Table 2. Averaged Vertical Velocity at the Central Bottom Region of the Droplet in Modes 2, 4, 6, and 8 mode no.
velocity [mm/s]
error [%]
2 4 6 8
0.36 0.79 3.87 3.61
1.35 2.5 1.37 1.29
Figure 8. Distribution of the velocity vector of the internal flow field in the vibrating water droplet.
Figure 7. Visualized flow pattern inside a droplet on the vibrating hydrophobic surface. The exposure time is 2 s.
flow direction is the same, except for the different size and generation location only. Figure 7c and d shows the images of modes 6 and 8, and each big elliptical vortex is created at both sides based on the centerline of the water droplet, which is slightly different from the previous modes (2 and 4). These results highlight the fact that as the number of lobe increases, the size of lobe decreases accordingly, and hence these lobes become smaller. This means that the wobbly surface of a droplet at modes 2 and 4 gradually becomes smoother at modes 6 and 8, and we observed this fact several times, so that this phenomenon seems to generate a large circulating vortex inside the droplet rather than the Y-shape. The internal velocity value of the water droplet being obtained through the visualization experiment was calculated by using Insight 4G after confirming that the vibration cycle of the lobe and that of the substrate of the water droplet are constant in each mode, as shown in the previously mentioned Figure 6. However, due to the water droplet distortion by shape vibration, the expansion or the sagging of the particles is taking place, and so, by taking photos in a way of magnifying high-speed camera frame from 2000 to 4000, the least distorted image was selected. In addition, Table 2 lists the values obtained from the velocity field in modes 2, 4, 6, and 8 by designating a specific area (for the diagram of the droplet in detail, see Figure 9). To obtain the velocity values as accurately as possible, the least distorted image, in which the water droplet shape was closest to the original hemisphere, was selected in each mode, but the complete velocity field of the water droplet was not successfully acquired due to the shape distortion.
Figure 9. Designated specific area of a water droplet.
Figure 8 shows the distribution of the velocity vector of the internal flow field inside the vibrating water droplet. The velocity fields are mostly well depicted except in the node point regions, which are transient regions between nearby up−down moving surfaces (i.e., lobes) making complicated flow patterns. As the mode increases, interestingly, the flow patterns become gradually well visualized because of the lower magnitude of lobes. However, mode 2, as an exception, has a better flow pattern inside the droplet (Figure 8a), which seems to have only two nodes on the droplet surface and a simple countervortex circulation region inside the droplet. The video files of Figure 8 are attached as a reference (please see the movie clips given in the Supporting Information). In particular, the flow features by each mode of 2, 4, 6, and 8 of the diversified forms could be clearly confirmed in the Supporting Information.
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CONCLUSIONS In this study, the mode shape and the internal flow pattern for different modes (2, 4, 6, and 8) of a deionized water droplet placed on a hydrophobic surface were identified when a forced vibration was applied to the hydrophobic surface. The hydrophobic surface used in this study had a high contact angle of 115°, and we obtained the actual resonant frequency values for each mode and compared them to each other in terms of the size of the lobes. As a result, the actual value of the resonance frequency was lower than the theoretical value of the 6745
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resonance frequency, and it was confirmed that as the mode number increased, the measurement uncertainty increased. In addition, the size of each lobe at the resonance frequency was relatively greater than that at the neighboring frequencies, and the internal flow of the nth order mode was observed through the flow visualization. Subsequently, a central bottom region of the droplet was chosen for the PIV measurements, and the velocity values were compared to each other. In all of the modes, a bisymmetrical flow pattern was generated about the central axis of the water droplet. In modes 2 and 4, a Y-shaped flow to the node point was observed, and in modes 6 and 8, a big elliptical vortex was confirmed. In addition, a small vortex was generated inside the lobe generated in each mode. Furthermore, it could be observed from the inner lobe that a small vortex is generated in each mode. In the case of the velocity field at the lower center of the water droplet, mode 4 exhibited higher velocity than mode 2, and the velocities of modes 6 and 8 were almost similar, but the velocity of mode 6 was approximately 7.2% higher than that of mode 8. Further research taking into account diversified variables affecting the water droplet pattern, such as forced vibration, temperature change, particle density inside water droplets, viscosity, and surface tension of water droplets, would be required for a better understanding of the behavior of water droplets.
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ASSOCIATED CONTENT
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AUTHOR INFORMATION
S Supporting Information *
Additional video files of Figure 8. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.5b02975. Corresponding Author
*Phone: +82 (0)51 5102302. Fax: +82 (0)51 5125236. E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy (no. 20114010203080). In addition, this research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2013005347).
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DOI: 10.1021/acs.jpcb.5b02975 J. Phys. Chem. B 2015, 119, 6740−6746
