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Shape Oscillation of a Drop in ac Electrowetting Jung Min Oh, Sung Hee Ko, and Kwan Hyoung Kang* Department of Mechanical Engineering, Pohang UniVersity of Science and Technology, San 31, Hyoja-dong, Pohang, 790-784, South Korea ReceiVed March 8, 2008. ReVised Manuscript ReceiVed May 3, 2008 A sessile drop oscillates when an ac voltage is applied in electrowetting. The oscillation results from the timevarying electrical force concentrated on the three-phase contact line. Little is known about the feature of drop oscillation in electrowetting. In the present work, the drop oscillations are observed systematically, and a theoretical model is developed to analyze the oscillation. It is revealed that resonance occurs at certain frequencies and the oscillation pattern is significantly dependent on the applied ac frequencies. The domain perturbation method is used to derive the shape-mode equations under the assumptions of a weak viscous effect and small drop deformation. The electrical force concentrated on the three-phase contact line is approximated as a delta function, which is decomposed and substituted into each shape-mode equation as a forcing term. The theoretical results for the shape and frequency responses are compared with experimental results, which shows qualitative agreement.
1. Introduction The electrical control of wettability, which is called electrowetting, is a versatile tool for handling micro- and nanoliter drops. The most commonly used configuration is the so-called electrowetting-on-a-dielectric (EWOD) in which a thin insulating layer is inserted between the liquid and the counter electrode to prevent the current flow. The electrowetting can be used as a very fast and efficient means to handle nearly any kind of drop with a relatively low electrical potential and power consumption. Potential applications of electrowetting have been demonstrated for the optical switch, variable focal lenses, micropumps, micromixers, and reflective displays. (See ref 1 and references therein.) The electrowetting phenomenon is induced by the Maxwell stress concentrated on the three-phase contact line (TCL).1–5 The Maxwell stress results from the Coulombic force acting on electrical charges induced around the TCL by the applied electric field. If we apply an ac electrical field, then the sign of the induced charge changes in harmony with the applied electric field, but the Maxwell stress is proportional to the square of the applied voltage; therefore, it is always exerted in the outward normal direction with respect to the drop surface for a conducting drop (Figure 1). Accordingly, the electrowetting phenomenon can be induced by ac voltage as well as by dc voltage. ac electrowetting has advantages of a decrease in contact angle hysteresis and a delay in contact angle saturation.4,5 The change in wetting tension in ac electrowetting can be decomposed into steady and timeharmonic parts. The steady part is responsible for the timeaveraged change in the apparent contact angle, which may be described by the Lippmann-Young equation.1–3 Little is known about the role of the time-harmonic part, even though it is expected to be closely related to the aforementioned advantages of ac electrowetting. * Author to whom all the correspondence should be addressed. Phone: +82-54-279-2187. Fax: +82-54-279-5899. E-mail:
[email protected]. (1) Mugele, F.; Baret, J.-C. J. Phys.: Condens. Matter 2005, 17, R705–R774. (2) Kang, K. H.; Kang, I. S.; Lee, C. M. Langmuir 2003, 19, 5407–5412. (3) Kang, K. H. Langmuir 2002, 18, 10318–10322. (4) Quilliet, C.; Berge, B. Curr. Opin. Colloid Interface Sci. 2001, 6, 34–39. (5) Blake, T. D.; Clarke, A.; Stattersfield, E. H. Langmuir 2000, 16, 2928– 2935.
Figure 1. Conceptual description of ac electrowetting.
Recently, several groups reported the existence of drop oscillations in ac electrowetting.6–11 Drop oscillations are expected to be related to the time-harmonic component of the Maxwell stress concentrated on the TCL. Two of us observed hydrodynamic flow in ac electrowetting and suggested that the shape oscillation of the drop surface is one of the causes of the flow.6 They showed that the oscillation pattern strongly depends on the forcing frequency. Baret et al. showed that the mixing of dye can be significantly enhanced by drop oscillation.7 They investigated the characteristics of drop oscillation with respect to forcing frequency. On the application side of electrowetting, a thorough understanding of the nature of the oscillation will be helpful in optimizing the mixing efficiency in drop-based microfluidics. More importantly, a fast, precise change in the meniscus shape from one state to another is required in electrowetting-based liquid lenses5,12 and reflective displays.13 A theoretical model should be developed to understand and analyze the drop (6) Ko, S. H.; Lee, H.; Kang, K. H. Langmuir 2008, 24, 1094–1101. (7) Mugele, F.; Baret, J.-C.; Steinhauser, D. Appl. Phys. Lett. 2006, 88, 204106. (8) Baret, J.-C.; Decre, M. M. J.; Mugele, F. Langmuir 2007, 23, 5173–5179. (9) Cooney, C. G.; Chen, C.-Y.; Emerling, M. R.; Nadim, A.; Sterling, J. D. Microfluid Nanofluid 2006, 2, 435–446. (10) Chung, S. K.; Zhao, Y.; Yi, U.-C.; Cho, S. K. IEEE Conference on Micro Electro Mechanical Systems; Kyoto, Japan (MEMS 2007), Kobe, Japan, 2007, pp 31-34. (11) Gunji, M.; Washizu, M. J. Phys. D: Appl. Phys. 2005, 38, 2417–2423. (12) Gabay, C.; Berge, B.; Dovillaire, G.; Bucourt, S. SPIE Proc. 2002, 4767, 159–165. (13) Hayes, R. A.; Feenstra, B. J. Nature 2003, 425, 383–385. (14) Takaki, R.; Adachi, K. J. Phys. Soc. Jpn. 1985, 54, 2462–2469.
10.1021/la8007359 CCC: $40.75 2008 American Chemical Society Published on Web 06/27/2008
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Figure 2. Instantaneous shapes of drops during one period of oscillation at 60 V: (a) f ) 33, (b) 100, and (c) 196 Hz.
Figure 3. Patterns of drop oscillation that are obtained by superposing more than 50 images at 60 V for various frequencies.
oscillation. Typically, the driving force of drop/bubble oscillation is either the surface force exerted on the surface of an entire drop or the body force acting on the entire drop volume.16–21 What distinguishes the drop oscillation in ac electrowetting is that the driving force of the oscillation can be approximated as a line force. This is because the Maxwell stress is concentrated on the (15) Yoshiyasu, N.; Matsuda, K.; Takaki, R. J. Phys. Soc. Jpn. 1996, 65, 2068–2071. (16) Plesset, M. S. J. Appl. Phys. 1954, 25, 96–98. (17) Plesset, M. S.; Prosperetti, A. Ann. ReV. Fluid Mech. 1977, 9, 145–185. (18) Prosperetti, A. Q. Appl. Math. 1977, 35, 339–352. (19) Prosperetti, A. J. Fluid Mech. 1980, 100, 333–347. (20) Lee, S. M.; Kang, I. S. J. Fluid Mech. 1999, 384, 59–91. (21) Oh, J. M.; Kim, P. J.; Kang, I. S. Phys. Fluids 2001, 13, 2820–2830.
extremely narrow region around the TCL.3 As far as we know, there has been no theoretical analysis of the drop oscillation induced by a line force. In the present work, the drop oscillation in ac electrowetting is investigated experimentally, and a theoretical model to describe the drop oscillation is developed. We consider a typical electrowetting system in which a tiny conducting drop is surrounded by air and placed on a dielectric layer. In this experiment, the oscillation pattern is observed by using a highspeed camera with a changing forcing frequency. In the theoretical analysis, the drop is modeled as a half-region of an isolated drop in an unbounded domain. A linear analysis is performed for the
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Figure 4. Patterns of drop oscillation that are obtained by superposing more than 50 images at 100 V for resonance frequencies, f, of (a) 95, (b) 189, and (c) 284 Hz.
Figure 5. Shape modes represented by r ) R [1 + 0.3 Pn(cos θ)] with some of the lowest-order Legendre polynomials.
oscillation on the basis of the domain perturbation method. Equations of motion for the shape oscillations are obtained by decomposing the surface motion of a drop into an infinite number of shape modes. The resonance frequencies and the oscillation patterns are predicted, and the results are compared with experiments.
2. Experiment The conventional setup for the electrowetting experiment is prepared to observe the drop oscillation. The Parylene-C layer of 5 µm thickness is deposited as an insulating layer on ITO-coated glass. On top of that, a Teflon AF1600 layer of about 100 nm thickness is spin coated to make the surface hydrophobic. An aqueous NaCl solution of 10-3 M is used, which has an electrical conductivity of 1.3 × 10-2 S/m, a drop of which having a base radius of about 1 mm is placed with a micropipette on the substrate. The temperature and the relative humidity are kept in the range of 25 ( 0.5 °C and 45 ( 5%, respectively. An electrical signal is generated by a function generator (33220A, Agilent) and is amplified 100-fold by a voltage amplifier (A800, FLC). An ac voltage with an rms value of 60-100 V is applied, and the frequency is varied from 20 to 1000 Hz. Throughout this article, what we call the voltage represents the rms value. The oscillating motion of a drop is captured by a high-speed camera (MotionXtra HG-100k, Redlake) at a frame rate of 3000-6000 fps. Figure 2 shows the instantaneous images at φ ) 0, π/4, π/2, 3π/4, and π for f ) 33, 100, and 196 Hz at 60 V, respectively, where φ denotes the phase angle of the input voltage. Supporting Information A and B contain movie clips that correspond to Figure 2a,c, respectively. The oscillation frequency is exactly twice the input frequency because the electrical force is proportional to the square of the voltage. Note that the input frequency denotes the frequency of the input electrical signal. The patterns of the shape oscillations are strongly dependent on the frequencies. For example, the drop shape changes from the half-oblate spheroid to the half-prolate spheroid repeatedly at f ) 33 Hz, and it alternates between twolobed and three-lobed shapes at f ) 100 Hz. It is interesting that time-periodic 1D movements of the TCL produce different patterns of drop oscillation for different input frequencies. The drop profiles at φ ) 0 and π are slightly different. In general, the static contact angle in electrowetting is dependent on the polarity of the
electrode,22,23 so the shape of a drop is not necessarily the same at φ ) 0 and π. Figure 3 represents the frequency dependence of oscillation patterns at 60 V. (See Supporting Information C for a movie clip.) Each image in Figure 3 is a superposition of over 50 images captured by the high-speed camera during one period of oscillation. The drop oscillates more vigorously at certain input frequencies of f ) 33, 61, 100, 196, and 289 Hz compared to oscillation at neighboring frequencies. Actually, as will be shown in sections 3 and 4, these frequencies correspond to the resonance frequencies. The amplitude of oscillation at the resonance frequency generally decreases with frequency. Another interesting feature observed in Figure 3 is that the oscillation patterns are significantly different for different resonance frequencies. That is, the number of node points at resonance frequencies of f ) 33, 61, 100, 196, and 289 Hz, which is indicated by the arrows in Figure 3, monotonically increases as 2, 2, 4, 6, and 8. We will discuss in sections 3 and 4 why the resonances appear at the particular frequencies with particular oscillation patterns. Figure 4 shows the oscillation patterns for an increased voltage of 100 V at some resonance frequencies. The oscillation patterns for the first and second resonances are not shown because they were too vigorous to capture stable images. The drops moved back and forth irregularly for these two cases. Compared with the cases of 60 V, the time-averaged contact angle is decreased, and the amplitude of oscillation generally increases. The time-averaged contact angles are about 114, 84.9, and 72° at 0, 60, and 100 V, respectively. At f ) 95 Hz, drop oscillation is vigorous enough to make the drop move around irregularly on the surface. As a result, even the electrode is vibrating in Figure 4a. Note that all of the resonance frequencies are shifted a little to lower frequencies compared with the cases of 60 V. The first resonance frequency shifted from 33 to 27 Hz, the third one shifted from 100 to 95 Hz, and the fourth one shifted from 196 to 189 Hz. This result was obtained repeatedly in experiments.
3. Theoretical Modeling We pursue an explanation of how the drop shapes become like those shown in Figure 2 and why the resonance occurs at particular (22) Seyrat, E.; Hayes, R. A. J. Appl. Phys. 2001, 90, 1383–1386. (23) Fan, S.-K.; Yang, H.; Wang, T.-T.; Hsu, W. Lab Chip 2007, 7, 1330– 1335.
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Figure 6. Coordinate system for theoretical modeling.
input frequencies in organized patterns with peculiar shapes. Consider a sessile drop oscillating on a flat plate. For a rigorous analysis of the shape oscillation, one has to solve an unsteady fluid dynamics problem numerically, which involves more or less tricky issues such as the dynamic contact angle, the free surface, and the boundary layer that may exist at the liquid-substrate surface. An analysis of the sessile drop seems impractical in the current stage; therefore, a simple model that meets our need is desired, instead of a complicated and sometimes indigestible numerical result. However, the method used to analyze the shape oscillation for an isolated spherical drop in an unbounded domain, which is based on the domain perturbation method, has been well established and successfully implemented to predict the shape oscillation of a bubble and drop in electromagnetic fields, sound fields, and so forth.16–21,24,25 In the method, the shape of a drop is represented by a linear combination of an infinite number of shape modes. If we set the origin at the center of drop, then the shape of a drop can be expressed as a linear combination of Legendre polynomials. ∞
r(θ, t) ) R + ε
∑ an(t) Pn(cos θ)
(1)
Figure 7. Temporal evolutions of shape modes of n ) 2, 4, and 6 at f ) 100 Hz with ε ) εdVrms2/2d γ.
n)2
Here, R denotes the volume-averaged radius of the drop, Pn denotes the Legendre polynomial of order n, and ε is a small parameter. The Legendre polynomial is the eigenfunction for the eigenvalue problem of the Laplace equation in axisymmetric spherical coordinates.26,27 The Legendre polynomials are mutually orthogonal in the interval of 0 e θ e π. Therefore, by the spectral representation theorem,26,27 any differentiable function defined within 0 e θ e π can be expressed as a linear combination of Pn. Each Pn represents a specific shape of a drop as shown in Figure 5 in which the dotted line represents a sphere (i.e., r ) R). The specific shape pattern represented by Pn is called the shape mode of Pn, which is recognized by the number of lobes (Figure 5). Here, the P1 mode is omitted because it corresponds to the translational motion of a drop.20,21 A brief introduction to the linear analysis method for the drop oscillation problem is provided in the Appendix. We will apply the method for the analysis of the drop oscillation by regarding the oscillating sessile drop as a half-region of a spherical oscillating drop in an unbounded domain (Figure 6). Then, the equator of the spherical drop corresponds to the TCL, and the cross section containing the equator becomes the contact surface. The substrate wall corresponds to the plane of symmetry for an isolated drop. (24) Strani, M.; Sabetta, F. J. Fluid Mech. 1984, 141, 233–247. (25) Leal, G. AdVanced Transport Phenomena. Fluid Mechanics and ConVectiVe Transport Processes; Cambridge University Press: New York, 2007; pp 250-282. (26) Arfken, G. B.; Weber, H. J. Mathematical Methods for Physicists, 6th ed.; Elsevier Academic Press: San Diego, CA, 2005; pp 634-638, 756-757. (27) Rade, L.; Westergren, B. Mathematics Handbook for Science and Engineering, 4th ed.; Springer-Verlag: Berlin, 1999; pp 259-298.
This is the key assumption of our analysis for the oscillation problem of the sessile drop. To regard a sessile drop as a half of an isolated drop in an unbounded domain, the substrate surface should be viewed as a slipping wall that has no tangential stress component. However, the substrate wall restricts the fluid motion by way of the no-slip condition that certainly violates the slip condition of the simple symmetric plane. However, for an oscillatory boundary-layer flow created near the substrate surface, the effect of viscosity is usually confined at a thin boundary layer with a thickness of ν/ω, which is O(10-4-10-5 m) under the present experimental conditions, in which ν is the kinematic viscosity of the fluid. Therefore, the boundary layer flow does not have any significant influence on the bulk oscillatory flow that is practically related to the shape evolution of the drop. The drop oscillation is driven by a line force acting on the equator of the drop, which corresponds to the TCL. The force acting on the equator (FTCL) can be represented as a sum of the electrical force (Fel), capillary force (Fc), and the contact-line friction (Ff):
FTCL ) Fel + Fc + Ff
(2)
For the sake of simplicity of analysis, the capillary forces of the air-substrate and liquid-substrate interface are neglected, together with the contact-line friction. Its consequences will be discussed later in this article. The Maxwell stress (or the electrical stress) is concentrated on the extremely narrow region around the TCL, which is on the
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order of the thickness of the dielectric layer.2,3 Therefore, the jump in the electrical normal stress acting on the TCL can be approximated by using the delta function at θ ) π/2.
[[n · (n · Te)]] )
εdV2 2 δ(η) 2d R
(3)
Here, η ) cos θ, εd is the permittivity, d is the thickness of the dielectric layer, and V is the applied voltage, respectively. The delta function is decomposed into an infinite number of shape modes as27 ∞
δ(η) )
∑ dnPn(η)
(4)
n)0
Here, the coefficient dn can be obtained by using the orthogonality of the Legendre polynomial as follows:26,27
dn )
2n + 1 2
∫-11 δ(η) Pn(η) dη ) 2n 2+ 1 Pn(0)
(5)
If eqs 4 and 5 are substituted into eq 3, then the jump in the electrical normal stress can be represented in terms of Pn as follows:
εdV2 2 ∞ 2n + 1 [[n · (n · T )]] ) Pn(0) Pn(η) 2d R n)0 2 e
∑
2
εdV 2dR
(7)
If eq 7 is applied to eq A7 in the Appendix, which is the standard form of the oscillation equation, then the oscillation equation for the nth mode is obtained to first-order accuracy as follows:
µ γ a¨n + 2(n - 1)(2n + 1) 2 a˙n + n(n - 1)(n + 2) 3 an FR FR εdV2(t) 1 ) n(2n + 1)Pn(0) (8) ε 2dFR2 Here, µ, F, and γ denote the viscosity, the density, and the surface tension of the drop, respectively. Equation 8 describes the temporal evolution of the amplitude of the nth shape mode. It should be noted that only the even modes are excited because Pn(0) ) 0 for n ) odd and all of the even modes can be excited by just the 1D oscillatory motion of TCL. Note that when only the even modes are excited the microscopic contact angle, which is the contact angle at the TCL, should be fixed at 90° because the even modes are symmetric at the substrate surface (Figure 6). Nevertheless, the apparent contact angle may change in time because we use an infinite numbers of (even-mode) Legendre polynomials. Here, the apparent contact angle represents the contact angle that is obtained by the macroscopic shape of the drop, without considering the microscopic deformation of the drop shape in the vicinity of the TCL. When the effect of viscous damping is weak, the approximate resonance oscillation frequency can be obtained from eq 8 as
γ FR3
(9)
Because the oscillation frequency is twice the input frequency, the resonance frequency (ωn) is half of the resonance oscillation frequency (ζn). When the input electrical voltage is set to be V(t) ) 2 Vrms sin ωt, a harmonic oscillation will result, and the amplitude of a shape mode can be decomposed into a steady part and an oscillatory part as follows
an ) a¯n + An cos(2ωt - Rn)
(10)
where ajn and An denote the steady component and the oscillation amplitude for the Pn mode, respectively, and Rn is the phase shift with respect to the input voltage. The steady part represents the time-averaged deformation of a drop. If the capillary force is considered, then the time-averaged deformation may be consistent with the Lippmann-Young equation. This deformation does not affect the oscillation within the framework of the present model. An analysis of the time-averaged deformation of the drop is beyond the scope of the present work and will be investigated in future work. If eq 10 is substituted into eq 8, then An and Rn are obtained as
An ) -
sn
√pn2 + qn2
Rn ) tan-1
(6)
As a result, the electrical (line) force is decomposed into an infinite number of modes that will drive each shape mode of motion. If eq 6 is compared with eq A6 in the Appendix (i.e., the spectral representation of the stress), then the coefficient fn is expressed as follows:
fn ) (2n + 1)Pn(0)
ζn2 ) n(n - 1)(n + 2)
qn pn
(11)
(12)
where
sn )
εd Vrms2 2n + 1 R Pn(0) ε (n - 1)(n + 2) 2dγ ω 2 pn ) 1 ωn
( )
qn )
ω µ (n - 1)(2n + 1) 2 ωn2 FR
(13a) (13b) (13c)
The maximum values of |An| are obtained at the resonance frequency.
|An|max ) |sn|
ωn R2 (n - 1)(2n + 1) µ
(14)
From eq 14, we can see that |An|max is inversely proportional to n1.5, which means that the amplitude of oscillation becomes smaller as the mode number increases.
4. Results and Discussion 4.1. Linear Analysis and Experimental Results. The fourthorder Runge-Kutta-Merson28 method is used to integrate the oscillation equations derived in the previous section. In the numerical simulation, we set ε ) εdVrms2/2d γ for convenience, and physical parameters are set as the values consistent with the experimental conditions. The temporal evolutions of a2, a4, and a6 are shown for the input frequency of f() ω/2π) ) 100 Hz in Figure 7. There is a transient period of time, what we call the stabilizing time, before establishing harmonic motions. The stabilizing time decreases with mode number. The oscillation of a2 is stabilized after more than 150 periods, whereas a4 and a6 become stable within 30 and 10 periods, respectively. The amplitude of each mode depends sensitively on the input (28) De Vahl Davis, G. Numerical Methods in Engineering and Science; Allen & Unwin: London, 1986; p 179.
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Figure 8. Frequency dependence of the dimensionless amplitude of the shape modes. Table 1. Comparison of Predicted Resonance Frequencies with Experiments fn ) ωn/2π (Hz) experiments n
linear analysis
60 V
100 V
2 3 4 6 8
38.5 74.6 115.6 211.1 322.5
33 61 100 196 289
27 60 95 189 284
frequency, and the shape of the drop is primarily determined by a dominant shape mode. The dominant mode at f ) 100 Hz is a4, as observed in Figure 7, and as a consequence, the oscillation pattern looks like that in Figure 3f. Figure 8 represents the frequency dependence of |An|, which is calculated by using eq 11. Figure 8 shows the resonance frequency and the relative amplitudes (|An|/R) of the shape modes. In Figure 8, the amplitude of the P2 mode has its maximum value at f ) 38.5 Hz and becomes smaller in the other frequency range. The input frequency of 38.5 Hz corresponds to the first resonance in the experiment (Figure 3b). The other resonances observed in the experiment can also be matched with one of the peaks in Figure 8 by comparing the shape patterns. Some of shape patterns are shown in Figure 5. Table 1 compares the resonance frequencies obtained by linear analysis and experiments. The resonance frequencies predicted by the theoretical analysis are greater by about 12% than those obtained by experiments. In experiments, the resonance frequency slightly decreases with an increase of voltage (Table 1), which cannot be predicted in the present linear model. The shift of the resonance frequency can be induced by the second-order interactions between shape modes.29,30 The maximum amplitude of the P2 mode can be calculated by using eq 14. The maximum amplitude of a shape mode is inversely proportional to n1.5, so it decreases rapidly with the mode number as shown in Figure 8. Figure 9 shows the oscillation patterns of the drop at the resonance frequencies corresponding to n ) 2, 4, 6, and 8 for 60 V. Each image is produced by superposing the drop shape at 25 equally divided steps in one period, and it is compared with the experimental results in Figure 3. As shown in Figure 9, the shape pattern changes significantly depending on the resonance frequency. It is observed that the overall shape of the drop oscillation is very close to the experimental results in Figure 3. The concave shape at the top of the drop in the theoretical analysis cannot be observed in the experiments. The needle electrode placed on top may somehow affect the shape of the drop as a result of the wettability change at the interface. Figure 8 shows that the oscillation in Figure 3b is that dominated by the P2 mode (29) Trinh, E.; Wang, T. G. J. Fluid Mech. 1982, 122, 315–338. (30) Trinh, E.; Zwern, A.; Wang, T. G. J. Fluid Mech. 1982, 115, 453–474.
and can be predicted by the present model. The other shape modes are kept small compared with the P2 mode at this resonance frequency so that the effect of other modes is hardly observed in the simulations and experiments. The same is true for other resonance oscillations. The amplitudes predicted by the present model are greater than those observed in experiments if Figure 9 is compared with Figure 3. One of the reasons will be the negligence of contactline friction. The contact line friction is a kind of damping force, so it can reduce the oscillation amplitude.31,32 An analysis to consider the effect of contact-line friction is undergoing in our group. 4.2. Nonlinear Effects. Figure 10 shows the response of a drop at the first resonance frequency of 27 Hz when the voltage is increased to 100 V, where the corresponding electrowetting number is ε ) 3.42 × 10-1. The oscillation is so vigorous that the position of the drop is not stable but moves around the electrode irregularly. When the amplitude of oscillation is large enough, the drop oscillation even becomes asymmetric. Furthermore, the center of mass of the drop moves up and down; as a consequence, the entire drop is sometimes even detached from the plate surface. A similar phenomenon is also observed for the second resonance frequency of f ) 60 Hz. The present theoretical model is derived under the assumption of small deformation, so when the deformation is significant, the present model is invalid to describe the shape of the drop. In the first resonance, for example, the dimensionless maximum amplitude, |A2|max /R, becomes greater than 65, which means that r in eq 1 can even have a negative value at large ε, which is unrealistic. Note that the second resonance frequency is not predicted in the theoretical analysis because the observed frequency corresponds to the odd mode of n ) 3. In principle, the odd modes cannot appear in the present model as mentioned in the previous section. We presume that the second resonance is induced by other effects because the odd mode is not activated by the line force. Interestingly, higher odd modes such as n ) 5 and 7 are not observed, and the amplitudes of the oscillations at f ) 51 and 85 Hz are relatively larger than those at nonresonance frequencies of 20, 161, and 268 Hz shown in Figure 3. This means that the P3 mode is influential in a broader frequency range compared with the other shape modes. Because there is no other external surface force acting on the drop surface except the surface tension force, we conjecture that the P3 mode is induced by the gravitational force acting on the volume of the drop. If the drop moves up and down by drop deformation (Figure 10 is an extreme case), the phenomenon is equivalent to the time-periodic pressure variation inside the droplet, which can be another driving force of drop oscillation.14,15 Further investigation is needed to identify the origin of the P3 mode clearly, which is beyond the scope of this article.
5. Conclusions Drop oscillation induced by ac electrowetting is investigated by experiment and theoretical analysis. In the experiments, it is observed that the drop oscillation becomes resonant at certain input frequencies and that the pattern of shape oscillation is different at each resonance frequency. The distinctive shape of the drop oscillation is visualized by superposing the instantaneous images. In the theoretical analysis, the oscillation equations are derived on the basis of the conventional perturbation method. The electrical force acting on the TCL is approximated as the line force, which is expressed by the delta function in the (31) Blake, T. D.; Haynes, J. M. J. Colloid Interface Sci. 1969, 30, 421–423. (32) Wang, K.-L.; Jones, T. B. Langmuir 2005, 21, 4211–4217.
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Figure 9. Drop oscillation calculated by linear analysis at resonance modes of n ) 2, 4, 6, and 8.
Figure 10. Instantaneous images of drop oscillation during two periods for f ) 27 Hz at 100 V.
theoretical modeling. The proposed model predicts characteristics such as the resonance frequency fairly well. The present model can be further developed to investigate the unsteady nature of electrowetting, which is important in the electrowetting-based liquid lens and reflective displays. Acknowledgment. This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korean government (MEST) (R0A-2007-000-20098-0).
Appendix: An Overview of the Conventional Method Used to Analyze the Oscillation of a Drop We provide here, for general readers, a brief overview on the methodology of the conventional linear analysis for drop oscillation. A more complete review of this issue can be found in Plesset,16 Plesset and Prosperetti,17 Prosperetti,18,19 Lee and
Kang,20 and Oh et al.21 Consider an isolated conducting drop in an unbounded domain. Our objective is to predict the shape oscillation of the drop. We assume that the shape deformation of the drop is axisymmetric, and we neglect the gravity force. Through the spectral representation of eq 1, the problem of finding the shape oscillation of a drop is transformed to determining the coefficient an’s. The normal stress condition, which balances the normal component of the total stress and the surface tension force, is given by
[[n · (n · T)]] ) [[n · (n · Th)]] + [[n · (n · Te )]] ) γ(∇ · n) (A1) where [[ • ]] ) [ • ]out - [ • ]in, and n and γ are the outward unit normal vector at the drop surface and the surface tension, respectively. The total stress (T) includes the Maxwell stress (or the electrical stress, Te) as well as the hydrodynamic stress (Th).
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When there is no electric force and fluid motion, eq A1 is reduced to the static balance between the pressure jump across the drop surface (∆p) and the capillary pressure of 2γ/R, i.e.,
∆p )
2γ R
(A2)
Under the assumption of weakly viscous flow, the flow field is regarded as a potential flow, and the effect of viscosity is included by the viscous pressure correction.18 As mentioned, the pressure (p) and velocity potential (φ) can be represented in a similar fashion to eq (1), since the Legendre polynomial is the eigenfunction of the Laplace equation. The coefficients in the spectral representation for pressure and velocity potential is related to an’s by way of the Navier-Stokes equation and the kinematic condition as follows: ∞
φ)
∑ φn(an, a˙n, a¨n)Pn(cosθ)
(A3)
n)0 ∞
p)
∑ pn(an, a˙n, a¨n) Pn(cos θ)
(A4)
n)0
The jump of the hydrodynamic normal stress across the drop surface is obtained from the velocity potential and the pressure as ∞
[[n · (n · Th)]] )
∑ hn(an, a˙n, a¨n) Pn(cos θ)
(A5)
n)0
The jump in the electrical normal stress is assumed to be independent of an and can be expressed by the spectral representation as follows:
[[n · (n · Te )]] )
∑ fnPn(cos θ)
(A6)
n)0
The unit normal vector is approximated in terms of an in a linear form with first-order accuracy (O(ε)), and the surface tension force (γ(∇ · n)) is also expressed in a form similar to eq A6. Then, if eqs A5 and A6 are applied to eq A1, then the normal stress condition gives a set of differential equations for an, which are called the shape mode equations or the oscillation equations as follows:
{
ε
}
1 ¨ 2(n - 1)(2n + 1) µ ˙ γ FRan + a + (n - 1)(n + 2) 2 an n n R n R ) fn (A7)
The first and second terms in parentheses originate from [[n · (n · Th)]] and represent the inertial and viscous contributions to the stress acting on the drop surface, respectively. The third term is associated with γ(∇ · n) in eq A1. The Maxwell stress on the right-hand side is the forcing term in the oscillation equation. Supporting Information Available: Movie clips A and B demonstrate drop oscillations at 33 and 196 Hz in slow motion, respectively; these correspond to the cases of Figure 2a,c, respectively. The playing speed is 0.01×. Movie clip C shows drop oscillation with forcing frequency varied from 20 to 289 Hz at Vrms ) 60 V, which serially shows the cases of Figure 3. The playing speed of the movie clip is 1×. This material is available free of charge via the Internet at http://pubs.acs.org. LA8007359
