pubs.acs.org/Langmuir © 2009 American Chemical Society
Capillary Spreading Dynamics of Electrowetted Sessile Droplets in Air Prosenjit Sen* and Chang-Jin “CJ” Kim Mechanical and Aerospace Engineering Department, University of California, Los Angeles (UCLA), California 90095 Received January 7, 2009. Revised Manuscript Received March 3, 2009 We report the contact line dynamics of sessile water droplets, 1.1-1.6 mm in radius, spread by electrowetting in air. Coplanar electrodes patterned on the substrate allow a true sessile condition with no wire into the droplet. The frequency response of the droplets is studied using 25 VAC ranging from 10 to 205 Hz. The effect of contact angle hysteresis is seen in form of stick-slip motion. A model developed provides a good match to the experimental result. Step response is studied with voltages in the range of 20-80 VDC. Two regimes of motion are observed. In the first regime, local flows cause the contact line speed to increase and reach a maximum while the contact angle is still changing. Global flows in the second regime cause the contact line to move with a reduced speed and attain the spherical shape pertaining to the new equilibrium contact angle. A model is used to describe the motion.
Introduction Many interesting devices have emerged that use actuation of sub-microliter droplets by electrowetting. Understanding the dynamics of such droplets will present opportunities to design new devices and improve performance of those existing. The vibration of free droplets has been studied extensively. The first mathematical model for inviscid droplet oscillation neglecting the external fluid was given by Rayleigh1 and accounting for the external fluid by Lamb.2 These oscillations sustain through the balance of inertial and surface energies, resulting in ω2n µ γ/FR3 (ωn, natural frequency; γ, surface tension; F, density; and R, radius). For viscid droplets, viscous dissipation leads to damping and amplitude decay. Compared to unconstrained droplets, significantly less work has been performed for oscillations of partially constrained droplets such as sessile droplets on solid substrates. An early theoretical study was motivated by crystal growth in microgravity and provided a solution for pinned droplets in contact with spherical substrates.3 Experimental study of gravity-flattened droplets (puddles) on a vibrating substrate showed contact line pinning at low excitation amplitudes due to contact angle hysteresis.4,5 As the amplitude was increased above a threshold, the contact line was freed, and the radius of the liquid-substrate contact area was observed to oscillate. At even higher amplitudes, parametric instabilities (subharmonic triplons) were further observed. Oscillation of liquid marbles has been recently reported.6 With the advent of microelectromechanical systems (MEMS), many devices involving droplets with moving contact lines, commonly actuated by electrowetting-on-dielectric (EWOD),7 have been developed for various applications (1) Rayleigh, L. Proc. R. Soc. London 1879, 29, 71. (2) Lamb, H. Hydrodynamics, 6th ed.; Cambridge University Press: Cambridge, UK, 1932. (3) Strani, M.; Sabetta, F. J. Fluid Mech. 1984, 141, 233. (4) Noblin, X.; Buguin, A.; Brochard-Wyart, F. Phys. Rev. Lett. 2005, 94, 166102. (5) Noblin, X.; Buguin, A.; Brochard-Wyart, F. Eur. Phys. J. E 2004, 14, 395. (6) McHale, G. Langmuir 2009, 25(1), 529. (7) Lee, J. Sens. Actuators, A 2002, 95, 259. (8) Cho, S. K.; Moon, H.; Kim, C.-J. J. Microelectromech. Syst. 2003, 12, 70. (9) Hayes, R. A.; Feenstra, B. J. Nature 2003, 425, 383.
4302
DOI: 10.1021/la900077u
including laboratory-on-a-chip,8 liquid display,9 and microswitches.10 Unlike the case of puddles,3 the droplet sizes in microdevices are typically smaller than the capillary length κ-1 = (γ/Fg)1/2 (g, gravitational acceleration). Unlike the inertial actuation by the vibrating substrate,3 the electrowetting actuation is by capillary forces and leads to a different local phenomenon near the contact line in which a rapid change near the contact line in short-term leads to a global shape change in long-term. Shape oscillations for an electrowetted droplet have been reported with a wire into the droplet which limits the true sessile condition.11 Further hydrodynamic flows inside electrowetted droplets have been explained using shape oscillations.12 The objective of this Letter is to study the spreading dynamics, that is, frequency and step response of true sessile droplets in the capillary regime under electrowetting actuation.
Experimental Section We observed the dynamics of electrowetted water droplets of volume 4, 8, and 12 μL on a glass substrate. A common practice is to bias the droplet through a thin conducting wire that touches the droplet on the top. In order to eliminate experimental error due to the wire, devices were designed with coplanar electrodes, as shown in Figure 1. The surface consisted of patterned gold electrodes covered with a dielectric layer of 3500 A˚ of silicon nitride, which was lithographically opened in the middle to expose the underlying gold. The entire surface was coated with 2000 A˚ of Teflon to make it hydrophobic. With no EWOD actuation, the static contact angle of the water droplet on the device shown in Figure 1 was ∼120° with advancing and receding angles of ∼125° and ∼115°, respectively, resulting in a contact angle hysteresis of ∼10°. When an electrical potential was applied between the droplet (via the middle electrode) and the outer electrode, the droplet spread according to the Lippmann-Young equation cos θV = cos θ0 + cV2/2γ (θV, contact angle at voltage V; θ0, intrinsic contact angle; c, capacitance per unit area).13 The frequency response was studied by applying sinusoidal signals ranging from 10 to 205 Hz with 5 Hz increments. Actuation with 25 VAC (10) (11) (12) (13)
Sen, P.; Kim, C.-J. J. Microelectromech. Syst. 2009, 18, 174. Oh, J. M.; Ko, S. H.; Kang, K. H. Langmuir 2008, 24(15), 8379. Ko, S. H.; Lee, H.; Kang, K. H. Langmuir 2008, 24(3), 1094. Moon, H. J. Appl. Phys. 2002, 92, 4080.
Published on Web 3/16/2009
Langmuir 2009, 25(8), 4302–4305
Letter
Figure 1. (Left) Schematic of electrowetting actuation of a sessile droplet in air on a coplanar EWOD device. The coplanar configuration eliminates the need for the conducting wire inserted into the droplet and allows an accurate study. (Right) Droplet radius (R), contact radius (r), and contact angle (θ) of a sessile drop in the capillary regime. ensured to free the contact line from pinning and make it slide. Step response, on the other hand, was studied by applying 20-80 V step signals with 10 V increments. With the system capacitance in the order of pF, the system charging time has a negligible effect on the interface response. The actuation sequence was captured from the side using a high-speed camera (2000 fps for frequency response and 10 000 fps for step response) and analyzed to extract the contact angle (θ), contact radius (r), and droplet radius (R) as defined in Figure 1. The droplet size range was selected to achieve a high pixel resolution (∼7 μm/pixel) for the available imaging setup. Experimental errors from droplet evaporation can be neglected, as the droplet radii were in the millimeter scale and each run required less than 10 s. This choice however leads to droplet diameters close to the capillary length, and the droplet shape deviates from the spherical cap (approximate rms error of 13 μm from a spherical fit for a 1.3 mm radius droplet) due to gravitational flattening.
Figure 2. Measured response of a droplet under varying driving frequencies, using 25 VAC EWOD actuation on an 8 μL water droplet. The top curve in blue is the contact angle, and the bottom curve in red is the contact radius. Contact angle and contact radius are in phase for (a) and (c). Response is 180° out of phase for (b). Subharmonic modes are present at some frequencies (d).
Results and Interpretation The frequency responses of a water droplet on the aboveprepared device are presented in Figures 2 and 3. Figure 2 shows the contact angle and contact radius for an 8 μL droplet, actuated by VAC = V0 sin ωt (ω, frequency of actuation signal; V0, voltage; t, time), which provides an actuation force of FA µ 0.5V20(1 - cos2ωt). The time invariant part of the actuation force lowered the static contact angle to 105°, on which oscillation was centered. At low frequency (20 Hz), the contact line and the contact angle were in phase, that is, θ decreased as r increased (Figure 2a). This behavior, where the r increased as θ decreased, is called in phase, as this is the normal behavior of the droplet when shape and volume are conserved. They were out of phase by 180° after the first resonance (Figure 2b) and in phase again as the second mode became dominant (Figure 2c). Similar to a mass-spring oscillator under dry friction,14 the effect of hysteresis was clearly seen in the form of stick-slip motion, which occurs when the net available force |FA - FS| drops below the resistance, where FS is the surface restoring force of a deformed droplet. Normal stops occurred when the contact line changed its sliding direction, and can be understood by nearly constant actuation force (dFA/dt ∼ 0 f ωt ∼ nπ/2, with n being an integer). Abnormal stops (i.e., stick) occurred when the difference in rate of change of the surface-restoring force and the actuation force over time (i.e., dFS/dt - dFA/dt) leads to a reduction of the net available force (i.e., |FA - FS|) below the resistance force leading to contact line stick, and the contact line resumes sliding in the same direction on release. Depending on frequency, two normal stops and zero to two stick-slips were observed per (14) Hong, H.-K.; Liu, C.-S. J. Sound Vib. 2000, 229, 1171.
Langmuir 2009, 25(8), 4302–4305
Figure 3. Frequency response of contact radius oscillation for 4, 8, and 12 μL water droplets under 25 VAC EWOD actuation. (Inset) Comparison of theoretical and measured resonance frequency for the three different droplet volumes. (Images) Droplet resonance modes shown with a 8 μL droplet. response cycle. Figure 2a shows an example of one stick-slip per cycle. However, these abnormal stops were not observed to occur near resonance. A nonlinear effect was visible at some frequencies in the form of subharmonic modes (Figure 2d). In the studied frequency range, two resonance modes were observed, as shown in Figure 3, which presents for all the three droplet volumes along with a few exemplary pictures for the 8 μL droplet. Below the first resonance, the response was quasi-static, and the droplet followed the excitation. Static friction due to hysteresis and nonlinear spring effects of the sessile droplet even for a relatively small deformation15 resulted in a nonflat response at low frequency. After the resonance, the response dropped rapidly and passed through a minimum, after which the second mode became dominant. Smaller droplets showed higher resonance frequency and smaller gain at the resonance, most apparently for the first mode. (15) Becker, E.; Hiller, W. J.; Kowalewiski, T. A. J. Fluid Mech. 1991, 231, 189.
DOI: 10.1021/la900077u
4303
Letter
Figure 4. Step response of an 8 μL droplet to different voltages. (Images) Captured images of a droplet at different stages of response. (Inset) Zoomed view to show the response near the origin.
The step responses of the droplet are presented with Figures 4 and 5. Figure 4 displays the contact line sliding over time for four different actuation voltages as well as a few exemplary pictures representing different stages of an actuated droplet. The responses indicate the droplet was underdamped. Higher voltages (60 and 80 V) produced an oscillation before the final state, indicating the overshoot had sufficiently deformed the free surface of the droplet so that its restoring force was large enough to make the contact line slide back against the contact angle hysteresis. With lower voltages (20 and 40 V), on the other hand, the contact line stayed near the overshoot peak, indicating the overshoot was not large enough to overcome the hysteresis. The inset presents the measured data in a magnified time frame to show the details during the initial 1 ms. Two regimes of contact-line sliding were observed for the step response with respect to the sliding speed, as shown in Figure 5a, which presents the speeds for four different actuation voltages (20, 40, 60, and 80 V) along with the contact angle for 80 V. The force per unit length driving the contact line for a given voltage V is γ(cos θV - cos θ0), where θV is the equilibrium contact angle at voltage V and θ0 is the static contact angle. In an initial regime (approximately up to 0.4 ms from motion initiation) while the apparent contact angle was still changing, the sliding velocity increased to a high value (e.g., 240 mm/s for 80 V) as shown in Figure 5a. As the contact angle stabilized, however, the sliding speed decreased to less than 150 mm/s. We believe electrostatic force near the contact line leads to an initial deformation dominated by local flows near the contact line, and the small mass involved with the local flows explains the acceleration observed in this regime. The local flow changes the local curvature near the contact line as seen in the 0.4 ms picture of Figure 4, giving rise to a capillary pressure gradient along the droplet surface. After the contact angle stabilizes, the pressure gradient acts to restore the droplet to a new spherical shape pertinent to the new contact angle. In this second regime, contact line deceleration is attributed to the global flow involving larger mass, development of a viscous boundary layer, and reduction of available force (FA - FS). 4304
DOI: 10.1021/la900077u
Figure 5. Step response of an 8 μL droplet to different voltages. (a) Measured contact line speed and contact angle. The top solid curve is the measured contact angle for 80 V. The dashed curves are contact line speeds for 20-80 V. (b) Maximum measured contact line speed as a function voltage.
Mathematical Model The Reynolds number calculated with approximate phase velocity and wavelength of the surface modes for the tested droplet is ∼60, signifying that the inertial effect is significant compared to the viscous loss, consistent with the underdamped response shown in Figure 4. Complex coupling between contact angle and contact radius has restricted the availability of an analytical solution to the case of a pinned droplet in contact with a spherical substrate only.3 A model neglecting the substrate and assuming hemispherical droplets has been presented ref 11. However, a simplified model assuming a spherical cap for the droplet shape can be derived for the first mode as follows. Surface energy stored in a deformed droplet is given by dES = γ{dS - (cos θ0)dA} (S, liquid-air area; A, liquid-solid area). For small oscillation under the assumption of a constant contact angle θ = θ0 Es ¼ γðR2 -R0 2 Þð2 þ cos θÞð1 - cos θÞ2
ð1Þ
For kinetic energy, assuming simple flows satisfying : the incompressibility condition (r 3 ν = 0), we use: νx = xR/R, : νy = - 2yR/R, where ν is the fluid velocity and R = dR/dt.5,16 The total kinetic energy is : Ek ¼ 3FΩ f ðθÞ R 2
ð2Þ
(16) Okumura, K. Europhys. Lett. 2003, 62, 237.
Langmuir 2009, 25(8), 4302–4305
Letter
where f(θ) = (0.4 - 0.25cos θ - 0.5cos3 θ + 0.35cos5 θ)/ (2 - 3cos θ + cos3 θ) and Ω is the droplet volume. At equilibrium, balancing dissipation from the viscous boundary layer (Lv)17 and hysteresis (Ls) with the rate of energy change, we have d(Es + Ek)/dt = -d(Ls + Lv)/dt, which gives : : :: R þ bR þ ωn 2 R þ hsgnðRÞ ¼ 0
ð3Þ
In the above equation, ω2n = γπ(2 + cos θ)(1 - cos θ)2/3FΩ f(θ), b = (ηFω)1/2πR20 sin4 θ/3FΩ f(θ), and h = 2πR0 sin2 θγ(cos θ0 - cos θa)/3FΩ f(θ), where θa is the advancing contact angle, η is the coefficient of viscosity, b is the viscous dissipation coefficient, and h is the static dissipation coefficient due to hysteresis. The frequency response of the droplet is understood by considering balance of inertial and capillary forces. The inset in Figure 3 shows the measured fundamental frequencies of the three droplets of different volumes along with the calculated frequencies. Even with our very simplified linear velocity field approximation, a good agreement was found between the measured and the calculated values. Resonant frequencies have a f µ R-1.5 fit with a numerical coefficient of 3.17 10-3 (R2 value for the fit is 0.9989) for the present sessile droplet, compared to 3.84 10-3 for a free droplet of the same radius range. The confidence in the fit is limited by the frequency resolution of 5 Hz and a small data set of three volumes. Variation of the viscous dissipation coefficient as b µ R-1 explains smaller resonance gain and larger deviation of the measured frequency from the theoretical values for smaller droplets due to larger viscous effects for smaller droplets. Shift of resonance frequencies due to viscous effects was not considered for calculated values in the inset of Figure 3. The step response of the droplet is understood by recognizing that the liquid mass in motion changes with time. Ignoring the effect of viscosity, we have d(mv)/dt = FA - kx, where m is the mass of liquid in motion per unit length of contact line, v is the contact line speed, k is the spring constant of the droplet surface, and x is the displacement of the contact line. In the early stages following,18 cylindrical approximations for shape evolution give m ≈ FR(x + δ), where δ accounts for the initial mass that is set in motion which leads to ( ) FR δk 2 2 3 x þ ð:::Þx þ ::: 2δx þ 1 þ ð4Þ t ≈ FA FA (17) Landau, L.; Lifshitz, M. Fluid Mechanics, 2nd ed.; ButterworthHeinemann: Oxford, UK, 1987. (18) Quere, D. Europhys. Lett. 1997, 39, 533.
Langmuir 2009, 25(8), 4302–4305
In the initial stages (x , 2δ/(1 + δk/FA)), there is a regime of acceleration where the speed increases linearly with time, as seen in Figure 5a. In the limit of small initial mass (δ , 2FA/k) for intermediate stages (2δ , x , 3FA/2k), by neglecting higher order terms, the solution gives a maximum velocity of umax = (FA/FR)1/2 for the inertial motion.18 In the later stages when the higher order terms become significant, the contact line decelerates and the speed reduces. The maximum velocity calculated for an 8 μL droplet actuated by 80 V is 240 mm/s, which is in good agreement with the measured value of 238 mm/s. For electrowetting, FA µ V2, which implies a linear relationship between the maximum speed and the applied potential (νmax µ V), consistent with the experimental data of Figure 5b. Interestingly, the time to reach the maximum speed is measured to be approximately 0.4 ms for voltages from 40 to 80 V for an 8 μL droplet. An order-of-magnitude calculation using R/νmax matches the 0.4 ms value and shows ∼0.1 ms (which is also our measurement resolution) variation between 40 and 80 V.
Summary Presented was an experimental study of the contact line dynamics for electrowetting actuation of a true (i.e., no penetrating wire) sessile droplet. The frequency response showed the effect of contact angle hysteresis in the form of stick-slip motion, leading to normal and abnormal stops of the contact line. The contact line and contact angle were in phase at the first and second resonance mode but out of phase as the response passed through a minimum between the resonances. A model was developed for the first mode, showing a good match with the experimental results. The step response revealed two regimes: a local flow followed by a global flow. In the initial regime, the contact angle changed and the contact line moved rapidly, ensuing flows local to the contact edge. In the subsequent regime, the droplet attained a new equilibrium shape pertaining to the new contact angle through flows global in the droplet. The initial regime was modeled as a system of increasing mass, predicting the maximum velocity of the contact line and its linear relationship with the actuation voltage. Acknowledgment. The authors would like to thank Mr. Wyatt Nelson for his discussions and help with the experimental setup. The current work was supported by DARPA HERMIT program.
DOI: 10.1021/la900077u
4305
