Stick Slip to Sliding Transition of Dynamic Contact Lines ...ˆ’Slip to Sliding Transition of Dynamic Contact Lines under AC Electrowetting D. J. C. M. ’t Mannetje, F. Mugele,

Stick−Slip to Sliding Transition of Dynamic Contact Lines under ACElectrowettingD. J. C. M. ’t Mannetje, F. Mugele, and D. van den Ende*

Physics of Complex Fluids, MESA+ Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, TheNetherlands

*S Supporting Information

ABSTRACT: We show that at low velocities the dynamics ofa contact line of a water drop moving over a Teflon-like surfaceunder ac electrowetting must be described as stick−slipmotion, rather than one continuous movement. At highvelocities we observe a transition to a slipping regime. In theslipping regime the observed dependence of the contact angleis well described by a linearization of both the hydrodynamicand the molecular-kinetic model for the dynamic contact linebehavior. The overall geometry of the drop also has a stronginfluence on the contact angle: if the drop is confined to a disk-like shape with radius R, much larger than the capillary length,and height h, smaller than the capillary length, the advancingangle increases steeper with velocity as the aspect ratio h/R issmaller. Although influence of the flow field near a contact lineon the contact angle behavior has also been observed in other experiments, these observations do not fit either model. Finally, inour ac experiments no sudden increase of the hysteresis beyond a certain voltage and velocity was observed, as reported by otherauthors for a dc voltage, but instead we find with increasing voltage a steady decrease of the hysteresis.

■ INTRODUCTION

Moving contact lines have been studied for decades, yet a fulldescription of the physical processes, especially the dissipationnear the contact line, remains elusive. Contact lines areinvolved in many applications, from oil extraction out of rocksto immersion lithography used to create computer chips. Tounderstand and improve contact line behavior in theseapplications, several local model descriptions exist. Whichmodel to use, however, remains an open question in manycases.1,2

Apart from discussion about which local model to use, bulkliquid flow and flow geometry can also affect the motion of thecontact line.3−6 Despite this experimental evidence, mostcontact line models do not include the large-scale flow. Thismeans each application will need its own model description asflow geometry varies greatly between, for example, immersionlithography and inkjet printing.For immersion lithography, research has focused on the issue

of contact line stability at high velocities as most practicallyrelevant.7,8 For lower velocities the setting can also be used tostudy contact line dynamics more generally.9−12 The results forimmersion lithography have also been directly linked to otherexperiments such as those on sliding drops.13,14 We thereforestudy an immersion-like system, not only as practically relevantbut also as a model system for contact line dynamics. Our focusis to study contact line physics in an immersion-like systemwhen an alternating current (ac) electric signal is applied to

improve the surface wetting by electrowetting (a schematicpicture is presented in Figure 1).15 Earlier research showed thatac electrowetting can reduce the contact angle hysteresis byreducing pinning forces at a dynamic contact line.16,17 In theseac experiments the motion of the contact line was drivenindependently of the applied electrowetting, and the contactline motion was only affected by the change in hysteresis. Thedynamics seemed otherwise independent of the applied voltage

Received: July 25, 2013Revised: October 30, 2013Published: November 12, 2013

Figure 1. A schematic image of our setup: a water drop slides on ahydrophobic electrowetting substrate (glass covered by ITO and thenCYTOP) while being held by a hydrophilic holder, of which only thelower part is shown (green, made of Nylon) which is a distance h(0.5−2 mm) above the surface.

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and frequency. For a dc electric signal a large increase inhysteresis was found instead at high velocities and appliedvoltages.18 In case the contact line is driven by the electricforce,19,20 large changes in contact line dynamics were observedwhen varying the applied voltage, beyond the effect of thedriving force variation.In this research we study the behavior of contact lines under

ac electrowetting and find that at low velocities the dynamics ofthe contact line must be described as stick−slip motion, ratherthan one continuous movement. At high velocities we observe atransition to a slipping regime, which can be described by ahydrodynamic or molecular-kinetic theory for the contact linelinearized around 90°; both give the same dependence ofcontact angle on velocity. In contrast with observations for dcelectrowetting,18 no sudden increase of the hysteresis beyond acertain voltage and velocity has been observed, but instead wefind a steady decrease with increasing applied voltage.Moreover, except for this decrease in hysteresis, the dynamicsof the contact line in the slipping regime does not change withapplied voltage. Finally, we observe that the geometry of thedrop has a surprisingly strong influence on the contact angle: ifthe drop is confined to a disk with radius R = 2 cm, i.e., muchlarger than the capillary length, and height h = 2−0.5 mm, i.e.,smaller than the capillary length, the advancing angle increasesmore rapidly with velocity as the aspect ratio h/R is smaller.

■ AC ELECTROWETTINGTo improve the wetting of a water droplet on a hydrophobicdielectric layer, one applies an electric field over the dielectriclayer between the droplet and an underlying conductor (seeFigure 1). This is called electrowetting.15 Because of this field,the contact angle is modified according to

θ θ η= +cos cos Y (1)

with η = 1/2(c/σ)U2, where θY is Young’s angle, c is the

capacitance per unit area of the dielectric layer, and σ is theinterfacial tension of the drop. The electrowetting number ηrepresents the ratio between the electrostatic energy and thesurface energy. Equation 1 also applies to ac fields if onesubstitutes the rms value for the voltage. However, under acelectrowetting the contact line hysteresis is also reduced asexplained by Li et al.16,17

■ EXPERIMENTOur experimental setup is depicted in Figure 1. A drop is confinedbetween a hydrophilic holder made of nylon and a hydrophobic diskwith a diameter of 300 mm, which rotates in the horizontal plane. Theholder is positioned at 250 mm from the center and a short distance h(0.5−2 mm) above the disk, giving a width to height aspect ratio of thedrop of 10−40. The drop is created by inserting a fixed amount ofliquid via syringe, not shown, through the center of the holder, fillingthe entire gap under the holder, giving an approximate drop volume of150−600 μL. We apply a sinusoidal ac voltage at 10 kHz with varyingamplitude; this high frequency is chosen to avoid resonances in themotion of the liquid−air interface. The disk is made of ITO glasscovered with a 600 nm layer of CYTOP CTL-809M, prepared by spin-coating from a 3% solution by Philips Miplaza (Eindhoven, TheNetherlands). CYTOP is an amorphous fluoropolymer similar toTeflon AF, with better dielectric breakdown characteristics. Therelatively low thickness of the CYTOP layer allows operation at lowvoltage; the equilibrium contact angle of a water drop on this substratecan be varied typically between 115° and 100°. The setup allows us tostudy the advancing and receding contact angle of a single drop atvarying sliding velocity, using a ccd camera that observes the dropletalong the radial direction of the rotating disk at a frame rate of 8

images/s. The contact angle is determined from a linear fit to the air−water interface near the contact line, found using a Matlab routine.The images are recorded and stored on a PC that also controls theapplied voltage and the rotational speed of the disk. To scan thevelocity range (0 < vplate < 1 m/s) during an experiment, a lowacceleration or deceleration of about a = 0.01 m/s2 is applied.

■ RESULTS AND DISCUSSIONI. High-Velocity Regime. We observed the dependence of

the contact angle on the disk velocity for several applied acvoltages (10 kHz frequency) at h = 1 mm. The angles at boththe advancing and receding side were recorded, and we found,as expected, a reduced difference between them for highervoltages.16 In Figure 2, the results for η ≈ 0, 0.2, 0.4, 0.6, and

0.8 have been given; test measurements at η = 1 show evidenceof contact angle saturation.15 These measurements are donewith a constantly increasing velocity. Measurements atvelocities higher than 0.6 m/s are affected by the recedingcontact line behavior, which becomes unstable, and seem to beunreliable (the advancing angle decreases with increasingvelocity). The instability is caused by the fact that at highvelocities the water−air interface at the receding contact linestretches out into a tail. Eventually, this tail becomes so longthat a Rayleigh-plateau-like instability develops which createsdrops from this side. In the literature this is called a pearlinginstability.21 Because of this instability the liquid volume underthe holder decreases, and eventually the advancing sidedetaches from the holder edge sliding away under the holder.However, it appears from our results that the behavior at theadvancing side is already affected by the pearling instability atthe receding line before the advancing contact line detachesfrom the holder edge.At the receding side of the droplet the noise level was too

high to determine the θ versus vplate relation accurately, but dueto the different shape of the interface near the receding contactline, we expect another θ versus vplate relation. Moreover, athigher speeds the drop creeps upward along this side of theholder prior to drop loss, effectively making the drop height at

Figure 2. Contact angle of a drop as a function of velocity for both theadvancing (right, positive velocity) and receding (left, negativevelocity) contact line. Each measurement is the average of five curves.Measurements are for drop height h = 1 mm; the curves represent(from top to bottom) applied electrowetting η ≈ 0 (red), 0.2 (blue),0.4 (green), 0.6 (magenta), and 0.8 (black).

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the receding contact line not well-defined; given the resultspresented in Figure 3, this means the θ(vplate) relation is ill-defined. Still, we can clearly see a reduction in the 0-velocityhysteresis, as seen in prior works.16,17

To investigate the influence of the height of the droplet, wemeasured the slope of the advancing θ versus vplate relation athigh velocity for several voltages. At 0 V the slope isapproximately 0.45 rad s/m for 0.5 mm, 0.24 rad s/m for 1mm, and 0.12 rad s/m for 2 mm high drops. These values canbe compared with those known from the literature. In thevelocity range that we consider the velocity dependence of thecontact angle of a nonconfined drop is known to be welldescribed by a linearization of the molecular kinetic22 or thehydrodynamic model:23

θ θ= + cvA,R A,R,0 plate (2)

In the molecular kinetic (adsorption−desorption) model c = ξ/σ,22 where ξ = kBT/(κλ

3) is the contact line friction, kBT isthermal energy, κ is the adsorption−desorption frequency, andλ is the distance between adsorption sites on the substrate. Wetake the molecular absorption frequency κ to be approximately1/t, where t is the time for a molecule-sized sphere to diffusethe distance λ equal to one molecular dimension (0.25 nm).This gives κ ≈ 14 MHz, and one obtains for c ≈ 0.26 rad (m/s)−1. In the hydrodynamic model with θA,R,0 ≈ π/2, c =9(θA,R,0)

−2Ca ln(Lm/a),23 where Ca = μvplate/σ is the capillary

number, Lm is the macroscopic length scale, and a is the sliplength; ln(Lm/a) is usually estimated to be about 10,corresponding to (sub)nanometer slip length and tens ofmicrometers for the macroscopic scale. This results in a typicalvalue for c = 0.28 rad (m/s)−1, which is the same as the valueobtained from the molecular kinetic model. The values for theslope we obtain for our confined drops are of the same order ofmagnitude, especially for the 1 mm high drop.As can be seen in Figure 3, the best-fitting high-velocity

slopes from Figure 2 are independent of the applied voltage;i.e., the contact line friction coefficient c in eq 2 is unchanged.However, this coefficient does depend on the height of thedrop; the mechanism for this height dependence is currently

unclear. As the liquid flow is driven at the two solid surfacesfrom the drop holder at the top and the moving plate at thelower side, we expect that the velocity gradient between thesetwo surfaces in the drop will be stronger when the drop is moreconfined. According to the work of Blake et al.,5,6 an increase inthe local velocity gradients near the contact line, imposed bythe global flow field, will change the contact angle.Unfortunately, their experiments show a nonmonotonouschange in contact angle as a function of the global flow field,which makes it impossible for us to make a prediction and testit against our results; later experimental4 and theoretical work24

suggests a decreasing advancing contact angle with increasingflow velocity near the contact line, which is the opposite of ourobservation. However, in those studies no distinction has beenmade between increasing flow velocity and increasing flowgradient (as the geometry remains the same), while in theexperiments presented here the flow gradient is likely highestfor the lowest height, while the flow volume is largest for thelargest height.

II. Low Velocity Regime: Stick−Slip to Sliding. For theadvancing angle, we observe in Figure 2 at low velocities atendency to a steeper slope in the θ(vplate) relation when avoltage is applied. For η = 0 (0 V), this effect is not observed.At η ≈ 0.2 (22 V) it extends to around 0.15 m/s before it levelsoff and at η ≈ 0.4 (31 V) to 0.3 m/s. At η ≈ 0.6 (43 V) it is notas clear, but there might still be a transition around 0.4−0.5 m/s. For η ≈ 0.8 (49 V) the velocity range may be too small toidentify such a region.To check if this effect is caused by the constant acceleration

of the disk (a = 0.01 m/s2) during the experiment, we calculatethe Bond number, Bo = ρaL2/σ, where a is the dropacceleration and L its length (2 cm) while ρ is the liquiddensity, giving a value of Bo = 0.06. Hence, surface tensionforces are much stronger than inertial forces. Tests at a = 0.002m/s2 show the same dependence of drop shape and contactangle on the velocity, confirming this conclusion. Furthermore,we performed experiments with drops of 1 and 2 mm height atη ≈ 0.2, in which the velocity was first (continuously) increasedand next decreased at a = 0.01 m/s2. The increasing anddecreasing velocity branch show the same contact angle versusvelocity behavior, as can be seen in Figure 4, which also clearlyshows the steeper slopes at low velocities. Therefore, weconclude that we indeed measure in all our experiments thesteady state contact angle versus velocity relation.

Figure 3. Slope of contact angle versus velocity curves as a function ofdrop height for η ≈ 0 (black squares), 0.2 (red circles), 0.4 (bluetriangles), and 0.6 (green circles). The slope does not depend on theapplied voltage.

Figure 4. Upon increasing and decreasing the drop velocity withconstant acceleration, both a 1 mm (blue) and 2 mm (black) highdrop show only a dependence of θ on contact line velocity and nodependence on acceleration (η ≈ 0.2).

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To understand the observed contact angle behavior of amoving droplet under electrowetting, we have to consider tworegimes. In the first regime the contact line will be pinned(vrelative = 0 with vrelative the difference in speed between contactline and plate), and the contact angle θ increases from θ > θR(receding contact angle) toward θA (advancing contact angle)while Δx = vplateΔt, where vplate is the disk speed under theholder and Δx the displacement of the advancing contact lineover the time Δt. In the second regime θ = θA or θ = θR whilethe contact line is sliding over the substrate, and vrelative (and soΔx(t)) is controlled by the momentary value of dθ/dt. Both θAand θR depend on the relative speed vrelative and the voltageU(t).In our analysis, we will use eq 2 for the velocity dependence

of θA,R, while assuming that only θA,R,0 changes with the appliedvoltage, i.e.

θ θ η= +U tcos ( ) cos (0) ( )A,R,0 A,R,0 (3)

The net result of this “two regime” behavior has beenschematically depicted in Figure 5. Here the receding andadvancing contact angles have been plotted as a function oftime at a given speed.

As long as the contact line is pinned, the contact anglefollows one of these curves. But at a certain moment the rate ofchange of the contact angle becomes too large, and the contactline cannot follow any more. This is indicated by the almoststraight lines in between both harmonic curves. At high voltage,a second effect becomes important. The minimum advancingangle is lower than the receding angle at 0 V. So, for very lowvelocities, the contact angle at the advancing side would belower than the receding angle for part of the electrowettingperiod. This leads to the contact line receding against theaverage disk motion, increasing the contact angle at vplate = 0while not increasing it at high velocities.To analyze the motion in this intermediate regime, we

consider the air−liquid interface near the contact line in moredetail. Because of the stick−slip character of the contact linemotion the liquid near the contact line is periodically driven bythe sum of the electric and pinning forces and a surface wave isgenerated that propagates along the interface,24 as can beobserved form Figure 6 (and the movies in the Supporting

Information) for a 1 kHz electric signal and 1 mm drop height.Similar experiments at 2 and 4 kHz show a decreasingwavelength and amplitude with frequency.Assuming harmonic potential flow, this wave can be

characterized as a capillary wave21 with a frequency ω = 2ωel(which is twice the electric driving frequency because thecontact angle depends quadratic on the applied voltage) and awavenumber q = 2π/λ = (ω2ρ/σ)1/3. Using q−1 as a length scalefor the penetration depth of this wave, we can relate thedisplacement x(t) of the contact line with the momentarycontact angle θ(t):

θ π− = qx ttan[ /2] ( ) (4)

where x is the displacement of the contact line on the substrate(see Figure 6). Equation 4 will be used to predict themomentary contact angle θ(t) when the contact line displace-ment is known or vice versa. If the contact line is pinned, werewrite eq 4 as

θ π= +t qx t( ) arctan[ ( )] /2 (5)

where x(t) = x(tp) + (t − tp)vplate and tp the moment of pinning.If θ = θA,R, we rewrite eq 4 as

θ π= −−x t q t( ) tan[ ( ) /2]1A,R (6)

Using eqs 5 and 6, we calculate θ(t) for given vplate and η(t) asexplained in the Supporting Information. Figure 5 shows anexample result for vplate = 0.16 m/s and ηrms = 0.8 while ω/2π =10 kHz.As a consequence, the time-averaged contact angle θ will be

smaller at low velocities than the average advancing angle; inFigure 5 it decreases from 1.7 to 1.6 rad. Thus, the averagecontact angle θ decreases more strongly with decreasing diskvelocity than indicated by eq 2.Analyzing the surface waves, of which Figure 6 shows a

snapshot, reveals to us a value for the wavenumber q oralternatively the wavelength λ = 2π/q. Measurements on a 1mm thick drop at 1, 2, and 4 kHz (frequency of the appliedvoltage) show a decreasing wavelength and amplitude withincreasing frequency: 160, 90, and 60 μm at 1, 2, and 4 kHz,respectively. From the definition of q we obtain wavelengths of480, 300, and 190 μm, about 3 times larger than the measuredwavelength. In view of the simplifying assumptions we made inour modeling the agreement is reasonable. For the 10 kHzsignals used in these experiments the oscillation is only a fewmicrometers in amplitude, and as such the wavelength is not

Figure 5. Calculated time trace (blue) of the instantaneous contactangle at the advancing side of the drop at vplate = 0.16 m/s and η = 0.8as a function of time. In black the advancing and in red the recedingangle (determined by the sinusoidal electric field) are shown. Forextremely low velocities the instantaneous angle is effectively constantduring transition from the advancing to the receding curve. For highvelocities it follows the black curve (= the advancing angle caused bythe applied EW). Contact line friction is taken into account to alter theadvancing/receding angle.

Figure 6. Oscillation of water (right on each picture, black) directlyabove the wafer (bottom of the picture) due to a 1 kHz appliedvoltage. (A) shows the maximum extension on the wafer and (B) theminimum. The oscillation is seen to travel up the liquid−air interface.Movies of this oscillation are shown in the Supporting Information.

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detectable with our optical system. In our calculations to fit theresults, which were obtained applying a voltage at 10 kHz, werely on the calculated value for the wavelength using thedefinition of q given four lines before eq 4.We fitted our model calculations of the time-averaged

advancing contact angle θA(vplate,Urms) to the experimental data,using the contact line friction coefficient ξ, the penetrationdepth q−1, and dielectric thickness d as fitting coefficients. Theresults have been presented in Figure 7. The behavior of thecontact angle versus velocity can be divided in three regimes:(1) high velocity, (2) low velocity and low voltage, (3) lowvelocity and high voltage. In regime 1 the velocity is highenough for the contact angle to follow the momentaryadvancing angle over the full cycle, and the electric contributionto the reduction of θ is just given by η(t) (as discussed in lastsection); no sticking of the contact line occurs in this regime. Inregime 2 the velocity is too low for the contact angle to followthe momentary advancing angle over the full cycle. It lacks themomentary advancing angle until this itself is sufficientlyreduced again (see Figure 5). Because of this, an additionalreduction occurs. In the limit for v → 0 the total reduction isdetermined by 2η(t).16 In regime 3, the momentary value of theadvancing contact angle becomes smaller than the 0 V recedingangle, so the lagging contact angle follows for some part of thecycle the receding angle (see Figure 5), suppressing furtherreduction of the contact angle.In regimes 1 and 2 (η = 0.0, 0.2, 0.4) the agreement between

our model and the experiment is remarkably good, taking theassumptions made in the modeling into consideration. Inregime 3, however (η = 0.6, 0.8), the contact angle reduction isoverestimated for v < 0.2 m/s. Moreover, the fitted values for ξ(0.017 and 0.008 Pa·s for 1 and 2 mm height, respectively) arein reasonable agreement with the value expected from themolecular kinetic model (0.015 Pa·s), while the fitted dielectricthickness d = 382 nm is also sufficiently close to 600 nm asexpected from the spin-coating process. The fitted inversewavenumber is q−1 = 10 μm, while we obtained for thetheoretical value q−1 = 26 μm. This is in agreement with thefactor of 3 difference for the calculated and experimentallyestimated wavelengths of the surface waves at 1, 2, and 4 kHz;see the discussion in context with Figure 6.

■ CONCLUSION

In conclusion, we show that at low velocities the dynamics ofthe contact line under ac electrowetting must be described as astick−slip motion. This motion is characterized by an

oscillatory motion coupled to the linear average motion ofthe contact line, which our model qualitatively describes. Athigh velocities we observe a transition to a full slipping regime.We show that a linearized hydrodynamic or molecular-kineticmodel for the contact line gives the same dependence of thecontact angle on velocity, and in the slipping regime theobserved contact angles are close to those predicted by thesemodels. Moreover, we find no sudden increase of the hysteresisbeyond a certain voltage and velocity as previously found byNelson et al.18 but instead observe a steady decrease as theapplied voltage increases. Finally, we observe that the geometryof the drop has an influence on the contact angle: the advancingangle increases more rapidly with velocity as a drop is confinedmore into a thin disk with a radius larger than the capillarylength but a height smaller than the capillary length. Wetentatively attribute this effect to hydrodynamic assist, butfurther research is needed to show it.

■ ASSOCIATED CONTENT*S Supporting InformationMovies displaying the oscillation shown in Figure 6 and thesimulation procedure for Figure 7. This material is available freeof charge via the Internet at http://pubs.acs.org.

■ AUTHOR INFORMATIONCorresponding Author*E-mail [email protected] (D.v.d.E.).

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis work is part of the research program “Contact LineControl during Wetting and Dewetting” (CLC) of the“Stichting voor Fundamenteel Onderzoek der Materie(FOM)”, which is financially supported by the “NederlandseOrganisatie voor Wetenschappelijk Onderzoek (NWO)”. TheCLC program is cofinanced by ASML and Oce.́

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Figure 7. Observed (lines) and calculated (symbols) contact angle (η ≈ 0 (red and dots), 0.2 (blue and triangles), 0.4 (green and squares), 0.6(magenta and diamonds), 0.8 (black and right triangles)) as a function of velocity for a drop height of 1 (left) and 2 (right) mm. The model fit is forinsulator thickness d = 382 nm (nominal 600 nm), penetration depth q−1=10 μm, and contact line friction coefficients ξ = 0.017 Pa·s (1 mm) and0.008 Pa·s (2 mm).

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