Langmuir 2008, 24, 11847-11850
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Phase Selection in Capillary Breakup in AC Electric Fields Florent Malloggi,* Dirk van den Ende, and Frieder Mugele Physics of Complex Fluids, Faculty of Science and Technology, IMPACT and MESA+ Institute, UniVersity of Twente, P.O. Box 217, 7500AE Enschede, The Netherlands ReceiVed May 19, 2008. ReVised Manuscript ReceiVed July 15, 2008 We study the detachment of conductive aqueous drops in ambient oil from an electrode in the presence of ac electric fields. Making use of the electrowetting effect, we determine the charge of the detached sessile drops. Drops are found to be discharged at high ac frequency in line with earlier predictions. At low frequencies, we find a wide but unexpected charge distribution displaying a frequency-dependent nonzero minimum charge. This observation is explained in terms of the stabilization of capillary bridges in electric fields, which prevents the hydrodynamic pinch-off for certain phases of the ac field.
1. Introduction The generation and manipulation of liquid drops has become a paradigm in microfluidics in recent years. While standard systems rely purely on hydrodynamic forces,1 electric field assisted approaches ranging from electrospray ionization2 to electrowetting3 have become increasingly popular owing to their enhanced flexibility and additional functionality. In particular, combinations of hydrodynamic and electrostatic approaches appear very promising.4-7 While the intricate coupling between hydrodynamic and electric forces has been the focus of any research in electrospraying and jetting since the early days of Zeleny8 and Taylor9 (for recent reviews, see refs 10-12), it has been largely ignored by the electrowetting (EW) community despite the fact that splitting and merging of drops in high electric fields are key functionalities of EW-based Lab-on-a-Chip systems.13,14 Only a few early studies point to the importance of local electric fields during the generation of small satellite drops at very high voltage.15,16 The early electrohydrodynamic literature focused on the analysis of the linear stability of perfectly conductive liquid cylinders in axisymetric electric fields. By considering the effect of the fluid viscosity, Saville17 showed that a tangential electric stress can (1) Song, H.; Chen, D. L.; Ismagilov, R. F. Angew. Chem., Int. Ed. 2006, 45, 7336–7356. (2) Fenn, J.; Mann, M.; Meng, C.; Wong, S.; Whitehouse, C. Science 1989, 246, 64–71. (3) Mugele, F.; Baret, J. C. J. Phys.: Condens. Matter 2005, 17, R705. (4) (a) Collins, R. T.; Jones, J. J.; Harris, M. T.; Basaran, O. A. Nature Physics 2008, 4, 149–154. (b) Basaran, O. A. AIChE J. 2002, 48, 1842–1848. (5) Ganan-Calvo, A. M. Phys. ReV. Lett. 2007, 98, 134503. (6) Link, D. R.; Grasland-Mongrain, E.; Duri, F.; Sarrazin, A.; Cheng, Z.; Cristobal, G.; Marquez, M.; Weitz, D. A. Angew. Chem., Int. Ed. 2006, 45, 2556– 2560. (7) Malloggi, F.; Vanapalli, S. A.; Gu, H.; van den Ende, D.; Mugele, F. J. Phys.: Condens. Matter 2007, 19, 462101. (8) Zeleny, J. Phys. ReV. 1917, 10, 1–6. (9) Taylor, G. I. Proc. R. Soc. London A 1964, 280, 383–397. (10) Saville, D. A. Annu. Fluid. Mech. 1997, 29, 27–64. (11) Lopez-Herrera, J. M.; Ganan-Calvo, A. M. J. Fluid. Mech. 2004, 501, 303–326. (12) Collins, R. T.; Harris, M. T.; Basaran, O. A. J. Fluid. Mech. 2007, 588, 75–129. (13) Cho, S. K.; Moon, H. J.; Kim, C. J. Microelectromech. Syst. 2003, 12, 70–80. (14) Fair, R. B. Microfluid Nanofluid 2007, 3, 245–281. (15) Vallet, M.; Vallade, M.; Berge, B. Europ. Phys. J. B 1999, 11, 583–591. (16) Mugele, F.; Herminghaus, S. Appl. Phys. Lett. 2002, 81, 2303–2305. (17) Saville, D. A. Phys. Fluids 1971, 14, 1095–1099.
stabilize the cylinder against long-wavelength perturbations promoting very long liquid jets.18,19 In a more recent series of studies, Mugele and co-workers observed self-excited oscillations in electrowetting experiments, during which liquid drops periodically detach from an electrode and reattach to it.20-22 It turned out that the occurrence of the phenomenon is controlled by the dynamics of the pinch-off in the presence of ac electric fields. For values of a dimensionless parameter R, measuring the relative importance of electrical and hydrodynamic time scales with respect to the applied ac frequency, f, above ≈1 drops oscillate, whereas for smaller values they do not (the exact definition of R will be given below). According to the model that was presented, oscillations occur only if the drops are discharged during the pinch-off. While the discharging of the drops for R > 1 was inferred convincingly from the occurrence of drop oscillations, experiments at low R displayed some evidence of a coupling between the pinch-off dynamics and the electric field that could not be explained.23 In this article, we measure directly the charge of the drops after detachment in ac electric fields. For R > 1, our results quantitatively confirm the predicted discharging mechanism. For R < 1, we find by analyzing the charge distribution of thousands of drops that the electric field affects the hydrodynamics of the pinch-off and that the electrohydrodynamic coupling thereby determines the charge of the detached drop.
2. Experimental Section We used a standard electrowetting configuration consisting of a conductive Si substrate, an insulating layer of 1 µm thick thermally grown SiO2 coated with a hydrophobic layer of Teflon AF (thickness d ≈ 100 nm; Dupont), and a conductive drop (deionized water with dissolved NaCl, σ ) 1 mS/cm), as sketched in Figure 1. An alternating (ac) voltage U ) 2Urms sin(2πft) with root-mean-square value Urms ) 0-50 V and frequency f ) 1-100 kHz is applied between the substrate and the drop via a Pt wire (radius re ) 125 µm) immersed into the drop. The setup is immersed in an ambient bath of silicone oil (Fluka AS4, viscosity η ≈ 6 mPa · s) in order to both prevent (18) Saville, D. A. Phys. Fluids 1970, 13, 2987–2994. (19) Mestel, A. J. J. Fluid. Mech. 1994, 274, 93–113. (20) Klingner, A.; Herminghaus, S.; Mugele, F. Appl. Phys. Lett. 2003, 82, 4187–4189. (21) Baret, J. C.; Mugele, F. Phys. ReV. Lett. 2006, 96, 016106. (22) Baret, J. C.; Decre´, J. M.; Mugele, F. Langmuir 2007, 23, 5173–5179. (23) Baret, J. C. Ph.D. Thesis, Faculty of Science and Technology, University of Twente, 2005.
10.1021/la801541z CCC: $40.75 2008 American Chemical Society Published on Web 08/30/2008
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Figure 1. Sketch of the experimental setup and equivalent {electrode + drop + dielectric} electrical circuit: (1) capacitor, (2) resistance (of capillary bridge) in series with capacitor, and (3) open circuit. Inset: Capillary bridge close to breakup. re is the radius of the electrode.
evaporation during the experiment and reduce contact angle hysteresis. The oil-water interfacial tension is γ ) 38 mN/m. While the wire is deeply immersed into the drop, the whole system {drop + dielectric + substrate} can be represented electrically by a simple capacitor, in which the drop forms a deformable electrode that adjusts its shape to minimize its free energy upon applying a voltage. Once the voltage is applied, the experiment consists of pulling up the Pt electrode vertically at a constant velocity (≈1 cm/s). During this process, a thin capillary bridge is formed, which eventually breaks and leaves a sessile drop behind on the substrate. While we pull out the wire, we measure the current in the system by recording the voltage across a 10 kΩ resistance in series with the drop. Finally, we measure the contact angle θ from side view images (recorded by a CCD camera) of the detached drop. All phases of the process along with the corresponding electric equivalent circuits are illustrated in Figure 1. While the electrode is immersed into the drop, the potential is well-defined and hence the apparent contact angle θ decreases with U following the well-known electrowetting equation:3
cos θ(η) ) cos θY + η
(1)
Here θY ≈ 170° is Young’s angle (at zero voltage), and η ) 0rUrms2/ 2dγ is the dimensionless electrowetting number which measures the relative strength of electrostatic and surface tension forces. 0 and r ()2) are the vacuum susceptibility and the dielectric constant of the insulating layer, respectively, and d is the dielectric thickness and γ the surface tension. After the pinch-off, the drop is electrically isolated with a fixed charge q. In this case, the equilibrium contact angle is given by
cos θ(q) ) cos θY +
q2d 2A2γ0r
(2)
where A is the solid-droplet interfacial area. Equation 2 implies that the charge of the sessile drop can be determined directly from the contact angle.
3. Results and Discussion Let us first focus on the distribution of contact angles (and hence charge) after the pinch-off. We performed a series of experiments with different voltages Urms and driving frequencies f. For each set of parameters (Urms, f) several hundred runs were recorded. In Figure 2, we show a typical normalized contact angle distribution P(θ) of the final contact angle for a rather low ac frequency (f ) 1 kHz, corresponding to R )0.19; see below for the exact definition of R). θ* indicates the equilibrium contact
Figure 2. Normalized contact angle distribution P(θ) for a fixed rootmean-square (rms) voltage Urms ) 35V. Experimental data: (circles) R ) 0.19; (solid line) phase distribution in case of equally distributed phase; (dots) sine-square phase distribution; (dashed line) (sine-square mulitplied with a gap gate) phase distribution (see Figure 5). Inset: P(θ) for intermediate frequencies and fixed voltage Urms ) 35 V (left and right vertical scales are different); from left to right: R ) 0.58, 0.77, 1.54, 1.93, 7.7.
angle according to eq 1 with the wire being immersed. Upon pulling out the wire, the contact angle sometimes increases and sometimes decreases, indicating that the charge of the detached drop can be either smaller or larger than the rms charge in the attached state. Overall, the distribution is rather broad (θ ≈ 90°-150°, for the present data) with one well-defined maximum. Interestingly, however, the maximum value of θ ) θmax is clearly smaller than θY, indicating that there is a minimum charge on the dropsan effect that we will analyze in detail below. For lower voltage, the distributions are shifted toward higher contact angles but display the same qualitative behavior, including in particular a maximum value θmax < θY. As we increase the frequency at constant voltage, both the maximum of P(θ) and θmax shift toward higher θ. Simultaneously, the probability distribution becomes narrower (see inset of Figure 2). For the highest frequency used (f ) 40 kHz; R ) 7.7), the distribution is sharply peaked at θY, implying that the residual charge of the drop is zero in that case. Under the latter conditions, we also find the drops oscillating periodically between the attached and detached state, as in ref 21. To summarize the results: we find that the drops are quasidischarged for high frequencies while there is always a finite charge remaining on the drop at low frequencies with a broad distribution corresponding to the observed range of contact angles (θ ∈ [θmin, θmax] < θY). Let us now compare the observed discharging of the detached drops with the prediction of the model presented in ref 21. Briefly, the model is based on the RC-circuit model (number 2 in Figure 1) with a time-dependent resistance of the meniscus Rn(t) ) R0(t/tc)-µ that diverges at the pinch-off (defined as t ) 0) due to the well-known algebraic singularity of the radius of the capillary bridge.24 (Here µ ≈ 1 and R0 ) (σre)-1, with re the electrode radius.21) The differential equation for the dimensionless charge Q ) q/CUmax, and dimensionless time ˜t ) 2πft reads
R1+µt˜ -µ
dQ + Q ) sin(t˜ + φ) dt˜
(3)
In this equation, φ is the phase of the electric field at the moment of pinch-off, which is not determined a priori. Here R (24) Eggers, J. ReV. Mod. Phys. 1997, 69, 865–926.
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Figure 3. Dimensionless droplet charge after breakup |Qmax| vs the dimensionless frequency R. Comparison of numerical result and experiments. Experiment: (triangles) Urms ) 25 V, bullet Urms ) 35 V. Model: (full line) µ ) 1. Inset: Dimensionless charge Q vs φ for several R (0.19, 0.77, 1.9, 7.7).
) 2πf(R0Ctµc )(1/(1+µ)) is the dimensionless control parameter that determines the behavior of the solution. It describes the relative magnitude of the characteristic time scales in the problem, the period 1/f of the ac field, the RC time constant R0C of the equivalent circuit, and the hydrodynamic time scale tc of the pinch-off. If we solve eq 3 for the charge of the drop after the pinch-off, the result will thus depend on the choice of φ; or equivalently, the choice of φ determines the charge of the detached drop. The inset of Figure 3 shows Q as a function of φ for a series of values of R. Obviously, the absolute value of Q varies between zero and a maximum value Qmax. However, the φ dependence is only practically relevant for R < 1. For larger R, Qmax decreases to zero such that the final charge is always small, independent of φ. The reduction of Qmax with increasing R is shown in the main panel of Figure 3. The origin of this discharging mechanism for R > 1 is discussed in detail in ref 21. The present experimental results extracted from Figure 2 and similar data for other voltages constitute a direct confirmation of that earlier prediction. Only for the largest values of R, we observe minor deviations from the model: the measured values of Qmax do not quite drop to zero. We attribute this deviation to the functional behavior of eq 2 and the large value of θY, for which small errors in θ translate into rather large uncertainties in q. In the following, we focus on the situation for R < 1, for which the detached drops carry a finite charge. We are mainly interested in the relation between the charge distribution and the associated distribution of φ. In the simplest case, one might assume that the phase of the electric field at the moment of pinch-off is completely random. In that case, for a sinusoidally varying applied voltage, one would expect a distribution of U at t ) 0 of P(U) ) 1/π((2Urms)2 - U2)-1/2,25 reflecting the fact that U(t) spends most of its time close to its extremal values. Combined with eq 1, this expression leads to a probability distribution of θ
P(θ) )
sin θ 2π√(cos θ - cos θY)(cos θmin - cos θ)
(4)
(In deriving this expression, we assumed that the capacitance C is constant during the pinch-off, which is justified since the global response time of the drop is long compared to the pinch-off time.22) The result of eq 4, integrated over typical experimental binning intervals of 5°, is shown as solid line in Figure 2. It (25) Lynn, P. A. An Introduction to the Analysis and Processing of Signals; Macmillan Press Ltd.: London, 1989.
Figure 4. Last period of the electrical current before the breakup for low R (0.19). Four different phase detachment φi ) 2πf∆ti are represented, φi is the specific phase offset for each one of the four curves. The current drops to zero when the drop detaches. The reference curve corresponds to the current without pinch-off (Pt electrode immersed in the drop).
Figure 5. Phase distribution P(φ) from experiment (R ) 0.19, circles), compared with the approximated distributions as used in the analysis: (dashed line) sine-square distribution, (dash/dotted line) (sine-square mulitplied with a gap gate) distribution. Inset: Preferential phase of the current at the pinch-off.
clearly fails to describe the experimental data. Hence we conclude that the value of φ is not randomly distributed,26 which implies that the hydrodynamics of the pinch-off process must be coupled to the electric field. To study the distribution of φ directly, we recorded the current I through a resistance R ()10 kΩ) in series with the drop (see Figure 1). Figure 4 shows a typical current curve as a function of the nondimensional time ˜t + φ with ˜t running from infinity to 0 at the pinch-off. Several periods before the pinch-off, the current is perfectly in phase with the applied voltage, i.e., φ ) 0. Upon approaching the breakup, however, an increasing phase shift φ appears, which results from the fact that the RC time constant of the system becomes comparable with 1/f. From curves such as the one in Figure 4, we extracted the values of φ. A typical distribution P(φ) for R , 1 is shown in Figure 5. P(φ) displays two maxima around φ ) 0 and around 180°; in between, around φ ) 90° and 270°, two pronounced gaps appear. The capillary bridges clearly avoid pinching off for these values of φ, the values that correspond to the maxima of the driving voltage (and hence the undisturbed current), see inset of Figure 5. (26) To check the validity of eq 4, we performed a series of independent experiments, in which we disconnected a drop with a deeply immersed wire from the ac power supply using an electronic switch actuated at random times (data not shown). In that case, we did indeed find a contact angle distribution with two peaks close to θmin and close to θY, in agreement with eq 4.
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Before interpreting this finding, we check the consistency between the phase distribution measured electrically and the contact angle distribution measured optically. To do so, we insert the electrically measured phase distribution into eq 3 and calculate the corresponding charge and contact angle distribution. To simplify the analysis, we approximated P(φ) by a sine-square function as indicated by the dotted line in Figure 5. The result, shown as the dotted line in Figure 2, shows that the electrical and the optical measurements are indeed consistent with each other. Both measurements indicate that the moment of breakup of a capillary neck in an ac electric field is synchronized with the phase of the ac field in such a way that the detached drops always carry a finite amount of charge. What is the origin of the phase selection mechanism observed here? Qualitatively, it is well-known that electric fields E exert a stress, described by the Maxwell stress tensor, on liquid surfaces.27 In electrowetting, it is usually assumed that the liquid in the drop is perfectly conductive, which implies that the electric stresses are oriented normal to the interface and are only relevant close to the three-phase contact line.3,28 In the present experiments with ac voltage, however, the decrease in the measured current upon approaching the pinch-off indicates that a substantial fraction of the applied voltage drops across the meniscus (see Figure 1). Hence, tangential electric fields must be present. If we combine the typical current amplitude of order 10-5 A with the conductivity of the aqueous phase and the typical dimensions of the capillary bridge (length, radius: O(re)), we find that the electric field along the capillary bridge is of order 10 V/10-4 m ) 105 V/m. (External normal fields are much weaker owing to the absence of a “counter electrode” in the radial direction.) The effect of longitudinal fields on the stability of perfectly conductive liquid cylinders can be understood in terms of a linear stability analysis,18 which takes into account the field distribution as well as the internal flow fields for a small sinusoidal perturbation of a cylindrical liquid jet. The surface charge, due to the conductivity, induces (27) Landau L. D.; Lifschitz, E. M. Lehrbuch der Theoretischen Physik. Band 8: Elektrodynamik der Kontinua; Akademie Verlag: Berlin, 1990. (28) Buehrle, J.; Herminghaus, S.; Mugele, F. Phys. ReV. Lett. 2003, 91, 086101.
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electrical shearing stress as soon as the interface is slightly deformed. The dielectric constant difference between the drop phase and the ambient oil leads to a normal stress. In the present experiments, it turns out that the electric field has indeed a stabilizing effect for perturbations of long wavelength. In fact, the stabilizing effect originates mainly from the strong dielectric contrast and the local normal electric fields resulting from the perturbation and the local distribution of charges. When the Maxwell stress is strongest, the stabilization is most efficient. Since the stress is proportional to E(t)2, one expects a sinesquare behavior for the stabilizing force in line with the experimental observation in Figure 5.
4. Conclusion The stabilization and phase selection mechanism described here is very general and it is expected to apply to any pinch-off process in the presence of ac electric fields, including electrowetting-driven drop generation devices7 and contact line instabilities,15,16 as well as ac electrospray ionization. However, two important requirements have to be met: first, R < 1 is required to prevent the drop discharging; second, the voltage drop across the capillary bridge has to be sufficiently large to generate a sufficiently strong Maxwell stress. The latter requirement is fulfilled in many electrowetting experiments, but not necessarily in other geometries, for which the capacitance between the detaching drop and the environment can be substantially smaller. In that case, the current through the capillary bridge (for a given geometry) is also smaller and may not be sufficient to stabilize the capillary bridge. A more detailed and quantitative analysis of the break up process including simultaneous current measurements and a direct visualization using high-speed video imaging of the breaking capillary bridge will be reported in a forthcoming publication. Acknowledgment. We acknowledge financial support by the Institute for Mechanics Process and Control Twente (Impact) as well MESA+ Institute for Nanotechnology at Twente University. LA801541Z
