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Capillary Bridges in Electric Fields Anke Klingner, Juergen Buehrle, and Frieder Mugele* Department of Applied Physics, University of Ulm, Albert-Einstein-Allee 11, 89081 Ulm, Germany Received October 31, 2003. In Final Form: May 7, 2004 We analyzed the morphology of droplets of conductive liquids placed between two parallel plate electrodes as a function of the two control parameters electrode separation and applied voltage. Both electrodes were covered by thin insulating layers, as in conventional electrowetting experiments. Depending on the values of the control parameters, three different states of the system were found: stationary capillary bridges, stationary separated droplets, and periodic self-excited oscillations between both morphologies, which appear only above a certain threshold voltage. In the two stationary states, the morphology of the liquid is modified by the electric fields due to electrowetting and due to mutual electrostatic attraction, respectively. We determined a complete phase diagram within the two-dimensional phase space given by the control parameters. We discuss a model based on the interfacial and electrostatic contributions to the free energy. Numerical solutions of the model are in quantitative agreement with the phase boundaries found in the experiments. The dynamics in the oscillatory state are governed by electric charge relaxation and by contact angle hysteresis.
Introduction There is an increasing desire to manipulate tiny amounts of liquid, which stems mainly from the demands of biotechnology and combinatorial chemistry. In both fields, a large number of chemical reactions need to be performed in order to obtain the desired information or product.1 It was shown that scaling down the volumes of material handled in these reactions increases the efficiency and reduces costs tremendously. Various approaches are being followed in order to satisfy this demand for a reliable liquid manipulation scheme on the micrometer scale.2-10 One of them is so-called open microfluidic systems, which are characterized by the presence of free fluid-fluid interfaces and three-phase contact lines. The most promising approach to manipulate liquid in this type of device makes use of electric fields, which can be applied via suitably patterned electrodes and switched at high rates with little power consumption.4-9 Frequently, actuation of liquid is achieved by means of the electrowetting effect, which describes the reduction of the contact angle as a function of an applied voltage:9 Consider a droplet of a conductive liquid sitting on a flat electrode, which is covered with a thin insulating layer of thickness δ and * Corresponding author. E-mail address: physik.uni-ulm.de.
frieder.mugele@
(1) Microsystem Technology in Chemistry and Life Science; Manz, A., Becker, H., Eds.; Springer: Berlin, 1998; Vol. 194. (2) Whitesides, G. M.; Stroock, A. D. Phys. Today 2001, 54, 42. Pfohl, T.; Mugele, F.; Seemann, R.; Herminghaus, S. ChemPhysChem 2003, 4, 1291. (3) Jakeway, S. C.; de Mello, A. J.; Russell, E. L. Fresenius’ J. Anal. Chem. 2000, 366, 525-539. (4) Prins, M. W. J.; Welters, W. J. J.; Weekamp, J. W. Science 2001, 291, 277. (5) Pollack, M. G.; Fair, R. B.; Shenderov, A. D. Appl. Phys. Lett. 2000, 77, 1725-1726. (6) Sandre, O.; Gorre-Talini, L.; Ajdari, A.; Prost, J.; Silberzan, P. Phys. Rev. E 1999, 60, 2964-2972. (7) Jones, T. B.; Gunji, M.; Washizu, M.; Feldman, M. J. J. Appl. Phys. 2001, 89, 1441-1448. (8) Cho, S. K.; Moon, H.; Kim, C.-J. J. Microelectromech. Syst. 2003, 12, 70-80. (9) Quilliet, C.; Berge, B. Curr. Opin. Colloid Interface Sci. 2001, 6, 1. (10) Pollack, M. G.; Shenderov, A. D.; Fair, R. B. Lab Chip 2002, 2, 96-101.
dielectric constant r. If a voltage U is applied between the droplet and the electrode, the contact angle θ(U) is decreased compared to Young’s contact angle θy. This is due to the additional electric free energy contribution of the capacitor formed by the solid-liquid interface and the flat electrode. The decrease of θ follows the Lippmann equation
cos θ ) cos θy +
0rU2 2σlvδ
(1)
Here, 0 and σlv are the electric susceptibility of vacuum and the surface tension of the liquid, respectively. For various practical reasons, it is useful to consider devices where the liquid is confined between two parallel solid substrates. In this case, the fluid typically forms capillary bridges spanning the gap between the two substrates.5,6,8,10 In the present work, we investigate the stability of such capillary bridges as a function of the voltage applied to the electrodes. After describing the experimental details briefly, we present the experimental results, which include a regime where capillary bridges break and re-form periodically.11 Subsequently, we present an electrostatic model. We show that the experimental results can be reproduced by minimizing the system’s free energy. Technical Details The experimental setup consists of a parallel plate capacitor as shown schematically in Figure 1a. The plates are made of glass, which is covered with a layer of conductive indium tin oxide (ITO). The ITO electrodes are covered with a thin electrically insulating layer (thickness δ ≈ 3-5 µm) of amorphous fluoropolymer (AF1601, r ) 1.93312). Insulating layers were produced by dip coating from a 6 wt % solution of AF1601 in FC-75 following a recipe described in ref 13. Before the experi(11) Klingner, A.; Herminghaus, S.; Mugele, F. Appl. Phys. Lett. 2003, 82, 4187-4189. (12) Teflon AF, Amorphous Fluoropolymer, DuPont, P.O. Box 80702, Wilmington, DE 19880-0702. (13) Seyrat, E.; Hayes, R. A. J. Appl. Phys. 2001, 90, 1383-1386.
10.1021/la036058j CCC: $27.50 © 2004 American Chemical Society Published on Web 07/02/2004
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domain software calculates numerically the shape of a free surface (represented by a triangulation grid) such that its free energy is at a minimum. Here, we used the Surface Evolver in order to calculate the electric field distribution. This can be achieved if the electrostatic field energy is represented by a specific free surface energy. In this case, the surface profile at the energy minimum represents a solution of the Laplace equation, that is, the electrostatic potential, for the given set of boundary conditions. More details are described in Appendix A. The numerical accuracy was tested by comparing numerical results for simple geometries (e.g. spherical and cylindrical capacitor) to known analytic solutions. The characteristic edge length of the triangulation grid was adjusted such that the accuracy of the field energy was better than 0.1%. Experimental Results
Figure 1. (a) Sketch of the experimental setup with the applied voltage U at the movable, parallel electrodes (black) covered by insulating layers (light gray). The left configuration shows the separated droplets (liquid in dark gray) of height h. The right one shows a capillary bridge connecting the two substrates. (b) Equivalent circuit diagram where the switch S changes between the two configurations shown in part a: separated droplets, S open; bridge configuration, S closed. C1 and C2 represent the capacitances of the insulating layers and the air gap, respectively (C1 . C2).
ments the two capacitor plates were aligned using the reflections of a laser beam. We placed various amounts (between 1 nL and 4 µL) of a 1:1 volume mixture of glycerol and a 2.25% NaCl solution in Millipore water inside the capacitor. (For simplicity, air was used as surrounding medium.) The mixture had an advancing contact angle of 100° ( 2°. The contact angle hysteresis was between 5° and 10°. Since the capillary length κ ) (σlv/gF)1/2 is 2.4 mm (σlv ≈ 65 mJ/m2, the density F ) 1.13 g/cm3, and g is the gravitational acceleration), gravity could be neglected, except for minor corrections for the largest droplets. The electrical conductivity of the liquid was 0.4 S/m, unless otherwise noted. It was thus considered as a perfect conductor. To prevent charge accumulation at the solidliquid interface, which may spoil reproducibility, we connected the ITO electrodes to an ac power supply. Typically, it was operated at a frequency between f ) 1 and 10 kHz with a root-mean-square amplitude between 0 and 2200 V. Droplets or capillary bridges were viewed from the side using a CCD camera with a zoom objective. An additional CCD camera allowed for imaging the solidliquid interface through the top glass plate. The separation d between the electrodes could be varied using a stepping motor with a nominal step width of 0.625 µm. The stepping motor, the CCD cameras, and the amplitude of the applied voltage were computer controlled via a custom written LabView program.14 Numerical calculations of the field energy were carried out using the Surface Evolver,15 version 2.14. This public (14) LabView Graphical Programming for Instrumentation from National Instruments. (15) Brakke, K. Exp. Math. 1992, 1, 141.
Two types of experiments were performed: either we kept the distance between the plates d constant and varied the applied voltage U, or we kept U constant and varied d. (a) Voltage Dependence. Let us consider the first case of zero voltage. We prepared two symmetric droplets of total volume V and adjusted d such that the gap between the two droplets adopted some desired value, typically of the order of the height h0 of the droplets or less. Since θy is close to 90°, the droplets are essentially half spheres at zero voltage. When the voltage on the electrodes was increased, the droplets became elongated (Figure 2a) along the field axis while the contact angle remained constant to within 5°. The liquid surfaces could be approximated with high accuracy by ellipsoids. This is in agreement with observations of free droplets in homogeneous electric fields.16 Parts b and c of Figure 2 show the drop height h(U) normalized to h0 versus U. The droplets are elongated up to a critical voltage Uc and critical elongation (h/h0)c. At higher voltage they become unstable and snap together. We investigated the dependence on the system size for a series of droplets of different volumes. Two typical examples of a more extended data set are shown in Figure 2b and c. The data points in Figure 2b represent three experiments, where we varied the volume and the plate separation simultaneously such that the normalized separation d ˜ ) d/V1/3 remained constant. (The solid lines show model results that will be discussed below.) Hence, the system looked geometrically similar every time. Under these conditions, the critical elongation (h/h0)c was essentially constant, while the critical voltage Uc increased with increasing system size. When we kept the drop volume constant instead and increased d ˜ , both (h/h0)c and Uc increased (Figure 2c). The reason is that the electrostatic attraction decreases with increasing d ˜ . Hence, the snap on is shifted to higher Uc. Moreover, we found that (h/h0)c increases linearly with Uc. When we prepared the system in the bridge state before increasing U from zero to a finite value, the contact angle was found to decrease substantially (see Figure 3). As we will explain below, this contact angle reduction is due to electrowetting, which takes place in the bridge state but not in the droplet state. As the contact angle decreased, the area of the solid-liquid interface increased and the minimum width of the bridge decreased due to volume conservation. Depending on the plate separation, capillary bridges became unstable and broke up into two separated droplets beyond some critical voltage (data not shown). (16) Garton, C. G.; Krasucki, Z. Proc. R. Soc. (London), Ser. A 1964, 280, 211-226.
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Figure 3. Contact angle vs applied voltage in the bridge state. Experimental data (square) and analytical curve (solid line) according to the Lippmann equation. Inset: Snapshots at two different voltages.
Figure 4. Phase diagram. Normalized plate separation vs normalized voltage. Symbols: 9, experimental results for separation of bridge rupture d ˜ off; O, experimental results for separation of bridge formation d ˜ on; 2 and 3, lower and upper boundaries of the oscillatory regime, respectively (δ ) 5 µm, f ) 10 kHz, V ) 30 nL, 1:1 mixture of glycerol and saltwater). Lines: model predictions (s, separation of bridge rupture d ˜ off; - - -, separation of bridge formation d ˜ on for V ) 30 nL; - - -, d ˜ on for V ) 30 pL). Figure 2. Deformation of liquid droplets vs applied voltage. (a) Snapshots at 0, 500, 1000, 1340 V (left to right) for V ) 0.3 µL and d ˜ ) 1.84. The dots represent an ellipsoidal fit of the droplet contours. (b) Droplet height h/h0 for d ˜ ) 1.84: V ) 0.009 µL (circles), 0.3 µL (triangles), and 2.2 µL (squares). (c) h/h0 for V ≈ 2.5 µL: d ˜ ) 1.29 (circles), 1.56 (triangles), and 1.84 (squares). Solid lines: numerical results. Arrows indicate the instability of the droplets.
(b) Dependence on Plate Separation. Let us now consider the second type of experiment. We prepared again two separate droplets at zero voltage and at large separation (left in Figure 1a). Upon reducing d ˜ , the two droplets approached each other and eventually snapped on to form a capillary bridge. We denote the critical ˜ on is separation for snap on as d ˜ on. At zero voltage, d essentially determined by geometric overlap of the two ˜ on ) 2[3(1 - cos θy)/[2π(2 droplets.17 The latter occurs at d ˜ 1, we made a surprising observation:11 At large separation, only separated droplets were stable, and at small separation, only the bridge phase was stable. At intermediate separations (within the region between the down and the up triangles in Figure 4), however, the system oscillated periodically between the bridge phase and the droplet phase at a rate between one and a few tens of cycles per second. For pure glycerol at an ac frequency of 1 kHz, the oscillations were slow enough to be monitored in detail by video microscopy (see Figure 5). We observed up to a few hundred oscillation cycles. At some point, the oscillations stopped, presumably due to progressive sample degradation, which is not uncommon in electrowetting experiments at high voltage.21,22 We also attribute the significant scatter of the boundary lines of the oscillatory region (up and down triangles) in Figure 4 to surface deterioration. Nevertheless, oscillatory behavior was observed for many different samples with volumes ranging from 0.03 to 1 µL. A close inspection of the video data, from which Figure 5 was extracted, shows that sparks appear on those video frames immediately preceding the formation of a new bridge. Given the dielectric strength of air (≈3 × 106 V/m at zero frequency) and the width of the air gap of 100 and 200 µm immediately prior to the jump to contact, this implies a voltage drop of several hundred volts between the droplets. (In contrast, no indications of sparks were seen when the bridges broke apart.) In Figure 6, we show a typical trace of the contact angle and of the normalized diameter w ˜ ) w/V1/3 of the solidliquid interface as a function of time within the oscillatory regime. It is clearly seen that the contact angle decreases rather abruptly upon bridge formation. Subsequently, the liquid slowly spreads essentially at constant contact angle. Once the capillary bridge breaks, the contact angle relaxes approximately exponentially toward Young’s angle while the contact line simultaneously recedes. During most of the relaxation phase, the liquid-vapor interface adopts a spherical cap shape; that is, it is mechanically relaxed. Model The experiments showed that both separated droplets and capillary bridges are destabilized upon increasing U. To understand the observations qualitatively, we consider the electrical equivalent circuit shown in Figure 1b. The (21) Vallet, M.; Vallade, M.; Berge, B. Eur. Phys. J. B 1999, 11, 583591. (22) The applied voltage in this region is typically around 1 kV or beyond, which is well within the saturation regime of electrowetting. After the experiment, the contact angle at zero voltage is typically reduced below 90°, which is a clear indication of surface degradation.
Figure 6. Contact angle θ and normalized diameter w ˜ of Asl vs time for bridges (squares) and droplets (circles). Vertical dashed and dotted lines indicate the change from bridge to droplets and vice versa.
capacitances of the insulating layer and the air gap are represented by C1 and C2, respectively. For separated droplets, corresponding to the situation where the switch S is open, the qualitative explanation is obvious: Each droplet has the same electrical potential as its adjacent electrode due to capacitive coupling across the thin insulating layer (capacitance C1 ≈ 10 pF . C2). Hence, the contact angle is equal to θy. Furthermore, there is a strong electric field between the droplets. Thus, the droplet surfaces attract each other and deform. The equilibrium deformation is determined by the balance between electrostatic and capillary forces. At the critical voltage, the electrical attraction exceeds the maximum restoring force due to surface tension and the droplet state becomes unstable. In the presence of a capillary bridge (switch S closed), the liquid assumes an electric potential that is half the applied voltage for symmetry reasons. Hence, there is a potential drop of U/2 across each insulating layer. This causes the observed reduction in contact angle due to the electrowetting effect (eq 1, Figure 3). As we will show below, destabilization of the bridges can be understood solely by considering this effect of the applied voltage on the contact angle. From a formal point of view, the equilibrium configuration of the liquid is found by minimizing the free energy functional F of the system at constant volume. The functional is given by
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F[A] ) Fif + Fel )
Klingner et al.
1
∑i σiAi - ∫2EBDB dV
(2)
The first term in the middle and on the right-hand side (rhs) represents the interfacial energy. It includes the energies of all interfaces: i ) sl (solid-liquid), lv (liquidvapor), and sv (solid-vapor). The second term in the middle and on the rhs stands for the electrostatic energy in the system. E B and D B represent the electric field and the displacement field, respectively. The integration extends over the total volume of the system. Note that the electric energy enters with the negative sign because of the contribution from the power supply.23 As noted above, we neglect the contribution of gravity because the typical system size is small compared to the capillary length. Furthermore, we consider the liquid as a perfect conductor; that is, the electric field vanishes inside the liquid. Additional simplifications that apply for the two different morphologies independently will be discussed below. (a) Separated Droplets. Above, we showed (cf. Figure 2a) that the shapes of the separated droplets in the experiments can be approximated by ellipsoids. In accordance with the experiments, we assume that the contact angle is independent of the applied voltage. With these approximations, the functional minimization of eq 2 reduces to a simple minimization of the free energy F as a function of one variable, which characterizes the elongation of the droplet along the field direction. We chose to parametrize the geometry by the normalized height h ˜ ) h/V1/3 of the droplet. Since the system size scales with V1/3, the contribution of the interfacial energies is given ˜ ), where fθ(h ˜ ) is a function of h ˜ only. by Fif ) σlvV2/3fθ(h The electric contribution Fel was computed numerically. We assumed that each droplet is at the potential of its own electrode. Then, we solved the Laplace equation ∇2φ ) 0 for the electrostatic potential φ everywhere outside liquid. More details of the computational method are described in Appendix B. We calculated Fel for various values of h ˜ and d ˜ and interpolated the numerical results by a continuous function. In Figure 8, we plot the total free energy F ) Fif + Fel versus h/h0 for a series of different voltages. It is clearly seen that F has a minimum at h ) h0 for zero voltage. For U > 0, the minimum shifts to larger values of h. For U f Uc, the minimum disappears and turns into an inflection point. At this voltage, the droplet phase becomes unstable and capillary bridges form spontaneously. We repeated the same calculation for the set of experimental parameters in Figure 2. The solid lines in Figure 2 represent the numerical results, which were obtained without any fitting parameter. The critical values of the instability are reproduced within a typical scatter of approximately 10%. The model predicts a higher elongation and a smaller critical voltage than seen in the experiments. Both errors can be explained consistently if we take into account the finite contact angle hysteresis. In the presence of hysteresis, the driving force for contact line motion must exceed a certain critical value before the line begins to move. This reduces the actual droplet elongation as compared to the calculated equilibrium shape. Furthermore, the critical elongation obtained experimentally is expected to be smaller, since both the finite step width of the voltage increments and possible mechanical perturbations reduce the experimental value for the critical elongation. The dashed line in Figure 4 represents the stability limit for the droplet phase obtained from this numerical (23) Landau, L. D.; Lifschitz, E. W. Theoretische Physik Vol. VIII: Elektrodynamik der Kontinua; Akademie-Verlag: Berlin, 1976.
Figure 7. Plot of ln(Asl2(cos θ - cos θy) vs time according to eq 6. The symbols refer to a series of consecutive relaxation cycles in the droplet state. The solid line is a fit to eq 6 with τ ) 0.4 s.
Figure 8. Free energy F vs droplet height h (V ) 1 µL, d ˜ ) 2.39): s, U ) 0 V; - - -, Uc/3; ‚‚‚, 2/3Uc; -‚-, Uc ) 3221.5 V.
analysis with θ ) 90°. At any point below this line, the droplet phase is unstable and bridges form spontaneously. The numerical result (again: no fitting parameters) agrees well with the experimental result up to U ˜ ≈ 1. The dotted line in Figure 4 shows the same stability limit for a droplet with a 1000 times smaller volume. Obviously, the stability limits for the droplet phase can be approximated by straight lines with slopes that depend on the liquid volume.24 (b) Capillary Bridges. If the liquid forms a capillary bridge, it assumes the potential U/2. Hence, the electric field between the footprints on the bridge on the insulating layers and the adjacent electrodes is very high. Since the thickness of the insulating layers is approximately 3 orders of magnitude smaller than the plate separation, most of the electric field energy is located inside the dielectric layers. For the computation of Fel in the bridge state, we included only this contribution; that is, we wrote
0rAsl U δ 2
Fbridge ) -2 el
2
()
(3)
This is the same approximation as in the classical derivation of eq 1.9 The first factor 2 on the rhs accounts for the two solid-liquid interfaces; the second one (in the denominator) accounts for the fact that only half the voltage drops across each insulating layer. The solid line in Figure 3 shows a plot of eq 1 for the specific parameters of our experiment. Obviously, the observed behavior of the contact angle in the bridge phase follows the Lippmann (24) The slopes can be shown to scale with V-1/6. The stability limits can be approximated by d ˜ c ≈ (6/π)1/3 + 12U/V1/6.
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equation up to U ≈ 300 V. The assumption, that only the field energy from the capacitor contributes is thus valid.25 Hence, the stability of capillary bridges as a function of U is mapped onto the stability of ordinary capillary bridges (without electric field) as a function of the contact angle. The latter problem, however, has been studied in detail in the literature; for example, see refs 26 and 27. For instance, it is well-known that the meridional profiles of rotational symmetric capillary bridges are either nodoids, catenoids, or unduloids, depending on whether the pressure is negative, zero, or positive, respectively. Depending on the contact angle, different instability modes control the stability of the bridge. For θ , 90°, bridges always break symmetrically. For θ ≈ 90°, however, an asymmetric mode dominates, which leaves behind a small and a large droplet on the two surfaces. This is consistent with our observations of asymmetric bridge rupture at small voltage.28 In the following, we will focus on the rupture of capillary bridges at relatively high voltage, that is, for θ , 90°. For this case, Willet et al.29 gave a simple expression for the rupture distance as a function of θ. Adapted to our geometry, it reads d ˜ off ) (1 + θ/2). If we combine this result with eq 4, we obtain the bridge instability line, which is given by
1 d ˜ off ) 1 + arccos(cos(θy) + U ˜ 2) 2
(4)
The result, which does not contain any fitting parameters, is plotted as a solid line in Figure 4. The curve is universal. In particular, it does not depend on the absolute drop volume. The initial decrease at low voltage reflects a decrease in contact angle following the Lippmann equation. For U ˜ . 1, the bridge instability line is horizontal at d ˜ off ) 1. This corresponds to the case of zero contact angle. In practice, this line should lie at d ˜ off ) 1 + θsat/2, where θsat is the saturation angle of electrowetting.9 For any point above the solid line, the bridge phase is predicted to be unstable. For 0.2 < U ˜ < 1, the agreement between experiment and model is satisfying. From the video images, we know that the two data points at lowest voltage correspond to a situation with asymmetric bridge rupture. The deviation from the model is thus not surprising. For the present volume, the bridge instability line (solid) and the droplet instability line (dashed) intersect around U ˜ ) 0.8 and d ˜ ) 1.4. Since bridges are unstable above the solid line and droplets are unstable below the dashed line, both phases should be (dynamically) unstable in the region between the two curves for U ˜ > 0.8. Indeed, this is the region where we observed the oscillatory behavior in the experiments. The deviations between the upper and the lower experimental boundaries of the oscillatory region and the model predictions are not surprising in view of both sample degradation at high voltage and contact angle saturation (see above). (c) Dynamics of Bridge Oscillations. Since surface tension is the driving force for the morphological relaxation, the oscillation rate (≈1 s-1 in Figures 5 and 6) should be compared to both the characteristic viscous relaxation (25) Additional field-induced effects are limited to a region of diameter δ around the contact line. For details, see: Buehrle, J.; Herminghaus, S.; Mugele, F. Phys. Rev. Lett. 2003, 91, 086101. (26) Langbein, D. W. Capillary surfaces: shape-stability-dynamics, in particular under weightlessness; Springer: Berlin, New York, 2002. (27) Boucher, E. A. Rep. Prog. Phys. 1980, 43, 498-546. (28) In the case of asymmetric rupture, we always found the larger droplet on the bottom electrode. This symmetry breaking was the clearest manifestation of gravity throughout the experiments. We also note that gravity increases the range of contact angles that is dominated by the asymmetric instability mode. (29) Willett, C. D.; Adams, M. J. Langmuir 2000, 16, 9396-9405.
time τvisc ) ηV1/3/σlv (η ) viscosity) and the characteristic inertial relaxation time τin ) (FV/σlv)1/2. If we insert the parameters for pure glycerol (as used in Figures 5 and 6), both time scales are ≈10 ms. Both the spreading of the droplet after bridge formation and the contact angle saturation after bridge rupture are much slower. Therefore, they must thus be governed by other relaxation processes. Let us first consider the contact angle saturation. When the capillary bridge breaks, the two remaining droplets are electrically insulated. In the case of dc voltage, the droplets would carry a charge of Q ) (C1(U/2), respectively. In this case, the contact angle is given by
cos(θ) ) cos(θy) + Q2/(2σlvCAsl)
(5)
To allow for the contact angle to relax back to the Young’s angle, the droplets must discharge. In the present case of ac voltage, the precise value of the initial charge is expected to depend on the details of bridge rupture. Yet, complete contact angle relaxation requires complete discharging of the droplets, independent of the initial charge. Most likely, discharging occurs across the insulating layers due to their finite conductivity, as indicated by the dashed resistors in Figure 1b. That is, Q(t) ) Q0 exp(-t/RC) with the time constant RC ) r0Fins. Here R ) Finsδ/Asl and C ) r0Asl/δ are the resistance and the capacitance between the liquid and the counter electrode, respectively. (Fins is the specific resistivity of the dielectric coating.) Inserting this into eq 5, we find for the time dependence of the contact angle
cos θ ) cos θy +
δQ02 20rσlvAsl
2
t exp -2 RC
(
)
(6)
Note that both θ and Asl depend on time. Figure 7 shows that the contact angle relaxation in the droplet state in Figure 6 is indeed consistent with eq 6. We obtain a time constant of 0.4 s. Using dielectric spectroscopy, we found that the tangent of the dielectric loss angle ∆ is approximately 0.01 for our insulator films at f ) 1 kHz. This gives rise to an RC time constant of τ ) RC ) (ω tan(∆))-1 ) 20 ms (for ω ) 2π(1 kHz)), which is about 1 order of magnitude faster than expected. Despite this slight discrepancy, which may be due to the fact that contact angle relaxation takes place at zero frequency, whereas the dielectric response was measured at 1 kHz, it is reasonable to assume that the relaxation in the droplet state is determined by the electrical discharging of the droplets. What determines the relaxation rate in the bridge phase? According to Figure 6, relaxation takes place essentially at constant contact angle. At first glance, this seems surprising. Usually, spreading (or dewetting) of liquid droplets involves a simultaneous relaxation of the contact angle due to the one-to-one correspondence between the droplet radius and the contact angle for spherical caps of fixed volume. For capillary bridges, however, there is an infinite manifold of morphologies for any given contact angle and volume, each characterized by a different pressure. Therefore, morphological relaxation at constant contact angle is possible.30 In the presence of finite contact angle hysteresis, however, it is very difficult for the system to move the contact line once the forces at the contact line are equilibrated. This is the qualitative explanation for the slow relaxation of the (30) In the present experiments, we found that the pressure of the capillary bridges (as determined from the mean curvature) was always positive; that is, the bridges are unduloids. During spreading, the pressure increased roughly by a factor of 2.
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bridges. A quantitative description of the relaxation process would involve a detailed modeling of both the liquid morphology and contact angle hysteresis, which is beyond the scope of the present paper.
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comparing the free energies for the separated droplets and the capillary bridge as a function of the applied voltage, the liquid volume, and the normalized plate separation. Appendix A
Discussion Overall, the model describes the experimental observations very well. In particular, the field-induced elongation of separated droplets, the occurrence of an oscillatory regime, and the contact angle relaxation after bridge rupture are reproduced correctly. From a theoretical point of view, the existence of an oscillatory regime may seem surprising. In our model, we used notions related to thermodynamic equilibrium. At low voltage, the droplet and the bridge state are thermodynamic phases. The boundary lines in Figure 4 represent the stability limits of these phases in a thermodynamic sense. However, the oscillatory regime does clearly not represent a thermodynamic equilibrium state. Energy dissipation is involved in the repeated creation and rupture of capillary bridges. This energy is supplied by the power supply. When the contact angle decreases immediately after the formation of a capillary bridge, electric work is done against surface tension. This energy is dissipated later when the droplets relax back to their original shapes after bridge rupture. The results presented here could be useful for various microfluidic applications. The most obvious application is to split a given amount of liquid, which is present in the form of a capillary bridge, into two separate droplets. Given a suitable manipulation scheme for individual droplets on planar electrodes, each droplet can be processed further individually. Bridge rupture can also be used to transfer material between the two surfaces. If, for instance, one surface has a small orifice through which liquid can be fed into the device, a large droplet can be formed on that surface. By applying a voltage, a bridge can first be formed and subsequently be broken into two droplets of equal size, thus leading to net material transfer. Even in the absence of net liquid transfer, the repetitive formation and rupture of capillary bridges between two droplets will lead to progressive mixing of the material in the two droplets. This process is expected to depend on the highly nonlinear hydrodynamic flow patterns during bridge oscillation. Experiments in that direction are underway in our laboratory. Conclusion In conclusion, we presented a detailed experimental and numerical study on the stability of capillary bridges of conductive liquids between planar electrodes covered with a thin insulating layer. In the bridge phase, the contact angle on the substrates was shown to decrease due to the electrowetting effect. For the case of two separated droplets, the electric field leads to a mutual attraction. At low voltage, the droplets become elongated along the field axis with the equilibrium elongation being determined by the balance of electric forces and surface tension. Beyond a geometry-dependent critical voltage, the droplet surfaces become unstable, snap together, and form a bridge. At high voltage and within a certain range of plate separations, the system was found to oscillate periodically between the bridge state and the droplet state, at rates between one and a few tens of cycles per second. The oscillation frequency is determined by slow relaxation processes, which involve electric charge relaxation after bridge rupture and morphological relaxations of capillary bridges in the presence of contact angle hysteresis. Experimental findings were reproduced numerically by
Here, we describe a method for how the Surface Evolver, which is usually used to calculate equilibrium shapes of free interfaces, can be adapted in order to solve partial differential equations in a more general context. The Surface Evolver performs a numerical variation of a specific free energy functional under appropriate boundary conditions and constraints. Interfaces are represented discretely by sets of vertexes (represented by three spatial coordinates x, y, and z), edges joining adjacent vertexes, and facets (bound by neighboring edges). The free energy is composed of integrals over the facets and the edges. For the standard case, when the Evolver is used to calculate the equilibrium surface of a given amount of liquid, the energy is simply given by an integral over the surface. Let us now consider a general functional F[Φ] ) ∫F(Φ,∂Φ/ ∂q1,∂Φ/∂q2,q1,q2) dq1 dq2 of a scalar potential Φ with coordinates q1 and q2. If the functional density F involves derivatives of Φ no higher than first order, it is well-known from variational calculus that F takes an extremal value if F obeys the Euler-Lagrange equation
∂ ∂Φ
F-
∂
∑i ∂q
∂F
i
( )
∂
)0
∂Φ ∂qi
Here, we want to solve the Laplace equation ∇2Φ ) (∂2Φ/∂q12 + ∂2Φ/∂q22) ) 0, which is the Euler-Lagrange equation arising from the minimization of the electrostatic free energy.31 To solve this partial differential equation, the potential Φ is represented as a surface, where we choose in particular x ≡ q1 and y ≡ q2 and for the potential z ) Φ(x,y). The normal vector b n at each point of the surface is then given by the gradient of z - Φ(x,y); that is
(
b n) -
∂Φ ∂Φ ,,1 ∂x ∂y
)
In the Surface Evolver representation, the length of the normal vector of each facet is given by the facet’s area. Minimization of the functional density F, expressed in terms of x, y, and z and the normal vector components nx, ny, and nz, yields a solution of the corresponding partial differential equation. The Surface Evolver integration method for such general free energy densities is called facet•general•integral. Considering the electrostatic potential Φ in the absence of charges (which is given by Poisson’s equation ∇2Φ ) 0 subject to boundary conditions), the corresponding free B 2 dV, where the electric energy is given by Fel ) (0/2)∫E field E B ) ∇Φ. Expressed in cylindrical coordinates (r,φ,ζ), Fel reads
Fel )
[
∫dr dφ dz r 20 (∂r∂ Φ)
2
+
2
(∂ζ∂ Φ)
+
1 ∂ Φ r2 ∂φ
(
)] 2
If the problem involves an azimuthal symmetry, ∂Φ/∂φ ) 0 and integration over φ yields an energy density (31) Note that the free energy is only one particular choice for the functional to be minimized.
Capillary Bridges in Electric Fields
0 ∂ 2 ∂ Φ + Φ F ) 2π r 2 ∂r ∂ζ
Langmuir, Vol. 20, No. 16, 2004 6777 2
[( ) ( ) ]
The minimum of the corresponding free energy solves the Laplace equation in polar coordinates (r,ζ), which is
1 ∂ ∂ ∂2 r Φ + 2Φ)0 r ∂r ∂r ∂ζ
(
)
With the coordinates x ≡ r and y ≡ ζ, we have
nx2 + ny2
F ) π0x
nz2
(In fact, the length of b n does not need to be unity.) This is the exact form of the free energy specified in the Surface Evolver macro program. Figure 9 shows a typical triangulation used in the Surface Evolver. Appendix B To save computational time, we made use of the symmetry of the problem and calculated the rotational symmetric potential field for the geometry indicated in Figure 9.32 We solved the Laplace equation ∆φ ) 0 for the electrostatic potential φ with the bottom electrode and the droplet at φ ) 0 and with φ ) 1/2 for the symmetry plane halfway between the plates. This corresponds to a dimensionless voltage of φ ) 1 and a dimensionless volume of V ) 1. For fixed values of h ˜, d ˜ , and θ, the field strength scales with V-1/3. Furthermore, it is proportional to U. The field energy, which is obtained by integrating the square of the field strength over the system volume, thus scales quadratically with the applied voltage and linearly with the characteristic length V1/3. If we denote the (32) We neglected the minor contribution due to the insulating layers.
Figure 9. Geometry for the simulations and a typical triangulation pattern used by the Surface Evolver. The white bar indicates a break in the horizontal axis.
normalized volume element and the normalized gradient operator by dV ˜ and ∇ respectively, Fel reads
Fel(h ˜ ,d ˜ ,θ,U ˜ ,V1/3) ) -
0U2V1/3 (∇φ˜ (h ˜ ,d ˜ ,θ))2 dV ˜ 2
∫
Acknowledgment. We thank Stephan Herminghaus for discussions. We are indepted to Renate Nikopoulos for help with the sample preparation and during experiments. This work was supported by the German Science Foundation within the priority program “Wetting and Structure Formation at Interfaces”. Supporting Information Available: Video sequences of the oscillatory regime. This material is available free of charge via the Internet at http://pubs.acs.org. LA036058J
