ARTICLE pubs.acs.org/Langmuir
Dual-Frequency Electrowetting: Application to Drop Evaporation Gauging within a Digital Microsystem Johannes Theisen† and Laurent Davoust*,‡ † ‡
Microfluidics Group, Laboratory of Geophysical and Industrial Fluid Flows (LEGI), University of Grenoble, 38041 Grenoble, France Electromagnetic Processing of Materials (EPM) Group, Materials and Processes Science and Engineering Laboratory (SIMaP), Grenoble Institute of Technology (Grenoble INP), 38402 Saint Martin d’Heres, France ABSTRACT: This paper addresses a method to estimate the size of a sessile drop and to measure its evaporation kinetics by making use of both Michelson interferometry and coplanar electrowetting. From a high-frequency electrowetting voltage, the contact angle of the sessile droplet is monitored to permanently obtain a half-liquid sphere, thus complying perfectly with the drop evaporation theory based on a constant contact angle (Bexon, R.; Picknett, R. J. Colloid Interface Sci. 1977, 61, 336350). Low-frequency modulation of the electrowetting actuation is also applied to cause droplet shape oscillations and capillary resonance. Interferometry allows us to measure a time-dependent capillary spectrum and, in particular, the shift in natural frequencies induced by drop evaporation. Consequently, diffusive kinetics of drop evaporation can be properly estimated, as demonstrated. Because of coplanar electrode configuration, our methodology can be integrated in open and covered microsystems, such as digital lab-on-a-chip devices.
’ INTRODUCTION Microfluidic applications using the electrowetting technique1,2 for displacement of droplets3,4 are subject to a number of difficulties: contact line friction during displacement,57 surface contamination by biological species (biofouling),8,9 and evaporation,10 which is obviously not shared with microchannel microfluidics.11 This last issue is one striking reason why, typically, in digital microsystems for bioassays, drop evaporation needs to be controlled by means of imaging techniques: the time-dependent profile of the drop is recorded; wetting radius and angle of the drop are extracted; and the data are compared to the available theory. The aim of the present paper is to furnish a fully integrated methodology for drop evaporation gauging within an automated digital microsystem. Among important objectives, there is the essential need to avoid (time-consuming) drop imaging and to address open and covered microsystems. Among typical non-imaging techniques for monitoring the evaporation of sessile drops, there is, for instance, the well-known quartz crystal microbalance (QCM), which is able to measure the mass variation of millimeter-sized drops.12,13 A different technique to monitor the evaporation rate of minuscule droplets in the 1 μm range was presented by Arcamone et al.:14 a microresonator is used to transduce the mass change into a frequency variation in a microelectromechanical system (MEMS). This method also bypasses imaging analysis. Another method uses a cantilever sensor developed by Liu and Bonaccurso,15 which is a non-imaging technique able to measure mass, radius, and contact angle simultaneously. This method is adapted to a 1 μm sized sessile droplet (diameter between 20 and 80 μm of pure liquids and mixtures). A further non-imaging r 2011 American Chemical Society
technique is developed by Gong and Kim,16 taking advantage of the capacitance measurement of a droplet in a capped lab-on-a-chip between top and bottom plates. When the footprint area of the drop is deduced, its volume may be estimated. Concerning digital lab-on-a-chips, the QCM and the capacity measurement can be easily integrated in state-of-the-art systems. These two techniques are at the same time adapted to common droplet volumes for lab-on-a-chips. For a summary of the above methodologies, please refer to Table 1. In this paper, we present a new methodology based on both electrowetting on dielectrics (EWOD) as a transduction mechanism and interferometry to measure evaporation of a sessile drop, with a volume ranging between 0.5 and 10 μL, typically. Here, our methodology is compatible with configurations of variable confinement (open or partially or fully closed systems) provided that the sessile drop remains free to oscillate. A dedicated electrowetting chip with coplanar electrodes4 is developed. Hydrophobic and dielectric layers on the electrodes acting in combination with electrowetting enable monitoring of an effective wetting angle for an aqueous drop at a constant value of θ = π/2 during the entire evaporation process. The drop is then made to oscillate by perturbation of the actuation voltage between the buried electrodes. The resonance spectrum of this capillary oscillation depends upon the radius, which may be calculated progressively in the course of evaporation. Received: September 17, 2011 Revised: November 2, 2011 Published: November 05, 2011 1041
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Table 1. Summary of the Available Different Mass/Volume Sensing Techniques technique
transduction
measurement
drop volume
QCM12,13
quartz resonance frequency
electrical
some microliters
microresonator14
resonator frequency change
electrical
attoliters
cantilever15
solid vibrations
optical
femtoliters
capacitance16
ring oscillator circuit
electrical
some microliters
dual-frequency EWOD (this paper)
capillary oscillations
optical
some microliters
Figure 1. Setup of the interferometry system. The laser beam crosses the optical system to form two first-order reflections on the drop apex and the solid substrate, which are detected by the APD. (Inset) Electrode pair in use for electrowetting actuation.
This technique offers the advantages of the QCM with respect to the volume range and is fully compatible with integration requirements for digital lab-on-a-chips, as well as offering the advantage of permanently adjusting the contact angle (at θ = π/2, for instance, in this paper) using EWOD at a high frequency in alternating current (AC) mode (15 kHz). Therefore, the two classical evaporation regimes17 can be applied: constant contact angle mode or constant contact radius mode. A more practical advantage compared to the imaging method is the possible measurement of evaporation in a confined environment with a drop distribution, which might hinder proper imaging. This situation is usually encountered in lab-on-a-chips, because parallelization of tasks is one of the main goals of these systems. This paper is focused on coplanar electrowetting of a sessile drop in contrast to a setup with needle18 or catenary19 counter electrodes, for both of which an interferometry measurement would be impossible. The coplanar electrode configuration is also a reference configuration for development of EWOD-based lab-on-a-chips. In this case, the imaging method is nevertheless developed for the purpose of validation of our methodology.
’ ELECTROWETTING AS A TRANSDUCTION MECHANISM Electrowetting on dielectrics is a technology commonly used in new microfluidic applications, such as liquid lenses20
and EWOD-based displays.21 A more recent application consists of a cooling methodology for active heat-transfer management on processor units.22 EWOD is also commonly used in digital lab-ona-chips for moving, cutting/merging drops, or mixing solubilized reagents among many operations. Although a large part of biotechnological applications in lab-on-a-chips may be performed with silicone oil as a filler medium, some applications, such as lyophilization or chemical noses, still require an ambient gas phase.23 In this case, we show how EWOD-induced drop shape oscillations can be provided without the need for any external needle electrode (popular configuration) but from a coplanar electrode configuration fully compatible with microsystem integration.24 We observe that these oscillations and the subsequent capillary resonance induced in low-frequency EWOD regime form a convenient transduction mechanism for gauging evaporative mass transfer. With regard to bioassay applications, for instance, the need should be pointed out for controlling drop solvent quantities when performing chemical reactions. This is particularly relevant when chemical and evaporation kinetics are of the same order. Coplanar electrowetting electrode geometry consists of two half moons with a 4 mm base width facing each other with a 100 μm gap (Figure 1). This geometry is used for all of the experiments. A second electrode geometry (base width of 2 mm and gap width of 3 μm) is used only for providing the data of Figure 2. The electrodes are covered by two coatings, an insulating layer 1042
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Figure 2. Wetting characteristics of a 1.3 μL drop in the coplanar electrode configuration. The wetting curves and wetting radii correspond to a 3 μm gap (O) and 100 μm gap (3) currently used in this paper. The black line corresponds to a fit of the LippmannBerge eq 4, with doil = 260 nm as a free parameter.
(Si3N4, silicon nitride, thickness dSi3N4 = 600 nm, and relative permittivity εSi3N4 = 6.3) and a hydrophobic layer (SiOC, thickness dSiOC = 1000 nm, and relative permittivity εSiOC = 2.75). This technology has been demonstrated to exhibit excellent properties even in the presence of high voltage,25 which is a prior feature as far as coplanar electrode designs are concerned. The electrical potential applied to the electrode pair consists of two AC voltages, U1 (frequency ν) and U2 (frequency f), provided by two frequency generators. These two signals are processed by an analog multiplier providing an output voltage, U3 = (1 + U2/10V)U1. Voltage U3 is further amplified by a highvoltage amplifier delivering dual-frequency electrowetting actuation, U = 100U3. The electrode pair has one electrode connected to the mass, while the other electrode is connected to voltage U. The purpose of working with a dual-frequency actuation is that, when a high frequency ν = 15 kHz is imposed on U1, its amplitude may be used to tune the contact angle (for instance, θ = π/2) in a range, which depends upon the composition of the underlying substrate and contact angle hysteresis. Because of the minimization of electrostatic energy, voltage contribution U1 means that the droplet straddles the underlying electrodes,4 making the droplet position far more stable than in standard geometries. Therein lies a second feature of prior importance, because interferometry measurements on a curved surface require a high degree of local accuracy to be feasible. Voltage modulation U2 is characterized by a low frequency in the range of f ∼ 101000 Hz. This voltage is used to oscillate radially the contact line of the drop, otherwise stabilized in position, and therefore to induce drop shape oscillations of a frequency fosc = 2f, thereby controlling a capillary wave network standing along the drop surface. As demonstrated below, from a change in the frequency fosc, it is easy to select capillary resonance with a given set of spherical standing wave modes along the drop surface. In Figure 2, the wetting curves (U2 = 0) are presented for the coplanar electrode geometry with two different gaps g. A first gap width of 3 μm exhibits a wetting curve quite similar to theoretical expectations. A second gap width of 100 μm shows a slightly different behavior; it seems that electrowetting is not as efficient for this geometry. This difference may be accounted for by the fact that a larger gap width involves a smaller reduction in surface
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Figure 3. Verification of the gap correction factor according to Yi and Kim24 for the approximate gap area (4, eq 6) and for the exact expression (O, eq 5). The corresponding drop volume is shown on the right axis. The data are taken from a drop of the initial volume of 4 μL deposited on an electrode pair with a 100 μm gap.
energy because of electrowetting, as explained by Yi and Kim.24 In their work, these authors explain the discrepancy between wetting curves for different gap widths by the modification of the Lippmann equation according to the difference in electrode area covered by the drop. Consequently, a correction factor K may be introduced in the LippmannBerge equation to take into account this area change cos θ cos θ0 ¼ K
cU 2 2γ
where θ0 is the contact angle at zero voltage, c is the capacity between the drop and the electrodes underneath where c = (dSi3N4/ε0εSi3N4 + dSiOC/ε0εSiOC + doil/ε0εoil)1, and ε0 is the vacuum permittivity. The additional capacity contribution doil/ ε0εoil is due to a silicone oil film with thickness doil = 260 nm and permittivity εoil = 2 (for details, please refer to the last paragraph in this section). To calculate the correction factor, K, the reader is invited to refer to Appendix 1. Note that the approximation, g , RB, valid for a gap much smaller than the wetting radius, is demonstrated to be fair even for a drop as small as 500 μm. In this work, the value of the electrowetting contact angle, θ, is constantly regulated to be equal to π/2. This allows us to consider that the mean radius of the sessile droplet and the base radius are both equal, because, moreover, gravity can be disregarded (drop size is small enough). In the experiments, one observes that the amplitude of U1 needs to be reduced in the course of evaporation. As explained above, the wetting voltage U1 needs to be corrected for large gap widths. The correct wetting voltage may be calculated via the correction factor K, such that KU21 is constant for all contact radii RB. As clearly observed in Figure 3, this is not the case for a 4 μL droplet under evaporation; another phenomenon needs to be considered. As an attempt to explain this discrepancy, the curvature of the electric flux lines between buried electrodes and the solid/liquid interface, just above the gap, is suspected to generate an additional capacity, cg, through the electrode gap. As a first consequence, the correction factor K for the voltage between the drop and the electrodes is no longer equal to 1 / 4 . A second consequence is that the voltage, V g , between the gap and the drop can no longer be considered as negligibly small, as initially assumed by Yi and Kim.24 The lack of a correct 1043
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Langmuir modeling for the voltage Vg makes a calibration of the system unavoidable, to correctly feed back the contact angle to a value of π/2. This calibration is performed via a curve U1(RB), generated from imaging, prior to dual-frequency EWOD experiments. Then, the voltage U1 is regulated according to the most recently calculated drop radius issued from the processed interferometry signal. The time delay between two measurements including postprocessing is approximately 80 s. The drops with an initial volume of 4 μL are derived from ultrapure water (resistivity of 18.2 MΩ) and pure glycerol (99%+, Alfa Aeser). One first solution (a) is pure water only. A second aqueous solution (b) contains 1/6 of glycerol by weight. A third and last aqueous solution (c) contains 1/4 of glycerol by weight. The surface tensions are measured to be 57, 47, and 39 mN/m for solutions a, b, and c, respectively. Surface tension measurements have been performed with the Wilhelmy technique using a NIMA tensiometer. Because of traces of contamination, surface tensions are found to be lower than those expected on the basis of theory. It is important to know the actual surface tension of the used liquids, to correctly calculate the drop radius from eq 3. The drops are deposited on the electrodes, which are covered by a very thin film of silicone oil (VWR Prolabo, viscosity of 50 mPa s), to avoid stickslip motion as well as an excessive value of contact angle hysteresis. The oil film after drainage is estimated to have a thickness of 260 nm, according to the fit in Figure 2. A droplet of oil is deposited on the chip, sponged up by a piece of paper (Whatman 3MM filter paper), and drained under capillarity for 24 h, while being protected from dust. Contact angle measurements have been taken from imaging with a camera Basler A622f, and the images are analyzed with the ImageJ-plugin DropSnake created by the Biomedical Imaging Group at EPFL Lausanne.26
’ DROPLET INTERFEROMETER A kind of Michelson interferometer is developed to detect surface motion at the apex of a sessile drop; a laser beam (wavelength λ = 532 nm) horizontally crosses a polarizer and a beam splitter and is directed onto a mirror, where it is reflected vertically onto the drop apex underneath a microscope objective for precise focusing of the laser beam and subsequent collection of the reflected light. Please refer to Figure 1 for a sketch of the optical setup. The liquid/gas interface results in two reflections: a first reflection of the laser beam at the drop apex and a second reflection at the solid substrate, on which the drop rests. These two reflected beams take the same path back to the beam splitter, where they are directed through a second microscope objective to collect maximum light intensity on the active surface of an avalanche photodiode (APD), which is located at the focal point. Further, higher order reflections of the beam at the drop scale are not observed to interfere with the primary reflections because they are deviated away from the microscope objective by the curved surface of the oscillating drop. The delivered light intensity I at the APD complies with the law27 pffiffiffiffiffiffi 8π~n ζðtÞ ð1Þ I µ Is þ Ia þ 2 Is Ia cos λ where ~n is the refractive index of the liquid and Is and Ia are the light intensities received from the solid substrate and the drop apex, respectively. The oscillation amplitude of the vertically moving apex is ζ(t) = ζ0(f)sin(4πft), where f is the low-frequency modulation of the EWOD actuation. The spectrum of the maxi-
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Figure 4. Capillary oscillation spectrum of a drop (initial volume of 10 μL) for actuation frequencies between 25 and 100 Hz. Two resonance peaks, f2 = 32 Hz and f4 = 84 Hz, are observed. (Inset) Interferometry signal as delivered from one period of the drop oscillation (initial drop volume = 10 μL, and excitation frequency f = 30 Hz).
mum oscillation amplitude ζ0(f) is related to the maximum frequency fm of the interferometry signal. ζ0 ¼
λ fm 16π~n f
The maximum frequency is evaluated via a routine, described and used by Picard and Davoust.28,29 The signal detected at the APD is delivered to an analog filter (Krohn-Hite model 3362, band pass 1 kHz/200 kHz, Butterworth type) prior to recording at a sampling frequency of 250 kHz. Before each measurement, a delay of 0.3 s is respected to ensure operation in the steady regime of drop oscillations, which are recorded for a duration of 10 periods of the electric signal, corresponding to 20 periods of drop shape oscillations. An example for one period of oscillation of a 10 μL droplet is given in Figure 4. The data are subsequently subjected to post-processing to calculate an entire spectrum on the frequency range f2 ( 50 Hz, for which the resonance frequency is evaluated. An example for a 10 μL droplet is given in Figure 4; two resonance frequency peaks for oscillation modes 2 and 4 are observed at f2 = 32 Hz and f4 = 84 Hz. The drop apex is well-suited for the interferometry measurement of the oscillation amplitude. This is due to two main reasons: (1) Capillary waves excited from the oscillating contact line are propagated radially inward up to the apex where a wave interaction is expected to occur, and as a result, excited waves are propagated radially outward. In steady conditions, a resonance mechanism is therefore expected at the drop apex where the amplitude of drop oscillations is largest and where oscillatory displacement must remain perfectly vertical and an antinode occurs. Finally, optimum sensitivity is expected at the drop apex. (2) The apex is the only location along the drop surface that remains parallel to the coplanar electrodes while translating vertically to them. Hence, use is made of the drop apex as a semi-reflecting mirror, while the electrodes underneath provide a second reflection of the laser beam. This has the advantage of directly working in the reference system of the electrodes and offers an elegant way of taking the interferometry measurement without the need for an external mirror. 1044
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With a precise optical alignment, i.e., a laser beam impacting at a right angle onto the electrodes, a reproducible and easy to process interferometry signal is obtained (Figure 4).
’ THEORETICAL FRAMEWORK Consider a sessile, half-sphere water droplet with contact angle θ = π/2 and radius R(t), whose initial value is R0. Evaporation acts to downsize the droplet radially inward with a constant contact angle. These conditions are experimentally performed with fair accuracy from the EWOD-based methodology proposed in this paper. A well-known model of constant contact angle regime was presented by Bexon and Picknett,17 supported by the experimental developments by McHale et al.30 From the literature, a common agreement has been found on the evaporation law for a sessile droplet with a constant contact angle. When the contact angle is θ = π/2, a model can be written as follows: ! t 2 RðtÞ ¼ R02 1 ð2Þ τevap where τevap = R20F/[2Dvap(c∞ c0)] is the characteristic evaporation time for diffusion-limited evaporation, F is the density of water, Dvap is the diffusion coefficient of the water vapor in the surrounding air, and c0 and c∞ are the vapor concentration at the interface and far from it, respectively. To estimate surface stretching because of EWODinduced drop shape oscillations and its potential impact on mass transfer, a calculation is presented in Appendix 2 for the deformation mode n = 2. Surface stretching induced by oscillating EWOD is found negligibly small, ΔS/S ≈ 1 106, where ΔS is the stretched surface area and S is the surface area without oscillations. Droplet shape oscillations depend upon surface tension and the averaged radius of the droplet. A sophisticated analysis by Kang et al.18 delivers a theoretical model for the oscillations of a sessile drop in AC electrowetting. The electrical force in the wetting plane is modeled as a δ function at the triple contact line (TCL), FTCL = cU2/8, through which harmonic motions are established. This force depends upon the applied voltage U and the resulting surface capacitance c of the dielectric layers. In this model, contact line friction and hysteresis are ignored. The amplitude of drop oscillations is consistently found to depend upon excitation frequency f. The capillary resonance is demonstrated to occur for a given series of resonance frequencies, fn, deduced from the classical dispersion relation rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 γ ð3Þ fn ¼ 4π λn ðθÞFRðtÞ3 where n = 2k and k ∈ N is the mode number of the oscillation of a sessile drop. In this paper, we introduce the factor λn(θ), which classically represents the eigenvalue n(n 1)(n + 2) for the oscillation of a free droplet but which is modified in this case with the theory by Strani and Sabetta,31 devoted to the oscillation of a sessile drop with a contact angle θ (refer to Appendix 3 for further details). Consequently, from the dispersion relation eq 3, it is clearly demonstrated that the variation in the radius can be converted into a resonance frequency shift, detected by means of interferometry.
’ RESULTS AND DISCUSSION To check the correct measurement of the dispersion relation eq 3 by interferometry, water drops of different volume have been tested with respect to their resonance response. For a drop volume of 10 μL, a typical spectrum can be observed from the de-
Figure 5. Dependence of resonance frequency upon droplet volume. The data are post-processed, including the theory of sessile drop oscillations by Strani and Sabetta.31 The downward error bars account for the evaporation kinetics during the measurement.
Figure 6. Evaporation of a droplet (initial volume of 4 μL) for 47 min. The color coding represents the oscillation amplitude in micrometers at resonance frequency f2, as measured by interferometry. The shift of frequency f2 is due to the diminishing drop radius, as induced by evaporation (ambient temperature T = 22.3 °C, and relative humidity j = 44.5%).
pendence of apex displacement upon actuation frequency. In Figure 4, the values of the two resonance frequencies, f2 and f4, depend upon the drop radius and, therefore, evaporation kinetics. The resonance frequencies for different volumes are also presented in Figure 5, taking into account the frequency correction proposed by Strani and Sabetta. Three successive oscillation modes, n = {2, 4, and 6}, are considered for comparison to theory. The four experimental points below the n = 2 curve (Figure 5) belong to the additional so-called “rocking” mode (n = 1) for sessile drops,31 whose presence basically originates from the underlying substrate. In Figure 6, the evolution of the resonance spectrum is shown for a mode number n = 2. In the course of time, the drop radius diminishes under evaporation, and as a result, resonance frequency f2 is found to increase. The resonance frequency may then be converted into the droplet radius, as illustrated in Figure 7, showing the evaporation-driven downsizing of a 4 μL droplet for 47 min at ambient temperature T = 22.3 °C and relative humidity j = 44.5%. The red triangles in Figure 7 show the experimental time dependence of R2, derived from the resonance frequency values f2, as expected from eq 3. This quantity 1045
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This means that the error on R2 scales with the square root of R thus decreases as evaporation advances, which is the case, as seen in Figure 7. The evaporation curves for the three different solutions a, b, and c are measured with our methodology (EWOD + interferometry) and are displayed in Figure 8. They are all non-dimensionalized acfit cording to R/Rfit 0 and t/τevap, where R0 is the initial radius of the fitted curve. Because of the non-dimensionalization, the three curves should match, a fact that is approximately checked. The evaporation time scales measured by linear fitting are (a) τevap = 2884 ( 31 s for pure water, (b) τevap = 3100 ( 39 s for the 5:1 water/glycerol mixture, and (c) τevap = 3739 ( 77 s for the 3:1 water/glycerol mixture.
Figure 7. Evaporation of a droplet (initial volume of 4 μL) for 47 min. Comparison between imaging results (O, τevap = 2488 s) and interferometry measurements (4, τevap = 2386 s) with ambient temperature T = 22.3 °C and relative humidity j = 44.5%.
Figure 8. Evaporation of three different drops consisting of water and glycerol, as measured by interferometry: (a) τevap = 2884 ( 31 s for (4) pure water, (b) τevap = 3100 ( 39 s for (O) 5:1 water/glycerol, and (c) τevap = 3739 ( 77 s for (0) 3:1 water/glycerol. Radius and time are non-dimensionalized by R0 and τevap, respectively (ambient temperature T = 22.3 °C, and relative humidity j = 44.5%).
is compared to image post-processing (green circles). A discrepancy between the two curves can be seen in Figure 7. This discrepancy could be due to the correction factors introduced by Strani and Sabetta and the existence of a mismatch between theory and experiment pointed out by the same authors.31 A linear fit is applied to the experimental data on the square of the radius, which gives a typical value of the evaporation time scale, τevap. The evaporation time scales are found to be 2488 ( 31 and 2386 ( 47 s from the interferometry and the imaging data, respectively, which represents a relative error of 4% between the two methods. The given error is calculated from the expression ((δR20τevap)2 + (R20τevap2δτevap)2)1/2, where δR20 and δτevap are the errors on R20 and τevap for a linear fit of the data, respectively. The discrepancy between the imaging and interferometry measurements probably results from an error in the relationship R(f), which may be calculated by eq 3. This error is quantifiable by pffiffiffi 4 δf δR2 ¼ R 2 µ R δf 3 f
’ CONCLUSION In this paper, oscillating coplanar EWOD is proven to be a convenient transduction mechanism to measure evaporation kinetics of sessile drops. Drop shape oscillation frequencies are shown to convert a time-dependent radius into a resonance frequency shift, as illustrated in our case from the evaporation regime at a constant contact angle. Furthermore, the addition of a high-frequency voltage, a regime that we refer to as dualfrequency electrowetting, allows for independent monitoring of either the contact angle or the base radius. It would also appear that the correction factor for large gap widths provided by Yi and Kim24 does not reflect the whole truth as far as wetting characteristics for non-ideal electrode configurations are concerned. As a means of drop shape oscillation detection, interferometry performed at the drop apex offers several advantages. In addition to accurate measurement of the apex motion of the order of one laser wavelength, this technique is adapted to situations in which imaging could not be conveniently performed. For instance, the methodology proposed can be performed in confined geometries, such as those in digital lab-on-a-chips. Moreover, it is possible to estimate the size of a drop (and, therefore, its volume) provided that the dependence of the contact angle upon the EWOD voltage has been calibrated. More generally, this methodology can be used to measure the local impact of evaporation on a population of drops involved throughout the different processes developed in a digital microsystem based on electrowetting (cooling 22 or biosensing 23 ). At the time being, the present methodology is extended to the measurement of the change in surface tension, γ, caused by the adsorption of biomolecules. 28,29 To this end, the contribution of the evaporation kinetics in eq 3 must be properly measured and subtracted. ’ APPENDIX 1: CORRECTION FACTOR FOR LARGE GAPS The correction factor K introduced by Yi and Kim24 takes into account a non-uniform actuation distribution in coplanar electrowetting. Given two electrodes facing each other with a gap width g, the LippmannBerge equation is cos θ cos θ0 ¼ K
c 2 U 2γ
ð4Þ
with the correction factor K = Ad/2At, assuming that the drop is equally positioned on both electrodes (electrostatic equilibrium). In this case, Ad is the area covered by the drop 1046
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on one electrode and At (=2Ad + Ag) is the total wetted area, with Ag being the area of the gap covered by the drop. Assuming that the contact line is perfectly circular, the areas Ad and Ag can be written as 1 Ad ¼ ðπRB 2 Ag Þ 2 Ag ¼ RB g
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðg=2Rb Þ2 þ 2Rb 2 arcsinðg=2RB Þ
¼ 2RB g þ O ðg=2RB Þ
ð5Þ
If the gap is considered to be much smaller than the base diameter, g , RB, which is a fair assumption at least in the early stages of the evaporation process, the previous eq 5 can be simplified as Ag ¼ 2RB g
ð6Þ
where the gap area is assumed rectangular, such that K = 1/4 g/ 2πRB. As seen in Figure 3, this approximation is fully justified in our experiments because the drop size is never smaller than 100 μm. When controlling the contact angle at a constant value, θ = π/2, the right-hand side of the LippmannBerge eq 4 should remain constant, especially the product KU21, a point which is nevertheless not experimentally checked (Figure 3).
’ APPENDIX 2: NEGLIGIBLE INCREASE OF THE SURFACE AREA UNDER DROPLET OSCILLATIONS Under the effect of surface oscillations, the liquidair interface is stretched and, thus, could offer more surface for mass transfer. Hence, evaporation kinetics may be enhanced from drop oscillations. To estimate the impact of drop shape oscillations, the analytic expression for the shape of a sessile droplet as given by Kang et al.18 could be considered, stating that the droplet shape d(t) at a given resonance frequency fn is given by the coefficients δ and an and the Legendre polynomials Pn dðtÞ ¼ R0 ð1 þ δ
∞
∑ an Pnðcos ϕÞÞ
n¼2
With regard to the capillary spectrum and the importance of the resonance frequency, f2, at mode n = 2, the amplitudes an for modes n > 2 are much smaller than a2, making it possible to ignore those contributions and reduce the previous sum to its first element n = 2. To estimate the surface increase because of shape oscillations, one may consider the maximum droplet amplitude measured by interferometry for this resonance mode, δa2 = 3 μm. The area of the drop surface, S, may be calculated via the surface integral over the droplet hemisphere S ¼ 2π
Z π=2
¼ 2πR02
0
dðϕÞ2 sin ϕ dϕ
Z π=2 0
2 δa2 1 þ P2 ðcos ϕÞ sin ϕ dϕ R0
which scales as S ¼ 2πR02 ð1 2:53 106 Þ f
jΔSj ¼ 2:53 106 2πR02
Hence, surface enhancement is found negligible for evaporation provided that oscillation amplitudes are small enough; this
last criterion is by far compatible with classical abilities of interferometry.
’ APPENDIX 3: CALCULATION OF THE DISPERSION RELATION ACCORDING TO STRANI AND SABETTA The eigenvalues λn(θ), used in the dispersion relation, eq 3, have been calculated by Strani and Sabetta.31 Ignoring the influence of the ambient phase on drop shape oscillations, these eigenvalues are obtained by calculating the determinant of the matrix Aij detðAij λi δij Þ ¼ 0 where i = 1, 2, ..., j = 1, 2, ..., and δij is the Kronecker symbol. The elements of the matrix Aij are defined by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2i þ 1Þð2j þ 1Þ Gij Aij ¼ Aji ¼ pffiffiffi 2 ijðjðj þ 1Þ 2Þ Gij ¼ Gji ¼ ðPj ð0Þð2 2
Z 1 0
Z 1 0
P1 ðτÞPi ðτÞdτ
P1 ðτÞPi ðτÞdτ
Z 1 0
Pj ðτÞdτ þ
Z 1 0
Pi ðτÞdτÞ
Z 1 0
Pi ðτÞPj ðτÞdτÞ for k 6¼ 1
1 1 G11 ¼ þ ln 2 3 2 where Pk(τ) is the Legendre polynomial of the kth order. This calculation is based on the assumption that the contact angle is equal to π/2. A slightly more complicated method must be used to calculate the eigenvalues for contact angles different from π/2.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
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