Uphill Movement of Sessile Droplets by Electrostatic Actuation

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Uphill Movement of Sessile Droplets by Electrostatic Actuation S. Datta,† A. K. Das,*,‡ and P. K. Das† †

Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721302, India Department of Mechanical and Industrial Engineering, Indian Institute of Technology, Roorkee 247667, India

Downloaded by CENTRAL MICHIGAN UNIV on September 12, 2015 | http://pubs.acs.org Publication Date (Web): September 10, 2015 | doi: 10.1021/acs.langmuir.5b02184

‡

ABSTRACT: The dynamics of uphill motion and the internal circulation of a sessile droplet by inducing asymmetric electrocapillarity were formulated and investigated numerically. We developed and analyzed a coupled electro-hydrodynamic model that includes conservative body and surface forces along with electrostatic effects. The interplay between gravity and electrostatic actuation is influenced by induction voltage, the inclination of the surface, and the droplet volume. Actuation voltage on the sessile drop causes an internal circulation which, upon increasing strength, overcomes the gravitational pull to climb uphill. As uphill droplet climbing is a spatiotemporal phenomenon, droplet volume plays a major role in accommodating the internal circulations and subsequent climb. Simultaneously, actuation due to electrostatic force behaves differently on different inclined surfaces, causing a roll down at higher inclination and an uphill climb at lower ones. A pattern map has been generated to identify favorable conditions for uphill movement based on the inclination, actuation voltage, and volume of the droplet.

1. INTRODUCTION The manipulation of a sessile droplet has received considerable attention over the last few decades and has been revamped particularly since the advancement of the μ-TAS system1 or labon-a-chip devices.2 The transportation of droplets as a discrete volume of chemical reagents by creating a dynamic wettability gradient with the employment of electricity (electrowetting) has attracted interest due to its higher reliability, precision, and controllability3 compared to those for thermocapillary-driven pumping,4 vapor-bubble-based pumping,5 and opto-electrowetting.6 Extensive experimental studies7−19 have been carried out to explore the viability of electrowetting on flat surfaces and understand the physics behind it. Pollack et al.5 experimentally achieved a droplet velocity of more than 10 cm/s on an open structure electrode arrangement by the application of a suitable voltage and switching frequency. Their study reveals an independence of size over droplet velocities for a range of voltage. Cho et al.,7 on the other hand, performed the merging, splitting, and creation of a droplet confined between two parallel plates by an electrowetting on dielectric technique (EWOD). They reported limiting design criterion beyond which the splitting of droplet is not achievable by EWOD. In order to reduce the complexity, cost, and time involved in experiments, analytical20 and numerical21−23 approaches also attracted researchers to comprehend the physics of electrowetting. Bahadur and Garimella20 developed a model based on energy minimization to analyze electrowetting as a tool for the thermal management of an electronic chip. Different numerical routes, i.e., free-energy-based lattice Boltzmann approach,21 a finite-volume-based volume of fluid approach,22 and a finite element approach23 are being used to compare the electro© XXXX American Chemical Society

dynamic actuation force for different geometries of electrodes in open-type electrowetting. A broad overview of studies on electrowetting can be obtained in the literature from Nelson et al.,24 Mugele and Baret,25 and Fair.26 In general, droplet manipulation by electrostatic actuation is practiced in single horizontal planes. However, translation on a nonhorizontal surface can facilitate multilayered or 3-D microfluidic device design, enhancing compactness and volumetric capacity compared to more planar ones.27 Abdelgaward et al.27 carried out all-terrain droplet actuation on a flexible surface which could take the shape of a vast range of complex geometry. They performed droplet translation by electrostatic actuation on inclined, twisted, vertical, and upturned surfaces. Recently, the manipulation of a droplet on an inclined surface has been demonstrated by Mannetje et al.28 Their study illustrates the trapping of a droplet on an inclined surface and translates it to the desired direction by an external electric field. Datta et al.29 investigated the upward climbing behavior of droplets by a lumped capacitance approach for a range of voltages and inclinations. However, in their analytical model a droplet is considered to be a nondeforming hemispherical liquid mass. In the current research, we formulated and investigated droplet motion along inclined surfaces by an electromechanical approach. The transient electric field is obtained by solving the Poisson and charge conservation equation, and the electrostatic effect is coupled with hydrodynamics by incorporating Maxwell’s stress term in a Received: June 15, 2015 Revised: August 9, 2015

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DOI: 10.1021/acs.langmuir.5b02184 Langmuir XXXX, XXX, XXX−XXX

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two-phase momentum equation. A volume of fluid (VOF) algorithm is used to track the boundary of a sessile drop for different actuation voltages, surface inclinations, and volumes.

2. PHYSICS OF ELECTROWETTING In the electromechanical approach, electrostatic force is considered to be the reason for contact line motion.30 The electrostatic force per unit volume due to charge (ρc) can be expressed by r r 1 felec = ρc E − E2∇ε (1) 2 where ρc, E, ε, and ρm are the free charge density, electric field, permittivity and mass density of the liquid medium. Momentum conservation confirms that the volumetric body force due to electricity is same as that obtained by integrating the momentum flux density over the surface confining the liquid. Thus, the total body force for a fluid volume can be estimated to be (Felec)i =

∫ (τm)ij nj dA

Figure 2. Schematic of the electrode arrangement for imparting droplet motion along the inclined plane.

translate a droplet along an inclined plane. Initially the sessile droplet is situated on an array of electrodes at the foothill of the inclined plane. For translation, electric potential is applied to the front electrode; below the drop, other electrodes are kept grounded. The gradient of the electric field after crossing a limiting value drives the droplet against the gravitational pull over the inclined surface. This motion seizes as the drop moves over the actuated electrode. A switching of electrodes is necessary to obtain continuous uphill movement. In the current research, we have simulated droplet uphill climbing for singleelectrode actuation.

(2)

where nj is the direction normal to the surface and (τm)ij is the Maxwell stress tensor expressed as ⎛ ⎞ 1 (τm)ij = ε0ε⎜EiEj − δijE2⎟ ⎝ ⎠ 2

(3)

3. COMPUTATIONAL FRAMEWORK An electro-hydrodynamic numerical simulation is performed utilizing an open-source Gerris flow solver,31−33 which solves the transient incompressible Navier−Stokes equation using a finite volume approach. Gerris is equipped with an electrohydrodynamic module which considers body force due to electricity. The governing equations coupled with the electric field can be expressed as (4) ∇u = 0

As a consequence of the electric field, the free charge and dipoles present in the liquid reorient themselves. If the liquid is considered to be perfectly conducting, then the electric field in the bulk of the droplet fades away and gathers at the liquid−gas or liquid−solid interface. For a sessile droplet, due to its wedge shape near the three-phase contact line, the charge density increases as compared to other parts of the interface. Electrostatic forces due to the presence of charge at the interfaces of the drop are shown in Figure 1. The vertical

⎛ ∂u ⎞ ρm ⎜ + u·∇u⎟ = −∇p + ∇(2μτv) + σλδsurn + felec ⎝ ∂t ⎠

(5)

For interface tracking in two-phase flow, Gerris uses a piecewise linear volume of fluid (VOF) advection scheme. There is a conservation equation of the liquid volume fraction (α) and constitutive equations where density and viscosity can be written as Figure 1. Schematic representation of electrostatic forces at solid− liquid and gas−liquid interfaces.

components of the force at the solid−liquid interface near the wedge are balanced by a stress at the solid surface. At the contact line, in a very thin region, the horizontal component of electrostatic force exists. This causes the contact line to move outward. If the electric potential is applied asymmetrically over a portion of the droplet, then local spreading occurs due to the presence of the horizontal component of the electromagnetic force in the actuated location, which perturbs its equilibrium state. To regain its minimum surface energy condition, the droplet reorients itself to take the shape of a spherical cap and thus causes an overall translation of mass in the direction of actuation. Figure 2 schematically shows a conventional opentype electrode arrangement. A similar electrode arrangement (detailed in section 4) is considered in the present study to

∂α + ∇(αu) = 0 ∂t

(6)

ρm = αρl + (1 − α)ρg

(7)

μ = αμ l + (1 − α)μg

(8)

where a density−viscosity pair of liquid and gas combinations are considered to be (ρl, μl) and (ρg, μg), respectively. In eq 5, σ and n are the surface tension coefficient and the normal to the interface between liquid and gas. λ is the mean local curvature of the interface estimated from the height-function method,34 and Gerris is second-order accurate from the VOF field. The contact angle is used as the boundary condition of the methodology. Dirac distribution function δsur ensures that the surface tension force acts at the interface only. Gerris considers a balance force implementation of the continuum surface force approach to formulate surface tension stress as a volumetric force. τv is the stress tensor due to viscosity and can be B


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Langmuir represented as τv = μ(∇u + ∇uT). To estimate the volumetric electric force in eq 5, a transient conservation equation for ρc and the Poisson equation for electric potential (ψ) are solved.

from the mesh size, the maximum level of refinement is varied and the velocity of the droplet is calculated for comparison. It is observed that upon changing the maximum level from 7 to 8, the velocity of the droplet differs by 9.3%. On the other hand, further refinement from the maximum level of 8 to 9 decreases the velocity by 2.1%. Since the deviation is less, the highest level of refinement is kept at 8 (minimum cell size near the interface and low vorticity region of 3.9 × 10−3 mm; 0.0039 is the time of the drop diameter). As the phenomena is governed by interfacial shape and droplet movement due to circulation, adaptation is applied on the basis of the vorticity field and the gradient of the volume fraction. Though the number of meshes will be varying from time to time and for different cases, around 80 000 cells are in each simulation with adaptation. Figure 3 shows the dynamic mesh refinement during the translation of a 0.262 μL droplet along a vertical plane.

∂ρc

S + ∇(ρc u) = − ρc ∂t ε

∇(ε∇ψ ) = −ρc

(9) (10)

Here, S is the conductivity, and the electrical phenomena can be described as

∇(ε∇E) = ρc

(11)

∇×E=0

(12)


Once the electric field and charge distribution are obtained at a particular time instant using Maxwell’s electromagnetic equations, the volumetric electric force is computed as felec = ∇τm = ρc E −

1 2 E ∇ε 2

(13)

Gerris utilizes a multilevel Poisson solver to tackle the governing equations numerically. Second-order accurate, staggered-in-time discretization along with a time-splitting projection method is used to propagate the solution with time. A second-order upwind scheme of Bell et al.35 is used for the descretization of advection terms in the momentum equation. A quadtree adaptive mesh refinement technique31 is employed for spatial discretization of the domain based on predefined criterion. In the next section, before detailing the results we have dicussed the domain and mesh thoroughly. Figure 3. Dynamic mesh adaption during translation along the actuated electrode. (a) t = 0.0 s and (b) t = 0.00223 s.

4. RESULT AND DISCUSSION In the present study, a droplet of deionized water in air (as a surrounding medium, Mo = ((gμc4Δρ)/(ρc2σ3)) = 1.65 × 10−11) is considered for analyzing uphill climbing behavior due to electrostatic actuation. The equilibrium contact angle of deionized water−air with the solid surface is assumed to be 90°. The density, viscosity, and surface tension are considered to be 1000 kg/m3, 0.00089 Pa·s, and 0.072 N/m, respectively. No slip, no penetration conditions are implemented on the surface on which the drop is resting. To avoid the effect of the wall, the droplet is placed in a 2D rectangular domain of height equal to twice the height of the droplet and length equal to six times the diameter of the droplet. All of the boundaries are considered to have no slip and no penetration. For actuation, the drop is asymmetrically placed over the electrode strip array (as shown in Figure 2), which has a span equivalent to the droplet radius. We considered a thin dielectric layer (∼1 μm) over the electrodes around which the drop in electric potential can be neglected. At the beginning of the simulation the droplet is placed over three electrodes in which the upper one is electrified and the other two are kept grounded (Figure 2). The quadtree adaptive mesh refinement technique31 is employed for spatial discretization of the domain. The refinement takes place hierarchically from a maximum cell size (level 4) to a predefined minimum cell size (maximum level). While refining from a parent cell to the next level it is divided into four equal daughter cells. This signifies that, for a refinement up to the nth level, the minimum cell size is 1/2n times that at the lowest level. A thorough analysis has been carried out to decide the size of the smallest scale (maximum level) and the basis of adaptation in order to capture the physics behind the phenomenon. To check the independence of the solution

To check the accuracy of the developed model we have compared the obtained velocity field along with experiments of Malk et al.36 They have shown a circulation pattern inside phosphate-buffered saline drops under the actuation of 90 V rms and 150 Hz frequency. Figure 4 shows the comparison

Figure 4. Comparison of velocity field inside droplet between experiment and present study. (a) Malk et al.36 and (b) the present study.

between experimental snapshot of Malk et al.36 and the observed velocity field of the numerical model. Both figures demonstrate similar internal circulation patterns. We have simulated droplet dynamics at different inclinations under 80 V actuation. Figures 5 and 6 depict phase contours of a droplet translating along 45° and 90° inclined surface at different time instants. Initially the contact line at the upper end of the droplet moves forward due to electric actuation. Bulk fluid on the other hand experiences gravitational pull. This results in the shape of the droplet being asymmetric (Figures 5(b),(c) and 6(b)) by gathering more liquid at the bottom end compared to the upper side. However, at latter stages inertia of C


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inclined plane. One can observe that due to the sudden application of an electric field from a static condition there is a rush of fluid in the advancing end. This causes a sharp decrease in the advancing angle. After that, when the drop is moving in the upward direction, the advancing contact angle shows a continuous increase and reaches its nominal value at 90°. At this position, the drop is already pinned at the end of the electrode, but due to inertia it continues to increase its advancing contact angle. As the receding end only drags the liquid mass along the surface, its angle is not varying much with time. A similar type of pattern in the change of advancing and receding contact angles is observed in other inclinations and drop volumes. A close look at the electric field distribution inside the drop reveals the reason behind motion toward the actuated electrode. Figure 8 represents the isolines of electric potential in the flow domain. The electric potential attains its maximum value at the actuated electrode and fades away in the bulk. This causes the body force due to the electric field to take a higher source term in the fluids over the actuated electrode and, as a consequence, a higher velocity. As the droplet behaves like a continuum, it follows the leading edge and moves uphill. This results in a change in the electric field with time as depicted in different time stamps at Figure 8. We consider the dielectric layer to be very thin and thus the potential across it to be small. To investigate the reason behind the droplet motion toward the actuated electrode, we tried to observe the velocity vectors during the uphill climb of the drop. This reveals the physics behind the overall motion of the droplet following the leading edge. Figure 9 illustrates the velocity vectors during the uphill motion of the drop. Due to the presence of an external electric field, the internal circulation of the liquid is initiated to balance the stress generated by a potential gradient. It can be observed from the figure that at the beginning two strong vortices are generated at the leading edge of the droplet over the actuated electrode and at the junction of the actuated and grounded electrodes. The upper one stretches and drags liquid toward the actuation whereas the second one feeds the liquid from the bulk. This results in the sliding of liquid along the solid surface and flow back along the liquid−gas interface. But the strength of sliding dominates the flow back and causes overall uphill movement over the inclined surface. It may be noted that the velocity vector confirms sliding motion rather than rolling behind the climbing up. We observed that the actuation force strictly depends on the applied voltage, droplet size, and inclination of the plane. If the


Figure 5. Phase contour of a 0.262 μL droplet moving along a 45° inclined plane under 80 V actuation. (a) t = 0.0 s, (b) t = 0.00148 s, (c) t = 0.00268 s, (d) t = 0.00428 s, and (e) t = 0.00714 s.

Figure 6. Phase contour of a 0.262 μL droplet moving along a 90° inclined plane under 80 V actuation. (a) t = 0.0 s, (b) t = 0.00223 s, (c) t = 0.00423 s, (d) t = 0.00663 s, and (e) t = 0.00873 s.

the bulk fluid in the direction of motion increases the apparent contact angle in the upper end. At the end of the actuated electrode the contact line stops due to a change in the direction of the electric field; inertia of the bulk, however, tries to move it further and thus the formation of a bulge (Figures 5(e) and 6(e)) occurs. Figure 7 shows the variation of advancing and receding contact angles during the translation of the drop along a 45°

Figure 7. Variation of contact angle for a drop on a 45° inclined plane under 80 V actuation. The droplet volume is 0.262 μL.

Figure 8. Electric field at different time instants during 0.262 μL droplet motion under 100 V actuation potential: (a) t = 0.0 s, (b) t = 0.00228 s, and (c) t = 0.00714 s. D


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Figure 10. Variation of minimum voltage required for upward translation.

Figure 9. Velocity vectors inside a 0.262 μL droplet during translation under 100 V actuation. (a) t = 0.00030 s, (b) t = 0.00228 s, (c) t = 0.00428 s, and (d) t = 0.00714 s.

circulation is not sufficient, then the droplet will flow downward. Hence, a threshold voltage is required to translate the drop over an inclined surface against gravity. The force required to translate a droplet along an incline increases with rising slope due to the augmentation of the component of gravitational pull along the surface. It is expected that a higher voltage is required to push a droplet in the upward direction on steep surfaces as compared to a lower inclination. The opposing force for uphill climbing will also increase with the size of the droplet, which needs a higher threshold voltage to sustain upward motion. In the present study at first the threshold voltage required for upward movement is obtained for different inclinations and sizes. Then the actuation voltage and inclinations are increased in steps to explore the interplay between the gravitational pull and electrostatic force. The maximum voltage is constrained to 100 V to make the approach realistic (with a consideration of the possibility of dielectric breakdown in a real-time experiment). Figure 10 represents the mutual dependencies of the minimum actuation voltage to sustain an upward droplet motion with droplet size as well as the inclination of the surface. It is evident from the figure that the droplet with a 0.033 μL volume while moving along a 15° inclination requires only 5 V to sustain an upward motion. But, for a droplet of 2.093 μL volume and 90° inclination, the requirement of minimum voltage is 30 V for upward translation. However, the full translation of the droplet along the actuated electrode needs a higher voltage than that shown in Figure 10. 4.1. Influence of Voltage. Above the threshold voltage (shown in Figure 10), an increase in actuation voltage causes an increase in the electric source term in the momentum equation, hence causing an increase in the velocity (Re) of climbing up. Figure 11(a) shows the variation of velocity of the center of mass of the droplet in terms of the Reynolds number (Re = ((ρvD)/μ)) as it moves over the actuated electrode for different

Figure 11. Variation of droplet Re with displacement for different actuation voltages; electrode size = 0.5 mm.

voltages. As the actuation force due to the electric field predominates over the opposing forces for higher voltages, Re of the droplet keeps on increasing until the contact line reaches the end of the actuated electrode. At the end of the actuated electrode direction of electrostatic force changes its direction, which restricts the motion of contact line in forward direction. However, the inertia of the bulk fluid tries to move it further and causes a protuberance of the interface in the forward direction beyond the actuated electrode. The deformation of the droplet causes the center of mass to move further. Figure 11(b) shows the shape and location of the liquid−gas interfaces relative to the actuated electrode (shown by the thick line) at the end of their forward motion, and it can be observed from Figure 11(b) that the higher the actuation voltage, the higher the bulge formation and thus the displacement. For actuation with 70 V, electrostatic force is not enough to overcome the opposing forces causing the contact line to stop before complete translation along the actuated electrode. The variation of velocity (Re) with time has been plotted in Figure 12 for different actuation voltages. For higher actuation voltages the droplet attains its maximum Re and reaches it faster than that of lower voltages. The duration of motion is maximum for a droplet with 70 V actuation, even though it does not traverse the full actuated electrode length. In summary, though actuation starts beyond the threshold voltage (Figure 10), complete translation before switching the E


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Figure 12. Variation of droplet Re with time for different actuation voltages; electrode size = 0.5 mm. Time is nondimensionalized by the total period of traverse.

Figure 14. Variation of droplet Re with time for different inclinations; electrode size = 0.5 mm. Time is nondimensionalized by the total period of traverse.

actuated electrode in the next one requires a certain minimum voltage. It depends on the droplet size and inclination. Moreover, as the electrode size is a dominating parameter here, we have not tried to report a voltage map for complete uphill climbing for different inclinations and sizes. 4.2. Effect of Inclination. To analyze the effect of inclination separately on droplet motion, displacement vs Re (velocity) has been plotted for inclination angles of 30°, 45°, 60°, and 90° in Figure 13, keeping the actuation voltage

As the opposing force is larger for higher inclinations, the time required for translation is also higher. However, these slow translations along uphill surfaces are not successful for continuous motion after switching. It may be that this situation can show continuous motion at some smaller electrode size. Hence, we have commented only on the velocity pattern (Re) and have not mentioned continuous motion for a particular setup. 4.3. Effect of Droplet Size. In this section we consider the droplet dynamics separately for different droplet sizes. It can be observed from numerical simulations that smaller drops attain higher Re (velocity) for a short span of time as compared to larger drops. By normalizing with the maximum Re (corresponding to the maximum velocity), a comparison of Re with displacement is shown in Figure 15. In this

Figure 13. Variation of droplet Re with displacement for different inclinations; electrode size = 0.5 mm.

unaltered (80 V). The volume of the droplet is considered to be 0.262 μL. For 30° and 45° inclined surfaces, gravitational force in the direction of motion is less as compared to the electrostatic actuation force. During uphill climbing in these cases, the maximum velocity of motion (Re) is observed at the end of the actuated electrode. However, for higher inclinations, force due to gravity predominates over electrostatic actuation after a certain displacement and hence does not reach the end of the electrode to sustain continuous motion after switching. The dependence of electrostatic force on displacement can be expressed as Felec(x) = (V2/2)((dC(x))/(dx)), where C(x) is the equivalent capacitance of the system.37 This causes a deceleration over steep surfaces of the droplet after attaining a maximum velocity (Re ≈ 50−60) at a somewhat intermediate location over the actuated electrode. A similar trend in curves can be found in the time vs Re (velocity) plot for different inclinations as shown in Figure 14.

Figure 15. Variation of nondimensional velocity with displacement for droplets with different sizes.

representation we have depicted the voltage and inclination as 100 V and 15°, respectively. For smaller droplets, the surface tension force dominates the body force, resulting in its motion mainly governed by the electrostatic force at the contact line. However, in the case of larger droplets the inertia of the bulk fluid also contributes to the forces causing the motion. This trend is observed for all inclinations under moderate voltages which will not cause dielectric breakdown. It can be observed that for a longer droplet there is a flat, uniform Re (velocity) throughout its traverse, whereas smaller droplet peak Re (velocity) builds slowly and reaches its maximum when arriving at the electrode end (Figure 15(b)). For a droplet with a 0.262 μL volume, higher Re (velocity) causes a larger protuberance at F


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(3) Bahadur, V.; Garimella, S. V. An energy-based model for electrowetting- induced droplet actuation. J. Micromech. Microeng. 2006, 16, 1494−1503. (4) Sammarco, T. S.; Burns, M. A. Thermocapillary pumping of discrete drops in microfabricated analysis devices. AIChE J. 1999, 45, 350−366. (5) Geng, X.; Yuan, H.; Oguz, H. N.; Prosperetti, A. Bubble-based micropump for electrically conducting liquids. J. Micromech. Microeng. 2001, 11, 270−276. (6) Chiou, P. Y.; Moon, H.; Toshiyoshi, H.; Kim, C. J.; Wu, M. C. Light actuation of liquid by optoelectrowetting. Sens. Actuators, A 2003, 104, 222−228. (7) Cho, S. K.; Moon, H.; Kim, C. J. Creating, transporting, cutting, and merging liquid droplets by electrowetting-based actuation for digital microfluidic circuits. J. Microelectromech. Syst. 2003, 12, 70−80. (8) Pollack, M. G.; Shenderov, A. D.; Fair, R. B. Electrowetting-based actuation of droplets for integrated microfluidics. Lab Chip 2002, 2, 96−101. (9) Pollack, M. G.; Fair, R. B. Electrowetting-based actuation of liquid droplets for microfluidic applications. Appl. Phys. Lett. 2000, 77, 1725−1726. (10) Kuo, J. S.; Mihalic, P. S.; Rodriguez, I.; Chiu, D. T. Electrowetting Induced Droplet Movement in An Immiscible Medium. Langmuir 2003, 19, 250−255. (11) Mohseni, K.; Dolatabadi, A. Electrowetting droplet actuation in micro scale devices. In 43rd AIAA Aerospace Science Meeting and Exhibit, 2005, p AIAA-677. (12) Baret, J. C.; Decré, M. M.; Mugele, F. Self-excited drop oscillations in electrowetting. Langmuir 2007, 23, 5173−5179. (13) Nelson, W. C.; Sen, P.; Kim, C. J. Dynamic contact angles and hysteresis under electrowetting-on-dielectric. Langmuir 2011, 27, 10319−10326. (14) Carpenter, K.; Bahadur, V. Electrofreezing of water droplets under electrowetting fields. Langmuir 2015, 31, 2243−2248. (15) Fan, S. K.; Hsu, Y. W.; Chen, C. H. Encapsulated droplets with metered and removable oil shells by electrowetting and dielectrophoresis. Lab Chip 2011, 11, 2500−2508. (16) Cooney, C. G.; Chen, C. Y.; Emerling, M. R.; Nadim, A.; Sterling, J. D. Electrowetting droplet microfluidics on a single planar surface. Microfluid. Nanofluid. 2006, 2, 435−446. (17) Paik, P.; Pamula, V. K.; Chakrabarty K. Adaptive Hot-Spot Cooling of Integrated Circuits Using Digital Microfluidics: Proceedings of IMECE 2005, p 81081. (18) Lee, J.; Moon, H.; Fowler, J.; Schoellhamer, T.; Kim, C. J. Electrowetting and Electro-Wetting-on-Dielectric for Microscale Liquid Handling. Sens. Actuators, A 2002, 95, 259−268. (19) Moon, H.; Kim, J. Using EWOD (electrowetting-on-dielectric) actuation in a micro conveyor system. Sens. Actuators, A 2006, 130131, 537−544. (20) Bahadur, V.; Garimella, S. V. Energy Minimization-Based Analysis of Electrowetting for Microelectronics Cooling Applications. Microelectron. J. 2008, 39, 957−965. (21) Aminfar, H.; Mohammadpourfard, M. Lattice Boltzmann method for electrowetting modeling and simulation. Comput. Methods in Appl. Mech. Engrg. 2009, 198, 3852−3868. (22) Arzpeyma, A.; Bhaseen, S.; Dolatabadi, A.; Wood-Adams, P. A coupled electro-hydrodynamic numerical modeling of droplet actuation by electrowetting. Colloids Surf., A 2008, 323, 28−35. (23) Abdelgawad, M.; Park, P.; Wheeler, A. R. Optimization of device geometry in single-plate digital microfluidics. J. Appl. Phys. 2009, 105, 094506. (24) Nelson, W. C.; Kim, C. J. Droplet Actuation by Electrowettingon-Dielectric (EWOD): A Review. J. Adhes. Sci. Technol. 2012, 26, 1747−1771. (25) Mugele, F.; Baret, J. C. Electrowetting: from Basics to Applications. J. Phys.: Condens. Matter 2005, 17, R705−R774. (26) Fair, R. B. Digital Microfluidics: Is a True Lab-on-a-Chip Possible? Microfluid. Nanofluid. 2007, 3, 245−281.


the end of the actuated electrode. As a result, the displacement of the center of mass is higher for droplets with a 0.262 μL volume than for droplets with a 2.093 μL volume. However, for droplets with a 0.033 μL volume, the predominance of the surface tension force prevents it from deforming much at the end of the actuated electrode. This causes a lower displacement in the center of mass than for droplets of other sizes as shown in Figure 15. A comparison between the magnitudes of velocities for differently sized droplets can be made from Figure 16, where the droplet velocity is plotted against time.

Figure 16. Variation of droplet velocity with time for different sizes.

5. CONCLUSIONS An electro-mechanical approach is adopted to numerically analyze the uphill climbing behavior of a droplet due to electrostatic actuation. The velocity field inside the droplet reveals the dominance of sliding instead of rolling during uphill climbing. Variation of the velocity of the center of mass of the droplet with displacement and time is analyzed for various parameters such as the actuation voltage, inclination, and droplet size. Result shows that, for high actuation voltages, the maximum velocity occurs at the end of the actuated electrode. Velocity variation along different inclinations reflects the interplay between the electrostatic force and gravitational pull. The study also reveals that the inertia of the bulk fluid for larger droplet influences the uphill climbing velocity. The present method can be useful to develop compact microfluidic devices where fluids have to be manipulated in more than one plane.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support of this work was received from the Department of Science and Technology, India (grant no. SB/ FTP/ETA-84/2013).

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REFERENCES

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Uphill Movement of Sessile Droplets by Electrostatic Actuation

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