Unravelling Electrostatic Actuation on Inclined and ... - ACS Publications

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Unravelling Electrostatic Actuation on Inclined and Humped Surfaces: Effect of Substrate Contact Angle Saikat Datta,† Arup K. Das,*,‡ and Prasanta K. Das† †

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

‡

S Supporting Information *

ABSTRACT: Application of external electric field has become a wellestablished technique to achieve dynamic wetting over a flat surface on which the drop is resting. However, an increase in electric field can overcome the gravitational pull and the drop can climb the substrate against contact-line friction. The present study investigates droplet up-climbing for different wettability regimes using the energy minimization method. We proposed a capacitive network around an up-climbing drop for evaluation of the resultant force on it. This method captures the influence of the orientation and geometry of the surfaces on the droplet motion. Motive forces for dynamic wetting and velocity over inclined surfaces (15−75°) are predicted. Furthermore, over a humped surface, a voltage map is obtained for steady droplet motion. Variation of the voltage map for different equilibrium contact angles of the hump is also presented. A onedimensional momentum balance equation is used to predict the temporal location of the drop and the forces associated. geometry. Jones27 used lumped parameter electromechanics to determine the net force on a liquid mass. The electric actuation force is calculated based on the derivative of the energy stored in the capacitor network formed in the EWOD system. Chen et al.31 proposed a theoretical model to investigate the electrowetting induced capillary rise inside a parallel plate channel considering the dynamic contact angle as a function of the square of the applied voltage and capillary number. An analytical model based on the energy minimization method (considering Lippmann’s equation for energy reduction) has been developed by Bahadur and Garimella24 to investigate the electrowetting of droplets, confined between two parallel plates. Effects of important parameters such as the thickness of the dielectric layer, change of droplet shape, etc. are reported in their study. Berthier et al.32 developed an analytical relation by taking into consideration the static contact angle hysteresis, to determine the minimum actuation potential required to mobilize a liquid droplet on an EWOD system. To investigate the heat absorption characteristics of a moving droplet due to electrowetting, Bahadur and Garimella25 proposed a modeling framework based on electromechanical energy minimization. The behavior of droplets on a rough surface due to external voltage is also investigated by their group. Their study provides an insight into the decisive factors for designing the thermal management system of electronic chips, by electrowetting. Microfluidic operations based on electrostatic actuations are commonly performed on single horizontal planes. The ability

1. INTRODUCTION Miniaturization of devices has become a common trend in the field of applied chemistry and engineering applications in order to enhance productivity and the reduction of lead time. As a consequence, there has been a growing research interest to develop and improve lab-on-a-chip and μ-total analysis systems, where chemicals are handled in the form of liquid droplets. The application of electric field has emerged as the most efficient technique to mobilize liquid drops in lab-on-a-chip devices and biological assays. This is commonly known as electrowetting. It was first explained by Lippmann1 and substantiated by Berge,2 who demonstrated the concept of electrowetting in an applicable form by providing an insulating layer over the bare metal surface and separated the liquid layer from the electrode. His concept is popularly known as electrowetting on dielectrics (EWOD). With the above breakthrough, EWOD has become an attractive area of investigation and a substantial amount of research has been carried out on translation,3−12 agglomeration,3,4,11,12 bifurcation,3,4,11,12 and dispensing3,4,11,12 of droplets on microassays in digital microfluidic devices. Detailed reviews on this popular technique can be found in Nelson et al.,13 Mugele and Baret,14 and Fair.15 Apart from experimental observations3−12,16−19 and numerical simulations,20−23 EWOD is explored and successfully analyzed with the aid of analytical approaches24−27 and explained based on several theoretical interpretations such as thermodynamic,1 electromechanical,28,29 and energy minimization.2 Ren et al.30 established a theoretical model to study the dynamics of electrically actuated droplet transport. Their study analyzes the mechanical energy dissipation during drop motion as a function of different operating parameters such as contact line friction, viscosity of filler fluid, interfacial tension, and © XXXX American Chemical Society

Received: November 26, 2015 Revised: March 7, 2016 Accepted: March 24, 2016

A

DOI: 10.1021/acs.iecr.5b04503 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

Figure 1. Physical configuration of the model: (a) 3-D view and (b) exploded view of electrode arrangement.

Figure 2. Electrode−droplet interface with the capacitance network.

to translate small fluid volume on nonhorizontal planes can provide more flexibility toward multilayered compact device design. The automation of complex chemical processes in these multilayered or three-dimensional (3-D) microfluidic devices can increase the volumetric capacity the same as planar ones. Droplet actuation by external electric field along different complex geometries such as twisted, inclined, upturned, and vertical surfaces was performed by Abdelgawad et al.33 Recently, the up-climbing of droplets was numerically investigated by Datta et al.34 Their study reveals that the upclimbing phenomena of droplets occur due to sliding motion instead of rolling. They also showed the range of voltage for which the drop can climb in upward direction for various inclinations. An analytical model is also developed to address the up-climbing behavior of droplets along inclined planes by Datta et al.35 All these preceding studies effectively demonstrate the dynamics of droplet motion on inclined surfaces under electrostatic actuation. However, they do not address the consequences of varying surface types (hydrophilic or hydrophobic) on droplet actuation. In the EWOD systems the droplet is always in contact with the solid surface. Thus, the equilibrium contact angle on a solid surface has a definite influence on the electrowetting actuation of droplets. The present investigation is an effort to comprehend the effect of

substrate contact angle on the droplet motion along inclined planes under external electric field. Based on the energy minimization principle, it has been also tried to keep a constant velocity of a drop over a curved surface. In section 2 we describe the methodology in brief and show the equivalent capacitance network. Based on the theoretical framework, droplet motion over inclined hydrophobic/hydrophilic surfaces is studied in section 3. Salient conclusions are mentioned in section 4.

2. METHODOLOGY In the present work an analytical framework based on the lumpcapacitance method has been adopted to investigate the electrowetting phenomena along an inclined plane. The energy minimization principle is applied to calculate the actuation force due to open type electrowetting. The physical configuration of the model is shown in Figure 1. At any instant of time the droplet is situated over two consecutive electrodes covered by a layer of dielectric material. In order to initiate motion, voltage is applied to the front electrode, whereas the rear electrode is kept grounded. The droplet is considered to be perfectly conducting. Fringing effects of the electric field from the sides of electrodes are neglected. The dynamic response of the droplet to the actuation is studied by B


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contact angle from its equilibrium value during the motion (dynamic contact angle) also causes loss in the capillary force. This opposing force is incorporated in the present model as the contact line friction by following the work of Ren et al.30 and Bahadur et al.24 Here, the contact line resistance is taken to be linearly proportional to the velocity (molecular kinetic

mass averaged velocity. The internal circulation and variation of the local flow properties inside the droplet are neglected. Thus, the droplet is considered to be a nondeformable single liquid mass moving under electric field in this simplified analytical model. In the present model the junction of the two neighboring electrodes is considered as the origin of the onedimensional reference frame for drop movement. A1 and A2 are the areas covered by the droplet on the front and rear electrodes, respectively (Figure 1b) and are functions of the displacement of the droplet front end from the end of the rear electrode. Considering the droplet as a spherical cap (or a portion of a sphere) and its footprint as a circle, the areas A1 and A2 can be calculated from the geometry as follows:

theory;36 v =

⎛ 5μ v ⎞ ⎛1 ⎞ Fopp = ⎜ l ⎟(πr 2) + ⎜ Cρf v 2⎟A + (ξv)(2πr ) + mg ⎝2 ⎠ ⎝ 2h ⎠ sin θ + 2l1γ(cos ϕth − cos ϕ)

(2)

Considering ϕ as the equilibrium contact angle, the radius (r) of the droplet can be written as ⎤1/3 ⎡ 3 ∀ sin 3 ϕ r=⎢ ⎥ ⎣ π (2 − 3 cos ϕ + cos3 ϕ) ⎦

C = 0.59 +

(10)

where ∀ is volume of the droplet. The capacitor formed due to the presence of the dielectric layer between the droplet and electrode is considered to be of parallel plate type in nature. The network of capacitor formed in the present configuration (Figure 1) is shown in Figure 2. The capacitance C1 and C2 can be calculated as kA1ε0 d

(4)

C2 =

kA 2 ε0 d

(5)

where Re is the Reynolds number. The Re is calculated as Re =

C1C2 C1 + C2

dC

dE 1 c 2 − c1 dc1 2 = V dx 2 c1 + c 2 dx

(11)

and h is the height of the droplet as expressed below: ⎛ 1 − cos ϕ ⎞ h = r⎜ ⎟ ⎝ sin ϕ ⎠

(6)

(12)

Details of the derivation of the projected area and droplet radius for a constant volume of liquid are presented in Appendix S1 (Supporting Information). As air is considered as the filler fluid, the contribution of drag force is low in the present case. However, the inclusion of the drag force term facilitates the model to be applicable for different filler fluids with various densities. Due to the presence of microdefects on the solid surface, the contact line gets pinned during the initiation of motion. Experimental studies38 show that contact angle raises at the advancing front and reduces at the retracting wetting front. The difference in the contact angle is known as contact angle hysteresis. This produces a loss in capillary forces. A threshold actuation voltage is required to initiate the movement of the droplet. Following the work of Berthier et al.32 and Ahmadi et al.,37 the resistance force due to contact angle hysteresis is included as the last term in eq 9. ϕth is the threshold contact

(7)

The actuation force (Fact) is originated from the tendency of the system to minimize its energy by relocating the droplet in the system. Hence, the actuation force can be obtained as Fact =

μf

⎛ h⎞ r 2 cos−1⎜1 − ⎟ − (r − h) r 2 − (h − r )2 ⎝ r⎠

For an applied voltage V, the energy (E) stored in the capacitor network (as a function of the displacement) for the present configuration can be found as follows 1 E(x) = Ceq(x)V 2 2

ρf vDm

where Dm is the diameter of the cross section of the spherical segment (droplet) having maximum area. Thus, for a hydrophilic surface it is the footprint diameter of the droplet. On a hydrophobic surface, Dm is considered as the diameter of the droplet which takes the shape of a portion of a sphere due to high contact angle. A is the projected area of the droplet in a plane transverse to the direction of motion, which can be expressed as

where k, d, and ε0 are the relative permittivity, the thickness of the dielectric layer, and the permittivity of vacuum, respectively. The equivalent capacitance (Ceq) for the network shown in Figure 2 can be expressed as Ceq(x) =

2 × 10−4Red 3.4 3.4 + − Red 0.89 Red 0.5 1 + (3.64 × 10−7Red 2)

(3)

C1 =

(9)

where v is the traversing speed, μl is the viscosity of the droplet, ρf is the density of filler fluid, and ξ is the proportionality constant for contact line friction with a value ξ = 0.04 N s/m2.24 The values of μl and ρf are considered to be 8.9 × 10−4 Pa·s and 1.2 kg/m3. C is the drag coefficient of a spherical cap inside filler fluid which is empirically estimated as37

(1)

A 2 (x) = πr − A1

Forces (Fopp) opposing the

motion of the droplet can be expressed summarily as

⎛ x⎞ A1(x) = r 2 cos−1⎜1 − ⎟ − (r − x) r 2 − (x − r )2 ⎝ r⎠ 2

γ(cos θs − cos θd) ). ξ

(8)

dC

where dx1 = − dx2 . The traversing droplet also encounters forces opposing its motion along an inclined surface. The opposing forces during up-climbing originate due to shear stress by the wall, drag force offered by the secondary medium, the gravitational pull, and force due to contact angle hysteresis. The divergence of the C


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of the droplet, an arrangement needs to be made for the continuation of the actuation force. An intelligent electrode design and the subsequent switching scheme is one option to sustain the actuation. In the present study, we have designed an array of stripped electrodes (as shown in Figure 5) and proposed a switching mechanism so that the droplet mass center, despite moving forward, will never reach the junction between the actuated and ground electrodes. It is assumed that, for a continuous movement, by switching the actuated electrodes, overlap of the drop over the actuated electrode is kept constant.

angle due to electrostatic actuation at the onset of motion which can be expressed as ⎡ r cos(ϕ − α) − l 2 cos(ϕ + α) ⎤ ϕth = cos−1⎢ ⎥−α l1 ⎦ ⎣

(13)

l1 and l2 are shown in Figure 3. α is the angle of hysteresis, which can be defined as the difference between the contact

3. RESULTS AND DISCUSSION The increase in the footprint area of the droplet on a hydrophilic surface enhances the actuation force. The contact line resistance also gets augmented at lower equilibrium contact angle. The interplay between the actuation force and the opposing forces is studied by solving the derived momentum equation (eq 14). However, the presence of the terms due to the actuation and drag force make the equation nonlinear. Thus, it is discretized in a finite difference scheme to attain the desired solution by time marching (as independent variable) in the forward direction. The first computational step is performed by a forward difference scheme and then a central differencing is applied to obtain the solution in an explicit manner. We considered that the droplet is initially resting dx ( dt = 0) with an initial displacement (x|t=0 = constant).

Figure 3. Droplet footprint on two consecutive electrodes.

angle made by the advancing and retracting wetting fronts at the same electric potential.32 Based on the experimental data presented by Berthier et al.,32 the value of contact angle is taken as 7°. Considering the upward direction is positive, the equation of motion of the droplet in along the incline can be expressed as

t=0

Extensive checks have been carried out to select the step size (Δt) in the time marching. The value of Δt is increased in steps starting from 2 × 10−3 s. The magnitudes of the droplet velocity at different specific voltages have been compared for various step sizes. As a representative case, the variation of the velocity of a 2.09 μL droplet (along a 60° incline) with actuation voltage for different step sizes is shown in Figure 6. The maximum deviation in the magnitude of the velocity is 0.536% between the cases with step sizes of 2 × 10−3 and 2 × 10−4 s (shown in the inset of Figure 6). For the step sizes of 2 × 10−5 and 2 × 10−6 the maximum difference in the velocity compared to the case with Δt = 2 × 10−3 s are 0.586 and 0.590%, respectively (i.e., the reduction of time step from 2 × 10−5 to 2 × 10−6 causes a maximum deviation of 0.0047% only). Further decrement of step size does not show much effect on the result. Thus, time marching with a step size of 2 × 10−5 s is used in the present study. The thickness of the dielectric layer is considered as 1 μm. The relative permittivity of the material is taken as 3. The validity of the model is checked by comparing the maximum transfer rate of the droplet as a function of voltage between the present study and experimental data presented by Pollack et al.4 Here, the maximum transfer rate is the reciprocal of the time period taken by the droplets to traverse a single electrode. Figure 7 shows the comparison of the present model and Pollack et al.4 where a 900 nL droplet of 0.1 M KCl is actuated by electric field on an array of electrodes with a length of 1.5 mm. The gap between the electrodes in the parallel plate arrangement is 0.3 mm. 3.1. Influence of Surface Contact Angle on Velocity. The velocity of the droplet is one of the primary influencing parameters in translation by EWOD. At the same electrostatic actuation, the velocity of the droplet can vary due to the change in the hydrophobicity of the solid surface. Thus, it is interesting to observe the effect of the surface type (hydrophobic or

d2x = Fact − Fopp (14) dt 2 where m is the mass of the droplet. Solving the above equation, the displacement and the velocity at a particular time instant can be determined. In Figure 4 we have shown the diminishing nature of the actuation force for a representative case as the droplet moves toward the junction of the electrodes. Hence the droplet starts to traverse toward the actuated electrode but slows down as its mass center moves toward the junction. Eventually it faces no actuation when the mass center is sitting over the junction. This seizes the droplet motion further. To continue the movement m

Figure 4. Variation of actuation force as the droplet traverses toward actuated electrode. D


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Figure 5. Schematic of array of narrow stripped electrodes. At any time instant the droplet will be situated on multiple electrodes.

Figure 8. Variation of velocity with nondimensional droplet displacement for different contact angles. Figure 6. Variation of velocity as a function of actuation potential for different step sizes in time marching; contact angle = 90°.

kept constant. For each surface type (characterized by equilibrium contact angle), the droplet starts its motion with an initial acceleration due the actuation force. As the actuation force decreases with the displacement, the acceleration of the droplet becomes zero at a position where the forces in the direction of motion are equal to the opposing forces. This situation appears when the center of the droplet reaches the junction of the electrodes; however, due to inertia it moves further. Droplets with lower equilibrium contact angle possess higher actuation force due to larger footprint area. This results in a higher velocity and displacement on the hydrophilic surfaces than the hydrophobic ones. In Figure 9a, the velocity of droplets with different equilibrium contact angles (at a particular time instant after the initiation of the actuation) is plotted as a function of the applied voltage. Here, the droplet is considered to be resting over multiple electrodes for a constant actuation. The angle of inclination is kept constant at 30°. Figure 9a accentuates to a requirement of lower actuation voltages for the droplets with smaller equilibrium contact angle. This is due to the increase in the actuation force (at the same electric potential) because of enlarged droplet footprint at the lower contact angles. However, the velocity of the droplets on hydrophobic surfaces becomes larger at the elevated actuation potentials (e.g., 100 V). To dig down further, the magnitudes of the actuation and opposing forces are plotted as a function of electric potential at Figure 9b and Figure 9c for contact angles 45 and 120°, respectively. In spite of having larger hysteresis force at 45°

Figure 7. Comparison of transport frequencies between the present study and Pollack et al.4 at different electric potentials.

hydrophilic) on the electrically triggered motion of a droplet. Figure 8 represents the variation of the velocity as a function of the displacement of a 2.09 μL droplet moving along a 45° incline with 80 V actuation. The equilibrium contact angle of the droplet (with the solid surface) is varied from 45 to 120°. The displacement is normalized by the radius of the droplet footprint. The electrode is arranged in a manner shown in Figure 1 with a size equal to the diameter of the droplet. In all the cases the initial displacement of the droplet front end is E


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Figure 9. Influence of equilibrium contact angle on drop motion along a 30° incline. (a) Voltage vs velocity plot for a drop volume of 2.09 μL for different contact angles. (b) Variation of forces involved in electrostatic actuation along a hydrophilic surface (ϕ = 45°) as a function of voltage. (c) Variation of governing forces with actuation potential for ϕ = 120°.

Figure 10. Variation of velocity with the actuation voltage for different contact angles: (a) volume = 2.09 μL; (b) volume = 16.75 μL.

hysteresis force due to the change of the surface type is more in the case of a droplet with larger volume. Thus, the augmentation in the threshold voltage required for the upward motion (due to the change of equilibrium contact angle) is high for a larger droplet (60−76 V) compared to a smaller one (46− 55 V). However, the trends of the curves remain the same in both cases. At 100 V, the velocity of a 2.09 μL droplet on a hydrophobic surface is higher than that on a hydrophilic one as discussed in section 3.1. For a droplet of 16.75 μL volume, the velocity on a hydrophobic surface remains less than the velocity on a hydrophilic surface at 100 V actuation. 3.3. Effect of Contact Angle on the Inclination of Upward Motion. Figure 11a represents the variation of threshold voltage (minimum voltage required for upward motion of the droplet) as a function of equilibrium contact angle. The angle of inclination is varied from 15 to 75°. At the same electric potential the actuation force reduces with the increment of equilibrium contact angle due to the reduction in the footprint area of the droplet. Higher voltage is required to

contact angle, the enhanced actuation force causes the initiation of motion at lower voltage; the actuation force remains higher at larger voltages. However, due to larger perimeter and velocity (for ϕ = 45°) the contact line friction force increases at a faster rate with actuation voltage. For the drop motion on hydrophilic surface the force due to viscosity also raises significantly at higher actuation potentials. Thus, the velocity at the contact angle of 45° increases at a slower rate (with voltage) compared to the case of the 120° contact angle. This results in a higher droplet velocity with 120° at larger actuation potential. 3.2. Contact Angle Sensitivity on Droplet Size. The change in the behavior of a droplet due to variation of size on different types of surfaces (i.e., hydrophobic or hydrophilic) is also worthwhile to know for better device design. Figure 10 depicts the variation of the velocity with voltage for two different size drops (2.09 and 16.75 μL volume) moving along a 30° inclined surface. For both cases, data has been plotted for surface contact angles of 120° (hydrophobic) and 45° (hydrophilic). The relative increment in the magnitude of the F


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Figure 11. Influence of contact angle on the inclination of upward motion. (a) Variation of minimum voltage required for up-climbing with equilibrium contact angles for a 2.09 μL droplet. (b) Magnitudes of governing forces at different contact angles. (c) Variation of actuation forces at different constant electric potentials as a function of contact angle.

Figure 12. Schematic representation of droplet motion along curved path. (a) Convex path. (b) Concave path.

overcome the same opposing force at higher contact angles. Thus, the threshold voltage rises with the contact angle in each inclination. As the velocity of the droplet is negligible at the initiation of motion, the threshold actuation force mainly depends on the gravitational and hysteresis forces. Figure 11b shows the variation of the governing forces with the contact angle during the initiation of motion along 15 and 75° inclines. In both the inclination the magnitude of the force due to hysteresis remains the same; it decreases with contact angle after showing some initial increment. However, larger gravitational pull (in the direction of motion) increases the requirement of threshold force at 75° inclination.

Figure 11c depicts the isopotential lines (variation of actuation force at the same voltage) in the equilibrium contact angle vs actuation force plot. Due to the nonlinear relation with V and r (eq 8), the actuation force varies at a higher rate for the larger values of electric potential. However, at low inclination (θ = 15°) the slope of the isopotential line is lower than the rate of change of the threshold actuation force with the contact angle (shown in Figure 11c). This results in a decreasing trend of threshold voltage at higher equilibrium contact angles at θ = 15° (Figure 11a). 3.4. Droplet Motion over a Circular Hump. Heat removal at a constant rate from the hot spots during the G


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Figure 13. Variation of actuation voltage with angular displacement to maintain velocity equal to 0.3 m/s. (a) Convex surface. (b) Concave surface.

Figure 14. Variation of actuation voltage with angular displacement and time to maintain constant droplet velocity (0.3 m/s). (a) Convex surface. (b) Concave surface.

the contact angle is also investigated for the above scenario. As the radius of the droplet is much smaller than that of the hump, the footprint of the droplet is considered as a flat surface. To characterize the present location of the drop over the surface, we define its angular position anticlockwise from the center of the circular hump as shown in Figure 12. 3.4.1. Voltage Map. Figure 13 depicts the variation of voltage to keep constant velocity (we have assumed 0.3 m/s), for droplet motion along a convex path and a concave path, respectively. Over the convex path the droplet requires higher voltage at the beginning of its motion as the component of the gravitational pull is maximum and opposite at this location. The requirement of voltage decreases with displacement; when ψ is more than π/2, the force due to gravity acts in the same direction of the actuation force and thus decreases the voltage requirement further. For the concave path the force due to gravity helps the motion up to ψ = π/2 and opposes the motion for the rest of the path. The magnitude of the actuation voltage for a constant velocity is only a function of angular displacement for a constant path radius (R); however, it changes with time as the radius of the curvature varies. Figure 14a shows the variation of the voltage requirement as a function of time and angular displacement for convex surfaces, whereas

thermal management of electronic chips or controlled reactions in biomedical microreactors by electrowetting may require the velocity of a droplet to be constant. However, the component of the gravitational pull in the direction of motion of the droplet changes with the geometry of the surface. As a consequence, the requirement of the actuation voltage also changes. In order to achieve a constant velocity of a droplet along a terrain path mimicking the geometry of the device, the magnitude of the actuation voltage has to be known at a particular time and location. Once the magnitude of the applied voltage is determined, the actuation of the electrodes can be programmed accordingly to have a constant velocity along its path. As any geometry can be a combination of circular curves, here in this article we have simplified a terrain structure by a circular hump and tried to maintain a constant velocity of droplet traverse by adjusting the applied voltage. In the present section, the voltage map has been determined for droplet movement along a convex and concave surface as shown in Figure 12. Figure 12a schematically shows the droplet motion along a convex surface, whereas Figure 12b represents the droplet motion along a concave surface. The electrodes (shown in red) are actuated with voltage and actuation of the electrode shifts with the droplet as it moves along the path. Influence of H


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Figure 15. Influence of the contact angle of the drop motion along circular terrain paths at a constant velocity (0.3 m/s). (a) Variation of actuation voltage with angular displacement along a convex path at different contact angles. (b) Requirement of electric potentials at different angular positions along a concave surface to attain constant velocity. (c) Variation of governing forces along a convex path at ϕ = 45°. (d) Changes in actuation and opposing forces at different angular positions of a hydrophilic (ϕ = 120°) concave surface during drop motion at constant velocity.

opposing forces dominate the motion. This causes the requirement of actuation (voltage) to be higher for lower contact angles (during downhill motion). For the droplet with 120° contact angle, after a displacement of 2.88 rad the requirement of actuation become 0 V. Thus, the motion of the droplet becomes fully governed by gravitational pull and other opposing forces. A similar but opposite trend of the voltage map can be observed for the concave path.

Figure 14b represents the same but for concave surfaces. The radius of curvature of the hump is varied from 2.5 to 7.5 cm in steps. The time required for the same angular displacement depends upon the radius of the path; higher radius involves more time than smaller ones. This is why the curves are divergent, while starting from the same initial point. 3.4.2. Influence of Contact Angle. Figure 15a,b represents the voltage maps for droplets with different contact angles translating along a convex path and a concave path at a constant velocity. At the same potential, the actuation force enhances on hydrophilic surfaces due to the larger droplet footprint area. The opposing forces also increase at lower equilibrium contact angle. The variation of the governing forces during the motion of the droplet is shown in Figure 15c and Figure 15d for the contact angles of 45 and 120°, respectively. Due to the change of contact angle from 45 to 120°, the contact line friction force reduces significantly. At the initiation of motion along a convex path, gravitational pull acts in the direction opposite to the motion. This requires a higher voltage to sustain the upward motion. At higher electric potential, actuation force increases sharply with the reducing contact angle (as discussed in section 3.3). Thus, droplets with smaller contact angle require lower voltage to maintain the constant velocity. However, during the downhill motion gravitational pull acts in the direction of motion and thus the necessity of actuation potential is reduced. At lower actuation voltages the contact line friction and other

4. CONCLUSION Influence of surface type (i.e., hydrophilic or hydrophobic) on electrowetting parameters is assessed by characterizing the surface with the equilibrium contact angle. The motion of the droplet is first analyzed for electrodes with dimensions equaling the diameter of the droplet. Droplets with lower equilibrium contact angles are identified to achieve higher velocity albeit of higher contact line friction and other opposing forces. The effect of the equilibrium contact angle on the translation of droplet overlapping multiple electrodes is studied for various parameters such as actuation voltage, droplet size, and inclination angles. The predominance of the opposing forces during drop motion over a hydrophilic surface at higher actuation potentials is identified. Results show that the requirement of threshold actuation potential for the onset of upward motion is less at hydrophilic surfaces. The study also I


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reveals higher sensitivity of smaller droplets to the nature of the surfaces. The increment in the threshold actuation potential with the decrease of surface contact angles is presented quantitatively. Voltage variations, in order to impart a constant velocity droplet motion along terrain paths, are analyzed. The present study could aid the design of control logic for droplet actuation in digital microfluidic systems.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.5b04503. Estimation of droplet footprint, overlap on electrodes, and projected area on the transverse plane (Appendix S1) (PDF)

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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 SB/ FTP/ETA-84/2013).

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REFERENCES

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