Control of Drop Impact and Proposal of Pseudo-superhydrophobicity

Article pubs.acs.org/IECR

Control of Drop Impact and Proposal of Pseudosuperhydrophobicity Using Electrostatics Subhamoy Pal,‡ Ansari M. Miqdad,‡ Saikat Datta,† Arup K. Das,*,‡ and Prasanta 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

‡

ABSTRACT: The phenomenon of droplet collision with a charged substrate is investigated numerically by a coupled electro-hydrodynamic model. A charge conservation equation and Poisson equation are solved to obtain the transient electric field. The divergence of Maxwell stress (due to the electric field) is included in the transient momentum equation as a volumetric force to couple the electrostatic force with the hydrodynamics. The interface between the two phases is tracked by volume of fluid method. The motion of the contact line on the solid substrate is controlled by concentric ring shaped charged regions. The electric stress in the vicinity of the contact line restrains its motion in the desired direction, which changes the impact behavior substantially. A hydrophilic surface shows superhydrophobic characteristics when actuated by a sufficient magnitude of electric potential. The phenomenon is analyzed with different parametric variations like electric potential, wetting nature of the substrate, and velocity of collision as it is governed by the mutual interaction between the inertia, electrostatic, and capillary forces. The present method can be used to develop engineering surfaces with tunable wetting nature.

1. INTRODUCTION The occurrence of droplet collision with solid surfaces is ubiquitous in nature such as the impact of rain drops on soil, as well as in a vast range of engineering applications like selfcleaning surfaces,1 spray cooling,2 inkjet printing,3 forensic sciences,4 anti-icing surfaces,5 and hybrid bioelectronic device fabrication.6 Thus, the collision behavior of a droplet has become a topic of fundamental importance for comprehension of its applicability in different engineering scenarios. Both experimental3,7−11 and numerical10−17 investigations have been performed to analyze the phenomenon of droplet impact. The experimental study of Rioboo et al.9 shows that for different parameters, like viscosity, drop diameter, surface roughness, and wetting nature, similarity of spreading pattern exists at early stages of the impact only, whereas the influencing factors dominate at the later stages. The influence of surface tension and contact angle on the impact behavior was investigated by Pasandideh-Fard et al.10 using both numerical and experimental techniques. Their analysis showed that the reduction of surface tension does not influence the early stages of the impact, but it increases the spreading diameter and decreases recoil height. Visser et al.11 investigated the high-speed collision of droplets with a solid surface. Their numerical model predicted the scaling laws for rim diameter and the transient boundary layer thickness during spreading. The numerical investigation of Hasan et al.12 provides the influence of the different parameters like Weber number, Reynolds number, and Froude number on droplet deformation after the impact. Philippi et al.13 performed numerical simulations to analyze the early stages of droplet © XXXX American Chemical Society

impact on a solid surface. From their study, it is obtained that the pressure and velocity field acquire a self-similar structure at the initial stages of collision. Recently Wildeman et al.14 studied the dissipation mechanisms during the collision of a droplet with solid surfaces by direct numerical simulations. Their study reveals that for a droplet colliding with a high Weber number, the total energy loss due to the surface deformation is approximately half of the initial kinetic energy irrespective of the properties of liquids. The manipulation of drop impact with the solid substrate is essential to obtain desired physical and chemical conditions on the treated surfaces. Various methods can be applied to control the droplet impact such as altering the wetting nature and roughness of the surface,5 use of surfactant,10 controlling the velocity of collision,9 etc. Electrostatic control of droplet generation and impact is also extensively used in inkjet printing technology3,18 and for fabrication of structured components of microelectrical devices.19 In this method, an electronic charging and deflection system is used to manipulate the liquid droplet. Here, after the generation and charging, the droplets are deflected by the electrostatic force to impact at desired location. Another popular method of electrostatic liquid handling is electrowetting.20 In this technique, the wetting property of a solid substrate is manipulated with the application of an Received: Revised: Accepted: Published: A

May 17, 2017 August 29, 2017 September 8, 2017 September 8, 2017 DOI: 10.1021/acs.iecr.7b02036 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research external electric field. After the initial investigation by Lippmann21 and the subsequent improvement of its applicability by Berge,22 electrowetting was studied extensively by both experimental23−26 and theoretical27−31 methods and implemented successfully in various microfluidic and lab-on-achip applications. A detailed survey of the studies on electrowetting can be obtained from Mugele and Baret,20 Fair,32 and Nelson et al.33 Electrowetting is commonly used to maneuver a small volume of liquids (in the form of droplets), either placed on a solid substrate or restricted inside a small confinement in various microfluidic applications like translation,24 agglomeration,25 dispensing,23 and deformation of liquid menisci.34 However, this technique could be advantageous to manipulate the contact line motion during the collision of a droplet with a solid surface. Lee et al.35 studied the impact behavior of a droplet with an electrically charged surface covered with a Teflon layer. The Teflon was stretched to manipulate the surface roughness. It was observed that at 3 kV of electric potential the droplet got stuck to the surface covered with stretched Teflon whereas in case of untreated Teflon it bounced partially. In their study, the electric field in the vicinity of the solid surface varied smoothly. However, a steep change in the direction of the electric field by applying an electric potential to specific regions could direct the contact line in a desired way and change the impact dynamics significantly. In the present study, electro-hydrodynamic simulations have been performed to investigate the phenomenon of drop impact on a concentric ring shaped electrode array. The volumetric electric force, estimated from the solution of charge conservation and Poisson equation is included in the momentum equation to couple the electric phenomena with the hydrodynamics. The influence of the wetting nature of the surface, electrical potential, and impact velocity on the collision behavior is analyzed by tracking surface with a volume of fluid (VOF) formulation.

Figure 1. Schematic of the electrostatic force in the vicinity of the contact line.

2. METHODOLOGY Electrowetting is a typical example of electro-hydrodynamic phenomenon. It can be characterized by the motion of fluids due to the electric stress generated when subjected to an external field. In electrowetting, when a droplet is placed in an electric field, the free ions and the dipoles present in the liquid reorganize themselves in accordance with the applied field. This results in the accumulation of charge near the interface regions. The sharp curvature in the wedge shaped contact line vicinity causes a high charge concentration build up in this region. Thus, the electric field produces higher stress near the contact line. The electrostatic force imparted on this region is schematically shown in Figure 1. The vertical component of this electrostatic force is balanced the by the stress generated at the solid surface. However, the horizontal components try to drag the interface outward. This reduces the apparent contact angle, keeping the equilibrium one unaltered. In the present study, the technique of electrowetting is adopted to manipulate a droplet after impact on a solid surface. The physical configuration of the present setup is depicted in Figure 2. To control the motion of the contact line, concentric ring shaped electrodes are considered. Electric potential is applied to the alternative electrodes (red rings in Figure 2). The electrodes between the actuated ones are kept grounded (blue rings in Figure 2). The droplet is allowed to impact at the center of the electrode array. The phenomenon is considered to be

Figure 2. Schematic of the simulation domain and the scheme of electrodes.

axisymmetric. No slip and no penetration conditions are assumed at all the walls. In the present study, the axisymmetric electro-hydrodynamic module of open-source Gerris flow solver36−38 is used to simulate the impact phenomenon. The migration of charge and the motion of fluids make the electro-hydrodynamic phenomenon intricate, as these, in turn, alter the electric field. Thus, a charge conservation equation is required to obtain the transient charge distribution inside the flow domain. Neglecting the generation inside the bulk, the charge conservation equation can be expressed as ∂qc

C + ∇·(qcu ⃗) = − qc ∂t ε

(1)

where ε, C, and qc are permittivity, electric conductivity, and volumetric charge concentration, respectively. The electric potential inside the domain as a result of the redistribution of the charges can be obtained at each time step by solving the Poisson equation as ∇·(ε ∇ϕ) = −qc B

(2) DOI: 10.1021/acs.iecr.7b02036 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research

Figure 3. Distribution of computational cells in the simulation domain: (a) t = 0.0 ms and (b) t = 20.7 ms.

where ϕ is the electric potential. After the estimation of charge and electric field (E⃗ ) distribution at a step in the time marching, the volumetric electric can be obtained by the divergence of Maxwell stress (τe ) as e ⃗ = ∇· τe = qcE ⃗ −

1 ⃗2 E ∇ε 2

scheme. Also, a second order accurate Crank−Nicolson type37 discretization is employed for the viscous term. Restructuring of the discretized momentum equation in a Helmholtz type equation enables Gerris to solve it by a multilevel Poisson solver. Quadtree spatial discretization technique37 is used to refine the mesh based on the interface gradient and vorticity. The distribution of the cell inside the computational domain is shown in Figure 3. The refinement is accomplished from a maximum cell size (minimum level) to a minimum one (maximum level) hierarchically. In Quadtree discretization, the parent cell is divided into four equal cells; that is, at the nth level, the minimum size of the cell would be 1/2n times the maximum cell size. Rigorous checks have been performed to determine the highest levels of refinement. In order to depict the mesh independence and validity of the simulation, a standard problem of a suspended droplet inside an electric field is considered. The hydrodynamic parameters are considered similar to Feng.39 However, the drop and the domain sizes are kept identical to the present study. Figure 4 shows the variation of the radial electric field along a line inclined at 45° with the horizontal axis. The simulated results are also compared satisfactorily with the theoretical solution of Taylor40 (considering the same parameters). The maximum

(3)

In Gerris, the momentum conservation equation incorporates the volumetric electrostatic force (e)⃗ as a source term to couple the hydrodynamics with the electric phenomenon. The incompressible transient mass and momentum conservation equation coupled with the electric phenomenon can be expressed as (4)

∇·u ⃗ = 0

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

(5)

Here, p and u⃗ are the pressure and velocity vector. τv is the viscous stress tensor. The force due to the surface tension is estimated based on the interface normal (n⃗), the local mean interface curvature (λ), and the coefficient of surface tension (σ). To ensure the area of application of the surface tension to be at the liquid−gas interface only, the Dirac-delta function (δsur) is included in the equation. λ is calculated based on the height-function formulation. Gerris uses a finite volume based piecewise linear volume of fluid (VOF) scheme to track the gas−liquid interface. Volume fraction (α) of the fluids at each cell of the computational mesh is taken as the order parameter of the VOF scheme. Considering the continuity of the fluid, the interface advection is obtained by ∂α + ∇·(αu ⃗) = 0 ∂t

(6)

ρm = αρw + (1 − α)ρa

(7)

μ = αμw + (1 − α)μa

(8)

Here, μw and ρw are the viscosity and density of the liquid phase, whereas μa and ρa are the same quantity for the gaseous phase. In Gerris, the volume fraction and the pressure are discretized in a second order accurate staggered-in-time

Figure 4. Variation of the radial electric field along a 45° inclined ray with horizontal for a droplet suspended inside the electric field. C

DOI: 10.1021/acs.iecr.7b02036 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research variation of the electric field (at the interface) between the results of theoretical calculation and simulation with level 6 is 9.27%. The deviation reduces to 3.6% and 1.2% with the maximum refinement level of 7 and 8, respectively. With further refinement to level 9, the deviation does not reduce much (0.6%). Thus, to make the computation economic, refinement level of 8 is used in all the simulations.

3. RESULTS AND DISCUSSION In the present study, the droplet is assumed to be composed of deionized water, and air is considered as the secondary medium. The surface tension is assumed to be 0.072 N/m. The viscosity of the air and water are kept constant at 0.00089 Pa·s and 1.81 × 10−5 Pa·s, respectively. Densities of the liquid and gas phases are considered as 1000 kg/m3 and 1.1644 kg/ m3. Water is assumed to have electric conductivity and permittivity of 5.5 × 10−6 S/m and 710 × 10−12 F/m. Electric conductivity and permittivity of the gas phase are considered as 8.0 × 10−15 S/m and 8.85 × 10−12 F/m, respectively. The diameter of the droplet is kept fixed at 1 mm. The width of each electrode is assumed to remain constant at 0.1 mm. Figures 5 and 6 depict the evolution of the phase contours during the drop impact with actuation voltages of 0 and 80 V.

Figure 6. Evaluation of phase contour during the collision of the droplet with actuation of 80 V: (a) t = 0.0 ms, (b) t = 17.0 ms, (c) t = 20.0 ms, (d) t = 20.7 ms, (e) t = 21.3 ms, (f) t = 21.7 ms, (g) t = 22.2 ms, (h) t = 22.7 ms, (i) t = 23.1 ms, (j) t = 23.3 ms, (k) t = 23.4 ms, and (l) t = 26.0 ms.

surface, the contact line moves in the outward directions (Figure 6c−e). However, the contact line ceases its outward motion. At the same time, the liquid at the bulk tries to flow in the transverse direction. This results in a stretching of the droplet above the plane of the solid surface (Figure 6f). The energy accumulated in the deformed droplet drives the liquid in upward direction. Subsequently, the droplet gets detached from the solid surface and starts an upward journey (Figure 6i−l). In order to investigate the reason behind the droplet bouncing, the electric field inside the domain is analyzed in detail. Figure 7 depicts the contour of electric potential inside

Figure 5. Evaluation of phase contour during the collision of the droplet without actuation potential: (a) t = 0.0 s, (b) t = 13.7 ms, (c) t = 19.9 ms, (d) t = 20.3 ms, (e) t = 20.5 ms, (f) t = 21.2 ms, (g) t = 21.7 ms, (h) t = 22.2 ms, (i) t = 23.29 ms, (j) t = 24.0 ms, (k) t = 25.4 ms, and (l) t = 26.0 ms.

The equilibrium contact angle is kept constant to 90° for both the situations. The initial height of the droplet mass center is 2.45 mm. For the case with no actuation potential, the droplet sticks to the solid surface upon impact. After the initial contact (Figure 5c), the motion of the bulk liquid gets constrained by the solid surface. As a result, it is directed away from the central axis and spreading occurs (Figure 5c−h). At this stage, the interface acts as a strained membrane; and the kinetic energy of the liquid is converted into potential energy of the distorted interface. After a certain period, the strained droplet pushes the liquid back toward the central axis. This causes the droplet to become elongated along the longitudinal axis (Figure 5j). The periodic distortion of the droplet along the two perpendicular directions continues until the energy gets dissipated by the viscosity and a final hemispherical shape is attained (not shown in the figure). The result of the droplet impact with the actuation of 80 V in the electrodes is different from the earlier situation. In this case, after the initial touch with the solid

Figure 7. Contour electric potential inside the simulation domain. Actuation potential = 80 V.

the simulation domain for a case of actuation with 80 V. It can be observed from the figure that the potential remains almost constant and relatively less in magnitude in the bulk of the droplet. However, closer to the solid surface it attains a higher value. This results in a larger electrostatic force near the solid surface. It is evident from the close look of the electric field in the vicinity of the solid surface (shown in the inset of Figure 7) that the electric field alters its direction at the midspan of each electrode. Moreover, it remains higher near the junctions of the D

DOI: 10.1021/acs.iecr.7b02036 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research

occurs in a cyclic manner until the spreading front recoils. However, in Figure 8, a single cycle is shown, considering the time of initiation (t′) of the cycle as zero. The velocity vectors inside the droplet during the impact with an actuation of 100 V are shown in Figure 9. Initially, the bulk of the droplet has downward velocity. After the hindrance from the solid surface, the bulk liquid starts moving in the transverse direction. Once the electric field near the solid surface restrains the contact line, the inertia of the bulk drives the liquid along the lower part of the droplet menisci toward the central axis. The motion of the liquid toward the core of the droplet gets hindered by the liquid flowing from the opposite direction. As a consequence, the liquid starts upward movement along the central axis. Eventually, the upward stream reaches the droplet menisci at the top affecting the bulk liquid too. Finally, the whole droplet starts moving upward. 3.1. Effect of Electric Potential. From the above discussion, it is evident that, due the presence of electric field, the impact of a drop with a solid surface results in a different situation in comparison to the case with no electric actuation. Thus, it is interesting to analyze the influence of the actuation potential on the drop impact behavior. Figure 10 shows the variation of the height of the droplet mass center with time for different actuation potentials. The initial height of the droplet is kept constant at 2.45 mm. As a consequence, the velocity of impact remains the same for all the cases with different actuation potential. For the case actuated with 60 V, the electrostatic force does not suffice to hold back the contact line during its motion away from the axis of symmetry. Thus, it overcomes the opposing electrostatic forces during spreading. This results in the sticking of the droplet with the surface after some initial oscillation. For the cases of the actuation with 80 and 100 V, the electrostatic force is sufficient to pin down the contact line. As a consequence, the droplets rebound from the surface resembling the behavior of impact with a superhydrophobic surface. Actuation with 100 V produces higher electrostatic force compared to 80 V potential. As a result, it

two consecutive electrodes. This also changes the direction of the electrostatic force that restrains the forward motion of the contact line. To have further insight, the pressure field near the contact line has also been analyzed during the impact. Figure 8

Figure 8. Temporal evolution of pressure near the contact line during its attempt at forward motion. The time (t′) of the initiation of pressure build up is considered as zero. Actuation potential = 80 V.

shows the variation of the pressure at the contact line with time. In our earlier communication,31 it was shown that there is a formation process of strong vortices at the junction of the two consecutive electrodes. Since the density of air is very low, the vortices do not have much impact on the pressure field outside the droplet. However, inside the liquid zone, the vortices create positive and negative high pressure zones near the electrode junctions. When the contact line reaches the junction of the electrodes, the heavier fluid is pushed toward the solid surface. As a result, pressure starts building up. However, due to the impermeable wall, the liquids of this high pressure region are directed upward. This causes a vortex formation at this region with a negative pressure zone at its core. This phenomenon

Figure 9. Velocity field inside computational domain during the drop impact with the solid surface. Electric potential = 100 V. (a) t = 10.9 ms, (b) t = 12.5 ms, (c) t = 13.3 ms, (d) t = 14.0 ms, (e) t = 14.8 ms, and (f) t = 15.6 ms. E

DOI: 10.1021/acs.iecr.7b02036 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research

augmented substantially. For the droplet initiated from a height of 3.675 mm, the inertia force dominates over the force generated due to the electric field. As a consequence, the wetting facade overcomes the opposing electrostatic force barrier, and the droplet gets attached to the surface. On the other hand, the droplets initiated from elevations of 1.225 mm and 2.45 mm rebound from the substrate. However, the latter one attains larger distortion upon collision and accelerates more rapidly due to the higher inertia. Figure 12 represents the

Figure 10. Variation of the mass center height of the droplet with time for different electric potential. ξ = Coefficient of restitution (the ratio of the velocity at the instant of initial touch and detachment of the droplet). h and Vz are the height and upward velocity of the droplet mass center.

restrains the contact line in advance as compared to the actuation with 80 V, causing a higher velocity and displacement after the impact. To characterize the drop rebound further, the coefficient of restitution (ξ) is estimated and mentioned in the inset of Figure 10. 3.2. Effect of Height. Figure 11 depicts the influence of the initial height of the droplet on the impact with an electrically

Figure 12. Variation of the minimum voltage (ϕthreshold) required for drop rebound with the initial height and impact velocity. Contact angle = 90°.

minimum requirement of the applied electric potential to observe pseudo-superhydrophobic behavior of the substrate at different initial elevations. Since, the velocity of impact differs with the initial height, the variation of the threshold voltage as a function of the impact velocity is shown in the inset of Figure 12. From the slope of the curve, it can be concluded that the threshold electric potential to obtain the pseudo-superhydrophobic behavior of the substrate is more sensitive at the lower elevations. 3.3. Effect of Contact Angle. At this point, it is worthwhile to analyze the effects of the wetting nature of the surface on the drop impact (with electric actuation). Figure 13 depicts the variation of the droplet mass center as a function of Figure 11. Temporal variation of the height of the mass center of the droplet initiated from a different elevation.

charged surface. To make a comparison, at the initiation of the impact and height and velocity of the center of mass of the droplet are normalized by the initial values. The time required for the droplet to reach the solid surface is used to scale the time coordinate. The electric potential and the contact angle are considered to be fixed at 70 V and 90°, respectively. During the downward motion, the droplet situated at higher altitude, gains higher velocity at a particular distance from the solid surface, in comparison with the cases with lower initial elevation. This results in the higher slopes in velocity variation for the cases with larger elevations during the downward motion. The electrostatic force generated in the vicinity of the substrate does not change in all the cases, due to the application of the same actuation potential (70 V). However, with the increase in the initial elevation, the inertia of the droplet is

Figure 13. Variation of the height of the droplet mass center with time for surfaces with different wetting nature. Actuation potential = 50 V. F

DOI: 10.1021/acs.iecr.7b02036 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research

(5) Mishchenko, L.; Hatton, B.; Bahadur, V.; Taylor, J. A.; Krupenkin, T.; Aizenberg, J. Design of Ice-Free Nanostructured Impacting Water Droplets. ACS Nano 2010, 4, 7699−7707. (6) FitzGerald, S. P.; Lamont, J. V.; McConnell, R. I.; Benchikhel, O. Development of a High-Throughput Automated Analyzer Using Biochip Array Technology. Clin. Chem. 2005, 51, 1165−1176. (7) Jiang, Y. J.; Umemura, A.; Law, C. K. An Experimental Investigation on the Collision Behavior of Hydrocarbon Droplets. J. Fluid Mech. 1992, 234, 171−190. (8) Lagubeau, G.; Fontelos, M. A.; Josserand, C.; Maurel, A.; Pagneux, V.; Petitjeans, P. Spreading Dynamics of Drop Impacts. J. Fluid Mech. 2012, 713, 50−60. (9) Rioboo, R.; Marengo, M.; Tropea, C. Time Evolution of Liquid Drop Impact onto Solid, Dry Surfaces. Exp. Fluids 2002, 33, 112−124. (10) Pasandideh-Fard, M.; Qiao, Y. M.; Chandra, S.; Mostaghimi, J. Capillary Effects During Droplet Impact on a Solid Surface. Phys. Fluids 1996, 8, 650−659. (11) Visser, C. W.; Frommhold, P. E.; Wildeman, S.; Mettin, R.; Lohse, D.; Sun, C. Dynamics of High-Speed Micro-Drop Impact: Numerical Simulations and Experiments at Frame-to-Frame Times Below 100 ns. Soft Matter 2015, 11, 1708−1722. (12) Hasan, M. N.; Chandy, A.; Choi, J. W. Numerical Analysis of Post-Impact Droplet Deformation for Direct-Print. Engineering Applications of Computational Fluid Mechanics 2015, 9, 543−555. (13) Philippi, J.; Lagrée, P. Y.; Antkowiak, A. Drop Impact on a Solid Surface: Short-Time Self-Similarity. J. Fluid Mech. 2016, 795, 96−135. (14) Wildeman, S.; Visser, C. W.; Sun, C.; Lohse, D. On the Spreading of Impacting Drops. J. Fluid Mech. 2016, 805, 636−655. (15) Eggers, J.; Fontelos, M. A.; Josserand, C.; Zaleski, S. Drop Dynamics After Impact on a Solid Wall: Theory and Simulations. Phys. Fluids 2010, 22, 062101. (16) Mehdi-Nejad, V.; Mostaghimi, J.; Chandra, S. Air bubble entrapment under an impacting droplet. Phys. Fluids 2003, 15, 173− 183. (17) Bussmann, M.; Mostaghimi, J.; Chandra, S. On a ThreeDimensional Volume Tracking Model of Droplet Impact. Phys. Fluids 1999, 11, 1406−1417. (18) Le, H. Progress and Trends in Ink-Jet Printing Technology. J. Imaging Sci. Technol. 1998, 42, 49−62. (19) Basaran, O. A.; Gao, H.; Bhat, P. P. Nonstandard inkjets. Annu. Rev. Fluid Mech. 2013, 45, 85−113. (20) Mugele, F.; Baret, J. C. Electrowetting: from Basics to Applications. J. Phys.: Condens. Matter 2005, 17, R705−R774. (21) Lippmann, G. Beziehungen Zwischen den Capillaren und Elektrischen Erscheinungen. Ann. Phys. 1873, 225, 546−561. (22) Berge, B. Électrocapillarité et mouillage de films isolants par l’eau. Comptes Rendus de l’Académie des Sciences. Série 2, Mécanique, Physique, Chimie, Sciences de l’univers, Sciences de la Terre 1993, 317, 157−163. (23) 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. (24) Pollack, M. G.; Shenderov, A. D.; Fair, R. B. ElectrowettingBased Actuation of Droplets for Integrated Microfluidics. Lab Chip 2002, 2, 96−101. (25) 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. (26) Kuo, J. S.; Spicar-Mihalic, P.; Rodriguez, I.; Chiu, D. T. Electrowetting Induced Droplet Movement in an Immiscible Medium. Langmuir 2003, 19, 250−255. (27) Bahadur, V.; Garimella, S. V. Energy Minimization-Based Analysis of Electrowetting for Microelectronics Cooling Applications. Microelectron. J. 2008, 39, 957−965. (28) Aminfar, H.; Mohammadpourfard, M. Lattice Boltzmann Method for Electrowetting Modeling and Simulation. Comput. Methods in Appl. Mech. Eng. 2009, 198, 3852−3868.

time. Here, the actuation potential and the initial height of the droplet are kept fixed at 50 V and 2.45 mm, respectively. For a hydrophilic surface (θ = 45°), the capillary force dominates over the force generated due to electric field. As a consequence, the contact line overcomes the electrostatic force barrier and the droplet remains attached to the surface. Moreover, the upward fluid stream arrives at the top of the droplet menisci earlier, in comparison, due the low drop height. This results in the formation of a daughter droplet as shown in inset of Figure 13. For the case with hydrophobic surface (θ = 120°), the capillary force for the recoiling of the droplet gets assistance from the electrostatic force. Thus, in spite of the low actuation (50 V), it rebounds from the surface.

4. CONCLUSION The influence of electric field on the dynamics of droplet collision with a solid substrate is studied through coupled electro-hydrodynamic numerical simulations. The study shows that with the application of electric potential to a ring shaped electrode array on a solid surface, the impact behavior mimics a case of drop collision with a superhydrophobic substrate. The electric field close to the solid surface shows that there is a change in the direction of the field at the midspan of the electrode and the high field strength at the junction restrains the contact line from moving forward. The interplay among the capillary force, electrostatic force, and inertia is investigated with various parameters like electric potential, initial drop height, and wetting nature of the substrate. The results reveal that the force due to the inertia dominates the electric force at low actuation potential and thus the spreading front overcomes the electrostatic resistance. The study also shows that the threshold potential required to retain pseudo-superhydrophobic behavior of the substrate is more sensitive to the lowvelocity collision. The present technique can be used to create tunable superhydrophobic surfaces in various engineering applications.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Arup K. Das: 0000-0002-2323-4745 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Financial support of this work was received from Department of Science and Technology, India (Grant No. SB/FTP/ETA84/2013).

■

REFERENCES

(1) Blossey, R. Self-Cleaning Surfaces Virtual Realities. Nat. Mater. 2003, 2, 301−306. (2) Kim, J. Spray Cooling Heat Transfer: The State of the Art. Int. J. Heat Fluid Flow 2007, 28, 753−767. (3) Van Dam, D. B.; Le Clerc, C. Experimental Study of the Impact of an Ink-Jet Printed Droplet on a Solid Substrate. Phys. Fluids 2004, 16, 3403−3414. (4) Laan, N.; De Bruin, K. G.; Bartolo, D.; Josserand, C.; Bonn, D. Maximum Diameter of Impacting Liquid Droplets. Phys. Rev. Appl. 2014, 2, 044018. G

DOI: 10.1021/acs.iecr.7b02036 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research (29) 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. (30) Abdelgawad, M.; Park, P.; Wheeler, A. R. Optimization of Device Geometry in Single-Plate Digital Microfluidics. J. Appl. Phys. 2009, 105, 094506. (31) Datta, S.; Das, A. K.; Das, P. K. Uphill Movement of Sessile Droplets by Electrostatic Actuation. Langmuir 2015, 31, 10190− 10197. (32) Fair, R. B. Digital Microfluidics: Is a True Lab-on-a-Chip Possible? Microfluid. Nanofluid. 2007, 3, 245−281. (33) Nelson, W. C.; Kim, C. J. Droplet Actuation by Electrowettingon-Dielectric (EWOD): A Review. J. Adhes. Sci. Technol. 2012, 26, 1747−1771. (34) Berge, B.; Peseux, J. Variable Focal Lens Controlled by an External Voltage: An Application of Electrowetting. Eur. Phys. J. E: Soft Matter Biol. Phys. 2000, 3, 159−163. (35) Lee, M. W.; Latthe, S. S.; Yarin, A. L.; Yoon, S. Dynamic Electrowetting-On-Dielectric (DEWOD) on Unstretched and Stretched Teflon. Langmuir 2013, 29, 7758−7768. (36) Popinet, S. Gerris: A Tree-Based Adaptive Solver for the Incompressible Euler Equations in Complex Geometries. J. Comput. Phys. 2003, 190, 572−600. (37) Popinet, S. An Accurate Adaptive Solver for Surface-TensionDriven Interfacial Flows. J. Comput. Phys. 2009, 228, 5838−5866. (38) López-Herrera, J. M.; Popinet, S.; Herrada, M. A. A ChargeConservative Approach for Simulating Electrohydrodynamic TwoPhase Flows Using Volume-Of-Fluid. J. Comput. Phys. 2011, 230, 1939−1955. (39) Feng, J. Q. Electrohydrodynamic Behavior of a Drop Subjected to a Steady Uniform Electric Field at Finite Electric Reynolds Number. Proc. R. Soc. London, Ser. A 1999, 455, 2245−2269. (40) Taylor, G. I. Studies in Electrohydrodynamics. I. The Circulation Produced in a Drop By an Electric Field. Proc. R. Soc. London, Ser. A 1966, 291, 159−166.

H

DOI: 10.1021/acs.iecr.7b02036 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX