Letter pubs.acs.org/Langmuir
Can Vibrations Control Drop Motion? Rodica Borcia,*,† Ion Dan Borcia,† and Michael Bestehorn† †
Lehrstuhl Statistische Physik/Nichtlineare Dynamik, Brandenburgische Technische Universität, Erich-Weinert-Strasse 1, 03046 Cottbus, Germany ABSTRACT: We discuss a mechanism for controlled motion of drops with applications for microfluidics and microgravity. The mechanism is the following: a solid plate supporting a liquid droplet is simultaneously subject to lateral and vertical harmonic oscillations. In this way the symmetry of the backand-forth droplet movement along the substrate under inertial effects is broken and thus will induce a net driven motion of the drop. We study the dependency of the traveled distance on the oscillation parameters (forcing amplitude, frequency, and phase shift between the two perpendicular oscillations) via phase field simulations. The internal flow structure inside the droplet is also investigated. We make predictions on resonance frequencies for drops on a substrate with a varying wettability.
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INTRODUCTION Since the 19th century, scientists have been fascinated by droplet behavior under vibrations. The first ones were Rayleigh,1 Lamb,2 Landau & Lifshitz,3 which analytically calculated by different methods the eigenfrequencies of small capillary oscillations at the free surface of an inviscid spherical drop of radius R: ω = (m(m − 1)(m + 2)σ/ρR3)1/2, where m denotes the vibration mode (m ≥ 2), ρ is the liquid density, and σ is the surface tension coefficient. Recently this topic is back in fashion. In the past years, a lot of investigations have been performed on shape deformations corresponding to different resonance modes at levitating drops4−8 or electrowetting-driven oscillating drops sandwiched between two substrates often found in lab-on-chip devices based on digital microfluidics.9 Mechanisms of droplet motion can be of huge interest for controlling small quantities of liquid on substrates. First experiments have been performed by Brunet and his collaborators. In ref 10, the droplet displacement on a patterned substrate is induced by ultrasonic surface acoustic waves, while in refs 11 and 12 a drop climbing uphill on an inclined surface against the gravity force has been realized under vertical vibrations. The last scenario has been also widely theoretically/ numerically investigated in refs 13−18. For both aforementioned mechanisms, symmetry breaking inside the system is responsible for the droplet displacement. In this work we discuss a mechanism of drop control under inertial effects induced by a vibrating solid plate. This mechanism can be of large interest for applications in microfluidics or microgravity, where the role of the gravity effects is negligible, while the role of the inertial effects induced by vibration (as the driving mechanism of the drop) becomes significant. The drop’s asymmetry along the substrate is realized by using two oscillations at the bottom plate perpendicular to each other with the same frequency (see Figure 1). We © XXXX American Chemical Society
Figure 1. Sketch about the system: a liquid droplet is sitting on a solid substrate. The sessile drop is surrounded by a gas and in partial contact at the substrate through a precursor film. The solid substrate oscillates harmonically in both the vertical and horizontal directions. The two driving vibrations are adjusted to the same frequency. The solid substrate can have arbitrary wetting properties, from hydrophobic substrates (less precursor film at the solid contact) to hydrophilic substrates (much precursor film at the solid boundary). The two perpendicular oscillations will initiate an asymmetry in the droplet behavior moving periodically back-and-forth along the substrate under inertial effects. A net driven motion of the drop along the solid substrate is induced. The substrate may have different wetting properties, and contact angles varying between 10° and 150°. For the simulations presented in this paper, the ratio of the forcing amplitudes in the horizontal and vertical directions is n = 5.0.
attribute the net driven motion to the deformations of the liquid droplet, which are different when the drop moves to the right or to the left. We reactivate in this way a mechanism experimentally realized some years ago19 but theoretically studied under a minimal model based on the analogy that a drop behaves like a spring with different resonant frequencies at which a small vibration induces a significant deformation. We deliver here numerical simulations coming from the hydrodynamic fundamental set of equations. We illustrate that such combined parallel and perpendicular vibrations allow one to Received: August 27, 2014 Revised: November 14, 2014
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Figure 2. Several snapshots illustrating the asymmetry of a droplet for a hydrophobic substrate, ρS = 0.3, θ = 125° (panels a and b) and for a hydrophilic substrate, ρS = 0.95, θ = 10° (panels c and d) subject to lateral and vertical harmonic oscillations in phase (ΔΦ = 0) with each other. The droplet topography looks different when the droplet moves to the right or to the left. Moving to the right, at t = 3 T/8 the droplet is pulled over and becomes taller (panels a and c, respectively). Moving to the left, at t = 7 T/8 it is pressed against the substrate and becomes smaller (panels b and d). The arrows in the background indicate the elongation of the triggering horizontal oscillatory motion at the solid plate.
stress. This term describes the contribution of capillary forces and substitutes in the phase field model the classical boundary condition for tangential stresses at the droplet interface.20−22 2 represents the square gradient parameter, which relates the surface tension coefficient σ with the interface gradient energy
control the droplets’motion. We systematically investigate the traveled distance on the oscillation parameters. We make predictions on resonance frequencies for drops on a substrate with a varying wettability.
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PHASE FIELD SIMULATIONS We perform phase field simulations in two spatial dimensions. Phase field models offer an elegant description of the problem. A phase field that contains information about the local state of the system composition enables a continuous treatment of the problem from one medium to the other, including the interfacial region. The phase field plays the role of an order parameter taking distinct, constant values in the different phases and varying continuously from one phase to the other with a rapid but smooth variation across the interface. The most natural phase field is the density ρ, scaled by the mean liquid density. Then ρ ≈ 1 denotes the liquid phase, and ρ ≈ 0 is the gas bulk. Complicated explicit boundary conditions along the interface are avoided and captured implicitly by gradient terms of ρ in the hydrodynamic basic equations. The model consists of the Navier−Stokes equation coupled with the continuity equation: ρ
dv ⃗ = −∇p + ρ∇(∇·(2∇ρ)) + ∇·(η∇v ⃗) dt
∂ρ + ∇·(ρv ⃗) = 0 ∂t
+∞
2(∂ρ /∂z)2 dz . term: σ = ∫ −∞ The momentum and continuity equations are discretized using a second-order central difference approximation for the spatial derivatives and an explicit Euler method for the time integration. The mesh is of 400 × 200 points for the simulations presented in this work under periodic boundary conditions in the horizontal plane and with no-slip condition for the velocity field at the wall boundaries (at the top and at the bottom). The distance between two mesh points is δx = δz = 2, and the integration time step is δt = 0.1. The scaled time is 2 × 10−5 s, and the scaled length is 10−5 m. The model was validated earlier for static contact angles,24 dynamic contact angles,25 and for studying the mixing of two component systems.26,27 The density field is ρ = 10−3 at the top boundary and ρ = ρS at z = 0, simulating in this way at z = 0 a solid support with the associated van der Waals long-range interactions at the liquid− solid interface. Pismen & Pomeau21 assumed that for shortranged solid−fluid interactions compared to the thickness of the diffuse interface, a supplementary energy term (a polynomial function of density) can be locally added in the free-energy functional in the vicinity of the wall. By minimizing this free-energy density, the (Dirichlet) boundary condition ρ = ρS has been obtained. ρS is a free parameter between 0 and 1, which represents the density at the substrate and describes the wettability properties at the bottom plate. 0 ≤ ρS ≤ 0.5 denotes hydrophobic surfaces and 0.5 ≤ ρS ≤ 1 hydrophilic ones. The
(1)
(2)
and is thermodynamically consistent for a liquid−gas interface with mass conservation for each phase.23 In eq 1, dv/⃗ dt represents the total derivative of the velocity field v,⃗ p is the thermodynamical pressure, and η is the dynamic viscosity. The second term on the right-hand-side denotes the Korteweg B
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static contact angle θ is related to the substrate density ρS through21 cos θ = −1 + 6ρS2 − 4ρS3
We start from a flat thin liquid layer with ρ = 1 in a gas atmosphere with ρ ≈ 0. The whole system is at rest (v ⃗ = 0, everywhere). The thin film breaks up in small droplets that nucleate until one drop remains. The transition from film to drop is illustrated, for example, in Figure 7 from ref 24. By changing ρS, one gets droplets of the same mass m = ∬ ρ ρ≥0.99
dx dz with different contact angle at the bottom plate. The droplet radius in two spatial dimensions is calculated as R = (m/π)1/2. Then, the bottom plate is simultaneously submitted to horizontal and vertical harmonic oscillations according to vx(z = 0) = Aω sin(ωt ),
vz(z = 0) = 0.2Aω sin(ωt + ΔΦ)
For drops of radius R = 0.78 mm and σ = 0.05 N/m, the frequency and amplitude of the excitation are adjustable in a wide range from 10 Hz to 103 Hz for frequency and 10−5 m to 1 mm for amplitude. Drops in two spatial dimensions can be considered infinite cylinders. Therefore, the eigenfrequencies for spherical drops should be replaced in 2D by the eigenfrequencies for a cylindrical jet: ωr = (m(m2 − 1)σ/ ρR3)1/2.1 We will refer in the following to the resonance frequency of the elongated mode (m = 2) for an inviscid cylindrical jet: ωr = (6σ/ρR3)1/2 = 8 × 102 Hz. One uses ωr for the frequency scaling because it incorporates the liquid droplet properties. Figure 2 presents several snapshots for hydrophobic and hydrophilic liquid droplets for ω/ωr = 0.13 and A = 1 mm. The drop moves forth and back during a period. Because of the two perpendicular oscillations acting at the solid plate, the motions to the left and to the right do not compensate each other. For ΔΦ = 0, moving to the right, at t = 3T/8 the droplet is pulled up (left-hand-side panels in Figure 2), while moving to the left (second-half-cycle), at t = 7T/8 the drop is pressed against the plate (right-hand-side in Figure 2, T = 2π/ω being the vibration period at the solid plate). In this way the symmetry of the droplet movement along the substrate under inertial effects is broken: the drop advances more to the right than to the left. Taller drops are faster, and a net motion to the right during one cycle results. In order to complete the understanding of the liquid flow at the substrate, the flow lines inside the drop are plotted for the cases when the drop moves to the left (Figure 3a) and to the right (Figure 3b). The representation of the velocity field inside the tiny droplet in the center-of-mass coordinate system emphasizes that the flow structure inside the drop is controlled from the vibrating plate, the droplet being pulled from the solid support.
Figure 3. A liquid droplet in its own vapor sliding along the shaking plate at the solid boundary, from right to left in panel (a) and from left to right in panel (b). The velocity field is represented in the center-ofmass coordinate system. The numerical simulations presented in this figure correspond to the hydrophilic case illustrated in Figure 2c,d.
stroboscopic method is used: we consider the drop position at the same phase of the forced oscillation. We investigate the dependency of Δx on the oscillation frequency ω, on the forcing amplitude A, and on the phase shift ΔΦ between the lateral and vertical harmonic oscillations. Figure 4 shows the dependence of the net driven displacement Δx on the oscillation amplitude A. The curves correspond to different contact angles, for ΔΦ = 0, for an excitation frequency close to the resonance frequency of the elongated mode ω/ωr = 0.87, and for forcing amplitudes
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DEPENDENCE ON AMPLITUDE, PHASE SHIFT AND VISCOSITY We consider the net driven motion Δx by following the location of the drop’s mass center: xCM =
∬ρ ≥ 0.99 ρx dx dz ∬ρ ≥ 0.99 ρ dx dz
Figure 4. Net driven displacement Δx over 10 cycles versus oscillation amplitude at the solid boundary. The curves correspond for different contact angles, for ΔΦ = 0, for an excitation frequency close to the resonance frequency of the elongated mode ω/ωr = 0.87, and for forcing amplitudes ranging between 10−5 m and 1 mm.
over 10 periods. The droplet experiences a combination between a balance and a unidirectional motion along the substrate. In order to eliminate the balancing effect, a C
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ranging between 10−5 m and 1 mm. One can see that the traveled distance Δx monotonically increases with the forcing amplitude A of the shaking plate and saturates for larger amplitudes, near the droplet resonance. For much larger amplitudes (than those considered in Figure 4), the droplet breaks up, and a part of it loses contact with the supporting plate rising into the gas atmosphere. We note that we concentrate our attention in this work only on droplets that are still connected with the controlling shaking plate. Our numerical simulations correspond to a perfect homogeneous substrate; no threshold occurs, as observed in the experiments presented in Figure 3b from ref 11 or Figure 3b from ref 19, where the drop (additionally) has to overcome the gravity and the surface defects on the substrate. For small amplitudes, one finds the scaling law Δx = aAb with a = 0.035, b = 1.84 for θ = 90°, a = 0.024, b = 1.68 for θ = 55°, and a = 0.016, b = 1.57 for θ = 38°. Figure 5 depicts the dependence of the net driven motion on the phase shift ΔΦ, between the two forced oscillations in the vertical and horizontal directions. Depending on ΔΦ, the drop travels to the right (Δx > 0) or to the left (Δx < 0). The dissipationinduced by the friction along the contact line and the viscosity of the mediumleads to shifted sinusoidal curves
of different amplitudes. One can verify that the function Δx(ΔΦ) satisfies: Δx(ΔΦ + π) = −Δx(ΔΦ) for each substrate density ρS (namely, for each contact angle θ) and for each kinematic viscosity ν (ν = η/ρ), a fact that has also been observed in experiment.19 The shifts and the maxima achieved by the lateral traveled distance Δx along the substrate depend both on the contact angle θ (Figure 5a) and on the kinematic viscosity of the medium ν (Figure 5b). For example, in Figure 5a (for ν = 5 cSt), the maximum of the net driven displacement appears at ΔΦ = 9π/5 for θ = 125° and ΔΦ = π/5 for θ = 10°.
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DEPENDENCE ON FREQUENCY For different wettabilities, Figure 6a evidences the droplet average velocity of advancing along the solid substrate versus
Figure 6. Influence of the excitation frequency at the controlling bottom plate for arbitrary wettabilities (ΔΦ = 0, A = 1 mm): (a) advancing average droplet velocity over 10 cycles versus reduced frequency ω/ωr; (b) for the case illustrated in panel a, one extracts the reduced frequency ω/ωr versus contact angle θ corresponding to the maximal advancing velocity.
the reduced excitation frequency at the bottom plate ω/ωr. The plot in Figure 6a corresponds to a phase shift ΔΦ = 0, i.e., a net driven motion to the right and positive velocities (v > 0). As one can see from Figure 6a, for a certain excitation frequency at the bottom plate, the drop response becomes maximal and the advancing velocity achieves a maximum. We will call this frequency the resonant frequency under partial wetting.
Figure 5. Net driven displacement Δx versus phase shift ΔΦ over 10 periods for (a) arbitrary wetting properties at the solid surface (ω/ωr = 0.13, A = 1 mm, ν = 5.0 cSt) and (b) different kinematic viscosities (ω/ωr = 0.13, A = 1 mm, θ = 55°). D
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(9) Mampallil, D.; Eral, H.B.; Staicu, A.; Mugele, F.; van den Ende, D. Electrowetting-Driven Oscillating Drops Sandwiched between Two Substrates. Phys. Rev. E 2013, 88, 053015. (10) Brunet, P.; Baudoin, M.; Matar, O.B.; Zoueshtiagh, F. Droplet Displacement and Oscillations Induced by Ultrasonic Surface Acoustic Waves: A Quantitative Study. Phys. Rev. E 2010, 81, 036315. (11) Brunet, P.; Eggers, J.; Deegan, R.D. Motion of a Drop Driven by Substrate Vibrations. Eur. Phys. J.: Spec. Top. 2009, 166, 11−14. (12) Brunet, P.; Eggers, J.; Deegan, R.D. Vibration-Induced Climbing of Drops. Phys. Rev. Lett. 2007, 99, 144501. (13) John, K.; Thiele, U. Self-Ratcheting Stokes Drops Driven by Oblique Vibrations. Phys. Rev. Lett. 2010, 104, 107801. (14) John, K.; Thiele, U. Transport of Free Surface Liquid Films and Drops by External Ratchets and Self-Ratcheting Mechanisms. Chem. Phys. 2010, 375, 578−586. (15) Benilov, E.S. Thin Three-Dimensional Drops on a Slowly Oscillating Substrate. Phys. Rev. E 2011, 84, 066301. (16) Benilov, E. S.; Billingham, J. Drops Climbing Uphill on an Oscillating Substrate. J. Fluid Mech. 2011, 674, 93−119. (17) Benilov, E.S.; Cummins, C.P. Thick Drops on a Slowly Oscillating Substrate. Phys. Rev. E 2013, 88, 023013. (18) Savva, N.; Kalliadasis, S. Droplet Motion on Inclined Heterogeneous Substrates. J. Fluid Mech. 2013, 725, 462−491. (19) Noblin, X.; Kofman, R.; Celestini, F. Ratchetlike Motion of a Shaken Drop. Phys. Rev. Lett. 2009, 102, 194504. (20) Anderson, D. M.; McFadden, G. B.; Wheeler, A.A. DiffuseInterface Methods in Fluid Mechanics. Annu. Rev. Fluid Mech. 1998, 30, 139−165. (21) Pismen, L. M.; Pomeau, Y. Disjoining Potential and Spreading of Thin Liquid Layers in the Diffuse-Interface Model Coupled to Hydrodynamics. Phys. Rev. E 2000, 62, 2480−2492. (22) Zoltowski, B.; Chekanov, Y.; Masere, J.; Pojman, J.A.; Volpert, V. Evidence for the Existence of an Effective Interfacial Tension between Miscible Fluids. 2. Dodecyl Acrylate−Poly(dodecyl acrylate) in a Spinning Drop Tensiometer. Langmuir 2007, 23, 5522−5531. (23) Borcia, R.; Bestehorn, M. Phase Field Modeling of Nonequilibrium Patterns on the Surface of a Liquid Film under Lateral Oscillations at the Substrate. Int. J. Bifurcation Chaos Appl. Sci. Eng. 2014, 24, 1450110. (24) Borcia, R.; Borcia, I.D.; Bestehorn, M. Drops on an Arbitrarily Wetting Substrate: A Phase Field Description. Phys. Rev. E 2008, 78, 066307. (25) Borcia, R.; Borcia, I.D.; Bestehorn, M. Static and Dynamic Contact Angles - A Phase Field Modelling. Eur. Phys. J.: Spec. Top. 2009, 166, 127−131. (26) Borcia, R.; Bestehorn, M. Different Behaviors of Delayed Fusion Between Drops with Miscible Liquids. Phys. Rev. E 2010, 82, 036312. (27) Borcia, R.; Menzel, S.; Bestehorn, M.; Karpitschka, S.; Riegler, H. Delayed Coalescence of Droplets with Miscible Liquids: Lubrication and Phase Field Theories. Eur. Phys. J. E 2011, 34, 24.
Extracting now the resonance frequency under partial wetting as a function of the contact angle θ, one obtains the diagram in Figure 6b. From this diagram one can see that the resonance frequency increases with increasing surface hydrophobicity and saturates around θ ≈ 50° to a value close to the resonance frequency of the elongated mode predicted by the linear theory given in ref 1 for an inviscid cylindrical jet. Benilov and Billingham16 find from a two-dimensional shallow-water model that the drop’s drift is caused by the interaction of even and odd (“spreading” and “swaying”) oscillatory modes induced by vibration. In the nonlinear regime, the driven velocity v shows an oscillatory dependence on frequency. The observed extrema become stronger for intermediate values of the velocity derivative along the contact line v′(θ). For the contact angles considered in Figure 6a, a similar behavior to that presented in Figure 2 from ref 16 has been found for θ = 55°. For θ > 55°, the “swaying” mode breaks up the drop before its velocity reaches v = 0, while, for θ < 55°, the “spreading” mode is dominant and spreads the shaken droplet.
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CONCLUSION We have numerically studied the mechanism of drop control and propulsion under combined parallel and perpendicular vibrations at the solid plate supporting the liquid droplet. The investigation of the internal flow structure inside the droplet shows that the drop is controlled from the vibrating substrate. The net driven displacement has been investigated for different parameters (forcing amplitude, frequency, and phase shift). For various contact angles, scaling laws for small amplitudes have been derived. We made predictions about resonance frequency for drops on an arbitrary wetting substrate, a problem that could be of large interest for drop control in microfluidics and microgravity.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
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