virginia palero 1*, julia lobera 1, philippe brunet 2, mª...

16th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 09-12 July, 2012

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3D characterization of the inner flow in an oscillating drop

Virginia Palero 1*, Julia Lobera 1, Philippe Brunet 2, Mª Pilar Arroyo 1

1 Grupo de Tecnologías Ópticas Laser (TOL), I3A-Universidad de Zaragoza C/ Pedro Cerbuna, 12, 50009-Zaragoza, España

2 Laboratoire Matière et Systèmes Complexes, Paris, France * correspondent author: [email protected]

Abstract In this work, measurements of the velocity field inside a vertically shaken drop are presented. The drop wetting condition is slightly hydrophobic. The drop is sandwiched between two horizontal glass plates in order to allow a good visualisation. Digital in-line holography is utilised to access the tridimensional (3D) flow inside the drop. When shaken, the drop experiences several transitions of its dynamics and shape: at low amplitude, its contact line is pinned and when the amplitude overcomes a threshold, it becomes unpinned. At even higher amplitude, the drop shows the faceted shape of a star. The obtained velocity vector maps are related to the transition in dynamics and shape. 1. Introduction In the seek for the dynamical properties of liquids drops, which are more and more used for microfluidics applications, various studies have been devoted to the determination of their eigen modes in the frame of their response to external forces. In the linear regime and for inviscid spherical drops, the resonance frequency has been theoretically predicted a long time ago by Rayleigh and Lamb [1]:

(1) where fn denotes the resonance frequency of the nth mode of oscillation, V is the drop volume, R its radius, σ and ρ the liquid surface tension and density. However, the ideal situation of a spherical drop without contact to any solid is barely encountered in reality. In many situations, in particular in more and more lab-on-chip microfluidics benches like DNA arrays [2], the drop of liquid sits on a substrate and adopts a general shape which is a crossover between a spherical cap (if the drop typical size is smaller than the capillary length lc = (γ/(ργ))1/2 ) and a pancake (if the size is much larger than lc). Along the contact-line, the liquid/air interface of the drop meets the substrate with a contact angle of θ, which value is set by the interfacial surface energies (Young’s law). This peculiar shape induces a complexity which makes the vibration mode non trivial for a pure analytical approach. Instead, the eigenmodes of sessile drops are determined either experimentally or numerically [3,4]. Furthermore, the substrate natural heterogeneity and roughness induce contact-line pinning and contact-angle hysteresis [5]: this is an additional source of complexity, which accounts for the liquid-solid interactions at the microscopic scales, and which has yet to be fully understood [6] and quantitatively determined. A common way to access the dynamical properties of a drop through its eigen modes, is to put it


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onto a mechanical shaker. As the liquid drop is subjected to a time-dependent acceleration [4,7-10], this allows for the observation of a host of phenomena from linear to strongly non-linear behaviour, i.e. from surface wave undulations to strong surface deformations up to atomization [7]. The pinning force modifies the vibration modes of a sessile drop, but reciprocally the vibrations can also influence the pinning behavior. For instance, the contact-line of a vibrated drop either can stay pinned under the vibrations, either can be intermittently or continuously unpinned [4]. Various authors evidenced and studied similar unpinning effect with low-frequency mechanical vibrations [4, 11-14], and it is expected that at large enough prescribed vibration, the contact-line be continuously unpinned. Furthermore, when shaken, a drop weakly pinned on its substrate can experience a shape transition from axisymmetric to that of finite azimuthal wave-number, giving it the shape of a star. This transition happens in various situations where a periodic field (acceleration, magnetic field, ...) can induce a parametric forcing in the drop. This type of instability can be utilized on purpose to induce constant mixing or particle resuspension, for instance to prevent ”coffee rings” due to evaporation at the periphery of the drop [15,16] that are detrimental for many lab-on-chip applications [2]. However, to access a precise visualization of the flow inside a shaken drop is far from being easy. First, due to its strong curvature, a sessile drop behaves like a lens which deflects the rays of light. Second, the flow inside the shaken drop is supposed to be fully tridimensional (3D), and most of the existing experiments so far only address a 2D cartography of the flow. Therefore, we have carried out experiments to visualize the inner flow with a 3D velocimetry setup, which utilizes high-speed digital in-line holography. The instantaneous 3D motion of the fluid particles in the drop is then determined for various states of the drop subjected to shaking. In order to get rid of the deflection of the rays of light, we have conducted our experiments with a large drop (puddle) that was sandwiched between two hydrophobic glass plates, on which water has a static contact-angle of about 90◦ after an ad-hoc chemical treatment. In section 2, we first remind the general features of what is expected for a drop under periodic vertical acceleration field, and we give a short review of situations where large drops can adopt a star shape when shaken. Then in section 3, we present our experimental setup in detail, including the optical technique and image post treatment. Section 4 presents qualitative and quantitative results, with the 3D fields for various situations. Finally, section 5 summarizes the main conclusions. 2 Break-up of axisymmetry in a vertically shaken liquid puddle 2.1 General features Here we give a qualitative understanding of how a drop can generate standing waves, associated to the shape of a star, as a response to a time-periodic excitation. We basically follow the analysis found in the paper by Yoshiyasu et al. [5]. Let us consider the simple case of a large drop (liquid density ρ and surface tension σ) sitting on a non-wetting substrate (figure 1). If the drop volume is large enough, it spreads like a puddle which radius R is much larger than the height h. For the convenience of the analysis, we assume the situation of a non-wetting substrate, and a contact angle θ equal to 180◦, although in practice θ is rarely larger than 160◦. The height h is equal to twice the capillary length lc:


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and it ranges between 2 and 3 mm for most liquids. If the substrate is shaken in the vertical direction, the drop is subjected to a time-periodic acceleration field a(t). The balance between this effective gravity and surface tension builds an effective capillary length lc

* that is also time dependent, which implies that the height of the puddle (h = 2 lc

* ) is time-periodic:

(2) Due to the volume conservation of the drop, the radius also fluctuates with a period of 1/fe. In the frame of some assumptions (no pinning and radius much larger than h), the eigen frequencies of a liquid puddle on a non-wetting substrate do not differ much from those of a spherical drop described by eq. (1). Takaki and Adashi’s analysis [17] yields:

(3)

It is clear from eq. (2) that the resonance frequency fn for the free oscillations are modulated in time. Therefore, this modulation leads to parametric forcing, by analogy with the classical case of a vertically shaken pendulum [27]. Yoshiyasu et al. show with simple arguments that the equation for the horizontal displacement of the drop periphery u(t) is governed by an equation similar to the Mathieu equation:

(4)

with ωn = 2πfn is the pulsation associated to the nth mode, and .

Figure 1. A liquid puddle on a non-wetting substrate vibrated in the vertical direction at a frequency fe. Due to a time-periodic acceleration a(t), the puddle radius R(t) and height h(t) are time-dependent. Experimentally indeed, for large enough forcing, the frequency of the drop oscillations fn is equal to half the prescribed frequency fe, which is expected for parametric forcing. Consequently, the drop


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exhibits a standing wave at its periphery, which wavelength λ has to match both the perimeter 2πR and the number of lobes n: λ= 2πR/n, n being fixed by the prescribed frequency: fn = fe/2. Therefore, the star shapes are specific to the peculiar geometry of a flat puddle, i.e. that R is much larger than h. Weak contact-line pinning between the drop and the substrate is required to obtain parametric forcing. For strong pinning, the drop radius remains constant and instead the contact-angle oscillates at the forcing frequency fe [4], due to the vertical oscillation of the drop’s center of mass: the drop simply responds harmonically. The mechanism for forcing then should emerge from coupling between the lateral interface and the inner flow. This is why we investigate here this internal flow below and above threshold for parametric forcing. 2.2 A brief review of previous experiments Although there exists various experiments in which star drops can be observed (see in [18] for a recent review), like in “Leidenfrost drops” levitating on hot plates [19], time-modulated acoustic levitation [20] or magnetically levitated liquid metal drops [21], to put a water puddle on a non-wetting and non-sticky vibrating surface is probably the easiest way to observe them. Yoshiyasu et al. [2] offers a discussion of the results and carried out the first qualitative experiments, using a teflon plate. The authors pointed out that it is particularly important that the water drop formed a contact angle θ larger than 120◦. However, Noblin et al. [8] evidenced that liquid stars - denoted there as ”triplons” to refer to the deformation of the triple (contact) line - could be observed on a substrate with contact-angle θ smaller than 90◦, providing that the contact angle hysteresis was small enough. The only difference is the existence of a threshold in acceleration (or shaking amplitude) below which the contact-line keeps pinned [8]. They generalized eq. (4) in the case of solid-like friction, and evidenced that, above the threshold for contact line unpinning, the instability turning an initial axisymmetric drop to a faceted star drop is associated to an exponential growth of the lobe amplitude versus time. Therefore, this feature contains the signature of a linear instability. A threshold in forcing amplitude for the depinning of the contact-line and for the appearace of star drops is the direct consequence of weak, but finite contact-line pinning. Recent experiments by Okada and Okada [23] produced more quantitative data using a more hydrophobic teflon plate. They focused on the mode n=3, and produced a phase diagram for this simple mode varying both amplitude and excitation frequency. They also checked that the response frequency scaled with V1/2 for the same mode number n, with V being the volume. 3 Digital in-line holography Digital holography [24, 25] is a well known technique for fluid diagnostics due to its great versatility as it allows the measurement of the 3D-3C velocity field in a fluid volume. Digital holography shares with optical holography the recording process, i. e., the light scattered by the object (tracers in a flow, droplets, bubbles,...) interferes with a reference beam in a photosensitive device (a CCD or CMOS sensor in digital holography). Therefore, both the amplitude and the phase of the object wave are captured. However, in digital holography, the hologram is mathematically reconstructed, avoiding the chemical processing needed for developing the holographic plate. Two main disadvantages remain when digital holography is applied. First, the low spatial resolution of the digital cameras does not allow the recording of high spatial frequencies and forces the use of


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small angles between the object and the reference beam. One immediate consequence is that the virtual and real images are not fully separated when the hologram is reconstructed. Secondly, the recording of a whole fluid volume requires a low particle concentration as the defocused particle images would add noise when a plane is analysed. There are several holographic techniques but, in this work we have chosen to use the in-line approach [26, 27, 28]. In this configuration a fluid volume is illuminated with a coherent collimated beam. Light diffracted by the particles in the flow forms the object beam, while the non-diffracted light forms the reference beam. The great attractiveness of the in-line holographic set-up is that it requires minimum optical equipment and laser coherence length. 3.1 Mechanical set-up A water drop, of 8 mm in diameter, seeded with 10 µm latex particles, was squeezed between two parallel glass plates separated by a distance of 2.25 mm. The glass was treated to be slightly hydrophobic (contact angle around 90º, with hysteresis around 10º). Both plates were attached to the vertical axis of a mechanical shaker connected to a function generator that produces vibrations in a wide range of frequencies. Figure 2a shows a picture of the system. Figure 2b shows a close up of the plates. The glass plates were attached to a plexiglass plate inserted in between, whose width defines the droplet height. The liquid was injected with a syringe between the two plates. Two other glass plates were used as counterweight, and were also attached to the plexiglass plate (left hand side of the image). The shaker frequency was fixed at 55 Hz for all the experiments, as it appeared that this frequency was adapted to a fair excitation of the drop eigen modes. The amplitude changed from 0 to a maximum of 1.4 mm.

(a) (b)

Figure 2. a) Mechanical set-up; b) glass plates close-up.

3.2 Optical set-up A drop was illuminated from below with a collimated laser beam (figure 3a), coming from a continuous laser (λ = 532 nm). A mirror above the droplet was used for redirecting the light scattered by the particles (object beam) and the direct light (reference beam) to a high speed camera (CMOS, 10 bits, 1024x1024, 3000 Hz maximum recording rate using the whole sensor size). A lens (f’ = 55m, f# = 2.8) in front of the camera formed the droplet image close to the camera sensor. This optical configuration was chosen because it maximizes the inside view of the drop and ensures uniform illumination. Besides, a lateral drop viewing would provide mainly information about the shaker oscillation, and would make difficult to extract information about the flow inside the drop. Series of 200 holograms were recorded at different oscillation amplitudes. The image magnification was set at 2.48. The camera acquisition rate varied between 1000 and 3000 Hz.


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Figure 3b shows an example of a typical hologram recorded with this set-up. The seeded drop is in the image right hand side, limited by a black area. This black fringe is the shadow cast by the movement of the contact line. Such movement is different in the drop upper part and in the drop bottom. In order to increase the spatial resolution, only a third of the drop was imaged.

(a) (b) Figure 3. a) Optical configuration; b) Hologram recorded with this set-up. 3.3 Holograms analysis

Commercial PIV analysis it is not suited in general for these droplet experiments, as the particle images become highly uncorrelated due to the bulk out-of-plane displacement of the droplet for two consecutive exposures. Furthermore a proper 3D reconstruction of the particle field for each exposure should be computed in order to recover a 3D velocity field.

Prior to the computation of the 3D velocity field, it is also convenient to remove the background illumination, and to normalise the particle intensity by this background illumination.

The in-line recording configuration implies the real and the virtual image of each particle are present in our reconstruction. However, care has been devoted to record the droplet being out-of-focus, as this would make that twin images be focused at different planes. This also determines the best performance of the PTV analysis from PIV analysis, as a smaller window would make the presence of out-of-focus particle images negligible in the cross-correlation calculation.

Thus a PTV analysis has been computed, in which the particle locations are based on the 3D intensity maxima locations, and the displacement calculation on a 3D cross-correlation of the complex amplitude [29]

4 Results 4.1 Visualization of the states for a shaken drop In order to analyse the different states of a shaken droplet, the maximum amplitude (A) in the mechanical shaker was progressively increased. The droplet undergoes different states: contact line


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pinned, contact line unpinned but axisymmetric drop, contact line unpinned and non-axisymmetric drop, contact line unpinned and ‘star’ shape and eventually, drop break-up at maximal amplitude. Figure 4 shows these states. The droplet is visualized from different viewpoints, for four different amplitudes. Images in the upper row correspond to lateral view, with the shadow of the drop shape appearing between the two horizontal glass plates. Images in the central line show the droplet top view and the bottom row shows the contact line movement over several periods. The images of the last row were obtained from the sums of 200 holograms corresponding at different amplitudes. The sum has been binarized, thus the resulting images show the time average over such a long duration. The contact line movement is evident. As the amplitude increases, so does the change in the contact line position. At low amplitude (A= 0,24 mm, figure 4a) the droplet contact line is pinned to the substrate and the droplet barely moves. As the amplitude increases (A= 0,85 mm, figure 4b) the contact line is unpinned but the drop is still axisymmetric. With A= 1mm the drop contact line is unpinned and the drop is axysimmetric. The star shape is obtained at an amplitude close to the maximum (A=1.34 mm).

(a) (b) (c) (d)

Figure 4. Droplet visualization from the side (top row), from the top (middle row), and contact line motion (bottom row) for a) pinned contact line, A = 0.24 mm; b) unpinned, A = 0.85 mm, axisymmetric drop; c) unpinned and faceted shape, A =1 mm, break-up of axisymmetry. d) Star-shaped drop. A= 1.34 mm. From averaged drop position, we deduced the contact-line motion versus amplitude of vibration (figure 5). This offers a quantitative picture of the depinning by vibrations: below A equal to about 0.6 mm, the contact-line averaged width is about 50 pixels, which simply corresponds to the width of the shadow, meaning no contact-line motion at all. Above A=0.6 mm, the width experiences a


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sharp increase with A until it reached a second plateau between A=1 and 1.2 mm. Beyond (A>1.2 mm), the width further increases, corresponding to the occurrence of stars.

Figure 5. Contact line motion vs amplitude for data taken at different times 4.2 Velocity field measurements of a vibrating drop in different regimes In the following we present the velocity vector maps corresponding to the different states for a shaken drop. Different facts have to be taken in consideration.

First, the velocity vector maps displayed are a 2D representation of a 3D flow. The contribution of all the particles in the drop volume is added in a plane. The drop seeding was low, giving a very sparse velocity vector distribution. Thus, this view improves the spatial resolution, allowing clear flow structure visualization. Second, the drops were slowly drifting, so from time to time the plate with the drop inside had to be re-positioned in order to centre the drop in the image.

(a) (b)

Figure 6. a) Vertical component of the velocity inside a shaken droplet, at small A (pinned contact line). b) Instantaneous velocity vector map for various depth z.

Drop contact line pinned. (A= 0,24 mm) Figure 6a shows the overlapping of 54 velocity vector maps (corresponding to the maps measured in a period with a camera acquisition rate of 3000 Hz). The colour code represents the vertical component of the velocity Vz. Practically no movement


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inside the droplet is measured. Only a small displacement near the contact line is obtained. In the figure 6b an instantaneous velocity vector map is shown, where the vector colour indicates the z position. A small separation between the upper and the bottom contact lines can be distinguished, which corresponds to a situation like the shot in the figure 4a: the bottom contact line is slightly outwards compared to the upper contact line.

(a) (b)

Figure 7. a) Vertical component of the velocity inside a shaken droplet, at moderate A (unpinned contact line). b) Instantaneous velocity vector map for various depth z. Drop contact line unpinned, axisymmetric. (A= 0,85 mm) In figure 7a the overlapping of 55 velocity vector maps is shown. The velocity vector map is similar to the previous case. The flow displacement is small in the droplet centre but is higher near the contact line. Several structures in the form of ellipse can be seen. In these experiments the vertical shaker axis was slightly inclined related with the OZ axis. Particles not moving with the drop flow (as dust in the glass plates) described these elliptical trajectories. Figure 7b shows an instantaneous velocity map. Particles inside the droplet with a z position bigger than 4.5 mm are moving outwards, while the particles bellow move in the opposite direction. Drop contact line unpinned, non-axisymmetric (A= 1 mm) In this case, clearly defined 2D flow structures are formed in the droplet bulk. Figure 8a shows an instantaneous velocity field corresponding to the image on the right side. In figure 8b two velocity vector maps have been added in order to improve the structure visualization. The drop shows azimuthal standing waves and oscillates between two different configurations in one period Te, the time between the two configurations being Te/2. Therefore, the frequency of response is equal to the excitation frequency. According to the usual picture of parametric instability, which holds for an oscillator of modulated resonance frequency (see section 2), this is surprising as one would have expected a response frequency equal to half of the forcing frequency. At least, the equality fn= fe/2 was noticed in previous experiments of free-standing drops levitating or sitting on super-hydrophobic substrates [17-23]. Here, our peculiar geometry of a drop sandwiched between two plates could explain this behavior as it could involve a more significant influence of the meniscus dynamics - this is striking from Figure 4c and d, which responds harmonically at the frequency of forcing.


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From eq. (3), the predicted resonance frequency of an azimuthal mode n=5, for R=4 mm, is 57.6 Hz, quite close to the frequency of forcing of 55 Hz. Indeed, the mode n=5 is that appearing in faceted drops (Figures 8) and star drops (Figures 9).

(a) (b)

(c) (d) Figure 8. Instantaneous velocity vector map for various depth z at high A (unpinned contact line, non-axisymmetric drop) showing a) the velocity structures at Te/2, and its correspondent hologram (b); c) the structures at Te, and its correspondent hologram (d) Drop contact line unpinned, non-axisymmetric, star shape. (A= 1,35 mm). Figure 9 shows two holograms of the droplet evolution at oscillation amplitude near the maximum. These images, together with the one on figure 4d, show the complex motion undergone by the drop when the star mode is excited. In this case no velocity fields could be obtained as the liquid seeding was too low.


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Figure 9. Two images of the drop oscillation at A=1.34 mm. 5. Conclusions In summary, this study showed for the first time how internal structures of the flow can be related to the appearance of the break-up of axisymmetry of a drop shape induced by external vertical vibrations. Although such faceted and star shapes are qualitatively predicted from a simple pendulum-like model, more specific mechanisms in a hydrodynamics point of view should benefit from such internal visualizations. Furthermore in an applied prospective, the prescription of external vibrations at appropriate frequency can be used as a simple way to induce mixing in weakly-pinned drops, thanks to the appearance of such star shapes that produces vortices in the fluid flow. Acknowledgments Authors wish to thank Spanish Ministerio de Ciencia e Innovación and European Comission FEDER program (project DPI2010-20746-C03-03), to Gobierno de Aragón – Fondo Social Europeo (project PI044/08 and Laser Optical Technology research group, T76) and COST-ESF (action P21). References

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virginia palero 1*, julia lobera 1, philippe brunet 2, mª...

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