Formation of vase-shaped drops

Martin Coux, Pierre Chantelot, Lucie Domino, Christophe Clanet, Antonin Eddi, and David Quéré

Phys. Rev. Fluids 5, 033609 – Published 16 March 2020

Abstract

Vibrated substrates are useful tools to move and reshape drops. Experiments usually involve periodical excitation at small amplitude but new effects can be generated by using larger and/or quicker movements. Recently, impulsive motion of a plate has been shown to enhance drop takeoff. In this article we discuss the beautiful, elusive shapes obtained when a water droplet deposited on a nonwetting substrate is subjected to a strong vertical impulse. Drops are highly reshaped to form truncated cones that eventually collapse. We report and discuss the evolution of the geometrical features of these so-called “vase-shaped droplets”. Our understanding of the physical phenomena at stake in the experiment allows us to model the evolution of the shape of the droplet. In particular, we show that the top and the bottom of the vase follow dynamics with different timescales, leading to the truncated cone shape. We show that it is possible to play with one of these dynamics by changing the wetting of the liquid on the rising substrate, and are thus able to change the shape of the vase from cylindrical to completely flat.

Received 13 September 2019
Accepted 4 February 2020

DOI:https://doi.org/10.1103/PhysRevFluids.5.033609

Physics Subject Headings (PhySH)

Contact line dynamics Drop breakup Surface tension effects Thin fluid films Wetting

Fluid DynamicsPolymers & Soft Matter

Authors & Affiliations

Martin Coux ^1,2, Pierre Chantelot^1,2,*, Lucie Domino¹, Christophe Clanet^1,2, Antonin Eddi¹, and David Quéré^1,2

¹Physique et Mécanique des Milieux Hétérogènes, UMR 7636 du CNRS, ESPCI Paris, PSL Research University, Sorbonne Université, Université de Paris, 75005 Paris, France
²LadHyX, UMR 7646 du CNRS, École polytechnique, 91128 Palaiseau, France

^*Corresponding author: p.r.a.chantelot@utwente.nl

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Vol. 5, Iss. 3 — March 2020

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Images

Figure 1
(a) Water droplet with radius $R_{0} = 1.8 mm$ after a kick of its nonwetting substrate (top) and substrate position (bottom). The plate rises until reaching its maximum speed $U = 1.90 \pm 0.07 m / s$ at time $τ = 5.6 \pm 0.2 ms$ . Letters refer to corresponding pictures. The droplet is strongly reshaped and forms a liquid vase, with a toroidal rim of maximal extension $2 R (t)$ , at height $h (t)$ above the takeoff altitude $z (τ^{*})$ (indicated by the white dashed line), and a circular liquid-solid contact of diameter $2 r (t)$ . (b) Time evolution of the radii $R (t)$ (black data) and $r (t)$ (blue data) for a droplet ( $R_{0} = 1.8 mm$ ) subject to a kick with $U = 1.50 \pm 0.03 m / s$ and $τ = 3.4 \pm 0.2 ms$ . The rim takes off at time $τ^{*} = 4.8 \pm 0.3 ms$ after which $r (t)$ linearly decreases with time with speed $| \dot{r} | = 0.80 \pm 0.05 m / s$ . $R (t)$ increases until its maximal extension $R_{\max}$ and then it decreases at constant acceleration $\ddot{R} = 72 \pm 2 m / s^{2}$ (red dashes). (c) Time evolution of the altitude of the peripheric rim for a drop with $R_{0} = 1.8 mm$ (black data) and for a propylene bead with similar size ( $R_{0} = 1.75 mm$ ) and density ( $ρ = 950 kg / m^{3}$ ) (red data) for a kick with $U = 1.70 \pm 0.02 m / s$ and $τ = 4.9 \pm 0.2 ms$ . Both objects take off at the same time $τ^{*}$ , with the same velocity $\dot{h} = 1.8 \pm 0.1 m / s$ . Dashes show a parabolic fit of the rim trajectory, $h (t) = \dot{h} t - \ddot{h} t^{2} / 2$ , with $\ddot{h} = 97 \pm 6 m / s^{2}$ . Inset: Sketch of a liquid vase, the rim has a minor diameter $d_{0}$ . The liquid-solid contact is a disk with thickness ε that meets the solid at an angle $θ_{r}$ .
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Figure 2
(a) Spreading parameter $R_{\max}$ / $R_{0}$ of the drop as a function of the Weber number $W e = ρ U^{2} R_{0} / γ$ , for drops of various radii $R_{0}$ . Dashes show a power law of exponent $0.25 \pm 0.03$ and prefactor $0.93 \pm 0.1$ . (b) Takeoff time $τ^{*}$ as a function of the time $t^{*}$ such that $\ddot{z} (t^{*}) = - g$ . Data for drops ( $R_{0} = 1 mm$ and $R_{0} = 1.8 mm$ ) and for polypropylene beads collapse on a line of slope $1.30 \pm 0.15$ . (c) Takeoff speed $\dot{h}$ as a function of the maximal kick velocity $U$ for four drop radii $R_{0}$ . The dotted line has a slope 1.
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Figure 3
(a) and (b) Deceleration $\ddot{h}$ of the rim height and acceleration $\ddot{R}$ of its closure as a function of the quantity $γ R_{\max} / ρ R_{0}^{3}$ derived by balancing the rim inertia with surface tension. Dashed lines are linear fits of slope $2.1 \pm 0.3$ and $2.1 \pm 0.2$ , respectively. Inset in (b): high-angle shot of a typical vase, which provides the ratio $2 R_{max} / d_{0}$ (7, here).
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Figure 4
Shape of water drops ( $R_{0} = 1.8 mm$ ) $3 \pm 0.2 ms$ after a kick with $U = 1.7 \pm 0.1 m / s$ and $τ = 4.9 \pm 0.2 ms$ , for different wetting conditions. (a) On silanized glass, the receding angle of water (which characterizes pinning on the substrate) is $85 \pm 3^{\circ}$ . (b) On a substrate covered with micrometric, hydrophobic pillars, it is $145 \pm 3^{\circ}$ . (c) On a substrate with hydrophobic nanobeads, it is $155 \pm 2^{\circ}$ . The vase shape gradually evolves from a cylinder to a cone. (d) For a nonwetting liquid marble (water covered with hydrophobic beads) or (e) for a Leidenfrost drop, the vase becomes a bowl or even a plate.
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