Self-Excited Motions of Volatile Drops on Swellable Sheets

Aditi Chakrabarti, Gary P. T. Choi, and L. Mahadevan

Phys. Rev. Lett. 124, 258002 – Published 25 June 2020

Abstract

When a volatile droplet is deposited on a floating swellable sheet, it becomes asymmetric, lobed and mobile. We describe and quantify this phenomena that involves nonequilibrium swelling, evaporation and motion, working together to realize a self-excitable spatially extended oscillator. Solvent penetration causes the film to swell locally and eventually buckle, changing its shape and the drop responds by moving. Simultaneously, solvent evaporation from the swollen film causes it to regain its shape once the droplet has moved away. The process repeats and leads to complex pulsatile spinning and/or sliding movements. We use a one-dimensional experiment to highlight the slow swelling of and evaporation from the film and the fast motion of the drop, a characteristic of excitable systems. Finally, we provide a phase diagram for droplet excitability as a function of drop size and film thickness and scaling laws for the motion of the droplet.

Received 29 September 2019
Revised 1 February 2020
Accepted 22 May 2020

DOI:https://doi.org/10.1103/PhysRevLett.124.258002

Physics Subject Headings (PhySH)

Pattern formation

Drops & bubbles Nonequilibrium systems Thin films

Polymers & Soft Matter

Authors & Affiliations

Aditi Chakrabarti ¹, Gary P. T. Choi ¹, and L. Mahadevan ^1,2,*

¹John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
²Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

^*Corresponding author. lmahadev@g.harvard.edu

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Vol. 124, Iss. 25 — 26 June 2020

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Images

Figure 1
(a) The experimental setup comprises of a thin elastic film ( $H \sim tens$ of microns) made of PDMS afloat aqueous glycerol. Acetone droplets undergo spontaneous rotation as demonstrated via a single cycle of (b) a bean mode with $n = 1$ undergoing clockwise rotation ( $3 μ l$ drop on $9 μ m$ PDMS film) and (c) a star shaped mode with $n = 5$ undergoing anticlockwise rotation ( $20 μ l$ drop on $19.5 μ m$ PDMS film), and (d) a $30 μ l$ drop of acetone undergoing oscillatory to and fro motion on a $28.5 μ m$ PDMS film. The scale bars in (b)–(d) denote 2 mm. (e) Peclet numbers [ $Pe \sim O (10)$ ] plotted as a function of buckling strain ( $ϵ_{b}$ ) for acetone droplets showing droplet instability. The dotted line indicates the critical value of $ϵ_{b}$ , which demarcates between spherical cap states ( $n = 0$ ) and lobed states ( $n = 1$ , 3, 4, 5, 6).
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Figure 2
1D excitable droplet motion. (a) A drop of volume $V = 20 μ l$ placed on a sagging clamped PDMS strip (5 mm wide, $100 μ m$ thick) oscillates back and forth while the film buckles periodically. (b) Schematic showing that the film locally swells in response to the solvent droplet causing the film to buckle and the droplet to slide away. The exposed swollen buckled region then loses solvent via evaporation causing the film to unbuckle. Simultaneously, the drop causes the film to swell at a new location and the process repeats. The sag in the film prevents the drop from running away and leads to an oscillating state of the drop with frequency $Ω$ . (Inset) The shape of the potential $W (x, h)$ accompanying each film state with the droplet position $x (t)$ shown with a blue dot. (c) A trace of the $x$ coordinate of the center of mass of an acetone droplet undergoing relaxation oscillations as a function of time, where the slow timescale is due to swelling of the PDMS film and simultaneous regeneration due to evaporation, and the fast timescale is that of the drop motion. The kinks correspond to the buckling of the film.
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Figure 3
(a) Phase space of instantaneous snapshots of droplets of acetone (volume $V$ ) when they are deposited on PDMS films (thickness $H$ ). Smaller droplets on thicker films are “pinned” in spherical cap state whereas larger droplets on thinner films spontaneously form lobes $n$ ( $= 1$ to 6) that undergo rotation. (b) Bifurcation diagram of the shapes comprising spherical pinned state ( $*$ ) with $n = 0$ and chiral spinning state depicting number of lobes $n$ as a function of the dimensionless elastocapillary number ( $\sqrt{ϵ_{cap}} / ϵ_{b}$ ). The color of each filled symbol depicts the amplitude of the lobes normalized by the radius of the drops ( $| b |$ ). There is a critical threshold around $\sqrt{ϵ_{cap}} / ϵ_{b} ≃ 3$ where the shapes bifurcate from spherical cap to lobed state. (Inset) An example of a fitted shape ( $n = 3$ ) with contact radius $a$ and amplitude $| b |$ labeled. (c) Frequency ( $Ω$ ) of the spinning drops in the 3D case scales as $ω / n$ , where $ω \sim 1 / \sqrt{τ_{x} τ_{s}}$ , where the black dotted line is a least squares fit to the data with slope 11 (Inset). Schematic of the solvent droplet ( $n = 5$ ) that swells the PDMS film leading to undulations, which are enhanced here for visualization, and $J$ is the evaporative flux from the film. The liquid in the lobes traverse from hills to valleys along the circumference, giving rise to coordinated spinning of the drop with frequency $Ω$ .
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