Self-Lifting Droplet Driven by the Solidification-Induced Solutal Marangoni Flow

Feng Wang, Li Chen, Yuqi Li, Peng Huo, Xi Gu, Man Hu, and Daosheng Deng

Phys. Rev. Lett. 132, 014002 – Published 2 January 2024

Abstract

Multicomponent droplets are pertinent to diverse applications ranging from 3D printing to fabrication of electronic devices to medical diagnostics and are typically inherent with the occurrence of the phase transition in the manifestation of evaporation and solidification. Indeed, the versatile transformations and fascinating morphologies of the droplets have been identified, which primarily arise from the evaporation-induced flow. Here, we report the self-lifting behavior of a frozen binary droplet, resulting in a nearly doubling in height, in a fashion that defies against the gravitational effect. This counterintuitive observation is attributed to an internal solutal Marangoni flow up to $1 mm / s$ , which is driven by the enriched solute concentration locally in the vicinity of the solidification front. Moreover, we perform theoretical analysis by incorporating the propagation of solidification front, and the calculated spatiotemporal evolution of droplet shape agrees with experiments excellently. The effects of several key physical parameters on self-lifting are elucidated quantitatively, providing guidance to control the self-lifting. These results will further advance our understanding of underlying physicochemical hydrodynamics in the multicomponent liquid systems subjected to heat transfer and phase change, consequently shedding light on the relevant technological applications.

Received 12 October 2023
Accepted 6 December 2023

DOI:https://doi.org/10.1103/PhysRevLett.132.014002

Physics Subject Headings (PhySH)

Drops & bubbles

Binary fluids

Fluid Dynamics

Authors & Affiliations

Feng Wang , Li Chen, Yuqi Li, Peng Huo, Xi Gu , Man Hu, and Daosheng Deng ^*

Department of Aeronautics and Astronautics, Fudan University, Shanghai 200433, China

^*dsdeng@fudan.edu.cn

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Vol. 132, Iss. 1 — 5 January 2024

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Images

Figure 1
Self-lifting of a frozen binary droplet on a supercooled surface. (a) Sketch for a frozen binary droplet for the initial state ( $t = 0$ ), the self-lifting state ( $0 < t < t_{f}$ ), and the final state ( $t = t_{f}$ ), respectively. $H_{0}$ , $R_{0}$ , $θ_{0}$ for the initial height, radius, and apparent contact angle of the binary droplet; $H_{t}$ , $H_{s}$ , $H_{f}$ for the time-dependent total height, the height of the solidification front, and the final height of the frozen binary droplet; $θ_{low}$ , $θ_{up}$ , and $Δ θ_{dyn}$ for the contact angle at the solid-gas interface, liquid-gas interface, and the dynamic growth angle. High-speed imaging and thermal imaging snapshots for (b) pure water droplet $c_{0} = 0$ , $T_{s} = - 23.5 ° C$ ; (c) binary droplet $c_{0} = 0.01$ , $T_{s} = - 22.5 ° C$ ; (d) binary droplet $c_{0} = 0.05$ , $T_{s} = - 21.5 ° C$ ; (e) binary droplet $c_{0} = 0.10$ , $T_{s} = - 20.1 ° C$ . The red dashed line for the solidification front in high-speed images, and the white dashed line for the isotherm $T = T_{f}$ (for the freezing point of the binary liquid) in thermal images. Scale bar for 1 mm.
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Figure 2
Dynamics of self-lifting for a binary droplet. (a) The time evolution of self-lifting ratio $H_{t} / H_{0}$ at different $c_{0}$ . (b) The final self-lifting ratio $H_{f} / H_{0}$ of a binary droplet at different $c_{0}$ . (c) The initial apparent contact angle $θ_{0}$ of a binary droplet at different $c_{0}$ . (d) The propagation of the solidification front with a $1 / 2$ scaling law at different $c_{0}$ . (e) The effective diffusivity of the solidification front ( $H_{f}^{2} / t_{f} = 4 D_{s}$ ) linearly proportional to $Δ T$ , but independent of $c_{0}$ . The shadow in (b) and (c) for the standard deviation of at least five measurements.
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Figure 3
Solutal Marangoni flow near the vicinity of solidification front in the liquid phase for a frozen binary droplet. (a) Sketch for the self-lifting behavior. $θ_{low}$ , $θ_{up}$ , and $Δ θ_{dyn}$ for the contact angle at the solid-gas interface, liquid-gas interface, and the dynamic growth angle; $H_{l}$ , $R_{l}$ for the height and radius of the remaining liquid phase during the droplet freezing; the green shadow for the concentration distribution of ethanol. During the propagation of the solidification front, the solute (ethanol) concentration at the front ( $c_{high}$ ) is accumulated and enriched, becoming greater than the concentration at the apex ( $c_{low}$ ). Consequently, the concentration-dependent surface tension (red arrow) generates a solutal Marangoni flow (yellow arrow), forming an upward solutal Marangoni flow $U_{M}$ along the interface. (b) PIV measurement of a frozen binary droplet, $c_{0} = 0.05$ , $T_{s} = - 37.8 ° C$ . (c) PIV measurement of a frozen water droplet, $c_{0} = 0$ , $T_{s} = - 33.6 ° C$ . (d) Flow field in (b). (e) Solutal Marangoni flow field obtained by simulation. (f) The evolution of $U_{m}$ and $Δ θ_{dyn}$ obtained from experiments. Scale for 1 mm.
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Figure 4
Dynamic growth angle of a frozen binary droplet. (a)–(c) The evolution of the apparent contact angle of the remaining droplet $θ_{up}$ and the ice substrate $θ_{low}$ at different concentration $c_{0}$ . (d) Evolution of the dynamic growth angle $Δ θ_{dyn} = θ_{up} - θ_{low}$ at different $c_{0}$ . (e) The time-averaged dynamic growth angle $\bar{Δ θ_{dyn}}$ for $t / t_{f} < 0.5$ in (d) at different $c_{0}$ , triangles for the experimental results, the solid line for the fitting result to guide the eye, and error bars for the standard deviation of at least five data points.
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Figure 5
Theoretical model for the self-lifting behavior. (a) For the spatiotemporal evolution of droplet profile, the numerical model consistent with the experiment [for $c_{0} = 0.01$ in Fig. 1]. (b) The calculated self-lifting height ratio, $H_{f} / H_{0}$ , dependent on density ratio $ρ_{s} / ρ_{l}$ and $Δ θ_{dyn}$ for a given $θ_{0} = 90 °$ . (c) Calculated $H_{f} / H_{0}$ (transparent curved surface) as a function of initial contact angle ( $θ_{0}$ ) and the dynamic growth angle ( $Δ θ_{dyn}$ ) for water $ρ_{s} / ρ_{l} = 0.917$ ; experimentally measured values (bars) and theoretically estimated values exhibit similar trends.
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