Contact line length dominance in evaporation of confined nonspherical droplets

Letter
Open Access

Contact line length dominance in evaporation of confined nonspherical droplets

Gun Oh, Jae-Hong Lim, Sung Hoon Kang, and Byung Mook Weon

Phys. Rev. Research 6, L012026 – Published 8 February 2024

Abstract

Free droplets are spherical within capillary lengths and become nonspherical when trapped in a confined space. Confined nonspherical droplets are as common as spherical droplets. Yet, their evaporation dynamics are not fully understood because of their geometrical complexity. We use monochromatic synchrotron x-ray microtomography to investigate the evaporation dynamics of confined nonspherical water droplets trapped by micropillars based on three-dimensional geometrical information with time. We find two types of confined nonspherical droplets: Wenzel droplets with single-sided air-water interfaces, and Cassie-Baxter droplets with double-sided interfaces. Both droplets show similar sphericity at large volumes but approach different ones at small volumes: Wenzel droplets follow a thin film limit, whereas Cassie-Baxter droplets follow a spherical sessile limit. Despite the geometrical complexity of confined nonspherical droplets, the vapor diffusion mechanism suggests that the evaporative flux is maximal at the contact line, which governs the evaporation dynamics, as proven by observations. The proportionality of the evaporation rate to the contact line length demonstrates the contact-line-length-dominant evaporation dynamics of confined nonspherical droplets. The findings of this study can unify the evaporation mechanism for spherical sessile and confined nonspherical droplets, even with geometric complexity.

Received 18 September 2023
Accepted 10 January 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.L012026

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Drop & bubble phenomena Evaporation

Interfaces Water

X-ray imaging X-ray tomography

Polymers & Soft MatterCondensed Matter, Materials & Applied PhysicsFluid Dynamics

Authors & Affiliations

Gun Oh ^1,2, Jae-Hong Lim³, Sung Hoon Kang ^2,*, and Byung Mook Weon ^1,4,†

¹Soft Matter Physics Laboratory, School of Advanced Materials Science and Engineering, SKKU Advanced Institute of Nanotechnology (SAINT), Sungkyunkwan University, Suwon 16419, Republic of Korea
²Department of Mechanical Engineering and Hopkins Extreme Materials Institute, Johns Hopkins University, Baltimore, Maryland 21218, USA
³Pohang Accelerator Laboratory, POSTECH, 80 Jigokro-127-beongil, Nam-gu, Pohang, Gyeongbuk 37673, Republic of Korea
⁴Research Center for Advanced Materials Technology, Sungkyunkwan University, Suwon 16419, Republic of Korea

^*shkang@jhu.edu
^†bmweon@skku.edu

Article Text

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Supplemental Material

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References

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Issue

Vol. 6, Iss. 1 — February - April 2024

Subject Areas

Fluid Dynamics

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Images

Figure 1
Confined nonspherical droplets with geometric complexity. (a) Wenzel droplets: a $1.5 μ L$ droplet sunk to the substrate bottom. (b) Cassie-Baxter droplets: a $1.0 μ L$ droplet floated among hexagonally arrayed micropillars. The left images were taken by x-ray microtomography (along view I), the middle schematic images were from view I, and the right were from view II. The formation of Wenzel or Cassie-Baxter droplets depends on the initial droplet volumes because the initial accommodable space surrounded by micropillars was $\sim 1.5 μ L$ (see Figs. S1 and S2 within the Supplemental Material [40]).
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Figure 2
Evaporation dynamics of confined nonspherical droplets by time-resolved x-ray microtomography. (a) The time evolution of a Wenzel droplet, taken by x-ray microtomography, shows a gradual height decrease for $t / t_{f} = 0.2 \sim 0.92$ with no detachments from the micropillars. (b) The time evolution of a Cassie-Baxter droplet, taken by x-ray microtomography, shows a gradual decrease of the height for $t / t_{f} = 0.2 \sim 0.3$ and detachments from some of the micropillars for $t / t_{f} = 0.4 \sim 0.99$ . The final small volume of a Wenzel droplet near $t_{f}$ is similar to a thin film with a specific upper surface area and a minimal thickness (a thin film limit). In contrast, a last tiny Cassie-Baxter droplet near $t_{f}$ is similar to a small sessile droplet on a curved surface (a spherical sessile limit).
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Figure 3
Cross-sectional schematic illustration. At the early stages, Wenzel droplets (upper) and Cassie-Baxter droplets (lower) have similar geometrical changes between the height ( $h$ ) and the width ( $w$ ), where $h$ gradually decreases with time while $w$ is invariant due to the contact line pinning during evaporation. In the late stages, they separately approach different geometrical limits. Wenzel droplets reach a thin film limit, where $h$ is ultimately small while $w$ is almost invariant. Cassie-Baxter droplets reach a spherical sessile limit, where $h$ and $w$ decrease simultaneously by retraction (by detaching from some of the micropillars), and thus eventually, spherical sessile droplets remain on a curved surface of micropillars.
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Figure 4
Contact-line-length-dominant evaporation dynamics. (a) The droplet mass decreases with time, showing the linear $M^{2 / 3} \propto t$ relations, from $- d M / d t = k M^{1 / 3}$ in the vapor diffusion mechanism of sessile droplets, where the $k$ values are similar for Wenzel and Cassie-Baxter droplets (marked by the open circle and square for volume data taken with x-ray microtomography and the lines for mass data). All experiments were conducted at $T = 25^{\circ} C$ and $R H = 0.3$ . (b) The linear $S \propto L$ relations highlight the critical geometry of confined nonspherical droplets (whereas $S \propto L^{2}$ for spherical sessile droplets, marked by the gray line). The thin film limit is the fate of Wenzel droplets (marked by the red star), and the spherical sessile limit is the fate of Cassie-Baxter droplets (marked by the blue star). (c) The linear $- d M / d t \propto L$ relations look identical between Wenzel and Cassie-Baxter droplets and significantly deviate from spherical sessile droplets (marked by the gray line). (d) The evaporation rates ( $- d M / d t$ ) depend on the sphericity ( $Ψ$ ) for confined nonspherical droplets, showing that Wenzel and Cassie-Baxter droplets separately approach different geometrical limits in the late stages.
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Figure 5
Contact line geometry. (a) The contact line trajectory of the Wenzel droplet, taken from Fig. 2 at $t / t_{f} = 0.4$ , has a hexagonal shape on a curved surface. (b) The contact line trajectory of the Cassie-Baxter droplet, taken from Fig. 2 at $t / t_{f} = 0.2$ [same volumes as (a)], has a summation of six elongated ellipses on a curved surface. Interestingly, the flux is almost the same regardless of the angle in the Cassie-Baxter droplet. The arbitrary position $r$ and $θ$ on a polar coordinate are defined along the air-water interface, and $r$ is normalized by the triple-phase contact line position $r_{0}$ . In both cases, the dimensionless evaporative flux $J$ calculated by $r$ and $θ$ in Eq. (1) shows that the evaporative flux diverges at the contact line, regardless of $θ$ .
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