Droplets on chemically patterned surface: A local free-energy minima analysis

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Droplets on chemically patterned surface: A local free-energy minima analysis

Yanchen Wu, Fei Wang, Michael Selzer, and Britta Nestler

Phys. Rev. E 100, 041102(R) – Published 24 October 2019

Abstract

Droplet wetting on solid surfaces is a ubiquitous phenomenon in nature and applications. The wetting behavior of droplets on homogeneous surfaces has been accurately elucidated by the quintessential Young's law. However, on heterogeneous substrates, due to the energy barriers and contact line pinning effect, more than one equilibrated droplet pattern exists, which is more close to reality. Here, we propose a concise mathematical-physical model to delineate the droplet patterns on chemically patterned surfaces: stripe, “chocolate,” and “chessboard.” The present concept is capable of predicting the number as well as the morphologies of the equilibrated droplets on chemically patterned surfaces. We anticipate that the current work can be applied to fabricate programmable surfaces involving droplet manipulation in integrated circuits, biochips, and smart microelectronics.

Received 24 June 2019

DOI:https://doi.org/10.1103/PhysRevE.100.041102

Physics Subject Headings (PhySH)

Coatings Patterning Surface tension effects Thermodynamics of computation Wetting

Statistical Physics & ThermodynamicsPolymers & Soft MatterNonlinear DynamicsFluid Dynamics

Authors & Affiliations

Yanchen Wu^* and Fei Wang ^†

Institute of Applied Materials–Computational Materials Science, Karlsruhe Institute of Technology, Straße am Forum 7, 76131 Karlsruhe, Germany

Michael Selzer and Britta Nestler

Institute of Applied Materials–Computational Materials Science, Karlsruhe Institute of Technology, Straße am Forum 7, 76131 Karlsruhe, Germany and Institute of Digital Materials Science, Karlsruhe University of Applied Sciences, Moltkestraße 30, 76133 Karlsruhe, Germany

^*yanchen.wu@kit.edu
^†fei.wang@kit.edu

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Issue

Vol. 100, Iss. 4 — October 2019

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Images

Figure 1
(a) A sessile droplet on chemically patterned surface. The dashed ellipse (with center $O_{1}$ semiaxes $a$ and $b$ ) is the droplet baseline. The dashed circular arc is on the liquid-gas interface with the circle center $O_{2}$ and curvature radius $r_{c}$ . $β$ depicts the polar angles on the surface of the yellow dashed arc. $θ_{e,1}$ to $θ_{e,4}$ describe the intrinsic contact angles on the pattern lattices. (b)—(d) Selected chemically patterned surfaces: stripe, “chocolate,” and “chessboard,” defined by the arrangement of the intrinsic contact angles.
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Figure 2
(a)–(c) Surface energy landscapes in the $a - b$ space and snapshots of sessile droplets on chemically striped patterned surfaces with different droplet sizes [(a) $R / L = 2$ , (b) $R / L = 5$ , (c) $R / L = 9$ ]. The chemical heterogeneities are described by $f_{1} (r_{b}, φ)$ in Eq. (1). At equilibrium, the droplet base center stays either on the center of the blue (hydrophilic) stripe (namely, position 1 and $i = 1$ ) or on the center of the red (hydrophobic) one (i.e., position 2 and $i = 0$ ). The energy landscapes are accordingly calculated when the droplet base centers are on these two different positions. The contour lines indicate the energy levels (red for low energy and blue for high energy states) and the color changes from blue to red illustrate the decrease of the energy. The energy minima are indicated by different numbers, corresponding to the snapshots labeled with the same number. The red dashed ellipses in (a)(III) denote the analytical results, which can be read from the corresponding energy landscapes. (d) Predicted droplet configurations for different droplet sizes. The filled and empty symbols describe the simulation and analytical results, respectively. The two dashed lines are guide lines to highlight a trend in the data.
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Figure 3
(a)–(c) Surface energy landscapes for chocolate-patterned surfaces. The chemical heterogeneities are described by $f_{2} (r_{b}, φ)$ in Eq. (1). The energy minima are indicated by different numbers, corresponding to the snapshots in (e) labeled with the same number. (d) Sketch of the surface topology and three base centers of droplets (gray, hydrophobic; white, hydrophilic). The droplet shapes in (a), (b), and (c) correspond to the positions $P_{1}, P_{2}$ , and $P_{3}$ . (e) Snapshots of equilibrated droplets through PFM (blue, hydrophilic; red, hydrophobic). The red dashed ellipses denote the analytical results. (f) The number $N$ of equilibrium shapes of droplets as a function of $R / L$ . The black dashed line is the guide line.
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Figure 4
(a)–(e) Surface energy landscapes for chessboard-patterned surfaces. (a) and (d) correspond to the situations when the directions of $a$ and $b$ are horizontal and vertical, respectively, as shown in (g)2. The chemical heterogeneities are described by $f_{3} (r_{b}, φ)$ in Eq. (1). (b), (c), and (e) describe the energy landscapes in a rotated coordinate system and the directions of $a$ and $b$ are shown in (g)5. In the rotated system, $δ_{1}$ and $δ_{2}$ in $f_{3} (r_{b}, φ)$ are substituted by $δ_{1}^{'}$ and $δ_{2}^{'}$ , respectively, with $δ_{1}^{'} = (δ_{1} + δ_{2}) / \sqrt{2}$ and $δ_{2}^{'} = (δ_{1} - δ_{2}) / \sqrt{2}$ . The energy minima are indicated by different numbers, corresponding to the snapshots in (g) labeled with the same number. (f) Schematic description of the surface topology and three base center points of equilibrated droplets (gray, hydrophobic; white, hydrophilic). The two lines indicate two possible directions of the semiaxis $a$ . (a) and (b), (d) and (e), and (c) correspond to $P_{1}, P_{2}$ , and $P_{3}$ , respectively. (g) Snapshots of equilibrated droplets via PFM (blue, hydrophilic; red, hydrophobic). The red dashed ellipses denote the analytical results. (h) The number $N$ of equilibrium shapes of droplets as a function of $R / L$ . The black dashed line is the guide line.
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Figure 5
The number $N$ of equilibrium droplet shapes as a function of the energy discontinuous line density $ρ : = n / {(2 L)}^{2}$ for different patterned surfaces, Here, $n$ is the total effective number of pinning lines within a square cell with an area of ${(2 L)}^{2}$ . The three insets indicate how $ρ$ is calculated for different patterned surfaces. The numbers “1” and “0.5” shown in the cells stand for the effective number of pinning lines. The lines with different symbols denote the results for droplets with different sizes (square, $R / L = 0.5$ ; circle, $R / L = 1.5$ ; triangle, $R / L = 2$ ).
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