Topography-Dependent Effective Contact Line in Droplet Depinning

The mobility of a fakir state droplet on a structured surface is fundamentally determined by the effective length of a microscopic contact line. However, it is largely unknown how the surface topography determines the effective contact line length. Based on the direct measurement of droplet adhesion force and the visualization of contact line, this work shows that effective contact line length is topography dependent as opposed to prior notion. On pored surfaces, contact line is not distorted, and the effective length approaches the droplet apparent perimeter regardless of pore dimensions. On pillared surfaces, the distortion of contact line is significantly dependent on the packing density of the pillar structures so that the effective length is as small as a pillar diameter on densely packed pillars and as large as a pillar perimeter on sparsely-packed pillars, while changing linearly between the two extremes.

Figure 1

(a) Scanning electron microscope images of pillared (i) and pored (ii) surfaces with varying dimensions, where $\lambda$, $d$, and $\mathrm{\Phi }$ represent the structure pitch, diameter, and solid fraction, respectively. The scale bar is $20\mu \mathrm{m}$. (b) Schematic of the measurement of adhesion force of a water droplet ($\sim 3\mu \mathrm{L}$) on a solid surface. (c) Force measured with respect to the vertical position of a pored surface of $\mathrm{\Phi }=0.65$. Insets illustrate the change in the droplet shape and the maximum adhesion force is denoted as $F_{\max }$. (d) The principal radii of curvatures measured on droplets in convex and concave shapes.

Figure 2

Measured maximum adhesion forces ($F_{\max }$) on pillared (filled circle), pored (circle), and smooth (filled diamond) surfaces with respect to solid fractions ($\mathrm{\Phi }$).

Figure 3

(a) The calculated contact line fraction ($\delta$) on pillared (filled circle), pored (circle), and smooth (filled diamond) surfaces with respect to solid fractions ($\mathrm{\Phi }$). The red-dotted line at $\delta =1$ is for the guide for reading. The dark-gray, light-gray, and yellow regimes show three distinct trends of $\delta$ on pillared surfaces (filled circle) with respect to $\mathrm{\Phi }$. (b)–(d) The shapes of the pinned contact line on the pored surfaces (orange-solid line) of $\mathrm{\Phi }=0.50$, 0.78, and 0.93, respectively, on the brink of detachment (i) and right after the detachment (ii).

Figure 4

(a) The ratio ($x$) of the effective contact line length ($l_i$) on a pillar to a pillar size ($d$) with respect to the packing density ($d/\lambda$) of pillars. The black-dashed line represents the best fitting line. (b)–(d) The effective contact lines on pillar tops (orange-solid lines) and the lowest local positions of the distorted liquid-gas interfaces in between pillars (yellow-dotted line) in the cases of $d/\lambda =0.18$, 0.41, and 0.80, respectively, on the brink of detachment (i) and right after the detachment (ii).