Self-similarity of contact line depinning from textured surfaces

Abstract

The mobility of drops on surfaces is important in many biological and industrial processes, but the phenomena governing their adhesion, which is dictated by the morphology of the three-phase contact line, remain unclear. Here we describe a technique for measuring the dynamic behaviour of the three-phase contact line at micron length scales using environmental scanning electron microscopy. We examine a superhydrophobic surface on which a drop's adhesion is governed by capillary bridges at the receding contact line. We measure the microscale receding contact angle of each bridge and show that the Gibbs criterion is satisfied at the microscale. We reveal a hitherto unknown self-similar depinning mechanism that shows how some hierarchical textures such as lotus leaves lead to reduced pinning, and counter-intuitively, how some lead to increased pinning. We develop a model to predict adhesion force and experimentally verify the model's broad applicability on both synthetic and natural textured surfaces.

Figure 1. ESEM fixture and movie frame.

(a) Experimental apparatus within an ESEM chamber. A water droplet is held against a superhydrophobic micropillar surface by a copper wire. The 10 μl drop is cooled by a Peltier device and swept across the surface (x direction) by rotating the ESEM stage about the y axis. (b) Illustration of electron beam imaging area with respect to droplet contact line. (c) Single frame from movie of water droplet receding along superhydrophobic micropillars in the positive x direction shows capillary bridge formation. Arrow indicates penultimate capillary bridge. Scale bar, 10 μm. For complete movie, see Supplementary Movie 1.

Figure 2. Image sequences of moving contact line.

(a) Receding and advancing contact line on micropillar array with 3.3 μm spacing. Arrow indicates direction of drop movement. Scale bar, 50 μm. For complete movie, see Supplementary Movie 2. (b) Receding contact line on micropillar array with 40 μm spacing. Arrow indicates direction of drop movement. Scale bar, 50 μm. For complete movie, see Supplementary Movie 1.

Figure 3. Capillary bridge geometry as a function of roughness feature spacing.

(a) Representative ESEM frame showing a water drop receding from the right at a velocity of 2 μm s⁻¹ on micropillars spaced 75 μm apart, with profile of capillary bridge highlighted; scale bar, 20 μm. (b) Representative ESEM frame showing a water drop receding from the right at a velocity of 2 μm s⁻¹ on micropillars spaced 15 μm apart, with profile of capillary bridge highlighted. Scale bar, 20 μm.

Figure 4. SEM micrographs of micro-contact line.

(a) SEM micrograph of smooth micropillar. Scale bar, 10 μm. (b) ESEM frame of water droplet receding on smooth micropillars with 15 μm spacing. Microscale receding contact angle θ_μ^r=86°±5° (approximately equal to that on a smooth surface: θ^r=90°±3°). Scale bar, 20 μm. (c) SEM micrograph of nanograss micropillar. Scale bar, 10 μm. (d) ESEM frame of water droplet receding along nanograss-covered micropillars with 25 μm spacing. Microscale receding contact angle θ_μ^r=140°±5° (approximately equal to that on a planar nanograss surface: =145°±3°). Scale bar, 20 μm. See Supplementary Movie 4.

Figure 5. Schematic of self-similar contact line pinning.

(a) A liquid droplet that rests in a Cassie–Baxter state on a hierarchical surface exhibits an apparent receding angle θ₀^r. (b) The apparent contact line of the drop is divided into many smaller first-level contact lines, each at the top of a first-level roughness feature with width w and spacing s. Each of these first-level contact lines sits at the base of a first-level capillary bridge, which has a local receding contact angle θ₁^r. (c) The apparent contact line of each second-level capillary bridge is further divided into smaller second-level contact lines, each atop a second-level roughness feature. Each second-level contact line sits at the base of a second-level capillary bridge, which has a local receding contact angle θ₂^r.

Figure 6. Pinned fraction schematics.

(a) A contact line projected on a textured surface will be pinned to a number of peripheral micropillars. (b) Effect of texture geometry on pinned fraction. Micropillars with sufficiently dense spacings will result in pinned fractions greater than 1.

Figure 7. Macroscopic adhesion measurements.

(a) Adhesion force versus spacing ratio for smooth micropillars, nanograssed micropillars and a N. nucifera leaf. (b) Ratio of measured adhesion force per unit length of the contact line of a macroscopic drop to that on a smooth surface versus total pinned fraction for smooth micropillars, nanograss micropillars, flat nanograss and a N. nucifera leaf. Deviation at large pinned fractions is due to interaction of capillary bridges. Dotted line indicates the ratio as predicted by equation (3). Error bars indicate s.e. (c) Adhesion force per unit length of projected contact line of a macroscopic drop normalized to remove the effect of the nanograss hierarchy, plotted against the pinned fraction at the first hierarchy. For the case of nanograssed micropillars, (F/l)₀ was obtained by measuring the adhesion force of a drop on a nanograss surface and normalizing with the projected length of the contact line. For the case of smooth micropillars, (F/l)₀ is simply . Dotted line indicates the normalized force as predicted by equation (4). Error bars indicate s.e.