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Self-similarity of contact line depinning from textured surfaces

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.


Supplementary Table S1 -Satellite drop deposition
a Velocity was not reported by the authors. The epoxy was rolled along a tilted surface, and since the viscosity of the epoxy is less than that of glycerol, is estimated to roll at least as fast as the glycerol drop, and most likely much faster. Therefore a capillary number Ca = 10 -6 represents a lower bound.  Table S1). The capillary numbers encountered in the current study are sufficiently low (Ca ~ 10 -8 ) as to avoid deposition. Although satellite drops have been observed on other studies, 42-44 this is most likely due to both significantly larger capillary numbers and/or smaller local receding contact angles, which lead to necking and eventual rupture of the capillary bridges. If the detachment velocity is too high, the capillary bridge will form a neck, and the contact line will not be able to recede before the neck ruptures. Furthermore, if the receding angle is much less than 90°, the capillary bridge will be able to form a neck and rupture without reaching depinning condition that is necessary for avoiding a satellite drops. Therefore, the condition for avoiding the deposition of satellite drops is that the detachment velocity is slow enough to allow the contact line to completely depin, and that the contact angle is high enough to depin before forming a neck.

Supplementary Note 2 -Vertical force balance
Considering a liquid drop sitting in a Cassie state on a textured surface, the contact patch will consist of an interior region that sags between the roughness features and a peripheral region that is pulled up between adjacent roughness features. If an imaginary plane is placed just above the tops of the roughness features so that it intersects the drop immediately above the contact line, then the interior portion of the plane will reside completely within the drop, and the peripheral region will intersect the air-water interface (Fig. S1). Gravity is assumed to be negligible for this drop, whose radius is much smaller than the capillary length: R << σ ρg and so there will be two principal forces acting on this plane. The first, due to the pressure within the drop, is equal to where σ is the surface tension of the liquid, R is the radius of the drop, and r b is the radius of the contact patch. The second force, due to surface tension acting along the peripheral roughness features, is equal to 2πr b Φ 0 σ sin θ r ( ) , where Φ 0 is the total pinned fraction, and θ r is the intrinsic receding contact angle. For a static drop, these two forces are equal, and for the experiments in the current study are of the order 1x10 -7 N.
When an external vertical force is applied to the drop, the drop responds by both receding to a smaller contact patch radius and a smaller local contact angle. Both of these effects act to increase the force due to surface tension above its original value for the unperturbed drop. We argue that since the force due to the internal pressure of the unperturbed drop is more than two orders of magnitude smaller than the final detachment force (F > 1x10 -5 N), the force due to pressure may be neglected in the vertical force balance leading to Equation 2.

Supplementary Note 3 -Error analysis of Equation 3
We have made a simplifying assumption that for the case of water on silanized silicon pillars, capillary bridges are formed on only the peripheral micropillars with a nearly uniform contact angle around the entire micropillar. As noted on pg. 7, the contact angle is not uniform all around the pillar, but differ by as much as 5°. The error in the adhesion force of a single capillary bridge is given by: The error is 7.1% when θ r = 86° and δθ r = 5°, which represents the worst-case error, and so we believe this assumption to be appropriate for the case of water on silanized silicon. However, this assumption will not be appropriate with other liquid/solid systems that have different contact angles, e.g. as seen with the epoxy drops imaged by Dufour et al. 2,3 In that case, the contact angle differs by almost 90° from the outside to the inside of the roughness feature, so we would need to evaluate the integral given in the general expression for pinning force.
The high humidity and low temperatures of the ESEM experiments were needed to minimize evaporation rates, and any error in comparing the results to the macroscale adhesion experiments conducted at room temperature need to be taken into account. The ESEM instrument used in this study is constrained to an upper operating pressure of 3000 Pa, but the poor resolution at this pressure warrants imaging at pressures as low as possible. 1000 Pa was chosen as a tradeoff between better image resolution at lower pressures and lower evaporation rates at higher pressures. The saturation temperature of water at 1000 Pa is 6.97°C, and since the ESEM chamber walls surrounding the sample are at room temperature (~20°C), both the water drop and the sample experience a radiative heat flux. To counter this positive heat flux, a negative heat flux is provided by means of a Peltier cooling stage. The drop is held by a copper wire so that the incident radiative heat flux may be conducted away into the Peltier stage. Likewise, the textured samples are in thermal contact with the Peltier stage so that the incident radiative heat flux is also conducted away. Despite these efforts, it was not possible to completely eliminate evaporation of the drop. The evaporation rate of a drop of water held in the ESEM under the reported humidity conditions (P = 1000 Pa, T Peltier = 2°C) was measured to be approximately 5 pL/s.
As a consequence of the low temperature, the surface tension of water is higher than at the room temperature conditions reported in most other studies. The contact angle will also change with temperature. However, both of these effects are quite small. The error in Equation (3) with respect to error in surface tension and contact angle is given by: The error in Equation (3) due to temperature difference between an ESEM temperature of 6.5°C (slightly below saturated temperature) and room temperature is 2.8%, and so temperature effects are considered to be negligible.