Mapping microscale wetting variations on biological and synthetic water-repellent surfaces

Droplets slip and bounce on superhydrophobic surfaces, enabling remarkable functions in biology and technology. These surfaces often contain microscopic irregularities in surface texture and chemical composition, which may affect or even govern macroscopic wetting phenomena. However, effective ways to quantify and map microscopic variations of wettability are still missing, because existing contact angle and force-based methods lack sensitivity and spatial resolution. Here, we introduce wetting maps that visualize local variations in wetting through droplet adhesion forces, which correlate with wettability. We develop scanning droplet adhesion microscopy, a technique to obtain wetting maps with spatial resolution down to 10 µm and three orders of magnitude better force sensitivity than current tensiometers. The microscope allows characterization of challenging non-flat surfaces, like the butterfly wing, previously difficult to characterize by contact angle method due to obscured view. Furthermore, the technique reveals wetting heterogeneity of micropillared model surfaces previously assumed to be uniform.


Supplementary Figure 2. Accuracy and repeatability of droplet adhesion forces.
Snapin and pull-off forces, respectively, on a,b 20 µm radius pillars, c,d 35 µm radius pillars and d,e 50 µm radius pillars. Error bars present standard deviation for ten repetitions on each pillar. Figure 3. Force curves for a, Silicon nanograss coated with fluoropolymer, b, Glaco on Si wafer, c, Hydrobead on Si wafer, d, striped blue crow butterfly wing and e, copper (II) hydroxide nanowires. Figure 4. Force curves for a, PDMS replica of silicon nanograss, b, golden bird butterfly wing, c, silicon nanograss coated with parylene, d, silicone nanofilaments on Si wafer, e, fluoroalkyl self-assembled monolayer on Si wafer and f, fluoropolymer on Si wafer. Note the volume loss visible in panel f, where the final force level is significantly higher than the initial value. The flat surface has a macroscopically measured receding contact angle of less than 90° (see Supplementary  Table 2) and leaves a droplet behind on the surface at pull-off.  Table 2). The measurements were carried out on 100 different locations on the sample. Approach and retraction speed are shown to have little or no effect on measured force.  Figure 3). b Mean of three measurements, two spots had snap-in force smaller than sensor noise. c The butterfly wings are curved and obscure the surface baseline, making it not feasible to measure advancing and receding contact angles (Supplementary Figure 7).

Supplementary Note 1. Computational model of the snap-in and pull-off forces.
To compute the adhesive force resulting from the droplet between the probe disk and the substrate, we follow the general approach of H.-J. Butt et al. [1]. The axially symmetric geometry of the problem is shown in Supplementary Figure 9. We assume that the probe-to-substrate distance is changing slowly so that the droplet interface is always at equilibrium and that gravity can be neglected.

Supplementary
A priori, we do not know the pressure Δ inside the droplet, but we know the volume (constant): where is the weight of the droplet, = 15 µN in our experiments, ≈ 1000 kg m -3 is the density of the liquid, and ≈ 9.81 m s -2 is the gravitational acceleration.
We consider two different, but closely similar, problems: 1) Fixed radius problem: Contact line on the substrate is pinned e.g. to the edge of a micropillar so that contact radius is fixed. Contact angle may vary. 2) Fixed angle problem: Contact angle is fixed, e.g. on a perfectly flat substrate. Contact line is free to move i.e. may vary.
We focus on solving the fixed radius problem; it is trivial to adapt the method to the fixed angle problem.

Problem statement
The fixed radius problem is a boundary value problem:  etc.

Shooting method
There are several ways to solve this boundary value problem; here we solve it using a shooting method, which is simple to implement, but numerically not very efficient. Instead of solving the boundary value problem, we consider the initial value problem The code to solve Supplementary Equation 12 and to compute the force using Supplementary Equation 9 is given in Supplementary Software 1. where ℎ cap is the height of the spherical cap and ℎ oxide is the thickness of the oxide cap of the pillars, ℎ oxide ≈ 1.2 µm for our pillars. Using Supplementary Equation 16, we get a much-improved match between the model and the data (Supplementary Figure 13). We conclude that the earlier discrepancy between the model and experimental data on the smallest pillars was an artefact of the undercut shape of the pillar tops; for flat samples, the model in Supplementary Fig. 12 is still expected to be more appropriate.
Supplementary Figure 13. Comparison between experimental snap-in force data and numerical results using (16). This is the same data as in Figure 2d.
Our experimental system could not observe small contact radii accurately, but the simulation model can be used to estimate the contact radius of the probed area for a given snap-in force, using the model in Supplementary Figure 12. The smallest snap-in force we have measured was 7.8 nN for the Hydrobead sample (Supplementary Table 2). Using the model in Supplementary Fig. 12, this corresponds to a contact radius of ~ 10 µm. Furthermore, the model can also be used to predict how these values would change with the volume of the droplet. Supplementary Figure 14 shows the snap-in force as a function of droplet weight for the case of = 100 µm. The relative change wx wy y x yz15 µN ≈ −0.55. During a typical measurement lasting for 20 s, the weight loss due to evaporation was observed to be around 1%, corresponding to an estimated increase of 0.55% in the snap-in force.

Pull-off forces on micropillars
We define the pull-off force as the maximum adhesive force with respect to ℎ i.e.
pull-off r, V = max | ℎ, , A naïve implementation of Supplementary Equation 17 is numerically extremely inefficient, as within each step of the numerical optimization, we need to solve (12). It is much better to solve these problems simultaneously, as a constrained optimization problem:  (17). This is the same data as in Figure 2e.

Fixed angle problem
The fixed angle problem is The only difference between the fixed angle problem and the fixed radius problem is that the contact angle, instead of contact radius, is defined at the substrate. The solution proceeds identically to the fixed radius problem: 1) non-dimensionalize; 2) find and Δ using a shooting method; and 3) use (2) to compute the force. We omit repeating these steps for brevity.
This problem was studied in ref.
[1] and our method recreates the numerical results from that paper (Supplementary Figure 16). There are some minor differences stemming from numerical inaccuracies, mostly at the limit when the meniscus becomes unstable. These inaccuracies do not affect the computed snap-in or pull-off forces. This model can be used to predict the relationship between the contact angle and snap-in / pull-off forces for ideal, smooth surfaces, using Supplementary Equation 14 for the snap-in distance. The model predicts that as the contact angle increases, both the snap-in force and the pull-off force decrease (Supplementary Figure 17).