Through-drop imaging of moving contact lines and contact areas on opaque water-repellent surfaces

A myriad of natural surfaces such as plant leaves and insect wings can repel water and remain unwetted inspiring scientists and engineers to develop water-repellent surfaces for various applications. Those natural and artificial water-repellent surfaces are typically opaque, containing micro- and nano-roughness, and their wetting properties are determined by the details at the actual liquid–solid interface. However, a generally applicable way to directly observe moving contact lines on opaque water-repellent surfaces is missing. Here, we show that the advancing and receding contact lines and corresponding contact area on micro- and nano-rough water-repellent surfaces can be readily and reproducibly quantified using a transparent droplet probe. Combined with a conventional optical microscope, we quantify the progression of the apparent contact area and apparent contact line irregularity in different types of superhydrophobic silicon nanograss surfaces. Contact angles near 180° can be determined with an uncertainty as low as 0.2°, that a conventional contact angle goniometer cannot distinguish. We also identify the pinning/depinning sequences of a pillared model surface with excellent repeatability and quantify the progression of the apparent contact interface and contact angle of natural plant leaves with irregular surface topography.


Computational model for contact angle
To compute the contact angle for each frame of the measurement we assume an axisymmetric droplet profile shown in Fig. S1. We also assume that the droplet shape is always in equilibrium and gravity can be neglected. Under these conditions the droplet's profile is given by Young-Laplace equation: (1 + ' 2 ) 3 2 where ; and are the first and second derivative of ; is the Laplace pressure = ( ) ' '' ( ) ∆ and the surface tension of water.
where the contact angle can be obtained as .
To solve for we frame the problem as an initial value problem defined by and .
We use a double shooting method to first find that solves and then find that solves branches, shown in blue and red in Fig. S1. The first branch is integrated using the inverse equation for (Eq. 3), from until , and from there on using (Eq. 2) until .
with ; and the first and second derivatives of .
The contact radius is measured from the top-view camera, by fitting a circle to the contact interface, which is identified through the machine vision algorithm, described in Algorithm S1. The disk radius was measured under the microscope to be . The volume is controlled ≃ 511 µ during the experiment to be approximately . In post-processing, the volume is measured from 1.5 µ side-view camera at the start and end of the experiment, to account for evaporation. The volume during the experiment is estimated by linear interpolation. We set the initial height to be that of an ideal spherical droplet attached to the disk, based on the initial volume at the moment of first contact. The sample-to-disk height is continuously measured with a laser-interferometer ℎ displacement sensor (Fig. S2), relative to the initial height.

Contact Line Irregularity
To calculate the contact line irregularity, , first the outline of the wetting interface is obtained through machine vision and a circle is fitted to it (see Fig. 2e). The deviations from the circle are integrated around the perimeter, , and normalized to the perimeter of the fitted circle : * = *

#(4)
This is analogous to the calculation of topographical roughness 1 .
where is the surface height profile, measured from the mean line and is the length along ( ) which the integral is taken. Where as is a measure of topographical height along in the axis, can be interpreted as the CL roughness along the wetting interface perimeter, in the plane of the interface .
Algorithm S1 -Pseudo-code for wetting interface identification Inputs: Video file containing frames ; index of highest sample stage position; and ,       The data is then processed to isolate the interface (d) and the meniscus (e). The data is levelled by a plane fitted to the interface. 300 radial slices are taken around the meniscus data, to each a quadratic fit is performed. The contact angle is measured as the angle of the quadratic fit at the intersection with the z=0 plane. The mean and standard deviation of the contact angles are presented in Table S2 as θ DHM for each type of nanograss. To determine the end of the transition phase, p 1 , the data was first filtered. Then p 1 was found as the point where the rate of change in area during receding was 5% larger than the rate of change for the equivalent area during advancing, p 2 .     Supporting Tables   Table S1 Surface properties of each different silicon nanograss type. Etch depth and spike tip radius were measured manually using side-view SEM measurements. Spike density was calculated from AFM images, with spike spacing being the square root of the inverse of spike densities.

Supporting Videos
Movie S1 (separate file) Contact area, contact line irregularity and contact angle on four different silicon nanograss substrates. The measurements were carried out for silicon nanograss #A, #B, #C, and #D. The green line shows the detected interface outline. Area and apparent contact line irregularity plots are shown in real-time. Video contrast and brightness were adjusted.
Movie S2 (separate file) Pinning and depinning on silicon pillars with different droplet-sample alignments. Three cases were measured: centered on one pillar, between four pillars, and between two pillars. Video contrast and brightness were adjusted.
Movie S3 (separate file) Contact area and contact line irregularity on Maranta and Musa plant leaves. The green line shows the interface outline. Area and apparent contact line irregularity plots are shown in realtime. Video contrast and brightness were adjusted.
Movie S4 (separate file) Lateral scan measurement on scratched silicon nanograss #A samples. Two cases were measured, a dot-like scratch and a line-like scratch. Video contrast and brightness were adjusted.