Droplet Memory on Liquid-Infused Surfaces

The knowledge of droplet friction on liquid-infused surfaces (LIS) is of paramount importance for applications involving liquid manipulation. While the possible dissipation mechanisms are well-understood, the effect of surface texture has thus far been mainly investigated on LIS with highly regular solid topographies. In this work, we aim to address this experimental gap by studying the friction experienced by water droplets on LIS based on both random and regular polysilsesquioxane nanostructures. We show that the available models apply to the tested surfaces, but we observe a previously unreported droplet memory effect: as consecutive droplets travel along the same path, their velocity increases up to a plateau value before returning to the original state after a sufficiently long time. We study the features of this phenomenon by evaluating the motion of droplets when they cross the path of a previous sequence of droplets, discovering that moving droplets create a low-friction trace in their wake, whose size matches their base diameter. Finally, we attribute this to the temporary smoothing out of an initially conformal lubricant layer by means of a Landau–Levich–Derjaguin liquid film deposition behind the moving droplet. The proposed mechanism might apply to any LIS with a conformal lubricant layer.


S1.1 Constant velocity validation
The study of droplet friction scaling relies on the assumption of an equilibrium between viscous dissipation and gravitational driving force, which results in a constant droplet velocity.
Therefore, only sufficiently constant velocity signals were considered for this analysis.
This was validated by verifying that the average of the relative velocity gradient along the axis did not exceed 1 % for at least 6 mm, that is: Moreover, even if Equation S1 was satisfied, the maximum allowed standard deviation of velocity was 4 % of its average value. Velocity signals that did not meet these conditions were excluded from this analysis. S1

S1.2 Determination of crossing position
The crossing position cross was chosen at least 20 mm from the deposition position of reference and probe droplets. Moreover, it was ensured that the motion of reference and probe droplets was recorded for at least 10 mm before they encountered the crossing point, in order to clearly observe their behavior at cross .
The exact position of cross , as well as the base diameter of the trace droplets trace , was extracted from still images of the trace droplets, as their motion was parallel to the optical axis of the camera.

S1.3 Droplet size in crossing experiments
The influence of probe and trace droplet size in crossing experiments was evaluated by varying their volume between 6 µL and 20 µL, and surface tilt was changed accordingly to maintain the same Ca. This was accomplished by noting that, for any two droplets 1 and 2, if Equation 2 is valid: In our experiments, 1 = 10°and 1 = 10 µL, and the values of and employed are summarized in Table S1. When setting trace , the values in Table S1 were rounded to the nearest integer, owing to the lower resolution of the custom tilting stage scale.

S2.1 Apparent water contact angle on LIS
The definition of an apparent contact angle on LIS app is not trivial, 1 owing to the presence of a lubricant wetting ridge around the test liquid droplet.  In our LIS, we did not observe prominent wetting ridges during water contact angle measurements, as shown in Figure S1. For the sake of completeness, we calculated app with both of the aforementioned methods, as summarized in Table S2. The analysis was performed with the Krüss ADVANCE software, using the Laplace-Young fitting routine for the extrapolation method and the Tangent routine for the inflection point method.
Results obtained with the two methods are in fairly good agreement, as expected for small wetting ridges. In the main text, we discuss only the values obtained with the inflection point method, due to their more straightforward physical interpretation. 4

S2.2 Reorientation of liquid molecules
The time required for liquid molecules to change their orientation at a given temperature can be estimated from the time required for the molecules to diffuse their own diameter: 5 where W is the molecular weight, is the density, A is the Avogadro constant, and B is the Boltzmann constant.
For the PDMS oil we used in our experiments, = 19 mPa s and = 0.95 g cm −3 . We can estimate the molecular weight from interpolation of literature data on W as a function of kinematic viscosity, 6 obtaining W ≈ 1800 g mol −1 . Equation S3 then yields a reorientation time ≈ 23 ns at 22°C.