Abstract
Droplets move on substrates with a spatio-temporal wettability pattern as generated, for example, on light-switchable surfaces. To study such cases, we implement the boundary-element method to solve the governing Stokes equations for the fluid flow field inside and on the surface of a droplet and supplement it by the Cox-Voinov law for the dynamics of the contact line. Our approach reproduces the relaxation of an axisymmetric droplet in experiments, which we initiate by instantaneously switching the uniform wettability of a substrate quantified by the equilibrium contact angle. In a step profile of wettability the droplet moves towards higher wettability. Using a feedback loop to keep the distance or offset between step and droplet center constant, induces a constant velocity with which the droplet surfs on the wettability step. We analyze the velocity in terms of droplet offset and step width for typical wetting parameters. Moving instead the wettability step with constant speed, we determine the maximally possible droplet velocities under various conditions. The observed droplet speeds agree with the values from the feedback study for the same positive droplet offset.
Highlights
If a droplet can go uphill and move in response to light, where are the limits of its motility? In their seminal experiment, Chaudhury and Whitesides[1] demonstrated that liquid droplets can be driven up a tilted plane against gravity by chemically treating the plane so it gradually becomes more wettable with height
The most well-known among them is the Cox–Voinov law[40,41] derived from hydrodynamic considerations: vcontact 1⁄4 9m lngðlgh=lÞÀydyn[3] À yeq3Á. It relates the difference of the cubes of dynamic and equilibrium contact angles, ydyn and yeq, to the velocity of the contact line vcontact
As a step we check the fitness of the mesh facing the substrate and if the maximum edge lengths exceeds 150% of the minimal edge length, the positions of all substrate vertices are adjusted by tangential displacements.**. Since this technique is completely derived from literature and by its very nature must not affect the dynamics of the droplet, we do not reproduce the algorithm in detail here
Summary
If a droplet can go uphill and move in response to light, where are the limits of its motility? In their seminal experiment, Chaudhury and Whitesides[1] demonstrated that liquid droplets can be driven up a tilted plane against gravity by chemically treating the plane so it gradually becomes more wettable with height. Ichimura et al.[2] showed that sessile droplets start moving in response to gradients in wettability, which are produced by a photo-chemical reaction These experiments essentially demonstrated an early use of structured light to create motion on the micron scale, an approach which has recently become the focus of intense research.[3,4,5] Light-driven fluid motion[6,7,8,9,10] is an especially favorable control mechanism because of its high precision and controllability through the established experimental methods in optics.[11] Notably, it can be used in combination with or as an alternative to electrowetting techniques.[12] Precise control of fluid motion is foundational for advanced lab-on-a-chip devices,[13,14,15] as well as self-cleaning surfaces,[16,17] and printing with sub-droplet precision.[4,18].
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