Spontaneous Movement of a Droplet on a Conical Substrate: Theoretical Analysis of the Driving Force

Abstract

Experiments and simulations have shown that a droplet can move spontaneously and directionally on a conical substrate. The driving force originating from the gradient of curvatures is revealed as the self-propulsion mechanism. Theoretical analysis of the driving force is highly desirable; currently, most of them are based on a perturbative theory with assuming a weakly curved substrate. However, this assumption is valid only when the size of the droplet is far smaller than the curvature radius of the substrate. In this paper, we derive a more accurate analytical model for describing the driving force by exploring the geometric characteristics of a spherical droplet on a cylindrical substrate. In contrast to the perturbative solution, our model is valid under a much weaker condition, i.e., the contact region between the droplet and the substrate is small compared with the curvature radius of the substrate. Therefore, we show that for superhydrophobic surfaces, the derived analytical model is applicable even if the droplet is very close to the apex of a conical substrate. Our approach opens an avenue for studying the behavior of droplets on the tip of the conical substrate theoretically and could also provide guidance for the experimental design of curved surfaces to control the directional motion of droplets.

Steering droplets on substrates using moving steps in wettability

“…Notably, for all cases studied with θ eq ranging from 30 • to 120 • , the droplets move in the direction of increasing curvature. The same behavior was recently confirmed for droplets on cylinders of varying size [43]. Indeed, when we place a droplet onto a substrate peak with θ eq = 120 • in our BEM simulations and then perturb the droplet, the position is unstable.…”

“…To start, we re-examine the nonlinear oscillator model of Eq. (43), where a spatially periodic force drives the droplet with a constant amplitude v c . For the deforming substrate we introduced a traction force F ⊥ in Eq.…”

Single Condensation Droplet Self-Ejection from Divergent Structures with Uniform Wettability

“…E surf,pore is expressed in eq (explicit form in eq S4.5). Under these hypotheses, E surf,pore is the potential of a driving surface force F surf (eq ), − positive as H b increases for the case considered here. E s u r f , p o r e = σ l v false( A normalc normala normalp , normalt + A normalc normala normalp , normalb − A normalt normalr normalu normaln normalc normala normalt normale normald .25em normalc normalo normaln normale .25em normalcos nobreak0em.25em⁡ ϑ normale normalq false) F s u r f = − d E s u r f , p o r e d H b …”

“…E surf,pore is expressed in eq (explicit form in eq S4.5). Under these hypotheses, E surf,pore is the potential of a driving surface force F surf (eq ), − positive as H b increases for the case considered here. …”

Quantitative nanopatterning of fg-scale liquids with dip-pen nanolithography

“…This is in contrast with conical probes where there is no stable configuration for a drop to rest on the conical probe-it will either advance to the base of the cone in a clamshell configuration if the probe is wetting, or wick to the tip of the cone and fall off if the probe is non-wetting. This phenomenon of drops spontaneously moving when placed on cones has been explored analytically, experimentally, and through simulation [32][33][34][35]. That said, the probe tip defines a location that pins the drop, which means that robust sensing can occur by only measuring a single mode.…”