Film deposition of a self-propelled droplet on a cone with slip

TL;DR

This study investigates the dynamics of a self-propelled viscous droplet on a conical substrate, revealing how cone geometry and slip length influence droplet motion and film formation, with results validated by molecular dynamics simulations.

Contribution

It introduces a comprehensive lubrication model for droplet motion on cones, deriving new scaling laws for film thickness and validating them against molecular dynamics simulations.

Findings

Droplet velocity increases with cone angle and slip length.

A film forms at the receding edge, with thickness decreasing as slip length increases.

Two distinct scaling laws describe the film thickness in different slip regimes.

Abstract

We study the dynamic wetting of a self-propelled viscous droplet using the time-dependent lubrication equation on a conical-shaped substrate for different cone radii, cone angles and slip lengths. The droplet velocity is found to increase with the cone angle and the slip length, but decrease with the cone radius. We show that a film is formed at the receding part of the droplet, much like the classical Landau-Levich-Derjaguin (LLD) film. The film thickness $h_f$ is found to decrease with the slip length $\lambda$. By using the approach of matching asymptotic profiles in the film region and the quasi-static droplet, we obtain the same film thickness as the results from the lubrication approach for all slip lengths. We identify two scaling laws for the asymptotic regimes: $h_fh''_o \sim Ca^{2/3}$ for $\lambda\ll h_f$ and $h_f h''^{3}_o\sim (Ca/\lambda)^2$ for $\lambda\gg h_f$, here $1/h''_o$ is a characteristic length at the receding contact line and $Ca$ is the capillary number. We compare the position and the shape of the droplet predicted from our continuum theory with molecular dynamics simulations, which are in close agreement. Our results show that manipulating the droplet size, the cone angle and the slip length provides different schemes for guiding droplet motion and coating the substrate with a film.