(363f) Shape and Roll-Off Angle of a Pinned Droplet On a Progressively Inclined Plate

Droplets hanging on a window are a common sight. A long standing problem is to predict the maximum size of a droplet that can hang on a window without rolling down. In this work we contribute to the solution of this problem for the model system of a droplet on an incline. We construct a model that allows us to a priori predict how a droplet deforms with increasing inclination angle and for what angle it rolls off.

The general approach to model this roll-off angle is to resolve the force balance between gravitational  and pinning forces: Bo sin α=kr(cos θ_r − cos θ_a). In this description, k is currently predicted to be a constant in contrast to recent experimental and numerical findings [1, 2]. In our model we take a more fundamental approach to determine the pinning force by predicting the shape of the droplet base and the contact angle distribution around the base. From this approach we predict the variable k. We find that it depends on the initial contact angle of the droplet and, surprisingly, also on the value of the Bond number and contact angle hysteresis (cos θ_r − cos θ_a) itself. Additionally, the model allows us to understand more generally how a pinned droplet will behave under given forcing.

[1] V. Berejnov and R. E. Thorne. Effect of transient pinning on stability of drops sitting on an inclined plane. Phys. Rev. E, 75:066308, 2007.
[2] M. J. Santos, S. Velasco, and J. A. White. Simulation analysis of contact angles and retention forces of liquid drops on inclined surfaces. Langmuir, 28(32):11819–11826, 2012.