# Contact line depinning from sharp edges

## TL;DR

This paper derives mathematical criteria for contact line depinning from sharp edges by analyzing equilibrium and stability of a liquid layer at a wedge vertex, identifying critical angles where depinning occurs.

## Contribution

It introduces a stability analysis based on variational methods to determine critical contact angles for depinning from sharp corners.

## Key findings

- Equilibrium contact angles are bracketed by the wedge's flank contact angles.
- Stability is lost at critical depinning angles via saddle-node bifurcations.
- Contact line depinning occurs beyond these critical angles.

## Abstract

With aim of finding mathematical criteria for contact line depinning from sharp corners, we have studied the equilibrium and stability of a semi-infinite planar liquid layer pinned at the vertex of a wedge. Equilibrium is compatible with a fan of apparent contact angles $\theta_0$ bracketed by the equilibrium contact angles of both flanks of the wedge, so the contact line could remain pinned if $\theta_0$ is within this fan. However, the analysis of the stability of these solutions, studied exploiting the variational structure of the problem through turning-point arguments, shows that the equilibrium becomes unstable at critical depinning advancing $\theta_0^a$ and receding $\theta_0^r$ contact angles, which are found as subcritical saddle-node bifurcations. Equilibrium is thus possible (stable) within the interval $\theta_0^a < \theta_0 <\theta_0^a$ but the contact line depins from the vertex beyond these critical angles if they occur within the equilibrium fan.