Catenoid stability with a free contact line

TL;DR

This paper analytically investigates the stability of liquid catenoids with a free contact line, providing criteria and phase diagrams that describe their equilibrium and stability properties relevant to nanometer-scale surface patterning.

Contribution

It introduces a variational approach to determine stability criteria for catenoids with a free contact line, including phase diagrams and analysis of equilibrium multiplicity.

Findings

All catenoids are stable against non-axisymmetric perturbations.

A critical volume exists below which catenoids become unstable to axisymmetric perturbations.

The stability regions are characterized by phase diagrams reflecting geometrical symmetry.

Abstract

Contact-drop dispensing is central to many small-scale applications, such as direct-scanning probe lithography and micromachined fountain-pen techniques. Accurate and controllable dispensing required for nanometer-resolved surface patterning hinges on the stability and breakup of liquid bridges. Here, we analytically study the stability of catenoids pinned at one contact line with the other free to move on a substrate subject to axisymmetric and non-axisymmetric perturbations. We apply a variational formulation to derive the corresponding stability criteria. The maximal stability region and stability region are represented in the favourable and canonical phase diagrams, providing a complete description of catenoid equilibrium and stability. All catenoids are stable with respect to non-axisymmetric perturbations. For a fixed contact angle, there exists a critical volume below which catenoids are unstable to axisymmetric perturbations. Equilibrium solution multiplicity is discussed in detail, and we elucidate how geometrical symmetry is reflected in the maximal stability and stability regions.