Energetics-based model for a diffusiophoretic motion of a deformable droplet

TL;DR

This paper develops a mathematical model describing the diffusiophoretic motion and deformation of a droplet on a liquid surface driven by surface tension gradients, analyzing stability and state transitions.

Contribution

It introduces a novel energetics-based model that accounts for droplet deformation and motion, explicitly deriving evolution equations and identifying stable states.

Findings

Identifies three stable states of droplet behavior.

Derives explicit equations for motion and deformation dynamics.

Discusses transitions between different stable states.

Abstract

We construct a mathematical model for a diffusiophoretic motion of a deformable droplet, which is floating on a liquid surface and is driven by the surface tension gradient originating from the surface concentration field of the chemicals that are emitted from the droplet. We define the free energy of the system by including the surface and line energies. From the calculation of the functional of the free energy, we obtain a mathematical model for the diffusiophoretic motion with deformation. By only considering the deformation of the second mode, we explicitly derive the time-evolution equations for the translational motion and the elliptic deformation. There are three stable states: an immobile circular droplet, an immobile elliptically deformed droplet, and a mobile droplet with the elliptic deformation in which the minor axis meets the motion direction, and we discuss the transition between these three stable states.