Free energy of colloidal particles at the surface of sessile drops

TL;DR

This paper investigates how finite system size affects the free energy of a colloidal particle on a sessile droplet surface, using analytical and numerical methods including a capillary image technique.

Contribution

It introduces an analytical approach based on Green's functions and capillary images to compute free energy variations for particles on sessile droplets, considering boundary conditions and volume constraints.

Findings

Particle free energy can be positive or negative depending on boundary conditions.

Identifies stable positions of particles at the drop apex or intermediate angles.

Reveals non-monotonic Green's function behavior affecting particle interactions.

Abstract

The influence of finite system size on the free energy of a spherical particle floating at the surface of a sessile droplet is studied both analytically and numerically. In the special case that the contact angle at the substrate equals $\pi/2$ a capillary analogue of the method of images is applied in order to calculate small deformations of the droplet shape if an external force is applied to the particle. The type of boundary conditions for the droplet shape at the substrate determines the sign of the capillary monopole associated with the image particle. Therefore, the free energy of the particle, which is proportional to the interaction energy of the original particle with its image, can be of either sign, too. The analytic solutions, given by the Green's function of the capillary equation, are constructed such that the condition of the forces acting on the droplet being balanced and of the volume constraint are fulfilled. Besides the known phenomena of attraction of a particle to a free contact line and repulsion from a pinned one, we observe a local free energy minimum for the particle being located at the drop apex or at an intermediate angle, respectively. This peculiarity can be traced back to a non-monotonic behavior of the Green's function, which reflects the interplay between the deformations of the droplet shape and the volume constraint.