Effective interactions and equilibrium configurations of colloidal particles on a sessile droplet

TL;DR

This paper analyzes the free energy landscape of two colloidal particles on a sessile droplet, deriving analytical and numerical solutions for their interactions and equilibrium configurations under deformation.

Contribution

It introduces an analytical method using electrostatic analogies for small deformations and a numerical approach for arbitrary contact angles to study particle interactions.

Findings

Analytical solution for small deformations at contact angle π/2.

Numerical calculations of tangential forces for arbitrary contact angles.

Identification of equilibrium configurations with minimal free energy.

Abstract

We study the free energy landscapes of a pair of submicron spherical particles floating at the surface of a sessile droplet. The particles are subjected to radial external forces resulting in a deformation of the droplet shape relative to the reference shape of a spherical cap. This deformation leads to tangential forces on the particles. For small deformations and for the contact angle $\theta_0$ at the substrate being equal to $\pi/2$, the corresponding linearized Young-Laplace equation is solved analytically. The solution is constructed by employing the method of images from electrostatics, where each of the particles plays the role of a capillary monopole and the substrate is replaced by a virtual drop with image charges and by imposing the conditions of fixed droplet volume and vanishing total force on the droplet. The substrate boundary conditions determine the signs of the image capillary charges and therefore also the strength of the tangential forces on the particles. In the cases of an arbitrary contact angle $\theta_0$ these forces are calculated numerically by employing a finite element method to find the equilibrium shape of the droplet for those configurations in which the particles are close to the local free energy minima.