Reply to the Comments on "Curvature capillary migration of microspheres" by P. Galatola and A. Wurger

TL;DR

This paper clarifies the capillary energy of microparticles on curved fluid interfaces, correcting previous literature errors through experiments and simulations, and discusses the differences between pinned and equilibrium contact lines.

Contribution

It provides a new expression for capillary energy considering both pinned and equilibrium contact lines, correcting prior literature and supporting findings with experiments and simulations.

Findings

Derived a new expression for capillary energy on curved interfaces.

Identified and corrected conceptual errors in previous comments.

Supported the new model with experimental and simulation evidence.

Abstract

We have studied microparticle migration on curved fluid interfaces in experiment and derived an expression for the associated capillary energy $E$ for two cases, i.e., pinned contact lines and equilibrium contact lines, which differ from expressions derived by others in the literature. In this problem, a particle of radius $a$ makes a disturbance in a large domain characterized by principal radii of curvature $R_1$ and $R_2$. Since $a$ is smaller than all associated geometric and physico-chemical length scales, analysis calls for a singular perturbation approach. We recapitulate these concepts, identify conceptual errors in the Comments about our work, and provide evidence from experiment and simulation that supports our view.