The Importance of Boundary Conditions for Fluctuation Induced Forces between Colloids at Interfaces

TL;DR

This paper investigates how boundary conditions at the contact line influence fluctuation-induced Casimir forces between colloids at fluid interfaces, revealing sensitive dependence and cancellations of leading terms.

Contribution

It introduces a functional integral approach to calculate fluctuation forces, emphasizing the role of boundary conditions and multipole expansion in understanding Casimir interactions at interfaces.

Findings

Force depends on boundary conditions at the contact line

Cancellations occur between repulsive and attractive contributions

Multipole expansion explains boundary condition effects

Abstract

We calculate the effective fluctuation induced force between spherical or disk-like colloids trapped at a flat, fluid interface mediated by thermally excited capillary waves. This Casimir type force is determined by the partition function of the system which in turn is calculated in a functional integral approach, where the restrictions on the capillary waves imposed by the colloids are incorporated by auxiliary fields. In the long-range regime the fluctuation induced force is shown to depend sensitively on the boundary conditions imposed at the three-phase contact line between the colloids and the two fluid phases. The splitting of the fluctuating capillary wave field into a mean-field and a fluctuation part leads to competing repulsive and attractive contributions, respectively, which give rise to cancellations of the leading terms. In a second approach based on multipole expansion of the Casimir interaction, these cancellations can be understood from the vanishing of certain multipole moments enforced by the boundary conditions. We also discuss the connection of the different types of boundary conditions to certain external fields acting on the colloids which appear to be realizable by experimental techniques such as the laser tweezer method.