How small can an equilibrium colloidal cluster on a liquid-air interface be?

Abstract

A vertical force on a pair of spherical colloids trapped at a liquid-air interface induces their logarithmic pairwise attraction. However, in clusters of size R much larger than the capillary length λ, the many-body interaction dramatically changes the capillary attraction: it is much stronger and no longer logarithmic [V. M. Pergamenshchik, Phys. Rev. E 79, 011407 (2009)]. In this paper, this theory is shown to describe also small clusters with R ≪ λ if they consist of sufficiently large numbers of colloids. The analytical solution for a small circular cluster with a short range powerlaw repulsion with arbitrary exponent n is found. Irrespective of n, the attraction force and the energy scale with the distance L between colloids as L−3 and L−2 for clusters of any size. We estimate the minimum size of a cluster in which the predicted many-body capillary attraction can prevent evaporation of colloids due to their thermal motion. For standard parameters, the minimum number of colloids in a stable nonevaporating cluster can be as low as 20–200. These numbers are in accord with the experimental findings on reversible clusterization on a liquid-air interface.