Stability and minimum size of colloidal clusters on a liquid-air interface

A vertical force applied to each of two colloids, trapped at a liquid-air interface, induces their logarithmic pairwise attraction. I recently showed [Phys. Rev. E 79, 011407 (2009)] that in clusters of size $R$ much larger than the capillary length $\lambda$, the attraction changes to that of a power law and is much stronger due to a many-body effect, and I derived two equations that describe the equilibrium coarse-grained meniscus profile and colloid density in such clusters. In this paper, this theory is shown also to describe small clusters with $R\ll$ $\lambda$ provided the number $N$ of colloids therein is sufficiently large. An analytical solution for a small circular cluster with an arbitrary short-range power-law pairwise repulsion is found. The energy of a cluster is obtained as a function of its radius $R$ and colloid number $N$. As in large clusters, the attraction force and energy universally scale with the distance $L$ between colloids as $L^{-3}$ and $L^{-2}$, respectively, for any repulsion forces. The states of an equilibrium cluster, predicted by the theory, are shown to be stable with respect to small perturbations of the meniscus profile and colloid density. The minimum number of colloids in a circular cluster, which sustains the thermal motion, is estimated. For standard parameters, it can be very modest, e.g., in the range 20–200, which is in line with experimental findings on reversible clusterization on a liquid-air interface.