Strong collective attraction in colloidal clusters on a liquid-air interface

It is shown that in a cluster of many colloids, trapped at a liquid-air interface, the well-known vertical-force-induced pairwise logarithmic attraction changes to a strongly enhanced power-law attraction. In large two-dimensional clusters, the attraction energy scales as the inverse square of the distance between colloids. The enhancement is given by the ratio $\eta =\left(\text{square}\text{of}\text{the}\text{capillary}\text{length}\right)$/(interface surface area per colloid) and can be as large as ${10}^5$. This explains why a very small vertical force on colloids, which is too weak to bring two of them together, can stabilize many-body structures on a liquid-air interface. The profile of a cluster is shown to consist of a large slow collective envelope modulated by a fast low-amplitude perturbation due to individual colloids. A closed equation for the slow envelope, which incorporates an arbitrary power-law repulsion between colloids, is derived. For example, this equation is solved for a large circular cluster with the hard-core colloid repulsion. It is suggested that the predicted effect is responsible for mysterious stabilization of colloidal structures observed in experiments on a surface of isotropic liquid and nematic liquid crystal.

Figure 1

(Color online) (a) Cluster of spherical colloids on a liquid-air interface, its exact height $h$ and slow envelope $H$ induced by the force $f$ directed upward. The magnitudes of $H$, $h$, and $\tilde{h}=h-H$ are greatly exaggerated: while the colloid diameter and the separation $L$ are of several micrometers, $H$ can be as large as a micrometer, whereas the fast modulation $\tilde{h}$ is just about a nanometer. (b) Single cell of the interface around the colloid at $X$ and its projection $S_1$ onto the reference plane $S$. (c) Top view on a hexagonal cluster of 13 colloids; the dashed line shows the boundary $\partial S_1$ of the single elementary cell $S_1$.