Coalescence of repelling colloidal droplets: a route to monodisperse populations.

Populations of droplets or particles dispersed in a liquid may evolve through Brownian collisions, aggregation, and coalescence. We have found a set of conditions under which these populations evolve spontaneously toward a narrow size distribution. The experimental system consists of poly(methyl methacrylate) (PMMA) nanodroplets dispersed in a solvent (acetone) + nonsolvent (water) mixture. These droplets carry electrical charges, located on the ionic end groups of the macromolecules. We used time-resolved small angle X-ray scattering to determine their size distribution. We find that the droplets grow through coalescence events: the average radius (R) increases logarithmically with elapsed time while the relative width σR/(R) of the distribution decreases as the inverse square root of (R). We interpret this evolution as resulting from coalescence events that are hindered by ionic repulsions between droplets. We generalize this evolution through a simulation of the Smoluchowski kinetic equation, with a kernel that takes into account the interactions between droplets. In the case of vanishing or attractive interactions, all droplet encounters lead to coalescence. The corresponding kernel leads to the well-known "self-preserving" particle distribution of the coalescence process, where σR/(R) increases to a plateau value. However, for droplets that interact through long-range ionic repulsions, "large + small" droplet encounters are more successful at coalescence than "large + large" encounters. We show that the corresponding kernel leads to a particular scaling of the droplet-size distribution-known as the "second-scaling law" in the theory of critical phenomena, where σR/(R) decreases as 1/√(R) and becomes independent of the initial distribution. We argue that this scaling explains the narrow size distributions of colloidal dispersions that have been synthesized through aggregation processes.