Growth of droplets on a substrate by diffusion and coalescence

An analytical and numerical study of the early stages condensation of three-dimensional (3D) droplets onto a partially wetting 2D substrate is presented. We show that when surface coverage is low, a mechanism involving Brownian diffusion of the droplets, leading to interaction through coalescences, can explain a number of experimental features (e.g., in breath-figure experiments). First, a motionless drop that incorporates diffusing droplets can asymptotically grow as ${\mathit{t}}^{1/3}$. In the second situation, we consider a constant flux of monomers diffusing on the substrate. These monomers coalesce and form bigger drops with mass conservation. These drops diffuse in turn with a lower diffusion coefficient. A numerical simulation of this process shows a rapid formation of a well-defined, monodisperse family of droplets. A scaling analysis of the simulation is able to predict a number of growth exponents. These laws are finally compared with recent experiments and their relevance is discussed.