Self-diffusion and ‘visited’ surface in the droplet condensation problem (breath figures)

We study experimentally and theoretically new aspects of condensation, growth and coalescence of droplets on a substrate. We address in particular the dynamics of a ‘marked’ droplet which undergoes coalescences with neighbouring droplets. We find that the number of coalescences of the droplet grows as log t (t is time) and that the traveled distance increases as 〈R(t)〉, the average radius of the droplets at time t. The fraction of the surface which was never covered by any droplet (an important quantity for applications, such as drug spreading or surface decontamination), decays as t−θ, showing that completion occurs only slowly. Heuristic arguments, accurate numerical simulations on simplified models (which neglect polydispersity) and an exact solution reported elsewhere [A. Bray, B. Derrida and C. Godréche, Europhys. Lett. 27 (1994) 175] strongly support these findings, and show that this power law decay is a generic feature, common to many different situations. Finally, the contour of the ensemble of ‘dry’ sites appears fractal with dimension dƒ ⋍ 1.22, an experimental result not reproduced by the simplified models.