Scaling description for the growth of condensation patterns on surfaces

Abstract

The growth of a droplet pattern condensing on a surface (e.g., breath figures) is analyzed for the case of arbitrary dimensionality. The growth of a droplet occurs through a two-step process: (i) continuous growth by condensation on its surface and/or on the substrate and (ii) coalescence with neighboring droplets. As long as the one-droplet growth remains scale invariant, the growth of the entire pattern is shown to be self-similar in time and very general growth laws can be deduced, whose exponents are functions only of the dimensionality of the space of the droplets, and of the condensing substrate. In particular, for the case of three-dimensional droplets condensing on a plane substrate (breath figures), the growth exponent of a single droplet should be (1/3, whereas the exponent corresponding to a mean droplet, averaged over the pattern, should be unity. Comparison with experiment is performed and possible deviations from the predictions are considered.