Stable Laws in Breath Figures on a One-Dimensional Substrate

Abstract

Breath figures are ensembles of droplets formed by condensation of water vapor on cold non-wetting surfaces. Mist on a window is an everyday example of this phenomenon. The formation of breath figures involves two mechanisms: growth of individual droplets by continual condensation of water and coalescences of pairs of droplets, when they touch, into single droplets. Extensive studies of breath figures^1,3 show that the radius of a droplet grows as a power law in time between coalescences

$${r_i}(t) \approx {t^\alpha }$$

(1)

where the exponent α is governed by the mode of condensation of water vapor into droplets. When two droplets touch, they coalesce very rapidly into one droplet. The new droplet is centered on the center of mass of it’s two parent droplets and assuming that the form of a droplet does not depend highly on its radius, the radius of the new droplet is given by conservation of mass

$${r_{1 + 2}} = {\left( {{r_1}^3 + {r_2}^3} \right)^{1/3}}$$

(2)

at the beginning of such an experiment initial droplets nucleate on the substrate. They are so small that there are hardly any coalescences observed. When the droplets grow to radii of the order of the separation between them, many coalescences are then observed. In this regime, which persists for a long time, the ensemble of droplets seems self similar in time. The distribution of droplet radii is scaling in time. The coverage of the substrate, i. e., the fraction of it covered by droplets, which is null in the begining, attains a constant value in time, hinting that the whole breath figure is indeed self-similar in time.