PHYSICAL REVIEW 8 VOLUME 42, NUMBER 1 1 JULY 1990 Two-dimensional ordering during droplet growth on a liquid surface A. Steyer, P. Guenoun, and D. Beysens Service de Physique du Solide et de Resonance Magnetique, Centre d'Etude Nucleaire de Saclay, F-9l 1 91 Gif su-r Yv-etre CEDEX, France C. M. Knobler Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90024 (Received 3 January 1990) In contrast to the breath-figure patterns that arise when ~ater condenses on solid surfaces, the condensation and growth of droplets of water on paraSn oil can produce distinct two-dimensional structures. These occur because, in contrast to droplets on a solid surface, droplets of an immisci- ble Auid on the surface of a liquid can interact by mechanisms other than coalescence. An exper- imental investigation of the translational and orientational order in the patterns enables a connec- tion to be made between their morphology and growth. I. INTRODUCTION The morphology of growing domains can provide clues to the nature of the processes by which they grow. We present here a study of the relation between the growth and the structure of the patterns of droplets that form when water condenses on the surface of paraffin oil, a liquid in which it is not soluble. Such two-dimensional droplet condensation patterns have been called breath figures (BF), and extensive measurements of the patterns that occur when fluids condense on solid surfaces have been carried out. ' We focus here on the BF that form on liquid surfaces; the interactions between droplets in this case are more complex than those when the growth occurs on a solid and more complex morphologies may be expect- ed. Our principal finding is that an ordered droplet struc- ture develops during growth that has the appearance of a hexagonal two-dimensional crystal with many defects. A close analogy can be made between the growth of BF on liquids and two-dimensional solidification. In both cases the system evolves from disorder to order continuously, which suggests that a hexatic phase3 might be an inter- mediate in the process. We begin by briefly reviewing the main characteristics of the growth of BF. We will then describe the evolution of the structures and characterize them; their topology will be correlated with the different stages of growth. the rate of droplet growth. (iii) Coalescence-dominated stage. Here the surface coverage is high and constant. Its value is about 0.55, that found on solid substrates. In this stage, droplet coales- cence markedly accelerates the growth and continuously rescales the pattern. The pattern remains self-similar and the average droplet radius grows as (R) ~ t. The differences between droplet growth on solids and liquids stem from differences in the nature of the interac- tions between the droplets. On a solid surface, the drop- lets have the form of sections of a sphere (they are essen- tially hemispherical on silanized glass). Macroscopic droplets move only because of coalescences. There can therefore be no rearrangement to increase the surface cov- I I i ] I I I I coord. :- 0. 2— II. GROWTH I. AliV FOR BF The growth of BF on liquids is found2 to evolve through several stages similar to those observed on solids and found in simulations. These characteristics are delineat- ed in Figs. 1 and 2 and described below. (i) Initial stage. Here the droplets are isolated and do not interact strongly. The surface coverage (the fraction of the oil surface occupied by the droplets) is low and the average droplet radius (R) varies as (R) a: t 't . (ii) Crossover stage. This stage of growth is character- ized by a high surface coverage and a marked increase in (c 0 l I i I l 10 t I i l i iiil t (s) FIG. 1. Time evolution of (a) the apparent surface coverage e; (b) the distribution P, of polygons with coordination num- bers n 5, 6, 7; (c) the entropy S of configuration per drop. 1086 1990 The American Physical Society