Super lattice formation of an array of volatile wetting droplets

For an ordered array of critical volatile wetting droplets the formation of a super lattice by an Ostwald-ripening like competition process is considered. The underlying diffusion problem is treated within a quasistatic approximation and to first order in the inverse droplets distance. The approach is rather general but a square lattice and a triangular lattice are studied explicitly. Dispersion relations for the super-lattice growth of these arrays are calculated. In a recent experiment C. Schäfle et al [1] nucleated diethylene glycol droplets out of a supersaturated vapor phase on a hexagonal array of hydrophilic (alkanethiol) patches on a hydrophobic substrate. The pressure of the system was reduced afterwards so that the droplets started to shrink by evaporation. During this process the hexagonal lattice split into two triangular super lattices in a sense that the droplets on one super lattice shrank faster than those on the other super lattice. The super lattice of faster shrinking droplets disappeared, finally. The remaining triangular droplet lattice appeared to be stable against further development of super structures. The appearance of the triangular super lattice from the hexagonal lattice is interpreted as an Ostwald-ripening type process. The system reduces its free energy by having few large droplets instead of many small [2, 3, 4]. Similar observations where made by the same authors [1] as well as by Lacasta et al [5] simulating CahnHilliard dynamics of droplet arrays in a concentration field. The study of super lattice growth on an array of wetting droplets is not only of fundamental interest as an example of pattern formation and the dynamics of wetting on structured surfaces, but it is also important to learn about the features of volatile droplet arrays since they can be used to build e.g. chemical sensors (see [6, 7]). The scope of this paper is the analytic discussion of the diffusional growth properties of a set of wetting droplets sitting on hydrophilic patches on a hydrophobic substrate. We are considering circular patches of equal radius a. Square and triangular lattice configurations of the patches will be studied explicitly. It is assumed that there is a wetting droplet sitting in the center of each patch. The different possible equilibrium configurations and phase transitions in such a system were discussed in great detail by Lenz and Lipowsky only recently [8, 9]. To discuss the diffusional growth of a wetting droplet out of the surrounding gas phase, one has to specify the boundary conditions of the diffusion equation on the substrate and on the droplets surface. Since there is no diffusion flux into the substrate, the normal derivative of the concentration field vanishes there. The concentration cs in the gas phase on top of the droplet is given by the Gibbs-Thomson relation. It is a constant along the droplets surface and is just a function of the radius of curvature R [4, 10]