Self-Organized Drops on the Surface of Liquids
Anyone who has ever painted a ceiling and used too much paint or too thin a paint knows the rub: The paint gathers at certain arbitrary points, becoming thicker and thicker until finally the dye ruptures and drains. Physicists calls this behavior an "instability".
What is the reason for this instability? And why is the flat paint layer unstable? Ideally, the dye layer is perfectly flat. In real life, however, there is always some albeit very small roughness, called disturbances. In the course of time these disturbances can either shrink or grow. The first situation belongs to a stable state of the flat, undisturbed surface, the latter to an unstable one. In our case we have an unstable state. For any given small disturbance where the layer is locally a little bit thicker, gravity pulls the free (under) surface of the paint a little bit downwards. Thereby dye flows in from the neighboring regions. And the thickness at position of the disturbance increases. As a consequence, gravity pulls even harder increasing the surface inhomogeneity further until the dye separates or ruptures and drains.
In the laboratory, one can perform a similar experiment using any liquid instead of paint. One can avoid drain and rupture by stabilizing an extremely thin fluid layer, for example, by applying a vertical thermal gradient by cooling from the top (the ceiling). The (under)surface of a very thin part of the layer is then closer to the ceiling and therefore colder. Since surface tension decreases with increasing temperature, surface tension is larger at thin parts and liquid is pumped in from the neighbourhood, avoiding a completely dry region and rupture. Without rupture, small droplets are created at the surface in the beginning. In the course of time these droplets merge and increase in size, an effect which is named "coarsening" (fig.1).
A similar process can be observed in the kitchen if oil and vinegar is mixed. At first oil is distributed in the form of many small bubbles in the vinegar. By and by, the oil droplets merge, and in the ideal case it remains only one large region with oil. This transition occurs because it minimizes the total size of the interface between oil and vinegar, leading to an energetically preferred state.
Our computations have shown that the dynamics of pattern formation changes completely if evaporation and condensation play a significant role. If the experiment is repeated with a volatile liquid, i.e. a liquid with a large evaporation rate, and if the ambient pressure and temperature have values such that evaporation and condensation is in balance if the film is flat, we predict the following scenario: For a disturbance where the layer is a little bit thicker than the equilibrium height, the surface is further from the cooling top and has a temperature that is a little higher than the equilibrium value. As a consequence, liquid evaporates there and the disturbance is reduced. On the contrary, surface segments over thinner regions are a bit colder, gas condenses and the liquid’s thickness is growing.
In this way evaporation/condensation acts stabilizing at the equilibrium. On the other hand, we found that coarsening and the formation of large drops and areas are now avoided. How come? The evaporation rate of a drop and its mass are both proportional to its surface area, for circular drops proportional to the square of the drop’s radius. Therefore large as well as small droplets evaporate with the same speed. But smaller ones may compensate their mass loss much better since the mass inflow from the neighboring regions is proportional to the size of the drop’s boundary and therefore proportional to the drop’s radius. Thus the ratio of evaporating mass and inflow increases with the radius of the drop and is smaller for small drops – large drops are destroyed faster than small ones. Very small droplets on the other hand are unfavorable because they need too much energy to be stored inside the deformed surface. Thus an optimal size for droplets exists which can be computed for each liquid. After a certain time, the surface will be covered with drops of this optimal size, and that in a very regular manner (fig.2).
As already known from crystal lattices of solids, there are several basic types of filling patterns, but in our case on a macroscopic scale of some millimeters. On a plane these are stripes, squares, and hexagons.
In conclusion we see that evaporation is responsible for two new important effects:
- Stabilization of a flat surface (equilibrium height) and prevention of rupture
- Selection of an optimal drop size instead of coarsening and formation of a very regular surface structure.
Our work is based on theoretical and numerical methods and its results can be considered as predictions. It remains a challenge for the experimentalist to verify (or falsify) it.