# Would a cup of a superhydrophobic material ever fill with water?

This is a thought experiment inspired by the Lotus Effect: some surfaces with patterned hydrophobic pillars have a contact angle for droplets which is pretty close to 180 degrees.

Imagine a tiny cup made of extremely hydrophobic material. Initially this cup is immersed in liquid water, but filled with nothing (ie vacuum). As a simplification, optionally assume a material which is perfectly hydrophobic, ie which has no electrostatic interactions with water, and also no London dispersion interactions, only (very short range) repulsive interactions. (If you don't need the simplification, think of the cup as made of a real material like PTFE instead). The cup is also quite tiny (a few micrometers to nanometers in diameter), to make it similar to a single space in between the pillars of a lotus leaf. Does the cup ever fill with water?

Trying to work this out:

1. Surface tension tries to minimize the total surface area of the water interface. Filling the cup with water creates a larger surface area of the water (corresponding to the walls) which would be energetically unfavorable; because our assumed cup is made of material with (almost) no attractive interactions with water, there is nothing to counter surface tension. The water would assume whatever shape minimizes area, in other words a flat interface accross the opening of the cup.

2. Pressure: Clearly as pressure increases, at some external pressure the cup would fill. The pressure would be whatever makes the work of pressure x volume to move the interface inwards (into the cup) more than the work of surface tension x incremental area (to create an interface with the walls). Since one side of this depends on volume, and the other on area, the results would depend on the size of the cup, with a threshold diameter below which the cup doesn't fill at all, and above which it fills completely, for a given pressure.

3. Evaporation: initially the cup is filled with vacuum. It would quickly equilibrate to water vapor instead, at the vapor pressure corresponding to the temperature of the system. This would work against the external pressure, but it would (at typical temperatures and pressures) be a small contribution.

4. Condensation: spontaneous clusters of water molecules in vapor can form into tiny droplets inside the cup, but (since the walls are non-attractive) there is nothing which would cause clusters to form preferrentially on the walls (as in the "fogged glass" effect). The clusters would exist in equilibrium with the vapor, but (I'm not sure how to reason about this part) the equilibrium would be such that clusters don't continue to grow into a single large drop which fills the cup.

5. Dissolved gases: if the water has any gases dissolved in it (eg oxygen and nitrogen from air) they would also equilibrate so their partial pressure in the cup is equal to the partial pressure in whatever gas volume the water is in equilibrium with. If the water is equilibrated with the atmosphere, the cup would end up full of air at 1 atmosphere (which counters external pressure).

The part that is perhaps least obvious to me is what would prevent droplets from growing spontaneously from the (saturated) vapor. Of course if they ever touch the bulk liquid interface they would merge with it due to surface tension, and the interface would flatten, but the growing droplets would seem to be able to fill the whole cup just before that. Droplet growth and shrinkage would be random (assuming the system is at thermal equilibrium), and also droplets can spontaneously aggregate or split, so the size of the droplets is a random walk. Very small droplets should stay suspended indefinitely through brownian motion; saturated vapor is probably a stable distribution of different size clusters as well as single molecules.

In conclusion: this suggests that void spaces in water (either filled with water vapor or water vapor + dissolved gases) can be stable indefinitely at equilibrium, if they are in a sufficiently small concave volume, and the applied water pressure (eg hydrostatic) is sufficiently small.

• I don't understand the "close" votes; I see nothing but mainstream physics being described, and the question invites a mainstream physics answer. Commented Jul 25 at 23:22
• @Chemomechanics Thank you!! I don't understand it either. I asked on meta physics.meta.stackexchange.com/questions/14736/… I was about to make a few edits and hope for a vote to reopen ;) Commented Jul 26 at 4:23
• Thank you for this link. I think you've gotten some good feedback to make the question even better suited for this site (although I think it was already well suited). Commented Jul 26 at 4:41
• @Chemomechanics Edits done! Please have a look and let me know if there's anything else that can be improved Commented Jul 26 at 4:56
• The question is now reopened. Commented Jul 26 at 5:09