Pinning and gas oversaturation imply stable single surface nanobubbles

Surface nanobubbles are experimentally known to survive for days at hydrophobic surfaces immersed in gas-oversaturated water. This is different from bulk nanobubbles, which are pressed out by the Laplace pressure against any gas oversaturation and dissolve in submilliseconds, as derived by Epstein and Plesset [J. Chem. Phys. 18, 1505 (1950)]. Pinning of the contact line has been speculated to be the reason for the stability of the surface nanobubbles. Building on an exact result by Popov [Phys. Rev. E 71, 036313 (2005)] on coffee stain evaporation, here we confirm this speculation by an exact calculation for single surface nanobubbles. It is based only on (i) the diffusion equation, (ii) Laplace pressure, and (iii) Henry's equation, i.e., fluid dynamical equations which are all known to be valid down to the nanometer scale. The crucial parameter is the gas oversaturation ζ of the liquid. At the stable equilibrium, the gas overpressures due to this oversaturation and the Laplace pressure balance. The theory predicts how the contact angle of the pinned bubble depends on ζ and the surface nanobubble's footprint lateral extension L. It also predicts an upper lateral extension threshold for stable surface nanobubbles to exist.