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 $\zeta$ 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 $\zeta$ and the surface nanobubble's footprint lateral extension $L$. It also predicts an upper lateral extension threshold for stable surface nanobubbles to exist.

Figure 1

(a) Sketch of a spherical cap-shaped surface nanobubbles and definition of the parameters describing the surface nanobubble's geometry, namely the nanobubble's footprint lateral extension $L$ (or its footprint radius $L/2$), its height $H$, its contact angle $\theta$ on the gas side, and its radius of curvature $R=(L^2+4H^2)/\left(8H\right)=L/(2sin\theta )$. (b) Sketch of the shrinking process of a pinned surface nanobubbles with initial contact angle ${\theta }_i$: Due to the pinning, $L$ is fixed. Shrinking thus implies a decrease in $\theta$ and $H$ and an increase in the radius of curvature. Therefore, the Laplace pressure inside the bubble reduces and eventually becomes too weak to further press out the bubble against the oversaturation. The contact angle ${\theta }_e<{\theta }_i$ of this stable equilibrium state depends on $\zeta$ and $L$. Surface nanobubbles with initially ${\theta }_i<{\theta }_e$ will grow towards ${\theta }_e$.

Figure 2

(a) Sketch of the stable equilibrium ${\theta }_e$ for pinned surface nanobubbles in the $\dot{\theta }$ vs $\theta$ phase space, cf. Eq. (10). A sketch of the bubble height phase space $\dot{H}$ vs $H$ looks the same. (b) Sketch of the unstable equilibrium $L_e$ for unpinned surface nanobubbles with constant contact angle in the $\dot{L}$ vs $L$ phase space, cf. Eq. (15). For bulk bubbles, the sketch looks the same, with the lateral extension $L$ then meaning the bubble diameter.

Figure 3

Equilibrium contact angle ${\theta }_e$ of a nitrogen bubble in water as function of the lateral footprint size $L$ for four different oversaturations $\zeta =0.25$, 0.5, 1.0, 2.0, bottom to top.

Figure 4

Time evolution of the contact angle for a single surface nanobubble for various initial contact angles ${\theta }_0=1^{\circ },5^{\circ },{10}^{\circ }$, and ${20}^{\circ }$. The material parameters are chosen for a nitrogen bubble: $D=2\times {10}^{-9}{\text{m}}^2/\text{s},c_s=0.017\text{kg}/{\mathrm{m}}^3,{\rho }_g=1.165\text{kg}/{\mathrm{m}}^3$. We picked the lateral extension of the pinning site as $L=100$ nm and a gas oversaturation of $\zeta =1$. Irrespective of the initial contact angle, the surface nanobubble grows towards the equilibrium contact angle ${\theta }_e=arcsin(\zeta L/L_c)=2.{02}^{\circ }$.