Why Surface Nanobubbles Live for Hours

We present a theoretical model for the experimentally found but counterintuitive exceptionally long lifetime of surface nanobubbles. We can explain why, under normal experimental conditions, surface nanobubbles are stable for many hours or even up to days rather than the expected microseconds. The limited gas diffusion through the water in the far field, the cooperative effect of nanobubble clusters, and the pinned contact line of the nanobubbles lead to the slow dissolution rate.

Figure 1

Sketch of a liquid layer of thickness $\ell$ in contact with a solid (left). The top of the liquid is exposed to atmospheric conditions. At the solid-liquid interface nanobubbles are present with typical internal contact angle $\theta$, height $h$, and radius of curvature $R$. They are not drawn to scale. The arrows indicate the gas flow direction. On the right a further enlargement of one nanobubble is shown.

Figure 2

Results of the calculations using the initial conditions from Eq. (5) and parameters described in the text. (a) Snapshots of the concentration profile at 1 h intervals. Because the bubbles are pinned, the radius of the curvature increases as the bubble drains, lowering the concentration at the bubble side ($z=0$). (b) Evolution of the nanobubble gas content (black curve) and the radius of curvature of the liquid-gas interface $R$ [red (gray) curve] as a function of time.

Figure 3

Numerical results for the initial conditions [Eq. (6)]. (a) Hourly snapshots of the concentration profile. Due to the initial conditions, some gas that is initially dissolved in the liquid first has to drain to the atmosphere until a concentration gradient exists near the bubble. As compared to the linear profile case (Fig. 2) the bubbles gain an additional few hours of lifetime because of this. (b) Evolution of the nanobubble gas content and the radius of curvature of the surface nanobubbles.