Dynamic equilibrium explanation for nanobubbles' unusual temperature and saturation dependence

The dynamic equilibrium model suggests that surface nanobubbles can be stable due to an influx of gas in the vicinity of the bubble contact line, driven by substrate hydrophobicity, that balances the outflux of gas from the bubble apex. Here, we develop an alternate formulation of this mechanism that predicts rich behavior in agreement with recent experimental measurements. Namely, we find that stable nanobubbles exist in narrow temperature and dissolved gas concentration ranges, that there is a maximum and minimum possible bubble size, and that nanobubble radii decrease with temperature.

Figure 1

(a) Schematic of the dynamic equilibrium mechanism for nanobubble stability. (b) Sum of influx and outflux, scaled by the concentration at the bubble surface [$j_{\mathrm{out}}=J_{\mathrm{out}}/C\left(R\right)$, $j_{\mathrm{in}}=J_{\mathrm{in}}/C\left(R\right)$], versus bubble radius for several different temperatures. The blue (solid) curves denote temperatures at which stable points are found.

Figure 2

(a) Stable bubble radius versus temperature according to dynamic equilibrium model, shown for different values of $A$ (dashed curves). (b) Stable bubble radius versus solute concentration for several different temperatures. Arrow indicates direction of increasing $T$.

Figure 3

Sum of influx and outflux versus both radius and contact angle for $T$ $=$ 27 °C. Points where the fluxes balance are shown by the solid curve. The dashed curve is Eq. (2), with ${\theta }_{\infty }$ $=$ 30° and $\delta$ $=$ 10 nm.