Stability, Dynamics, and Tolerance to Undersaturation of Surface Nanobubbles

The theoretical understanding of surface nanobubbles—nanoscale gaseous domains on immersed substrates—revolves around two contrasting perspectives. One perspective, which considers gas transport in the nanobubbles’ vicinity, explains numerous stability-related properties but systematically underestimates the dynamical response timescale by orders of magnitude. The other perspective, which considers gas transport as the bulk liquid equilibrates with the external environment, recovers the experimentally observed dynamical timescale but incorrectly predicts that nanobubbles progressively shrink until dissolution. We propose a model that couples both perspectives, which is capable of explaining the stability, dynamics, and unexpected tolerance of surface nanobubbles to undersaturated environments.

Figure 1

Two perspectives of nanobubble stability. (a) The global perspective considers transport from bulk liquid to the external world through a semi-infinite pool of thickness $\ell$. (b) The local perspective focuses on gas transport surrounding a nanobubble. We assume that the substrate possesses a hydrophobic attraction $\varphi$ with interaction distance $\lambda$; note that $c(z\gg \lambda )=c_0$. (c) Numerical solution of the global problem [Eqs. (1)–(3)], for $\ell =1\mathrm{mm}$ and $c_i=c_{\mathrm{sat}}$. Concentration profiles $c(z)$ are captured every 60 s, from top to bottom. (d) Temporal evolution of $c_0/c_{\mathrm{sat}}$, the leftmost point of each curve in (c). (e) For common potentials such as van der Waals (green line) or the short-range hydrophobic attraction (red line), $\lambda \sim 1\mathrm{nm}$. (f) The dynamics of a gas nanobubble with $L=100\mathrm{nm}$ under atmospheric conditions ($c_0/c_{\mathrm{sat}}=1$ or $\zeta =0$) with short-range hydrophobic attraction ${\varphi }_0/k_{B}T=-1$ (red line), van der Waals (green line), or zero potential (Lohse-Zhang model [10], dashed line). The gas concentration in the far field is exactly saturated $c_0=c_{\mathrm{sat}}$. Note the substantial (8 orders’) difference in timescales between (d) and (f).

Figure 2

The long-time dynamics of nanobubbles under equilibration and undersaturation as predicted by the combined model. We assume the short-ranged hydrophobic potential ${\varphi }_h$, for ${\varphi }_0/k_{B}T=-2$, $\lambda =1\mathrm{nm}$, and $L=200\mathrm{nm}$. (a) Equilibration to atmospheric conditions is investigated with the liquids initially supersaturated to various degrees $1<c_0/c_{\mathrm{sat}}<3$. (b) The nanobubbles’ response to undersaturation is investigated with the liquids at a common supersaturation of $c_0/c_{\mathrm{sat}}=2$ and compelling the liquid to relax to various degrees of undersaturation $0<c_i/c_{\mathrm{sat}}<1$. In all cases, except the bottommost curves in (b), the nanobubble attains an equilibrium contact angle after ${\tau }_G$ and remains there indefinitely.

Figure 3

How surface nanobubbles ($L=200\mathrm{nm}$) resist undersaturated conditions. (a) The equilibrium angle of a nanobubble as a function of the dissolved gas concentration coupled from the external environment, $c_i/c_{\mathrm{sat}}$, for different interactions: short-range hydrophobic attraction ${\varphi }_h$ for ${\varphi }_0/k_{B}T=0,-1,-2$ and van der Waals ${\varphi }_v$. The tolerance to undersaturation, $c_i^{*}$, is defined as the largest $c_i$ for which the equilibrium angle ${\theta }_e=0°$; $c_i^{*}$ for each potential is indicated by arrows. (b) Systematically adjusting the strength of ${\varphi }_h$ resolves the development of tolerance as a function of hydrophobicity (${\varphi }_0<0$ for a hydrophobic substrate). Nanobubbles immediately dissolve under any degree of undersaturation for zero potential (Lohse-Zhang theory) or hydrophilic potentials.