# Dynamic equilibrium Mechanism for Surface Nanobubble Stabilization

## Overview

**Authors:** Michael P. Brenner & Detlef Lohse

**Source:** arXiv:0810.4715v1

**Soft matter key words:** nanobubbles, Laplace pressure, hydrophobic surface

## Abstract

In this publication the authors provide a theory to account for the stabilization of surface nanobubbles that form on submerged hydrophobic surfaces. The nanobubbles have significant technological importance since they can augment fluid slip in hydrophobic surfaces and reduce fluid dissipation for flows in small devices. Classically, small gaseous bubbles are expected to dissolve due to their large Laplace pressure which causes a diffusive outflux of gas. The authors suggest that the bubbles are actually stabilized by a continuous influx of gas near the contact line, due to the gas attraction towards hydrophobic walls. With the influx balancing the outflux, a metastable equilibrium is reached.

## Soft Matter Snippet

The mechanism that accounts for the stabilization of surface nanobubbles on submerged hydrophobic surfaces is that the gas outflux from the bubble is compensated by gas influx at the contact line. Namely the authors propose that this influx doesn't have to occur uniformly across the bubble, but can be spatially concentrated, especially near the contact line where intermolecular forces are more significant. The diffusive volume outflux is:

<math>j_{out} (R) = \pi R D (1-\frac{c_{\infty}}{c(R)}) </math>

Where R is the bubble radius, D is the diffusion constant of the gas in the liquid and c(R) is the concentration of gas in the bubble. The compensatory gas volume influx is:

<math>j_{in} (R) = \frac{2 \pi s D R}{tan{\theta} (R)}</math>

Here s is the relative strength of the attraction potential, highlighting the physical meaning of the influx: a potential attracting solute molecules to the bubble wall. Plotting the two equations gives rise to fig.2. The crossing point between the two curves defines the equilibrium radius <math>R^*</math>, for which outflux equals influx and the bubble is in dynamic equilibrium. Based on this, the authors note that if the slope of <math>j_{out}</math> at R=0 is larger than that of j_{in}, no surface nanobubbles can emerge.
In terms of the attraction potential, for stable nanobubbles to occur the following condition must be met:

<math>s > \frac{1}{2} tan{\theta} (1- \frac{c_{\infty}}{c(R \rightarrow 0)}) \approx \frac{1}{2} \tan{\theta}_0</math>