# Dynamic equilbrium mechanism for nanobubble stabilization

### Dynamic equilibrium Method for Surface Nanobubble Stabilization

M P Brenner and D Lohse

Phys. Rev. Lett. **101**, 214505 (2008)

**Keywords:** nanobubble, diffusion, Laplace pressure

## Summary

A variety of recent studies has demonstrated of nano-scale bubbles of air on the surface of hydrophobic surfaces. Based on the Laplace pressure alone, these bubbles should dissolve rapidly, but yet they are observed over the course of several houra. Previous theories to account for this include: a reduction in surface tension at small length scales, oversaturation of gas in the liquid near the bubble surface, stabilization due to contaminants in the liquid, and induced charges around the bubble interface. The authors propose how the outflow of gas from the surface into the liquid is balanced by uptake of gas around the perimeter.

## Soft matter aspects

This paper touches on one of the central themes of the course: how the dynamics at a three-phase interface (i.e. surface, liquid, and gas) can determine the behavior of a system. In this case, the authors are studying the presence of nanobubbles at a surface. The outward pressure of the bubbles is calculated from the radius of curvature. In a typical example, the radius R = 50 nm and the contact angle <math>\theta</math> = 10 degrees or a radius of curvature of 250 nm. This basic setup is shown below:

Given a surface tension of 73 mN/m (at 20 C), this corresponds to a pressure of <math>2 \gamma/R = 0.58 MPa</math>. According to Henry's Law, the concentration of gas at the bubble surface is given by <math>c(R) = c_0 P_gas / P_0</math>, where <math>c_0</math> is the saturated concentration of gas in the liquid, <math>P_gas</math> is the gas bubble pressures, and <math>P_0</math> is atmospheric pressure. Since the pressure increases as the bubble size decreases, this causes the concentration of gas at the bubble surface to increase, leading to faster diffusion at the interface until the system diverges.

To calculate the outflow of gas from the system, the authors integrate the product of the diffusion constant and the concentration gradient over the surface of the bubble. By assuming that most of the outflow is perpendicular to the wall surface <math>\big(\nabla_c \approx \delta_z c\big)</math>, this is given by:

<math>J_{out}(R) \approx 2 \pi F \int_{0}^{R} r (\delta_z {c}) dr</math>.

Similarly, the rate of gas flow into the bubble around the perimeter is given by:

<math>J_{in}(R) \approx \frac{2 \pi R c(R)\Delta \phi}{\eta \tan(\theta)}</math>,

where <math>\Delta \phi</math> is the energy gain when a solute molecule adsorbs to the surface and <math>\eta</math> is the motility of the solute molecule.

By comparing these two expressions as a function of radius, a crossover point (R*) is reached, in which the inflow and outflow are matched, so the system is in dynamic equilibrium:

Since this is a non-equilibrium process, some external input (e.g. a thermal gradient or the measurement apparatus itself) is needed to maintain the bubble size. Eventually the bubble must dissolve, in accordance with the second law of thermodynamics, but experiments have yet to be done over such long time scales.

- Naveen Sinha