# Dynamic Equilibrium for Surface Nanobubble stabilization

**Introduction**

There is ample evidence for the presence of stable surface nanobubbles on hydrophobic surfaces, despite the predicted bubble dissolution from classical bubble stability theory. Presence of nanobubbles on hydrophobic surfaces is strongly suggested by the presence of soft, spherical, domains when imaged using an AFM tip. Furthermore, the size of these structures increases with dissolved gas concentration, suggesting the existence of surface nanobubbles. However, the observation of long-lived surface nanobubbles is inconsistent with classical bubble stability theory, which predicts bubble dissolution. Presence of nanobubbles significantly alters the hydrodynamics near the surface, therefore it is of interest to understand the nature in which these bubbles can exist. Understanding the mechanism in which these bubbles exist has applications to microfluidic design, and for interface science in general. For instance, nanobubble theory has been used to explain the presence of an apparent "slip" boundary condition near surfaces, which reduces the total viscous dissipation.

Previously published theories on nanobubble stabilization explain the bubble stability by suggesting contaminant stabilization, charge stabilization, and the contributions of large curvatures that results in a reduction of surface tension. In this work, the authors explain bubble stability by noting that there is both gas influx and gas outflux at the contact line.

**Theory**

The authors explain nanobubble stability on a hydrophobic surface by assuming a flux balance between diffusive outflux due to the Laplace pressure across the surface of a bubble, and a diffusive influx at the contact line due to high gas concentrations near the surface. This scenario is illustrated below in Figure 1.

Solving the steady state diffusion equation in the liquid yields a volume flux rate from the bubble to the fluid which is given by
(1) <math>j_{out}(R) = \pi RD (1-\frac{c_{\infty}}{c(R)})</math>

where D is the diffusion coefficient of the gas, R is the radius of curvature of the bubble, c(R) is the concentration of the gas at the bubble surface, and

<math>c_{\infty}</math> is the gas concentration in the bulk. Henry's law gives the concentration of the gas at the bubble surface as

<math> c(R) = c_0 P_{gas}/P_0 </math>

where <math>c_0</math> is the gas concetration at pressure <math>P_0</math>

<math>P_{gas}</math> is the Laplace pressure across the bubble interface (<math>2 \gamma /R</math>

If we just were to consider diffusive outflux, then the volume of the bubble would decrease over time, resulting in complete dissolution. However, if we consider diffusive influx, bubbles do not dissolve. The authors note that gas tends to accumulate near walls because of an attractive potential of the solute to the surface. Assuming that the bubble is nearly flat, the volume influx is given by

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

where <math>\theta</math> is the contact angle and s is a relative scaling for the depth of the potential well (in other words, the attraction strength of the gas to the wall).

It is known that contact angle decreases with R, so the authors assume the contact angle has an R dependence of

(3) <math>cos(\theta) = cos(\theta_{\infty}) - \frac{cos(\theta_{\infty} - cos(\theta_0)}{1 + (R/\delta)}</math>

Where <math>\theta_0</math> is the contact angle as R approaches 0. <math>\delta</math> allows for microscopic corrections and has units of length.

When <math>j_{in} = j_{out}</math>, the bubble is in dynamic equillibrium.

Figure 2 below shows the volume flux rates for typical parameters in equation (3).

For stable nanobubbles to form, the influx must exceed the outflux as R->0. For small gas concentrations, the condition for gas bubble formation reduces to

(4) <math> s > \frac{1}{2} tan(\theta_0)</math>

Therefore, if the strength of attraction between the solute and the wall is strong enough, the bubbles are able to grow. As shown in Figure 3 below, the equilibrium radius of the bubble is strongly dependent on the strength of interaction of the solute with the walls (s) and the relative gas concentration (<math>c_{\infty}/c_0</math>. Therefore, the equilibrium radius should increase with increasing interaction potential and should increase with decreasing <math>\theta_{infty}</math>, consistent with experimental observation.

**Conclusions**

The authors proposed a dynamic equillibrium model to explain stability of nanobubbles on hydrophobic surfaces. Their theory is consitent with the observations that nanobubbles do not form under certain conditions, likely because the surface does not meet the criterion for stable bubble growth (equation (4)). This paper provides an initial insight to stability theory of bubbles. However, the authors do not confirm the model of the contact angle with radius in this work, and do not present quantitative measurements to direct;y verfiy the dependence of stability with gas concentration and wall attraction strength. These results would be interesting to verify experimentally. Furthermore, it would be illuminating to derive the stability theory of nanobubbles in the presence of viscous stresses since the stability criterion may be shifted in the presence of applied stress.

**Reference**

M. P. Brenner and D. Lohse, Phys. Rev. Lett. 101, 214505 (2008).