The dynamic equilibrium hypothesis posits that gas lost from the apex of a surface nanobubble is balanced by an influx of gas near the three-phase contact line due to surface hydrophobicity (Fig 2.1). We investigate the feasibility of this mechanism for nanobubble stability from a purely diffusive, continuum perspective. First, we consider the case of a hydrophobic boundary immersed in a liquid that contains a dissolved gas, without any bubbles present in the system. Assuming a simple, experimentally-motivated hydrophobic potential of the form [43]
Bz,11
we can obtain an analytical expression for the equilibrium dissolved gas distribution C z
(unperturbed due to the presence of a bubble) near the surface from the time-independent equation for diffusion in a field,/ .
D C C U (2.2)
Here, is the solute mobility, and D is the diffusion constant of the gas molecules in the liquid. Solving Eq.(2.2) for the simple one-dimensional case with the potential in Eq.(2.1) gives
exp
. B U z C z C k T (2.3)Hence, there is an enhancement of the concentration immediately next to the substrate due to the potential given by C z
0
Cexp
A k TB
. Here, C is the concentration infinitelyfar away and k is Boltzmann’s constant; the concentration at the interface is thus B augmented from the bulk value by the field through an exponential factor that depends on the strength of the attraction A, and the system temperature T. Clearly, increasing the factor A results in greater enhancement of the gas concentration near the surface. Similarly, increasing the system temperature acts to decrease the enhancement by giving gas molecules additional thermal energy, which allows them to more readily overcome the attractive influence of the potential U z
. This increase in the gas concentration near the boundary due to its hydrophobicity is known as a gas-enrichment layer [14,44], and is believed to be closely linked to the phenomenon of nanobubbles.Next, we place a single nanobubble on top of a hydrophobic boundary that features a gas-enrichment layer described by Eq.(2.3). The concentration of dissolved gas in the liquid at the air-water bubble interface (on the liquid-side) C R
depends on the internal bubble12
pressure, and is therefore specified through a combination of Henry’s law and the Young- Laplace equation,
0 2 sin , c H C R k P R (2.4)where
c is the bubble contact angle, gives the surface tension, kH is the Henry’s lawconstant, P0 is the ambient pressure, and R is the bubble footprint radius. By requiring that
the concentration at the solid-liquid interface due to gas enrichment is greater than the concentration at the bubble surface in order to drive an influx of gas, C z
0
C R
, we obtain an order-of-magnitude estimate for the minimum possible value of the hydrophobic interaction strength, 0 2 sin ln c 1 . B A k T P R (2.5)Using realistic values for the parameters in this expression, we find that for a typical nanobubble with R = 50 nm and = 25° under normal conditions (Tc 0 = 30 °C, P0 = 1 atm), A > 1.8 kcal/mol. Note that this estimate is greater than the actual value required, since we assumed that the influx is driven indirectly by the hydrophobic interaction through a concentration gradient. In principle, this influx occurs purely by the attraction of the gas molecules to the substrate, which is due to a combination of the hydrophobic attraction, as well as van der Waals interactions (ignored altogether in this analysis) that become important over small length scales.
This approximate analysis can be made more precise by explicitly placing a nanobubble on top of a surface featuring a gas-enrichment layer and hydrophobic potential, and numerically calculating the full concentration field, holding the concentration at the bubble surface and substrate constant. Once the steady-state distribution of concentration
13
across the system is determined, it is possible to evaluate the flux of gas along the bubble surface due to concentration gradients. In order to obtain the value of A necessary for balanced flux (i.e. zero net flux across the bubble surface) in this simple diffusive picture, we develop a 2D transport model involving a full solution to the equation for diffusion in a field using the COMSOL package. In this approach, an attractive potential [an approximation for the hydrophobic attraction, Eq.(2.1)] acts on the gas solute molecules in the liquid. If a bubble with fixed concentration at its surface is placed on top of such a hydrophobic substrate, it is possible to drive an influx of gas into the bubble near the contact line if the concentration increase at the surface due to the external field is sufficiently high. Here, there are two free parameters: (1) the ratio of the concentration at the hydrophobic surface to the concentration at the bubble air-water interface (which is related to the parameter A),
0 /
C z C R , and (2) the decay length for the exponential potential acting on the solute
molecules (non-dimensionalized by the bubble width), 1
/ 2
B R . Provided that
0 /
1C z C R , there will be an influx of gas into the bubble near the contact line
(assuming that a continuum level description is sufficient and that concentration gradients alone are responsible for mass transfer in the system).
By varying the two free parameters in this numerical model, we develop a state-space diagram that indicates parameter combinations giving zero total flux across the bubble surface (Fig. 2.2). The diagram is for a bubble with fixed geometry and a gas-side contact angle of 45°. The insets show the concentration profile in the vicinity of the bubble for two different balanced flux points featuring a different decay length for the hydrophobic potential, with red hues corresponding to regions of high dissolved gas concentration. As expected, the stable points asymptote as C z
0 /
C R
1, since the concentration14
difference, and hence driving force for influx, disappears in this limit. As the concentration difference approaches unity, the decay length necessary for balanced flux increases to compensate. Conversely, if the decay length is shortened, the magnitude of the concentration at the substrate surface C z
0 /
C R
necessary for stability dramatically increases in order to provide adequate influx over a smaller region of the bubble surface.From this diagram, we can obtain an estimate for the strength of the hydrophobic attraction necessary for balanced flux. The decay length for the hydrophobic interaction has been empirically determined to be ~1 nm [43]. Therefore, for a nanobubble with a radius of 50 nm, we have B-1/2R = 0.01. A linear interpolation using the data in Fig. 2.2 suggests that for a bubble of this size, C z
0 /
C R
~ 1.9 for zero net flux across the bubble surface. The value of A that yields C z
0 /
C R
1.9 for a typical nanobubble (T0 = 30 °C, P0 = 1atm) is 2.4 kcal/mol, slightly above the theoretical minimum of 2.1 kcal/mol given by
Fig. 2.2. Combinations of allowed parameters for balanced flux in the purely diffusive numerical model. The insets show the concentration profile near the bubble for two of these points, with red corresponding to high gas concentration and blue to low concentration.
15
Eq.(2.5) for a nanobubble with = 45°. It is important to note that if a smaller contact angle c was used in this analysis, the value of A required for balanced flux would be lower since more bubble surface area would be situated in the high-gas region, giving considerably greater influx across the bubble surface. In addition, decreasing the contact angle while keeping the bubble footprint radius constant leads to a reduction in the bubble internal pressure, and therefore lowers the value of C R
. If C R
is reduced, this implies that the value of C z
0
(and therefore A) necessary for influx is decreased as well. Generally, nanobubble contact angles are smaller than the 45° angle used in these simulations, which was selected for numerical reasons.We can compare this result to values for the strength of the hydrophobic attraction measured from experiment and simulation. Potential of mean force values for two hydrophobes (e.g. methane molecules) associating in water has been found from simulation to be around 0.5 kcal/mol [45]. On the other hand, the energy change per unit area in bringing together two like surfaces (type I) in a medium II equals twice the surface tension of the I-II interface [43]. Thus, we can also approximate the hydrophobic strength parameter as
2 I II
A
a, where a is the area of contact. The experimental surface tension for a typical hydrocarbon in water is on the order of 50 mJ/m2 [43]. Assuming a molecular area of contact of a ≈ 10 Å2, we find that A ≈ 0.72 kcal/mol using the above-formula. The values of A from our diffusive model are approximately three times as large (A~2.4 kcal/mol), although choosing a smaller contact angle in the calculations would further reduce the required value of A and bring it closer to these estimates. Ultimately, this numerical study demonstrates that given a gas-rich region near the substrate, it is possible to drive an influx of gas into a bubble, even if we neglect molecular effects (though molecular contributions such as disjoining16
pressure effects have been recently considered [46], including specifically in the context of the dynamic equilibrium model [47,48]). As discussed previously, this diffusive analysis overestimates the necessary strength of this parameter and only provides an order-of- magnitude assessment. If we assume the hydrophobic potential draws solute molecules across the liquid-gas interface near the contact line, rather than relying on a concentration gradient to drive gas transport, the value required for the parameter A is lower, which we demonstrate in the following sections.