We may use the Blasius solution of the Prandtl equation, since, by comparing the size of the nanobubble to the size of the water laminae we may approximate the relative curvature to be negligible, as well as the flow being laminar due to the fluid itself being static, and hence the flat plate approximation may apply. Thus, the approximate thickness of the boundary layer is obtained with equation as before in section 4.4. From Figures 2 and 3 for the same assumed conditions, we obtain a boundary layer thickness of about 60 microns. The exact volume of water available to interact with the nanobubble is, therefore about 9 × 10-19 litres. This, at pH 7, contains even less than one hydroxide ion, assuming uniform distribution of ions before the nanobubble is formed. Thus, drawing on the number of ions derived previously, even adding one ion to this volume significantly decreases the Debye length of the hydroxyl ions within the boundary layer. This in turn opens the possibility of pH being as high as 15 within the boundary layer, and the possibility of the number of ions being adsorbed being far larger. At pH 15, using the same equations as before, we obtain the Debye length to be 0.01 nm, with a corresponding surface charge of 2426 C, and the corresponding number of ions adsorbed to the surface being about 1.51 × 1022. This, however, exceeds the number of hydroxyl ions that there is room for on the surface, which is only about 1 × 10. This allows us to consider that the nanobubble surface might, in fact, be fully saturated, which,bato bucket gives the Debye length a value of 0.2 nm, a corresponding surface charge of 1.21 × 10-18C, and a pOH of 0.26, corresponding to a pH of 13.74.

The inter-ionic distance, x, can also be found using equation , by substituting the same values used earlier for radius and the number of ions. This gives a value for x to be about 85 nm. Comparing with the Debye length of the hydroxyl ions at pH 7, which is given by equation , it is shown to be well within the range of electrostatic effects of the hydroxyl ions in solution. This also implies that any movement of the ion which would disturb it from its equilibrium position, such as the diffusion of gas out of the bubble, will have a high activation energy, thus reducing the rate of diffusion and providing an explanation for the long lifetimes of bulk nanobubbles. The inter-ionic distance for the completely saturated nanobubble is assumed to be zero, with ions being in direct contact with each other. While this is an extreme case, it remains possible. In this case, then, the ions would completely block the diffusion of the gas within the bubble to the bulk fluid by simple steric repulsion, giving the nanobubble a very long lifetime. However, since we do have a limit to the lifetime, it is clear that this extreme case does not exist, but it is likely that the reality approaches it, and that the pH of the boundary layer surrounding the nanobubble is significantly higher than the bulk solvent outside it. Calculating the forces generated by ionic force and surface tension, using equation and for the first case of a stationary nanobubble results in a value of the ionic force, for a nanobubble of diameter 200 nm and with 20 ions on its surface to be approximately 4 × 10-16 N, corresponding to a pressure of 3 × 10-3 Pa, and for surface tension, the calculated value is 4.5 × 10-8 N, corresponding to a pressure of 3.6 × 105 Pa. For external pressure, assuming a water head height of 1m, the value is about 1 × 10Pa.

Substituting these values in equation , an internal pressure of 5.6 × 105 Pa is obtained. At 25°C, the number of moles of gas contained within the nanobubble determined from equation , n, to be 9.5 × 10-19 moles, or approximately 573,000 molecules. However, for the case of a nanobubble in motion, with a diameter of 200 nm and the number of hydroxide ions adsorbed being 1 × 10, with the external conditions and surface tension being the same, the repulsive force is 2.5 × 10-10 N, corresponding to 2 × 103 Pa. This gives a value for the internal pressure to be the same, about 9.4 × 10-19 moles, or approximately 569,000 molecules. Both of these numbers represent reasonable values for the number of moles of gas contained within the nanobubble. In a second possible case, the force acting to shrink the nanobubble may be considered to be equal and opposite to the force of repulsion between hydroxyl ions that are adsorbed to the surface. In this case, considering the same zeta potential and radius as the previous calculations, the equations yield a value of approximately 2.1 × 109 ions which is clearly not possible, with a Debye length of 8.5 × 10-15 m, which also gives an inter-ionic distance of 8.1 × 10-12 m. Since this distance is even smaller than the diameter of a hydroxyl ion, it is apparent that such a case cannot exist. Furthermore, the maximum number of ions that can be accommodated is approximately 1 × 10, whose force of repulsion is 2.6 × 10-10 N, which is not comparable to the force due to surface tension. Thus, it is concluded that hydroxyl ions adsorbed to the surface do not assist significantly in balancing the internal and external pressures of the nanobubble. As stated in section 4.1, the other contributing factor to the change in the rate of diffusion is the effect of the hydroxide ions adsorbed to the surface.

The mechanism of their actual inhibition would, conceivably be due to the steric hindrance imposed by them for an oxygen molecule attempting to leave the nanobubble. However, the spacing between the ions calculated in section 3.3.2 is far too high for any significant barrier to the diffusion. However, oxygen in gaseous state, that is to say, the oxygen molecule, is highly electronegative, and may offer significant repulsion to the hydroxide ions, as may other electronegative gases such as nitrogen. This would mean that the repulsive forces would, in theory require the ions adsorbed to the surface to change the spacing between them in order to permit the gas molecule to diffuse through an area free of the repulsion that force it to stay inside the nanobubble. This also implies that any movement of the ion which would disturb it from its equilibrium position, such as the diffusion of gas out of the bubble,dutch bucket hydroponic will have a high activation energy, thus reducing the rate of diffusion and providing an explanation for the long lifetimes of bulk nanobubbles. This possibility is analysed in this chapter, and can apply to oxygen, nitrogen and air, as it is a mixture of the two. Takahashi et al. report the zeta potential of micro-bubbles to be constant and independent of size, which, since the surface charge density is directly proportional to the zeta potential, implies that surface charge density is also constant. This indicates that the micro-bubble, as it shrinks, releases adsorbed ions from its surface in order to maintain the same surface charge density. We can assume that the shrinkage is thus opposed by the tendency of hydroxide ions to be de-adsorbed, since hydroxide ions appear to be in a lower energetic state when adsorbed to the surface of the nanobubble, than in solvation, which appears to be at a higher energy state. They would, therefore be forced to go into solution if the nanobubble cannot accommodate them on its surface due to shrinkage. However, for the nanobubble to shrink, the gas molecules contained within must escape, and to do so they must have sufficient momentum to provide the energy needed for the hydroxide ions adsorbed on the surface to be de-absorbed. Thus, the gas molecules require sufficient kinetic energy, which, when transmitted to the ions, must permit them to be de-adsorbed. We can also characterise this with a change in the force of repulsion between ions adsorbed on the surface and a change in the inter-ionic distance. The hydroxide ions, then, clearly have a role in inhibiting diffusion of the gas molecules contained within the nanobubble into the bulk fluid. The mechanism for this inhibition is assumed to be by means of ion-lone pair repulsion, between the hydroxide ions adsorbed to the surface and the lone pairs of the oxygen atoms within the nanobubble.

The surface of the bubbles, as shown earlier, is proposed to contain adsorbed hydroxide ions arranged in rhomboid unit cells, and by vector addition it is clear that the least repulsion to oxygen molecule to diffuse through the hydroxide ions would be at the centre of each rhombus, which would be limited to a greatly reduced area for the diffusion to occur through. This restriction would significantly increase the time needed for the gas to diffuse outward, causing the bubble to shrink at a much lower rate. Thus, the electrostatic repulsion would, in theory, be the weakest at the centre of each rhombus, and would presumably permit the number of oxygen molecules that can fit through it, as well as who have the requisite kinetic energy, to diffuse outward. However, the number of the ions adsorbed to the surface causes difference to the limitation of outward diffusion. If, as in the second case, hydroxide ions are assumed to completely saturate the surface, then the diffusion is inhibited by the steric repulsion or steric hindrance of the hydroxide ions on the surface. This in turn will reduce the diffusion to nearly negligible levels, giving the nanobubbles highly increase lifetimes. While both the cases of stationary and moving nanobubbles represent two opposite sides of the spectrum of possible cases, it is clear that the trend of increasing number of adsorbed ions correlates to a decrease in the outward diffusion of gas and thus increased lifetimes of bulk nanobubbles.The repulsion of the ions and the gas molecules is, essentially, a case of repulsion in aqueous solution, however, within the nanobubble, the case of the purely aqueous solution must be replaced with the case that the gas itself is a second medium with an interface. Thus, the solvent within the nanobubble becomes the oxygen gas and the ions are at the interface of the second medium. If the Gouy-Chapman theory of double layers is used, then the Debye length for the oxygen medium will approach infinity, and the effect of ionic repulsion extends throughout the nanobubble, allowing the hydroxide ions to repel oxygen molecules away from the interface and keeping them within the nanobubble and enabling them to balance the external pressure. The strength of the repulsive force would not be of the same level as the repulsion between, for example, two hydroxide ions, since the oxygen molecule is not charged, but the oxygen molecule also has two lone pairs in the valence shells of its constituent atoms, which can be repelled albeit much more weakly than an ion. If this conjecture is true, however, it will remain a valid mechanism for the inhibition of outward diffusion of electronegative gases from their respective nanobubbles. This hypothesis is supported by the work of Meegoda and co-workers, who report increasing size and zeta potential with increasing electronegativity of the gas contained within the nanobubble. They report the largest size and the highest zeta potential for nanobubbles composed of ozone, followed by oxygen, followed by air and lastly of nitrogen. Their electronegativity is directly related to the number of lone pairs of electrons they possess. Thus, it is reasonable to suppose that the nature of the bond formed is a stronger version of the standard hydrogen bond between water molecules, due to the dipole moment of the hydroxide ion. At the same time, however, the gas within the nanobubble also is repelled by the oxygen atom, the mechanism of which is by means of ion-lone pair repulsion, which would force the gas molecules to stay within the nanobubble, and hence severely limiting diffusion of the gas into the solvent.