As a concentration gradient is then formed between the water layers at the nanobubble surface and the bulk fluid, more hydroxide ions begin diffusing from the bubble through this diffusion layer to the surface of the bubble. The thickness of this layer can be found by Prandtl’s equation, where the fluid velocity is the velocity of the Brownian motion of the bubble as predicted by the Langevin equation, using the Ornstein-Uhlenbeck process. The same layer also acts a diffusion region for protons, which diffuse in from the bulk layer once they are depleted or their concentration changes, and must also be affected by the distance they must diffuse through to reach the nanobubble surface. These phenomena are further examined in the following sections. At the same time, protons from the diffusion layer also reach the hydroxide ion-rich surface, but much more slowly, at a rate about five times slower than the hydroxide ions. Upon reaching the surface, they start eliminating the hydroxide ions into water molecules, which further increases the dilution of both ions, and encourages diffusion from the bulk layer to the interface, which is probably a monolayer. When the three processes, of hydroxide diffusion, proton diffusion, and hydroxide elimination by protons, are in steady state or in dynamic equilibrium, we have a fixed amount of area which is not covered by hydroxide ions, however temporarily and will allow the diffusion of gas into the water. Taking an average, we can define a percentage of surface area of the nanobubble,chicken fodder system which will remain available for diffusion, which will be in proportion to the radius of the bubble.
This can be done by taking the size of one hydroxide ion, then finding the capacity of a nanobubble’s surface to adsorb hydroxide ions, correlating it with the number of ions being eliminated, and taking a ratio with the capacity which is a function of area, which is a function of radius. The rate at which the adsorption of the hydroxide ion takes place would then depend on two separate phenomena: firstly, the repulsion by the hydroxide ions already physisorbed onto the surface, which would force the ion to move along the surface until it finds a location that is unoccupied, and secondly, the velocity of the hydroxide ion as it travels through the hydrodynamic layer of the nanobubble. The velocity can be found by calculating the surface charge on the nanobubble, and using it and the initial distance between a particular ion to find the potential that drives it to move. The potential for the hydroxide ion to move to the nanobubble surface decreases as the surface charge increases, and thus the rate will eventually dwindle down to zero as the bubble achieves stability, and the potential will reach a constant value. The rate for the elimination of the physisorbed hydroxide ions, on the other hand, will only increase the surface charge density increases, since the elimination is accomplished by positively charged protons attracted to a negatively charged surface. The same equations for ionic mobility can be used to calculate the velocity of travel for the protons, but there is no equation needed for the rate of adsorption, as they simply react with the adsorbed hydroxide ion to give two molecules of water. The balance between these two rates thus depends on the time at which the reactions are taking place, which will ultimately determine the area needed for the diffusion of the gas into the water. To find the ion mobility, we first consider an ideal case where a newly-formed and shrinking bubble has no hydroxide ions physisorbed onto its surface, and is formed in pure water with a pH of 7. This gives us, assuming a perfectly uniform distribution of ions in the water, a concentration of 10-7 moles of hydroxide and protons each in the surrounding hydrodynamic layer.
Thus, the amount of both available to be physisorbed can be found by simply taking a section of the hydrodynamic layer up to the distance from the surface where we wish to find the concentration and time needed to reach the surface for the ions present at that distance from the surface. We take the volume of this section and multiply by molarity and Avogadro’s number to get the actual number of ions present, as shown below. In the derivation of the force balance presented in section 3.2, the formula takes into account the contribution of the repulsion between hydroxide ions adsorbed to the surface of the nanobubble. This section estimates the number of the ions adsorbed to the surface of the nanobubble, and uses the terms associated wit their arrangement to calculate this contribution and to examine the possibility that they can, indeed help to balance the inward and outward pressures exerted on the nanobubble surface and thus provide an explanation for their stability. Both possibilities of stationary and nanobubbles in motion are assumed and calculated to provide estimates for the repulsive force, and are substituted along with representative values in the derived equation for the force balance, and the result is presented. However, the stationary nanobubble is an ideal case, and in actual situations the bulk nanobubble is usually in motion due to Brownian motion, which also prevents it from rising to the surface. Thus, it can be established that the bulk solvent for a bulk nanobubble in motion, only consists of the boundary layer that moves with the nanobubble as it moves through the solvent. It also must supply the ions needed to stabilise it, and must contain the ions that are adsorbed. 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 d 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,fodder systems for cattle 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, 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.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.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, 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 microbubbles 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 microbubble, 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.