Several applications have been discovered, such as for wastewater treatment, fish farming, shrimp breeding, and hydroponics. These are further substantiated by Agarwal and coworkers , for such specific issues as the disinfection of infected surfaces, the degradation of organic compounds, and the disinfection of the water itself. The effects of increased yield of fish due to higher dissolved oxygen content are summarised by Endo et al. . The usage of hydrogen nanobubbles in gasoline to improve the calorific yield is also reported by Oha et al. . Other projected uses include the use of nanobubbles as contrast agents for the ultrasound imaging of tumours, as reported by Cai and co-workers, as well as reduction and removal of deposits of calcium oxalate, which is similar to the composition of kidney stones in rat kidneys, as presented by Hirose et al. Another application of the nanobubble’s ability to permit salts to crystallize is the design of self-cleaning membranes for desalination of water, which use nanobubbles as electrically conductive spacers and pass current through them to force the salts to crystallize on the nanobubble surface,nft channel which will permit easy removal of the accumulated salts. This was demonstrated and presented by Abida et al.
The pressure balance of the nanobubble is considered to be given by the Young-Laplace equation, which, as explained above, equates the internal pressure, external pressure and the surface tension. The first of the four forces that we consider in the Young-Laplace equation is internal pressure. It is proportional to the surface area of the nanobubble, and is assigned a positive sign since it acts to increase surface area. The second is the external pressure, given by the hydrostatic pressure acting on the surface of the bubble, which also decreases the surface area and is negative. The third is the surface tension, which acts along the surface area at the molecular level. The surface tension acts to decrease the surface area, hence the radius and size, and can also be assigned the negative sign. However, a fourth force which is thought to be integral to the stability is the electrostatic repulsion between hydroxide ions adsorbed to the surface of the nanobubble, or, possibly in the cloud surrounding the surface. This repulsion seeks to reduce the contact between the ions on the surface of the bubbles, which also acts to increase the distance between the ions, thus increasing the surface area, and therefore results in a positive pressure. The nature of the interaction between ions can be characterized by the expression for Coulombic repulsion. Since one hydroxide ion is of the order of 1 nanometre in diameter, and most nanobubbles are two orders of magnitude greater in size, we can ignore the curvature of the distance between them and take it to be linear.
The repulsion should, in theory, affect all neighbouring hydroxide ions, but is assumed to be insignificant beyond the nearest neighbours. We also assume the spatial arrangement of these ions over the surface to be close-packed in nature, since the repulsion is equal in all directions, and they would ideally assume a close-packed formation. This arrangement of ions is shown schematically below, in Fig. 1a, and as shown in Fig 1b it is assumed, due to close-packing, that they assume the formation of a rhomboidal unit cell, of side and diagonal length denoted by x, which will be referred to subsequently as the inter-ionic distance. That the nanobubble shrinks due to outward diffusion of the gas contained within is, of course, undisputed, but the precise methods and the rate of diffusion are highly debated. Previous theoretical studies have always assumed a model with a higher mass transfer coefficient, or longer time scales for the process to account for the reduced rate and the high lifetime of the nanobubble. However, it is reasonable to suggest that the change in the rate of diffusion can be attributed to two things: the velocity due to the Brownian motion of the nanobubble, and the inhibition of the diffusion due to the adsorbed hydroxide ions on the surface. In this chapter, the possible effects of Brownian motion are examined for the effect on the rate of diffusion that they may possess. Earlier studies have shown that nanobubbles can be formed by supersaturation, where the solubility limit of the gas, when surpassed will permit the gas to precipitate and form bulk nanobubbles as reported by Matsuki and co-workers .
The shrinkage of nanobubbles has so far been thought to be governed by Fick’s Laws, since it is a case of how fast the gas can dissolve into the surrounding fluid. Thus, according to the first law, it must be directly proportional to the outward gas flux, but the constant is still the diffusion constant D0 for the diffusion of the gas into water. However, this only holds true where the surface area of the nanobubble remains constant. It is, however, possible, that the outward diffusion is a case of Fick’s second law, since the surface area that is available to the gas to diffuse outward also changes according to size, and that this surface area determines the rate of shrinkage and thus the lifetime of the bulk nanobubble. It is then reasonable to suppose that the cause of the change of surface area available for diffusion is the change in the surface area occupied by hydroxide ions combined with the decreasing radius of the bulk nanobubble. The rationale for the assumption that the hydroxide ions adhere to and are released the nanobubble surface is based on two observations, as mentioned before. Firstly, the observation that all interfaces formed by water are negatively charged, and we consider nanobubbles to be a special case of a gas-water interface which may be charged in the same way. Secondly, the zeta potentials measured for nanobubbles are all negative, indicating that a negative ion present in pure water is responsible for the negative charge, which by elimination is the hydroxide ion. Further observations also indicate higher negativepotentials for more electronegative gases, such as oxygen and nitrogen, than for other reported gases such as argon and xenon as reported by Ushikubo et. al.. That nano- and micro-bubbles release hydroxide ions as they shrink is a well-known phenomenon. The stabilization and the shrinkage can be considered to be related to the same phenomenon; thus, the ideal case can be taken to be a nanobubble that is newly formed with no hydroxide ions at the surface at the instant of its formation of an interface. Here, the hydroxide ions present in the water immediately surrounding the bubble, in the hydrodynamic layer, adhere almost instantaneously, the time taken for the adsorption to occur being too small in comparison to the overall timescale to be important. As a concentration gradient is then formed between the water layers at the nanobubble surface and the bulk fluid,hydroponic nft 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.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, 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.