Dynamics of Rear Stagnant Cap formation at the surface of spherical bubbles rising in surfactant solutions at large Reynolds numbers under conditions of small Marangoni number and slow sorption kinetics.

On the surface of bubbles rising in a surfactant solution the adsorption process proceeds and leads to the formation of a so called Rear Stagnant Cap (RSC). The larger this RSC is the stronger is the retardation of the rising velocity. The theory of a steady RSC and steady retarded rising velocity, which sets in after a transient stage, has been generally accepted. However, a non-steady process of bubble rising starting from the initial zero velocity represents an important portion of the trajectory of rising, characterized by a local velocity profile (LVP). As there is no theory of RSC growth for large Reynolds numbers Re ¼ 1 so far, the interpretation of LVPs measured in this regime was impossible. It turned out, that an analytical theory for a quasi-steady growth of RSC is possible for small Marangoni numbers Ma « 1, i.e. when the RSC is almost completely compressed, which means a uniform surface concentration Γ(θ)=Γ(∞) within the RSC. Hence, the RSC angle ψ(t) is obtained as a function of the adsorption isotherm parameters and time t. From the steady velocity v(st)(ψ), the dependence of non-steady velocity on time is obtained by employing v(st)[ψ(t)] via a quasi-steady approximation. The measurement of LVP creates a promising new opportunity for investigation of the RSC dynamics and adsorption kinetics. While adsorption and desorption happen at the same localization in the classical methods, in rising bubble experiments desorption occurs mainly within RSC while adsorption on the mobile part of the bubble surface. The desorption flux from RSC is proportional to αΓ(∞), while it is usually αΓ. The adsorption flux at the mobile surface above RSC can be assumed proportional to ßC0, while it is usually ßC0(1-Γ/Γ(∞)). These simplifications may become favorable in investigations of the adsorption kinetics for larger molecules, in particular for globular proteins, which essentially stay at an interface once adsorbed.

Surfactant accumulation within the top foam layer due to rupture of external foam films.

The analysis of processes taking place in a steady pneumatic (dynamic) foam shows the possibility of different modes of surfactant accumulation within the top layers of bubbles due to rupture of external foam films. An increasing surfactant concentration within the top layers promotes the stabilisation of bubbles and the foam as a whole. Considering the balance of surfactant and water during the bursting of films it is possible to estimate the accumulated surfactant loss caused by a downwards flow through the Plateau borders of the subsurface bubble layer. This effect depends on the particular conditions, especially on the surfactant activity and concentration of the surfactant, water volume fraction in the foam and size of foam bubbles. The process of surfactant accumulation in the top foam bubble layer can be complicated due to the removal of part of the accumulated surfactant through transport with droplets spread out during bubble bursting.

Development in modeling submicron particle formation in two phases flow of solvent-supercritical antisolvent emulsion.

The use of a supercritical Solvent (S)-Antisolvent (AS) process (SAS) for fine particle production is finding widespread industrial applications. The perfection of this technology requires insight into many basic laws of interface and colloid science. In SAS the solute is dissolved in an organic solvent and the solution is sprayed into a near critical AS stream. SAS is a complex process involving the interaction of jet hydrodynamics, droplet formation, mass transfer, phase equilibrium, intra-droplet nucleation, and microcrystal growth. A complete description would have to take into account all of these processes; however, such a model is not currently available. In the two-phase flow of an S/AS emulsion, S diffuses from droplets into AS, while AS dissolves inside the S droplets. S replacement by AS (Supercritical CO2) causes solute supersaturation in the droplets. When it occurs near the critical point of the S/AS emulsion (80 bar, 32 degrees C), intra-droplet nucleation and precipitation of the solute occurs. The possibility of solute particle production and the particle size is controlled by the droplet size and by the interrelationship between three time scales. These are the droplet mass transfer time tau N, the nucleation time tau N, i.e., the time necessary for one particle nucleus to form in one droplet, and the droplet residence in the supersaturated stream tau res. An approximate analytical theory for intra-droplet nucleation is developed and the conditions necessary for nanoparticle production are established. The smaller the droplet dimension and the lower the solute concentration, the smaller the particle dimension that is obtained. The recent success in membrane emulsifying may be used for the production of micron-sized droplets. After the AS stream is saturated with S due to partial dissolution of the droplets, a quasi-equilibrium between the droplets and AS stream occurs and a steady and uniform zone with intra-droplet supersaturation is formed downstream. But tau res>tau N is necessary for one nucleus formation per droplet, i.e., tau res has to be much longer than that reported in the literature (10(-3) s), because tau N increases with decreasing droplet dimension. Accordingly, a long residence time version of the SAS process (tau res approximately 1 s) is necessary. However, a long tau res is problematic because of micro-droplet turbulent coagulation. Since an increase in tau res is difficult, a decrease in tau N by means of an increase in S becomes significant. This is achieved by using a phenomenon which we call supersaturation of the second kind S2 In the literature attention is paid only to a decrease in the equilibrium solute concentration, when solvent and antisolvent are mixed. However, S2 occurs due to an actual increase in concentration of solute within the droplets as they shrink due to S dissolution. The smaller the ratio of solvent to antisolvent flow rate, the larger the droplet shrinkage and the higher the S2 achieved. Due to large S2, nanoparticle production becomes possible even for solutes with high surface tension sigma and large molecular volume V o, while earlier it was impossible because of the exponential increase of tau N with increasing V o and sigma. Combining a long tau res and variable and precisely controllable supersaturation, which is uniform in space and enhanced due to S2, creates an opportunity for standardization of characterizing different solutes through their tau N, which is the key solute property affecting nanoparticle production by SAS.

Gravity as a factor of aggregative stability and coagulation.

Gravity is a potential factor of aggregative stability and/or coagulation for any heterogeneous system having a density contrast between the dispersed phase and its dispersion medium. However, gravity becomes comparable to other stability factors only when the particle size becomes large enough. Since the particle size may grow in time due to various other instabilities, even nano-systems may eventually become susceptible to gravity. There have been many attempts in the last century to incorporate gravity in the overall theory of aggregative stability, but the relevant papers are scattered over a wide variety of journals, some of which are very obscure. Reviews on this subject in modern handbooks are scarce and inadequate. No review describes the role of gravity at all three levels introduced by DLVO theory for characterizing aggregative stability, namely: particle pair interaction, collision frequency and population balance equation. Furthermore, the modern tendency towards numerical solutions overshadows existing analytical solutions. We present a consistent review at each DLVO level. First we describe the role of gravity in particle pair interactions, including both available analytical solutions as well as numerical stability diagrams. Next we discuss a number of works on collision frequency, including works for both charged and non-charged particles. Finally, we present analytical solutions of the population balance equation that takes gravity into account and then compare these analytical solutions with numerical solutions. In addition to the traditional aggregate model we also discuss work on a fractal model and its relevance to gravity controlled stability. Finally, we discuss many experimental works and their relationship to particular theoretical predictions.

Characterization of fractal particles using acoustics, electroacoustics, light scattering, image analysis, and conductivity.

Fractals are aggregates of primary particles organized with a certain symmetry defined essentially by one parameter-a fractal dimension. We have developed a model for the interpretation of acoustic data with respect to particle structure in aggregated fractal particles. We apply this model to the characterization of various properties of a fumed silica, being but one example of a fractal structure. Importantly, our model assumes that there is no liquid flow within the aggregates (no advection). For fractal dimensions of less than 2.5, we find that the size and density of aggregates, computed from the measured acoustic attenuation spectra, are quite independent of the assumed fractal dimension. This aggregate size agrees well with light-scattering measurements. We applied this model to the interpretation of electroacoustic data as well. A combination of electroacoustic and conductivity measurements yields sufficient data for comparing the fractal model of the particle organization with a simple model of the separate primary particles. Conductivity measurements provide information on particle surface conductivity reflected in terms of the Dukhin number (Du). Supporting information for the zeta potential and Du can also be provided by electroacoustic measurements assuming thin double-layer theory. In comparing values of Du from these two measurements, we find that the model of separate solid particles provides much more consistent results than a fractal model with zero advection. To explain this, we first need to explain an apparent contradiction in the acoustic and electroacoustic data for porous particles. Although not important for interpreting acoustic data, advection within the aggregate does turn out to be essential for interpreting electrokinetic and electroacoustic phenomena in dispersions of porous particles.

Wetting film stability and flotation kinetics.

Single bubble experiments performed with different size fractions of quartz particles and different, but known, contact angles revealed two modes of flotation dynamics in superclean water. (1.) A monotonic increase of collection efficiency Ecoll with increasing particle size was observed at high particle hydrophobicity and, correspondingly, a low wetting film stability (WFS). (2.) At low particle hydrophobicity and, correspondingly, high WFS, an extreme dependence of Ecoll on particle size was observed. The use of superclean water in our experiments prevented the retardation of bubble surface movement caused by surfactants or other impurities that is usual for other investigations and where particle-bubble inertial hydrodynamic interactions are suppressed. In the present study the free movement of the bubble surface enhances particle-bubble inertial interaction, creating conditions for different flotation modes, dependent on WFS. At the instant of inertial impact, a particle deforms the bubble surface, which may cause its rebound. Where the stability of the thin water film, formed between opposing surfaces of a bubble and a particle, is low, its rupture is accompanied with three phase contact line extension and contact angle formation before rebound. This prevents rebound, i.e. the first collision is accompanied by attachment. A high WFS prevents rupture during an impact. As a result, a contact angle does not arise and rebound is not prevented. However, rebound is accompanied by a second collision, the kinetic energy of which is smaller and can cause attachment at repetitive collision. These qualitative considerations are confirmed by the model quantification and comparison with measured Ecoll. For the first time the Sutherland equation (SE) for Ecoll is confirmed by experiment for smaller particle sizes and, correspondingly, very small Stokes numbers. The larger the particle size, the larger is the measured deviation from the SE. The SE is generalized, accounting for the centrifugal force, pressing hydrodynamic force and drainage in the low WFS case and, correspondingly, attachment occurs at first collision or during sliding. The derived generalized Sutherland equation (GSE) describes experimental data at low WFS. However, its application without account for possible rebound does not explain the measured extreme dependence in the case of high WFS. The theory for drainage during particle impact and the beginning of rebound enables conditions for either attachment or rebound in terms of the normal component of the impact velocity and the critical film thickness to be derived. Combining this condition with the GSE allowed the equation for Ecoll to be derived, accounting for attachment area shrinkage and attachment during a repetitive collision. This equation predicts the extreme dependence. Thus the WFS determines the modes of flotation dynamics and, in turn, probably affects the mechanisms, which control the flotation domain. At low WFS its upper boundary is controlled by the stability of the particle-bubble aggregate. At high WFS the upper boundary can be controlled by rebound because the latter reduces the attachment efficiency by a factor of 30 or more even with repetitive collision.

Microflotation Suppression and Enhancement Caused by Particle/Bubble Electrostatic Interaction.

The processes of attachment and detachment of small or medium-sized particles to relatively large bubbles during microflotation are considered in terms of the heterocoagulation theory. Calculations are made for the conditions that the surface potentials are of similar sign and constant, that one of the surface potentials is small, that hydrophobic attraction is absent, and that there are no surface deformations. Under these conditions bubble-particle aggregates may form as a result of an electrostatic attraction which exceeds the repulsive van der Waals force at intermediate distances. Next to electrostatic and van der Waals forces, hydrodynamic and gravitational forces are considered. These forces may overcome the electrostatic repulsion at large distances and promote particle bubble attachment. Strong electrostatic attraction at small distances, arising at a large difference of the surface potentials of the bubble and the particle and of low electrolyte concentrations, can prevent subsequent detachment by hydrodynamic and gravitational forces. With increasing electrolyte concentration the electrostatic barrier increases and the attractive electrostatic force diminishes. As a result, a critical electrolyte concentration for microflotation exists. Above this concentration attachment may still occur but it is followed by detachment. At lower electrolyte concentrations the electrostatic attractive force prevents the detachment. The dependence of the critical electrolyte concentration on the values of the bubble and particle potentials and the Hamaker constant is calculated. The critical concentration does not depend on particle or bubble size if the absolute values of the total detachment force and the total pressing force coincide, which is the case for Stokes and potential flow. For every electrolyte concentration lower than the critical value there are two critical particle sizes that limit the flotation possibility. For small particle sizes attachment is impossible because the pressing force is smaller than the electrostatic barrier. For large particle sizes detachment cannot be prevented because the detachment force exceeds the maximum electrostatic attraction. A microflotation domain of intermediate particle sizes exists in which irreversible heterocoagulation occurs. Copyright 2001 Academic Press.

Dynamics of Rear Stagnant Cap Formation at Low Reynolds Numbers.

The rising of a gas bubble in a surfactant solution is considered within the framework of the rear stagnant cap (r.s.c.) model at low Reynolds number. The adsorption is supposed to be the slowest process in the system. The bubble velocity is predicted to depend on time due to the nonstationary process of surface stagnant zone formation. Various types of bubble behavior are discussed. It is shown which information on the characteristics of adsorption one can extract from the time dependency of the bubble velocity. Copyright 2000 Academic Press.