Diffusion-controlled flocculation

The flocculation of colloidal particles can be divided into two main types (i) diffusion-controlled flocculation (ii) agitation-induced flocculation. The diffusion-controlled flocculation has been analyzed theoretically by von Smoluchowski (5). 

Fig. 6.2. Electron micrographs of Au clusters formed by diffusion-controlled flocculation. (Adapted, with the permission of Elsevier Science Publishers, from D. A. Weitz and J. S. Huang.7)...

Equation 6.18 is graphed in Fig. 6.6 for the cases q = 1, 2, 3. The number density of primary particles, pj(t), decreases monotonically with time as these particles are consumed in the formation of floccules. The number densities of the floccules, on the other hand, rise from zero to a maximum at t = (q - l)/2KDp0, and then decline. This mathematical behavior reflects creation of a floccule of given size from smaller floccules, followed by a period of dominance, and finally consumption to form yet larger particle units as time passes. Both experimental data and computer simulations, like that whose visualization appears in Fig. 6.1, are in excellent qualitative agreement with Eq. 6.18 when they are used to calculate the pq(t).13,14 Thus the von Smoluchowski rate law with a uniform rate coefficient appears to capture the essential features of diffusion-controlled flocculation processes. 

Meakin, R Diffusion-controlled flocculation the effect of attractive and repulsive interactions. J. Chem. Phys. 1983, 79(5), 2426-2429. 

The choice of chemical is usually based on trial-and-error procedures hence, demulsifier technology is more of an art than a science. In most cases a combination of chemicals is used in the demulsifier formulation to achieve both efficient flocculation and coalescence. The type of demulsifiers and their effect on interfacial area are among the important factors that influence the coalescence process. Time-dependent interfacial tensions have been shown to be sensitive to these factors, and the relation between time-dependent interfacial tensions and the adsorption of surfactants at the oil-aqueous interface was considered by a number of researchers (27, 31-36). From studies of the time-dependent tensions at the interface between organic solvents and aqueous solutions of different surfactants, Joos and coworkers (33—36) concluded that the adsorption process of the surfactants at the liquid-liquid interface was not only diffusion controlled but that adsorption barriers and surfactant molecule reorientation were important mecha-... 

The fast flocculation kinetics was investigated by Smoluchowski [12], who considered the process to be represented by second-order kinetics and the process to be simply diffusion-controlled. The number of particles n at any time t may be related to the initial number (at t = 0) by the following expression ... 

Another process of flocculation that occurs under shearing conditions is referred to as orthokinetic, to distinguish it from the diffusion-controlled perikinetic process. In this case, the rate of flocculation is related to the shear rate by the expression ... 

Philipse (5) also assumed that fast hydrolysis created an active monomer bulk. He studied the growth of silica nuclei, already synthesized, after extra addition of different amounts of TES with static light scattering. To explain his growth curves (radius versus time), he used a diffusion-controlled particle growth in a finite bulk of monomers or subparticles. The model contained equations from classical flocculation theories. It takes into account the exhaustion of the monomer bulk and the retarding influence of an (unscreened) electrostatic repulsion between growing spheres and monomers. 

Cowell et al. also found that the maximum in the instability (i.e. in X/W) corresponded to the free polymer being of a molecular weight comparable to that of the anchor polymer. Even imder these conditions, however, the dispersions only flocculated at a rate of about one-third of the hypothetical Smoluchowski diffusion controlled rate. This rate is significantly less (ca one-half) than that expected when allowance is made for hydrodynamic interactions between the approaching particles. This rate reduction, if real, may indicate the presence of a small repulsive energy barrier that must be surmounted during the flocculation process, as predicted by the theories of depletion stabilization. 

We therefore find that under stagnant and neutral buoyancy conditions, the relative rate of diffusion controlled collision is dominated by the radius of the antifoam drops—the larger the drops, the slower the rate of collision because both the number concentration and the diffusion coefficients of drops are lower. Increasing viscosity of the detergent liquid and decreasing the volume fraction of the antifoam also both decrease the rate of flocculation. 

Surfactants are also essential for prevention of flocculation of emulsions. In the absence of an emulsifier, the droplets undergo rapid flocculation (diffusion controlled process), and this results to strong flocculation. The latter is due to the van der Waals attraction which increases rapidly with decrease of separation distance between the droplets. To... 

Transport control of flocculation is realized in an especially direct way in the process known as diffusion-limited cluster-cluster aggregation5 (aggregation as used in this term means flocculation in the present chapter). In this process, which is straightforward to simulate and visualize on a computer, particles undergo Brownian motion (i.e., diffusion) until they come together in close proximity, after which they coalesce instantaneously and irreversibly to form floccules (or clusters ). The clusters then diffuse until they contact one another and combine to form larger clusters, and so on, until gravitational... 

However, if the ionic strength is enhanced, the thickness of the electrolyte cushion is rendered thinner. A larger fraction of particles may then collide per unit time, leading to a coagulation of the particles to a dense sediment. For a O.l-mol/dm 1 1 electrolyte solution, the repulsive electrostatic barrier (Tables 8.3 and 8.4) may be considered to be effectively removed and the particle coagulation is controlled by diffusion alone. Sometimes a balance is reached at intermediate distances (3 to 5 nm), which is characterized by a weak negative minimum. Then, the particles form a loose flocculated structure that is easily redispersed. To a great extent the association kinetics follows the same principles as for the ions discussed above, but at a slower rate. - - " ... 

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