Bridging flocculation model

Given the volume fraction of the polymer inside the gel, we are now able to propose a quantitative model of bridging flocculation. Because Crawford et al. [4] studied the contraction of the interlayer spacing as a function of uniaxial stress for the same system without any added polymer, we are able to convert the observed d-values to effective uniaxial pressures caused by the bridging polymers. If we assume that we have one polymer bridge when the end-to-end polymer distance l (calculated according to Equation 12.1) exactly matches the d-value with the... 

This section describes a three-dimensional model to mimic bridging flocculation in which the size of the polymer chains is much larger than that of the interacting particles (a situation often encountered in natural waters). 

Figure 2.11. A schematic model for the bridging flocculation mechanism.

The process of adsorption of polymers from solution may take several hours to reach equilibrium conditions, during which time the tendency to flocculate may change radically. Consequently many systems undergoing bridging flocculation are often in a non-equilibrium state with respect to polymer adsorption, which makes the process difficult to model. 

In contrast to bridging flocculation, depletion flocculation occurs when the added polymer cannot adsorb to the surfaces of the droplets. Under these conditions a layer around the droplets exists where the polymer concentration is depleted as compared to the bulk solution (see the depleted volume in Figure 4.3). The origin of the depleted volume lies with a geometrical restraint imposed by the finite voliune of the polymer molecule. This may be understood by modelling the polymer molecule as a sphere of radius r, whose position is defined by its centre. Since the polymer may not adsorb to the droplet surface, the closest it may approach the droplet is to touch the surface. In this arrangement the centre of the polymer will be a distance rg from the droplet surface. It follows that polymer centres cannot get closer than this to the droplet surface and consequently do not contribute to the segment density there. 

Schmitt, A., M.A. Cabrerizo-Vflchez, R. Hidalgo-Alvarez, and A. Femandez-Barbero. 1998. On the identification of bridging flocculation An extended collision efficiency model. Progress in Colloid and... 

The two major theories of flocculation, the bridging model (1) and the electrostatic patch model (2, 3 ), provide the conceptual framework for the understanding of polymer-aided flocculation, but they do not directly address the kinetics of the process. Smellie and La Mer (4) incorporated the bridging concept into a kinetic model of flocculation. They proposed that the collision efficiency in the flocculation process should be a function of the fractional surface coverage, 0. Using a modified Smoluchowski equation, they wrote for the initial flocculation rate... 

FIGURE 4.35 Bridging model for the flocculation of a colloidal particle by lyophilic polymers. 

It is well known that polymers may serve as bridges between colloidal particles to form floes nonetheless, very little quantitative information is available about their structure and formation, despite the fact that they play key roles in environmental systems [1], Particles may not only be bridged by polymers, but may also facilitate the formation of larger aggregates due to the adsorption of several polymer segments on the same particles. This process can be seen as an example of the CCA model, where polymer conformation, reactivity and total length play important roles. Computer models once again constitute a valuable tool that allows for predictions of flocculation processes. 

Cationic Polymers., The relation between zeta potential and flocculation by a polymer has been studied by Rjes (3IS), who pointed out that as soon as a colloidal particle is coated with polymer it bears the same charge as the polymer and is redispersed. Similar studies by Ries and Meyers (316) involved the use of microphoresis and electron microscope observations of model colloids and polymeric flocculants. Polyamine type flocculants appeared to extend out from the particle surface for a distance of 20-300 A. Flocculation occurs simultaneously through charge neutralization and bridging of polymer chains from particle to particle then excess polymer reverses the potential and redispersion occurs. Adsorption of poly [(1,2-dimethylvinylpyridinium) methylsulfate] on silica was similarly studied by Shyluk (317), who concluded that the polymer chains lay flat along the surface when no excess polymer was present. 

Runkana V, Somasundaran P, Kapur P (2006) A population balance model for flocculation of colloidal suspensions by polymer bridging. Chem Eng Sci 61 182-191... 

Runkana et al. [23, 25] developed population balance models (PBMs) for polymer-induced flocculation by two well-known mechanisms, simple charge neutralization [23] and bridging [25]. They assumed that polymer adsorption on oppositely charged particle surfaces is very fast and equilibrium conformation is achieved before collisions between particles take place. It was also assumed that polymer adsorbs uniformly and polymer surface coverage and adsorbed layer thickness are the same for all particles. The composite polymer-coated particle radius was estimated by adding adsorbed layer thickness to the soM particle radius. 

In a later development, Somasundaran et al. ]57] developed a PBM for aggregation by polymers in shear environments. The D LVO theory was extended for this case, as discussed in the previous section, by using the modifled expression for van der Waals attraction for particles covered with polymers and the expression for bridging attraction or steric repulsion derived from the scaling theory [25]. Their model was tested qualitatively with experimental data for the flocculation of colloidal alumina suspensions in the presence of PAA and was found to reproduce the observed experimental trends [60] reasonably well. 

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