Aggregation stability ratio

In slow coagulation, particles have to diffuse over an energy barrier (see the previous section) in order to aggregate. As a result, not all Brownian particle encounters result in aggregation. This is expressed using the stability ratio IV, defined as... 

A combination of equation (C2.6.13), equation (C2.6.14), equation (C2.6.15), equation (C2.6.16), equation (C2.6.17), equation (C2.6.18) and equation (C2.6.19) tlien allows us to estimate how low the electrolyte concentration needs to be to provide kinetic stability for a desired lengtli of time. This tlieory successfully accounts for a number of observations on slowly aggregating systems, but two discrepancies are found (see, for instance, [33]). First, tire observed dependence of stability ratio on salt concentration tends to be much weaker tlian predicted. Second, tire variation of tire stability ratio witli particle size is not reproduced experimentally. Recently, however, it was reported that for model particles witli a low surface charge, where tire DL VO tlieory is expected to hold, tire aggregation kinetics do agree witli tire tlieoretical predictions (see [60], and references tlierein). 

In order to take particle-particle interactions into account, a stability ratio W is included which relates the collision kernel /So to the aggregation kernel /3agg. The stability ratio W depends on the interaction potential aggregation rate without to the rate with interactions additional to the omnipresent van der Waals forces. For Brownian motion as dominant reason for collisions, the stability ratio W can be calculated according to Eq. (6) taken from Fuchs [ 10]. In case of shear as aggregation mechanism, the force dip/dr relative to the friction force should rather be considered instead of the ratio of interaction energy relative to thermal energy. 

Darling and van Hooydonk (1981) also considered how to reduce the diffusional collision rate to obtain slow coagulation and used the classical approach of Fuchs (Reerink and Overbeek, 1954), whereby an activation energy is computed from the pair interaction free energy of the aggregating particles. The reaction kernel is given by Eq. (6) divided by the stability ratio W,... 

The solubility of a block copolymer in water decreases as the concentration of electrolyte increases. When the concentration of the polymer is larger than its solubility, the polymer molecules precipitate onto the surface or form aggregates that remain dispersed into the colloidal system or deposit on the wall of the vessel that contains the colloidal dispersion. Let us start from two parallel plates, and then calculate the stability ratio of the system for spheres using the Detjaguin approximation. 

Application of Smoluchowski s equations typically results in an overestimation of the growth rate of aggregates due to the assumption that all collisions result in permanent attachment. Recognition of the importance of surface properties in the aggregation of small particles prompted Fuchs [6] to develop expressions to modify Smoluchowski s equations. Fuchs described the effect of the repulsive electrostatic interaction between two particles, which is a function of the particle separation distance, as a reduction in particle coagulation rate, Wy, termed the stability ratio. 

This simplification is realized when the population of nuclei is much larger than the population of aggregates, which is the case at high levels of saturation where nucleation plays a dominant role. This simplification is also justified by the fact that the colloidal stability ratio, IF, is large for aggreate-aggregate collisions (i.e., is small) but small for... 

For the CSTR in problem 6 the average colloid stability ratio is 1.0 due to the high concentration of NH4CI in the solution. Determine the particle size distribution produced by the precipitator if aggregation is also considered. Use the viscosity of water 1 cp. 

This expression was obtained by setting the shear aggregation rate to the Brownian aggregation rate and solving for the size a to which this corresponds assuming the colloid stability ratio, W is unity. 

When there is an energy barrier to aggregation, only a fraction 1/W of encounters lead to attachment. The variable W is the stability ratio, W = k2 /k2- Using W gives slow coagulation (hindered) times. In this case, the interaction energy and hydrodynamic viscous drag forces must be considered (J6). 

The stability ratio, w, of a dispersion is defined as the ratio of the rate constants for aggregation in the absence, and the presence, of an energy barrier, respectively ... 

If a significant repulsion exists in the interaction potential then only a small percentage of particle-particle collisions will result in a stable bond. The characteristic time for aggregation is extended by a factor W, called the stability ratio, which takes into account this likelihood of reaction ... 

For very large stability ratios, where only a tiny fraction of collisions between particles result in a stable bond, the process is termed reaction limited aggregation (RLA). 

In the case of slow coagulation the number of collisions between particles leading to their aggregation decreases due to the presence of a potential barrier, which prevents the particles from approaching each other. The fraction of successful (i.e. leading to aggregation) collisions is referred to as the collision efficiency, a. Inverse of collision efficiency is the stability ratio, W= la, which is equal to the ratio of true coagulation rate constant to that predicted by the Smoluchowski equation (VII.29). The stability ratio... 

First, the observed dependence of stability ratio on salt concentration tends to be much weaker than predicted. Second, the variation of the stability ratio with particle size is not reproduced experimentally. Recently, however, it was reported that for model particles with a low surface charge, where the DLVO theory is expected to hold, the aggregation kinetics do agree with the theoretical predictions (see [60], and references therein). 

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