Rapid Flocculation Theory

The birth and death functions predict the importance of particulate aggr ation on the final particle size distribution. The key concepts for the development of these two functions come from Smoluchowski s rapid flocculation theory, which was derived in Section 6.6.3  

fig is the viscosity of the gas and q is the sticJdng factor. The ith moment is described as follows  

Appl5ing the new birth and death functions given in equations (7.53) and (7.55), respectively, the governing differential equation for the population of particles, rj(r, t), becomes 

An analytical solution to this integro-differential equation is impossible without some simplifying assumptions. The birth fimction can be 

With this simplifsnng assumption in mind (i.e., equation (7.57)), the new population balance becomes 

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The rate of flocculation of selenium sols by solutions of potassium or barium chloride of various concentrations at temperatures between 15° and 20° C. has been determined.8 Results show that a very high concentration of these electrolytes is necessary for rapid flocculation. Smoluchowski s theory holds when the velocity of flocculation is not far removed from that obtaining when the colloidal particles are totally discharged. The results, however, deviate largely from this theory when the concentrations of the electrolytes are lower.3... 

We must now relate the energetics to the kinetics. The theory of rapid flocculation was first proposed by Smoluchowski , who treated the problem as one of diffusion (Brownian motion) of the spherical particles of an initially monodisperse dispersion, with every collision, in the absence of a repulsive force, leading to a permanent contact. In general, we may express the rate of flocculation, i.e. the rate of decrease of the total number of particles, as... 

Flocculation is a kinetic process and the rate at which a colloidal suspension flocculates forms one of its most important characteristics. Smoluchowski (1917) distinguished between rapid flocculation and slow flocculation and developed a theory based on the rate of collision between the particles (2). Rapid flocculation is considered to take place in the absence of a potential barrier and is limited only by the rate of diffusion of the particles towards one another. The flocculation time, defined as the time tia required for the number of particles to be reduced by one-half of the initial value is given by... 

Once formed, particles in solution interact with each other because of Brownian motion. The theory of rapid flocculation was first proposed by Smoluchowski,... 

For example, van den Tempel [35] reports the results shown in Fig. XIV-9 on the effect of electrolyte concentration on flocculation rates of an O/W emulsion. Note that d ln)ldt (equal to k in the simple theory) increases rapidly with ionic strength, presumably due to the decrease in double-layer half-thickness and perhaps also due to some Stem layer adsorption of positive ions. The preexponential factor in Eq. XIV-7, ko = (8kr/3 ), should have the value of about 10 " cm, but at low electrolyte concentration, the values in the figure are smaller by tenfold or a hundredfold. This reduction may be qualitatively ascribed to charged repulsion. 

Two diverse views of non-specific adhesion processes form the bases for contemporary theories introduced to rationalize observations of colloidal stability and flocculation in solutions of macromolecules (see 16-18 for general reviews). The first view is based on adsorption and cross-bridging of the macromolecules between surfaces. Theories derived from this concept indicate a strong initial dependence on concentration of macromolecules there is a rapid rise in surface adsorption for infinitesimal volume fractions (32) followed by a plateau with gradual attenuation of surface-surface attraction because of excluded volume effects in the gap at larger volume fractions (19-20). The interaction of the macroinolecule with the surface is assumed to be a snort range attraction proportional to area of direct contact. The second - completely disparate - view of non-specific adhesion is based on the concept that there is an exclusion or depletion of macromolecules in the vicinity of the surface, i.e. no adsorption to the surfaces. Here, theory shows that attraction is caused by interaction of tne (depleted) concentration profiles associated with each surface which leads to a depreciated macrornolecular concentration at the center of the gap. The concentration... 

We now consider the kinetics of the flocculation of a system for which there is an energy barrier. The rate is a function of the probability of particles having sufficient energy to overcome this barrier. The problem is again one of diffusion and was first solved by Fuchs . His theory leads to a factor W by which the rapid rate (Smoiuchowski) is reduced by the presence of a repulsive force W is called the stability ratio and is related to the height of the potential energy barrier by... 

Given the appropriate potential energy diagrams from the DLVO theory, the stability ratio may be calculated by graphical or numerical integration and then compared with experimental values of W=kyk, the ratio of the experimental rate constants for rapid and slow flocculation. Such a comparison is a severe test of the applicability of theory to experiment, and the observed deviations, although often not appreciable, reflect the assumptions and approximations which are necessary in the calculation of the potential energy terms. An advanced treatment of these issues will be found in Russel et al.- . 

According to the theory of von Smoluchowski the most rapid aggregation will occur when every contact leads to the adherence of one particle to another. So the rate of perikinetic flocculation or aggregation is given by... 

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