Smoluchowski fast coagulation rate

The so-called "Smoluchowski fast coagulation rate" may be characterized by a half-life independent of particle size (36) ... 

In the absence of a barrier to coagulation, and if the primary minimum is deep, every collision between a particle and a floe will lead to the growth of the floe. The rate of coagulation is then controlled entirely by the kinetics of the diffusion process leading to particle-particle collision. The theory of fast coagulation was developed originally by Smoluchowski (1918) and elaborated by Muller (1926). The rate equation has the same form as that for a bimolecular reaction ... 

If we remove the electrostatic barrier, e.g. by ion adsorption or by adding electrolyte, we can obtain a fast coagulation that is controlled by the difiusion mechanism and where we can assume that all collisions lead to adhesion of molecules. Smoluchowski has derived the following equation for the rate constant in the case of fast coagulation ... 

Plots of log W versus log C are shown in Figure 7.11. The condition log W = 0 (W = 1) is the onset of fast flocculation, and the electrolyte concentration at this point defines the critical coagulation concentration (ccc). Above the ccc, W will be less than 1 (due to the contribution of van der Waals attraction which accelerates the rate above the Smoluchowski value), whereas below the ccc, W will be greater than 1, and wiU increase in Hne with a decrease in electrolyte concentration. The data in Figure 7.11 also shows that the ccc decreases with an increase of valency. At low surface potentials, ccc oo 1/Z this referred to as the Schultze-Hardy mle. 

Note also that for imaginary particles, which experience neither long-range surface forces (JJi j = 0) nor hydrodynamic interactions (P = 1), Equation 5.325a yields a collision efficiency = 1 and Equation 5.321 reduces to the von Smoluchowski " expression for the rate constant of the fast irreversible coagulation. In this particular case. Equation 5.319 represents an infinite set of nonlinear differential equation. If all flocculation rate constants are the same and equal to Up the problem has a unique exact solution " ... 

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