Fuchsian kinetics

A more mechanistic approach to reaction controlled flocculation entails representing the sticking probability in terms of a Boltzmann factor (hat 

The effect of the potential energy V(r) on the second-order rate coefficient kmn can be described mathematically in a simple way by modifying the expression for the diffusional flux in Eq. 6.15a to include the Boltzmann factor  

Thus the collision frequency CF is reduced by the Boltzmann factor when cluster interactions are repulsive and is increased when they are attractive. The result corresponding to Eq. 6.15b is then 

The integral on the left side of Eq. 6.50b defines the stability ratio Wmn 28 

It follows from Eqs. 6.15c, 6.16, and 6.51 that the rate coefficient kmn now becomes 

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Equation 6.52 exhibits the inverse of the stability ratio playing the role of the sticking probability coefficient that reduces kmn below its value for pure transport control of flocculation whenever cluster interactions are repulsive. (Note that WM1I) 1 when V(r) vanishes, according to Eq. 6.51.) liquation 6.52 is a model for kni(1 known as Fuchsian kinetics. ... 

Fuchsian kinetics lead to the model form of the von Smoluchowski rate law that is obtained by introducing Eq. 6.52 into Eq. 6.10 ... 

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