Kramers-type rate theory

This analysis has been extended to allow for the de-grafting dynamics in the case of aggregated diblock copolymers ". The resulting Kramers type rate theory enables the analysis of the relaxation dynamics of copolymeric micelles". Related studies were reported by Ligoure and Liebler and by Johner and Joanny who respectively considered copolymer adsorption out of non-selective and selective solvents. 

The critical nucleus of a new phase (Gibbs) is an activated complex (a transitory state) of a system. The motion of the system across the transitory state is the result of fluctuations and has the character of Brownian motion, in accordance with Kramers theory, and in contrast to the inertial motion in Eyring s theory of chemical reactions. The relationship between the rate (probability) of the direct and reverse processes—the growth and the decrease of the nucleus—is determined from the condition of steadiness of the equilibrium distribution, which leads to an equation of the Fourier-Fick type (heat conduction or diffusion) in a rod of variable cross-section or in a stream of variable velocity. The magnitude of the diffusion coefficient is established by comparison with the macroscopic kinetics of the change of nuclei, which does not consider fluctuations (cf. Einstein s application of Stokes law to diffusion). The steady rate of nucleus formation is calculated (the number of nuclei per cubic centimeter per second for a given supersaturation). For condensation of a vapor, the results do not differ from those of Volmer. 

An additional but very important type of quenching process involves the photodissociation of weak bonds, such as found for tetraphenylhydrazine (Scheme 1) where the rate of dissociation into diphenylaminyl radicals shows a strong viscosity dependence. This reaction has been considered in terms of the high damping regime of the Kramers-Smoluchowski theory and it appears that, in addition to the activated process, there is an activationless dissociative pathway that favours formation of free radicals at infinite viscosity. 

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