Smoluchowski model

Smoluchowski theory [29, 30] and its modifications fonu the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in tenus of difhision effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological difhision equation in the absence of external forces. In the standard picture, one considers a dilute fluid solution of reactants A and B with [A] [B] and asks for the time evolution of [B] in the vicinity of A, i.e. of the density distribution p(r,t) = [B](rl)/[B] 2i ] r(t))l ] Q ([B] is assumed not to change appreciably during the reaction). The initial distribution and the outer and inner boundary conditions are chosen, respectively, as... 

The weakness of this boundary condition is being able to justify a large enough distance to be comparable with an infinite distance, or perhaps 1000R. In practice, this would require B to be in excess over A by about 109 times A more reasonable approach to this outer boundary condition would be to require that there be no loss or gain of matter over this boundary, as there is an approximately equal tendency for the B reactants to migrate towards either A reactant upon each side of the boundary. The proper incorporation of this type of boundary condition into the Smoluchowski model leads to the mean field theory of Felderhof and Deutch [25] and is discussed further in Chap. 8 Sect. 2.3 and Chap. 9 Sect 5. 

On the right-hand side, the terms represent the rate of loss A and all m quencher molecule density by diffusion of A and of each of the Q quencher molecules and, finally, by reaction of each quencher with A. The boundary conditions on the density n are different from those of the Smoluchowski model. No loss of any particle can occur on. the outer surface (Vnj , -> 0 etc.), i.e. a closed system such as a glass beaker Where the quencher and fluorophor can interpenetrate each other, there is no net... 

First, the diffusional radical reactions in solutions whose rates are proportional to I) sometimes have rate constants that are much smaller than their contact estimate for the isotropic black sphere, ki> = 4naD. It was proved that this is the result of chemical anisotropy of the reactants. Partially averaged by translational and rotational diffusion of reactants, this anisotropy manifests itself via the encounter efficiency w < 1, which enters the rate constant kn = w4nak) [249]. Even the model of white spheres with black spots is more appropriate for such reactions than the conventional Smoluchowski model. 

The difference between the Markov model lineshapes and those from the Smoluchowski model is particularly pronounced when the diffusion coefficient is of the order of the quadrupole coupling constant. In the limit of large diffusion coefficients, the two models converge, and in the limit of low diffusion coefficients, the spectra are dominated by small-amplitude oscillations within potential wells, which can be approximately modelled by a suitable Markov model. This work strongly suggests that there could well be cases where analysis of powder pattern lineshapes with a Markov model leads to a fit between experimental and simulated spectra but where the fit model does not necessarily describe the true dynamics in the system. 

We start with the two-body Smoluchowski model (2BSM) the details of the formulation (matrix and starting vector) are discussed in Section II.C. A stochastic system made of two spherical rotators in a diffusive (Smoluchowski) regime has been used recently to interpret typical bifurcation phenomena of supercooled organic liquids [40]. In that work it was shown that the presence of a slow body coupled to the solute causes unusual decay behavior that is strongly dependent on the rank of the interaction potential. 

The next model that we have treated in order of complexity is a three-body Smoluchowski model (3BSM). A field X has been included, coupled exclusively through first rank (dipole-field) interactions to the two spherical rotators. No direct coupling has been taken to exist... 

Marcus12 and others13 extended this model to include reactions in which electron transfer occurred during collisions between the donor and acceptor species, that is, between the short-lived Dn—Am complexes. In this context, electron transfer within transient precursor complexes ([Dn — A" in Scheme 1.1) resulted in the formation of short-lived successor complexes ([D(, + — A(m 1)] in Scheme 1.1). The Debye-Smoluchowski description of the diffusion-controlled collision frequency between D" and A " was retained. This has important implications for application of the Marcus model, particularly where—as is common in inorganic electron transfer reactions—charged donors or acceptors are involved. In these cases, use of the Marcus model to evaluate such reactions is only defensible if the collision rates between the reactants vary with ionic strength as required by the Debye-Smoluchowski model. The requirements of that model, and how electrolyte theory can be used to verify whether a reaction is a defensible candidate for evaluation using the Marcus model, are presented at the end of this section. 

Illustrative Cases. Three cases are illustrated in Figure 9, marked by the circles labeled A, B, and C. Case A refers to classical experiments by Swift and Friedlander (27) on the coagulation of monodisperse latex particles (diameter = 0.871 Jim) in shear flow and in the absence of repulsive chemical interactions. Considering a velocity gradient of 20 s-1, HA is 0.0535, log HA is — 1.27, and djd- is 1.0 for these experimental conditions. The circle labeled A in Figure 9 marks these conditions and indicates that the hydrodynamic corrections to Smoluchowskis model predict a reduction of about 40% in the aggregation rate by fluid shear. The experimental measurements by Swift and Friedlander showed a reduction of 64%. This observed reduction from Smoluchowskis rectilinear model was therefore primarily physical or hydro-dynamic and consistent with the curvilinear model. 

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