Diffusion Smoluchowski model

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. 

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. 

Smoluchowski theory [29, 30] and its modifications form the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in terms of diffusion effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological diffusion 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,f) = i[B](r( if))/[B]j ([B] is assumed not to change appreciably during the... 

The number of models that has been presented in literature to account for the complicated diffusion control of bimolecular termination reactions is overwhelming. The majority of these efforts are based on the Smoluchowski model [202], which was originally developed for colloidal systems. The Smoluchowski model describes the rate coefficient for a diffusion-controlled reaction and is based on the appearance of concentration gradient in a homogeneous mixture. This gradient is formed as a result of the chemical reaction between species, radicals in this case, which causes a local depletion of the reactants. Consequently, a net flux of reactants is induced towards those loci which are poor in radicals. Von Smoluchowski was the first to use this treatment for diffusion-controlled reactions. ... 

Mahabadi and O Driscoll s starting point was somewhat different to the one used by Horie et al. The microscopic rate coefficient for bimolecular termination was now expressed as the product of (i) the rate coefficient for translational diffusion and (ii) the probability of reaction, both being dependent upon the separation distance between the two polymer coils. This reaction probability was supposed to be determined by the rate coefficient of segmental diffusion of the chain ends (Smoluchowski model [202]) and the time available for a termination reaction. The latter parameter effectively cancels out the influence of the translational diffusion coefficient, making the process only dependent upon the rate of segmental diffusion. The final equation that Mahabadi and O Driscoll obtained for k was simplified for practical purposes and expressed as a product of two functions ... 

In recent times, the translational diffusion model with a constant capture radius has been given new elan by Russell [97, 119, 130, 131, 214, 215], With computer simulations Russell has shown that a translational diffusion model can account for almost all experimentally observed phenomena (see section 2.3.5). He applied the Smoluchowski model (equation 2.15) in which the diffusion coefficients A were defined according to ... 

Model 3 Smoluchowski model [46] with chain-length dependent translational diffusion coefficients [45] ... 

The atbove-described Debye-Smoluchowski model is subject to severe limitations (Wilemski and Fixroam, 1973). First, the choice of the coordinate system is not self-evidently valid. Second, the mutual diffusion coefficient is assumed to be constant, even for short separation distances between A and B, where some vairiations are expected. Third, the diffusion ecjuation is only valid for low concentrations. A last limitation is due to the method of describing the reaction process in which it is... 

Perrin model and the Johansson and Elvingston model fall above the experimental data. Also shown in this figure is the prediction from the Stokes-Einstein-Smoluchowski expression, whereby the Stokes-Einstein expression is modified with the inclusion of the Ein-stein-Smoluchowski expression for the effect of solute on viscosity. Penke et al. [290] found that the Mackie-Meares equation fit the water diffusion data however, upon consideration of water interactions with the polymer gel, through measurements of longitudinal relaxation, adsorption interactions incorporated within the volume averaging theory also well described the experimental results. The volume averaging theory had the advantage that it could describe the effect of Bis on the relaxation within the same framework as the description of the diffusion coefficient. 

Diffusion occurs when there is a concentration gradient of one kind of molecule within a fluid. In terms of random walk model, the average distance, x, after an elapsed time, t, between molecule collisions in a diffusion movement is characterized by the Einstein-Smoluchowski relation,... 

The experimental and simulation results presented here indicate that the system viscosity has an important effect on the overall rate of the photosensitization of diary liodonium salts by anthracene. These studies reveal that as the viscosity of the solvent is increased from 1 to 1000 cP, the overall rate of the photosensitization reaction decreases by an order of magnitude. This decrease in reaction rate is qualitatively explained using the Smoluchowski-Stokes-Einstein model for the rate constants of the bimolecular, diffusion-controlled elementary reactions in the numerical solution of the kinetic photophysical equations. A more quantitative fit between the experimental data and the simulation results was obtained by scaling the bimolecular rate constants by rj"07 rather than the rf1 as suggested by the Smoluchowski-Stokes-Einstein analysis. These simulation results provide a semi-empirical correlation which may be used to estimate the effective photosensitization rate constant for viscosities ranging from 1 to 1000 cP. 

Alternatively, one can make the reactivity of groups dependent on the size and shape of the reacting molecule. In such a way, for instance, the effect of steric hindrances, cyclization, and diffusivities of the molecules can be modeled using generalized Smoluchowski coagulation differential equations. 

A probabilistic kinetic model describing the rapid coagulation or aggregation of small spheres that make contact with each other as a consequence of Brownian motion. Smoluchowski recognized that the likelihood of a particle (radius = ri) hitting another particle (radius = T2 concentration = C2) within a time interval (dt) equals the diffusional flux (dC2ldp)p=R into a sphere of radius i i2, equal to (ri + r2). The effective diffusion coefficient Di2 was taken to be the sum of the diffusion coefficients... 

Now, everything falls into place We set out to study the laws of random walk by using the simple model of Fig. 18 and found the Bernoulli coefficients. We then saw that for large n (which is equivalent to large times), the Bernoulli coefficients can be approximated by a normal distribution whose standard deviation, a, grows in proportion to the square root of time, tm (Eq. 18-3). And now it turns out that the solution of the Fick s second law for unbounded diffusion is also a normal distribution. In fact, the analogy between Eqs. 18-3b and 18-17 gave the basis for the law by Einstein and Smoluchowski (Eq. 18-17) that we used earlier (Eq. 18-8). The expression (2Dt)U2 will also show up in other solutions of the diffusion equation. 

Over molecular length scales, the diffusion distances become very short (< 1 nm) so that only very rapid events can be influenced by these short diffusion times. Necessarily, this limits the number of systems to only relatively few, where the rate at which the reactants can approach one another is slow or comparable with the rate at which the reactants react chemically with each other. Some typical systems which have been studied are discussed in Sect. 2. The Smoluchowski [3] theory of reactions in solution, which occur at a rate limited solely by how fast the reactants can approach each other (sufficiently closely to react chemically almost instantaneously) is discussed in Sect. 3. If the chemical reaction is not so rapid, the observed rate of reaction may be influenced by both the rate of approach and the rate of subsequent chemical reaction. Collins and Kimball [4], and later Noyes [5], have extended the Smoluchowski theory (1917) to consider this situation (Sect. 4). In light of these quantitative theoretical models of diffusion-limited rate processes, some of the more recent and careful experiments on diffusion-controlled reactions in solution are considered briefly in Sect. 5. As the Smoluchowski theory... 

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

Bagchi and co-workers [47-50] have explored the role of translational diffusion in the dynamics of solvation by employing a Smoluchowski-Vlasov equation (see also Calef and Wolyness [37] and Nichols and Calef [42]). A significant contribution to polarization relaxation is observed in certain cases. It is found that the Onsager inverted snowball model is correct only when the rotational diffusion mechanism of solvation dominates the polarization relaxation. The Onsager model significantly breaks down when there is an important translational contribution to the polarization relaxation [47-50]. In fact, translational effects can rapidly accelerate solvation near the probe. In certain cases, the predicted behavior can actually approach the uniform continuum result that rs = t,. 

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