Smoluchowski kinetics

Fig. 6. Normalized time-dependent rate coefficient. Solid line corresponds to Effective Medium Approximation, and dotted line shows the usual Smoluchowski kinetics (without any obstruction). [Reproduced with permission from Ref. 41, Copyright 2003 American Chemical Society].

Particle number concentrations for various half-lives according to Smoluchowski kinetics... 

Both modes can occur in precipitation processes, but in a stirred precipitator orthokinetic agglomeration clearly predominates. From the Smoluchowski kinetic expressions for perikinetic and orthokinetic agglomeration it can be deduced (Sohnel and Mullin, 1991) that the relationship between agglomerate size D and time t may be expressed by... 

At very low polymer volume fractions, the entry follows Smoluchowski kinetics Sm = 1), whereas for polymer volume fractions greater than 0.1% Smis increased (Figure 25.4). The BD simulations reveal an almost linear dependence of the radical capture rate coefficient on the polymer volume fraction in the dispersion, cf. Equation 25.26 ... 

For rapid coagulation induced by the addition of 1.0 mol dm KQ, the Smoluchowski kinetic scheme gave an excellent description for the rapid coagulation of colloidal dispersions (Figures 12 and 13). This was shown by the linearity of the second order rate constant plots the lack of any concentration effects the closeness of the final rate constants to the diffusion-limited value of 6.1 X 10" cm s ... 

The rate of collision of antifoam drops will be determined by a combination of diffusion and buoyancy. Here we simplify by considering these two processes separately in order to illustrate their relative significance. Under stagnant and neutral buoyancy conditions the rate of collision will be determined by diffusion and can be described by so-called Smoluchowski kinetics [91]. If we assume that the emulsion is monodisperse we can therefore write for the initial rate of colloision dA p/df, where Aaf is the concentration of antifoam drops per unit volume of detergent liquid, that... 

The pair correlation function is assumed to obey the Kirkwood-Smoluchowski kinetic equation (Hess 1980). In the case of Couette flow, the equation is... 

Obviously, these fines can interact with other components in the papermaking suspension, such as colloids, polymers and polyelectrolytes. Despite their complicated shapes and polydispersity in size and shape, the homoflocculation of fines subjected to shear follows Smoluchowski kinetics, since flocculation times were found to follow adequately the predictions of Smoluchowski s theory [11]. Hence heteroflocculation is expected to follow these predictions as well. Heteroflocculation between papermaking fines and day, treated with PEI, has been studied experimentally, as was that between MCC (microcrystalline cellulose) and clay [1 Ij. The behavior of these systems is very similar to the heteroflocculation of clay with latex (discussed above). When the clay concentration was below monolayer coverage of clay on fines (or MCC), the fines flocculated, whereas when the clay concentration was above monolayer coverage, the system was stable, that is it consisted of fines, fully coated by clay. 

There are two approaches to the kinetics of emulsion flocculation. The first stems from a relationship due to Smoluchowski [52] for the rate of diffusional encounters, or flux ... 

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... 

For a more complete understanding of colloid stability, we need to address the kinetics of aggregation. The theory discussed here was developed to describe coagulation of charged colloids, but it does apply to other cases as well. First, we consider the case of so-called rapid coagulation, which means that two particles will aggregate as soon as they meet (at high salt concentration, for instance). This was considered by von Smoluchowski 1561 here we follow [39, 57]. 

There are two general theories of the stabUity of lyophobic coUoids, or, more precisely, two general mechanisms controlling the dispersion and flocculation of these coUoids. Both theories regard adsorption of dissolved species as a key process in stabilization. However, one theory is based on a consideration of ionic forces near the interface, whereas the other is based on steric forces. The two theories complement each other and are in no sense contradictory. In some systems, one mechanism may be predominant, and in others both mechanisms may operate simultaneously. The fundamental kinetic considerations common to both theories are based on Smoluchowski s classical theory of the coagulation of coUoids. 

Smoluchowski, M.V., 1917. Mathematical theory of the kinetics of coagulation of colloidal systems. Zeitschrift fur Physikalische Chemie, 92, 129-168. 

All models described up to here belong to the class of equilibrium theories. They have the advantage of providing structural information on the material during the liquid-solid transition. Kinetic theories based on Smoluchowski s coagulation equation [45] have recently been applied more and more to describe the kinetics of gelation. The Smoluchowski equation describes the time evolution of the cluster size distribution N(k) ... 

Galina, H. Mean-Field Kinetic Modeling of Polymerization The Smoluchowski Coagulation Equation. Vol. 137, pp. 135-172. 

In an early attempt, Mozumder (1968) used a prescribed diffusion approach to obtain the e-ion geminate recombination kinetics in the pure solvent. At any time t, the electron distribution function was assumed to be a gaussian corresponding to free diffusion, weighted by another function of t only. The latter function was found by substituting the entire distribution function in the Smoluchowski equation, for which an analytical solution was possible. The result may be expressed by... 

The foundations of the theory of flocculation kinetics were laid down early in this century by von Smoluchowski (33). He considered the rate of (irreversible) flocculation of a system of hard-sphere particles, i.e. in the absence of other interactions. With dispersions containing polymers, as we have seen, one is frequently dealing with reversible flocculation this is a much more difficult situation to analyse theoretically. Cowell and Vincent (34) have recently proposed the following semi-empirical equation for the effective flocculation rate constant, kg, ... 

The two major theories of flocculation, the bridging model (1) and the electrostatic patch model (2, 3 ), provide the conceptual framework for the understanding of polymer-aided flocculation, but they do not directly address the kinetics of the process. Smellie and La Mer (4) incorporated the bridging concept into a kinetic model of flocculation. They proposed that the collision efficiency in the flocculation process should be a function of the fractional surface coverage, 0. Using a modified Smoluchowski equation, they wrote for the initial flocculation rate... 

The Smoluchowski theory for diffusion-controlled reactions, when combined with the Stokes-Einstein equation for the diffusion coefficient, predicts that the rate constant for a diffusion-controlled reaction will be inversely proportional to the solution viscosity.16 Therefore, the literature values for the bimolecular electron transfer reactions (measured for a solution viscosity of r ) were adjusted by multiplying by the factor r 1/r 2 to obtain the adjusted value of the kinetic constant... 

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. 

Kinetic methods describing the evolution of distributions of molecules by systems of kinetic differential equations (obeying either the classic mass action law of chemical kinetics or the generalized Smoluchowski coagulation process). 

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... 

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