Smoluchowski equation modified

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... 

It is important to notice that both the original and the modified Fokker-Planck equations give the probability distribution of a particle as a function of time, position and velocity. However, if we are interested in time intervals large enough compared to jS 1, the Fokker-Planck equation, equation (3), can be reduced to a diffusional equation for the distribution function w, frequently called the Smoluchowski equation (Chandrasekhar, 1943) ... 

In order to establish the validity condition of a diffusion like equation for the probability of escape of a particle over a potential barrier, the solution of the modified Fokker-Planck equation is compared to the solution of the modified Smoluchowski equation. Since the main contribution to the determination of the escape probability comes from the neighborhood of the maximum in the potential energy (x = x J, the potential energy function was approximated by a parabolic function and the original Fokker-Planck equation was approximated at the vicinity of xmax by (Chandrasekhar, 1943) ... 

A few corrections have been proposed to modify the Helmholtz-Smoluchowski equation (3, 16, 28). Under certain experimental conditions many of these corrections become too small to be considered (6). However, these corrections are not as important as those that have to be introduced when potentials are calculated from mobility data (4, 5, 20, 21, 32, 33). 

On closer inspection, the combination rate constants are about 1/4 of the estimated diffusion-controlled rate constant. For acetonitrile, for example, fcjj - 2.9 X 10 L mol" s from the von Smoluchowski equation wiA a diffusion coefficient from a modified version of the Stokes-Einstein relation, D - fcT/4jiT r. Owing to the restriction to singlet state recombination, an experimental rate constant 1 /4 of is quite reasonable. On the other hand, for these heavy metals, the spin restriction may not apply, in which case one would argue that the geometrical and orientational requirements of these large species could well give recombination rates somewhat below the theoretical maximum. 

The kinetics of flocculation in a fluidized bed are described by the equations in section 4.8. The simplified Smoluchowski equation 4.42 can be modified by the special bimodal distribution of the fluidized bed (floe blanket) particles ... 

This analysis concerns for instance the formation of the twisted intramolecular charge transfer state, the photoisomerization processes, etc... in solution for which the intramolecular motion is related with the solvent motion. The unimolecular reaction -the passage from the reactant well to the product well in the Kramers treatment- is modeled by a sink term depending on the reaction coordinate. In the high-viscosity case the motion is governed by the modified Smoluchowski equation [6 ]... 

What is quite surprising in these plots is not the (solution-like) Stem-Volmer behavior that is found, but the diffusion coefficients which can be calculated for MV2+. From the luminescence lifetime of Ru(bpy)32+ in the absence of quencher, the kgv values can be converted to bimolecular quenching rate constants kq. The diffusion coefficient is derived from kq by means of a modified Smoluchowsky equation (equation 1), where Dq is the diffusion coefficient of MV2+, Rq is its radius (taken to be 6.7 A), R is the average... 

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. 

In some cases, particularly in the growth of aerosol particles, the assumption of equilibrium at the interface must be modified. Frisch and Collins (F8) consider the diffusion equation, neglecting the convective term, and the form of the boundary condition when the diffusional jump length (mean free path) becomes comparable to the radius of the particle. One limiting case is the boundary condition proposed by Smoluchowski (S7), C(R, t) = 0, which presumes that all molecules colliding with the interface are absorbed there (equivalent to zero vapor pressure). A more realistic boundary condition for the case when the diffusion jump length, (z) R, has been shown by Collins and Kimball (Cll) and Collins (CIO) to be... 

Application of Smoluchowski s equations typically results in an overestimation of the growth rate of aggregates due to the assumption that all collisions result in permanent attachment. Recognition of the importance of surface properties in the aggregation of small particles prompted Fuchs [6] to develop expressions to modify Smoluchowski s equations. Fuchs described the effect of the repulsive electrostatic interaction between two particles, which is a function of the particle separation distance, as a reduction in particle coagulation rate, Wy, termed the stability ratio. 

Fredrickson3 has formulated expressions for the concentration depolarization of fluorescence in the presence of molecular rotation. A theoretical examination of diffusion influenced fluorescence quenching by nearest possible quenching neighbours in liquids has been made35. A modified version of Smoluchowski - Collins - Kimball formulation of the Stern - Volmer equations has been matched with experimental data for quenching of anthraquinone derivatives by N,N-dimethyl- -toluidine. Another paper discusses this work on the basis of the kinetics of partly diffusion controlled reactions3 . 

In 1809, Reuss observed the electrokinetic phenomena when a direct current (DC) was applied to a clay-water mixture. Water moved through the capillary toward the cathode under the electric field. When the electric potential was removed, the flow of water immediately stopped. In 1861, Quincke found that the electric potential difference across a membrane resulted from streaming potential. Helmholtz first treated electroosmotic phenomena analytically in 1879, and provided a mathematical basis. Smoluchowski (1914) later modified it to also apply to electrophoretic velocity, also known as the Helmholtz-Smoluchowski (H-S) theory. The H-S theory describes under an apphed electric potential the migration velocity of one phase of material dispersed in another phase. The electroosmotic velocity of a fluid of certain viscosity and dielectric constant through a surface-charged porous medium of zeta or electrokinetic potential (0, under an electric gradient, E, is given by the H-S equation as follows ... 

For large values of Ka, i.e., for high electrolyte concentrations, Smoluchowski modified the Henry equation as... 

Noyes [8], Wilemski and Fixman [11] have pointed out that it is not strictly correct to apply Smoluchowski [1] or radiation [7] boundary conditions to the diffusion equation to model bimolecular chemical reactions. Both Teramoto and Shigesada [12] and Wilemski and Fixman [11] have proposed a modified diffusion equation by introducing a sink term to represent the reaction rate at a set of relative phase-space coordinates of two reacting species. Let a simple diffusive process be described as... 

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