Smoluchowski effect

Fig. 34. Work-function change during deposition of Cu onto Cu (the substrate of the Cu layers is Pt(lll)). Temperature dependence of the observed oscillations at almost identical deposition rates (a) 420 K (b) 400 K (c) 370 K (d) 340 K. Upon lowering the deposition temperature from 420 K, where besides an initial decrease no periodic variation is found, to 340 K an oscillatory behavior sets in and the amplitude of the work-function-change oscillations increases. This experiment nicely demonstrates the influence of surface roughness on work function (Smoluchowski effect). From [95N].

Here, as in the discussion of the Smoluchowski effect, we have made expHdt reference to the discrete, periodic stracture of the lattice. The symmetry ofa periodic lattice is substantially lower than the continuous translational and rotational symmetry of the uniform jeUium model. Only translations involving a Bravais lattice vector and a discrete number of point symmetry operations remain and the constant potential has to be replaced by a crystal potential, which reflects the lowered symmetry. This has profound consequences for the electronic structure. 

Xd takes into account the Smoluchowski effect. X is the heat conductivity of the continuum gas, a the mean free path of the molecules, y he accomodation coefficient and xp is the average gap width between the particles with respect to heat conduction in a ratified gas and is to be calculated from i=n f j... 

The simple treatment of this and of other electrokinetic effects was greatly clarified by Smoluchowski [69] for electroosmosis it is as follows. The volume flow V (in cm /sec) for a tube of radius r is given by applying the linear velocity V to the body of liquid in the tube... 

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... 

Smoluchowski see von Smoluchowski) Solvent cage, 198, 202 Solvent effects. 197-199, 204—206 Specific acid-base catalysis,... 

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 order to describe the effects of the double layer on the particle motion, the Poisson equation is used. The Poisson equation relates the electrostatic potential field to the charge density in the double layer, and this gives rise to the concepts of zeta-potential and surface of shear. Using extensions of the double-layer theory, Debye and Huckel, Smoluchowski,... 

Illustration Effect of flow type on shear induced collisions in homogenous linear flows. The collision frequency for a general linear flow [Eq. (15)] is obtained following Smoluchowski s (1917) approach as (Bidkar and Khakhar, 1990)... 

The Smoluchowski-Levich approach discounts the effect of the hydrodynamic interactions and the London-van der Waals forces. This was done under the pretense that the increase in hydrodynamic drag when a particle approaches a surface, is exactly balanced by the attractive dispersion forces. Smoluchowski also assumed that particles are irreversibly captured when they approach the collector sufficiently close (the primary minimum distance 5m). This assumption leads to the perfect sink boundary condition at the collector surface i.e. cp 0 at h Sm. In the perfect sink model, the surface immobilizing reaction is assumed infinitely fast, and the primary minimum potential well is infinitely deep. 

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, ... 

Studies on orthokinetic flocculation (shear flow dominating over Brownian motion) show a more ambiguous picture. Both rate increases (9,10) and decreases (11,12) compared with orthokinetic coagulation have been observed. Gregory (12) treated polymer adsorption as a collision process and used Smoluchowski theory to predict that the adsorption step may become rate limiting in orthokinetic flocculation. Qualitative evidence to this effect was found for flocculation of polystyrene latex, particle diameter 1.68 pm, in laminar tube flow. Furthermore, pretreatment of half of the latex with polymer resulted in collision efficiencies that were more than twice as high as for coagulation. 

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... 

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