Einstein-Smoluchowski equation effects

Some general comments might be useful, however, before considering the individual methods. First, the techniques may be divided into (i) macroscopic methods, which are used to measure the effect of long-range motion of atoms and (ii) microscopic methods, which are used to measure the effect of jump frequencies of atoms [210, 212]. In principle, for a simple jump process via point defects in a solid, the two are interconnected by the classical Einstein-Smoluchowski equation [204] ... 

It is interesting to inquire why there should be an effect of (n) on the rate of reaction of a small molecule with a polymer coil. We are guided by the discussion of Behzadi and Schnabel (41) concerning the reaction of 0H (produced by pulse radiolysis) with poly(ethylene oxide), poly(vinylpyrrolidone) and dextrane. First, we assume that the coil-small molecule reaction rate can be described by the Einstein-Smoluchowski equation (where Ap 0 and PqE = 1 is assumed for simplicity) ... 

Our aim in this study was to evaluate the impact of gelatin on IL diffusion. For that purpose we have used a correlation to separate the mobility, i, and the effective number density of charge carriers, n, from conductivity (oq) obtained from the dielectric measurements, allowing also to estimate the diffusion coefficient of migrating charges, D, which is done by considering the following Einstein-Smoluchowski equation ... 

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. 

The Peclet number compares the effect of imposed shear (known as the convective effect) with the effect of diffusion of the particles. The imposed shear has the effect of altering the local distribution of the particles, whereas the diffusion (or Brownian motion) of the particles tries to restore the equilibrium structure. In a quiescent colloidal dispersion the particles move continuously in a random manner due to Brownian motion. The thermal motion establishes an equilibrium statistical distribution that depends on the volume fraction and interparticle potentials. Using the Einstein-Smoluchowski relation for the time scale of the motion, with the Stokes-Einstein equation for the diffusion coefficient, one can write the time taken for a particle to diffuse a distance equal to its radius R, as... 

Another reason Einstein s equation is inaccurate is because of the electrical double layer surrounding the particles in aqueous solution. The presence of a double layer gives an electro-viscous effect which causes (1) an extra force to be needed to move two similarly charged double layers past one another and (2) a larger effective volume of the particle, due to its double layer of immobilized solvent molecules. Von Smoluchowski [16] derived an equation for the primary electro-viscous effect... 

This concept which is based on a random walk with a well-defined characteristic waiting time (thus called a discrete-time random walk) and which applies when collisions are frequent but weak leads to the Smoluchowski equation for the evolution of the concentration of Brownian particles in configuration space. If inertial effects are included (see Note 8 of Ref. 2, due to Fiirth), we obtain the Klein-Kramers equation for the evolution of the distribution function in phase space which describes normal diffusion. The random walk considered by Einstein [2] is a walk in which the elementary steps are taken at uniform intervals in time and so is called a discrete time random walk. The concept of collisions which are frequent but weak can be clarified by remarking that in the discrete time random walk, the problem [5] is always to find the probability that the system will be in a state m at some time t given that it was in a state n at some earlier time. 

Application of the Smoluchowski equation (1.3) requires a knowledge of the sizes and diffusion coefficients of molecules. An estimate of the effective values of r can usually be made from molecular volumes. Diffusion coefficients present more of a problem, since not very many have been experimentally determined over a range of temperature. They may, however, be eliminated from Equation (1.3) by using the Stokes-Einstein relation, which is theoretically derived from the same molecular model and is often in fairly good accord with experimental data it expresses D in terms of the viscosity of the solvent (t)) and the... 

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. 

EmStoN s equation has been extended by SmolUchowski for the case where the dispersed particles are the carriers of an electric double layer The outermost layers extending into the liquid, which arc carried along by the moving liquid, experience a drag through the opposite charge on the particles, whereby an extra increase of the internal friction occurs (electroviscous effect) The modihed Einstein equation then takes on the form... 

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