Smoluchowski equation translational motion

For example, in the case of PS and applying the Smoluchowski equation [333], it is possible to estimate the precipitation time, fpr, of globules of radius R and translation diffusion coefficient D in solutions of polymer concentration cp (the number of chains per unit volume) [334]. Assuming a standard diffusion-limited aggregation process, two globules merge every time they collide in the course of Brownian motion. Thus, one can write Eq. 2 ... 

Self-diffusion is the random translational motion of ensembles of particles (molecules or ions) originating from their thermal energy. It is well known that diffusion, which is closely related to the molecular size of the diffusing species, is given by the Einstein-Smoluchowski equation, Eq. (6.1) [8] ... 

It is straightforward to include the translational motion into the Smoluchowski equation. Again the hydrodynamics is considered first. Suppose the rod is moving with the velocity F in a quiescent fluid (see Fig. 8.4). If the rod moves along , the rod will feel a hydrodynamic drag, which is parallel to V and is written as C F. On the other hand if F is perpendicular to u, the drag is again parallel to V and is written as Cj. V-... 

We can now write down the Smoluchowski equation which includes both the rotational and translational motions. Let V(R, u r) be the probabO-... 

The fundamental rate expression to be considered is the Smoluchowski relation k = 4n iVDAB AB (Equation (2.1)). The derived expression ART/r] (Equation (2.3a)), is a useful approximation, but deviations from it are observed, because the Stokes-Einstein equation which is involved is derived by hydrodynamic theory for spherical particles moving in a continuous fluid, and does not accurately represent the measured values of translational diffusion coefficients in real systems. Although the proportionality Da 1 /rj is indeed a reasonable approximation for many solutes in common solvents, the numeral coefficient 1 /4 is subject to uncertainty. In the first place, this theoretical value derives from the assumption that in translational motion there is no friction between a solute molecule and the first layer of solvent molecules surrounding it, i.e., that slip conditions hold. If, however, one assumes instead that there is no slipping ( stick conditions), so that momentum is... 

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