Einstein relations, kinetic coefficients

Flocculation kinetics can be described in different ways. Here we introduce a treatment first suggested by Smoluchowski [547], and described in Ref. [538], p. 417. The formalism can also be used to treat the aggregation of sols. A prerequisite for coalescence is that droplets encounter each other and collide. Smoluchowski calculated the rate of diffusional encounters between spherical droplets of radius R. The rate of diffusion-limited encounters is SttDRc2, where c is the concentration of droplets (number of droplets per unit volume). For the diffusion coefficient D we use the Stokes-Einstein relation D = kBT/finr/R. The rate of diffusion-limited encounters is, at the same time, the upper limit for the decrease in droplet concentration. Both rates are equal when each encounter leads to coalescence. Then the rate of encounters is given by... 

The above-mentioned computer simulation and experimental studies have addressed various aspects of mass dependence, but they all show that the selfdiffusion coefficient of a tagged molecule exhibits a weak mass dependence, especially for solutes with size comparable to or larger than the size of the solvent molecules. Sometimes this mass dependence can be fitted to a power law, with a small exponent less than 0.1 [99]. This weak mass dependence has often been considered as supportive of the hydrodynamic picture. In hydrodynamics the diffusion of a solute is conventionally described by the well-known Stokes-Einstein (SE) relation, which predicts that the diffusion is totally independent of the mass of the solute. Kinetic theory, on the other... 

In the first part of the argument, Einstein derived a relationship between the diffusion coefficient and other physical quantities. On the basis of the molecular kinetic theory of heat, he asserted that particles suspended in a liquid will experience the same osmotic pressure that molecules do. If an external force K acts on a suspension of Brownian particles, then in equilibrium this force will be balanced by osmotic pressure forces given by the relation... 

The first cumulant is related to the mean particle diffusion coefficient by = D)q. Once the mean diffusion coefficient is determined, the average hydrodynamic size may be calculated using the Stokes-Einstein relationship. The homogeneity exponent. A, is then determined from the kinetic... 

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