Brownian diffusion relative motion between particles

We begin by examining the rate of collision of suspended spherical particles in a static fluid due to Brownian motion. This theory was first put forward by the great Polish physicist M. Von Smoluchowski, to whom we have often referred. In consequence of the equivalence between diffusion and Brownian motion, we consider the relative motion between the particles as a diffusion process. The particles are assumed to be in sufficiently dilute concentration that only binary encounters need be considered. To further simplify the calculation, we consider the suspension to be made up of only two different-sized spherical particles, one of radius a, and the other of radius... 

Here Do is the coefficient of Brownian diffusion determined by the formula (8.70), and corresponding to the free Brownian motion of particles. The factor X depends on the relative distance between approaching particles and can be determined from the resistance law F = 6nftaUX r/a), which is applicable to the relative motion of particles along their line of centers with the velocity U (see expression (8.36)). 

The dynamic friction term gives the correlation between the Brownian displacement in each time interval and the driven motions of the same particle at later times. As was first shown by Mazo(33), these correlations are nonzero and serve to retard diffusion. Correlations between Brownian and driven motions arise because each particle in solution is surrounded by its radial distribution function. The particle s Brownian displacements momentarily carry it off-center relative to the spherically symmetric distribution of neighboring particles. Until the radial distribution relaxes to the new location of the particle, which does not occur instantaneously, the neighboring particles tend to drive the particle of interest back to its initial location. The contributions of dynamic friction to D (34), Ds(35), the drag coefficient fo for motion at constant velocity(36), and r (37) have been obtained. 

Agglomeration of Pt crystallites due to Brownian motion can really be observed and it can also be shown that, indeed, the interaction between the Pt particles and the supporting soot in the presence of the electrolyte, phosphoric acid, is weak enough to allow for relatively free movement of the Pt particles. This fast process obviously is also the reason for the nonobservability of slower surface diffusion-induced Ostwald ripening. Fortunately alloy catalysts composed of platinum and nonnoble metals seem to show a reduced tendency to agglomeration as their deterioration and activity loss is much slower than that of the pure platinum catalyst. 

Particles, chains, aggregates and floes were allowed to move in space according to their respective diffusion coefficients recomputed at each step from their masses and conformations. Brownian motion was represented by random walks. The number of random walks during a unit of (relative) time was proportional to the diffusion coefficients of the moving entities. The link between the relative and physical time was made by correlating the mean displacement of a reference particle (not participating in the aggregation process) and its diffusion coefficients. 

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