Smoluchowski mechanism

There are two general theories of the stabUity of lyophobic coUoids, or, more precisely, two general mechanisms controlling the dispersion and flocculation of these coUoids. Both theories regard adsorption of dissolved species as a key process in stabilization. However, one theory is based on a consideration of ionic forces near the interface, whereas the other is based on steric forces. The two theories complement each other and are in no sense contradictory. In some systems, one mechanism may be predominant, and in others both mechanisms may operate simultaneously. The fundamental kinetic considerations common to both theories are based on Smoluchowski s classical theory of the coagulation of coUoids. 

By comparing time-resolved and steady-state fluorescence parameters, Ross et alm> have shown that in oxytocin, a lactation and uterine contraction hormone in mammals, the internal disulfide bridge quenches the fluorescence of the single tyrosine by a static mechanism. The quenching complex was attributed to an interaction between one C — tyrosine rotamer and the disulfide bond. Swadesh et al.(()<>> have studied the dithiothreitol quenching of the six tyrosine residues in ribonuclease A. They carefully examined the steady-state criteria that are useful for distinguishing pure static from pure dynamic quenching by consideration of the Smoluchowski equation(70) for the diffusion-controlled bimolecular rate constant k0,... 

We have in this way obtained a generalization of Einstein s theory of the interaction between matter and radiation including multiple photon processes and involving transition probabilities. But there is a basic difference. The operator definite positive. We no longer have a simple addition of transition probabilities. This corresponds exactly to the interference of probabilities discussed in Section IV. The process is not of the simple Chapman-Smoluchowski-Kolmogoroff type (Eq. (11)) the operator transition probability. As the result, the second of the two sequences discussed above may decrease the effect of the first one. It is very interesting that even in the limit of classical mechanics (which may be performed easily in the case of anharmonic oscillators) this interference of probabilities persists. This is in agreement with our conclusion in Section IV. 

Also Smoluchowski equation , but we shall use that name in a more special sense, see VIII.l. An earlier reference to Bachelier is given by E.W. Montroll and B.J. West, in Studies in Statistical Mechanics VII (E.W. Montroll and J.L. Lebowitz eds., North-Holland, Amsterdam 1979) p. 76. 

Bagchi and co-workers [47-50] have explored the role of translational diffusion in the dynamics of solvation by employing a Smoluchowski-Vlasov equation (see also Calef and Wolyness [37] and Nichols and Calef [42]). A significant contribution to polarization relaxation is observed in certain cases. It is found that the Onsager inverted snowball model is correct only when the rotational diffusion mechanism of solvation dominates the polarization relaxation. The Onsager model significantly breaks down when there is an important translational contribution to the polarization relaxation [47-50]. In fact, translational effects can rapidly accelerate solvation near the probe. In certain cases, the predicted behavior can actually approach the uniform continuum result that rs = t,. 

If the von Smoluchowski rate law (Eq. 6.10) is to be consistent with the formation of cluster fractals, then it must in some way also exhibit scaling properties. These properties, in turn, have to be exhibited by its second-order rate coefficient kmn since this parameter represents the flocculation mechanism, aside from the binary-encounter feature implicit in the sequential reaction in Eq. 6.8. The model expression for kmn in Eq. 6.16b, for example, should have a scaling property. Indeed, if the assumption is made that DJRm (m = 1, 2,. . . ) is constant, Eq. 6.16c applies, and if cluster fractals are formed, Eq. 6.1 can be used (with R replacing L) to put Eq. 6.16c into the form... 

The actual mechanism by which the ions constituting the ionic atmosphere are dispersed is none other than the random-walk process described in Section 4.2. Hence, the time taken by the ionic cloud to relax or disperse may be estimated by the use of the Einstein-Smoluchowski relation (Section 4.2.6)... 

Modeling Many researchers, including Goto et al. [140], Wang et al. [141], Fujinawa et al. [145], and Hano et al. [150], have attempted to quanhfy the rate of demulsihcation. In a recent development, Ichikawa et al. [151] and Ichikawa and Nakajima [152] proposed a theory for electrostatic demulsihcation of O/W emulsions (by low-external electric helds) based on the Poisson-Smoluchowski equahon. Their theory revealed that the applied held induced demulsihcahon according to the following mechanisms [152] ... 

Howarth (HI5) assumed a mechanism analogous to Brownian motion. But his derivation for collision rates is wrong because he introduced a variable particle diffusion coefficient in the Smoluchowski equation (06, S27) which had been derived on the basis of a constant coefficient. 

Prince and Blanch [92] modeled bubble coalescence in bubble columns considering bubble collisions due to turbulence, buoyancy, and laminar shear, and by analysis of the coalescence probability (efficiency) of collisions. It was assumed that the collisions from the various mechanisms are cumulative. The collision density resulting from turbulent motion was expressed as a function of bubble size, concentration and velocity in accordance with the work of Smoluchowski [111] ... 

Intuitively, bubble coalescence is related to bubble collisions. The collisions are caused by the existence of spatial velocity difference among the particles themselves. However, not all collisions necessarily lead to coalescence. Thus modeling bubble coalescence on these scales means modeling of bubble collision and coalescence probability (efficiency) mechanisms. The pioneering work on coalescence of particles to form successively larger particles was carried out by Smoluchowski [109, 110]. 

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