Smoluchowski s equation

In a homogeneous system, tire rate of mixing is governed by Smoluchowski s equations [77], according to which tire diffusion-limited association rate of S and L (equation (C2.14.13)), supposed uncharged, equals tliat of tire flux and is... 

Smoluchowski s equation, like the fragmentation equation, can be written in terms of the scaling distribution. Furthermore, general forms may be determined for the tails of the scaling distribution—limits of small mass, xls(t) < 1, and large mass, x/s(t) > 1. The details can be found in van Dongen and Ernst (1988). 

This can be done by developing equations for the moments—for example, multiplying Smoluchowski s equation by xfdx, integrating from 0 to infinity, and manipulating the limits of integration yields (Hansen and Ottino, 1996b) ... 

As the particle size increases further, a becomes unimportant unless it is very small and (1.18) reduces to Smoluchowski s equation... 

R. M. Ziff, E. D. McGrady, and P. Meakin, On the validity of Smoluchowski s equation for cluster-cluster aggregation kinetics, J. Chem. Phys. 82 5269 (1985). 

Smoluchowski s equation was extended up to Kn < 0.25, by introducing a slip correction factor C(Kri) in the expression for the diffusion coefficient. The coagulation coefficient is given for Kn < 0.25 by the expression... 

Application of Smoluchowski s equations typically results in an overestimation of the growth rate of aggregates due to the assumption that all collisions result in permanent attachment. Recognition of the importance of surface properties in the aggregation of small particles prompted Fuchs [6] to develop expressions to modify Smoluchowski s equations. Fuchs described the effect of the repulsive electrostatic interaction between two particles, which is a function of the particle separation distance, as a reduction in particle coagulation rate, Wy, termed the stability ratio. 

The majority of solutions of von Smoluchowski s equation have been numerical approximations due to the lack of general analytical solutions or satisfactory approximations. This section will review a number of approximations that have been generated numerically, and will discuss the difficulty associated with achieving an approximation for large aggregate structures. 

Electrophoresis of nonconducting colloidal particles has been reviewed in this chapter. One important parameter determining the electrophoretic velocity of a particle is the ratio of the double layer thickness to the particle dimension. This leads to Smoluchowski s equation and Huckel s prediction for the particle mobility at the two extrema of the ratio when deformation of the double layer is negligible. Distortion of the ion cloud arising from application of the external electric field becomes significant for high zeta potential. An opposite electric field is therefore induced in the deformed double layer so as to retard the particle s migration. 

For the case of a cylindrical particle, the electrophoretic mobility depends on the orientation of the particle with respect to the applied electric field. When the cylinder is oriented parallel to the applied electric field, its electrophoretic mobility fi is given by Smoluchowski s equation (1), namely. 

Note that except for the case of A 00, which corresponds to Smoluchowski s equation (21.1), the mobility tends to a nonzero limiting value. Only for the case of... 

Czapski G, Peled E. (1973) The scavenging of eaq- and on the possible breakdown of Smoluchowski s equation at high concentrations of solutes. J Phys Chem 77 893-897. 

Smoluchowski s equation in its simplest form describes rates of cluster formation and reduction ... 

At the above optimal concentration, the rate of coagulation of the aerosol happens to be rather small. According to Smoluchowski s equation, the particle number concentration decreases like... 

Smoluchowski s equation (4.13) is reasonably rigid, but in the derivation of (4.16) several assumptions were made. Putting the hydrodynamic radii of the molecules equal to their collision radii is especially questionable. As discussed above, Eq. (4.15) would only apply in homogeneous dilute systems. 

Strictly speaking, Smoluchowski s equation only applies if each encounter leads to reaction. If the chance of a reaction event is small, as is the case for large AG, the concentration gradient of reactant B near a molecule of reactant A will be smaller, and this will upset the relations on which Eq. (4.13) is based. 

For a particle radius of 200 A, using Smoluchowski s equation (with k( jff = 2 x 10-5 cm-1), Meisel calculated a rate constant of k = 3.0 x 1011 M 1s 1 for the encounter of 1-hydroxy 2-ethyl radicals with the Au sols. 

Methods. Determination of Agglomeration Rate. The kinetics of PEO agglomeration may be described in a wide initial range (up to 50-70% of total agglomeration) by all three forms of Smoluchowski s equation, in which the reciprocal particle number (1 /N), the mean particle volume (u), and the reciprocal cube of the particle surface (1/23) vary linearly with time (t) (14,... 

Figure 15 The potential of nitrofurantoin as a function of AICI < concentration at pH 4, 7. and 8. Open symbols Smoluchowski s equation full symbols O Brien and White method, (from Ref. 59. with permission.)...

Although the coalescence influence on the Brownian coagulation rate coefficient can be neglected, its influence on the final equations of the Smoluchowski theory remains. It can be shown that Smoluchowski s equation for the total number of particles ... 

According to the general rules of physicochemical kinetics the slowest process is rate controlling. If flie coagulation step is rate controlling, namely, when condition (34) is valid, then the coalescence is rapid and flie general equation of the theory in Ref 38 is reduced to second-order kinetics, i.e., to Smoluchowski s equation [Eq. (27)]. Floes composed of three, four, etc., droplets cannot be formed, because of rapid coalescence within the floe. In this case the structure of the floes becomes irrelevant... 

The ratio between the collision rate coefficient obtained by B D simulation (kj and the ideal rate coefficient given by Smoluchowski s equation (kc ) (Equation 25.24, which holds for a single particle at infinite dilution) is given by Equation 25.25 and... 

Fragmentation allows the particles, or aggregates, to escape after sticking. This effect is different from that obtained by simply considering a sticking probability. In this case, some reversibility is allowed, which can be chemically justified if the attractive secondary minimum of the DLVO potential is sufficiently weak. In von Smoluchowski s equation, it was assumed that aggregation was irreversible. Von Smoluchowski s equation can, however, be extended to include fragmentation processes ... 

The rate of the colhsion, ki, can be estimated using Smoluchowski s equation ... 

Equation 11 is Smoluchowsky s equation including an additional friction force due to the polymer. v(x) is the fluid velocity parallel to the particle surface at location x, the first term on the left is the viscous drag, the second is the force imparted to the fluid by the monomer beads, f is a friction coefficient and r(x) is the monomer density. The right hand term is the electrical volume force which is proportional to the field E and to the net charge p(x) per unit volume. The electrophoretic mobility Ue, which is the limit of v(x)/E for x > > lc can be expressed in the following form (see Appendix) ... 

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