Von Smoluchowski rate

The approximate rate constant in Eq. 6.16d is independent of floccule size, which results in a considerable simplification of the von Smoluchowski rate law ... 

Equation 6.18 is graphed in Fig. 6.6 for the cases q = 1, 2, 3. The number density of primary particles, pj(t), decreases monotonically with time as these particles are consumed in the formation of floccules. The number densities of the floccules, on the other hand, rise from zero to a maximum at t = (q - l)/2KDp0, and then decline. This mathematical behavior reflects creation of a floccule of given size from smaller floccules, followed by a period of dominance, and finally consumption to form yet larger particle units as time passes. Both experimental data and computer simulations, like that whose visualization appears in Fig. 6.1, are in excellent qualitative agreement with Eq. 6.18 when they are used to calculate the pq(t).13,14 Thus the von Smoluchowski rate law with a uniform rate coefficient appears to capture the essential features of diffusion-controlled flocculation processes. 

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 special case of Eq. 6.27 that obtains where m = n, that is, k°n= 2K , is trivially scale invariant, so this property ought to be implicit in the corresponding solution of the von Smoluchowski rate law, given in Eq. 6.18. That this is the case can be seen by noting the large-time limit of p,(t),... 

Equation 6.28 represents the simplest effect of scaling on the second-order rate constant in the von Smoluchowski rate law no effect whatsoever. More generally, if kmn satisfies the homogeneity condition... 

Given that Eq. 6.1 (with D 2) applies to reaction-controlled flocculation kinetics, Eq. 6.54 implies that MM(t) [or MN(t)] must also exhibit an exponential growth with time. Therefore, by contrast with transport-controlled flocculation kinetics, a uniform value of the rate constant kmn cannot be introduced into the von Smoluchowski rate law, as in Eq. 6.17, to derive a mathematical model of the number density p,(t). Equations 6.22 and 6.24 indicate clearly that a uniform kinil leads to a linear time dependence in the... 

Fuchsian kinetics lead to the model form of the von Smoluchowski rate law that is obtained by introducing Eq. 6.52 into Eq. 6.10 ... 

If the rate of a reaction is governed by the encounter frequency, it is said to be diffusion-controlled. This frequency imposes an upper limit on the rate of reaction that can be evaluated by the use of Fick s laws of diffusion. The mathematical expression of this phenomenon was first presented by von Smoluchowski.2 We shall adopt a simple approach,3,4 although more rigorous derivations have been given.5... 

The temperature dependence of a diffusion-controlled rate constant is very small. Actually, it is just the temperature coefficient of the diffusion coefficient, as we see from the von Smoluchowski equation. Typically, Ea for diffusion is about 8-14 kJ mol"1 (2-4 kcal mol-1) in solvents of ordinary viscosity. 

The foundations of the theory of flocculation kinetics were laid down early in this century by von Smoluchowski (33). He considered the rate of (irreversible) flocculation of a system of hard-sphere particles, i.e. in the absence of other interactions. With dispersions containing polymers, as we have seen, one is frequently dealing with reversible flocculation this is a much more difficult situation to analyse theoretically. Cowell and Vincent (34) have recently proposed the following semi-empirical equation for the effective flocculation rate constant, kg, ... 

Note that the particle diffusion term is ignored, just like particle dispersion due to SGS motions (this was found justified in a separate simulation). The shape of the sink term in the right-hand term of this equation is due to Von Smoluchowski (1917) while the local value of the agglomeration kernel /i0 is assumed to depend on the local 3-D shear rate according to a proposition due to Mumtaz et al. (1997). 

A constant reaction kernel is to be expected in the absence of enzyme reaction if the aggregation rate is determined by the Brownian motion of spherical particles which coalesce to form larger spheres. To a first approximation, the increased collisional cross-section is then compensated for by the decrease in diffusion rate (von Smoluchowski,... 

There is essentially a single modeling approach that has been developed, referred to here as the von Smoluchowski approach, and this method will be presented first. The von Smoluchowski approach requires analytical expressions to represent particle collision rates, to calculate collision efficiencies, and to dictate aggregate structure formation. These individual components are discussed in the subsequent sections, followed by analytical and numerical techniques of solving the von Smoluchowski equation. 

Tambour and Seinfeld [36] developed a more general, albeit approximate, solution than that of von Smoluchowski, and provided three analytical solutions for arbitrary initial particle size distributions and differing constraints. The solutions require the following form for the collision rate coefficients... 

The rate at which particles coagulate when the interaction energy between the particle is 0 was first investigated by von Smoluchowski... 

The set of differential equations of von Smoluchowski has been preserved with the exception of the first equation above, which accounts for the creation of base particles with the rate a. 

On closer inspection, the combination rate constants are about 1/4 of the estimated diffusion-controlled rate constant. For acetonitrile, for example, fcjj - 2.9 X 10 L mol" s from the von Smoluchowski equation wiA a diffusion coefficient from a modified version of the Stokes-Einstein relation, D - fcT/4jiT r. Owing to the restriction to singlet state recombination, an experimental rate constant 1 /4 of is quite reasonable. On the other hand, for these heavy metals, the spin restriction may not apply, in which case one would argue that the geometrical and orientational requirements of these large species could well give recombination rates somewhat below the theoretical maximum. 

In a hindamental paper von Smoluchowski (1917) presented a theoretical model for the kinetics of the coagulation process. He showed that in the initial stages of coagulation the rate of disappearance of the primary particles, ije.. those present as single particles in the original dispersion, could be written as... 

Note also that for imaginary particles, which experience neither long-range surface forces (JJi j = 0) nor hydrodynamic interactions (P = 1), Equation 5.325a yields a collision efficiency = 1 and Equation 5.321 reduces to the von Smoluchowski " expression for the rate constant of the fast irreversible coagulation. In this particular case. Equation 5.319 represents an infinite set of nonlinear differential equation. If all flocculation rate constants are the same and equal to Up the problem has a unique exact solution " ... 

To illustrate the effect of the reverse process on the rate of flocculation, we solved numerically the set of Equations 5.319, 5.331, and 5.332. To simplify the problem, we used the following assumptions (1) the von Smoluchowski assumption that all rate constants of the straight process are equal to Up (2) aggregates containing more than M particles cannot decay (3) all rate constants... 

Equation 5.320. We see that in an initial time interval all cnrves in Figure 5.55 touch the von Smoluchowski distribution (corresponding to h = 0), bnt after this period we observe a redaction in the rate of flocculation which is larger for the curves with larger values of b (larger rate constants of the reverse process). These S-shaped curves are typical for the case of reversible coagulation, which is also confirmed by the experiment. 

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