Coagulation kinetics, Smoluchowski theory

The simplest class of bimolecular reactions involves only one type of mobile particles A and could result either in particle coagulation (coalescence, fusion) A + A —> A, or annihilation, A + A — 0 (inert product). Their simplicity in conjunction with the simple topology of d = 1 allows us to solve the problem exactly, which makes it very attractive for testing different approximations and computer simulations. In the standard chemical kinetics (i.e., mean-field theory, Section 2.1.1) we expect in d = 2 and 3 for both reactions mentioned trivial behaviour quite similar to the A+B — 0 reaction, i.e., tia( ) oc t-1, as t — oo. For d = 1 in terms of the Smoluchowski theory the joint density obeys respectively the equation (4.1.56) with V2 = and D = 2Da. 

For a more complete understanding of colloid stability, we need to address the kinetics of aggregation. The theory discussed here was developed to describe coagulation of charged colloids, but it does apply to other cases as well. First, we consider the case of so-called rapid coagulation, which means that two particles will aggregate as soon as they meet (at high salt concentration, for instance). This was considered by von Smoluchowski 1561 here we follow [39, 57]. 

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. 

Smoluchowski, M.V., 1917. Mathematical theory of the kinetics of coagulation of colloidal systems. Zeitschrift fur Physikalische Chemie, 92, 129-168. 

All models described up to here belong to the class of equilibrium theories. They have the advantage of providing structural information on the material during the liquid-solid transition. Kinetic theories based on Smoluchowski s coagulation equation [45] have recently been applied more and more to describe the kinetics of gelation. The Smoluchowski equation describes the time evolution of the cluster size distribution N(k) ... 

Kinetics is concerned with many-particle systems which require movements in space and time of individual particles. The first observations on the kinetic effect of individual molecular movements were reported by R. Brown in 1828. He observed the outward manifestation of molecular motion, now referred to as Brownian motion. The corresponding theory was first proposed in a satisfactory form in 1905 by A. Einstein. At the same time, the Polish physicist and physical chemist M. v. Smolu-chowski worked on problems of diffusion, Brownian motion (and coagulation of colloid particles) [M. v. Smoluchowski (1916)]. He is praised by later leaders in this field [S. Chandrasekhar (1943)] as a scientist whose theory of density fluctuations represents one of the most outstanding achievements in molecular physical chemistry. Further important contributions are due to Fokker, Planck, Burger, Furth, Ornstein, Uhlenbeck, Chandrasekhar, Kramers, among others. An extensive list of references can be found in [G.E. Uhlenbeck, L.S. Ornstein (1930) M.C. Wang, G.E. Uhlenbeck (1945)]. A survey of the field is found in [N. Wax, ed. (1954)]. 

The kinetic theory of fast irreversible coagulation was developed by von Smoluchowski. Later the theory was extended to the case of slow and reversible coagulation. In any case of coagulation (flocculation), the general set of kinetic equations reads... 

In the absence of a barrier to coagulation, and if the primary minimum is deep, every collision between a particle and a floe will lead to the growth of the floe. The rate of coagulation is then controlled entirely by the kinetics of the diffusion process leading to particle-particle collision. The theory of fast coagulation was developed originally by Smoluchowski (1918) and elaborated by Muller (1926). The rate equation has the same form as that for a bimolecular reaction ... 

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 kinetics of coagulation were first analyzed by Smoluchowski [32], who assumed that no repulsive barrier was present, and that aggregation occurred by the attachment of single particles to clusters (ignoring cluster-cluster aggregation). The theory was further developed by Fuchs [33], who showed that a repulsive potential F(r) would reduce the coagulation rate by the faetor JV, called the stability ratio ... 

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