Coagulation equation

Drake, R.L., 1972. A general mathematical survey of the coagulation equation. In Topics in current aerosol research, Pt. 2. Eds. G.M. Tidy and J.R. Brocks, New York Pergamon. 

Galina,H. Mean-Field Kinetic Modeling of Polymerization The Smoluehowski Coagulation Equation, Vol. 137, pp. 135-172. 

Analytical approaches to obtain the agglomerate size distribution are possible for well-mixed systems and when the rate of aggregation of clusters is defined by simple functions. In general, irreversible aggregation in well-mixed systems is described by Smoluchowski s coagulation equation, which... 

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) ... 

For orthokinetic conditions the ratio of adsorption halftime to coagulation halftime can be calculated by integrating the adsorption Equation (6) and the coagulation Equation (2). 

It Is to be remarked that the process described by the Infinite set of kinetic (coagulation) equations can be simulated by Monte-Carlo methods ( ). The Information on the number of molecules of the respective size Is stored In the computer memory and weighting for selection of molecules Is applied given by the number and reactivity of groups In the respective molecule. 

The overall rate of agglomeration of any suspension, that is prepared experimentally and consists of small and large colloids, is obtained by adding the expressions derived for perikinetic and orthokinetic coagulation (Equations 2 and 5) ... 

Since ky does not depend on i or j, it can be absorbed, together with the symmetry factor a (equal to 4 for RA2 or 2 for ARB), into time units and the Smolu-chowski coagulation equation would assume its simplest form with the kernel equal to one ... 

The Smoluchowski coagulation equation has effectively been applied to model random irreversible homopolymerization of the monomer types presented in Table 4. As can be seen, in all cases the coagulation kernel has the bilinear form ... 

Table 4. The types of monomers undergoing homopolymerization and corresponding bilinear kernels in the Smoluchowski coagulation equation [21]. The functional groups A react with each other in models 1 and 2 and with groups B only in models 3 through 5...

The Smoluchowski coagulation equation with the kernel given by Eq. (60)... 

Table 5. Moments and explicit distributions generated by the Smoluchowski coagulation equation (prior to gel point) for models 1 and 2 in Table 4. [For both models Mi=l]...

In principle, the Smoluchowski coagulation equation and its modifications provide the correct distributions of aggregate sizes at positive values of the parameter t. For some kernels the solutions are limited to the range of 0 < f < tc. At tc the second and higher moments of distribution diverge. In this section the changes in distribution for t > tc are discussed. 

The Smoluchowski coagulation equation describes the rate of formation of acyclic aggregates. Only then it describes the evolution of a Markovian distribution [34]. Strictly speaking the Smoluchowski equation simply disregards any cycle formation. For polymers this is true for models with high functionality... 

To circumvent the problem of many equations, it is possible to represent the coagulation equation as a nonlinear integrodifferential equation of the form... 

Using K0 as the coagulation constant and neglecting the second term in Eq. 18.9 yield the usual form of the coagulation equation ... 

The kinetic method can be extended to include cyclization reactions but no suitable procedures have been developed as yet In recent years, much progress has been reached in the appUcation of coagulation equations to various cluster formation and aggregation processes... 

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