Scaling the von Smoluchowski Rate Law

Cluster fractals that are created by diffusion-limited flocculation processes are described mathematically by power-law relationships like those in Eqs. 6.1 and 6.5. These relationships are said to have a scaling property because they satisfy what in mathematics is termed a homogeneity condition 22 

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), 

The napierian logarithm of the right side of Eq. 6.29b has the large-time limit  

The scaling properties of the q-moments (Eq. 6.11) can be deduced from those of the pq(t)  

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