Reversible polymerization an application of a moment closure technique

2 Reversible polymerization an application of a moment closure technique 

In the analysis of most polymerization systems, the resulting set of moment equations is closed, i.e. complete, and consequently readily solved using 

This balance equation needs some explanation since the form is not immediately evident. The reverse step-growth reaction does not depend on the polymer concentration but on the number of AB linkages. Species P has n + m repeat units and n + m - 1 linkages. The macromolecule can therefore break in n -h m — 1 places. Since every molecule has two sites at which it can break to form two molecules one of size m, and the other of size n, the fraction of the events leading to such scission is 2/(n -h m — 1). The reverse reaction for step growth is proportional to the number of linkages, so that the rate expression includes the term 

Although it is possible to generate the moment equations by multiplying equation (3.72) by n and summing over all terms, it is easier to use the generating function formalism and differentiate them using equation (3.33). 

Using Table 3.1, the generating function representation of equation (3.72) is  

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