Systems with one type of branch unit

Systems with one type of branch unit A statistical approach was used by Flory (1941,1943, 1953) and Stockmayer (1943,1953) to derive an expression for the prediction of the extent of reaction at the gel point. In this approach, it is assumed that (1) the reactivity of all functional groups of the same type is the same and independent of molecular size and (2) there are no intramolecular reactions between functional groups of the same molecule. 

Consider polymerizatiori of bifunctional molecules A—A, B—B and trifunctional molecule (—A)s in a mixture, not necessarily in equimolar quantities. This will lead to trifunctionally branched network polymer (VI)  

A portion of the trifunctionally branched network (VI) of gelled polymer is schematically represented in Fig. 5.7, showing a series of contour lines or envelopes on which the branches end, either in a branch point or in a terminal bi-functipnal group (nonbranch unit). For convenience, the chain sections are shown as being equal in length. However, this will not affect our following analysis. 

Consider the chain section in envelope 1 (Fig. 5.7) having a branch point at each end. The four new chains originating from the two branch points on envelope 1 happen to lead further to three new branch points and one nonbranch unit on envelope 2. The resulting six new chain sections that emanate from envelope 2 happen to lead to three branch points and three nonbranch units on envelope 3, and so on. The probability that a branch arising at a branch point leads via bifiinctional units to another branch point rather than to a nonbranch unit is referred to as the branching probability, a. So (1 - a) is the probability that a branch arising from the branch point leads to a nonbranch unit. 

Moving on to an /th envelope from the randomly selected chain section enclosed in envelope 1 in Fig. 5.7, suppose there are T branch points on the zth 

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