Randomly Branched f-Functional Polycondensates

Inserting this result into Eq. (C.26) and using 0n = pn, which holds for Gaussian sub-chains, we finally arrive at90 

Polycondensates from Monomers with Unlike Functional Groups104 105  

However, such a rigorous approximation need not be made in the FS theory as will now be shown with several examples. In fact, the mean-field approximation would lead in most cases to grossly incorrect results. 

Here we treat in some detail the tri-functional polycondensate with three unlike functional groups105. In general, there will be 9 different probabilities of reaction i.e. a3, a2, a3, j8j, /J2, yS3, ylt y2, y3 where au a2, and a3 denote the fraction of A groups which have reacted with another A group, another B group, or another C group, respectively etc. These nine probabilities may conveniently be written as a matrix 

The number of units in the n-th generation can now in principle be found in a similar manner as outlined for the random polycondensates with equal reactive groups. We first notice that the result depends on how many of the units in the n-th generation are linked with their A, or B or C groups to a unit in the preceding generation. Therefore, it will be useful to introduce a population vector N (n) 

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Fig. 18. Reciprocal particle-scattering factors of star-molecules with polydisperse rays, where f denotes the number of rays per molecule. The same functions are obtained also for the ABC-type polycondensates, where nb denotes the number of branching points per molecule. The case f = 1 or nb = 0 is identical to linear chains obeying the most probable lengths distribution. It also represents the scattering behaviour of randomly branched f-functional polycondensates 1...

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