Cascade substitution

The basis of model calculations for copolymerization, branching and cross-linking processes is the stochastic theory of Flory and Stockmayer (1-3). This classical method was generalized by Gordon and coworkers with the more powerful method of probability generating functions with cascade substitution for describing branching processes (4-6). With this method it is possible to treat much more complicated reactions and systems (7-9). 

In this study computational results are presented for a six-component, three-stage process of copolymerization and network formation, based on the stochastic theory of branching processes using probability generating functions and cascade substitutions (11,12). 

The Compound Distribution and Cascade Substitution. Another type of composed distribution may be constructed when the number of participants is not fixed but follows a distribution of the sums. The resultant distribution is called a compound distribution. For illustration, the distribution of units in the n-th generation for a f-functional polycondensate may be considered (see Fig. 11). 

The cascade substitution for the construction of the path-weight generating function is achieved by... 

The rest of the work consists in the cascade substitution for the construction of a path-weight generating function which is done as usual. Gordon and Parker have focussed their interest on the position of the gel point which became shifted from ac = 0.5000, in the random case, to ac = 0.5733, for the steric hindrance case. It will certainly be of interest to investigate whether... 

In Ref. 114 the set of generating functions obtained from Eq. (C.119) after cascade substitution, were solved explicitly without making use of the matrix formalism. 

The distribution of molecules (trees) according to their degree of polymerization is obtained by so-called cascade substitution, in which the variable z is substituted by a generating function. Thus, the weight fraction distribution function W(z) is given by... 

The weight-fraction gf W(z) is obtained by cascade substitution analogous to Eqs. (6-8) for a single-component system. Depending on the type of root, the components of the wfgf Wa(z) and We(z) read... 

In a similar manner as used for 1,2-dione/diamine condensations, Kaupp carried out stoichiometric mechanochemical reactions of o-mercaptoanihne 86 with ninhydrin in the three reaction cascade (substitution, cyclization, and elimination), which gave the salt of 10a-hydroxyindeno[2,l-( ]benz[l,4]thiazin-l l(10a//)-one 89 (Schane4.23) [15], Washing of product with NaHC03 was the simple woricup in this efQcient sol-vent-free synthetic protocol. 

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