The 28 Fundamental Scenarios

The second column, Presence or Absence of Events that Determine the Number of Polymer Chains (N), comprises 6 sub-columns and shows the 28 scenarios selected for examination. Among the very large number of theoretically possible scenarios we have selected for analysis only those which seemed to be closest to real-life situations. Specifically, we have examined the effects on N of fast and slow initiation (and in the case of slow initiation, the respective effects of slow ion-generation and slow cationation), the effect of initiation by adventitious protic impurity (i.e., HX = H20 ), and the effects of chain transfer to monomer, specifically both monomolecular or zero-order-in-monomer chain transfer and bimolecular or first-order-in-monomer chain transfer. The presence or absence of these effects is indicated by the symbols, 1 or 0 in column two. The organization of the 28 scenarios is as follows  

The third column, Mathematical Descriptions of Individual Scenarios, shows the differential equations of N and M together with the initial conditions in terms of t, N, Ntr, Ntotal, and M (see legend to Table 1). 

The fourth column. Resulting Expressions, is divided into four subcolumns which contain the N vs Conv., Mn vs Conv., — In (1 — Conv.) vs Time, and Conv. vs Time formulas, and the numerical equations for presentation of the 

In scenarios A2b, A2c, and A3b the second terms in the — dM/dt expressions can be neglected because they are small relative to the first terms (the negligible terms are underlined by dotted lines). In a few scenarios (A3a, A3b, C3a, C3b) explicit solutions could not be derived and graphical methods were applied to obtain the equations from computer fitted curves. 

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Table 1 summarizes and defines the 28 fundamental scenarios considered, their mathematical description, and the expressions derived for the above-mentioned plots. The very large amount of information assembled in the table is subdivided into four main columns the first column, Symbols, shows abbreviations for the scenarios specificed in column two and will be used in the discussion. 

For the sake of simplicity and because of the large number (28) of fundamental scenarios considered, the rate constants in Scheme 1 are considered to be true constants, although we have established (see parts 2.3 and 3) that in many instances they are in fact not constants (they should preferably be termed rate coefficients) and are strongly influenced by the equilibria dormant living bionic. By disregarding these equilibria and assuming that the rate coefficients are constant, we were able to derive differential equations with rather simple analytical solutions. The detailed consideration of these multi-part equilibria would have lead to numerical solutions of the differential equations (see e.g., [45a, b]) and thus to an enormous increase in the length of this treatment. 

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