Nonlinear Step-Growth Reactions

For a completely bifunctional system such as an equimolar mixture of phthalic arid and ethylene glycol,/ = 2, and = 20 forp = 0.95. If, however, a trifimctional alcohol, glycerol, is added so that the mixture is composed of 2 mol diacid, 1.4 mol diol, and 0.4 mol glycerol, increases to 

The value of x is now 200 after 95% conversion, but only a small increase to 95.23% is required for x to approach infinity — a most dramatic increase. This is a direct result of incorporating a trifunctional unit in a linear chain in which the nnreacted hydroxyl provides an additional site for chain propagation. This leads to 

If the stoichiometry of the system is nnbalanced, the definition of the average functionality mnst be modified and becomes 

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During isothermal polymerization below Tg, the molecular weight and T, increase, and eventually T, will equal Tjure The main purpose of this section is to discuss the calculation of the time to vitrification, where vitrification is defined to occur when Tj, equals T ure- The concepts of vitrification and the TTT cure diagram are extended to linear systems for both step growth and chain reaction mechanisms, although most of the discussion will focus on the nonlinear step growth case, of which the cure of epoxy resins is an example. 

For the nonlinear step growth case above, eiTg, the crosslink density must be related to p. A relevant model, based on calculating the probabilities of finite chains being formed, has been published For the reaction of A -1- 2B2 (e.g., tetra-functional amine -b difunctional epoxy), A4 is considered to be an effective cross-linking site if three or more of its arms lead out to the infinite network. The probability of finding an effective crosslink is related to one minus the probability of a randomly chosen A leading to the start of a finite chain, which in turn is related to the extent of reaction. Application of this procedure to the system of Fig. 15 has been presented in detail The more complicated reaction of a tetrafunctional amine with a trifunctional epoxy was also considered. ... 

The time to vitrification, as a function of reaction temperature, can now be solved for each of the three cases considered. The only case for which experimental data are available for t j, is the nonlinear step growth case. Combining Eqs. (13)-(16), (19), and those relating the crosslink dmsity to p, results in the plot of T vs. tyj, shown in Fig. 16. The system used was the same one used in Fig. IS. Different values of the reaction order (n) were used in Fig. 16. The value of k obtained for n = 1 was used for all values of n. The fit is not entirely satisfactory, but the lack of an accurate kinetic model mitigates against a good fit. The calculated time to vitrification curve is S-shaped, as is seen experimentally. 

Fig. 16. Reaction tenqterature vs. time to vitrify for nonlinear step-growth polymerisition (A 4- 2B2) nth-order kinetics for n = t to 3 in inoements of 0.5 using the following parameters T, = -19 X ,T,= SOX T, = 166 X E. = 12.6 kcal/mole A = 4.51 x 10 min- M = 210 gm/mole Mg = 382 gm/mole. Data (squares) are from the study of Epon 82 /PACM-20 (See Fig. 4 caption for description of materials.) [Aronhime, M. T., Gillham, J. K. J. Coat. Tech. S6 (718), 35 (1984)]...

Fig. 17. Extent of reaction at vitrification vs. reaction temperature for nonlinear step-growth polymerization (A + 2B2). All kinetic orders have the same p at vitrification. For mpdel parameters and system, see Fig. 16 caption. [Aronhime,...

Molecular weight is used for linear systems, and for thermosetting systems that have not crosslinked (i.e., below g.iT,). There are four cases of importance-linear systems for step growth and chain reaction mechanisms, and nonlinear systems for step growth and chain reaction mechanisms — but only examples of the first three are discussed here. 

The last step in this second model is to relate the extent of reaction at vitrification to time. For the step growth mechanism, for both linear and nonlinear systons, nth order kinetics were assumed... 

UnUke Flory s combinatorial approach, a Markovian analysis such as tot proposed by Macosko and Miller leads to easy derivations of expressions for M and for nonlinear polymers. Before generalizing to a reaction implying multivalent molecules having valence v, the case of a step-growth polymerization involving X4 tetravalent molecules and Ny Y- -Y molecules wiU be considered. 

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