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Support for this mechanism comes from 13 C NMR examination of the con guration of the rst monomer unit inserted into the Ti active site [Ammendola R et al, 1986; Tritto et al, 1986; Zambelli et al, 1982a,b] The NMR analysis is enhanced by using 13 C-enriched alkyl groups in the group I III metal component There is some minimum.

2-64

H2 O K M 0 X n X n 1 2-65

A major source of variation, even between different implementation of the same numerical minimization routine, is the line search for the function minimum in the chosen direction. Minimization algorithms may use: an exact; an approximate; or no line search routine.

Equation 2-65, which applies equally to A polymerizations, indicates that [H2O] must be B greatly reduced to obtain high X n values. [H2O] is inversely dependent on essentially the square of X n since X n 1 is close to X n for large values of X n . It is also seen that the level to which the water concentration must be lowered to achieve a particular degree of polymerization increases with increasing K and increasing initial concentration of the reactants. Table 2-6 shows the calculated [H2O] values for selected values of K and X n at

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TABLE 2-6 Effect of Water Concentration on Degree of Polymerization in Open, Driven System K 0.1 Xn 1.32b 20 50 100 200 500 2b 20 50 100 200 500 5b 20 50 100 200 500 10 b 20 50 100 200 500 20 b 50 100 200 500 [H2O]a (mol L 1) 1.18b 1:32 10 3 2:04 10 4 5:05 10 5 1:26 10 5 2:00 10 6 2.50b 1:32 10 2 2:04 10 3 5:05 10 4 1:26 10 4 2:01 10 5 4.00b 0.211 3:27 10 2 8:10 10 3 2:01 10 3 3:21 10 4 4.50b 1.07 0.166 4:09 10 2 1:02 10 2 1:63 10 3 4.75b 0.735 0.183 4:54 10 2 7:25 10 3

6

M 0 5 M. A concentration of 5 M is fairly typical of a step polymerization that is often carried out with only the reactant(s) present (without solvent). The lowering of [H2O] to achieve a particular X n is less the more favorable the equilibrium (that is, the larger the K value). Thus the synthesis of polyamides (with typical K values > 102) is clearly easier from the equilibrium viewpoint than polyester synthesis (K $ 0:1-1). Polyesteri cation requires a greater lowering of [H2O] than does polyamidation. It should be understood that simply to lower [H2O] as much as possible is not the desired approach. One needs to control the [H2O] so as to obtain the desired degree of polymerization.

the derivative - can be plotted a t the current model during the iterative dp 5 fitting process - this is referred to the "empirical the decision boundary (where mlp(z) = 0.5) can be plotted a t some point in the minimization process or for successive iterations; the SC can be calculated and plotted. However, we will see, each of these three methods (as well as the final model) are dependent on the starting values and the minimization routine. Hence we are only exploring a subset of the possible solutions to these three simple experiments.

Although reversible or equilibrium polymerizations would almost always be carried out in an irreversible manner, it is interesting to consider the kinetics of polymerization for the case in which the reaction was allowed to proceed in a reversible manner. (The kinetics of reversible ring-opening polymerizations are discussed in Sec. 7-2b-5). Consider the polyesteri cation of Eq. 2-58 under stoichiometric conditions, where k1 and k2 are the rate constants for the forward and back reactions. The initial carboxyl and hydroxyl group concentrations are [M]0. The values at any time are [M], which is given by (1 p) [M]0. The concentrations of the products, [COO] and [H2O], at any time are equal because of the stoichiometry and are given by p M 0. The polymerization rate is the difference between the rates of the forward and back reactions

d M M 0 d 1 p k1 1 p 2 M 2 k2 p2 M 2 0 0 dt dt 2-66

dp M 0 k1 1 p 2 k2 p2 dt 2-67

We consider a small data set consisting of 10 points in 5 points in each of 2 classes (Figure 9.1) and we fit an M L P model of size 1 . 1 . 1 (so that there are four weights)

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