POLYMERIZATION PROBABILITY AND STATISTICS

The chain length distribution of free radical addition polymerization can also be derived from simple statistics. Thus, for polymer formed at any given instant, the distribution will be the most probable and will be governed by the ratio of the rates of chain growth to chain termination. 

An excellent way to treat such data is to use reaction probability models.(1,2) In the NMR analysis of tacticity, it is frequently possible to distinguish whether the configuration is chain-end controlled or catalytic-site controlled during polymerization. Various statistical models have been proposed. The chain-end controlled models include Bemoullian (B), and first- and second-order Markovian (Ml and M2) statistics.(1) The simplest catalytic-site controlled model is the enantiomorphic site (E) model.(3) The relationship between the chain-end and catalytic-site controlled models and possible hybrid models have been delineated in a recent article.(4)... 

Analysis of the poly(methyl methacrylate) sequences obtained by anionic polymerization was undertaken at the tetrad level in terms of two different schemes (10) one, a second-order Markov distribution (with four independent conditional probabilities, Pmmr Pmrr, Pmr Prrr) (44), the other, a two-state mechanism proposed by Coleman and Fox (122). In this latter scheme one supposes that the chain end may exist in two (or more) different states, depending on the different solvation of the ion pair, each state exerting a specific stereochemical control. A dynamic equilibrium exists between the different states so that the growing chain shows the effects of one or the other mechanism in successive segments. The deviation of the experimental data from the distribution calculated using either model is, however, very small, below experimental error, and, therefore, it is not possible to make a choice between the two models on the basis of statistical criteria only. 

As is well-known, this pair of expressions will not be valid for the most general case of a second order fluid, since p22 — tzi must not necessarily vanish for such a fluid. Eq. (2.9) states that the first normal stress difference is equal to twice the free energy stored per unit of volume in steady shear flow. In Section 2.6.2 it will be shown that the simultaneous validity of eqs. (2.9) and (2.10) can probably quite generally be explained as a consequence of the assumption that polymeric liquids consist of statistically coiled chain molecules (Gaussian chains). In this way, the experimental results shown in Figs. 1.7, 1.8 and 1.10, can be understood. 

The crystalline state configuration of polymeric selenium is a helix in which all the rotational angles are of the same sign with a value of 78°. This is fairly close to the expected value of 90°.52 The barrier to rotation about the Se-Se bond is thought to be roughly the same as that about S-S bonds. No reliable experimental results are available on the statistical properties of these chains. Some values of the glass-transition temperature of polymeric selenium have been reported15 and these could at least provide a measure of the dynamic flexibility of the chains. However, these results are probably compromised by the presence of cyclic selenium molecules that act as plasticizer. 

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