Statistical models of propagation

The purpose of this chapter, therefore, is to describe the basic concepts of the statistical analysis of copolymer sequence distribution. The necessary relationships between various comonomer sequence abundances are introduced, along with simple statistical models based on monomer addition probabilities. The relationships between the statistical models and propagation models based on reactivity ratios are discussed. The use of these models is then illustrated by means of selected examples. Techniques for extracting sequence information from in situ NMR measurements are also described. Finally, the statistical analysis of chemically modified polymers is introduced with examples. 

As reactants transfonn to products in a chemical reaction, reactant bonds are broken and refomied for the products. Different theoretical models are used to describe this process ranging from time-dependent classical or quantum dynamics [1,2], in which the motions of individual atoms are propagated, to models based on the postidates of statistical mechanics [3], The validity of the latter models depends on whether statistical mechanical treatments represent the actual nature of the atomic motions during the chemical reaction. Such a statistical mechanical description has been widely used in imimolecular kinetics [4] and appears to be an accurate model for many reactions. It is particularly instructive to discuss statistical models for unimolecular reactions, since the model may be fomuilated at the elementary microcanonical level and then averaged to obtain the canonical model. 

A unified statistical model for premixed turbulent combustion and its subsequent application to predict the speed of propagation and the stmcture of plane turbulent combustion waves is available (29—32). 

A general purpose program has been developed for the analysis of NMR spectra of polymers. A database contains the peak assignments, stereosequence names for homopolymers or monomer sequence names for copolymers, and intensities are analyzed automatically in terms of Bernoullian or Markov statistical propagation models. A calculated spectrum is compared with the experimental spectrum until optimized probabilities, for addition of the next polymer unit, that are associated with the statistical model are produced. 

Monomer concentrations Ma a=, ...,m) in a reaction system have no time to alter during the period of formation of every macromolecule so that the propagation of any copolymer chain occurs under fixed external conditions. This permits one to calculate the statistical characteristics of the products of copolymerization under specified values Ma and then to average all these instantaneous characteristics with allowance for the drift of monomer concentrations during the synthesis. Such a two-stage procedure of calculation, where first statistical problems are solved before dealing with dynamic ones, is exclusively predetermined by the very specificity of free-radical copolymerization and does not depend on the kinetic model chosen. The latter gives the explicit dependencies of the instantaneous statistical characteristics on monomers concentrations and the rate constants of the elementary reactions. 

This is the simplest of the models where violation of the Flory principle is permitted. The assumption behind this model stipulates that the reactivity of a polymer radical is predetermined by the type of bothjts ultimate and penultimate units [23]. Here, the pairs of terminal units MaM act, along with monomers M, as kinetically independent elements, so that there are m3 constants of the rate of elementary reactions of chain propagation ka ]r The stochastic process of conventional movement along macromolecules formed at fixed x will be Markovian, provided that monomeric units are differentiated by the type of preceding unit. In this case the number of transient states Sa of the extended Markov chain is m2 in accordance with the number of pairs of monomeric units. No special problems presents writing down the elements of the matrix of the transitions Q of such a chain [ 1,10,34,39] and deriving by means of the mathematical apparatus of the Markov chains the expressions for the instantaneous statistical characteristics of copolymers. By way of illustration this matrix will be presented for the case of binary copolymerization ... 

With the experimental techniques available at present, rate constants of diad formation cannot be determined directly. There is however a way to calculate the rate constants from the experimentally determined triad, diad, etc. fractions if the rate constant of propagation and the statistical model are known (e.g., a one-way mechanism, a two-way mechanism, enantiomorphic site model) (9, JO). Very few rate constants of propagation are available, however. 

J. P. Vigier, Model of quantum statistics in terms of a fluid with irregular stochastic fluctuations propagating at the velocity of light A derivation of Nelson s equations, Lett. Nuovo Cimento 24(8) (Ser. 2), 265-272 (1979). 

Table A2.6 Selected statistics of the PBLx intake distribution obtained from different model variance propagation methods. ...

The work of DiMarzio and Rubin (DiMarzio, 1965 Rubin, 1965 DiMarzio and Rubin, 1971) began the development of a related but more powerful approach. Rather than calculating microstructural details from a presumed architecture, Rubin s matrix method concentrates on the effect of local interactions on the propagation of the chain, thereby deriving the statistical properties of the random walk and the structure of the entire chain. This formalism is the foundation for several subsequent models, so some details are reviewed here. The notation is transposed into a form consistent with the contemporary models discussed below. 

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