Second-order Markov mechanism

In order to distinguish between Markov second order and Markov first order mechanisms, at least tetrads must be known. 

As a technique for the analysis of sequence distribution in copolymers, high-resolution NMR spectroscopy is particularly useful when the spectral resolution is sufficient to resolve the resonances of the specific sequences. A number of copolymer structural problems can be elucidated by using NMR spectroscopy. The composition of the copolymer can be quantitatively determined. The detection of compositional dyads can be used to determine the distribution of composition, that is, whether the sample is a mixture of homopolymers, a block copolymer, an alternating copolymer, or a random copolymer (see Chapter 1). If resonances are resolved due of the triad sequences of the copolymer, sequence distributions can be determined, and the mechanism of the copolymerization can be tested in terms of Bemoullian, first-order Markov, second-order Markov, or non-Markovian statistics. In rare circumstances, the tactic nature of the copolymer can be determined if distinguishable syndio- and isotactic n-ad resonances are resolved. Such an analysis has been carried out for copolymers of methyl methacrylate-methacrylic acid, for which the a-CHs resonances of all 20 triads have been assigned and have been used to determine the cotacticity of the copolymer [22]. 

More complex schemes have been proposed, such as second-order Markov chains with four independent parameters (corresponding to a copolymerization with penultimate effect, that is, an effect of the penultimate member of the growing chain), the nonsymmetric Bernoulli or Markov chains, or even non-Maikov models a few of these will be examined in a later section. Verification of these models calls for the knowledge of the distribution of sequences that become longer, the more complex the proposed mechanism. Considering only Bernoulli and Markov processes it may be said that at the dyad level all models fit the experimental data and hence none can be verified at the triad level the Bernoulli process can be verified or rejected, while all Markov processes fit at the tetrad level the validity of a first-order Markov chain can be confirmed, at the pentad level that of a second-order Maikov chain, and so on (10). 

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. 

It can be shown that every theoretical mechanism is automatically nth-order reversible for n = 1, 2, 3, 4 since in any polymer all n(ads) of these orders are automatically equifrequent with their reversals. Also, all first-and second-order Markov models are completely reversible. Third (and higher)-order Markov models are not completely reversible in general. 

Coleman and Fox published an alternative mechanism [82], According to these authors, the propagating centres exist in two forms, each of which favours the generation of either the m or r configuration. When both centres are in equilibrium, and when this equilibrium is rapidly established, the chain structure can be described by a modified Bernoulli statistics [83, 84]. The configurations of some polymers agrees better with this model than with first-or even second-order Markov models [84, 85]. 

With Bernoulli mechanisms, the ultimate unit of the growing chain has no influence on the linkage formed by a newly polymerized unit. With first-order Markov mechanisms, the ultimate unit does exert an influence, and in second-order Markov mechanisms, the penultimate, or second last, unit exerts an influence. In third-order Markov mechanisms it is the third last unit that exerts the influence on the linkage of newly joined units. Thus, Bernoulli mechanisms are a special case of Markov mechanisms, and could also be called zero-order Markov mechanisms. Second- and higher-order Markov mechanisms cannot be stated with confidence to occur in polyreactions, and, so, will not be discussed further. In addition, the discussion will be confined to binary mechanisms, that is, polyreactions where the unit possesses only two reaction possibilities. 

Similarly, for second-order Markov mechanisms, the transition probabilities Paa/a,Paa, B,Pba/a,Pba, b,Pab/a,Pab/b,Pbb/a, andpbb/b have to be considered, whereas, for Bernoulli mechanisms, only the transition probabilities ofp a and Pb need be considered. 

A description of the microstmcture by NMR spectroscopy of these copolymers, as well as a detailed understanding of the processes and mechanisms involved in these copolymerizations, proved difficult to achieve. A number of groups took on this challenge using various methodologies, which included synthesis of model compounds, NMR pulse sequences, synthesis of series of copolymers with different norbomene content and using catalysts of different symmetries, synthesis of copolymers selectively C-enriched, chemical shift prediction, and ab initio chemical shift computations. Such assignments enabled detailed information to be obtained on copolymerization mechanisms by Tritto et al. [24]. They employed a computer optimization routine, which allows a best fit to be obtained for the microstmctural analysis by NMR spectra in order to derive the reactivity ratios for both first- and second-order Markov models (Ml and M2, respectively). 

More detailed information on copolymerization mechanisms was obtained by Tritto et (Table 8). They used a computer optimization routine, which allows to best fit the microstmaural analysis by C-NMR spectra, to derive the reactivity ratios for both first- and second-order Markov models (Ml and M2, respectively). Hie theoretical equations relating copolymer composition and feed composition were fitted to the corresponding experimental data. The reactivity values agree with the reports that E-N copolymers obtained with IV-l/MAO are mainly alternating (ri x T2 1), the norbomene diad fraction is very low, and there are no norbomene triads or longer blocks (f2=0). 

The Bemoullian model and the first-order and second-order Markov models of the copolymerization mechanism can be tested by using the observed distribution of the triads. The results for the three models are shown in Table 7.15 for the sample with VA = 0.31%. 

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