Second order Markov model

A second-order Markov model has also been described to show the effect on stereochemistry of the monomer unit behind the penultimate unit [Bovey, 1972],... 

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]. 

For a second-order Markov model, one would assume that the prediction of response for the fourth month is independent of the response obtained in the first month, given the responses in the second and third months. Such a model could similarly be fitted in S-Plus as... 

The second-order Markov model requires the specification of eight conditional probabilities. This is due to the influence of the last three pseudoasymmetric centers of the growing chain. The details of the second-order Markov model were described by Bovey. For details, the reader is advised to consult the reference. For convenience, the eight conditional probabilities are designated by Greek letters ... 

Other statistical models. The second-order Markov model is sometimes also applied to copolymers. Here, the probability of addition of a given monomer depends not only on the identity of the chain end monomer, but also on the nature of the preceding or penultimate monomer unit. As there are then four possible types of chain end to consider (namely, -AA, -AB, -BA, and -BB), there are eight addition probabilities which describe addition of the A and B monomers (e.g. Pg g represents the probability of B adding to a -BA chain end). As with the first-order Markov case, only half of these are independent because (Paaa + aab) = ( aba + abb) = Equations for... 

In the case of the unsubstituted C -symmetric metallocenes 8 and 11, copolymerization proceeds under control of the last inserted monomer unit (chain-end control), that is, it can be described by a second order Markov model. Ethylene is inserted with these zirconocenes three times faster than norbornene. No norbornene block sequences longer than two (NN units) are formed, in agreement with parameters calculated for 8 (rsE = 2.40, r E = 4.34, rEN = 0.03, and rm = 0.00). This result easily explains the maximum observed Xn = 0.66. 

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). 

The ranges of the reactivity ratios obtained at the lowest [N]/[E] feed ratio are ri = 2.34-4.99 and r2 = 0.0-0.062. The r2 values are in general smaller than those obtained for propene copolymerization. The highest r x 2 values found for the copolymers prepared with catalyst 1-4 confirmed its tendency to give more random copolymers. The values of ri, r2, and ri x r2 for the E-N copolymers obtained with catalysts IV-1 and 1-5 are comparable with those of alternating ethene-propene copolymers with metallocene catalysts. The results of the second-order Markov model also showed that all rn values, as r, are similar to those found for ethene and propene copolymerization with metallocene catalysts with low reactivity ratios. Differences in ri2 and in r22 are illuminating, since they clearly show the preference of the insertion of ethene or norbomene into E-N-Mt (Mt = Metal) and N-N-Mt, respectively. Parameter ri2 increases in the order IV-1 < 1-5 I-l < 1-2, opposite to the tendency to alternate the two comonomers [88]. 

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 f22 values are in general lower than those obtained for propene or other a-olefms, in agreement with the low homopolymerization activity of norbomene. The f22 value for catalyst 1-5 is much greater than ri2 this shows the tendency of this catalyst to insert a third norbomene after the second one. It was clear that the next-to-last E or norbomene monomer unit exerts an influence on the reactivity of the propagating Mt-E or Mt-N species, which depends upon the catalyst stmc-ture. The second-order Markov model must be used to describe E-N copolymerizations promoted by metallocenes I-l, 1-2, and IV-1. A third-order or a more complex model may be required to fit the experimental data obtained with catalyst 1-6, where more sterically hindered indene substitutions are dominant. At higher norbomene concentrations, copolymers with all catalysts may need more complex models. These results allowed the conclusion that E-N copolymerization is dominated by the bulkiness of the norbomene monomer and of the copolymer chain. 

Ncop mol. % First-order Markov model Second-order Markov model Finemann-Ross References... 

Thus, most authors restrict themselves to first- or second-order Markov models when dealing with Ziegler-Natta-catalysis for copolymerization, and this is elaborated below. 

The seeond-order Markov model gives four copolymeiiza-tion parameters, which are gained from the sequence distribution (full triad distribution) with a high degree of reliabihty. The calculated triad distribution from the second-order Markov model is optimized by varying the reaction probabilities Pp until the best fit is reached. The four copolymerization parameters are calculated from reaction probabilities as follows (13- 15) ... 

If the experimentally determined triad distribution is eompared with the ealculated values based on a first-order or alternatively on a second-order Markov model (Mi or M2 of the table), both models give a reasonable good fit up to a monomer/eomonomer ratio in solution of 1.5. Beyond a ratio of 3, however, the second-order Markov model proves to be the better approximation as indicated by the sum of the error squares divided by the number of experimental values R of the table). 

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%. 

---