Markov model copolymerization

The experimental copolymer composition data for styrene(Mi)-fumaroni-trile(M2) give a good fit to Eq. (7.86) with rj = 0.072 and r[ = 1.0 [33], but deviate markedly from the behavior predicted by the st-order Markov model with ri = 0.23. Penultimate effects have been observed in a number of other systems. Among these are the radical copolymerizations of ethyl methacrylate-styrene, methyl methaciylate-4-vinyl pyridine, methyl acrylate-l,3-butadiene, and other monomer pairs. 

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. 

The styrene and acrylonitrile can be copolymerized by free radical methods using a continuous stirred tank reactor (CSTR). The reactivity ratios r,2 and rj, can be taken as 0.04 and 0.41, respectively. Construct a first-order Markov model using the dyad probabilities derived in Section 11.1. 

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. 

Various mathematical models have been applied to copolymerization reactions induced by Ziegler-catalysts. Wall (1) assumed the velocity of the monomer addition M j and M2 to be independent of the previously integrated monomeric unit (zero-order Markov model) ... 

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. 

Independent of the analytical methods the copolymerization parameters derived from the Mayo-Lewis equation have some major drawbacks, but do enable a part of the information contained in each copolymerization experiment to be extracted, namely the integral composition of the copolymer. A series of experiments is necessary to obtain the copolymerization parameters. Furthermore, using this approach it cannot be decided whether the application of the first-order Markov model is valid or not. One may even obtain a nearly perfect Fineman-Ross plot, where other methods show that a first-order Markov model cannot be applied. 

As mentioned above, a knowledge of the copolymerization parameters enables all the sequence distributions to be calculated. Or inversely, e q)erimentally determined sequence distributions yield the copolymerization parameters. In the simplest case, intensities of appropriate single sequences can be compared. The results are even more reliable, if the determination of the copolymerization parameters is based on the maximal accessible information on the microstructure of the copolymer. To derive the r-parameters from the overall triad distribution is the safest way (13). This may be done by calculating a triad distribution based on a first-order Markov model and optimizing by variation of the reaction probabilities Py imtil the best fit between calculated and experimental data... 

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) ... 

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%. 

The instantaneous composition of a copolymer X formed at a monomer mixture composition x coincides, provided the ideal model is applicable, with stationary vector ji of matrix Q with the elements (8). The mathematical apparatus of the theory of Markov chains permits immediately one to wright out of the expression for the probability of any sequence P Uk in macromolecules formed at given x. This provides an exhaustive solution to the problem of sequence distribution for copolymers synthesized at initial conversions p l when the monomer mixture composition x has had no time to deviate noticeably from its initial value x°. As for the high-conversion copolymerization products they evidently represent a mixture of Markovian copolymers prepared at different times, i.e. under different concentrations of monomers in the reaction system. Consequently, in order to calculate the probability of a certain sequence Uk, it is necessary to average its instantaneous value P Uk over all conversions p preceding the conversion p up to which the synthesis was conducted. 

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 ... 

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). 

In addition to the above mentioned dynamic problems of copolymerization theory this review naturally dwells on more traditional statistical problems of calculation of instantaneous composition, parameters of copolymer molecular structure and composition distribution. The manner of presentation of the material based on the formalism of Markov s chains theory allows one to calculate in the uniform way all the above mentioned copolymer characteristics for the different kinetic models by means of elementary arithmetical operations. In Sect. 3 which is devoted to these problems, one can also find a number of original results concerning the statistical description of the copolymers produced through the complex radical mechanism. 

The consistent kinetic analysis of the copolymerization with the simultaneous occurrence of the reactions (2.1) and (2.5) leads to the conclusion that the probabilities of the sequences of the monomer units M, and M2 in the macromolecules can not be described by a Markov chain of any finite order. Consequently, in this very case we deal with non-Markovian copolymers, the general theory for which is not yet available [6]. However, a comprehensive statistical description of the products of the complex-radical copolymerization within the framework of the Seiner-Litt model via the consideration of the certain auxiliary Markov chain was carried out [49, 59, 60]. 

A generalization of the theory of the binary copolymerization for multicomponent systems in the case of the terminal model (2.8) is not difficult since the copolymer microstructure is still described by the Markov chain with states S corresponding to the monomer units Mj. The number m of their types determines the order of... 

The general formulae (5.1), (5.3), and (5.7) are still valid under the transition to the more complicated models described in Sect. 2. In the case of the penultimate model it concerns also the dynamic Eqs. (5.2) into which now one should substitute the dependence j (i) obtained after the solution of the problem of the calculation of the stationary vector tE(x) of the Markov chain corresponding to this model. Substituting the function X1(x1) obtained via the above procedure (see Sect. 3.1) into Eq. (5.2) for the binary copolymerization we can find its explicit solution expressed through the elementary functions. However, this solution is rather cumbersome and has no practical importance. It is not needed even for the classification of the dynamic behavior of the systems, which can be carried out only via analysis of Eq. (5.5) by determining the number n = 0,1,2, 3 of the inner azeotropes in the 2-simplex [14], The complete set of phase portraits of the binary... 

A key facet of copolymerization is the possible disparity of reactivities of the monomers. Traditional procedure is to assume, at least as an approximation, that the reactivity of a growing propagating center depends only on the identity of its reactive end unit (i.e., the last monomer added), not on the composition and length of the rest of its chain [124-126] (first-order Markov or terminal model see also... 

The simple copolymer model is a first-order Markov chain in which the probability of reaction of a given monomer and a macroradical depends only on the terminal unit in the radical. This involves consideration of four propagation rate constants in binary copolymerizations, Eqs. (7-2)-(7-4). The mechanism can be extended by including a penultimate unit effect in the macroradical. This involves eight rate constants. A third-order case includes antepenultimate units and 16 rate coefficients. A true test of this model is not provided by fitting experimental and predicted copolymer compositions, since a match must be obtained sooner or later if the number of data points is not saturated by the adjustable reactivity ratios. 

To predict the course of a copolymerization we need to be able to express the composition of a copolymer in terms of the concentrations of the monomers in the reaction mixture and the relative reactivities of these monomers. In order to develop a simple model, it is necessary to assume that the chemical reactivity of a propagating chain (which may be free-radical in a radical chain copolymerization and carbocation or carboanion in an ionic chain copolymerization) is dependent only on the identity of the monomer unit at the growing end and independent of the chain composition preceding the last monomer unit [2-5]. This is referred to as the first-order Markov or terminal model of copolymerization. 

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