Markov first-order mechanism

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. 

In the general case of a binary first-order Markov mechanism, the mole fractions of A and B are given from equations (15-29) and (15-32) as... 

For this reason, first-order Markov mechanisms for the general (asymmetric) case are described in terms of two transition probabilities, e.g., Pa/b and pb/a. These two transition probabilities can be calculated from experimentally determined mole fractions using Equations (15-35)-( 15-37). They may not simultaneously be zero. From Equations (15-27) and (15-28) it follows thatPb/b >Pa/b whenpa/a >Pb/a, and there is a tendency to form both long A chains and long B chains (see Figure 15-2). 

Isotactic diads of both dd and ll units are formed in the copolymerization of d and l monomers. Thus, for a first order Markov mechanism, the following is obtained from equations (15-33) and (15-35) with respect to the monomeric units ... 

Four different elementary reactions occur in first-order Markov mechanisms, and their rate constants can be calculated using the experimentally determined diad and triad concentrations. A steady state must exist for each diad type in the case of infinitely long polymer chains. A new type is formed by propagation cross-reaction for every type that disappears, i.e., for the stereocontrol... 

Thus, according to mechanism, the ratios of mole fractions of different kinds of diads, triads, etc., lead to very different rate constant combinations (Table 15-7). For example, the mole fraction ratio of iso- and syndiotactic diads gives the ratio of rate constants for iso- and syndiotactic linking in the case of Bernoulli mechanisms, but gives the ratios of the rate constants for the cross-steps and not a mean of the rate constants in the case of first-order Markov mechanisms. 

Styrene-SQ., Copolymers. I would now like to discuss two systems which illustrate the power of C-13 nmr in structural studies. The first is the styrene-SO system. As already indicated, this is of the type in which the chain composition varies with monomer feed ratio and also with temperature at a constant feed ratio (and probably with pressure as well.) The deviation of the system from simple, first-order Markov statistics, —i.e. the Lewis-Mayo copolymerization equation—, was first noted by Barb in 1952 ( ) who proposed that the mechanism involved conplex formation between the monomers. This proposal was reiterated about a decade later by Matsuda and his coworkers. Such charge transfer com-... 

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). 

Having established that a particular polymerization follows Bemoullian or first-order Markov or catalyst site control behavior tells us about the mechanism by which polymer stereochemistry is determined. The Bemoullian model describes those polymerizations in which the chain end determines stereochemistry, due to interactions between either the last two units in the chain or the last unit in the chain and the entering monomer. This corresponds to the generally accepted mechanism for polymerizations proceeding in a nonco-ordinated manner to give mostly atactic polymer—ionic polymerizations in polar solvents and free-radical polymerizations. Highly isoselective and syndioselective polymerizations follow the catalyst site control model as expected. Some syndioselective polymerizations follow Markov behavior, which is indicative of a more complex form of chain end control. 

More convenient and entirely sufficient for the present purpose is the calculation of ratio of rate constants. The calculation will be reviewed for a one-way first-order Markov process. A one-way mechanism is chosen because it is intuitively the most appropriate model for a free radical mechanism. Furthermore it has some experimental support. The assumption of a first-order Markov process does not rule out higher Markov processes. The differentiation between a first-order Markov process and higher order Markov processes is however possible experimentally in very rare cases because it involves the determination of tetrad, pentad, etc. fractions (11, 12, 13, 14). A Bemoullian process is ruled out by the analysis of the data of Table I. 

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. 

From the Ah-NMR analysis of poly(3,8-d2-styrene oxide) obtained using ZnEt2/H20 as initiator Figure 2), the formation of the partly crystalline fraction can be described by first order Markov statistics, while that for the amorphous fraction follows Bemoulllan statistics. Different chain propagation mechanisms are, therefore, responsible for the formation of the two different polymer fractions obtained from this particular catalyst. Consequently, the existence of two different active centers, responsible for the two polymerization mechanisms and for formation of fractions I and II, are clearly indicated. 

Figure 15-2. Relationship between the various diad fractions Xij (that is, jr aa, Jr ab or orbb ) and the monad fraction jca for asymmetric Bernoulli (B) and first-order Markov (M) mechanisms, in the latter case, for pb/a = 0.5 pa/a-...

The stereocontrol of most free radical polymerizations appears to be governed by an end-controlled mechanism. It generally follows first-order Markov statistics with respect to diads (see also Section 16.5.2.3). The tactic-ity of the formed polymer is also influenced by the solvent used. The cause of this solvent control effect is unclear, and possibly is due at least partly to different degrees of solvation. A compensation effect (see Section 16.5.4.) exists in the relationship between the activation entropies and enthalpies for diad formation in various solvents. The compensation temperature TJj varies with monomer constitution (Table 20-11). The compensation enthalpies AAHI vary strongly according to both monomer and placement type. 

Several examples of NMR studies of copolymers that exhibit Bernoullian sequence distributions but arise from non-Bernoullian mechanisms have been reported. Komoroski and Schockcor [11], for example, have characterised a range of commercial vinyl chloride (VC)/vinylidene chloride (VDC) copolymers using carbon-13 NMR spectroscopy. Although these polymers were prepared to high conversion, the monomer feed was continuously adjusted to maintain a constant comonomer composition. Full triad sequence distributions were determined for each sample. These were then compared with distributions calculated using Bernoullian and first-order Markov statistics the better match was observed with the former. Independent studies on the variation of copolymer composition with feed composition have indicated that the VDC/VC system exhibits terminal model behaviour, with reactivity ratios = 3.2 and = 0.3 [12]. As the product of these reactivity ratios is close to unity, sequence distributions that are approximately Bernoullian are expected. 

To prove the existence of a Bernoullian sequence, only triad information is needed, but tetrad information is required to verify a first-order Markov sequence. More complicated stereochemical mechanisms are possible of course (such as reactions controlled by penultimate configurations), and these would have to be fitted by more complex statistical analyses requiring knowledge of more than two probabilities. However, most detailed analyses to date on the tactleities of polymers obtained in homogeneous free radical or ionic polymerization reactions have been found to conform to either Bernoullian or first-order Markovian statistics. 

By comparing the NMR resonance intensities with the predicted values given by the Bernoullian and first-order Markov (or higher), one can investigate the stereospecific polymerization mechanism. This approach has been used to define the mechanism for the free radical and anionically polymerized PMMA. The peak intensities for the measured stereosequences in PMMA from a free-radical polymerization are shown in Table 7.8. Using a probability of isotactic addition (P ,) equal to 0.24, the fraction of the triad, tetrad and pentads was compared with the measured values. The agreement suggests that the Bernoullian mechanism applies. 

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]. 

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. 

Stereocontrol of free radical polymerization is influenced by monomer constitution, solventy and temperature. Most polymerizations seem to follow at least a Markov first-order one-way mechanism. Ratios of the four possible rate constants ki/iy ki/8, k8/i, and k8/8 can be calculated from the experimentally accessible concentrations of configurational triads and diads. With increasing temperature, more heterotactic triads are formed at a syndiotactic radical whereas the monomer addition at an isotactic radical favors isotactic and not heterotactic triads. Compensation effects exist for the differences of activation enthalpies and activation entropies for each of the six possible combinations of modes of addition. The compensation temperature is independent of the mode of addition whereas the compensation enthalpies are not. 

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]. 

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). 

Analytical procedures for calculating the non-Gaussian probability density function of the response are generally based on the assumption of Markov processes. Therefore, in a preceding step, the equation of motion (6) has to be transformed by methods of classical mechanics into a set of first order differential equations. From this equation a parabolic partial differential equation, the so-called Fokker-Planck-equatlon, can be derived ... 

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). 

Tabie 2. Ratios of Rate Constants for Markov First Order Mechanisms... 

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

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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