Markov chain mechanism, first order

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. 

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. 

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. 

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). 

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... 

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]. 

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