Markov mechanism

The range of influence of units of the chain on the growth center classi-fles single mechanisms into Bernoulli or Markov mechanisms. The names derive from the mathematicians responsible for the statistics used to treat the... 

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. 

The influence of the chain units is given in transition probabilities for the linkage or placement step. In binary Markov mechanisms, the last unit. A, may become linked to a new unit. A, with the probability,pA/a, or with a new unit B, with the probability Pa/b. Since, for the unit being considered, only two possibilities are, by definition, available, the normalized sum of the transition probabilities is... 

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. 

It follows from this that Bernoulli and Markov mechanisms differ in whether the transition probabilities of the crossover, or hetero, steps are the same as or different from the homo steps (see Table 15-6). In addition, both types of mechanism can be subclassified as to whether the transition probabilities for the homo linkages are symmetric or asymmetric. In copolymerization, a symmetric Bernoulli mechanism with constitutionally different monomers is called azeotropic copolymerization with configurationally different monomers, it is called random flight polymerization and in stereocontrolled polymerization with nonchiral monomers, it is also called ideal atactic polymerization. ... 

Table 15-6. Classification of Bernoulli and Markov Mechanisms According to Transition Probabilities, p and Resulting Mole Fractions x on Assuming Infinitely Long Chains...

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. 

Green M S 1954 Markov random processes and the statistical mechanics of time-dependent phenomena. II. Irreversible processes in fluids J. Chem. Phys. 22 398... 

The sample labeled atactic in Fig. 7.10 was prepared by a free-radical mechanism and, hence, is expected to follow zero-order Markov statistics. As a test of this, we examine Fig. 7.9 to see whether the values of p, P, and Pj, which are given by the fractions in Table 7.9, agree with a single set of p values. When this is done, it is apparent that these proportions are consistent with this type... 

Electrolytic conductivity has also been measured in many binary systems. " Although data on conductivities in binary mixtures are very useful for practical purposes, the information from such data alone is limited from the viewpoint of elucidation of the mechanism. For example, the empirical Markov rule is well known for the electrical conductivity of binary mixtures. However, many examples have been presented where this rule does not hold well. 

Importantly, the value of the results gained in the present section is not limited to the application to actual systems. Eq. (4.2.11) for the GF in the Markov approximation and the development of the perturbation theory for the Pauli equation which describes many physical systems satisfactorily have a rather general character. An effective use of the approaches proposed could be exemplified by tackling the problem on the rates of transitions of a particle between locally bound subsystems. The description of the spectrum of the latter considered in Ref. 135 by means of quantum-mechanical GF can easily be reformulated in terms of the GF of the Pauli equation. 

Due to system complexity and paucity of information about the reaction mechanism(s), a Markov process model is proposed with probability pk = a cxpf -X)/k of the production of k ions of the anionic species X is the mean of the postulated Poisson distribution. The model also stipulates a probability q of reaction of the anionic species during two successive equal time periods of At each. 

Harhaji L, Isakovic A, Raicevic N, Markovic Z, Todorovic-Markovic B, Nikolic N, Vranjes-Djuric S, Markovic I, Trajkovic V (2007) Multiple mechanisms underlying the anticancer action of nanocrystalline fullerene. Eur. J. Pharmacol. 568 89-98. 

Isakovic A, Markovic Z, Todoiovic-Marcovic B, Nikolic N, Vranjes-Djuric S, Mirkovic M, Dramicanin M, Harhaji L, Raicevic N, Nikolic Z, Trajkovic V (2006b) Distinct cytotoxic mechanisms of pristine versus hydroxylated fullerene. Toxicol. Sci. 91 173-183. 

In addition to isolating complex 5, Markovic and Hartwig performed kinetic studies on the amination of methyl cinnamyl carbonate with aniline. The proposed mechanism involves reversible dissociation of product, reversible, endothermic oxidative addition of the allylic carbonate to form a 7i-allyliridium species, and irreversible nucleophilic attack on the 7i-allyliridium intermediate, as depicted in... 

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). 

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. 

The polymer stereosequence distributions obtained by NMR analysis are often analyzed by statistical propagation models to gain insight into the propagation mechanism [Bovey, 1972, 1982 Doi, 1979a,b, 1982 Ewen, 1984 Farina, 1987 Inoue et al., 1984 Le Borgne et al., 1988 Randall, 1977 Resconi et al., 2000 Shelden et al., 1965, 1969]. Propagation models exist for both catalyst (initiator) site control (also referred to as enantiomorphic site control) and polymer chain end control. The Bemoullian and Markov models describe polymerizations where stereochemistry is determined by polymer chain end control. The catalyst site control model describes polymerizations where stereochemistry is determined by the initiator. 

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. 

In many copolymerizations growth is influenced by the terminal (active) monomer unit. This can be described by Markov trials Zero order (or Bernoullian mechanism) means that the terminal unit of the growing chain does not influence the addition (rate, stereoregularity, etc.) of the next monomer molecule. Such copolymerizations often are called random . 

Markovic Z, Todorovic-Markovic B, Kleut D et al (2007) The mechanism of cell-damaging reactive oxygen generation by colloidal fullerenes. Biomateiials 28(36) 5437-5448... 

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