Markov statistics

Distribution of A and B units Bernouilli statistics Markov statistics - formation of Bn blocks Markov statistics isolated B units between Ajj blocks... 

When in polymerization the addition probability of a monomer unit is independent of the type of the last unit in the growing chain (as is the case in most radical polymerizations), then the chain growth is governed by the two probabilities, Pj and P2, of the addition of monomer 1 or 2. Since these two probabilities are bound by the relation Pj +P2 = 1, the polymer statistics is determined by a single parameter p = Pi = 1 — P2. This is the so-called Bernoulli statistics. When the probability of monomer addition depends on the type of the last monomer unit in the growing chain, the system is characterized by four transition probabilities, P -, P 2-> 21> and 22> bound by the conditions f ii + i2 = l d P21 + P22 = 1, defining two independent parameters (e.g., P21 = 1 — P22 and P12 = 1 — Pn). This is the so-called first-order Markov statistics. Markov... 

Increased specialization in the discipline by sophistication of the techniques, e.g., redundancy modeling, Bayesian statistics, Markov chains, etc., and by the development of the concepts of reliability physics to identify and model the physical causes of failure and of structural reliability to analyze the integrity of buildings, bridges, and other constructions... 

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... 

Use zero-order Markov statistics to evaluate the probability of isotactic, syndio-tactic, and heterotactic triads for the series of p values spaced at intervals of... 

If the fractions of triads could be measured, they either would or would not lie on a single vertical line in Fig. 7.9. If they did occur at a single value of p, this would not only give the value of p (which could be obtained from the fraction of one kind of triad), but would also prove the statistics assumed. If the fractions were not consistent with a single p value, higher-order Markov statistics are indicated. 

Figure 7.9 Fractions of iso, syndio, and hetero triads as a function of p, calculated assuming zero-order Markov (Bernoulli) statistics in Example 7.7.

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... 

In the research described in the preceding problem, Randall was able to assign the five peaks associated with tetrads in the C-NMR spectrum on the basis of their relative intensities, assuming zero-order Markov (or Bernoulli) statistics with Pm = 0.575. The five tetrad intensities and their chemical shifts from TMS are as follows ... 

On the basis of these observations, criticize or defend the following proposition Regardless of the monomer used, zero-order Markov (Bernoulli) statistics apply to all free radical, anionic, and cationic polymerizations, but not to Ziegler-Natta catalyzed systems. 

While static Monte Carlo methods generate a sequence of statistically independent configurations, dynamic MC methods are always based on some stochastic Markov process, where subsequent configurations X of the system are generated from the previous configuration X —X —X" — > with some transition probability IF(X —> X ). Since to a large extent the choice of the basic move X —X is arbitrary, various methods differ in the choice of the basic unit of motion . Also, the choice of transition probability IF(X — > X ) is not unique the only requirement is that the principle... 

Where the nature of the preceding dyad is important in determining the configuration of the new chiral center (Scheme 4.2), first order Markov statistics... 

Bcrnoullian statistics do not provide a satisfactory description of the tacticity. 6 This finding is supported by other work.28" 38 First order Markov statistics provide an adequate fit of the data. Possible explanations include (a) penpenultimale unit effects are important and/or (b) conformational equilibrium is slow (Section 4.2.1). At this stage, the experimental data do not allow these possibilities to be distinguished. 

It seems likely that other polymerizations will be found to depart from Bemoullian statistics as the precision of tacticity measurements improves. One study12 indicated that vinyl chloride polymerizations are also more appropriately described by first order Markov statistics. However, there has been some reassignment of signals since that time. 4 25... 

A general purpose program has been developed for the analysis of NMR spectra of polymers. A database contains the peak assignments, stereosequence names for homopolymers or monomer sequence names for copolymers, and intensities are analyzed automatically in terms of Bernoullian or Markov statistical propagation models. A calculated spectrum is compared with the experimental spectrum until optimized probabilities, for addition of the next polymer unit, that are associated with the statistical model are produced. 

A portion of the database for this polymer is shown in Figure 6. Literature reports that this polymer follows second-order Markov statistics ( 21 ). And, in fact, probabilities that produced simulated spectra comparable to the experimental spectrum could not be obtained with Bernoullian or first-order Markov models. Figure 7 shows the experimental and simulated spectra for these ten pentads using the second-order Markov probabilities Pil/i=0.60, Piv/i=0.35, Pvi/i=0.40, and Pvv/i=0.55 and a linewidth of 14.8 Hz. 

While other programs require modification of the actual code in changing the polymer, spectra, or model, only changes in the user database is required here. Changes in the program since a brief report (22) in 1985 include improvement of the menu structure, added utilities for spectral manipulations, institution of demo spectra and database. Inclusion of Markov statistics, and automation for generation of the coefficients in Equation 1. Current limitations are that only three models (Bernoul llan, and first- and second-order Markov) can be applied, and manual input Is required for the N. A. S. L.. 

For a number of copolymers, whose kinetics of formation is described by nonideal models, the statistics of alternation of monomeric units in macromolecules cannot be characterized by a Markov chain however, it may be reduced to the extended Markov chain provided that units apart from their chemical nature... 

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 ... 

Upon expressing from the equilibrium condition the complex concentration M12 through the concentrations of monomers, and substituting the expression found into relationship (21) we obtain, invoking the formalism of the Markov chains, final formulas enabling us to calculate instantaneous statistical characteristics of the ensemble of macromolecules with colored units. A subsequent color erasing procedure is carried out in the manner described above. For example, when calculating instantaneous copolymer composition, this procedure corresponds to the summation of the appropriate components of the stationary vector jt of the extended Markov chain ... 

An exhaustive statistical description of living copolymers is provided in the literature [25]. There, proceeding from kinetic equations of the ideal model, the type of stochastic process which describes the probability measure on the set of macromolecules has been rigorously established. To the state Sa(x) of this process monomeric unit Ma corresponds formed at the instant r by addition of monomer Ma to the macroradical. To the statistical ensemble of macromolecules marked by the label x there corresponds a Markovian stochastic process with discrete time but with the set of transient states Sa(x) constituting continuum. Here the fundamental distinction from the Markov chain (where the number of states is discrete) is quite evident. The role of the probability transition matrix in characterizing this chain is now played by the integral operator kernel ... 

As the result of theoretical consideration of polycondensation of an arbitrary mixture of such monomers it was proved [55,56] that the alternation of monomeric units along polymer molecules obey the Markovian statistics. If all initial monomers are symmetric, i.e. they resemble AaScrAa, units Sa(a=l,...,m) will correspond to the transient states of the Markov chain. The probability vap of transition from state Sa to is the ratio Q /v of two quantities Qa/9 and va which represent, respectively, the number of dyads (SaSp) and monads (Sa) per one monomeric unit. Clearly, Qa(S is merely a ratio of the concentration of chemical bonds of the u/i-ih type, formed as a result of the reaction between group Aa and Ap, to the overall concentration of monomeric units. The probability va0 of a transition from the transient state Sa to an absorbing state S0 equals l-pa where pa represents the conversion of groups Aa. 

Noteworthy that all the above formulated results can be applied to calculate the statistical characteristics of the products of polycondensation of an arbitrary mixture of monomers with kinetically independent groups under any regime of this process. To determine the values of the elements of the probability transition matrix of corresponding Markov chains it will suffice to calculate only the concentrations Q()- of chemical bonds (ij) at different conversions of functional groups. In the case of equilibrium polycondensation the concentrations Qy are controlled by the thermodynamic parameters, whereas under the nonequilibrium regime of this process they depend on kinetic parameters. 

For the interbipolycondensation the condition of quasiideality is the independence of the functional groups either in the intercomponent or in both comonomers. In the first case the sequence distribution in macromolecules will be described by the Bernoulli statistics [64] whereas, in the second case, the distribution will be characterized by a Markov chain. The latter result, as well as the parameters of the above mentioned chain, were firstly obtained within the framework of the simplified kinetic model [64] and later for its complete version [59]. If all three monomers involved in interbipolycondensation have dependent groups then, under a nonequilibrium regime, non-Markovian copolymers are known to form. 

In some very recent work by Karssenberg et al. [130], attempts have been made to improve the analytical ability of a technique like NMR spectroscopy to effectively predict the distribution of sequence lengths in polyethylene-alkene copolymers. They analyzed the entire [ C-NMR spectrum for homogeneous ethylene-propene copolymers. They used quantitative methods based on Markov statistics to obtain sequence length distributions as shown in Figure 22 [130]. The... 

Thus, as can be inferred from the foregoing, the calculation of any statistical characteristics of the chemical structure of Markovian copolymers is rather easy to perform. The methods of statistical chemistry [1,3] can reveal the conditions for obtaining a copolymer under which the sequence distribution in macromolecules will be describable by a Markov chain as well as to establish the dependence of elements vap of transition matrix Q of this chain on the kinetic and stoichiometric parameters of a reaction system. It has been rigorously proved [ 1,3] that Markovian copolymers are formed in such reaction systems where the Flory principle can be applied for the description of macromolecular reactions. According to this fundamental principle, the reactivity of a reactive center in a polymer molecule is believed to be independent of its configuration as well as of the location of this center inside a macromolecule. 

For many synthetic copolymers, it becomes possible to calculate all desired statistical characteristics of their primary structure, provided the sequence is described by a Markov chain. Although stochastic process 31 in the case of proteinlike copolymers is not a Markov chain, an exhaustive statistic description of their chemical structure can be performed by means of an auxiliary stochastic process 3iib whose states correspond to labeled monomeric units. As a label for unit M , it was suggested [23] to use its distance r from the center of the globule. The state of this stationary stochastic process 31 is a pair of numbers, (a, r), the first of which belongs to a discrete set while the second one corresponds to a continuous set. Stochastic process ib is remarkable for being stationary and Markovian. The probability of the transition from state a, r ) to state (/i, r") for the process of conventional movement along a heteropolymer macromolecule is described by the matrix-function of transition intensities... 

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