Markovian statistics

It should be emphasized that for Markovian copolymers a knowledge of the values of structural parameters of such a kind will suffice to find the probability of any sequence Uk, i.e. for an exhaustive description of the microstructure of the chains of these copolymers with a given average composition. As for the composition distribution of Markovian copolymers, this obeys for any fraction of Z-mers the Gaussian formula whose covariance matrix elements are Dap/l where Dap depend solely on the values of structural parameters [2]. The calculation of their dependence on time, and the stoichiometric and kinetic parameters of the reaction system permits a complete statistical description of the chemical structure of Markovian copolymers to be accomplished. The above reasoning reveals to which extent the mathematical modeling of the processes of the copolymer synthesis is easier to perform provided the alternation of units in macromolecules is known to obey Markovian statistics. 

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. 

Eigenvalues of the operator Qr are real while the largest of them, Af, equals unity by definition. As a result, in the limit n-> oo all items in the sum (Eq. 38), excluding the first one, Q Q f = Xr/Xfh will vanish. In this case, chemical correlators will decay exponentially along the chain on the scale n 1/ In AAt values n < n the law of the decay of these correlators differs, however, from the exponential one even for binary copolymers. This obviously testifies to non-Markovian statistics of the sequence distribution in molecules (see expression Eq. 11). The closer is to unity, the greater are the values of n. The situation when n 1 corresponds to proteinlike copolymers. 

Note An example of a statistical copolymer is one consisting of macromolecules in which the sequential distribution of monomeric units follows Markovian statistics. 

Statistical copolymers are copolymers in which the sequential distribution of the monomeric units obeys known statistical laws e.g. the monomeric-unit sequence distribution may follow Markovian statistics of zeroth (Bemoullian), first, second or a higher order. Kinetically, the elementary processes leading to the formation of a statistical sequence of monomeric units do not necessarily proceed with equal a priori probability. These processes can lead to various types of sequence distribution comprising those in whieh the arrangement of monomeric units tends towards alternation, tends towards... 

In spite of the fact that the composition of copolymers for the template and non-template set of experiments is not exactly the same, one can see from the data presented in Table 5.7 that the fraction of MMM triads is much higher for the copolymers obtained in the template system. Assuming the first order Markovian statistics, the following set of equations can be applied ... 

This indicates that a close contact of the carbanion with the counterion favours isotactic placements as well as short sequence length (corresponding to persistence ratios below 1). In the system Cs/THF the marked non-Bernoullian behaviour can be described by Markovian statistics rather than the Coleman-Fox model, i.e. the penultimate monomer unit influences the stereochemistry of the monomer addition (29). This effect can be interpreted by decreasing external solvation (III,IV) and increasing intramolecular solvation (I,II). 

The observed number average sequence lengths deviate from those predicted from Bernoullian statistics, i.e., n = 1/[VC1], However, corresponding calculations based on first order Markovian statistics are in excellent agreement. 

To calculate the number average sequence length using first order Markovian statistics, it is necessary to estimate p, ... 

The data shown in Tables HI and TV show that the 13C nmr spectra of vinyl chloride-vinylidene chloride copolymers have a redundancy of structural relationships. By analyzing a range of compositions, this system has been found to yield a reasonable description of both monomer composition and monomer sequence distribution. The data also show that this copolymer is a good example of a system best described by first order Markovian statistics as compared to Bernoullian statistics. 

Carbon-13 nuclear magnetic resonance was used to determine the molecular structure of four copolymers of vinyl chloride and vinylidene chloride. The spectra were used to determine both monomer composition and sequence distribution. Good agreement was found between the chlorine analysis determined from wet analysis and the chlorine analysis determined by the C nmr method. The number average sequence length for vinylidene chloride measured from the spectra fit first order Markovian statistics rather than Bernoullian. The chemical shifts in these copolymers as well as their changes in areas as a function of monomer composition enable these copolymers to serve as model... 

When the product of monomer relative reactivity ratios is approximately one r x r2 = 1), the last inserted monomeric unit in the chain does not influence the next monomer incorporation and Bernoullian statistics govern the formation of a random copolymer. When this product tends to zero (r xr2 = 0), there is some influence from the last inserted monomeric unit (when first-order Markovian statistics operate), or from the penultimate inserted monomeric units (when second-order Markovian statistics operate), and an alternating copolymer formation is favoured in this case. Finally, when the product of the reactivity ratios is greater than one (r x r2 > 1), there is a tendency for the comonomers to form long segments and block copolymer formation predominates (or even homopolymer formation can take place) [448],... 

A single step of the polymerization is analogous to a diastereoselective synthesis. Thus, to achieve a certain level of chemical stereocontrol, chirality of the catalytically active species is necessary. In metallocene catalysis, chirality may be associated with the transition metal, the ligand, or the growing polymer chain (e.g., the terminal monomer unit). Therefore, two basic mechanisms of stereocontrol are possible (145,146) (i) catalytic site control (also referred to as enantiomorphic site control), which is associated with the chirality at the transition metal or the ligand and (ii) chain-end control, which is caused by the chirality of the last inserted monomer unit. These two mechanisms cause the formation of microstructures that may be described by different statistics in catalytic site control, errors are corrected by the (nature (chirality) of the catalytic site (Bernoullian statistics), but chain-end controlled propagation is not capable of correcting the subsequently inserted monomers after a monomer has been incorrectly inserted (Markovian statistics). 

The terminal and penultimate models then correspond to first- and second-order Markovian statistics, respectively. But you don t actually have to know this, in the sense that we can just proceed using common sense. For example, the probability of finding the sequence ABABA in a system obeying first-order Markovian statistics (i.e., copolymerization where the terminal model applies) is given in Equations 6-31. 

We ll leave it to you to show what the same probability would be if second-order Markovian statistics applied (i.e., the penultimate model). 

Just as in the derivation of the copolymer equation for the terminal model, we start with a reversibility relationship P3 AAB = P3 BAA. Now we must use second-order Markovian statistics to write this in terms of conditional probabilities (Equation 6-64) ... 

D. Express the pentad sequence PS BABAB in terms of first order Markovian statistics. 

If the stereochemistry of addition does depend upon the configuration found at the end of the chain, whether it is m or r, then we have a terminal model, or first-order Markovian statistics. At minimum we need tetrad data from NMR—i.e., data for (mmm), (mmr), etc.—to test for the terminal model. Remember, we can always calculate triad data from tetrad data using the relationships previously given in Equations 7-26. Equations 7-34 relate the relevant conditional probabilities to observable tetrad and triad sequences. 

First-order Markovian statistics (the terminal model) are followed if Equations 7-35 are obeyed. 

Are these results consistent with Bemoul-lian or Markovian statistics ... 

A rigorous analysis of the same problem has been given by San Miguel and Sancho. They emphasized that the reduction process produces unavoidably non-Markovian statistics. Nevertheless they noted that the existence of Fokker-Planck equations does not conflict with the non-Markovian character of the stochastic process, since the corresponding solution, in harmony with ref. 19, is valid only to evaluate one-time averages and is of no use in multitime averages. 

In other words, if we are exploring the low-friction regime, the interplay of non-Markovian statistics and external field renders the system still more inertial, thereby widening the range of validity of the formula provided by Kramers for the low-friction regime provided that y be replaced by... 

The statistics of co-polymerization are rather complicated most of the co-polymerizations do not follow simple Bernoullian statistics, but are better described by terminal (first-order Markovian) or penultimate (second-order Markovian) statistics.59 574... 

In order to determine whether a system obeys first-order Markovian statistics, a knowledge of pentad tacticity is required to a high degree of experimental accuracy. This is very difficult for the case of radically polymerized PMMA, since the fraction of mm triads is usually less than 5% of the total. 

Non-Bernoulian behaviour of tacticity was also reported for poly(vinyl chloride) obtained at 5°C and — 30°C.164 After careful assessment of the precision and accuracy in tacticity determination, triad and tetrad fractions were found to be well-explained by the first-order Markovian statistics. 

The correctness of each model for a given copolymer system can be tested and confirmed by experimental observation. In general, when Bernoullian statistics do not describe the sequence distribution, Markovian statistics do. 

Okada et al. correlated the proportion of dyads and triads using Markovian statistics which can be applied only to the simplest four-parameter system [e.g. Scheme (15-1)]. In such a system the proportion of 3-triads (DDT+TDD) is equal to the geometrical average of a-(DDD) and y-(TDT) triads (P = 2 /a y). Analysis of Okada s data gives large deviations from this rule, e.g. the observed proportion of the P-dyad is half of that calculated by Markovian statistics (poh8d = 0.21, Poalcd = 0.47). Evidently this statistics cannot be applied, probably because of reversibility and the diversity of active centers in the system. 

Enantiomorphic Site with Chain-End Control. In the case of less stereoselective Cz-symmetric metallocene catalysts, the magnitude of chain-end control can be comparable to that of site control. In this case, obviously, the former has to be added to the model using Markovian statistics. The probability parameters are the same found for pure chain-end control p si re), i.e., the probability of insertion of a si monomer enantioface after a monomer inserted with the re face, p re si), p si si), and p re re). In this case, the metallocene chirality prevents the equiprob-ability of the si olefin insertion after a re inserted monomer (see structure on the left in Scheme 36) and re olefin insertion after a si inserted monomer (see structure on the right in Scheme 36). 

Statistical copolymers are those in which the monomer sequence follows a specific statistical law (e.g., Markovian statistics of order zero, one, two). Random copolymers are a special case of statistical copolymers in which the nature of a monomeric unit is independent of the nature of the adjacent unit (Bernoullian or zero-order Markovian statistics). They exhibit the structure shown in Figure 6.1. If A and B are the two monomers forming the copolymer, the nomenclature is poly (A-stat-B) for statistical copolymers and poly (A-ran-B) for the random case. It should be noted that sometimes the terms random and statistical are used indistinctly. The commercial examples of these copolymers include SAN poly (styrene-ran-acrylonitrile) [4] and poly (styrene-ran-methyl methacrylate) (MMA) [5]. 

Table m. Summary of two-component, 1 order Markovian statistical analysis of alginate fraction data obtained from SEC-NMR. 

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