Bemoullian statistics

Many radical polymerizations have been examined from the point of view of establishing the stereosequence distribution. For most systems it is claimed that the tacticity is predictable within experimental error by Bemoullian statistics [i.e. by the single parameter P(m) - see 4.2.1],... 

MMA polymerization is one of the most studied systems and was thought to be explicable, within experimental error, in terms of Bemoullian statistics. Moad et ai.jb have made precise measurements of the configurational sequence distribution for PMMA prepared from 13C-labeled monomer. It is clear that... 

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... 

The most commonly used technique for making C-13 NMR spectral assignments for vinyl homopolymers has been through the use of Bemoullian statistics (29) ( ). If one knows the relative concentrations of either m or r, any particular "n-ad" distribution can be calculated because... 

Note In a random copolymer, the sequence distribution of monomeric units follows Bemoullian statistics. 

The configurational-conformational characteristics of PP are discussed by considering every polymer chain as constituted by the periodic repetition of a sequence of monomeric units in a given configuration. Calculations are presented for the special case in which mesa and racemic diads are distributed according to Bemoullian statistics. Numerical results show that the characteristic ratio of atactic PP reaches an asymptotic value of 5.34 when the size of the periodic sequence corresponds to six monomeric units. 

Fig. 15. Comparison between observed and calculated steric triad fractions of polypropylenes obtained with soluble V-based catalysts. Dashed curves are calculated by the symmetric Bemoullian statistics (Eq. 30), Observed values are taken from Table 1...

The steric pentad distributions of the polypropylene with structure (14) are in accord with the symmetric Bemoullian statistical model (Eq. 30) based on stereochemical control by the last propylene unit of a growing chain end. 

The Pi (sometimes denoted Pnl) is for the probability of forming a new z -dyad the value is also obtained by comparison of the experimentally determined tetrad integrals with the probability that a particular tetrad would occur. Ps (sometimes denoted Pr) is the probability of forming a new. v-dvad. The probabilities of a particular tetrad stereochemistry can be calculated using Bemoullian statistics (see Table 2). For a value of 0 < Pi < 0.5, the polymerization proceeds preferentially through a... 

Fig. 8 Methine region (homonuclear decoupled 1H NMR spectrum) of moderately isotactic PLA (Pi = 0.69) Table 2 rac-Lactide tetrad probabilities based on Bemoullian statistics ...

You can now test the models. So far we have only talked about Bemoullian statistics, where the stereochemistry of the addition of a monomer does not depend upon the configuration at the end of the chain. Just to make sure you understand, from NMR triad information—i.e., (mm), (mr) and (rr) data—we can readily calculate the conditional probabilities P(r/m), P(m/m), P(m/r) and P(r/r) as shown in Equations 7-32. Bemouillian statistics are followed if Equations 7-33 are obeyed. 

You are given an 19F NMR spectrum of poly(vinylidene fluoride) (PVDF) that is sensitive to triad information. What would you expect the normalized intensity of the most intense line in the spectrum to be if 20% of the structural units are incorporated backwards into the chain. (Assume Bemoullian statistics.)... 

For any given polymer, if Bemoullian, the mm, mr, rr sequence frequencies, as estimated from the relative areas of the appropriate peaks in NMR spectra (see below), should lie on a single vertical line in Fig 2.15, corresponding to a single value of Pm (see example below). If this is not the case, then the polymer s configurational sequence deviates from the simple Bemoullian statistics. 

Determine whether the polymerization in each case can be determined by a single value of Pm in agreement with Bemoullian statistics. 

Determine the probability and test for consistency with Bemoullian statistics. What do the values of obtained in the two cases signify [Ans. Polymer 1 = 0.06 Polymer 2 Pm = 0.4 ]... 

All the data included in Table 12 indicate that the values of 4(mm)(rr) / (mr)2, which is expected to be unity if stereoregulation obeys Bemoullian statistics, 61 are less than unity as reported by several investigators.3,4,10 38 160 162163... 

The microstructure of the racemic polylactide has been studied by H- and 13C-NMR methods 15,29). The Bemoullian statistics of the dyad, triad and tetrad... 

It should be noted that, in some studies, deviations of 5-10% in e,xpected and measured NMR peak intensities have been ascribed to experimental error. Such error is sufficient to hide significant departures from Bemoullian statistics, ... 

The stereoiegularity pdy(meth)d methacrylate) obtained in THF by fluor-enjd- or cun caedum was fmind to be described not by Bemoullian statistics, but by first order Maikov statistics. ... 

Equations 2.48 through 2.50 allow one to generate theoretical mass spectra, which show identical peak intensity patterns for any random copolymer of a given composition, a remarkable result. Figure 2.20 shows the theoretical mass spectrum for an equimolar AB copolymer, and Figure 2.21 reports those for three and four component copolymers that follow Bemoullian statistics. 

Abate et al. analyzed the MALDI-TOF mass spectrum of a copolyester of microbial origin. They applied the Bemoullian statistics and performed a comparison between the experimental and the calculated MS intensities, as discussed in Chapter 10. 

The distribution of B blocks, which are included mostly in nonadsorbed chain sections, decays exponentially and thus should obey Bemoullian statistics that correspond to a zeroth-order Markov process. The average length of such blocks is dose to 2, which is the same as that of a RCP. In the case of A blocks, the distribution function /a(7) also decays exponentially in the initial region, which conesponds to short blocks induded in the random chain sections. For longer A blocks, however, the distribution becomes significantly broader and has a local maximum at 10. Hence, one... 

Another approach to creating materials with properhes different from any pure homopolymer is to use copolymers. Since the different mers are covalently attached to the macromolecule, they cannot undergo macroscopic phase segregation. The local structure of a bulk copolymer depends on the sequence distribution of the constituent subunits. Consider a copolymer composed of two t5q es of subunits, a so-called AB copolymer. The total munber of mers, N, is the sum of type A subunits, N, and type B subunits, N = N + Ng. The intramolecular composition is specified by the fractions x = N,j/N and Xg = Ng/N. A random copolymer is characterized by a sequence distribution that follows Bemoullian statistics the probabilities depend only on and Xg. 

Let us assume that the probability of genemting a meso sequence when a new monomer unit is added at the end of a growing chain can be denoted by a single parameter. Pm (and similarly Pr for a racemic sequence). Implied in it is an assumption that the polymer obeys Bemoullian statistics, that is, the probability of forming an /n or r sequence is independent of the stereochemical con guration of the chain already formed. Since addition can be only mot r type, it is obvious thatPm + Pr =1. 

Table 6.3 Comparison of Observed Realtive Areas of Methylene Tetrad Peaks with Those Calculated by Bemoullian Statistics ...

Matrix-assisted laser desorption ionization MS is well represented in this area [56, 85-94]. Cox et al. [88] used MALDI-MS to characterize low molecular weight polyolefin copolymers of isobutylene and paramethylstyrene and to extract composition information from spectra. Comparison of experimental oligomer distributions to a Bemoullian statistical model revealed severe overrepresentation of oligomers with higher relative amounts of paramethylstyrene. However, good agreement between the model and experimental data was obtained by the introduction of an ionization efficiency term in the model used. The differences between average composition determined by MALDI and NMR were discussed. 

If the reaction center adopts a preferred configuration with respect to the configuration of the penultimate unit in the chain (Scheme 25 fem fer)> then Bemoullian statistics apply. The stereochemistry of the chain is characterized by the single parameter, P(m) or P(r) (=1-P(m)]. The n-ad concentrations can be calculated simply by multiplying the concentrations of the component diads. Thus the relative triad concentrations are given by the following expressions (eqns [14]-[16]) ... 

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