Statistics Bernoullian

If the reaction center adopts a preferred configuration with respect to the configuration of the penultimate unit in the chain (Scheme 4.1 km kr) then Bernoullian statistics apply. The stereochemistry of the chain is characterized by the single parameter, P m) or P r) [= 1 The -ad concentrations can be... 

It should be noted that, in some studies, deviations of 5-10% in expected and measured NMR peak intensities have been ascribed to experimental error. Such error is sufficient to hide significant departures from Bernoullian statistics. 8 29... 

The mechanism of B polymerization is summarized in Scheme 4,9. 1,2-, and cis- and trews-1,4-butadiene units may be discriminated by IR, Raman, or H or nC MMR speclroseopy.1 92 94 PB comprises predominantly 1,4-rra//.v-units. A typical composition formed by radical polymerization is 57.3 23.7 19.0 for trans-1,4- c7a -1,4- 1,2-. While the ratio of 1,2- to 1,4-units shows only a small temperature dependence, the effect on the cis-trans ratio appears substantial. Sato et al9J have determined dyad sequences by solution, 3C NMR and found that the distribution of isomeric structures and tacticity is adequately described by Bernoullian statistics. Kawahara et al.94 determined the microslructure (ratio // measurements directly on PB latexes and obtained similar data to that obtained by solution I3C NMR. They94 also characterized crosslinked PB. 

The relative reactivities (in free-radical copolymerizations) of TBTM and MMA are 0.79 and 1.00 respectively (15). With equal concentrations of monomer, an excess of MMA in the polymer would be expected. In the following discussion A will represent the MAA or TBTM unit and B will represent the MMA unit. For A-centered triads four different arrangements are possible AAA, AAB, BAA, and BAB. Analogous sequences apply to the B-centered triads. For a random compositon, Bernoullian statistics should apply (14). With P (the proportion of TBTM) equal to 0.5, the probabilities of each of the A-centered triads is P 2 or 0.25. The AAB and BAA triads are indistinguishable and appear as a single resonance. 

Table I Tacticities of samples obtained in DME and THF, using mono- and bifunctional initiators and Na+ and Cs+ as the counterions. I,H,S isotactic (ami), heterotactic (mr/rm) and syn-diotactic (rr) triads m,r meso and racemic dyads p 2mr/H persistence ratio ( p 1 for Bernoullian statistics).

The influence of the chain expansion produced by excluded volume on the mean-square optical anisotropy is studied in six types of polymers (PE, PVC, PVB, PS, polylp-chlorostyrene), polylp-bromostyrenel. RIS models are used for the configuration statistics of the unperturbed chains. The mean-square optical anisotropy of PE is found to be insensitive to excluded volume. The mean-square optical anisotropy of the five other polymers, on the other hand, is sensitive to the imposition of the excluded volume if the stereochemical composition is exclusively racemic. Much smaller effects are seen in meso chains and in chains with Bernoullian statistics and an equal probability for meso and racemic diads. 

The tacticity of anionically prepared polystyrenes has been the subject of extensive study by a number of groups of workers, mostly by means of 13C-NMR spectroscopy. From a study of the aromatic Cl resonances, Matsuzaki and coworkers found 234) that there is a tendency towards syndiotacticity when using -butyl-Iithium in toluene as initiator. From the sensitivity of the CMR spectrum to the nature of the solvent employed it was concluded that the polymerization did not conform to Bernoullian statistics. Randall examined the methylene resonances in the CMR spectrum and concluded that butyllithium initiated polystyrene is essentially atactic 235) and that propagation is Bernouillian. Uryu et al.236) examined polystyrene... 

The distribution of B blocks, which are included mostly in nonadsorbed chain sections, decays exponentially and thus should obey Bernoullian statistics that correspond to a zeroth-order Markov process [10]. The average length of such blocks is close to 2, that is, the same as that of a random copolymer. In the case of A blocks, the distribution function/a (f) also decays exponentially in the initial region, which corresponds to short blocks included in the random chain sections. For longer A blocks, however, the distribution becomes significantly broader and has a local maximum at i 10 [95]. Hence, one can conclude that the distribution of A blocks strongly deviates from that known for random sequences. 

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. 

There is another way to confirm that the triad distribution cannot result from a random polymerization. The observed relative proportion of the triad structure in Figure Id is (l) (.93) (.26). The corresponding distribution for the triad structure for, 266 mole fraction VCI2 would be (l) (,36) (.13) Obviously, this comparison confirms from just a consideration of the triad distribution that the polymer does not conform to Bernoullian statistics. 

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. 

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 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... 

When these results are compared with those from the polymerization of the other monosubstituted oxiranes (9), it should be emphasized that the partly crystalline fraction of poly(styrene oxide), I, obtained in the presence of the ZnEt2/H20 catalyst is so far the only polyoxirane reported which does not follow Bernoullian statistics. Only one other case, namely that of the polymerization of phenylthiirane by a coordination catalyst system, is known until now which follows first-order Markov statistics (10). 

Let us assume that the probability of generating a meso sequence when a new monomer unit is added at the end of a growing chain can be denoted by a single parameter, which we shall call Pjn (and similarly Pr for a racemic sequence). This implies an assumption that the polymer obeys Bernoullian statistics, that is, the probability of forming an m or r sequence is independent of the stereochemical configuration of the chain already formed. It follows of course that Pm + Pr =1. because addition can be only m or r type. 

It has been reported that the tacticity of PMMA prepared with benzoyl peroxide (BPO) deviates slightly from Bernoullian statistics while that of PMMA... 

The oligomers of methyl vinyl ether were synthesized by the successive addition of methyl vinyl ether to the corresponding dimethyl acetal in the presence of BF3OEt2 in toluene the steric course of each addition step was studied by H NMR spectroscopic analysis of the oligomers from dimer to pentamer.248 The meso-diad probability of each addition step was found to be almost constant, indicating that the stereoregulation obeys Bernoullian statistics. 

Table IV shows the reactivity ratios rG and r, derived from the probabilities in Table III in accord with a first-order Markov model (2), where it is assumed that the more likely propagating terminal radical structure is 1 (—CHF-) and not 0 (—CH2). This assumption is consistent with gas phase reactions of VF with mono-, di-, and trifluoromethyl radicals, which add more frequently to the CH2 carbon than to the CHF carbon (20). The reactivity ratio product is unity if Bernoullian statistics apply, and we see this is not the case for either PVF sample, although the urea PVF is more nearly Bernoullian in its regiosequence distribution. Polymerization of VF in urea at low temperature also reduces the frequency of head-to-head and tail-to-tail addition, which can be derived from the reactivity ratios according to %defect — 100(1 + ro)/(2 + r0 + r,). Our analysis of the fluorine-19 NMR spectrum shows that commercial PVF has 10.7% of these defects, which compares very well with the value of 10.6% obtained from carbon-13 NMR (13). Therefore the values of 26 to 32% reported by Wilson and Santee (21) are in error.

As expected, fig increases and fi decreases with increasing extent of reaction. For the samples prepared using aqueous NaOH, the observed sequence lengths are in reasonable agreement with those expected for a random copolymer displaying Bernoullian statistics (Table III) (2). The random copolymer sequence lengths were calculated using Equation 6, where Pg is the mole fraction of dichlorocyclopropane units. 

For the other polymers the fig values are higher than calculated using Bernoullian statistics, indicating the presence of larger fractions of blocked dichlorocyclopropyl-containing units in polymers not prepared with aqueous NaOH than in comparable polymers prepared with aqueous NaOH. Comparison of samples 3 and... 

The intensity of each signal of the spectrum measured at 150°C was determined for polystyrenes prepared with benzoyl peroxide, n-butyllithium, and trifluoroboron etherate catalysts. The observed relative intensities of the signals were in good agreement with the calculated values assuming Bernoullian statistics with Pr of 0.54, 0.56, and 0.45 for the radical, anionic, and cationic polystyrenes, respectively (Table II). The Py> value of the radical polystyrene was nearly the same as that reported by Shepherd et al. (10). 

Assuming Bernoullian statistics, V=(l-a)2/a2, where a is the probability of a meso placement in the completely equilibrated polymer (or in polystyrene). V values of 0.62-0.65 thus imply that polystyrene can be characterized by a a value of 0.44. 

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. 

It seems likely that other polymerizations will be found to depart from Bernoullian statistics as the precision of tacticity measurements improves. One study " 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."" " ... 

As no effect of the penultimate inserted unit is taken into account, the formation of r or m diads is a random process that follows the Bernoullian statistic. Therefore, the symmetric first-order Markovian model becomes Bernoullian when diad formation is considered. This fact explains why this model is usually called Bernoullian. 

Figure 7. i4 125.8-MHz NMR spectrum of the methane carbon region in polyvinyl chloride, 10% in 1,4-dioxane-da at 370 K. Configurational assignments are based on the relative triad intensities assuming Bernoullian statistics ( Sj. 

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