Bernoullian probability

Once creation of the PV A database is complete, optimized probabilities may be calculated for the experimental spectrum at hand. Since the iterative procedure is restricted to a 2048 data point region, zoom cursors are displayed and set by the user until this condition is satisfied. In this case, the methylene region was selected and an initial guess for the Bernoullian probability (Pr=0.5) and linewidth (13.0Hz) were given. Optimized values for the probability and linewidth were Pr=0.52 and 12.8Hz, respectively. 

A triad sequence involves two monomer additions the Bernoullian probabilities of mm, mr, and rr are therefore P, 2Pm l — Pm) and (1 — Pm), respectively. A plot of these relations is shown in Fig. 2.15. It will be noted that the proportion of mr, that is, heterotactic units, rises... 

Table 10. Pentad Analysis, Bernoullian Probability of Enantioselective Insertion (b), and Probability of ...

Bovey and Tiers [20]. These parameters represent the (Bernoullian) probability of generating a meso dyad between a growing chain ending in x and incoming monomer y. Thus, for example, the probability of finding an isotactic MMM triad is simply given by first-order Markov probability of finding any MMM triad multiplied by... 

Designation Projection Bernoullian probability Designation Projection Bernoullian probability... 

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. 

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. 

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 distribution of the cis and turns double bonds in a given polymer chain may be expressed in terms of the ratios rt = (tt)/(tc) and rc = (cc)/(ct). If the probability of formation of a cis double bond is independent of the configuration of the previous double bond, the distribution will be random (Bernoullian) and characterized by a single parameter rt = 1 /rc. This is the case for polymers of norbornene with less than 35% cis content, but for polymers with more than 50% cis content the distribution is generally somewhat... 

Quantitative spectra can give an estimate of the probability parameters according to the copolymerization theory. Poly-(2-methylpentadiene-co-4-methylpentadiene) possesses a Bernoullian distribution of monomer units, moreover polymer composition is the same as that of the monomer mixture used for the formation of the inclusion compound. It can therefore be considered as an ideal azeotropic copolymer. In other instances, we observed that p + p is often lower than 1, (or that the product r. r is greater... 

A particularly interesting result in Table V is seen for the solutions containing about 66% methylene chloride. At this point a sharp break occurred in several properties (1) the syndiotactic content decreased and the isotactic content increased, (2) the yield dropped drastically, (3) the molecular weight decreased, and (4) the molecular weight distribution broadened. These changes all point to a change in the reaction mechanism, probably from one of solvated ion pairs to one of predominantly contact ion pairs at that solvent concentration. Nevertheless, the tacticities of all the polymers formed in this series were found to fit on a Bovey plot, which indicates that they were formed by a Bernoullian process and no penultimate effect was present. 

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. 

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. 

The probabilities of the regiosequence pentads for commercial PVF and urea PVF are shown in Table III. For the former sample it is apparent simply by inspection that the regiosequence distribution is not Bernoullian, since Pobs(C5) and Pobs (D5) are different (2). The distributions conform to first-order Markov statistics, characterized by two reactivity ratios r0 and r 5 where r0 = k /lq, and rj — ku/k10 (kjj is the rate constant for monomer addition to terminal radical i which generates the new terminal radical j). The present pentad data is insufficient to check the validity of this model, but it is unlikely that there is any deviation, as the same model has been tested and found adequate to describe the regiosequence distribution in PVF2 (2). 

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.

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. 

Bovey adopts a two-state model where the preceding unit is the monomer unit immediately preceding the active chain end. The Bovey-Bernoullian model is now described at dyad levels by 4 reactions defined by the 4 probabilities of occurrence of the 4 dyads ... 

In this case the probability of insertion of a monomer with a given enantioface at site 1 is equal to the probability of insertion of a monomer with the opposite enantioface at site 2. This probability is indicated with the parameter a. The resulting expressions for pentad distribution for the syndiotactic Bernoullian model are reported in Table 15. 

If Pr is the probability of a racemic placement for a growing vinyl polymer chain, deduce an expression for the ratios of the intensities of the C NMR peaks for the rr, mr and mm a-carbon triads and evaluate the ratios for Pj = 0.6. Assume that Bernoullian statistics apply, i.e. that the probability of a racemic placement is independent of the nature of the previous placements. (This is not necessarily true for a real system.)... 

Table 1.8 Tacticity sequence probabilities expressed in terms of propagation probabilities for Bernoullian and first-order Markov statistics the parameters u and v are defined in the text...

Other propagation models are also possible. An example is an adaptation [48,55,65] of the Coleman-Fox [66] mechanism in which the polymerisation is considered to take place at two active sites, each with a different Bernoullian propagation probability (see chapter 2). 

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