Copolymer Bernoullian

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. 

Simulation of the (n-Bu)3SnH reduction of PVC is carried out in a manner similar to that described for TCH. Instead of beginning with 100 TCH molecules we take a 1000 repeat unit PVC chain that has been Monte Carlo generated to reproduce the stereosequence composition of the experimental sample of PVC used in the reduction to E-V copolymers (2), ie. a Bernoullian PVC with P =0.45. At this point we have generated a PVC chain with a chain length and a stereochemical structure that matches our experimental starting sample of PVC. 

As to quantities 0 (Eq. 39), they are equal to Xa8Vfl, where 8Vfl is the Kro-necker symbol. In view of this, only the first item in the sum (Eq. 38), XaXp, is distinct from zero. This corresponds to the absence of chemical correlations along the macromolecule on all scales. In other words, the distribution of units for such copolymers is Bernoullian [2]. 

The ROMP of [2.2]paracyclophane-l,9-diene (128) yields poly(p-phenylenevinylene) (129) as an insoluble yellow fluorescent powder. Soluble copolymers can be made by the ROMP of 128 in the presence of an excess of cyclopentene387, cycloocta-1,5-diene388 or cyclooctene389. The UV/vis absorption spectra of the copolymers with cyclooctene show separate peaks for sequences of one, two and three p-phenylene-vinylene units at 290, 345 and about 390 nm respectively, with a Bernoullian distribution. The formation of the odd members of this series must involve dissection of the two halves of the original monomer units by secondary metathesis reactions. 

Thus, copolymers of the same composition can have qualitatively different sequence distributions depending on the solvent in which the chemical transformation is performed. In a solvent selectively poor for modifying agent, hydrophobically-modified copolymers were found to have the sequence distribution with LRCs, whereas in a nonselective (good) solvent, the reaction always leads to the formation of random (Bernoullian) copolymers [52]. In the former case, the chemical microstructure cannot be described by any Markov process, contrary to the majority of conventional synthetic copolymers [ 10]. 

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 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],... 

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... 

The triad distribution in the copolymer was found to be bernoullian and is accounted for by a one-parameter model therefore, chain end control of the stereochemistry was assumed [80]. For the systems based on bipyridine, no influence of the counter-ion on the stereochemistry of the produced copolymers was found however, the catalytic activity was highest with the weakest coordinating anion [81]. Bis-chelated complexes of 2,2 -bipyridine or 1,10-phenanthroline [ Pd(N-N)2 X2 ] were efficient catalyst precursors, particularly when used in 2,2,2-tri-fluoroethanol as the solvent [82] under these conditions stabilization of the catalytic system by an oxidant is unnecessary, and very high molecular weights were obtained [83]. 

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. 

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. 

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]. 

The Bernoullian process is therefore defined for a copolymer by two transition probabilities, and Pg, which reflect the mole fractions of monomers A and B within the resulting copolymer. Given the two addition probabilities, the mole fraction of any given sequence can be calculated straightforwardly. Thus, for example, the abundance of an A A dyad is given by P, whilst that of an AB dyad is equal to 2P P (the factor of two arises because the AB dyad represents both AB and BA sequences). Table 2.2 shows the set of Bernoullian expressions for the three dyad and six triad sequences in an A/B copolymer. 

It is important to note here that random copolymers which show conformity to Bernoullian statistics can result from non-Bernoullian processes under some sets of circumstances. In order to distinguish between random copolymers resulting from Bernoullian and non-Bernoullian processes, it is necessary to have access to additional information, e.g. comonomer feed information, copolymer composition versus conversion data, and so on. Examples of random copolymers resulting from non-Bernoullian processes are given in section 2.3. 

The four comonomer addition probabilities prove to be useful characteristics since they define the tendency of a copolymer towards either block formation or alternation. For example, in the case of bb ba aa monomers tend to add to the growing chain in an alternating fashion. If the above inequalities are reversed, like sequences of additions are preferred such that the comonomers tend to form blocks. Finally, there is a third case if P g = Pgg and Pg = P, the system reduces to Bernoullian statistics. 

Note, in addition, that the special case = rg = 1 corresponds to a random copolymer formed by a Bernoullian process. 

As stated earlier, copolymers whose sequence distributions can be described by Bernoullian statistics (i.e. so-called random or statistical copolymers) can result either from Bernoullian or non-Bernoullian processes. Those formed by genuine Bernoullian mechanisms are relatively rare, reflecting the likeli-... 

Several examples of NMR studies of copolymers that exhibit Bernoullian sequence distributions but arise from non-Bernoullian mechanisms have been reported. Komoroski and Schockcor [11], for example, have characterised a range of commercial vinyl chloride (VC)/vinylidene chloride (VDC) copolymers using carbon-13 NMR spectroscopy. Although these polymers were prepared to high conversion, the monomer feed was continuously adjusted to maintain a constant comonomer composition. Full triad sequence distributions were determined for each sample. These were then compared with distributions calculated using Bernoullian and first-order Markov statistics the better match was observed with the former. Independent studies on the variation of copolymer composition with feed composition have indicated that the VDC/VC system exhibits terminal model behaviour, with reactivity ratios = 3.2 and = 0.3 [12]. As the product of these reactivity ratios is close to unity, sequence distributions that are approximately Bernoullian are expected. 

Figure 2.2 Proportions of acrylamide-centred triad sequences for a series of copolymers of sodium acrylate (A) and acrylamide (M), prepared in inverse microemulsions, (a) High conversion polymers (b) low conversion polymers. The curves represent triad proportions calculated using Bernoullian statistics symbols represent experimental triad data (a) AMA (-I-) AMM ( ) MMM is the mole fraction of M in the copolymer. Reprinted with permission from [10]. (1986) American Chemical Society.

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