Bemoullian distribution

Random copolymers. A random copolymer is a special case of a statistical copolymer. It is a statistical copolymer in which the probability of finding a given monomeric unit at any given site in the chain is independent of the nature of the neighbouring units at that position (Bemoullian distribution). In other words, for such a copolymer, the probability of finding a sequence. ..ABC... of monomeric units A, B, C..., i.e. / (...ABC...], is given by... 

The ROMP of [2.2]paracyclophane-1,9-diene (73) yields poly(p-phenylene-vinylene) (74) as an insoluble yellow fluorescent powder. Soluble copolymers can be made with an excess of cyclopentene (Thom-Csanyi 1992b), cycloocta-1,5-diene (Thom-Csanyi 1993a), or cyclooctene (Thom-Csanyi 1994c). The UV-visible absorption spectra of the copolymers with cyclooctene show separate peaks for sequences of one, two, and three /7-phenylene-vinylene units at 290, 345, and about 390 nm, respectively, with a Bemoullian distribution. The formation of the odd members of this series evidently involves dissection of the two halves of the original monomer units by secondary metathesis reactions. 

Random copolymers in which the comonomer distribution follows Bemoullian or zero-order statistics are formed by ideal copolymerizations in which rjr2=l. However, a truly random distribution of the two units results only if the Bemoullian distribution is symmetric (i.e. when ri = r2=l). In this case, the two monomers have equal probability of reacting with a given active center, regardless of the monomer it is derived from, and the copolymer composition equals the comonomer feed composition at all conversions. Random copolymers are generally formed by radieal eopolymeriza-tions, whereas ionic copolymerizations tend to favor propagation of one of the comono-... 

In a Bemoullian distribution the sequence of comonomer units is completely random, since all the comonomers enjoy the same probability of being added to the growing chain. In the Markoffian distribution the probability of... 

This copolymerization reaction is peculiar, since the synthetic route adopted to produce the copolymer is such that the abimdance of the triad ABB and of the pentads BBBAB, BBABA, ABABB, BBABB is zero, and therefore the Bemoullian distribution is not followed. ... 

Other infixes are -ran- (Bemoullian distribution), -alt- (strictly alternating sequence), -block- (linear arrangement of blocks), and -graft- (a grafted arrangement describing a polymeric offspring from the main chain). 

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... 

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. 

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... 

For the statistical copolymer the distribution may follow different statistical laws, for example, Bemoullian (zero-order Markov), first- or second-order Markov, depending on the specific reactants and the method of synthesis. This is discussed further in Secs. 6-2 and 6-5. Many statistical copolymers are produced via Bemoullian processes wherein the various groups are randomly distributed along the copolymer chain such copolymers are random copolymers. The terminology used in this book is that recommended by IUPAC [Ring et al., 1985]. However, most literature references use the term random copolymer independent of the type of statistical distribution (which seldom is known). 

The copolymer described by Eq. 6-1, referred to as a statistical copolymer, has a distribution of the two monomer units along the copolymer chain that follows some statistical law, for example, Bemoullian (zero-order Markov) or first- or second-order Markov. Copolymers formed via Bemoullian processes have the two monomer units distributed randomly and are referred to as random copolymers. The reader is cautioned that the distinction between the terms statistical and random, recommended by IUPAC [IUPAC, 1991, in press], has often not been followed in the literature. Most references use the term random copolymer independent of the type of statistical process involved in synthesizing the copolymer. There are three other types of copolymer structures—alternating, block, and graft. The alternating copolymer contains the two monomer units in equimolar amounts in a regular alternating distribution ... 

The polymer stereosequence distributions obtained by NMR analysis are often analyzed by statistical propagation models to gain insight into the propagation mechanism [Bovey, 1972, 1982 Doi, 1979a,b, 1982 Ewen, 1984 Farina, 1987 Inoue et al., 1984 Le Borgne et al., 1988 Randall, 1977 Resconi et al., 2000 Shelden et al., 1965, 1969]. Propagation models exist for both catalyst (initiator) site control (also referred to as enantiomorphic site control) and polymer chain end control. The Bemoullian and Markov models describe polymerizations where stereochemistry is determined by polymer chain end control. The catalyst site control model describes polymerizations where stereochemistry is determined by the initiator. 

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. 

Actually polymerizations in the second group above give the most highly syndiotactic polymers and the sequence distribution is not the simple Bemoullian one expected from such a mechanism. No attempt seems to have been made to formulate a transition state for this type of polymerization. It would obviously lead to difficulties if expressed in terms of penultimate effects as the solvents used are strongly solvating and secondary interaction with penultimate units would not be favoured. 

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. 

A value of unity (or nearly unity) for the monomer reactivity ratio signifies that the rate of reaction of the growing chain radicals towards each of the monomers is the same, i.e. kn ki2 and 22 — A 2i and the copolymerization is entirely random. In other words, both propagating species and M2 have little or no preference for adding either monomer. The copolymer composition is the same as the comonomer feed with a completely random placement of the two monomers along the copolymer chain. Such behavior is referred to as Bemoullian. Free-radical copolymerization of ethylene and vinyl acetate and that of isoprene and butadiene are examples of such a system, but this is not a common case. Random monomer distributions are obtained more generally in a situation where both types of radicals have exactly the same preference for the same type of monomer as represented by the relationship... 

The distribution of the cis and trans double bonds in a given polymer chain may be expressed in terms of the ratios r( = (tt)/(tc) and r. = (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 (Bemoullian) and characterized by a single parameter r,= l/r. Figure 11.7 shows that this is the case for polymers of norbomene with less than 35% cis content, but for polymers with more than 50% cis content the distribution is generally somewhat blocky, with r,rc reaching values of 8 or more in some cases. 

In the early 1940s when the polymerization theory was developed, tiie ideal, terminal, and penultimate models for fhe copolymerization were established also the possible distribution laws for the monomer sequence along the copolymer chains were defined Bemoullian, firsf- and second-order Markoffian. ... 

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 (Bemoullian) copolymers. In the former case, the chemical microstmcture cannot be described by any Markov process, contrary to the majority of conventional synthetic copolymers. ... 

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. 

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