Polymerization statistics

When the concentrations of A and B may be varied independently (Eq. 2.2), the stoichiometric ratio of functionalities is defined by r = A0/B0, where A0 and B0 are the initial concentrations of functional groups A and B. As will be shown in Chapter 3, this ratio is very important in designing and controlling a step-growth polymerization. Statistical parameters at any... 

G. Donegani in Novara, Italy, where he begun his activity on group 4 metallocene research. After the relocation of the group to the Centro Ricerche G. Natta of Montell in Ferrara, he shifted his research interests to the NMR analysis of polymers and polymerization statistics. 

This model can be applied to evaluate the pentad distribution from NMR spectra of metallocene-based isotactic polypropenes in which overlapping with peaks from end group or regioirregular units (2.1 and 3.1) occurs. In this case only the peaks of mmmm, mmmr, mmrr, and mrrm pentads can be obtained from direct spectrum integration. Furthermore. the mmmr peak overlaps with the mmmm base, and the mmrr has to be correct by subtracting the contribution from the 2,1-erythro unit. The total pentad distribution is calculated under the hypothesis that the polymerization statistic follows a pure enantiomorphic site control, using the expressions reported in Table 13 as follows ... 

Family [6] defined three states of polymeric statistical fiactals depending on the system statistics extended, compensated and collapsed ones. The two main factors, influencing fiactal branching degree, were pointed out. The first Irom them is cluster concentration in system—if there are many clusters in the system, and then they occupy the entire volume. Therefore, other clusters availability restricts a fractal branching degree it is more branched in isolation (very diluted solution), than in concentrated solution. 

One of the advantages of NMR in polymer solution studies is the wealth of information available. A NMR spectrum may contain information on polymer microstructure, polymerization mechanism, side reactions, conq>ositional heterogeneity, and (sometimes) molecular weight. Frequently, the challenge is to interpret the spectrum, to extract the relevant information, and to maximize the information content. One way whereby this can be accomplished is through theoretical modeling. This can be carried out, for example, for the polymer structure, polymerization statistics, and reaction kinetics. 

Since the composition of a copolymer chain is critical for determining its properties, understanding how the kinetics of the copolymerization impact the composition is an important consideration. To cut to the chase, the composition of a copolymer formed under a particular set of conditions depends on the relative rates of the four principal reactions. This in turn depends on the rate constants (e.g., those shown in Figure 12) and concentrations of reactive species and monomers. Using either a steady-state approximation or a polymerization statistics derivation, one can determine the following expression for the relative amount of monomer A in copolymer given the amount in the feed through equation [12] where Fa is the mole fraction of A in the copolymer and/a is the mole fraction of A in the feed ... 

As we can see, the y-gauche effect prediction of NMR chemical shifts in vinyl polymers permits assignment of their NMR spectra, provides an opportunity to test or derive an RIS model description of their conformational characteristics, and may also permit a test of their polymerization statistics. 

Our purpose in this introduction is not to trace the history of polymer chemistry beyond the sketchy version above, instead, the objective is to introduce the concept of polymer chains which is the cornerstone of all polymer chemistry. In the next few sections we shall introduce some of the categories of chains, some of the reactions that produce them, and some aspects of isomerism which multiply their possibilities. A common feature of all of the synthetic polymerization reactions is the random nature of the polymerization steps. Likewise, the twists and turns the molecule can undergo along the backbone of the chain produce shapes which are only describable as averages. As a consequence of these considerations, another important part of this chapter is an introduction to some of the statistical concepts which also play a central role in polymer chemistry. 

In Chaps. 5 and 6 we shall examine the distribution of molecular weights for condensation and addition polymerizations in some detail. For the present, our only concern is how such a distribution of molecular weights is described. The standard parameters used for this purpose are the mean and standard deviation of the distribution. Although these are well-known quantities, many students are familiar with them only as results provided by a calculator. Since statistical considerations play an important role in several aspects of polymer chemistry, it is appropriate to digress into a brief examination of the statistical way of describing a distribution. 

In this section we turn our attention to two other questions raised in Sec. 5.2, namely, how do the molecules distribute themselves among the different possible species and how does this distribution vary with the extent of reaction Since a range of species is present at each stage of the polymerization, it is apparent that a statistical answer is required for these questions. This time, our answer begins, On the average. . . . ... 

The statistical nature of polymers and polymerization reactions has been illustrated at many points throughout this volume. It continues to be important in the discussion of stereoregularity. Thus it is generally more accurate to describe a polymer as, say, predominately isotactic rather than perfectly isotactic. More quantitatively, we need to be able to describe a polymer in terms of the percentages of isotactic, syndiotactic, and atactic sequences. 

On the basis of these observations, criticize or defend the following proposition Regardless of the monomer used, zero-order Markov (Bernoulli) statistics apply to all free radical, anionic, and cationic polymerizations, but not to Ziegler-Natta catalyzed systems. 

This model then leads us through a thicket of statistical and algebraic detail to the satisfying conclusion that going from small solute molecules to polymeric solutes only requires the replacement of mole fractions with volume fractions within the logarithms. Note that the mole fraction weighting factors are unaffected. 

When we discussed random walk statistics in Chap. 1, we used n to represent the number of steps in the process and then identified this quantity as the number of repeat units in the polymer chain. We continue to reserve n as the symbol for the degree of polymerization, so the number of diffusion steps is represented by V in this section. 

Manufacturing processes have been improved by use of on-line computer control and statistical process control leading to more uniform final products. Production methods now include inverse (water-in-oil) suspension polymerization, inverse emulsion polymerization, and continuous aqueous solution polymerization on moving belts. Conventional azo, peroxy, redox, and gamma-ray initiators are used in batch and continuous processes. Recent patents describe processes for preparing transparent and stable microlatexes by inverse microemulsion polymerization. New methods have also been described for reducing residual acrylamide monomer in finished products. 

Warshawsky and coworkers have recently reported the synthesis of a class of compounds which they call polymeric pseudocrown ethers . A chloromethylated polystyrene matrix is used here as in 6.6.2, but instead of adding a crown to the backbone, a strand of ethyleneoxy units is allowed to react at two different positions on the chain, thus forming a crown. Such systems must necessarily be statistical, and the possibility exists for forming interchain bridges as well as intrachain species. Nevertheless, polymers which could be successfully characterized in a variety of ways were formed. A schematic representation of such structures is illustrated below as compound 30. ... 

To understand the global mechanical and statistical properties of polymeric systems as well as studying the conformational relaxation of melts and amorphous systems, it is important to go beyond the atomistic level. One of the central questions of the physics of polymer melts and networks throughout the last 20 years or so dealt with the role of chain topology for melt dynamics and the elastic modulus of polymer networks. The fact that the different polymer strands cannot cut through each other in the... 

POLYMERIC ALLOYS MODEL MATERIALS FOR THE UNDERSTANDING OF THE STATISTICAL THERMODYNAMICS OF MIXTURES... 

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 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 simple statistical treatment of radical polymerization can be traced back to Schultz27 Texts by Flory2S and Bamford et al.29 are useful references. 

Statistical copolymers are formed when mixtures of two or more monomers are polymerized by a radical process. Many reviews on the kinetics and mechanism of statistical copolymerization have appeared1 9 and some detail can be found in most text books on polymerization. The term random copolymer, often used to describe these materials, is generally not appropriate since the incorporation of monomer units is seldom a purely random process. The... 

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