Distributions of molecular weight

While the problem of molecular weight distribution is serious with synthetic polymers, it is not so with proteins and nucleic acids. However, biological polymers in aqueous solutions under certain conditions often form dimers and trimers thus, the solution may not be homogeneous either for example, most bovine semm albumin (BSA) samples contain 10% of dimers. Knowledge about the distribution of molecular weight may apply equally to biological polymers. Since molecular weight is directly related to the size of the chain, the approach to the distribution problem is statistical in nature. 

Physical Chemistry of Macromolecules Basic Principles and Issues, Second Edition. By S. F. Sun ISBN 0-471-28138-7 Copyright 2004 John Wiley Sons, Inc. 

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Before we can explore how reactor conditions can be chosen, we require some measure of reactor performance. For polymerization reactors, the most important measure of performance is the distribution of molecular weights in the polymer product. The distribution of molecular weights dictates the mechanical properties of the polymer. For other types of reactors, three important parameters are used to describe their performance ... 

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. 

The proof that these expressions are equivalent to Eq. (1.35) under suitable conditions is found in statistics textbooks. We shall have occasion to use the Poisson approximation to the binomial in discussing crystallization of polymers in Chap. 4, and the distribution of molecular weights of certain polymers in Chap. 6. The normal distribution is the familiar bell-shaped distribution that is known in academic circles as the curve. We shall use it in discussing diffusion in Chap. 9. 

In the next section we shall examine the distribution of molecular weights for polymerization which follows the chain-growth mechanism. 

Throughout this section we have used mostly p and u to describe the distribution of molecular weights. It should be remembered that these quantities are defined in terms of various concentrations and therefore change as the reactions proceed. Accordingly, the results presented here are most simply applied at the start of the polymerization reaction when the initial concentrations of monomer and initiator can be used to evaluate p or u. The termination constants are known to decrease with the extent of conversion of monomer to polymer, and this effect also complicates the picture at high conversions. Note, also, that chain transfer has been excluded from consideration in this section, as elsewhere in the chapter. We shall consider chain transfer reactions in the next section. 

That the Poisson distribution results in a narrower distribution of molecular weights than is obtained with termination is shown by Fig. 6.11. Here N /N is plotted as a function of n for F= 50, for living polymers as given by Eq. (6.109). and for conventional free-radical polymerization as given by Eq. (6.77). This same point is made by considering the ratio M /M for the case of living polymers. This ratio may be shown to equal... 

In the research described in the last problem, the authorst determined the following distribution of molecular weights by a chromatographic procedure ... 

For preparative purposes batch fractionation is often employed. Although fractional crystallization may be included in a list of batch fractionation methods, we shall consider only those methods based on the phase separation of polymer solutions fractional precipitation and coacervate extraction. The general principles for these methods were presented in the last section. In this section we shall develop these ideas more fully with the objective of obtaining a more narrow distribution of molecular weights from a polydisperse system. Note that the final product of fractionation still contains a distribution of chain lengths however, the ratio M /M is smaller than for the unfractionated sample. 

Figure 8.5 illustrates the sort of separation this approach predicts. Curve A in Fig. 8.5 shows the weight fraction of various n-mers plotted as a function of n. Comparison with Fig. 6.7 shows that the distribution is typical of those obtained in random polymerization. Curve B shows the distribution of molecular weights in the more dilute phase-the coacervate extract-calculated for the volumes of the two phases in the proportion 100 1. The distribution in the concentrated phase is shown as curve C it is given by the difference between curves A and B. 

AH three processes give perfluoropolyethers with a broad distribution of molecular weights. They are typically separated into fractions by vacuum distillation. 

A mass of polymer will contain a large number of individual molecules which will vary in their molecular size. This will occur in the case, for example, of free-radically polymerised polymers because of the somewhat random occurrence of ehain termination reactions and in the case of condensation polymers because of the random nature of the chain growth. There will thus be a distribution of molecular weights the system is said to be poly disperse. 

Polymers and copolymers of acrylamide (obtained by copolymerization or postreaction of polyacrylamide) with different values of the molecular weight, composition, distribution of molecular weight and compositions, linear and cross-linked have different functions and are used in many fields. The main functions and applications of acrylamide polymers are shown in Table 4. 

We have prepared a copolymer-bearing amino side group and used it either alone or in combination with BP to initiate the photopolymerization of MM A [89]. The gel permeation chromatography (GPC) plot of PMMA initiated by the former system showed a bimodal distribution of molecular weight because both the radicals produced initiate polymerization as follows ... 

The FTS mechanism could be considered a simple polymerization reaction, the monomer being a Ci species derived from carbon monoxide. This polymerization follows an Anderson-Schulz-Flory distribution of molecular weights. This distribution gives a linear plot of the logarithm of yield of product (in moles) versus carbon number. Under the assumptions of this model, the entire product distribution is determined by one parameter, a, the probability of the addition of a carbon atom to a chain (Figure 4-7). ... 

Fujita [38] showed that for a log-normal distribution of molecular weights (the usual case for polysaccharides) Mz/Mw = Mw/M . 

Figure 4. Typical computer-generated plot of weight differential and cumulative distributions of molecular weight...

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