The Number Average Molecular Weight

The number average molecular weight usually is defined by one or the other of the alternative expressions 

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This result shows that the square root of the amount by which the ratio M /M exceeds unity equals the standard deviation of the distribution relative to the number average molecular weight. Thus if a distribution is characterized by M = 10,000 and a = 3000, then M /M = 1.09. Alternatively, if M / n then the standard deviation is 71% of the value of M. This shows that reporting the mean and standard deviation of a distribution or the values of and Mw/Mn gives equivalent information about the distribution. We shall see in a moment that the second alternative is more easily accomplished for samples of polymers. First, however, consider the following example in which we apply some of the equations of this section to some numerical data. 

The phenomena we discuss, phase separation and osmotic pressure, are developed with particular attention to their applications in polymer characterization. Phase separation can be used to fractionate poly disperse polymer specimens into samples in which the molecular weight distribution is more narrow. Osmostic pressure experiments can be used to provide absolute values for the number average molecular weight of a polymer. Alternative methods for both fractionation and molecular weight determination exist, but the methods discussed in this chapter occupy a place of prominence among the alternatives, both historically and in contemporary practice. 

In the next section we shall describe the use of Eq. (8.83) to determine the number average molecular weight of a polymer, and in subsequent sections we shall examine models which offer interpretations of the second virial coefficient. 

The number-average molecular weight of most commercially available acetal resins is between 20,000 and 90,000. Weight-average molecular weight may be estimated from solution viscosities. 

T is the glass-transition temperature at infinite molecular weight and is the number average molecular weight. The value of k for poly(methyl methacrylate) is about 2 x 10 the value for acrylate polymers is approximately the same (9). A detailed discussion on the effect of molecular weight on the properties of a polymer may be found in Reference 17. 

Molecular weights of polymers are determined by the weight—average molecular weight, and the number—average molecular weight, M. The... 

The width of molecular weight distribution (MWD) is usually represented by the ratio of the weight—average and the number—average molecular weights, MJM. In iadustry, MWD is often represented by the value of the melt flow ratio (MER), which is calculated as a ratio of two melt indexes measured at two melt pressures that differ by a factor of 10. Most commodity-grade LLDPE resias have a narrow MWD, with the MJM ratios of 2.5—4.5 and MER values in the 20—35 range. However, LLDPE resias produced with chromium oxide-based catalysts have a broad MWD, with M.Jof 10—35 and MER of 80-200. 

The number-average molecular weight is adjusted in the 12,000—15,000 range for apparel fibers, >20, 000 for high strength yams for tires and industrial end uses. 

For higher molecular weight polydimethyl siloxanes (Af > 2500), the number-average molecular weight is related to the bulk viscosity by the foUowiag formula, where the viscosity units are mm /s(=cSt). 

The number-average molecular weight of dimethylsiloxane can also be determined from the intrinsic viscosity [Tj, dL/g] (extrapolated to zero viscosity) ia toluene or methyl ethyl ketone according to the foUowiag equatioa (339,340) ... 

This can best be explained by taking a somewhat extreme theoretical example. Let us consider a tiny sample of polymer consisting of 1 molecule with a molecular weight of 100000 and 999 molecules with a molecular weight of 100. In this case the number average molecular weight will be... 

Fig. 30. Evolution of the glass transition temperature of polychloroprene-aromatic hydrocarbon resin blends (17 wt% resin content) as a function of the number average molecular weight of the hydrocarbon resin. Tg values obtained from DSC experiments.

The poly(styrene-b-isoprene) (P(S-b-IP)) and poly(-styrene-b-2-vinyl pyridine) (P(S-b-2VP)) block copolymers with narrow molecular weight distributions for blending with the microspheres were also synthesized using the additional anionic polymerization technique. The number-average molecular weights (Mns) and PS contents are also shown in Table 1. 

The number average molecular weight (M ) is related to the number of particles present in a sample. It is calculated by dividing the sum of the weights of all the species present (monomers, dimers, trimers, and so on) by the number of species present ... 

The number-average molecular weight of a polymer may be controlled by offsetting the stoichiometry of two dissimilar mutually reactive difunctional monomers. The polymer will have the same endgroup functionality as that of the monomer used in excess. For a generic polymer made from a difunctional monomer AA with A functional groups and an excess of difunctional monomer BB widt B functional groups, r is defined as... 

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