Average Molar Masses

In addition to the determination of molar mass distributions and various molar mass averages there are many experiments, requiring sometimes sophisticated data evaluation, that can be carried out with an analytical ultracentrifuge. Examples are the analysis of association, the analysis of heterogeneity, the observation of chemical reactions, and protein characterization, to mention only a few. A detailed discussion is beyond the scope of this article, but there is excellent literature available [77-79,81,87-89]... 

In Ref. [30] the consistency of the approach is demonstrated by recalculating the respective molar mass averages Mw and Mn of the nine polymer fractions from the SEC elugrams after universal calibration, which agree very well with the... 

Table 4 Molar mass averages (in kg/mol) calculated from the molar mass distributions obtained by SEC, short and long exposure TDFRS, and DLS...

Most authors found that for narrow-MMD polymer (Mw/Mn < 1.10) the agreement is within 10% to 15% or even better and therefore it is excellent. As a matter of fact, polymers with a narrow MMD can be obtained by anionic or cationic polymerization and their molar mass averages can be readily measured by traditional methods for MM determination. 

Note 1 An infinite number of molar-mass averages can in principle be defined, but only a few types of averages are directly accessible experimentally. The most important averages are defined by simple moments of the distribution functions and are obtained by methods applied to systems in thermodynamic equilibrium, such as osmometry, light scattering and sedimentation equilibrium. Hydrodynamic methods, as a rule, yield more complex molar-mass averages. 

Note 2 Any molar-mass average can be defined in terms of mass fractions or mole fractions. In this document only a few of the important molar-mass averages are given in terms of the mass fractions, Wi, of the species with molar mass M. These definitions are most closely related to the experimental determination of molar-mass averages. 

FIGURE 16.1 Schematic differential (—) and integral (—) representations of amount of polymer with certain molar mass present in the sample. The narrow, broad, and bimodal molar mass distributions are shown. The typical positions of weight, viscosity, number, and z-molar mass averages are depicted. 

As explained in Sections 16.3.4, 6.4.1, and 16.4.2, SEC is a nonabsolute method, which needs calibration. The most popular calibration materials are narrow molar mass distribution polystyrenes (PS). Their molar mass averages are determined by the classical absolute methods—or by SEC applying either the absolute detection or the previously calibrated equipment. The latter approach may bring about the transfer and even the augmentation of errors. Therefore, it is recommended to apply exclusively the certified well-characterized materials for calibrations. These are often called PS calibration standards and are readily available from numerous companies in the molar mass range from about 600 to over 30,000,000g moL. Their prices are reasonable and on average (much) lower than the cost of other narrow MMD polymers. Other available homopolymer calibration materials include various poly(acrylate)s and poly(methacrylate)s. They are, similar to PS, synthesized by anionic polymerization. Some calibration materials are prepared by the methods of preparative fractionation, for example, poly(isobutylene)s and poly(vinylchloride)s. 

The time it takes the molecules of a gas to effuse through an opening or diffuse through another gas is directly proportional to the square root of their molar mass. The average speed of molecules in a gas is directly proportional to the square root of the temperature and inversely proportional to the square root of the molar mass average speed oc (T/M)m. 

In this contribution, the experimental concept and a phenomenological description of signal generation in TDFRS will first be developed. Then, some experiments on simple liquids will be discussed. After the extension of the model to polydisperse solutes, TDFRS will be applied to polymer analysis, where the quantities of interest are diffusion coefficients, molar mass distributions and molar mass averages. In the last chapter of this article, it will be shown how pseudostochastic noise-like excitation patterns can be employed in TDFRS for the direct measurement of the linear response function and for the selective excitation of certain frequency ranges of interest by means of tailored pseudostochastic binary sequences. 

This section is concerned with the information that can be obtained by TDFRS about the molar mass and size distribution and the various molar mass averages. 

Molar Mass Distribution and Molar Mass Averages... 

Frequently, not the fine structure of a distribution is of interest but certain molar mass averages, corresponding to the lower moments of the distribution, especially Mw and Mn. They can be computed directly from P(T) and Eq. (39) ac-... 

Table 1 summarizes the molar mass averages Mn, MH and Mz for the different experimental techniques. Good agreement is found for MH less for Mn and Mz with particularly strong deviations for the PCS results. 

Fig. 2.3 shows a MMD curve. In this figure also the characteristic molar mass averages are indicated their definition is given in Table 2.5. 

Whilst a knowledge of the complete molar mass distribution is essential in many uses of polymers, it is convenient to characterise the distribution in terms of molar mass averages. These usually are defined by considering the discontinuous nature of the distribution in which the macromolecuies exist in discrete fractions i containing N. molecules of molar mass M. 

Higher molar mass averages sometimes are quoted. For example, certain methods of molar mass measurement (e.g., sedimentation equilibrium) yield the z-average molar mass (Afx) which is defined as follows... 

Further, accurate measurement of the small physical effects observed at small concentration of polymer calls for sensitive experimental methods. An addition complication arises from the different types of molar mass averages obtained by different methods. 

The sample amount can be easily determined from the injection volume and the sample concentration, and no information from a concentration detector is required. With this approach, the Mn value of any polymer sample can be determined by SEC using only a viscosity detector. Other molar mass averages, however, cannot be determined. The advantage of the Goldwasser Mn method is that it can access much wider molar mass ranges than other existing methods like osmometry or end-group methods. 

Gel permeation chromatography (GPC) is the established method for the determination of molar mass averages and the molar mass distributions of polymers. GPC retention is based on the separation of macromolecules in solution by molecular sizes and, therefore, requires a molar mass calibration to transform elution time or elution volume into molar mass information. This kind of calibration is typically performed with narrow molecular mass distribution polymer standards, universal, or broad calibration methods or molar-mass-sensitive detectors like light-scattering or viscosity detectors. 

The calculation of copolymer molar mass averages and so on, and copolymer polydispersity D is done as in conventional GPC calculations using the copolymer molar mass calculated from Eq. (3). 

Higher-order molar mass averages, such as Af, and Af-+1 emphasize the high molar mass tail of the molar mass distribution. Molecular theories of polymer dynamics predict these higher-order averages are important, but... 

In this section, two examples of binary mixtures of two different mono-disperse chain lengths are used to better understand the various molar mass averages. 

Different molar mass averages are defined as ratios of consecutive... 

Figure 1.6. Illustration of the discontinuous distribution of polymer chain lengths (a) and the apparently continuous distrihution and molar-mass averages (b) from a polymerization ...

D19.2 First, try to understand why different molar mass averages might give different numerical values. Polymers are unlike small molecules in that all small molecules of the same species have nearly identical masses. Polymers vary widely in mass because they can vary in the number of monomeric units they contain. Depending on how a polymer is synthesized and purified, it is entirely possible for one macromolecule to contain 2 monomer units and another 100. We call a polymer sample polydisperse if there is a large variation in mass among the molecules of the sample conversely, a sample is monodisperse if its range of masses is narrow. 

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