Polymerization mass average

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. 

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 mathematical model of network formation in the pregel stage will focus on the prediction of the gel conversion and the evolution of number-and mass-average molar masses, Mn and Mw, respectively. For chainwise polymerizations, calculations will be restricted to the limit of a very low concentration of the polyfunctional monomer (A4 in the previous example). Thus, homogeneous systems will always be considered. 

NUMBER AND WEIGHT (MASS) AVERAGE DEGREES OF POLYMERIZATION... 

The mass average degree of polymerization, Pw, is defined as the first moment with respect to zero of the differential mass distribution... 

In PVA-coated capillaries it was possible to separate at pH 2.5 the standards of poly-2-vinylpyridinium hydrochloride (p(2-VPy)) in the molecular mass range between 1500 and 1,730,000 g mol-1 with dextran T70 as sieving matrix [20]. An example is shown in Fig. 4, where a 5% solution of dextran T70 has been used. The efficiency of the monomolecular basic marker 4-aminopyridine is excellent, demonstrating the exclusion of secondary adsorptive effects at the capillary surface. Hence, the broad peaks of the polymeric standards are due to their polydispersity. As in CE the width of the peaks depends on their migration velocity through the detection window, no direct comparison of broadness of the individual peaks and analyte polydispersity is possible. However, for each individual peak the methods applied in SEC for calculation of the different molecular mass averages can be applied. 

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 ...

Another problem with GPC of condensation resins is the calibration of the columns. Because in the ohgomeric and polymeric regions of the resins no compounds with a special and singular molar mass and a clear molecular structure are available, similar or chemically related substances have to be used as calibration standards. However, differences in the hydrodynamic volumes even at the same molar mass cannot be excluded totally. This uncertain cahbration of the columns also induces a great uncertainty in the calculation of molar mass averages on the basis of the chromatograms obtained. 

The term weight average has now been superceded by the term mass average. Often, average is replaced by mean. Not only degrees of polymerization and molar masses, but also other properties occur mostly as averages. 

Thus, the standard deviation of the molar distribution of the degree of polymerization can be calculated from the number and mass averages. Further, the standard deviation is an absolute measure of the width of a Gaussian distribution (and only of a Gaussian distribution), since the molar fraction of 0.6826 lies within the limits Xn lon, and the molar fraction 0.9544 lies within Xn 2on. Thus, for Xw= 3170 and Xn = 3000 in the case of the example shown in Figure 8-2, On = 714, according to Equation (8-22). 

Macroscopic amounts of natural and synthetic polymers comprise a huge number of macromolecules (unless they consist of a completely cross-linked network). The individual polymeric molecules are different in the degree of polymerization, i. e. in molecular mass, as another principal difference from the low-molecular materials having a definite molecular mass common to all molecules. A polymer can be characterized by an average degree of polymerization or average molecular mass on a number or mass basis. 

If, for example, 2 mol of diamine is reacted until complete conversion (pcooH = 1) with a mol of dicarboxyUc acid (ro = 0.5), then the number-average degree of polymerization is X = 3, in accordance with equation (17-24), and the mass-average degree is Xw— 5, in accordance with equation (17-38). Following from Equation (17-40), however, the diamide will only be formed up to 25% by weight (w, = 0.25), so that the yield of this compound can only reach 25% of the theoretical formula molar conversion (Table 17-4). 

As can be seen from the numerical example (Table 18-4) for a kinetic chain length = 10, the degree of polymerization distribution is exceptionally narrow. Since the mass-average degree of polymerization Xw is obtained from Xw = Xwii, this gives, with Equation (18-45) and X = p I... 

In polydisperse primary chains, therefore, the critical concentration of cross-link-carrying monomer units depends on the mass average degree of polymerization, and not on the number average degree. 

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