Polymerization weight-average degree

FIGURE 16.24 Inulin isolated from mature Topinambur tubers sparated on Bio-Gel P-6 (140 X I. Scm) flow rate 0.33 ml/min eluent H20(dest) + 0.002% NaNs detection Waters 403 R differential refractive index detector, sensitivity 8X applied sample I ml of a 20-mg/ml aqueous solution weight average degree of polymerization dp = 9 number average degree op polymerization dp = 2 polydisper-sity dp /dp = 4.5. 

Figure I 6.34 Degree of polymerization distribution molar distribution ( ) with number average degree of polymerization (dp = 140 glucose unimers) mass distribution ( ) with weight average degree of polymerization dp = 1242 glucose unimers.

The instantaneous number and weight average degrees of polymerization are given by... 

The average values of for each seed were determined by averaging the values of ( )j) required to match the experimental number and weight average degrees of polymerization. The value of (j)j for each run was found by solving the equation for in... 

The weight average degree of polymerization (see Eq. VII-27) is similarly obtained ... 

If we regard it as a cross-linked polymer, we require ywj the weight average degree of polymerization if all cross-linkages were severed. This operation would amount to replacing each tetrafunctional unit, or pair of cross-linked units, with two bifunctional units. The extent of reaction p would not be affected. Hence, according to Eq. (VIII-8)... 

It remains to evaluate the summations required for the calculation of the weight fraction Wx of x-mer and the weight average degree of polymerization Xw. These are of the type... 

The broadening of the distribution as the reaction proceeds toward completion is shown also by the expressions for the number and weight average degrees of polymerization. Both go to infinity as a— ac, of course, but Xw gets there faster—so much so that the ratio of Xw/xn goes to infinity also (see Eq. 29). 

The weight average degree of polymerization Xw may be derived as follows, where it is assumed that a ac. We begin by defining... 

Fig. 72.—Number and weight average degrees of polymerization prior to gelation and for the sol after gelation vs. the over-all degree of cross-linking (p) for the rectangular distribution. ...

The commonly observed, and technically troublesome, formation of gel in the polymerization of dienes testifies to the occurrence of cross-linking processes during their polymerization. According to Eq. (13), gel should begin to form when p reaches the critical value pc = l/y , where is the weight average degree of polymerization of the primary molecules. Thus... 

It should be noted that the product of a step polymerization is a mixture of polymer molecules of different molecular weights. The molecular weight distribution is characterized by the number-average and weight-average degrees of polymerization, X and Xw> respectively, defined by... 

Fraction 1 (number) Yield in percent of amylose Reducing value as percent glucose Average molecular weights Average degrees of polymerization... 

Figure 5.7 Calculated dependence of the weight-average degree of polymerization of molecules, (P)w, and hard clusters, (Pc)w, on conversion in a stoichiometric Adh) + B2(h) + B2(s) system (h - hard, s - soft). The system corresponds to a mixture of short and hard chains crosslinked with a tetrafunctional crosslinking agent...

Higher-order averages are also important for network formation the number average value of functional groups per weight-average degree of polymerization is defined by... 

These observations require a detailed explanation. After several unsuccessful attempts a satisfying answer was finally found. A first step was made by the ingenious derivation of the molar mass distributions of randomly branched or randomly cross-linked materials [14]. The equation, that was later rederived by Elory [13], will be given in the next section. Here it suffices to point out that the width of the distribution, or the polydispersity index MJM , increases asymptotically with the weight average degree of polymerization... 

Fig. 19. Weight fraction molar mass distributions w(x) of the Schulz-Zimm type for various numbers of coupled chains in a double logarithmic plot. Note fory=l the Schulz-Zimm distribution becomes the most probable distribution in the limit of/ l the Poisson distribution is eventually obtained. In all cases the weight average degree of polymerization was 100. The narrowing of the distribution with the number of coupled chains is particularly well seen in the double logarithmic presentation...

The three types of distributions are plotted in Fig. 19. Actually the molar mass would be increased by the number of chains, but to make the narrowing more evident the curves were normalized to the same weight average degree of polymerization x,, =100. 

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