Molar mass Schulz-Zimm

The molar mass distribution of branched materials differ most significantly from those known for Hnear chains. To make this evident the well known types of (i) Schulz-Flory, or most probable distribution, (ii) Poisson, and (iii) Schulz-Zimm distributions are reproduced. Let x denote the degree of polymerization of an x-mer. Then we have as follows. 

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 Schulz-Zimm distribution would be found for/end-to-end coupled linear chains which obey the most probable distribution, as well as for/of such chains which are coupled onto a star center. This behavior demonstrates once more the quasi-linear behavior of star branched macromolecules. In fact, to be sure of branching, other structural quantities have to be measured in addition to the molar mass distribution. 

The molar mass distributiMi for two polymers. The first polymer (continuous trace) follows the Flory-Schulz MMD with Mn = 40000 Mw is the double, i.e., Mw = 80000). The second polymer (dotted trace) follows the Schulz-Zimm MMD with Mn = 40000 and Mw = 120000. The molar mass distribution is diplayed as the molar fraction (a) and weight fraction (b) vs. mass. 

In this case, the polydispersity index is (a + 2)/ + 1). Figure 2 reports the Schulz-Zimm MMD for a polymer sample with Mn = 40000 and Mw = 120000. Figure 2.2a reports the number fraction versus mass, whereas Figure 2.2b reports the weight fraction. The two plots are quite different due to the fact that short chains are very abundant but their weight is small. Nevertheless, the two plots are connected, since the weight fraction is the product of the molar fraction times the mass. For brevity, the descriptions of the Log-normal MMD and the Generalized Exponential MMD are omitted. (They can be found elsewhere. )... 

Important differential mass-distribution functions (probability density function of mass-distribution) are the most probable distribution (Schulz-Flory), the Schulz-Zimm distribution, the Poisson distribution, Tung distribution, and logarithmic normal distribution (Wesslau distribution) [08IUP2]. Methods for the determination of distribution functions of molar mass are listed in Table 4.1.4. 

Fig. 1.3. Molar mass distributions of the Schulz-Zimm type for /3 = 2 left) and of the Poisson type right). Both correspond to the same number average degree of polymerization, A n = 10 ...

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