Schulz-Zimm distribution

The calculations are applied to a polymer sample following a Schulz-Zimm distribution with Mw = 1.03 x 10s and Mw/N4n = 1.017. These values are representative for the polymer fractions used in most of the experiments in transient elongational flow [147, 155], To visualize the evolution of the degradation, it is convenient to make a distinction between the polymer fraction N from the starting material which remains intact and the fraction Nf newly formed following the degradation process... 

To generate the necessary distribution functions, the ratio of is used to approximate the true molecular weight distribution by a Schulz-Zimm distribution. It is also assumed that the reactive functional groups are distributed randomly on the polymer chain. The Schulz-Zimm parameters used to calculate distribution functions and probability generating functions (see below) are defined as follows ... 

PRINT "OF MW AND MN. THE RATIO OF MW TO MN IS USED TO GENERATE AN" 1220 PRINT "EQUIVALENT SCHULZ-ZIMM DISTRIBUTION FOR THE CROSSLINKER." 1230 PRINT 1240 PRINT 1250 INPUT 1260 PRINT 1270 PRINT... 

The equations relating Mn and Mw to radiation dose which are most frequently used apply to all initial molecular weight distributions for Mn, but only to the most probable distribution (Mw/Mn = 2) for Mw. However, equations have been derived for other initial distributions, especially for representation by the Schulz-Zimm distribution equation. 

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. 

Note 1 For large values of x, the most probable distribution converges to the particular case of the Schulz-Zimm distribution with b. ... 

The Schulz-Zimm distribution is called by statisticians the negative binomial distribution see Ref. 99, p. 291... 

The first approach assumes that the sample is composed of a discrete number or a specific distribution of components. Then the deviations from linearity for the proposed model can be included as higher order c-orrections in Equations 25 or 27. Current techniques are limited to the consideration of two discrete size populations of particles or to a size profile defined by two parameters such as the Schulz-Zimm distribution (7,26,27,28,29,30). If there is evidence that such a model describes the sample, it is certainly the best way to proceed. This approach has been applied recently to polydispersity analysis based on the assumption of a particular distribution model by Chen et al. (35, 36) and McDonnell et al. (26,37). 

Another distribution function such as the Schulz-Zimm distribution is asymmetric with a tail toward larger values of r ... 

The factors for the common Schulz-Zimm distribution function are listed in Table 8.2. For other distribution functions the reader is referred to [47,48]. 

A theoretical description of the depletion interaction between two plates caused by ideal polymer chains with a finite width of a Schulz-Zimm distribution was published by Tuinier and Petukhov. They derived an exaa expression for the potential... 

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. 

In eq 9.44 the distribution function is called generalized Schulz-Flory distribution (or Schulz-Zimm distribution) because of its introduction by Schulz in 1935, and by Flory in 1936. " To evaluate the polydispersity of polymers U = Mw/M - 1 is an important quantity. The quantity k of the Schulz-Flory distribution is given by k= IU. The two parameters f and k of the... 

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