Distribution function Schulz-Zimm

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]. 

One can then derive the relationship between the intrinsic viscosity and various molecular weight averages, if a distribution function is specified. For example, for the Schulz-Zimm distribution function (Eq. 2.61), we obtain the following relationships between [rj] and the weight and number average molecular weights. 

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 ... 

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

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... 

Theoretical evaluations of for v x have proved elusive, but comparisons of A and A j have been made on the basis of approximations for this function, and the use of the Schulz-Zimm distribution of M, comparing the functions... 

The two distribution functions are presented in Fig. 1.3, choosing /3 = 2 for the Schulz-Zimm distribution, and equal values of Nn = 10" ) in both cases. 

As shown in the function, the fraction with k crosslinks can be given by the Schulz-Zimm distribution. The total distribution is given by its sum and takes the form of a supergeometric function. 

These parameters are used calculate the site and mass distribution functions assuming a Schulz-Zimm molecular weight distribution. The Schulz-Zimm parameters are calculated in lines 930-950. The weight fraction of diluent (as a fraction of the amount of polymer) is then sought. If there is no diluent enter 0. If there is a diluent, the functionality and molecular weight of the diluent is requested (line 1040). The necessary expectation values are computed (lines 1060-1150). 

The choice of distribution function is best made on the basis of theoretical expectations, e.g., for the length distribution of cylindrical micelles an asymmetric distribution such as the Schulz-Zimm or log-normal distribution function is expected to be suitable [75]. 

Tanaka and Sole used both Schulz-Zimm and logarithmic-normaF distributions to calculate the effects of polydispersity on second virial coefficients for the Fade approximant expression. For the Schulz-Zimm function, A 2 /A increases from unity with increasing dispersion, as in the case of hard sphere interactions, but it approaches a finite limit and the ratio of. 4 2 to A2 for the monodisperse species matching M passes through a maximum with increasing dispersion and may then become less than unity. The trends exhibited in these results suggest that careful conventional fractionation procedures applied to typical noncrystalline polymers should diminish the effects of polydispersity on the second virial coefficient to the extent that a solution of a polymer fraction can safely be treated as a binary system. 

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