Weight Distribution Functions

Theoretically, it is also possible to obtain the volume weighted distribution function g(l) of the column length / ... 

Tung, L. H., Method of calculating molecular-weight-distribution functions from gel permeation chromatogram, /. Appl. Polym. Sci., 10, 375, 1966. 

More recently, the same author [41] has described polymer analysis (polymer microstructure, copolymer composition, molecular weight distribution, functional groups, fractionation) together with polymer/additive analysis (separation of polymer and additives, identification of additives, volatiles and catalyst residues) the monograph provides a single source of information on polymer/additive analysis techniques up to 1980. Crompton described practical analytical methods for the determination of classes of additives (by functionality antioxidants, stabilisers, antiozonants, plasticisers, pigments, flame retardants, accelerators, etc.). Mitchell... 

Molecular weight distribution function for the case where the length of the growth stage is short compared to the residence time in reactor. (Reprinted with permission from Chemical Reactor Theory, by K. G. Denbigh and J. C. R. Turner. Copyright 1971 by Cambridge University Press.)... 

Hence, is the statistics average Molecular weight calculated from the number Molecular weight distribution function, N(M). Mn is the statistical average Molecular weight calculated from the weight... 

Molecular weight distribution function, W (M). is the statistical average Molecular weight... 

The normal distribution function, also referred to as the Flory-Schulz distribution, relates the fraction of an x-mer (a polymer molecule consisting of x repeat units) in the entire assembly of molecules to its formation probability. It can be defined either as a number distribution function or as a weight distribution function. The number of moles of an x-mer (Nx) is given by the normal number distribution as follows ... 

Equations 2-86 and 2-89 give the number- and weight-distribution functions, respectively, for step polymerizations at the extent of polymerization p. These distributions are usually referred to as the most probable or Flory or Flory-Schulz distributions. Plots of the two distribution functions for several values of p are shown in Figs. 2-9 and 2-10. It is seen that on a... 

Table V. Weight Distribution Function of HEC After 24 Hr of Enzymic Hydrolysis Calculated from the Gauss—Laguerre Numerical Computation Method Using the Four First Terms of Equation 30 with the Scaling Factors...

Figure 6. Gauss-Laguerre weight distribution function of HEC after one day of enzymolysis, calculated by Lagrangian interpolation of the W(M) values (O) of Table V.

The prediction of the MWD of emulsion polymers proved to be a relatively intractable problem even after the advent of the Harkins-Smith-Ewart theory. Perhaps the most successful early attack on the problem was that of Katz, Shinnar and Saidel (2). They considered only two microscopic events entry and bimolecular termination by combination. Their theory resulted in a set of partial integrodifferential equations, whose numerical solution provided the lower moments of the molecular weight distribution function. Other attempts to predict the MWD of emulsion polymers include those of Parts and Wat ter son (3 ), Sundberg and Eliassen (4), Min and Ray (5) and Gardon (6). 

One advantage of the procedure delineated above is that it permits the complete molecular weight distribution function to be calculated, sometimes analytically, whatever the termination mechanism. Of course, the lower moments of the distribution function can also be readily calculated ... 

He derived the following equation for the differential weight distribution function ... 

Figure 5-5a shows the number distribution of degrees of polymerization in a linear (/ = 2) step-growth polymerization for several extents of reaction p. This is a plot of Eq. (5-22). The corresponding weight distribution function (Eq. 5-26) is plotted in Fig. 5-5b. The former figure shows that monomer is the most prevalent species on a mole basis at any extent of reaction. The corresponding weight fraction of monomer is negligible, however, at any p > 0.99[wi/ai = ( — p) from Eqs. (5-22) and (5-26)].

The relaxation modulus is the core of most of the viscoelastic descriptions and the above expression can be checked from experimental viscoelastic functions such as the complex shear modulus G (co) for instance. In addition to the molecTilar weight distribution function P(M), one has to know a few additional parameters related to the chemical species the monomeric relaxation time x,... 

This is the statistical weight-distribution function for a linear polycondensation reaction at the extent of reaction p. The number-distribution and weight-distribution functions are illustrated in Figs. 1 and 2 for values of p. 

Problem 5.13 Show mathematically that the maximum of the weight distribution function at high conversions in a most probable distribution is located at the number average degree of polymerization... 

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