Weight fraction Poisson

Fig. 2.8 Weight-fraction Poisson distribution of chain lengths for various values of x in a polymerization without termination. For comparison, the broken line represents the most probable distribution of molar mass when x = 11 (after Flory).

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...

Fig. 33. Poisson distribution of molecular weights. Plot of weight fraction vs. degree of polymerization. The average degrees of polymerization of the three curves shown in die figure are 50,100, 500...

Fig.4. Simulated distribution curves of weight fraction vs degree of polymerization having average degree of polymerization of 1000 and various M Mjj values. All the distribution curves were calculated assuming Gaussian distributions except for the Poisson distribution curve (MJM = 1.001)...

Fig. y.Weight fraction (wj) vs degree of polymerization (i) plot of SEC (dashed lines) and TGIC (solid lines) characterization results of a PS set synthesized under identical condition. Theoretical Poisson distributions matching the most probable value of MWD are also plotted with dotted lines. It clearly shows the band broadening of SEC elution peaks. Reproduced from [93] with permission... 

Figure 1.56. Number fraction and weight fraction of the Poisson distribution. Af and M are also indicated.

Figure 1.57 compares the weight fraction distribution of the Poisson distribution and the exponential distribution with the same M . The abscissa is on a logarithmic scale therefore, the ordinate is the weight fraction X i. The difference in the width between the two distributions is large. 

All the above formulae allow the direct and easy calculation of the effective elastic moduli and Poisson ratios of dense alumina-zirconia composites for a given zirconia volume fraction (f). Since usually the composition of alumina-zirconia composites (in the sequel abbreviated as AZ composites ) is given in weight percent (weight fractions ) the volume fraetions usually to be ealculated via the (theoretical) density. In the... 

Figure 8 Weight fraction (W ) versus degree of poiymerization (i) plots of different fractionations of a set of PS synthesized under identical conditions, SEC fractionation (dashed) and TGiC fractionation (solid), theoreticai Poisson distribution (dotted). Reprinted from Chang, T. Adv. Polym. Sci. 2003, 163,1, with kind permission of Springer Science+Business Media.

The distinguishing feature of such a mechanism occurs in the fact that the growth of all polymer molecules proceeds simultaneously under conditions affording equal opportunities for all. (This will hold provided the addition of monomer to the initiator is not much slower than succeeding additions.) These circumstances are unique in providing conditions necessary for the formation of a remarkably narrow molecular weight distribution—much narrower than may be obtained by polymer fractionation, for example. Specifically, they are the conditions which lead to a Poisson distribution of the number and mole fraction, i.e. ... 

Fig. 10. The fraction r0/yei for random crosslinking of high molecular weight primary chain distribution. 1 monodisperse or Poisson, 2 most probable, 3 self-convoluted random [Dobson and Gordon (4/)]...

The general expression [Eq. (8.53)] for the mole fraction of x-mer in the polymer is Poisson s distribution formula [6j. From the nature of the problem it is obvious, without the above derivation, that the numbers of molecules of various sizes must be represented by Poisson s distribution. Since >P = i oo + 1 at the completion of reaction [cf. Eq. (8.36)], the number average molecular weight will be given by... 

Problem 1.20 Find the molecular weight of the polymer that maximizes the number fraction in the Poisson distribution (Eq. 1.93). Assume that i 1. 

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