Number fraction distribution

Fig Number-fraction distribution of chain molecules at different extents of reaction in polycondensation... 

Fig. 2-9 Number-fraction distribution curve for linear polymerization. Plot 1, p = 0.9600 plot 2, p = 0.9875 plot 3, p = 0.9950. After Howard [1961] (by permission of Iliffe Books, London and Elsevier, Oxford).

Nj. is synonymous with the mole- or number-fraction distribution. When x is an odd number, there are x pathways and the (x — 1) term in Eq. 3-190 should be replaced by x. For polymerizations yielding high polymer, the difference between x and (x + 1) is negligible and can be ignored. [Some derivations of size distributions show exponents y and z, respectively, in Eqs. 3-186 and 3-187 instead of (y — 1) and (z — 1), which results in an exponent of x instead of (x — 1) in Eqs. 3-189 and 3-190). The differences are, again, unimportant for systems that yield high polymer.]... 

Potassium nitrate crystals have a density of 2.11 x 1(T12 g//um3, which allows for the determination of the estimated crystal numbers on each sieve in Table II. A cumulative number distribution, N(L), and a cumulative number fraction distribution, F(L), can be calculated using methods similar to those for calculating M(L) and W(L). 

When the bimolecular terminations are highly diffusion controlled, the termination reactions are dominated by interactions between radicals with short and long chain lengths even in bulk polymerization, and the MWD of the longer polymer radicals tends to follow the most probable distribution [287, 292]. Under such conditions, oligomeric chains that can be observed only in the number fraction distribution may be formed via disproportionation termination irrespective of particle size. Figure 13 shows the effect of particle size on the instantaneous chain length distribution where the bimolecular terminations are from disproportionation [265]. 

A simple alternative method was proposed by Gilbert et al. [296, 297] to determine the chain transfer constants based on the chain length distribution (CLD). If the dominant chain termination mechanism is chain transfer to monomer, the instantaneous numerical MWD (the number fraction distribution) is given by ... 

In a zero-one system in which a radical has just entered a polymer particle containing one polymer radical, and is terminated instantaneously, the number fraction distribution in the absence of a polymer transfer reaction is given by ... 

The instantaneous MWD of the primary chains formed during Interval II in a zero-one system (assuming combination termination) is given by the most probable distribution, whose number fraction distribution is given by ... 

Figure 4. Number fraction distributions of calculated mobilization pressures. Case (a) corresponds to bubble trains containing 100 bubbles case (b) to 1000 bubbles.

It is convenient to define the th moment of the number fraction distribution as the sum of the products of the number fraction of molecules... 

The zeroth (k = 0) moment is equal to unity because the number fraction distribution is normalized ... 

In order to characterize the molar mass distribution, several average molar masses are defined that emphasize different parts of this distribution. The number-average molar mass is defined as the ratio of the first (A = 1) to zeroth k = 0) moments of the number fraction distribution ... 

The weight-average molar mass is the ratio of the second and the first moments of the number fraction distribution ... 

Substituting this result into Eq. (1.45) we obtain the number fraction distribution for linear condensation polymers ... 

Comparison of the most-probable weight fraction distribution (0.991) with experimental nylon 66 data from G. B. Taylor, J. Am. Chem. Soc. 69, 638 (1947). The dashed line is the number fraction distribution W/v (0.991). 

The number- and the weight-average molar masses are calculated from the moments of the molar mass distribution. The first moment of the number fraction distribution is the number-average molar mass Mn. ... 

In the case of addition polymerization without termination, the number fraction distribution function (the probability that a given chain has degree of polymerization N) is given by the Poisson distribution function ... 

Synthetic polymers are often polydisperse, containing a mixture of molecules with different molar masses. This mixture is described by a distribution—number fraction (or mole fraction) rij of molecules with molar mass M. The distribution is characterized by its moments, with the kth moment of the number fraction distribution function defined as a sum over the distribution ... 

Show that the -moment of the number fraction distribution jv is related to the (k - l )-moment of the weight fraction distribution nv [derive Eq. (1.54)]. 

A linear polymer (A-mer) is a cluster of N monomers (sites) connected by N — 1 bonds and containing one unreacted A group and one unreacted B group. The number fraction distribution mole fraction ofN-mers) is given by the probability that a chosen unreacted A group is part of an N-mer. This number fraction of A-mers is the probability of A - 1 formed bonds (j) ) and one unreacted B group (1 p) [Eq. (1.52)] ... 

Comparing Eqs (6.3) and (6.9), we see that for /> 3, hyperbranched polymers have a quite different form for their number fraction distribution than linear polymers. The large number of unreacted B groups (Fig. 6.12) on large hyperbranched iV-mers broadens the distribution because the larger molecules have a higher probability of growing than smaller molecules do. 

In order to evaluate different moments, this number fraction distribution is rewritten as... 

This number fraction distribution has a form of a power law with an exponential cutoff at the characteristic degree of polymerization N = e ... 

Similar methods can be used to construct universal plots for molar mass distributions of linear and hyperbranched condensation polymers. The number distribution function n p, N) for linear condensation polymers is obtained from the number fraction distribution [Eq. (1.66)] ... 

Prove that the number fraction distribution for hyperbranched polymers is properly normalized [i.e., show that Eq. (6.9) satisfies Eq. (6.12)]. 

The stress relaxation of polymer networks on long time scales is believed to be due to arm retraction of dangling ends (Fig. 7.7). The polydispersity of dangling ends is determined during crosslinking. Assume the number fraction distribution of linear dangling ends [Eq. (1.52)]... 

Figure 5.5 Mole fraction or number fraction distribution of reaction mixture in linear step-growth polymerization for conversions of 95%, 98%, and 99% [Eq. (5.70)]. (From Ref. 15 with the permission of the American Chemical Society, Washington, D.C.)...

Problem 6.46 Derive weight-fraction distribution of degree of polymerization, tUi, from the number fraction distribution, n, described by Eq. (P6.44.23). 

---