Polymer mole fraction distribution

Fig. 51.—Mole fraction distribution of chain molecules in a linear condensation polymer for several extents of reaction p.

FIGURE 8.1 Number- (mole-) fraction distributions for linear step-growth polymers at different conversions (Eq. 8.3). 

Equation (2.44) describes the mole fraction distribution for the polymer formed and may be used to derive the number-average molar mass from... 

For a polymer possessing the most probable distribution, the mole fraction Na is given by Eq. (1) hence... 

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. ... 

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.]... 

The mole fraction or probability P of anionic polymer of degree of polymerization is described by the Poisson distribution... 

There is another way to confirm that the triad distribution cannot result from a random polymerization. The observed relative proportion of the triad structure in Figure Id is (l) (.93) (.26). The corresponding distribution for the triad structure for, 266 mole fraction VCI2 would be (l) (,36) (.13) Obviously, this comparison confirms from just a consideration of the triad distribution that the polymer does not conform to Bernoullian statistics. 

Flory Statistics of the Molecular Weight Distribution. The solution to the complete set (j - I to j = 100,000) of coupled-nonlinear ordinary differential equations needed to calculate the distribution is an enormous undertaking even with the fastest computers. However, we can use probability theory to estimate the distribution. This theory was developed by Nobel laureate Paul Floty. We have shown that for step ipolymeiization and for free radical polymerization in which termination is by disproportionation the mole fraction of polymer -with chain length j is... 

It can b<5 shown that in the absence of termination by combination, the mole fractions yj and weight fraction Wj are exactly the same as those for step polymerization. That is we can determine the dead polymer concentrations and molecular weight distribution of dead polymer in free radial polymerization for the Flory distributions. For example, the concentration of dead polymer of chain length n is... 

The samples used in the present study are (1) a copolymer consisting of p-hydroxybenzoic acid (HBA) and 6-hydroxy-2-naphthoic acid (HNA) in the mole fraction ratios 73 27 and (2) a polymer which contains HBA, HNA, terephthalic acid (TA) and hydroquinone (HQ) in the ratios 57 41 1 1. It will be shown later that the incorporation of the small amount of TA and HQ into the latter structure makes the material less shear—thinning than the copolymer containing only HBA and HNA, when examined theologically. It is generally believed that a random sequence distribution of monomers occurs in both... 

Since the molecular weight of the polymer is often undetermined or has a wide distribution, the use of mole fraction activity coefficients has many inherent difficulties. Patterson et al. (27) proposed the use of weight fraction activity coefficients, which is now standard practice. It is recommanded that all IGC data on polymers be reported this way. 

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 ... 

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)] ... 

The number distribution of degree of polymerization for i> = 300 and different values of ktd / ktc is plotted according to Eq. (P6.44.23) in Fig. 6.16(b). It is seen that all the distribution curves pass through the same point This means that there is a value of x for which the mole fraction of polymer with DP = x is independent of ktd / c- To determine this value of x analytically, Eq. (P6.44.23) may be differentiated with respect to ktd / fkc and equated to zero. This gives x = V + 1. 

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