Molecular Weight Distributions in Nonlinear Polymerizations

The molecular size distribution functions for three-dimensional polymers are derived in a manner analogous to those for linear polymers, but with more difficulty. The derivations have been discussed elsewhere [Flory, 1946, 1953 Somvarsky et al., 2000 Stockmayer, 

and only their results will be considered here. The number Nx, number or mole fraction N x, and weight fraction wx of x-mer molecules in a system containing mono-mer(s) with / 2 are given, respectively, by 

These equations are general and apply equally for multifunctional reactions such as that of Af with B, or that of Ay with A—A and B B. Depending on which of these reactant combinations is involved, the value of a will be appropriately determined by the parameters r,f, p, and p. For convenience the size distributions in the reaction of equivalent amounts of trifunctional reactants alone, that is, where a p, will be considered. A comparison of Eqs. 2-89 and 2-166 shows that the weight distribution of branched polymers is broader than that of linear polymers at equivalent extents of reaction. Furthermore, the distribution for the branched polymers becomes increasingly broader as the functionality of the multifunctional reactant increases. The distributions also broaden with increasing values of a. This is seen in Fig. 2-17, which shows the weight fraction of x-mers as a function of a for the polymerization involving only trifunctional reactants. 

The broadening of the distribution with increasing a can also be noted by the XwfXn value. Equations 2-167 and 2-169 show that the difference between the number- and weight-average degrees of polymerization increases very rapidly with increasing extent of reaction. At the gel point the breadth of the distribution Xw/Xn is enormous, since Xw is infinite, while X has a finite value of 4 (Fig. 2-19). Past the gel point the value of Xw/Xn for the sol fraction decreases. Finally, at a = p — 1, the whole system has been converted to gel (i.e., one giant molecule) and Xw/Xn equals 1. 

As mentioned previously, the behavior of systems containing bifunctional as well as trifunctional reactants is also governed by the equations developed above. The variation of wx for the polymerization of bifunctional monomers, where the branching coefficient a is varied by using appropriate amounts of a trifunctional monomer, is similar to that observed for the polymerization of trifunctional reactants alone. The distribution broadens with increasing extent of reaction. The effect of unequal reactivity of functional groups and intramolecular 

The number- and weight-average degrees of polymerization are given by 

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Tobita H. Molecular-weight distribution in nonlinear emulsion polymerization. J Polym Sci B Polym Phys 1997 35 1515-1532. 

The molecular weight distribution in this type of nonlinear polymerization will be much narrower than for a linear polymerization. Molecules of sizes very much different from the average are less likely than in linear polymerization, since this would require having the statistically determined / branches making up a molecule all very long or all very short. The distribution functions for this polymerization have been derived statistically [Peebles, 1971 Schaefgen and Flory, 1948], and the results are given as... 

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... 

They are discrete transforms and can therefore operate directly on the separate equations for each species, reducing them to one expression. Nonlinear terms arising from condensation polymerization can be handled and, with some difficulty, so can realistic terminations in free radical polymerization. They are a special case of the generating functions and can be used readily to calculate directly the moments of the distribution, and thus, average molecular weights and dispersion index, etc. Abraham (2) provided a short table of Z-transforms and showed their use with stepwise addition. 

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