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at r 1 for various values of s1 and s2 with
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where k, k0 , and k00 are the rate constants for the reactions of A, A0 , and A00 groups, respectively, with B groups. pc increases as s1 or s2 becomes larger than 1. If both s1 and s2 are larger than 1, the increase in pc is greater the larger is the difference between s1 and s2 . Plots such as Figs. 2-15 and 2-16 can be used to calculate the expected pc for a system where the reactivities of the functional groups are known or the relative reactivities (s in the rst
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Fig. 2-16 Effect of s1 and s2 on pc for polymerization of B with A 0 . pc values are shown on the B A plot. After Miller and Macosko [1978] (by permission of American Chemical Society, Washington, DC).
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system, s1 and s2 in the second system) in a particular reaction system can be determined from the measurement of pc . Some theoretical evaluations of the effect of intramolecular cyclization on gelation have been carried out [Gordon and Ross-Murphy, 1975; Kilb, 1958; Kumar et al., 1986; Stafford, 1981]. The main conclusion is that, although high reactant concentrations decrease the tendency toward cyclization, there is at least some cyclization occurring even in bulk polymerizations. Thus, even after correcting for unequal reactivity of functional groups, one can expect the actual pc in a crosslinking system to be higher than the calculated value. Various relationships have been derived to describe the crosslinking density of a system past pc [Argyropoulos et al., 1987; Durand and Bruneau, 1982a,b; Dusek, 1979a,b; Dusek et al., 1987; Flory, 1953; Miller and Macosko, 1976] (see also Sec. 2-11). The density of crosslinks in a reaction system increases with conversion and with increasing functionality of reactants. For a system such as
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the crosslink density increases as r increases. Varying the crosslink density by changing r is of practical interest in that it allows one to control the exibility or rigidity of the nal crosslinked polymer. 2-11 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 dif culty. The derivations have been discussed elsewhere [Flory, 1946, 1953; Somvarsky et al., 2000; Stockmayer,
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1943, 1952, 1953], 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 monomer(s) with f > 2 are given, respectively, by
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! fx x !f ax 1 1 a fx 2x 2 x! fx 2x 2 ! ! fx x !f ax 1 1 a fx 2x 2 Nx x! fx 2x 2 ! 1 af =2 ! fx x !f ax 1 1 a fx 2x 2 wx x 1 ! fx 2x 2 ! Nx N0 2-164 2-165 2-166
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The number- and weight-average degrees of polymerization are given by
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1 1 af =2 1 a Xw 1 f 1 a X w 1 a 1 af =2 1 f 1 a Xn Xn 2-167 2-168 2-169
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These equations are general and apply equally for multifunctional reactions such as that of A B. Af with Bf or that of Af with A and 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 r. 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.
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Fig. 2-17 Molecular weight distribution as a function of the extent of reaction for the polymerization of trifunctional reactants where a p. After Flory [1946] (by permission of American Chemical Society, Washington, DC).
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