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reacted with a nonbranch A unit is pB 1 r . Therefore the probability of obtaining a A segment of the type in Eq. 2-144 is given by pA pB 1 r pA n pB r. Summation of this over all values of n and then evaluation of the summation yields
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Either pA or pB can be eliminated by using the ratio r of all A groups to all B groups to substitute pA rpB into Eq. 2-146 to yield
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Equation 2-147 is combined with Eq. 2-145 to yield a useful expression for the extent of reaction (of the A groups) at the gel point
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Several special cases of Eqs. 2-147 and 2-148 are of interest. When the two functional groups are present in equivalent numbers, r 1 and pA pB p, Eqs. 2-147 and 2-148 become
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When the polymerization is carried out without any A molecules r 1 with r < 1 the A equations reduce to
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If both of the conditions stated are present, that is, r r 1, Eqs. 2-147 and 2-148 become
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These equations do not apply for reaction systems containing monofunctional reactants and/or both A and B type of branch units. Consider the more general case of the system
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A1 + A2 + A3 + + Ai + B1 + B2 + B3 + + Bj crosslinked polymer
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where we have reactants ranging from monofunctional to ith functional for A functional groups and monofunctional to jth functional for B functional groups [Durand and Bruneau, 1982a,b; Miller et al., 1979; Miller and Macosko, 1978; Stockmayer, 1952, 1953; Stafford, 1981.] The extent of reaction at the gel point is given by
pc 1 fr fw;A 1 fw;B 1 g1=2 2-156
where fw;A and fw;B are weight-average functionalities of A and B functional groups, respectively, and r is the stoichiometric imbalance (Sec. 2-6b). The functionalities fw;A and fw;B are de ned by
fw;A fw;B
2 fAi NAi fAi NAi 2 fBj NBj
2-157 2-158
fBj NBj
The summations in Eq. 2-157 are for all molecules containing A functional groups with NAi representing the number of moles of reactant Ai containing fAi number of A functional groups per molecule. The summations in Eq. 2-158 are for all molecules containing B functional groups with NBj representing the number of moles of reactant Bj containing fBj number of B functional groups per molecule. The utility of Eq. 2-156 can be illustrated for the system: 4 40 2 3 mol mol mol mol A1 A2 A3 A4 2 50 3 3 mol mol mol mol B1 B2 B3 B5
r, fw;A , fw;B , and pc are calculated as
1 4 2 40 3 2 4 3 0:80952 1 2 2 50 3 3 5 3 12 4 22 40 32 2 42 3 2:25490 1 4 2 40 3 2 4 3 12 2 22 50 32 3 52 3 2:41270 1 2 2 50 3 3 5 3 1 f0:80952 1:25490 1:41270 g1=2 0:83475
fw;B pc
2-161 2-162
Keeping in mind that pc is the gel point conversion for A groups since B groups are in excess (A groups are limiting), consider the effects of adding more monofunctional reagent to the system. The effect is different for adding A1 versus adding B1. For example, increasing A1 from 4 to 18 mol increases r to 0.92063 and decreases fw;A to 2.10345, with fw;B unchanged. The changes in r and fw;A cancel each other and Eq. 2-156 calculates that pc is unchanged at 0:83475. On the other hand, increasing B1 from 2 to 16 mol decreases r to 0.72857 and also decreases fw;B to 2.27143, with fw;A unchanged. The changes in r and fw;A reinforce each other and pc increases to 0.92750. The effect of increasing A1 is different from increasing B1 because B groups are in excess. The additional A groups added by increasing A1 results only in changing some of the end groups of the polymer chains from B groups to A groups, with no effect on pc . When B1 is increased, it tends to degrade the crosslinking process. The bifunctional and polyfunctional A reactants depend on the polyfunctional B reactants to build up the crosslinked network. Monofunctional B acts as a capping agent for chain ends to limit the extent of crosslinking and increase pc . 2-10c Experimental Gel Points
The two approaches to the problem of predicting the extent of reaction at the onset of gelation differ appreciably in their predictions of pc for the same system of reactants. The Carothers equation predicts the extent of reaction at which the number-average degree of polymerization becomes in nite. This must obviously yield a value of pc that is too large because polymer molecules larger than X n are present and will reach the gel point earlier than those of size X n . The statistical treatment theoretically overcomes this error, since it predicts the extent of reaction at which the polymer size distribution curve rst extends into the region of in nite size. The gel point is usually determined experimentally as that point in the reaction at which the reacting mixture loses uidity as indicated by the failure of bubbles to rise in it. Experimental observations of the gel point in a number of systems have con rmed the general utility of the Carothers and statistical approaches. Thus in the reactions of glycerol (a triol) with equivalent amounts of several diacids, the gel point was observed at an extent of reaction of 0.765 [Kienle and Petke, 1940, 1941]. The predicted values of pc , are 0.709 and 0.833 from Eqs. 148 (statistical) and 2-139 (Carothers), respectively. Flory [1941] studied several systems composed of diethylene glycol f 2 , 1,2,3-propanetricarboxylic acid f 3 , and either succinic or adipic acid f 2 with both stoichiometric and nonstoichiometric amounts of hydroxyl and carboxyl groups. Some of the experimentally observed pc values are shown in Table 2-9 along with the corresponding theoretical values calculated by both the Carothers and statistical equations.
TABLE 2-9 Gel Point Determinations for Mixture of 1,2,3-Propanetricarboxylic Acid, Diethylene Glycol, and Either Adipic or Succinic Acida Extent of Reaction at Gel Point (pc ) CO2 H r OH 1.000 1.000 1.002 0.800
r 0.293 0.194 0.404 0.375
Calculated from Eq. 2-139 0.951 0.968 0.933 1.063
Calculated from Eq. 2-148 0.879 0.916 0.843 0.955