Critical extent of reaction

As an example of the quantitative testing of Eq. (5.47), consider the polymerization of diethylene glycol (BB) with adipic acid (AA) in the presence of 1,2,3-propane tricarboxylic acid (A3). The critical value of the branching coefficient is 0.50 for this system by Eq. (5.46). For an experiment in which r = 0.800 and p = 0.375, p = 0.953 by Eq. (5.47). The critical extent of reaction, determined by titration, in the polymerizing mixture at the point where bubbles fail to rise through it was found experimentally to be 0.9907. Calculating back from Eq. (5.45), the experimental value of p, is consistent with the value =0.578. 

An important consequence of Eq. 2-138 is its limiting form at the gel point where the number-average degree of polymerization becomes inhnite. The critical extent of reaction pc at the gel point is given by... 

BB] is the concentration of BB, and [B] = 2[BB], The number of crosslinks is simply the number of BB monomer molecules in which both B double bonds are reacted, that is, //[BB. The number of polymer chains is the total number of A and B double bonds reacted divided by the degree of polymerization, ([A] + [B])ppCw (the weight-average degree of polymerization is employed for reasons previously described in Sec. 2-10). The critical extent of reaction at the gel point pc occurs when the number of crosslinks per chain is and thus is given by... 

For such a system the copolymer will consist of copolymerized A and B groups with pendant, unreacted C groups until the later stages of reaction. Crosslinking does not occur until relatively late in the reaction due to the low reactivity of the C group. The critical extent of reaction at gelation in this case is given by... 

Problem 5.36 Since at the gel point the infinite network begins to form, the probability of a finite chain should provide an independent means for determining the gel point. Starting with Eq. (5.225) for the simple Af (/ > 2) homopolymerization, derive a relation between the critical extent of reaction for gelation and the monomer functionality /. 

At the critical extent of reaction pc for the onset of gelation, the number of cross-links per chain is (see Problem 7.19), whence the critical extent of reaction at the gel point pc is obtained as... 

Since the number-average degree of polymerization of gel can be considered to be infinite (see Problem 1.5), i.e., Xn 00, Eq. (5.61) gives the extent of reaction at the gel point, called the critical extent of reaction or critical conversion... 

The first theory that attempted to derive the divergences in cluster mass and average radius accompanying gelation is that of Flory [52] and Stockmayer [53]. In their model, bonds are formed at random between adjacent nodes on an infinite Cayley tree or Bethe lattice (see Figure 47.7). The Flory-Stockmayer (FS) model is qualitatively successful because it correctly describes the emergence of an infinite cluster at some critical extent of reaction and... 

Table 17-8. Experimentally Observed Critical Extents of Reaction for Gel Formation (pA)crit in the Polycondensation of 2 mol of Glycerol with 3 mol of Phthalic Anhydride...

Since the critical extent of reaction is given by a = 1/2, the cubic power near above the sol-gel transition point is evident. 

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