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where the two functional groups in the B 0 reactant have a difference in reactivity by a B factor of s. s is the ratio of the rate constants for B and B0 groups reacting with A groups. pc is the value of p at which the curve becomes essentially vertical (i.e., M w ! 1). The curves shift to the right with increasing s, although the shift becomes progressively smaller with increasing s. The extent of reaction at the gel point increases as the B and B0 groups differ in reactivity. The more reactive B groups react to a greater extent earlier in the overall process, but gelation does not occur without reaction of the less reactive B0 groups. Overall there is a wastage of the more reactive B groups and pc is increased. There is an upper limit to this wastage with increasing s and pc levels off with increasing s. Figure 2-16 shows a pc contour map for the system

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

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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Macaulay to construct an American corporate bond yield average free from changes in the average quality of the bond issues composing the average His adjusted average of high-grade railroad bond yields, 1857 1937 (528), was quoted in our nineteenth-century tables The objective, however, should not be taken literally No railroad bonds were likely to exist in 1860 of quality equal to that carried by dozens in 1900 and 1925 Macaulay collected monthly prices and yields of all the principal railroad bond issues from 1857, discarded those that were of short maturity or otherwise unsuitable, and constructed an unadjusted average as a chain index of yields down to 1937 These were typical yields, and not highgrade yields Then, by studying the scatter between the lower and higher yields, he constructed his adjusted average to reflect the yields of bestquality bonds.

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

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,

Parts Open It Up! Circuit at First Glance Step-by-Step Bending Choosing a Cool Control Layout Case Considerations Marking the Board Drilling Holes Painting Control Mounting Soldering Testing Reassembly Musicality 285 285 285 286 286 287 287 287 287 287 288 288 288 288.

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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