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Although phenol itself is used in the largest volume, various substituted phenols such as cresol (o-, m-, p-), p-butylphenol, resorcinol, and bisphenol A are used for specialty applications. Some use is also made of aldehydes other than formaldehyde acetaldehyde, glyoxal, 2-furaldehyde. The exact composition and molecular weight of the resole depend on the formaldehyde to phenol ratio, pH, temperature and other reaction conditions [Maciel et al., 1984; Werstler, 1986]. For example, higher formaldehyde compositions will yield resoles containing more of the mononuclear compounds. The resole prepolymer may be either a solid or liquid polymer and soluble or insoluble in water depending on composition and molecular weight. Molecular weights are in the range 500 5000; most are below 2000. The resoles are usually neutralized or made slightly acidic. The pH of the prepolymer together with its composition determine whether it will be slow curing or highly reactive. Crosslinking (curing) is carried out by heating at temperatures up to 180 C. The crosslinking process involves the same chemical reactions as for synthesizing the prepolymer the formation of methylene and ether bridges between benzene rings to yield a network structure of type XXIV. The relative



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J v + w2 }

importance of methylene and ether bridges is dependent on temperature [Lenz, 1967]. Higher curing temperatures favor the formation of methylene bridges but not to the exclusion of ether bridges.





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(2.155) Before we try to get a closed-form expression from (2.155), let us pause to think about an import physical point that has been overlooked so far. No material known to us can reflect all frequencies equally well. Moreover, when the frequency becomes large, all materials become transparent. In particular, our plates cannot remain perfect reflectors all the way to infinite frequency. So we must include a cutoff in (2.155). However, we do not want to know the details of the fall-off of reflectivity with frequency. This is possible if the relevant physics happens for frequencies much lower than the ones where the reflectivity starts to fall off. We will see later that that is exactly what happens here, but for now let us just assume that it is so. In that case, we can use a mathematical trick called regularization to obtain a closed expression for (2.155) without having to specify precisely the general form of the cutofffunction. Let us define a cutoff function la(w/c) that falls off sufficiently rapidly when the frequency becomes large and that depends on a parameter a in such a way that when a -+ 0 the cutoff function becomes 1 for any frequency:

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The polymerization and crosslinking of phenol formaldehyde is a highly useful industrial process. However, the reactions that take place are quite dif cult to handle in a quantitative manner for a number of reasons. The assumption of equal reactivity of all functional groups in a monomer, independent of the other functional groups in the molecule and of whether the others are reacted, is dubious in this polymerization. Consider, for example, the routes by which trimethylolphenol (XXIb) can be produced in this system:

lim la(k)

k2

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2-173

lim la(k)

TABLE 2-10 Reaction of Phenol with Formaldehyde Catalyzed by Basea; b Reaction k1 0 k1 k2 0 k2 00 k2 k3 0 k3

(2.156)

Rate Constant 104 (L mol 1 s 1) 14.63 7.81 13.50 10.21 13.45 21.34 8.43

36Notice that N n disappears as we go to the continuum because then w = 0 is a point of measure zero. 37 As long as the plates are brought from infinity in a adiabatic way. In 4 we will see what happens when cavity walls are moved in a nonadiabatic way, an effect that is often called the dynamic Casimir effect.

Table 2-10 shows the rate constants for the various reactions. It is apparent that there are signi cant differences in the reactivities of the different functional groups in phenol (i.e., in the different positions on the ring). The reaction between phenol and formaldehyde involves a nucleophilic attack by the phenolate anion on formaldehyde (Eq. 2-174) (or its hydrated form). The electron-pushing methylol groups generally increase the reactivity of phenol toward further substitution although steric hindrance modi es this effect. It is dif cult to quantitatively discuss the reasons for the speci c differences in reactivity, since there is no general agreement on even the relative values of the seven different rate constnats. The evaluation of the different rate constants is dif cult from both the viewpoints of chemical analysis of the mono-, di-, and trimethylolphenols and the mathematical analysis [Katz and Zwei, 1978] of the kinetic data.

(2.157) Let us examine the integrals over v in (2.157). They have the general form (2.158) Changing the variable of integration from v to Vi simpler expression,

2-174

(2.159)

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