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space procedure that simplified electrodynamics in free space can be generalized for a perfectly conducting closed cavity (a perfect cavity for short). We will see that the plane waves appearing in the Fourier transforms in free space can just be replaced by a different sort of wave, whose shape is determined by the boundary conditions of the cavity. These waves are the cavity modes. They are extremely useful, especially for quantizing the electromagnetic field, because Maxwell equations for the electromagnetic field in the cavity involve only fields at the same point in mode space. From this point of view, the plane waves that appeared in the Fourier transforms in Section 2.2 are just modes offree space. Let us consider a closed cavity of any shape without any matter inside it, and made up of perfectly conducting walls. Any electromagnetic radiation inside it can be described by Maxwell equations without sources:



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and the dependence of Rp on monomer increases to second-order. For a polymerization using an isolated metallocenium borate (I*), there is no equilibium corresponding to Eq. 8-57, and the Rp expressions become

Rp kp I* M 8-62

V E=O, V B=O,

Rp K2 kp I* M 2 8-63

(A.I) (A.2) (A.3)





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in place of Eqs 8-60 and 8-61, respectively Various studies with different metallocenes and coinitiators show a range of behaviors from rst- to second order dependence of Rp on monomer for the polymerization of propene and other 1-alkenes [Frauenrath et al, 2001a; Herfert and Fink, 1993; Jungling et al, 1995; Liu et al, 2001c; Resconi et al, 2000; Zhao et al, 2000] The observation of an intermediate order such as 3 may indicate two types of propagating species; thus, metallocenes are not 2 simple single-site initiators One possibility is propagation by a combination of tight and loose ion pairs This possibility is supported by the effect of solvent on polymerization Polymerization is faster in methylene chloride than in toluene (Free-ion-propagating species are excluded from consideration by the absence of a common ion effect on rate.

13 14

(AA)

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) A different possibility is a decrease in reactivity of a propagating species after a 2,1-addition of monomer, with reactivity subsequently restored by a 1,2-addition If the two propagating species show different dependencies on monomer concentration, such as rst- and second-order, the overall effect is an intermediate order of dependence of Rp on monomer The observation that the dependence of Rp on monomer for the polymerization of 1-hexene by racMe2 Si(H4 Ind)2 ZrCl2 /MAO changes from rst-order to 14-order in going from 0 to 50 C adds support for either of the possible mechanisms [Zhou et al, 2001] The dependence of Rp on zirconocene is observed as rst-order in a range of polymerization systems, in line with both Eqs 8-60 and 8-62 The order of dependence of Rp on MAO is unclear.

c8t ' 18 VAB=--E c8t '

There is a minimum [MAO]/[Zr] ratio below which polymerization is not observed and a maximum [MAO]/[Zr] ratio above which no changes in Rp are observed The minimum may correspond to the amount of MAO required to destroy deleterious impurities The.

with the boundary conditions)

maximum may correspond to the amount of MAO that competely converts the metallocene to the metallocenium cation. Thus, an observation of the effect of polymerization rate on MAO can be carried out only in the concentration range between the minimum and maximum. The minimum and maximum [MAO]/[Zr] ratios are 150 and 600 for the polymerization of 1-hexene by rac-Me2 Si(H4 Ind)2 ZrCl2 /MAO. This narrow range limits the precision and accuracy of kinetic experiments, but within these limits Rp was approximately rst-order in MAO. Activation energies are generally in the range 40 60 kJ mol 1 ; some lower values have been observed, in a range similar to that for traditional Ziegler Natta initiators.

(A.5) (A.6)

8-5h-2

n Bls =0,

The degree of polymerization is given by the same expression that described traditional Ziegler Natta polymerizations (Eq 8-50) In the absence of H2 and other transfer agents, polymer molecular weight is limited by various b-hydride transfers from normal (1,2-) and reverse (2,1-) propagating centers, before and after rearrangement [Lehmus et al, 2000; Resconi et al, 2000; Rossi et al, 1995, 1996; Zhou et al, 2001] (Sec 8-4i-2) Vinylidene, vinylene, and trisubstituted double-bond end groups are formed in 1-alkene polymerizations, vinyl and vinylene in ethylene polymerization [Vinyl groups are also produced in some 1-alkene polymerizations, not by b-hydride transfer, but by b-alkyl transfer (Sec 8-4i-2)] The relative amounts of the different double-bond end groups vary with the metallocene and reaction conditions For example, consider the polymerization of 1-hexene by racMe2 Si(H4 Ind)2 ZrCl2 /MAO [Zhou et al, 2001].

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