Constant propagation

4 Propagation Constants and Characteristic Impedance 1.3.4.1 Propagation Constants 

Let us consider the meaning of the attenuation constant using the semiinfinite line case as an example. From Equation 1.62 and the boundary conditions, 

The earlier equation shows that the attenuation constant gives the attenuation of voltage after it travels for a unit length. 

we find propagation constants for a line with losses, that is, a line whose R and G are positive. From Equation 1.63, 

From earlier equations, the following results are obtained  

A major difference between the two methods of initiation is that the solvent in y-ray studies is almost inevitably the monomer itself, and these generally have lower dielectric constants than the chlorocarbon solvents most often used in the chemically initiated systems. As a result, it is not possible to compare the values of kp +) obtained from each technique without accounting for this difference in solvation. Classically, propagation involves charge dispersion in forming the transition-state complex and hence a reduction in the polarity of the system. Thus media of lower solvation power should favourably influence the process. (See reference 114 for more detailed discussion.) Experimentally the values of kp(+) from radiation-induced polymerizations are consistently higher than those obtained using stable salts as initiators, and this simplistic picture therefore seems to be confirmed. Dunn has recently carried out a detailed compilation of the available data on / p(+) and readers will find this an excellent distillation of the current position. 

Subira, G. Sauvet, J. P. Vairon, and P. Sigwalt, /. Polymer ScL, Part C, Polymer Symposia, 1976, 56, 221. 

in Developments in Polymerization , ed. R. N. Howard, Applied Science Publishers, London, 1978. 

The recent data for kp(+) for A-ethyl-3-vinylcarbazole allows a further interesting correlation of absolute reactivity with structure. The cation derived from this monomer is a substituted p-aminobenzyl type and as such its reactivity might be expected to lie between that of the propagating cation from N-vinylcarbazole and those from isobutylvinyl ether and/7-methoxystyrene. Indeed this seems to be so as the value obtained, 2 x lO s S slots neatly between the figures of 5 x 10 s and 5 x 10 M s as predicted. 

The other highly important feature of the polymerizations of this monomer is that they proceed with little if any termination and, perhaps more importantly, with little transfer. As a result, polymer degrees of polymerization approach closely to the theoretical value dictated by the molar ratio of monomer to catalyst and encourage the idea that by suitable substitution of the monomer it may yet be possible to arrive at a truly living carbocationic polymerization of an olefin. 

---

Inhibitors are characterized by inhibition constants which are defined as the ratio of the rate constant for transfer to inhibitor to the propagation constant for the monomer in analogy with Eq. (6.87) for chain transfer constants. For styrene at 50°C the inhibition constant of p-benzoquinone is 518, and that for O2 is 1.5 X 10. The Polymer Handbook (Ref. 3) is an excellent source for these and most other rate constants discussed in this chapter. 

Each of these reactions is characterized by a propagation constant which is labeled by a two-digit subscript The first number identifies the terminal repeat unit in the growing radical, and the second identifies the adding monomer. The rate laws governing these four reactions are... 

In this section we have seen that the copolymer composition depends to a large extent on the four propagation constants, although it is sufficient to consider these in terms of the two ratios ri and r2. In the next section we shall examine these ratios in somewhat greater detail. 

Table 7.2 lists a few cross-propagation constants calculated by Eq. (7.20). Far more extensive tabulations than this have been prepared by correlating copolymerization and homopolymerization data for additional systems. Examination of Table 7.2 shows that the general order of increasing radical activity is... 

Table 7.2 Values of the Cross-Propagation Constants k 12 for Four Monomer-Radical Combinations...

In summary, we can rank these reactions in terms of their propagation constants as follows ... 

If resonance effects a/one are considered, it is possible to make some sense of the ranking of various propagation constants. 

Methacrylates in general have modestly slower propagation constants, and higher glass transition temperatures than acrylates. 

A similar association may result from intermolecular interaction of two growing chains. Of course, the degree of such an association should depend on the concentration of growing polymers, i.e. the observed propagation constant, kobs, is given then by the relation ... 

Kinetic studies of the polymerization of mono-functional polymethyl methacrylate led to the determination of the propagation constants, k , of the sodium, potassium, and cesium salts 29- 35 36) of polymethyl methacrylates anions. Surprisingly, they... 

Table 1. Propagation constants of polymethyl methacrylate ion-pairs in THF at —98 °C 43)...

In dimethoxyethane DME, a more powerful solvating agent than THF, solvation by solvent molecules competes with the intramolecular solvation, increasing the reactivity of ion-pairs. Indeed, the propagation constants of Na+ and Cs+ salts of polymethyl methacrylate are higher in that solvent than in THF, although again both salts are nearly equally reactive 39) as shown in Fig. 5. 

Only fragmented data are available on polymerization of other methacrylates. Propagation constants and the respective Arrhenius parameters for the homopolymerization of various methacrylates initiated by sodium metallo-organics were reported recently +3,56) and are given in Table 2. 

The exceptionally low propagation constants of t-butyl and of phenyl methacrylate are notable. The polymerization of the former monomer was thoroughly examinedS5). At temperatures even as high as 25 °C this reaction, when performed in THF in the presence of salts depressing dissociation of ion-pairs, yields polymers of highly uniform size. The reaction is strictly first order in growing polymers and in monomer, and no... 

Only a few quantitative data are available on copolymerization of methacrylates. Direct determination of the cross-propagation constants is readily achieved in living polymer systems whenever the absorption spectra of the two propagating species are different. Unfortunately, this is not the case in the methacrylate series. A new approach to this problem was developed by Muller 43). A mixture of two monomers is copolymerized, the reaction is interrupted at various times, and the concentrations of the residual monomers are determined as functions of time. The pertinent differential equations include 4 constants ku, k12, k21, and k22. Since kn and k22 were independently determined, the remaining cross-propagation constants are obtained by computer fitting the experimental conversion curves to the calculated ones. 

The degree of aggregation of polystyryl alkali salts in hydrocarbons, as well as the reactivity of their respective unassociated pairs, decrease along the series Li +, Na+, K +, Cs+ (Ref.Il, pp. 20 21). For example, the propagation constant of the lithium pair in benzene at 25 °C is estimated to be greater than 100 M "1 sec- while those of K +, Rb+, and Cs+ were determined as 47, 24, and 18 M-1 sec-1, respectively. Such a gradation contrasts with that of the reactivities of tight pairs in ethereal solvents,... 

The absolute value of the propagation constant was determined89). The detailed mechanism of this complex reaction and the factors determining the isotacticity were discussed in a following paper90). 

The observations discussed above suggest that the kinetic order of lithium poly-isoprene propagation should vary with the living polymer concentration. The effect is imperceptible in aliphatic hydrocarbons, but is observed in benzene solutions. The apparent propagation constants of lithium polyisoprene (MW 2 2 10 ) were determined in benzene and the results are displayed in Fig. 16 in the form of a plot of log kapp vs log c, c denoting the total living polymer concentration. 

Under these conditions the maximum propagation constant, kpc = 750 M-1 sec-1, gives the absolute rate constant of the monomer addition to the complexed unassociated lithium polystyrene, a value obviously larger than that of the unassociated but also uncomplexed polymer. 

Note that equation 88 is based on the pseudo-homopolyirerlzatlon jproacli It reduces to liie sinple ptroduct of moncmer concentration by a suitably conposltlon averaged propagation constant and the total nuntoer of active chains in the particles. 

The results of chain transfer studies with different polymer radicals are compared in Table XIV. Chain transfer constants with hydrocarbon solvents are consistently a little greater for methyl methacrylate radicals than for styrene radicals. The methyl methacrylate chain radical is far less effective in the removal of chlorine from chlorinated solvents, however. Vinyl acetate chains are much more susceptible to chain transfer than are either of the other two polymer radicals. As will appear later, the propagation constants kp for styrene, methyl methacrylate, and vinyl acetate are in the approximate ratio 1 2 20. It follows from the transfer constants with toluene, that the rate constants ktr,s for the removal of benzylic hydrogen by the respective chain radicals are in the ratio 1 3.5 6000. Chain transfer studies offer a convenient means for comparing radical reactivities, provided the absolute propagation constants also are known. 

Propagation constants for butadiene and isoprene were determined from rate of polymerization per particle in emulsion polymerization. 

---