Propagation coefficients

The primary disadvantage of these detection methods is that the interaction of the field with the surrounding environment is weak, and therefore, the influence of the analyte on the propagation coefficient is small. As a result, achieving high... 

Fig. 6. The polymerization of y-benzyl-L-ghitamate NCA in dioxane at 25° C, initiated by n-bexyl-amine. The values of [NCA] initially and at time t are denoted by A0 and At respectively, 1 is the initial concentration of n-hexylamine. kz a, kt i are the apparent propagation coefficients, in mole-11 secrK [Reprinted from paper by R. D. Lundberg and P. Doty J. Am. Chem. Soc. 79 3961 (1957)]...

The normalized irreducible characters Xy express the way in which these specific features propagate when N and S change. Therefore they axe called propagation coefficients [65]. 

The propagation coefficient for polyisoprenyllithium in heptane has been estimated [65] by working at concentrations as low as 5 x 10" M. The reaction order was found to increase below 10" M (see Fig. 10) and became first in polyisoprenyllithium at about 5 x 10" M. Assuming this corresponds to complete dissociation of the aggregates, a value of 0.65... 

Figure 14 illustrates the effect of solvent on the propagation rate of polystyrylsodium at 25—30°C. The ordinate represents the apparent propagation coefficient (feapp J p/[monomer] [active centres]) i.e. assuming a simple bimolecular reaction... 

Fig. 17. Variation of ion-pair propagation coefficients, fep, with temperature in styrene polymerization (counter-ion, sodium). Solvents ( ) oxepane [105] ( ) tetrahydropyran [116] (o) tetrahydrofuran [111] ( ) tetrahydrofuran [110] ( ) dimethoxyethane [112] ( ) dimethoxyethane [110]. More data are now available over a wider range of experimental conditions, see ref. 233.

Fig. 20. Variation of the apparent propagation coefficients with reciprocal square root of concentration of active centres in the copolymerization of styrene (1) with p-methylstyrene (2). Solvent tetrahydrofuran. (o) kn, ( ) feji ( ) 22 ( ) 12 [146].

This compares with fep for free cationic growth of 3.5 x 10 1 mole sec at 15°C and act 0- It is worthwhile to consider why the styryl cation reacts about 10 times faster with its monomer than does the styryl radical. A similar comparison is possible in the case of JV-vinylcarbazole for which the reported [106] free radical propagation coefficient is 6.0 Imole sec ... 

The polymerization rate will then depend on the relative proportions and activities of Cf, C, etc. (which will be determined by A/T and Kj, Kf, etc., and the propagation coefficients.) Thus, it has been found with (7r-CsH5)2TiEtCl/AlEtCl2 that the rate of ethylene polymerization accelerates with time as active species are formed and with increase in Al/Ti ratios the rate rises to a steady value at constant Ti concentration and [C ] [Ti]o, and passes through a maximum at constant metal alkyl halide concentration. 

Coombes and Katchalski [29] have considered a slightly more complex version of this mechanism in which a second propagation coefficient operates above a critical degree of polymerization. Katchalski et al. [30] calculated the molecular weight distribution obtained in a system following scheme (12) but also including a bimolecular termination step. Various authors have analysed more complex systems in which the initiator is a polymeric species. Thus Gold [31] has shown that initiation by a poly a-amino acid with a Poisson distribution leads to a polymeric product with an over-all Poisson distribution, and Katchalski et al. [32] demonstrated that in multichain polymers synthesized from polyfunctional initiators Poisson distributions also arise. 

Fig. 5. Propagation coefficients (fe) at 25°C for the polymerization of 7-benzyl-glutamate NCA in NiN-dimethylformamide as a function of D/L composition [M]o/[I]o = 20. The solid line is calculated on the assumption that the L—NCA does not react with polymers ending in D-residues (absolute preference). Initiation by n-hexylamine. Lundberg and Doty [12].

From the measured consumption of the two hydrocarbons, the values of rl and r2, the ratios of the propagation coefficients as defined earlier, may be determined. At low conversions... 

Here co is the radian frequency of the field, a the attenuation coefficient, and the propagation coefficient. 

Here Zo is commonly called the characteristic impedance of the transmission line, and / is its propagation coefficient defined as... 

At microwave frequencies (/ > 10 Hz) a polymer material is regarded as a distributed circuit having a complex propagation coefficient y = a ifi, where a and p are the electrical attenuation and phase factors of the material. These are measured using microwave techniques and e and e" follow as... 

Table 3 Propagation coefficients (kp) and activation energies (E of different para-substituted (X) styrenes. 

In Eqs.(5) the propagation coefficient k is referred to the gation coefficient of vacuum, k. The complex quantity ti ... 

For a sound wave propagating in porous textiles, the porous gas-filled medium is often treated as an equivalent uniform medium for analysis purposes, so a propagation factor can be used to describe the dependence of the propagating wave on time t and the propagation coordinate x (Bies and Hansen, 2009). Here, m=Inf is. the angular frequency and /is the wave frequency. The propagation constant y is also called the propagation coefficient, which is a complex number and can be expressed as ... 

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