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Since the amount of light scattered by a polymer solution increases with molecular weight, this method becomes more accurate for higher polymer molecular weights. There is no upper limit to the molecular weight that can be accurately measured except the limit imposed by insolubility of the polymer. The lower limit of M w by the light scattering method is close to 5000 10,000. Below this molecular weight, the amount of scattered light is too small to measure accurately. 3. Solution viscosity is also useful for molecular-weight measurements. Viscosity, like light scattering, is greater for the larger-sized polymer molecules than for smaller ones. However, solution viscosity does not measure M w since the exact dependence of solution viscosity on molecular weight is not exactly the same as light scattering. Solution viscosity measures the viscosity-average molecular weight M v de ned by



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Substituting (2.76) into (2.75), then mUltiplying the resulting equation by (1/21r) exp( -ikx) and integrating over x along the entire infinite string, we find that 2 2 e- ikxo

1-22

y" + v k Y.:

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where a is a constant. The viscosity- and weight-average molecular weights are equal when a is unity. M v is less than M w for most polymers, since a is usually in the range 0.5 0.9. However, M v is much closer to M w than M n , usually within 20% of M w . The value of a is dependent on the hydrodynamic volume of the polymer, the effective volume of the solvated polymer molecule in solution, and varies with polymer, solvent, and temperature. More than one average molecular weight is required to reasonably characterize a polymer sample. There is no such need for a monodisperse product (i.e., one composed of molecules whose molecular weights are all the same) for which all three average molecular weights are the same. The situation is quite different for a polydisperse polymer where all three molecular weights are different if the constant a in Eq. 1-22 is less than unity, as is the usual case. A careful consideration of Eqs. 1-15 through 1-22 shows that the number-, viscosity-, and weight-average molecular weights, in that order, are increasingly biased toward the higher-molecular-weight fractions in a polymer sample. For a polydisperse polymer

= ~F,,(t),

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Fig. 3-2 First-order dependence of the polymerization rate Rp of methyl methacrylate on the monomer concentration [M]. The initiator is the t-butyl perbenzoate-diphenylthiourea redox system. After Sugimura and Minoura [1966] (by permission of Wiley-Interscience, New York).

(2.77)

is discussed in Sec. 3-4f.) The equivalent result arises if the second step of the initiation reaction (Eq. 3-14) were to become the rate-determining step instead of the rst step (Eq. 3-13). This occurs when kd > ki or when [M] is low. It is also frequently encountered in polymerizations initiated photolytically or by ionizing radiation (Secs. 3-4c and 3-4d) and in some redox-initiated polymerizations (Sec. 3-4b). In some systems it appears that the initiation step differs from the usual two-step sequence of Eqs. 3-13 and 3-14. Thus in the t-butyl hydroperoxide-styrene system only a minor part of the initiation occurs by the rst-order homolysis reaction (Eq. 3-26f), which accounts for the complete decomposition of t-butyl hydroperoxide in the absence of styrene. Homolysis of the hydroperoxide occurs at a much faster rate in the presence of styrene than in its absence. The increased decomposition rate in the t-butyl hydroperoxide styrene system occurs by a molecule-induced homolysis reaction which is rst-order in both styrene and hydroperoxide [Walling and Heaton, 1965]. The initiation reaction may be written as

3-35

19This "trick," known as regularization, is used again in this chapter, in the last section, when we compute the Casimir force between two parallel plates.

and results in a 3-order dependence of Rp on [M]. This initiation is probably best considered 2 as an example of redox initiation (Sec. 3-4b). Other exceptions to the rst-order dependence of the polymerization rate on the monomer concentration occur when termination is not by bimolecular reaction of propagating radicals. Second-order dependence of Rp on [M] occurs for primary termination (Eq. 3-33a) and certain redox-initiated polymerizations (Sec. 3-4b-2). Less than rst-order dependence of Rp on [M] has been observed for polymerizations (Sec. 9-8a-2) taking place inside a solid under conditions where monomer diffusion into the solid is slower than the normal propagation rate [Odian et al., 1980] and also in some redox polymerizations (Sec. 3-4b-2) [MapundaVlckova and Barton, 1978].

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