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Theoretical considerations indicate that kc would be very large, about 8 109 L mol 1 s 1 , in low-viscosity media (such as bulk monomer) for the reaction between two radicals. The rate constants for reactions of small radicals (e.g., methyl, ethyl, propyl) are close to this value (being about 2 109 L mol s 1 ) [Ingold, 1973]. Experimentally determined kt values for radical polymerizations, however, are considerably lower, usually by two orders of magnitude or more (see Table 3-11). Thus diffusion is the rate-determining process for termination, kc ) k4 , and one obtains

J.L24 ,



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Rt k1 k3 M 2 k2 k3 3-182

In the Find what field, type the text that you want to find. Type the replacement text in the Replace with field. Click Find Next.

(3.74) (3.75)

by assuming steady-state concentrations of both XXXVIII and XXXIX. Two limiting cases of termination arise. For the case of slow translational diffusion, k3 ) k2 , and

Rt k1 M 2 3-183

They also share the properties (3.44) and (3.45) of F and A, as you can easily check for yourself (see Problem 3.2):

Rt k1 k3 M 2 k2 3-184

4 t = -4





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Thus the experimentally observed termination rate constant kt corresponds to k1 and k1 k3 =k2 , respectively, for the two limiting situations. Segmental diffusion and translational diffusion are expected to be affected differently with conversion [Dionisio and O Driscoll, 1980; Mahabadi and O Driscoll, 1977a,b; Mahabadi and Rudin, 1979]. With increasing conversion the polymerization medium becomes a poorer solvent because of the increased polymer concentration. The size of the randomly coiled up propagating radical in solution (referred to as coil) becomes smaller and there is an effective higher concentration gradient across the coil. Segmental diffusion of the radical end out of the coil to encounter another radical is increased. Simultaneously, the increasing polymer concentration decreases translational diffusion as the reaction medium becomes more viscous and, at suf ciently high concentrations, the polymer radicals become more crowded and entangled with each other. Chain entanglement leads to a faster decrease in translational diffusion relative to the decrease with increasing viscosity. Stage IA behavior corresponds to the situation observed for many monomers where the increase in segmental diffusion is apparently exactly counterbalanced by the decrease in translational diffusion (i.e., kt remains constant). When the initial increase in segmental diffusion is greater than

(3.76)

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the decrease in translational diffusion, kt increases and the polymerization rate decreases (stage IB) Moderate stage IB behavior has been observed in several polymerizations (styrene, methyl methacrylate [Abuin et al, 1978; High et al, 1979]) At this time, no system has been observed with more than a moderate decrease in Rp = M I 1=2 with conversion At some point, translational diffusion decreases faster than the increase in segmental diffusion and rapid autoacceleration occurs (stage II): the gel effect Increased conversion results in increased polymer chain entanglements because of two effects: increased viscosity and increased polymer chain length [Lachinov, 1979] The quantitiative effects of polymer chain length on kt (and also kp ) have been discussed [Beuermann, 2002; Beuermann and Buback, 2002; Buback et al, 2000, 2002; O Shaughnessy and Yu, 1994, 1998; Smith et al, 2003] Termination becomes increasingly slower.

But Wand 4 transform in a different way from F and A. If they transformed in the same way as the fields, w t W + J.L24 t 4 would not be a four-vector, just as Ft F + J.L2 At A is not. Cook [120], Hammer and Good [268], and Nelson and Good [463] worked out how they transform from one Lorentz frame to another when J.L -+ o. The general case is discussed by Weaver et al. [624]. The next condition that our four-vector candidate to probability four-current must satisfy is the continuity equation. As we have mentioned before, the continuity

Although propagation is also hindered, the effect is much smaller, since kp values are smaller than kt values by a factor of 104 105 Termination involves the reaction of two large polymer radicals, while propagation involves the reaction of small monomer molecules and only one large radical High viscosity affects 1=2 the former much more than the latter Therefore, the quantity kp =kt increases and the result in accordance with Eq 3-25 is an increase in Rp with conversion A second consequence of this effect is an increase in molecular weight with conversion as required by Eq 3-95 These conclusions have been veri ed by the quantitative evaluation of the kp and kt values as a function of the percent conversion Thus, Table 3-17 shows data on the polymerization of methyl methacrylate [Hayden and Melville, 1960].

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