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weight of a polymer sample as well as the total number of initiator fragments contained in that sample. Termination by coupling results in two initiator fragments per polymer molecule, while disproportionation results in one initiator fragment per polymer molecule. The fractions of propagating chains, a and (1 a), respectively, which undergo termination by coupling and disproportionation can be related to b, the average number of initiator fragments per polymer molecule. For a reaction system composed of n propagating chains, coupling yields an initiator fragments and an/2 polymer molecules, while disproportionation yields 1 a n initiator fragments and 1 a n polymer molecules. The average number of initiator fragments per polymer molecule, de ned as the total initiator fragments divided by the total number of polymer molecules, is given as

Hence the magnitude of the influence of the position in feature space is bounded, in cases 1 and 2 above, by lx I This means that the y!~function for the bias term is 4 bounded by kl 5 1/4 the bias input is always 1, while for case 2, the Icz constant depends on the value of zpr which is unbounded. Hence we conclude that4

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Equations 3-96 through 3-98 show a signi cant characteristic of radical chain polymerizations. The kinetic chain length is inversely dependent on the radical concentration or the polymerization rate. This is of great practical signi cance any attempt to increase the polymerization rate by increasing the radical concentration comes at the expense of producing smaller polymer molecules. The kinetic chain length at constant polymerization rate is a characteristic of the particular monomer and independent of the method of initiation. Thus, for any monomer, n is independent of whether the polymerization is initiated by thermal, redox, or photochemical means, whether initiators are used, or of the particular initiator used, if [R ] or Rp is the same. The general relationship between the degree of polymerization and the kinetic chain length is

Although experimental data are not available for all monomers, most polymer radicals appear to terminate predominately or entirely by coupling (except where chain transfer predominates). Studies with small, aliphatic radicals clearly predict this tendency [Tedder, 1974]. However, varying extents of disproportionation are observed depending on the reaction system. Disproportionation increases when the propagating radical is sterically hindered or has many b-hydrogens available for transfer. Thus, whereas styrene, methyl acrylate, and acrylonitrile undergo termination almost exclusively by coupling, methyl methacrylate undergoes termination by a combination of coupling and disproportionation [Ayrey et al., 1977; Bamford, 1988; Bonta et al., 1976; Hensley et al., 1995]. Increased temperature increases the extent of disproportionation, with the most signi cant effect for sterically hindered radicals. The extent of disproportionation in methyl methacrylate increases from 67% at 25 C to 80% at 80 C [Bamford et al.,1969b]. There appears to be a tendency toward disproportionation for highly reactive radicals such as those in vinyl acetate and ethylene

polymerizations. However, the effect is relatively small as the extent of disproportionation in these two polymerizations is no greater than 10% at most [Bamford et al., 1969a,b].

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where IklJ 5 m/4 for W O ,

In many polymerization systems the polymer molecular weight is observed to be lower than predicted on the basis of the experimentally observed extents of termination by coupling and disproportionation. This effect is due to the premature termination of a growing polymer by the transfer of a hydrogen or other atom or species to it from some compound present in the system the monomer, initiator, or solvent, as the case may be. These radical displacement reactions are termed chain-transfer reactions and may be depicted as

3-102

where XA may be monomer, initiator, solvent, or other substance and X is the atom or species transferred. Chain transfer to initiator was referred to earlier as induced initiator decomposition (Sec. 3-4g-1). The rate of a chain-transfer reaction is given by

(8.6)

Rtr ktr M XA 3-103

5

where ktr is the chain-transfer rate constant. Chain transfer results in the production of a new radical A , which then reinitiates polymerization

3-104

for some constants k l and kz which depend on the path e. For the single perceptron = 1 while for MLP models it will take other values (see below). We see that the IC for a simple perceptron displays a variety of behaviors and, except in the case = 1, the single perceptron is not robust as the IC is unbounded. 8.5.2

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