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Radical polymerizations involve free propagating species. The planar or nearly planar trigonal carbon atom does not have a speci ed con guration since there is free rotation about the terminal carbon carbon bond (as indicated in Eq. 8-25 by the circular arrows). The con guration of a monomer unit in the polymer chain is not determined during its addition to the radical center but only after the next monomer unit adds to it. The situation can be depicted in Eq. 8-25 where the two placements syndiotactic (Eq. 8-25a) and isotactic (Eq. 8-25b) take place with the rate constants kr and km , respectively. Whether the same placement is propagated through successive additions of monomer units determines the stereoregularity of the nal polymer molecule. The amount and type of stereoregularity are determined by the value of kr =km . Isotactic polymer is produced if this ratio is zero, syndiotactic polymer if it is in nity, and atactic polymer if it is unity. For kr =km values between unity and in nity, the polymer is partially atactic and partially syndiotactic. For kr =km values between zero and unity, the polymer is partially atactic and partially isotactic.

The value of kr =km is determined by the difference Gz in the free energies of activation between the syndiotactic Gz and isotactic Gz placements r m

A Statistical Approach to Neural Networks f o r Pattern Recognitton by Robert A . Dunne Copyright @ 2007 John Wiley & Sons, Inc.

& ' kr Gz Sz H z exp exp km RT R RT 8-26

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z where Gz , Hr , Sz respectively are the activation free energy, enthalpy, and entropy for r r z syndiotactic placement and Gz , Hm , Sz are the corresponding quantities for isotactic m m placement. Stereoregularity should be temperature-dependent with ordered structures favored at low temperatures [Huggins, 1944]. Calculations based on small molecules indicated that the differences in activation enthalpy H z and entropy Sz between syndiotactic and isotactic placements are both expected to be small; H z is about 4 to 8 kJ mol 1 and Sz about 0 to 4 J mol 1 K 1 [Fordham, 1959]. Syndiotactic placement is favored over isotactic placement primarily because of the enthalpy difference. The small energy differences between syndiotactic and isotactic placements have been con rmed for several monomers by studying the temperature dependence of tacticity [Bovey, 1972; Pino and Suter, 1976, 1977]. Thus, H z and Sz are 4.5 kJ mol 1 and 4.2 J mol 1 K 1 , respectively, for methyl methacrylate and 1.3 kJ mol 1 and 2.5 J mol 1 K 1 , respectively, for vinyl chloride [Bovey et al., 1967; Fox and Schnecko, 1962]. The slight preference for syndiotactic placement over isotactic placement is a consequence of steric and/or electrostatic repulsions between substituents in the polymer chain. Repulsions between R groups, more speci cally, between R groups on the terminal and penultimate units of the propagating chain, are minimized in the transition state of the propagation step (and also in the nal polymer molecule) when they are located in the alternating arrangement of syndiotactic placement. The steric and electrostatic repulsions between R groups are maximum for isotactic placement. This mechanism (driving force) for syndioselective polymerization is referred to as polymer chain end control.

2

Fig. 8-9 Dependence of syndiotacticity on temperature for the radical polymerization of vinyl chloride. After Talamini and Vidotto [1967] (by permission of Huthing and Wepf Verlag, Basel and Wiley-VCH, Weinheim).

The difference in activation free energies for syndiotactic and isotactic placements leads to an increasing tendency toward syndiotacticity with decreasing polymerization temperature Figure 8-9 shows the increase in the fraction of syndiotactic dyads, (r), of poly(vinyl chloride) from 051 to 067 as the reaction temperature decreases from 120 to 78 C [Talamini and Vidotto, 1967] Corresponding data for methyl methacrylate show an increase in (r) from 064 at 250 C to 087 at 78 C [Fox and Schnecko, 1962; Isobe et al, 2000; Otsu et al, 1966] With decreasing temperature, the energy difference between syndiotactic and isotactic placements exerts a progressively increasing in uence on the stereoselectivity of the polymerization; at high temperatures, their effects are progressively diminished Since radical polymerizations are generally carried out at moderately high temperatures, most of the resulting polymers are highly atactic.

Now as 1x1, the conditional variance of the class labels does not depend on the inputs to the network, we only need to minimize the first term. I t can be seen that this term is minimized when the network gives a least squares estimate of the expected value of the targets, conditional on the observations X. If we consider target values with a one of & encoding (Section 2.1, p. 9) then,

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