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MLPs are in the same category as least squares linear regression in that they make no assumptions about the underlying error distribution. They simply minimize an objective error function, p, that is chosen as being sensible, and produce a mapping parameterized by a finite set of weights. Thus, while they are distributionfree, they are parametric procedures in the sense that a finite number of parameters are estimated during the fitting (or learning) process. Now if we make distributional assumptions, then the objective function p may be recovered via a likelihood equation. In this case our procedure, either linear regression or MLP modeling, may be derived as a maximum likelihood (ML) estimator. It is well known, for example, that least squares is not a robust method. However, the robustness or otherwise of the MLP is not immediately obvious; this question will be pursued in this chapter.

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range. The Ep value is 25, 47, 61, 49, 75, and 86 kJ mol 1 for epichlorohydrin, oxetane, THF, 1,3-dioxolane, oxepane, and 1,3-dioxepane, respectively [Chien et al., 1988; Dreyfuss and Dreyfuss, 1976; Sims, 1966]. The ERp value is 72 kJ mol 1 for 3,3-bis(chloromethyl)oxetane [Chapiro and Penczek, 1962; Penczek and Penczek, 1963]. As with ionic polymerizations of alkenes, the activation energies can vary considerably with reaction conditions. Thus ERp for 3,3-bis(chloromethyl)oxetane is only 12 16 kJ mol 1 for radiation-initiated polymerization in the crystalline state. The lower activation energy may be a consequence of a favorable orientation of the monomer for propagation in the crystalline state. The effect of temperature on the degree of polymerization is more complex. For most polymerizations, increasing the temperature decreases the polymer molecular weight as a result of increases in the rates of transfer and termination relative to propagation. Table 7-2 shows this effect for the polymerization of oxetane by boron tri uoride [Rose, 1956]. (The intrinsic viscosity is a measure of polymer molecular weight.) The decrease in polymer molecular weight in this polymerization is due primarily to an increase in intramolecular chain transfer to polymer as indicated by the corresponding increase in the yield of cyclic tetramer with increasing temperature. In other polymerizations the termination and transfer reactions may not be appreciably affected by increasing temperature, while the propagation reaction rate increases. The result is an increase in polymer molecular weight with increasing temperature. Figure 7-3 shows a THF polymerization in which the polymer molecular weight increases initially with increasing temperature up to about 5 C and then

Fig. 7-3 Effect of temperature on polymer molecular weight in the BF3 polymerization of tetrahydrofuran. After Burrows and Crowe [1962] (by permission of Wiley-Interscience, New York).

ln 1 ! ln 1 p kd t ln 1 p1 2 3-94

The value of kd is then found from the slope of a plot of the left side of Eq. 3-94 versus time. Figure 3-4 shows the plot for the AIBN-initiated polymerization of isoprene at three temperatures [Gobran et al., 1960]. Since kd is determined, f can be obtained from either Eq. 3-32 or 1=2 3-93 if the ratio kp =kt is known from other studies.

A brief discussion of robust statistics is given here. A full discussion is given in Hampel et al. (1986), Huber (1981) and Staudte and Sheather (1990). More introductory readings can be found in Goodall (1983) and Hogg (1979). Following Huber (1981), a functional T on a space is Gdteaux differentiable at if there is a function such that VG E D, T((1 - t)F t-0 lim t = /a(z)dc(s).

Fig. 3-4 Dead-end polymerization of isoprene initiated by azobisisobutyronitrile. After Gobran et al. [1960] (by permission of Wiley-Interscience, New York).

2

Equation 3-92 is also useful for determining the time needed to reach different extents of conversion for actual polymerization systems where both [M] and [I] decrease with time. 3-4h Other Aspects of Initiation

There are two possible modes of addition of a primary radical from initiator to the double bond of a monomer: head and tail additions, corresponding to Eqs 3-8 and 3-9, respectively Tail addition is favored over head addition for the same reasons head-to-tail propagation is favored over head-to-head propagation (Sec 3-2a) This expectation is generally veri ed in the few systems which have been experimentally examined [Bevington, 1988; Moad et al, 1982, 1984; Solomon and Moad, 1987] Tail addition occurs exclusively in the AIBN and tbutyl peroxide initiated polymerizations of styrene and the AIBN-initiated polymerization of vinyl chloride; there is no detectable head addition The ratio of tail : head additions varies from 20 : 1 to 13 : 1, depending on the reaction conditions, for the benzoyl- peroxide-initiated polymerization of styrene.

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