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Figure B.1 This is a circuit diagram of a. van der Pol oscillator. The triode compensates for the ohmic losses. keeping the oscillations going without attenuation. Having a pure sinusoidal wave rather than a damped one is vital fol' the transmission of human voice and music. The triode made commercial radio a reality.

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terms in the equation of motion oscillating with the different frequencies -3wo. -woo and 3wo will also be negligible. This is called the rotating-wave approximation. With these two approximations. (8.1) yields the following first-order differential equation for V: (8.3) where 1]' == 31]/4. For simplicity. let us regard V as real. Equation (8.3) has two steady-state solutions. obtained by setting V = O. One is Vs = 0 (Le . the oscillator does not oscillate at all). The other is Vs = Va/1]'. Let us see what happens when V deviates only a little bit from a steady-state solution. Let us take V = Vs + 15 and substitute it in (8.3). only keeping terms up to first order in 15. For the first steady-state solution. V. = O. we find that

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Initiation in activated lactam polymerization may involve contributions from both the imide dimer (XXXVI) and N-acyllactam (XXXIX). However, in the usual case the latter is by far the most important and the former may be ignored. Few detailed kinetic studies of these lactam polymerizations have been achieved, but some of the main reaction characteristics are indicated [Kuskova et al., 1978; Sebenda, 1989; Tani and Konomi, 1966]. The polymerization rate is dependent on the concentrations of base and N-acyllactam, which determine the concentrations of activated monomer and propagating chains, respectively. The degree of polymerization increases with conversion and with increasing concentration of monomer and decreasing N-acyllactam concentration. These characteristics are qualitatively similar to those of living polymerizations, but lactam polymerizations seldom are living. There is a fairly rapid decay of both the activated monomer and propagating chain concentrations due to a complex set of side reactions [Roda et al., 1976; Sebenda and Kouril, 1971, 1972; Sekiguchi, 1984]. These side reactions include hydrogen abstraction from C bonds H a- to amide carbonyls (which result in the formation of various b-ketoimides and b-ketoamides via nucleophilic substitution by the carbanions on lactam end groups of propagating

(8.4)

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chains) and subsequent Claisen-type condensation reactions of the b-ketoimides and b-ketoamides. The latter reaction yields water as a by-product, and water reacts with and decreases the concentrations of both activated monomer and N-acyllactam. Lactam polymerizations (nonassisted as well as assisted) are usually complicated by heterogeneity, usually when polymerization is carried out below the melting point of the polymer [Fries et al., 1987; Karger-Kocsis and Kiss, 1979; Malkin et al., 1982; Roda et al., 1979]. (This is probably the main reason why there are so few reliable kinetic studies of lactam polymerizations.) An initially homogeneous reaction system quickly becomes heterogeneous at low conversion, for example, 10 20% conversion (attained at a reaction time of no more than 1 min) for 2-pyrrolidinone polymerization initiated by potassium t-butoxide and N-benzoyl-2-pyrrolidinone. The (partially) crystalline polymer starts precipitating from solution (which may be molten monomer), and subsequent polymerization occurs at a lower rate as a result of decreased mobility of N-acyllactam propagating species. The MWD is usually broader than the most probable distribution as a result of branching, which occurs in the later stages of reaction. As the monomer and lactam anion concentrations decrease, there is an increasing tendency for the polymeric amide anion XLII to attack the

As J is proportional to 15 with a positive proportionality constant. if Vs = 0 is given even a tiny perturbation 15. be it positive or negative. the tendency will be to deviate even more from Vs = 0 with time. In other words. the steady-state solution Vs = 0 is unstable. For the second steady-state solution. V. = Va/1]'. we find that

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(8.5)

lactam end group of another polymer chain. Further, branching, and to some extent crosslinking, result from the Claisen-type condensations mentioned above. The MWD is considerably broader when an N-acyllactam is not used, since initiation is slow and polymer chains initiated early grow for longer times than those initiated later [Costa et al., 1979].

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We can see from this equation that as the time derivative of 15 is proportional to 15 with a negative constant of proportionality (because a is positive). it is always of opposite sign to 15; the tendency is to restore the oscillator to its Vs = va/1J' steady state.

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