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So to obtain the functional derivative in equation (5.102), we have to calculate the change in the Hamiltonian (5.98) when A. varies by oA.

5-134

d k (A + eoA) (211')3/2



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Function SUM INT ROUND ROUNDDOWN COUNT AVERAGE MIN MAX MEDIAN PMT RATE TODAY IF Category Math & Trig Math & Trig Math & Trig Math & Trig Statistical Statistical Statistical Statistical Statistical Financial Financial Description Adds values Rounds down to the nearest integer Rounds a number to a specified number of digits Rounds a number down Counts the number of cells in a range that contain data Averages a series of arguments Returns the smallest value in a series Returns the largest value in a series Returns the middle value in a series Finds the periodic payment for a fixed loan Returns an interest rate Syntax =SUM(number1,number2,...) =INT(number) =ROUND(number,number_digits) =ROUNDDOWN(number,number_digits) =COUNT(value1,value2,...)





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Cationic polymerization of acrolein has been reported to yield polymers containing both types of repeating units, although one would not expect the alkene group to be reactive toward cationic initiation. Dimethylketene has been polymerized by anionic initiators. Three different repeat units are formed (XLVII, XLVIII, XLIX) corresponding to polymerization through the alkene

Ok e~ 'r",

double bond, carbonyl double bond, and the two double bonds in an alternating manner, respectively [Pregaglia et al., 1963; Yamashita and Nunamoto, 1962]. Polymerization

+ ~ Jd 3 k {II*. II + c2 k 2 A* (A+ eoA)} - H 411'

through the more polar carbonyl group to form the polyacetal is favored by solvents of high polarity. Polymerization of the less polar alkene group, either completely to form the polyketone or partially to form the polyester, is favored by nonpolar solvents. Polyketone formation is favored by Li, Mg, and Al counterions, and polyester by Na and K. 5-7b Hydrogen-Transfer Polymerization of Acrylamide

The polymerization of acrylamide (or methacrylamide) by strong bases such as sodium, organolithium compounds, and alkoxides yields a polymer structure

(5.105)

5-135

.7l.. (k)

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which is quite unexpected [Bush and Breslow, 1968; Camino et al., 1970, 1977; Otsu et al., 1976]. The polyamide structure of the polymer has been con rmed by the hydrolysis of the polymer to 3-aminopropanoic acid. The polymerization mechanism involves addition of the anionic initiator to the carbon carbon double bond of the monomer

iI:(k, t) =

5-136

.71 ,.(k, t) =

=AVERAGE(number1,number2,...) =MIN(number1,number2,...) =MAX(number1,number2,...) =MEDIAN(number1,number2,...) =PMT(interest_rate,number_of_ periods,present_value,future_value,type) =RATE(number_of_periods,payment, present_value,future_value,type,guess) =TODAY() =IF(logical_text,value_if_true,value_if_false)

followed by proton abstraction from another monomer molecule to form the amide anion (L)

(5.107)

5-137

_k 2 A(k, t),

The amide anion initiates polymerization via a similar reaction sequence by addition of L to monomer. Propagation follows in a like manner:

(5.108)

(n + 1)

5-138

we find that equation (5.107) is just the complex conjugate of the Maxwell equation for the time derivative of the transverse component of the electric field in reciprocal space (5.80). The other Hamilton equations yield the following: the equations of motion of r", (5.99) and of A (5.102) only recover the definitions (5.96) and (5.97) of the conjugate momenta p", and II, respectively. The remaining Hamilton equation (5.100), however, yields (5.109) Substituting equation (5.99) or (5.96) and its total time derivative in equation (5.109), we obtain

This polymerization follows a path that is quite different than any of the previously discussed polymerizations. The propagating center is not an ion or a radical but is the carbon carbon double bond at one end of the propagating species LI. In addition, it is not the monomer but

m",r""j = q",EU,j(r"" t)

the monomer anion L that adds to the propagating center. The evidence for this polymerization mechanism comes from analysis for the alkene end group. This mechanism is very similar to that for the anionic ring-opening polymerization of cyclic amides (Sec. 7-3c). By analogy to the latter, the monomer anion can be referred to as activated monomer. The overall polymerization reaction is often referred to as a hydrogen-transfer polymerization. Polymerization of acrylamide does not proceed entirely through the hydrogen-transfer route under all reaction conditions. Polymers with mixed structures, resulting from concurrent hydrogen-transfer polymerization and normal polymerization through the carbon carbon double bond, occur in some cases. The relative extents of the two modes of propagation are affected by the initiator, solvent, monomer concentration, and temperature [Imanishi, 1989]. The polymerization of N,N-dimethylacrylamide proceeds exclusively through the carbon carbon double bond because there are no amide hydrogens for transfer [Kobayashi et al., 2000; Nakhmanovich et al., 2001]. 5-7c Polymerization and Cyclotrimerization of Isocryanates

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