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vector out of its original one-dimensional space in the line of real numbers into a new second dimension. Moreover, as 90 0 + 90 0 = 1800 , two consecutive multiplications by x must make this vector rotate by 1800 Thus, x 2 = -1, so that x cannot be a real number but must be the imaginary unit i = A. To rotate our vector by an arbitrary angle, the straightforward approach would be just to calculate the appropriate root of -1. But a much more powerful method which will also strengthen this analogy between algebra and geometry emerges from the following very simple argument. Consider a complex number z forming a vector of length p, making an arbitrary angle () with the line of real numbers as in Figure C.2. From basic trigonometry, it follows that



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public DataSet GetExcel(string fileName) { Application oXL; Workbook oWB; Worksheet ... Value); // get WorkSheet object oSheet = (Microsoft.Office.Interop.

Optically active polymers are rarely encountered. Most syndiotactic polymers are optically inactive since they are achiral. Most isotactic polymers, such as polypropene and poly(methyl methacrylate), are also inactive (Sec. 8-1a-1). Optically active polymers have been obtained in some situations and these are discussed below. 8-14a Optically Active Monomers

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Figure C.2 We can think of any point, z = x + iy, on the complex plane as a vector x 2 + y2 and which fonns an angle from the origin to that point, whose length is p == () == arctan{yfx) with the x-axis.





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Isoselective polymerization of one enantiomer or the other of a pair of enantiomers results in an optically active polymer [Ciardelli, 1987; Del ni et al., 1985; Pino et al., 1963]. For example, polymerization of (S)-3-methyl-1-pentene yields the all-S polymer. The optical activity of the polymer would be maximum for the 100% isotactic polymer. Each racemic placement of the S-monomer decreases the observed optical activity in the polymer.

Now consider the Taylor series of cos () and of sin () around ()

8-75

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There are a few instances of polymer optical activity arising from a chiral conformation. Many isotactic polymers; for instance polypropene, are composed of equimolar amounts of right- and left-handed helices, since the initiator is composed of equal numbers of the two enantiomeric states. This is also the case for polymerization of chloral (trichloroacetaldehyde) in hexane solution at 58 C with an achiral initiator such as lithium t-butoxide. The bulky CCl3 group forces monomer units placed in the meso arrangement in the rst few propagation steps to form a helical conformation; the helical conformation is subsequently propagated throughout the isoselective polymerization. The product is optically inactive since there is an equal probability of propagating via right- and left-handed helices. Polymerization of the bulky monomer chloral yields an optically active product when one uses a chiral initiator, e.g., lithium salts of methyl ( )- or ( )-mandelate or (R)- or (S)octanoate [Corley et al., 1988; Jaycox and Vogl, 1990; Qin et al., 1995; Vogl, 2000]. The chiral initiator forces propagation to proceed to form an excess of one of the two enantiomeric helices. The same driving force has been observed in the polymerization of triphenylmethyl methacrylate at 78 C in toluene by initiating polymerization with a chiral complex formed from an achiral initiator such as n-butyllithium and an optically active amine such as ( )-1-(2-pyrrolidinylmethyl)pyrrolidine [Isobe et al., 2001b; Nakano and Okamoto, 2000; Nakano et al., 2001]. Such polymerizations that proceed in an unsymmetrical manner to form an excess of one enantiomer are referred to as asymmetric polymerizations [Hatada et al., 2002]. Asymmetric polymerization has also been observed in the radical

(C.I)

polymerization of the bulky monomer (1-methylpiperidine-4-yl) diphenylmethyl methacrylate with an achiral initiator if polymerization is carried out in the presence of an optically active additive such as ( )-menthol [Nakano et al., 2001]. ( )-Menthol complexes with the achiral, bulky monomer and induces asymmetric polymerization. 8-14c Enantiomer-Differentiating Polymerization

= 0,

The isoselective polymerization of a racemic mixture of monomers can proceed in two ways depending on initiator, monomer, and reaction conditions. Racemate-forming enantiomerdifferentiating polymerization involves both the R and S monomers polymerizing at the same rate but without any cross-propagation [Hatada et al., 2002]. A racemic monomer mixture polymerizes to a racemic mixture of all-R and all-S polymer molecules [Pino, 1965;

(C.2) (C.3)

8-76 8-77

= () -

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