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conventional radical initiator such as AIBN and a transition metal in its higher oxidation state (e.g., Cu2 ) [Wang and Matyjaszewski, 1995b]. The initiator generates radicals which react with Cu2 to establish the equilibrium with RX and Cu . Simultaneous reverse and normal ATRP, using a mixture of radical initiator, RX, Cu2 , and Cu , has also been successful. Reverse ATRP and reverse plus normal ATRP may allow a ner control to obtain high polymerization rate without loss of narrow polydispersity [Gromada and Matyjaszewski, 2001]. Thermal self-initiation is also present for styrene (Sec. 3-4e) in all ATRP processes. ATRP allows the synthesis of well-de ned polymers with molecular weights up to $150,000 200,000. At higher molecular weights normal bimolecular termination becomes signi cant especially at very high conversion and results in a loss of control. There also appears to be slow termination reactions of Cu2 with both the propagating radicals and polymeric halide [Matyjaszewski and Xia, 2001]. 3-15b-3 Complex Kinetics



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where we have used the fact that (o-n)lq = -i lnq and that e3 Aeql can also be written as I:m em m3q" Substituting (3.149) in the last line of (3.155), we finally find that (3.156)





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The kinetics described in Sec. 3-15b-1 apply for conditions in which propagating radical concentrations are suf ciently low that normal bimolecular termination is negligible. This corresponds to a quasi-steady-state condition for propagating radicals. It is not quite a true steady state, but the change in radical concentration over the course of the polymerization is small, suf ciently small to escape detection. This condition also corresponds to a quasisteady-state high concentration of deactivator. To simplify matters, the term steady state is used in subsequent discussions. Theoretical considerations as well as experimental studies show that deviations occur when this steady-state condition is not met [Fischer, 1999; Shipp and Matyjaszewski, 1999, 2000; Souaille and Fischer, 2002; Yoshikawa et al., 2002; Zhang et al., 2001, 2002]. The polymerization rate under non-steady-state conditions, speci cally, conditions far from steady state, is given by

that is, 0- . (pIp) in the new coordinate system coincides with the 0ld37 0-3 :

& '1=3 M 0 3kp K I 0 Cu 0 t2=3 M 2 3kt 3-229

Click and drag the cell s fill handle across or down the number of cells that you want to fill. The changes to .

(3.157)

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Compare Eq. 3-229 with 3-224. The decay in monomer concentration depends on the 13 orders of both initiator and activator initial concentrations with no dependence on deactivator concentration and varies with t2=3 under non-steady-state conditions. For steady-state conditions, there are rst-order dependencies on initiator and activator and inverse rst-order dependence on deactivator and the time dependence is linear. Note that Eq. 3-229 describes the non-steady-state polymerization rate in terms of initial concentrations of initiator and activator. Equation 3-224 describes the steady-state polymerization rate in terms of concentrations at any point in the reaction as long as only short reaction intervals are considered so that concentration changes are small. Whether steady-state or non-steady-state conditions apply depends on the reaction system. Higher concentrations of deactivator decrease normal bimolecular termination by decreasing the concentration of propagating radicals. Steady-state low concentrations of radicals occur when the deactivator/activator ratio is about equal to or greater than 0.1. Non-steady-state conditions occur when the ratio is lower than 0.1. Non-steady-state means non-steady-state conditions for both propagating radicals and deactivator. Under steadystate conditions, both radical and deactivator concentrations are at steady-state, where the radical concentration is lower and the deactivator concentration is higher than for nonsteady-state conditions.

In the new coordinates, the eigenvalue problem is given by (3.158) Now then, for helicity ~ = -Ii, this yields 'l/J~ = -i'l/J1 and 'l/J~ = 0, where 'l/J1' 'l/J~, and 'l/J~ are the components of Analogously, for helicity ~ = Ii, (3.158) yields 'l/J~ = i'l/Ji and 'l/J~ = O. Last but not least, for helicity ~ = 0, (3.158) yields ...., ...., 'l/J1 = 'l/J~ = O. So for ~ = Ii, 'l/J is transverse. But for ~ = 0, 'l/J is longitudinal (i.e., parallel to the momentum). ThiS is forbidden by the transversallty condition (3.123), which in momentum space reads simply 'l/J~ = O. Thus, due to the transversality condition, the photon can have only two helicities ~ = Ii despite having spin I. This is a general consequence of vanishing rest mass, as is proved in the group theory treatment (see the recommended reading at the end of the chapter). In fact, we can see from what was done earlier in this section that when J.L does not vanish, the divergence does not either, allowing ~ = O.

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