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The equilibrium betweem dormant covalent species and active ion pairs and free ions is driven further toward the dormant species by the common ion effect by adding a salt that supplies the same counterion as supplied by the Lewis acid Free ions are absent in most systems; most of the species present are dormant covalent species with much smaller amounts of active ion pairs Further, the components of the reaction system are chosen so that there is a dynamic fast equilibrium between active and dormant species, as the rates of deactivation and activation are faster than the propagation and transfer rates The overall result is a slower but more controlled reaction with the important features of living polymerization (Sec 3-15).



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'8a t (k').

In addition to the choice of Lewis acid, added common ion salt, and temperature, the fast equilibrium between active and dormant species can be fostered by including additional nucleophiles (separate from the nucleophilic counterion) in the reaction system and by variations in solvent polarity Nucleophiles act by further driving of the dynamic equilibrium toward the covalent species and/or decreasing the reactivity of ion pairs Nucleophilic counterions and added nucleophiles work best in nonpolar solvents such as toluene and hexane Their action in polar solvents is weaker because the polar solvents interact with the nucleophiles and nucleophilic counterions, as well as the ion pairs Polar solvents such as methylene.





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(9.38) Now, if the perfect intracavity and free-space modes that we have used in expansions (9.24), (9.25), (9.30), and (9.31) were together able to describe any physically attainable configuration of the fields, an, b( k), and their Hermitian conjugates would span the space of the global operators a( k) and at (k) completely. In other words, we should have

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chloride are not conducive to LCP because the equilibrium between active and dormant species is driven more toward the active species, which leads to faster and less controlled polymerization. Added salts are useful in polar solvents for driving the equilibrium back the other way, back to the dormant covalent species. It is not an absolute necessity for LCP to have no free ions. If free ions are present, LCP is possible only if there are fast equilibria between free ions, ion pairs, and covalent species. If the equilibrium between free ions and ion pairs is slow, the result is a bimodal distribution. Further, to have any possibility of LCP with free ions present, the concentration and reactivity of the free ions should not be such that the reaction is too fast. The speci c combination of initiator, coinitiator, solvent temperature, and additives needed to achieve LCP varies with the monomer. The components of the reaction system need to be ne-tuned differently for different monomers (Sec. 5-2g-3). 5-2g-2 Rate and Degree of Polymerization

loco dk' {,BI(k, k')b(k') + ~(k, k')bt(k')} ,

5-54

(9.39)

in the absence of free ions and for the case where initiation is very fast (i.e., the concentration of propagating ion pairs equals the concentration of ion pairs derived from initiator and coinitiator). The concentration of ion pairs is rarely directly determined, but one can relate it to the concentration of initiator (ZY) and coinitiator (I). Two equilibria are involved

Right-click a blank area of the desktop and click New and then Shortcut. The Create Shortcut dialog box appears.

Y+(IZ )

where, as in the dressed operator version of Fano diagonalization [37, 38, 306, 307],

5-55

anI (k) = [a(k), a~]

Y IZ K1 K2 I ZY 5-56

= _1_ (ksin([k-kn]L)e-ikL *(k)

5-57

..;;LV k;.

Some living polymerization show a greater than rst-order dependence of Rp on [M] indicating that the initiation rate is slow and dependent on monomer. The number-average degree of polymerization for a living cationic polymerization is de ned as the concentration of monomer consumed divided by the total concentration of all propagating chains (dormant and active)

~ --- --:i-----

Xn p M 0 ZY 0 5-58

:-112-:

where [M]0 and [ZY]0 are initial concentrations of monomer and initiator, respectively, and p is the fractional conversion of monomer at any time in the reaction. The number-average degree of polymerization is M 0 / ZY 0 at complete conversion. Equation 5-58 is the LCP analog of Eq. 3-225 for ATRP living polymerization. The PDI for LCP is given by the

3 4 5

Figure 4.5 Schematic representation of a coherent state. The noise is represented by a circle, because any quadrature has the same variance of 1/4.

Poisson distribution [Flory, 1940; Peebles 1971; Szwarc, 1968], the same for ATRP and other living polymerizations

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