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transfer to counterioin is zero-order in monomer. Chain transfer to monomer is usually the dominant termination reaction compared to chain transfer to counterion. Chain transfer of a b-proton to other basic substances in the reaction mixture is also possible. The various b-proton transfer reactions limit polymer molecular weight, but do not terminate the kinetic chain.

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5-2c-2

= coshr, = sinh r,

HMnM +(IZ)

(4.80) (4.81)

5-20

(4.82)

occurs, for example, in the tri uoroacetic acid initiated polymerization of styrene [Throssell et al., 1956]



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Now, we notice that the minimum uncertainty squeezed state 1,6}).., which we obtain when S).. is applied to a coherent state 1,8}, is a right-hand-side eigenstate of the operator h; that is,

H CH2CH CH2CH (OCOCF3)

b)..I,8})..

5-21

S)..aS! (S)..I,8}) ,8S)..I,6}

Alternately, the propagating ion may combine with an anionic fragment from the counterion, for example

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(4.83)

5-22

5-23

= S)..al,6}





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Termination by combination differs from the other modes of termination in that the kinetic chain is usually terminated, since the concentration of the initiator coinitiator complex decreases. Equations 5-22 and 5-23 indicate the complexity of cationic polymerization even when seemingly similar initiators such as BCl3 and BF3 are used. Termination in the BCl3 -initiated polymerizations of isobutylene and styrene occurs almost exclusively by combination with chloride [Kennedy and Feinberg, 1978; Kennedy et al., 1977]. For BF3 -initiated polymerization, chain transfer to monomer is the major mode of chain breaking with a minor contribution by combination with OH. The differences are explained by the order of bond strengths: B > B > B [Jolly, 1984]. A similar situation is found in the polymerization of F O Cl styrene by trityl salts. Polymerization occurs readily when the counterion is SbF but poorly 6 with the corresponding hexachloroantimonate counterion [Johnson and Pearce, 1976]. Chloride ion easily transfers from the counterion to terminate the propagating center, while uoride is inactive toward transfer. Combination with counterion is also important when aluminum alkyl alkyl halideinitiating systems are used [DiMaina et al., 1977; Kennedy, 1976; Reibel et al., 1979].

= = 1,6})..,8

CH2 C(CH3)2(R3AlCl)

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This analogy with coherent states can be taken even further. Consider bib)... As [h, bi] = 1, bib).. behaves as a generalized number operator and b).., as generalized annihilation and creation operators. The eigenstates of b).. are generalized number states, and 1,8}).. has an expansion in terms of these generalized number states which is formally identical to that of coherent states in terms of number states, with a Poissonian distribution of generalized quanta. This particular type of squeezed state, which is also a minimum uncertainty state, has been named a two-photon coherent state by Yuen [660]. The coherent-state analogy mentioned above explains why this type of squeezed state can also be seen as a sort of generalized coherent state. The reason that this generalization can be regarded as a two-photon generalization, with ordinary coherent states being onephoton coherent states, has to do with the form of the squeezing operator. If we compare S).. and D(a), we see that S).. is D(a) with a -+ (1/2) inA and 0,-+ 0,2. So while the argument of the exponential in D(a) is linear in a and at, in S).. it is quadratic. In 5 when we treat the interaction with matter, you will realize that S).. and D( a) can be seen as time evolution operators associated with interaction Hamiltonians between a single mode of the radiation field and a classical currentY In the case of D(a), the interaction is linear, describing a one-photon transition. In the case of S).., the interaction is nonlinear (quadratic), describing a two-photon transition.

5-24

9This might seem a bit conlradictory at first sight, but such a "classical" two-photon current can actually be realized using the nonlinear dielectric permittivity of certain crystals. This method was in fact used in the first demonstration of squeezing in a parametric oscillator 1569].

or hydridation:

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