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Although the derivation above involves the steady-state assumption, the copolymerization equation can also be obtained by a statistical approach without invoking steady-state conditions [Farina, 1990; Gold nger and Kane, 1948; Melville et al., 1947; Tirrell, 1986; Vollmert, n 1973]. We proceed to determine the number-average sequence lengths, "1 and "2 , of monon mers 1 and 2, respectively. "1 is the average number of M1 monomer units that follow each n other consecutively in a sequence uninterrupted by M2 units but bounded on each end of the sequence by M2 units. "2 is the average number of M2 monomer units in a sequence n uninterrupted by M1 units but bounded on each end by M1 units. The transition or conditional probability p11 of forming a M1M1 dyad in the copolymer * * chain is given by the ratio of the rate for M1 adding M1 to the sum of the rates for M1 adding M1 and M2, that is



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We can do the integral in equation (6.10) in the following way: As(O, t) =

p11 R11 R11 R12 6-16

. { dt'

where R11 and R12 are the rates of Reactions 6-2 and 6-3, respectively. Substitution of the expressions for R11 and R12 into Eq. 6-16 yields

~ e(k) . 1r(t')e(k)

p11 r1 r1 M2 = M1 6-17

4q = --2

Similarly, the transition probabilities p12 , p21 , and p22 for forming the dyads, M1M2, M2M1, and M2M2, respectively, are given by

d dt1 1r(t') -JM(t - t 1 ) dt' 4qJM(O) () 2q .( )

p12 p21 p22 R12 M2 R11 R12 r1 M1 M2 R21 M1 R21 R22 r2 M2 M1 R22 r2 M2 R21 R22 r2 M2 M1 6-18 6-19 6-20

(6.11)





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where R21 and R22 are the rates of Reactions 6-4 and 6-5, respectively. * The sum of the transition probabilities of addition to M1 and M* are each, separately, 2 equal to 1:

Select the range of data that you want to chart. You can include any headings and labels, but do not include subtotals or totals.

6-22

where we have introduced the cutoff frequency WM, on the order of mc2 Iii, mentioned in Section 5.2, with JM(r) being a truncated delta function, given by

where N 1 x is the mole fraction of a sequence of M1 units of length x and the summation is over all-sized sequence lengths from 1 to in nity. N 1 x is the probability of forming such a sequence and is given by

(6.12)

N 1 x p11 x 1 p12 6-23

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The first term on the last line of equation (6.11) will lead to the same classical mass renormalization as the sum of the Coulomb self-energies of the particles. Nonetheless, because this is not a relativistic theory, we cannot include contributions from the interactions with the high frequency-modes, so this mass correction turns out to be only 4/3 of the mass correction obtained from the Coulomb self-energy [131]. If we now substitute equations (6.11) and (6.8) in equation (6.4), we obtain the quantum version of the classical Abraham-Lorentz equation [314]: [m .. + 4q2JM(O)] r = 3mc3

The meaning of Eq. 6-23 can be seen by considering, for example, the probability of forming a sequence M1M1M1. The probability of forming such a sequence is the probability p11 of * M1 adding M1 multiplied by the probability p11 of a second addition of M1 multiplied by the probability p12 of addition of M2 or p2 p12. 11

2q2... + 3c3 r -

n "1 p12 1 2p11 3p2 4p3 11 11 6-24

(6.13)

For p11 < 1, which holds in a copolymerization, the expansion series in Eq. 6-24 is 1= 1 p11 2 and Eq. 6-24 becomes

The coefficient mUltiplying the term on the left is the renormalized classical mass of the electron. The first term on the right-hand side is the instantaneous Coulomb force of the proton on the electron. The next term is the radiation reaction force that damps the motion of the electron as it radiates. The last term is absent in the classical

" n1 p12 1 p11 2 1 r1 M1 M2 p12 M2 6-25

2 3 4

theory when there is no incident external radiation on the atom. In the quantum theory, however, it is the electric field operator of the vacuum. The fact that this last term cannot be put equal to zero is the source of most of the differences between the classical and the quantum theory. Next, we will see that the vacuum fluctuations introduced by this term are in fact responsible for the existence of a stable ground state. As an approach based on the quantum Lorentz equation (6.13) presents many difficulties [131l, we will proceed in a different way. The rate of decay of atomic energy is given by

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