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884 Macromolecules, Structure

The dyad probabilities (i.e., frequencies of occurrence) are
given by
[m 1 m 1 ] = F 1 P 11 , (52)

[m 1 m 2 ](or [m 2 m 1 ]) = 2F 1 P 12 = 2F 1 (1 − P 11 )
= 2F 2 P 21 = 2F 2 (1 − P 22 ), (53)
[m 2 m 2 ] = F 2 P 22 . (54)
Here, as we have seen, F 1 and F 2 are the overall mole
fractions of m 1 and m 2 , respectively.
2
[m 1 m 1 m 1 ] = F 1 P , (55)
11
[m 1 m 1 m 2 ](or [m 2 m 1 m 1 ]) = 2F 1 P 11 (1 − P 11 ), (56)
[m 2 m 1 m 2 ] = F 2 (1 − P 22 )(1 − P 11 ).
(57)
The quantity P 11 expresses the conditional probability that
a chain ending in m 1 will add another m 1 , and P 22 ex-
presses the corresponding probability for m 2 . Here, P 12 is FIGURE 27 The 60-MHz proton spectra of (a) polyvinylidene
the probability that a chain ending in m 1 will add m 2 , equal chloride, (b) polyisobutylene, and (c) vinylidene chloride (m 1 ):
isobutylene (m 2 ) copolymer containing 70 mol % m 1 . Peaks are
to the probability that it will not add m 1 (i.e., 1 − P 11 ). Cor-
identified with monomer tetrad sequences (a) m 1 m 1 m 1 m 1 ; (2)
responding to the four rate constants k 11 , k 12 , k 21 , and k 22 m 1 m 1 m 1 m 2 ; (3) m 2 m 1 m 1 m 2 ; (4) m 1 m 1 m 2 m 1 ; (5) m 2 m 1 m 2 m 1 ; (6)
are the four probabilities P 11 , P 12 , P 21 , and P 22 , related by m 1 m 1 m 2 m 2 ; (7) m 2 m 1 m 2 m 2 . [From Hellwege, K. H., Johnsen, U.,
and Kolbe, K. (1966). Kolloid-Z. 214, 45.]
P 11 + P 12 = 1, (58)
P 21 + P 22 = 1, (59) methylene protons (a); the homopolymer of isobutylene
since a growing chain has only two choices. We choose to (which can be prepared with cationic but not with free
employ P 11 and P 22 ; it can be shown that they are given radical initiators) gives singlet resonances of 3:1 inten-
in terms of monomer feed mole fractions and reactivity sity for the methyl and methylene protons. The spectrum
of a copolymer, prepared with a free-radical initiator and
ratios by
containing 70 mol% vinylidene chloride is shown in (c).
r 1 f 1 The methylene resonances are grouped in three chemical
P 11 = , (60)
1 − f 1 (1 − r 1 ) shift ranges: m 1 m 1 -centered peaks at low field; peaks of
methylene protons in
r 2 f 2
P 22 = , (61)
1 − f 2 (1 − r 2 )
Cl CH 3
from which
C CH 2 C
(1 − f 1 )[m 1 m 1 ]
r 1 = , (62) Cl CH 3
f 1 (F 1 − [m 1 m 1 ])
m 1 m 2 -centered units near 3 ppm; and CH 2 and CH 3 reso-
(1 − f 2 )[m 2 m 2 ]
r 2 = . (63) nances of m 2 -centered sequences at high field. It is evident
f 2 (F 2 − [m 2 m 2 ])
that tetrad sequences are involved (assignments given in
Entirely analogous relationships apply to triad and tetrad the figure caption). If only dyad sequences were distin-
sequences. guished, there would be only three methylene resonances
The system vinylidene chloride (m 1 ): isobutylene (m 2 ) corresponding to m 1 m 1 , m 1 m 2 (or m 2 m 1 ), and m 2 m 2 se-
is appropriate to consider since the copolymer has no quences. The upfield isobutylene peaks show considerable
asymmetric carbons and no vicinal J coupling. The proton overlap and assignments here are less certain, but these
NMRspectrumthusconveysonlycompositionalsequence resonances are not required for the analysis.
information. In Fig. 27, 60-MHz proton NMR spectra of From dyad resonances r 1 and r 2 may be calculated. The
the homopolymers (a) and (b) are shown. The homopoly- relative intensity of the m 1 m 1 resonances centered near
mer of vinylidene chloride gives a single resonance for the 3.6 ppm gives [m 1 m 1 ] as 0.426, normalized over all dyad