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Macromolecules, Structure 883

d[M 1 ] [M 1 ] r 1 [M 1 ] + [M 2 ] the more reactive comonomer. As a result, the higher the
= , (48)
d[M 2 ] [M 2 ] r 2 [M 2 ] + [M 1 ] monomer conversion the more heterogeneous the product.
This in no way affects the determination of the overall
where M 1 and M 2 represent the two comonomers. The left composition or microstructure of the product but makes
side of this equation gives the ratio of the rates at which the it more difficult to interpret in terms of relative reactiv-
two monomers enter the copolymer, which in turn must ities. It is therefore customary in fundamental studies to
represent the composition of the copolymer being formed limit the conversion to about 5% or less, although drifts in
at any instant. The ratio [M 1 ]/[M 2 ] is the mole ratio of composition can be dealt with mathematically. In copoly-
monomers in the feed. The quantities r 1 and r 2 are the mer production on a practical scale, it is common prac-
reactivity ratios, defined as the ratios of propagation rate tice to achieve greater structural regularity by adjusting
constants: the monomer input as the reaction proceeds. This usually
means withholding the more reactive monomer.
r 1 = k 11 /k 12 , r 2 = k 22 /k 21 .
The detailed discussion of various copolymerization
Here, k 11 is the rate constant for the addition of monomer cases will not concern us here. Traditionally, the determi-
1 to a growing chain ending in a monomer 1 unit; k 12 is the nation of reactivity ratios, which provide important infor-
rate constant for the addition of monomer 2 to the grow- mation concerning the behavior of monomers and grow-
ing chain ending in monomer 1; k 21 and k 22 are the corre- ing chains, has required the determination of the overall
sponding terms for growing chains ending in a monomer comonomer composition of copolymers prepared from a
2 unit. series of feed ratios. Elemental analysis is most commonly
Equation (48) is the copolymer equation in terms of the used. A number of computational and graphic methods
molar concentrations of the monomers. It is usually more are employed to do this. It was realized very early that
convenient to express this relationship in terms of the mole the theoretical treatment that predicts overall composition
fraction in both feed and copolymer. The feed mole ratio also predicts the frequency of occurrence of comonomer
for monomer 1 is given by sequences, but at that time there was no way to observe
and measure these. This can now be readily done by NMR,
[M 1 ]
f 1 = 1 − f 2 = . (49) and the older and cruder methods are giving way to this
[M 1 ] + [M 2 ]
more powerful approach. By NMR it is also possible to
The instantaneous copolymer composition is given by observe copolymer stereochemistry (never considered in
earlierwork)andthepresenceofanomalousunits.Onecan
d[M 1 ]
F 1 = 1 − F 2 = , (50) also more readily detect deviations from the simple model
d[M 1 ] + d[M 2 ]
employed here, in which only the terminal residue of a
from which growing chain determines its reactivity; the effect of the
2 penultimate unit, if any, may be clearly observed. Finally,
r 1 f + f 1 f 2
1
F 1 = 2 2 . (51) it should be noted that by sequence measurements one can
r 1 f + 2 f 1 f 2 + r 2 f
1 2 determine reactivity ratios from only a single copolymer
(A parallel, but redundant, equation expresses F 2 .) provided the feed ratio is known. It may still be desirable
We have stated that these relationships deal with the to observe a range of compositions to assist in resonance
instantaneous composition of the copolymer. Since the assignments but it is not in principle essential.
comonomers generally do not enter the polymer in the For a random copolymerization, dyad, triad, and tetrad
same ratio as in the feed, the latter will drift in composi- sequences may be represented (ignoring stereochemistry)
tion as copolymerization proceeds, becoming depleted in as

Dyads
m 1 m 1 m 1 m 2 (or m 2 m 1 ) m 2 m 2
Triads
m 1 m 1 m 1 m 2 m 2 m 2
m 1 m 1 m 2 (or m 2 m 1 m 1 ) m 1 m 2 m 2 (or m 2 m 2 m 1 )
m 2 m 1 m 2 m 1 m 2 m 1
Tetrads
m 1 m 1 m 1 m 1 m 1 m 1 m 2 m 1 (m 1 m 2 m 1 m 1 ) m 2 m 2 m 2 m 2
m 1 m 1 m 1 m 2 (m 2 m 1 m 1 m 1 ) m 1 m 1 m 2 m 2 (m 2 m 2 m 1 m 1 ) m 2 m 2 m 2 m 1 (m 1 m 2 m 2 m 2 )
m 2 m 1 m 2 m 1 (m 1 m 2 m 1 m 2 )
m 2 m 1 m 1 m 2 m 2 m 1 m 2 m 2 (m 2 m 2 m 1 m 2 ) m 1 m 2 m 2 m 1