# Polymer Chemistry. The Basic Concepts - Himenz P.C.

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(7 .A) becomes

kin

-MjMj' + Mi--------> -MjMjMi-

(7.E)

^211

-+ -----*¦ -

(7.F)

and Eq. (7.1) becomes

Rp.m = k.nlM.lMlM,]

(7.5)

Rp ,211 " k211 [M2 Mj •] [Mj ]

(7.6)

when the effect of the next-to-last, or penultimate, unit is considered.

For now we shall restrict ourselves to the simpler case where only the

terminal unit determines behavior, although systems in which the

penultimate effect is important are well known.

426

Polymers with Microstructure

It is the magnitude of the various ê values in Eqs. (7.1)-(7.4) that

describes the intrinsic kinetic differences between the various modes of

addition, and the k's plus the concentrations of the different species

determine the rates at which the four kinds of additions occur. It is the

proportion of different steps which determines the composition of the

copolymer produced.

Monomer Mi is converted to polymer by reactions (7.A)and (7.C);

therefore the rate at which this occurs is the sum of Rp n and Rp 21:

Likewise, reactions (7.B) and (7.D) convert M2 to polymer, and the rate

at which this occurs is the sum of Rp 12 and Rp 22 :

The ratio of Eqs. (7.7) and (7.8) gives the relative rates of the two

monomer additions and, hence, the ratio of the two kinds of repeat units

in the copolymer:

We saw in the last chapter that the stationary-state approximation is

applicable to free-radical homopolymerizations, and the same is true of

copolymerizations. Of course, it takes a brief time for the stationary-

state radical concentration to be reached, but this period is

insignificant compared to the total duration of a polymerization

reaction. If the total concentration of radicals is constant, this means

that the rate of crossover between the different types of terminal units

is also equal, or that Rp 21 = Rp 12 :

Combining Eqs. (7.9) and (7.11) yields the important copolymer

composition equation:

(7.7)

(7.8)

d[Mj _ knlMrHMj +k21[M2-] [MJ d[M2] ê12[Ìã][Ì2] + k22 [M2 •] [M2]

(7.9)

k12[Mr][M2] = ê^Ìà-ÈÌË

(7.10)

or

[M,] = kai[M.] [MH k12[M2]

(7.11)

d[Mi] = [MJ (k j j /ê i2) [M t ] + [M2] d[M2] [M2] (k22/k21

)[M2 ] + [MJ

(7.12)

Copolymer Composition

427

Although there are a total of four different rate constants for

propagation, Eq. (7.12) shows that the relationship between the relative

amounts of the two monomers incorporated into the polymer and the

composition of the monomer feedstock involves only two ratios of

different pairs of these constants. Accordingly, we simplify the notation

by defining

r.

(7.13)

ê i2

and

k2o

r2 = -i!

(7.14)

k2i

With these substitutions, Eq. (7.12) becomes

d[Mj] _ [MJ rJMj+IMa] _ 1 +r1[M1]/[M2] d[M2] [M2] r2[M2] + [Mj]

1+r2[M2]/[M!]

(7.15)

Mayo and collaborators were among the earliest workers to clarify the

relationship between copolymer and monomer solution compositions.

The ratio d [Mi ] /d [M2 ] is the same as the ratio of the numbers of

each kind of repeat unit in the polymer formed from the solution

containing Mj and M2 at concentrations [M^ and [M2], respectively.

Henceforth we shall designate this ratio as nj/n2. Since the composition

of the monomer solution changes as the reaction progresses, Eq. (7.15)

applies to the feedstock as prepared only during the initial stages of

the polymerization. Subsequently, the instantaneous concentrations in the

prevailing mixture apply unless monomer is added continuously to replace

that which has reacted and maintain the original composition of the

feedstock. We shall assume that it is the initial product formed that we

describe when we use Eq. (7.15) so as to remove uncertainty as to the

monomer concentrations.

As an alternative to the ratios n1/n2 and [Mj]/[M2] in Eq. (7.15), it

is convenient to describe the composition of both the polymer and the

feedstock in terms of the mole fraction of each monomer. Defining F; as

the mole fraction of the ith component in the polymer and fj as the mole

fraction of component i in the monomer solution, we observe that

d[Mj]

Fj = 1 - F2 = -------------------

(7.16)

d[Mj]+d[M2]

and

[MJ ! ' [M i ] + [Mj]

(7.17)

428

Polymers with Microstructure

Combining Eqs. (7.15) and (7.16) into (7.17) yields rjfj2 + fjf2

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