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

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the fraction of ij sequences out of all possible sequences defines p^:

number of ij sequences

. .

number of ij sequences + number of ii sequences

This equation can also be written in terms of the propagation rates of

the different types of addition steps which generate the sequences:

Rj: êí[ÌÃ] [Ì.]

Ðí = --- - -------------------------- ------- (7-

31)

Rij + Rii êö[Ì,-] [M,] + ê"[Ì,-] [Mt]

For the various possible combinations in a copolymer, Eq. (7.31) becomes

k"|Mr]|M,] r, [M,]

P" ê,, (M, •] [M, ] + k13 [M,-] [M2] r,[M,]+[M,]

[M2]

P12 = ---------------

(7.33)

r,[M,] + [M,]

k2J[M2-][M,] r,[M,]

p = ---------------------------------= -----------------

(7.34)

k22[M2][M2] + k21[M2][M1] r2[M2] +[M,]

[Mj]

P" = "rn TkTi

C7'35*

r2 [M2] + [Mj]

Note that pn + p12 = p22 + p21 = 1. In writing these expressions we make

the assumption that only the terminal unit of the radical influences the

addition of the next monomer. This same assumption was made in deriving

the copolymer composition equation. We shall have more to say below about

this so-called terminal assumption.

Next let us consider the probability of finding a sequence of repeat

units in a copolymer which is exactly vx units of M! in length. This may

be represented as M2(M1)[;1M2. Working from left to right in this

sequence, we note the following:

A Closer Look at Microstructure

449

1. If the addition of

monomer Mi to a radical ending with M2 occurs L times

in a sample, then there will

be a total of L sequences, of unspecified length,

of Mi units in the sample.

2. If vx - 1 consecutive Mi monomers add to radicals capped

by Mi units,

the total number of such sequences is expressed in terms

of pa to be

Lp,,1'1"1.

3. If the sequence contains exactly vx units of type Mi, then the next

step

must be the addition of an M2

unit. The probability of such an addition

is given by p!2, and the number of such sequences is Lpn ^"^Pn •

4. Note that we use the symbol vi to indicate the number of

M- units in a

particular sequence. This should be distinguished from np which

gives the total number of Mj units in the copolymer without regard to

their distribution in various sequences.

Since L equals the total number of Mi sequences of any length, the

fraction of sequences of length ó i, 0 , is given by

The similarity of this derivation to those in Secs. 5.4 and 6.7 should be

apparent. Substitution of the probabilities given by Eqs. (7.32) and

(7.33) leads to

A similar result can be written for ô"2. These expressions give the

fraction of sequences of specified length in terms of the reactivity

ratios of the copolymer system and the composition of the feedstock.

Figure 7.3 illustrates by means of a bar graph how <pVl varies with vx

for two polymer systems prepared from equi-molar solutions of monomers.

The shaded bars in Fig. 7.3 describe the system for which rir2 = 0.03,

and the unshaded bars describe rir2 = 0.30. Table 7.5 shows the effect of

variations in the composition of the feedstock for the system rir2 = 1.

The following observations can be made concerning Fig. 7.3 and Table 7.5:

1. In all situations, the fraction 0^ decreases with increasing vx.

2. Figure 7.3 shows that for rir2 = 0.03, about 85% of the Mi units are

sandwiched between two M2's. We have already concluded that low values of

the rir2 product indicate a tendency toward alternation.

3. Figure 7.3 also shows that the proportion of alternating Mi units

decreases and the fraction of longer sequences increases as rir2

increases. The 50 mol % entry in Table 7.5 shows that the distribution of

sequence lengths gets flatter and broader for rt r2 = 1, the random case.

= Pi/1 1 P12

(7.36)

(7.37)

450

Polymers with Microstructure

Figure 7.3 Fraction of nj sequences of the indicated length for

copolymers prepared from equimolar feedstocks with rj r2 = 0.03 (shaded)

and rjr2 = 0.30 (unshaded). [Data from C. Tosi, Adv. Polym. Sci. 5:451

(1968).]

4. Table 7.5 also shows that increasing the percentage of Mj in the

monomer solution flattens and broadens the distribution of sequence

lengths. Similar results are observed for lower values of r^, but the

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