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

Himenz P.C. Polymer Chemistry. The Basic Concepts - Copyright, 1984. - 736 p.
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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
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
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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