Chemistry Reference
In-Depth Information
substitution of p
0
for the overall extent of reaction. Thus, for example Eq. (
5.18
)
for the weight fraction of chains should be replaced by Eq. (
5.19
)''
Þ
xp
ðÞ
x
1
Þ
2
w
cx
¼ 1
w
r
ð
ð
1
p
ð
5
:
19
Þ
w
cx
is the weight fraction of chains x-mer referred to the mixture of chain and ring
species, and p
0
, the conversion of the chains alone, is correlated with p according
to Eq. (
5.14
).
''Likewise, p in other formulas given in the preceding sections (e.g., Eq. (
5.16
))
of the present chapter should be replaced by p
0
. For polymers formed (or equili-
brated) in the absence of diluent, p
0
will be only slightly less than p, hence the
revision is usually scarcely significant. The dashed curve shown in Fig.
5.3
has
been calculated from Eq. (
5.19
).In order to obtain the complete size distribution,
including both, ring and chain species, the curves for the latter should be added to
the dashed curve. This revision would greatly increase the portion of very low
polymers over the amounts previously considered, and it would cause the com-
bined distribution for chains and rings to pass through a minimum at a low value of
x… Although, the proportion of rings in equilibrium in an undiluted polymer may
be scarcely significant on a weight basis, they may make a relatively large con-
tribution to the total number of molecules, and hence, markedly lower the DP. This
will be particularly true if p is large and the number of chains is small.''
In 1966, Flory and Semlyen [
6
] described an improved version of the JS
approach to the calculation of cyclization equilibrium constants (K
x
in all papers of
Flory and Semlyen) and said about their intention: ''The cyclization for larger
rings may in principle, serve as measure for the statistical configuration of chain
molecules… The theory of J ? S requires minor amendment in order to achieve a
close comparison with recent configuration studies on this polymer (PMDS). Their
treatment was set forth in terms of the hypothetical, freely jointed or random flight
chain model, a model which has been used extensively despite its unrealistic
failure to take account of the correlations of directions of successive bonds
imposed by bond angles and by hindrance of bond rotation. Realization of the full
potential of their theory as a basis for relating macrocyclization equilibria to chain
configuration statistics requires that it be recast in terms relevant to real molecular
chains. This we attempt below. The basic assumption underlaying their method
will be examined critically as well. Then the results of Brown and Slusarzuk will
be compared with theory, and implications of the comparison will be pointed out''
M
y
M
y
x
þ
c
M
x
ð
5
:
20
Þ
c
M
x
K
x
¼
M
y
x
½
=
M
y
ð
5
:
21
Þ
K
x
¼
ð
3
=
2p
Þ
3
=
2
=
2
5
=
2
\h
2
[
3
=
2
N
A
x
ð
5
:
22
Þ
K
x
¼
ð
3
=
2p
Þ
3
=
2
=
2
5
=
2
C
3
=
2
cr
x
5
=
2
N
A
ð
5
:
23
Þ
with: \ h2[ 0 mean square end-to-end distance
