Chemistry Reference
In-Depth Information
the numerical factor of 10
4
would have to be substituted for the value of 10
6
in
eqn (5.72). A variation in the units used and symbols is quite common in the
literature and one should be wary of this. Combining eqns (5.65) and (5.72) we
obtain an expression related to the overlap parameter:
c
¼
1
:
08
½
ð
5
:
73
Þ
This demarcates the boundary between the dilute behaviour of a polymer and the
semidilute regime. There can be slight variations in the value of the constant
relating these properties, depending upon the assumptions used. Once the
polymer is in the semidilute regime, the coils overlap and interpenetrate. They
do not necessarily form strong entanglements. In the region where the coil
overlap begins, the expanding effect of a good solvent becomes screened by
segments from neighbouring coils and the chains begin to collapse back toward
their y dimensions. This was investigated by de Gennes using scaling arguments
34
and he suggested the chain dimensions in the semidilute regime would reduce as
1
=
4
ðÞ¼
R
2
c
c
R
2
ð
5
:
74
Þ
A scaling argument does not describe the full functional form of the dependence
of the chain dimensions on concentration. We would expect the dimensions of
the polymer to be a smoothly changing function between c* and the concentra-
tion c
w
where the coils reach their limiting radius equivalent to the y dimensions.
However, to a first approximation we can suppose that eqn (5.73) applies up to
the concentration c
w
.
4
4
R
2
R
2
R
2
R
y
c
w
c
¼
¼
ð
5
:
75
Þ
c
ðÞ
The ratio of the root mean square lengths is termed the chain expansion factor:
c
w
c
¼
a
e
ð
5
:
76
Þ
and represents the dimension of the chain relative to y conditions. We begin by
combining eqns (5.65), (5.67), (5.70), (5.73) with (5.76) to obtain an expression
for c
w
:
K
5
=
2
R
K
Ry
c
w
¼
1
:
08
6
3
=
2
F
ð
5
:
77
Þ
which indicates the c
w
does not depend upon molecular weight. However, this
relies on the chain attaining its ideal configuration, i.e. n
¼
3/5, which is often not
the case experimentally. eqn (5.77) can be difficult to evaluate directly. We can
follow the argument of Graessley
25
and that of Berry et al.
26
to introduce the
