Chemistry Reference
In-Depth Information
to formulate these ideas into a fully coherent description of polymer dynamics.
A major lateral step was taken when de Gennes
34
applied scaling concepts to
polymers. We can illustrate this idea by closely following the argument given by
Doi and Edwards.
28
The properties of a Gaussian chain are such that its dimen-
sions and statistical distribution of the segments do not depend upon the length
of an individual segment for a large number N. Suppose we reduce the number
of segments by a factor l, then the new number of segments is N
0
¼
N/l. If the
chain is the same length, it is clear from eqn (5.84) that the link length must
change as b
l
p
. We can represent this transformation from the old chain to the
new chain as
and b
!
b
l
p
N
!
N
=
l
ð
5
:
107
Þ
If we knew how a physical property changed as we altered our chain we could
deduce how this property depended upon b and N. There are several statistical
measures of the length of the chain, for example the radius of gyration:
R
g
¼
b
N
p
6
p
ð
5
:
108
Þ
or the root mean square end-to-end length:
R
¼
b
N
p
ð
5
:
109
Þ
We could say quite generally that the average dimension of a chain is linear in
the link length but is some function of N, f(N):
average dimension
¼
f
ð
b
ð
5
:
110
Þ
So if we apply the transformation in (5.107):
b
l
p
N
l
f
ð
b
¼
f
ð
5
:
111
Þ
This can only be true when
f
ð
b
¼
constant
b
N
p
ð
5
:
112
Þ
This transformation applies equally well to a non-Gaussian chain, for example
in a good solvent where n
¼
0.5:
N
!
N
=
l
and b
!
bl
n
ð
5
:
113
Þ
So eqn (5.112) becomes:
average dimension
¼
constant
bN
n
ð
5
:
114
Þ
The constant in eqn (5.112) cannot be readily evaluated using scaling theory.
Our transformation applies equally well to the radius of gyration or the root
mean square end-to-end length, only the numerical constant changes. We
would like to be able to apply this idea to the role of concentration in semidilute
