Chemistry Reference
In-Depth Information
where the subscript zero refers to the unperturbed dimensions. The expansion
coefficient
α
η
,
which will be introduced in
Section 3.3
in connection with the ratio of intrinsic
viscosities of a particular polymer in a good solvent and under theta conditions.
If a random coil polymer is strongly solvated in a particular solvent, the molecu-
lar dimensions will be relatively expanded and
α
can be considered to be practically equal to the coefficient
α
will be large. Conversely, in a
very poor solvent
can be reduced to a value of 1. This corresponds to theta con-
ditions under which the end-to-end distance is the same as it would be in bulk
polymer at the same temperature (
Section 3.1.4
).
α
1.14.2.5
The Equivalent Random Chains
[10]
The real polymer chain may be usefully approximated for some purposes by an
equivalent freely oriented (random) chain. It is obviously possible to find a ran-
domly oriented model which will have the same end-to-end distance as a real
macromolecule with given molecular weight. In fact, there will be an infinite
number of such equivalent chains. There is, however, only one equivalent random
chain which will fit this requirement and the additional stipulation that the real
and phantom chains also have the same contour length.
If both chains have the same end-to-end distance, then
d
2
d
e
i
5
σ
e
l
e
h
i
5
h
(1-33)
where the unsubscripted term refers to the real chain and the subscript e desig-
nates the equivalent random chain. Here, l
e
is usually referred to as Kuhn length.
Also, if both have the same contour length D, then
D
5
D
e
5
σ
e
l
e
(1-34)
From
Eqs. (1-33) and (1-34)
:
d
2
l
e
5
h
i=
D
(1-35)
and
D
2
d
2
σ
e
5
=h
i
(1-36)
EXAMPLE 1-3
Calculate the Kuhn length of polyethylene at 140
C. At 140
C, C
N
5
6.8. And l
0.154 nm
5
d
2
l
2
/D, where D
and
θ
5
70
32
0
. Here, l
e
5
h
i
/D
C
N
σ
5
σ
lcos(
θ
/2). Therefore, l
e
5
C
N
l/cos
5
(
θ
/2)
1.3 nm.
5
