Chemistry Reference
In-Depth Information
Fig. 2.16
Illustration
of a molecular coil
defines this relationship [
7
]. To put it in other words, it is the square of the distances between various
masses and the center of the mass:
,
X
2
X
2
S
¼
m
i
s
i
m
i
i
i
n
o
X
2
2
S
¼
ms
av
:
ð
1
=NmÞ
where
is the vector distance from the
center of the mass to the terminal chain bond. The size of randomly coiled polymer molecules is
commonly designated by the root-mean square distance between the ends,
m
is the mass associated with each of the
N
chain bonds, and
S
2
. A molecular coil is
R
illustrated in Fig.
2.16
.
The distance between the chain ends is often expressed in terms of
unperturbed dimensions
(
S
0
or
R
0
)
and an
expansion factor
(
a
) that is the result of interaction between the solvent and the polymer
2
¼ S
0
2
2
2
¼ R
0
2
2
S
a
and
R
a
The unperturbed dimensions refers to molecular size exclusive of solvent effects. It arises from
intramolecular polar and steric interactions and free rotation. The expansion factor is the result of
solvent and polymer molecule interaction. For linear polymers, the square of the radius of gyration is
related to the mean-square end-to-end distance by the following relationship:
2
2
S
¼ R
=
6
is greater than unity in a good solvent where the actual
“perturbed dimensions” exceed the unperturbed ones. The greater the value of the unperturbed
dimensions the better is the solvent. The above relationship is an average derived at experimentally
from numerous computations. Because branched chains have multiple ends it is simpler to describe
them in terms of the radius of gyration. The volume that a branched polymer molecule occupies in
solution is smaller than a linear one, which equals it in molecular weight and in number of segments.
The volume that these molecules occupy in solution is important in determinations of molecular
weights It is referred to as the
hydrodynamic volume
. This volume depends upon a variety of factors.
This follows from the expansion factor,
a
