Chemistry Reference
In-Depth Information
Thus all dot products in
Eq. (1-26)
vanish except when each vector is multi-
plied with itself. We are left with
*+
5
σ
X
σ
d
2
l
i
l
2
h
i
5
(1-28)
i
5
1
The extended or contour length of the freely oriented macromolecule will be
1/2
from the above
equation. It can be seen that the ratio of the average end-to-end separation to the
extended length is l
σ
l. Its root mean square (rms) end-to-end distance will be l
σ
1/2
. Since
will be of the order of a few hundred even for
moderately sized macromolecules, d will be on the average much smaller than the
chain end separation in the fully extended conformation.
The average chain end separation which has been calculated gives little infor-
mation about the magnitudes of this distance for a number of macromolecules at
any instant. When this distribution of end-to-end distances is calculated, it is
found, not surprisingly, that it is very improbable that the two ends of a linear
molecule will be very close or very far from each other. It can also be shown that
the density of chain segments is greatest near the center of a macromolecule and
decreases toward the outside of the random coil.
For real polymer chains, there exist local correlations between bond vectors
with restricted bond angles and steric hindrance (see more sophisticated models
described below). Therefore, the actual mean square end-to-end distance of real
polymer chains must be larger than the one calculated using the freely oriented
chain model. Flory introduced the concept of characteristic ratio to signify such
difference. The characteristic ratio is chain length dependent when
σ
σ
100.
However, for infinite long chains, the characteristic ratio (i.e., C
N
) is given by
σ
,
d
2
real
=σ
l
2
C
N
5
(1-29)
The numerical value of C
N
depends on the flexibility of the polymer chain.
Here, C
N
is always greater than 1 and, for flexible polymers, typical values vary
from 5 to 10. For example, at 140
C, the characteristic ratio of polyethylene is
6.8 while that of poly(methyl methacrylate) is 9.0.
1.14.2.2
Freely Rotating Chains
The random-flight model used in the previous section underestimates the true
dimensions of polymer molecules, because it ignores restrictions to completely
free orientation resulting from fixed valence bond angles and steric effects. It also
fails to allow for the long-range effects that result from the inability of two seg-
ments of the chain to occupy the same space at the same time.
The effects of fixed rather than unrestricted bond angles can be readily com-
puted, and it is found that the random flight relation of
Eq. (1-28)
is modified to
d
2
l
2
h
i
5
σ
ð
1
1
cos
θÞ=ð
1
2
cos
θÞ
(1-30)
