Chemistry Reference
In-Depth Information
0
exp 2
R
413
3
5
0
:
48
0
R 3 413
2; 100
R 3 413
exp 2
1 2 exp 2
J
mol K ). Therefore, the fraction of
the bonds in either one of the gauche states is (1
Here, R is the universal gas constant (8
:
314
2
0.48)/2
5
0.26. And
h
cos
φi 5
0.48 cos (0 )
0.26 cos (120 )
0.26 cos (240 )
1
1
5
0.22. Using the calcu-
lated
h
cos
φi
and the corresponding
θ
(70 32 0 )in Eq. (1-31) , the chain end-
to-end distance at 140 C is thus expanded by a factor of
p for this reason. And
the corresponding C N value is 3. It is clear that by incorporating more structural
details of the monomer into the calculation of C N , its value becomes closer to
that of the experimental values (i.e., 6.8 at 140 C).
1.14.2.4 Rotational Isomeric State Model [9]
In the hindered rotation chains, the barriers to the rotational motion of individual
bonds are assumed to be independent. However, in the rotational isomeric state
model, such barriers are considered to be dependent for two consecutive bonds.
For example, in the case of polyethylene, a consecutive sequence of two bonds in
the g
has high energy, thereby a low probability to occur while two
consecutive bonds with g
1
and g
2
, a relatively lower energy combination, are
more probable. A full description of the rotational isomeric state chains is beyond
the scope of this textbook. Interested readers should refer to reference [9] for fur-
ther details. Nevertheless, the rotational isomeric state model has been incorpo-
rated into computer programs along with interatomic potentials, bond angles, and
so on to model the lowest energy conformations of macromolecules in specified
environments. Such molecular simulation studies have shown that the lowest
energy state of polymers in their “melt” condition is not necessarily that with the
highest entropy. In particular, molten polyethylene molecules do not resemble a
bowl of spaghetti. Rather, the overall conformation with the lowest energy is one
that comprises a significant fraction of shapes in which the chains are folded back
on themselves in an expanded version of the polyethylene crystal structure
described in Chapter 4.
When the average end-to-end distance of a macromolecular coil is described by
the rotational isomeric state model, the polymer is said to be in its “unperturbed”
state. Its dimensions then are determined only by the characteristics of the molecule
itself (i.e., bond lengths, bond angles, and barriers of rotation angles), not the inter-
atomic potentials. In general, the end-to-end distance of a dissolved macromolecule
is greater than that in its unperturbed state because the polymer coil is swollen by
solvent. If the actual average end-to-end distance in solution is
and g
1
1
d 2
1/2 ,then
h
i
h d 2
1 = 2
5 h d 0 i
1 = 2
i
α
(1-32)
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