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Macromolecules, Structure 889

TABLE IV Branching in High-Pressure Polyethylene where l is the length of a monomer unit and x is the num-
¯ M n = 18,400, ¯ M w = 129,000
ber of monomer units. This relationship is derived from
Number of branches per random walk statistics and assumes that bond rotation is
Types of branch 1000 backbone carbons free and that the bonds between two adjacent monomer
units can assume any angle. This model is called the freely
CH 3 0.0
jointed chain and is clearly unreasonable for a polymer,
CH 2 CH 3 1.0
where bond hybridization imposes restrictions on avail-
CH 2 CH 2 CH 3 0.0
able bond angles. However, it is qualitatively correct in
CH 2 CH 2 CH 2 CH 3 9.6
that any correlation between two monomer units must de-
CH 2 CH 2 CH 2 CH 2 CH 3 3.6
crease as the distance between the units is increased.
Hexyl and longer 5.6
We can improve this model by imposing a restriction
Total 19.8
on our hypothetical chain. If we allow the chain segments
to rotate freely, but now fix the valence angles, we have
what is known as a freely rotating chain. In this model,
2
significant because some of them—those with a tertiary ¯ r is given by
chlorine—are believed to be sites for the initiation of
1 − cos θ
2
2
the ¯ r = xl , (68)
1 + cos θ
Cl
where θ is the angle between successive bonds in a chain.
◦
2
2
CH 2 C CH 2 For a tetrahedral chain, θ = 109 and ¯ r = 2xl .
A further refinement is required to obtain a reasonable
CH 2
modelforapolymerchain.Intherestrictedrotationmodel,
CHCl
we take into account the fact that there are intrachain hin-
drances in a real polymer chain. The bond angles are fixed
by virtue of hybridization, and the chain rotation is limited
chemical decomposition of the polymer. Poly(vinyl chlo-
to a few rotational isomeric states. We shall first illustrate
ride) is a valuable material, used in very large volume
chain conformations with a simple model compound, and
for molding and electrical insulation. It is fire resistant
then extend our treatment to a segment of a polyethylene
and relatively cheap, but when heated is subject to loss
chain.
of hydrogen chloride, leading to the formation of deeply
Referring to Fig. 32, we see several conformations
colored polyene structures:
of butane, generated by rotation about the C 2 C 3 bond.
+
These are called gauche , trans, and gauche − confor-
·· CH 2 CHClCH 2 CHCl ·· →
−
mations and are usually abbreviated as g , t, and g .
+
·· CH CHCH CH ·· + 2HCl The available conformations are most easily visualized
in Newman diagrams, shown in Fig. 32(a). Looking down
The reaction is more complicated than this simple repre-
the C 2 C 3 bond, the substituents are projected onto planes
sentation suggests and is believed to be sensitized by the
parallel to the paper. The C 2 carbon is in front, and the
presence of labile centers such as a tertiary chlorine.
Branches are an integral part of other polymers such C 3 carbon, in this presentation, is obscured by the C 2
carbon. The Newman diagrams clearly show that in the
as star polymers and dendrimers. Star polymers can have
trans arrangement, the bulky groups are opposite each
a variable number of arms with a controlled molecular
other, separated by a dihedral angle of 180 . The gauche +
◦
weight. Dendrimer can have a high molecular weight and −
and gauche conformations are formally generated by ro-
many functional end groups.
tations about the C 2 C 3 bond. The bulky groups in the
gauche conformations are separated by a dihedral angle
◦
of 60 .
IV. CHAIN CONFORMATIONS Figure 32(b) shows how the torsional potential of n-
OF MACROMOLECULES butane varies as a function of conformation. The lowest
energy conformation occurs when n-butane is in the trans
A. Theoretical Introduction conformation. The population of n-butane in the eclipsed
conformations, corresponding to the tops of the energy
We have already seen in Eq. (13) that the rms end-to-end
barriers, is negligible. The trans and gauche conformers
distance of a polymer chain is given by
interchange very rapidly with each other, having lifetimes
2 1/2 1/2 2 2 −10
¯ r = lx or ¯ r = l x , of approximately 10 sec at room temperature.