AN INTRODUCTION TO PLASTICS - Plastics and Environmental Sustainability - page 79

anAFMimageofapolyelectrolytepolymermoleculedepositedfromadilute
solution onto a smooth surface (Kiriy et al., 2002).
Figure 3.2 Left : A ball and stick model of a section of a PP chain. Right :
An AFM image of a single polymer chain suggesting flexibility. Reprinted
with permission from Kiriy et al., (2002). Copyright (2002) American
Chemical Society.
3.1.1 Size of Polymer Molecules
Given such flexibility, the contour length of the polymer chain is not a good
measure of the size of a polymer chain made up of n segments each of
length ℓ linked end to end. The common metric of molecular size is based
on the distance between ends of the chain R . 3 But polymer molecules can
rearrange in different geometries, each with adifferent end-to-end distance,
R , and its mean value is used to quantify macromolecular size. The freely
jointed model of polymers therefore yields only a mean value for square 4 of
the end-to-end distance as . But in real chains, the bonds are also able to
rotate in any axis, allowing the chain to take diverse conformations in 3D
space. The above expression, when modified to take free rotation about the
covalent bonds into account, yields a slightly more complicated expression
3.1 for the of a very long chain (Kawakatsu, 2004; Rubinstein and Colby,
2003; Strobl, 2007):
(3.1)
where Θ is the valance angle defined by the bond vectors and C ∞ is called
the characteristic ratio. The value of C ∞ is essentially a measure of the
stiffness of polymer chains; for PE, it is 5.0, whereas for PS, it is greater
than 10 and for DNA, it is approximately 600! The expression is further
refinedbyalsoincludingperturbationsandlimitationsimposedbytorsional
angles on the different chain conformations. All these torsional angles are