Modern Polymer Spect..

3.9 Moving Towards Reality: From Order to Disorder 125
quickly mentioned here. We restrict our discussion to the most connnon case of 0-
bonded chains, which, as discussed in Section 3.5, do not permit long-range electronic
coupling.
Let y~ be the number of identical chemical units in a chain segment (different
chemical groups at either ends are neglected) and let each chemical unit consist of n
atoms; we treat here the case of a model molecule with free ends [18]. Let us take
one chemical unit and its 3 x n oscillators (which, for sake of clarity, we describe
here as internal (R) or group coordinates II). Because of elastic coupling with its
neighbor at either side, each oscillator couples with its identical neighbors and
generates 11 normal modes which can be described as waves characterized by a
phase coupling y = jn/ll (where j = 0,1,2,. . . y - 1). Their 11 frequencies y(y) lie at
finite y values along the dispersion curve of the corresponding infinite polymer with
identical chemical, stereochemical, and conformational structure. Each vibration
corresponds to a given phase coupling defined by the number of chemical repeat
unit which make up the finite chain. The introduction of phase coupling implies the
fact that within such short chains, nomal modes describe quasi regular waves (quasi
phonons) similar to those which propagate in an infinite chain. The remaining
modes which do not lie on the branches of the dispersion curves must be associated
with the vibrations localized on the end groups and can be identified by the constant
values of the frequencies when chain length is changed. It follows that:
(i) if I? is the number of repeat units making up the finite chain (neglecting the two
end groups) one expects to find a sequence of bands (or band progression)
which lie on each of the dispersion branches of the corresponding polymer.
Let us take for example the n-alkane n-nonadecane (CH~(CH~)I~CH~). Let us
focus on the vibrations of the segment consisting of y~ = 17 CH2 groups all in
trails conformation. Each CH2 generates 3 x 3 = 9 vibrations (CH2 antisymmetric
and symmetric stretch, CH? bending, wagging, twisting, rocking,
antisymmetric and symmetric C-C stretch, CCC bending, and C-C torsion).
Each of these ‘oscillators’ generates 17 waves, each of which is characterized by
a phase coupling nj/ 17. The corresponding frequencies lie on the dispersion
curves of a single chain of infinite all-trans polyethylene. The band progressions
observed for n-nonadecane will be discussed in Section 3.19.
(ii) The frequency range spanned by each band progression depends on the extent
of intramolecular coupling. If the dispersion of the frequency branch is small,
the progression is squeezed, many bands overlap (e.g., the stretching vibrations
of C-H groups), and no progression can be observed experimentally. When
intramolecular coupling is large, band progressions are clearly observed
(-1 100-720 cm-’ is the frequency range covered by CH2 rockings in all-trans
n-alkanes).
(iii) The intensity in infrared and/or Raman of each of the bands within the progressions
depends on the dipole (polarizability) changes associated with the
corresponding vibration (quasi phonon with y, z jn/y). We have previously
seen that for an infinite chain only k = 0 phonons are active and all other are
inactive, as dipoles or polarizability changes cancel out in the case of perfect
phonon waves. In going from short to longer chains, optical selection rules