Modern Polymer Spect..

100 3 T%xitionul Spectin as a Pvohe of Straictuval Order.
Let p be the number of atoms in the chemical repeat units and let the indices n
and n’ label the sites of the chemical repeat units along the polymer chain and i and
1 label the 3p internal co-ordinates within the repeat unit. The potential energy can
be formally written in a way similar to Eq. (3-13) [SO].
11; n‘
i. 1
(3-37)
where
(FR)llll’i, = (FR)ni”li (3-39)
let s : In - 11‘1 define the distance of interaction, which (see Section 3.6’1 is very
important in determining the vibrations of the chain; it follows that because of
translational symmetry
Substituting Eq. (3-40) into Eq. (3-37), one obtains
In Eq. (3-41), the first term gives the contribution of the intramolecular potential
by the reference unit at the n-th site, and the second and third terms collect the
contribution of intramolecular coupling of the n-th unit with the neighboring units
at distances s and -s respectively.
A similar equation can be written for the kinetic energy in a way similar to what
has been done for the finite molecule (Eq. (3-14)). Since the number of oscillators
is taken to be infinite (the chain is considered infinite) the dynamical problem results
in the writing of an infinite number of second-order differential equations, the
solutions of which are of the type
R”+S.
, - A, exp 1-i (At + sv,)] (3-42)
It should be noted that A, is independent of n, v, is the phase shift between two
adjacent roto-translationally equivalent internal co-ordinates, and A refers to the
vibrational frequency as in the case of small molecules (see Eq. (3-5)). Physically,
Eq. (3-42) tells us that the j-th coordinate at the n + s site oscillates with frequency /?
with an amplitude equal to the amplitude of the j-tli coordinate in the reference cell
at site n multiplied by a phase factor exp (-isq). Attention should be paid to the