Modern Polymer Spect..

3.8 Dtvz.sit,v of L’ihrntioizal Strrtes aid Nezrtroiz Scutteriny 1 19
We will introduce the basic principle and use of neutron-scattering spectroscopy
by first introducing an important dynamical quantity, namely the density of vibrational
states g(i1).
Let g(v) be the fraction of normal modes with frequencies in the interval v and
v + dv with dv 0. g(v) can be calculated analytically only for very simple atomic
chains which are never encountered when realistic polymeric materials must be
studied.
Let us take a very large molecule and let 3N z C be the number of vibrations of
the whole system; if c is the number of frequencies in the interval AV = v + dv then
go)) = c(Av)/C (3-50)
The total density of states is evaluated as the sum over all the r branches of the
dispersion relation (Eq. (3-43)). Numerical methods such as the ‘root sampling
method’ have been proposed for simple one- or tridimensional lattices [511. When
the systems become more complex, as in the cases discussed in this chapter, the
easiest numerical method is that proposed by Dean based on the application of the
Negative Eigenvalues Theorem (NET)[79]. The method is extremely useful and has
been extensively used in the laboratory of the writer for many polymeric cases. The
details of the method are discussed in the next section.
NET is a way to calculate g(v), i.e., it provides the number (or the fraction) of
vibrational energy states or normal modes in a chosen y (or k) interval. g(v) can be
plotted as an histogram whose accuracy depends on the frequency interval Ait chosen
in the numerical calculation (numerical resolution of the computer experiment).
This will be used later in this chapter in the study of disordered systems.
Let us consider g(v) of an ideal infinite polymer and examine the case of singlechain
polyvinylchloride in its three possible structures (threefold helix isotactic,
trans-planar syndiotactic, folded syndiotactic) [go]. Foi- easier understanding, Figure
3-12 shows both dispersion curves and g(v) for this polymer in its three possible
structures. It is apparent that the shape of g(v) is determined by the shape of the
dispersion curves which in turn depend on the vibrational force field used in the
calculation of ~(y). Flat sections of v(q) give rise to strong ‘singularities’ in g(v).
The shape of the dispersion curves and of electronic and/or vibrational g(v) for
both electronic bands and vibrational bands have been the subjects of extensive
theoretical treatments [51]. For infinite and perfect systems, the shape of g(i1) is
characterized by different types of singularities which are related to the dimensionality
of the systems and to many other physical factors. We restrict our analysis to
the case of polymeric materials which can be considered as one-dimensional lattices.
Our discussion can only be sketchy and qualitative, but it is aimed at pointing out
the features most relevant to the problems of disorder treated in this chapter.
T
Figure 3-12. Phonon dispersion curves v(p) (a, c). i(k) (b) and density of vibrational states gii,) from
0 to 1500 cn-’ of three possible models of a chain of polyvinylchloride showing the lR-Rainan
active and the inactive ‘singularities’ corresponding to flat sections of the disperion CUI-vcs (from
[801).