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Macromolecules, Structure 873

pass through but not the polymer. The solvent diffuses E = (n + 0.5)h /2π(k /m r ) 1/2 = (n + 0.5)hcv, (43)
through the membrane into the polymer solution in an at-
where h is Planck’s constant. The transition energies are
tempt to equalize the solvent pressure on both sides of the
given by hcv. Only transition between adjacent levels
membrane. This sets up a pressure difference that can be
are allowed in a quantum mechanical harmonic oscillator.
measured by a transducer or other appropriate methods.
Second, the molecular vibrator is not strictly harmonic but
rather anharmonic, with vibrational levels becoming more
closely spaced and transitions somewhat smaller in energy
III. MICROSTRUCTURE OF as n is increased. Another consequence of anharmonicity
MACROMOLECULES is that selection rules are relaxed, permitting transitions
to levels higher than the next immediately higher one.
We have seen (Section I.B) the types of structural iso- Transitions from n = 0 to n = 2 correspond to the appear-
merism of which polymer chains are capable—in par- ance of weak but observable first overtone bands having
ticular the occurrence of various types of stereochemical slightly less than twice the frequency of the fundamental
isomerism, branching and cross-linking, head-to-tail ver- band.
sus head-to-head:tail-to-tail isomerism, and monomer se- The appearance of a vibrational absorption band in the
quence isomerism in copolymers. We now describe briefly infrared region requires that the impinging radiation sup-
the two principal forms of spectroscopy that are used to ply a quantum of energy E just equal to that of the vibra-
observe and measure these structural features. tional transition hcv. It is also necessary that the atomic vi-
bration be accompanied by a change in the electric dipole
moment of the system, thus producing an alternating elec-
A. Vibrational Spectroscopy tric field of the same frequency as the radiation field. This
condition is often not met, as for example in the vibrations
The spectroscopic method that has the longest history for
of homopolar bonds such as the carbon–carbon bonds in
the study of macromolecules is infrared. More recently
paraffinic polymers.
applied and very closely related is Raman spectroscopy.
The Raman spectrum can give much the same informa-
Both deal with relatively high-frequency processes that
tion as the infrared spectrum, but they are in general not
involve variation of internuclear distances (i.e., molecu-
identical and can be usefully complementary. In Fig. 15 we
lar vibration). (Rotational and translational processes will
see at the left the Rayleigh scattering process, in which the
not concern us in polymer spectra.) As a first approxima-
molecule momentarily absorbs a photon, usually of visible
tion we may imagine that these molecular vibrators can
light, and then reradiates to the ground state without loss
be considered as classical harmonic oscillators. For a di-
of energy. However, the excited molecule may also return
atomic molecule of unequal masses m 1 and m 2 connected
to a higher vibrational state—the next highest in Fig. 15
by a bond regarded as a spring with a force constant k, the
frequency of vibration expressed in wave numbers (i.e., (center)—and then the reradiated photon will be of lower
cm −1 or reciprocal wavelength) is given by frequency by ¯ν. In a complex molecule there will be
many such states, and so the Raman spectrum, like the
v = (1/2πc)(k /m r ) 1/2 , (41)
where c is the velocity of light and m r the reduced mass,
given by
∼
m r = m 1 m 2 /(m 1 + m 2 ) = m 1 if m 2 m 1 . (42)
Thus, a small mass, such as hydrogen or deuterium, vi-
brating against a larger one, such as a carbon or chlorine
atom, will have essentially the frequency characteristic of
the smaller one. Most molecular vibration frequencies of
interest for polymer characterization will be in the range
of 3500 to about 650 cm −1 or in wavelength 2.5 to 15 µm.
Actual molecular vibrators differ from the classical os-
cillator in two respects. First, the total energy E cannot
have any arbitrary value but is expressed in terms of inte-
gral quantum numbers n: FIGURE 15 Energy level diagram for Raman scattering.