Modern Polymer Spect..

198 3 JJiber-ulioiiiil Spectra as a Probe oj Structural Ordeer-
Figure 3-61. Projection onto the ab face of the structuic
I 1
of the orthorhombic cell of polyethylene (two molecules
per unit cell, setting angle o = 42").
twiston is estimated to be -10 kcal/mol. The twiston model implies for a polymer:
(i) the initial formation of a disordered region near the surface of the crystal; (ii) the
propagation of the twiston through the crystal along the chain axis; and (iii) the
disappearance of the twiston at the opposite surface leaving the chain rotated by
180" and translated by one CH2 unit.
The idea is formally very appealing since, from the spectroscopic viewpoint,
it implies a collective mobility of the polymethylene chain which accounts for the
observed longitudinal mobility, surface melting, or disordering and transport of
matter without requiring large-amplitude libro-torsional motions which are not
detected by vibrational spectroscopy, as previously discussed.
The twiston model has been further expaiided by Mansfield [I671 and later by
Skinner and Wolynes [168]. We shortly mention the physics behind since it may be
applied to other cases in polymer physics. The whole problem of polynielhylene
chains is reviewed in [169].
Let us take the structure of the orthorhombic unit cell of polyethylene (Figure
3-61) and consider the CH2 units as point masses of mass 117 at a distance ci from the
axis of the molecule. The units can rotate by an angle 0 about the molecular axis.
When they rotate they are subject to the elastic response C by the torsions between
two adjacent (CH?) units. From Figure 3-61 it is apparent that when the chain
rotates as a rigid body about its axis in an orthorhombic lattice it probes a twofold
periodic potential V(0) from the crystal field.
The Hamiltonian proposed by Mansfield has the following form
In Eq. (3-61) the first term represents the contribution from the crystal potential,
the second term the kinetic energy, and the third is the quadratic elastic contribution.
In a gentle twist it can be assumed that Oi do not change much from one unit
to the other. According to Mansfield one can make the continuous approximation
and reach a solution of the type:
@(x - xo - vt) = 4 arctan exp [ k y(x - xo - vt)] (3-62)