Characterization of Liquid Crystalline Polymers - Liquid Crystalline Polymers - page 233

Figure 4.30. The radial (r) and azimuthal ( ϕ ) positions of two diffraction maxima M
and M ; ∆r and ∆ ϕ respectively is the radial width and azimuthal width.

Shibaev and Plate, 1984; 1985; De Vries, 1985; Noel, 1985; Azaroff, 1987;
Noel, 1989). For powder samples of low mass liquid crystals, the X-ray
diffraction pattern can be divided into inner rings (inner maxima) at small
angles (2 θ ∼ 20 ◦ ).

With an increasing of the degree of molecular orientation, the rings will
decay into arcs with decreasing azimuthal width. The location and size of
the diffraction maxima are described by the radial and azimuthal positions
as well as the radial and azimuthal widths (Figure 4.30). The radial posi-
tion refers to the distance r of the maximum to the center of the X-ray
diffraction pattern; the azimuthal position refers to the angle ϕ made by
a reference line through the center and the line from the maximum to the
center.
From the radial position one can obtain the angle between the diffracted
beam and the incident beam (Figure 4.30, right). This angle is related to
the average distance between the long axes of adjacent molecules if the
angle 2 θ is about 20 ◦ (outer maxima). It is related to the molecular length
or layer thickness if 2 θ is only a few degrees (inner maxima). The radial
width of the diffraction maxima is related to the regularity of the molecular
packing. The more constant is the spacing between adjacent molecules (or
layers), the smaller is the radial width of the outer (or inner) maxima. Thus,
no Bragg diffraction is obtained for a nematic phase because of the lack of
positional order. The X-ray pattern of a nematic phase (Figure 4.31(a)) is
characterized by only the diffused inner (corresponding to molecular length)
and outer (corresponding to average lateral spacing between molecules)
rings. The intermolecular spacing can be estimated from the radial posi-
tion of the outer ring using the equation 2d sin θ =1 . 117 λ based on the

3 ◦ ) and outer rings (outer maxima) at large angles (2 θ ∼