Chemistry Reference
In-Depth Information
600
[110]
500
400
300
[111]
200
[200]
[211]
[220]
[221]
100
0
0.1
0.15
0.2
Q (Å
−1
)
0.25
0.3
Figure 4.6
Radially integrated SAXS profi le for the surfactant phytantriol in excess
water at 20°C recorded at the Australian Synchrotron SAXS/WAXS beam line by the
authors. The corresponding Miller indices for the Pn3m cubic phase are identifi ed from
the relative peak spacings as shown in Table 4.1.
Radial integration of the resulting two-dimensional (2D) intensity versus
angle data yields the one-dimensional scattering function
I
(
Q
), where
Q
is the
length of the scattering vector, defi ned as the difference between the wave
propagation vectors for incident (
k
) and scattered (
k
′
) radiation:
(
)
4
πθ
λ
sin
/
2
Qkk
=−′ =
with
the scattering angle.
The long-range orientational order possessed by many LLC systems results
in classical intense peaks in the
I
(
Q
) vs.
Q
plot, as shown in Figure 4.6. These
peaks are precisely analogous to the Bragg peaks measured for crystalline
systems using X-ray diffraction; however, the peak positions found for LLCs
generally occur at much lower
Q
values (corresponding to lower scattering
angles), since liquid crystalline structure is supramolecular, with typical unit
cell dimensions in the nanometer rather than Angstrom range.
In general, for LLC systems, the position of the peaks in the
I
(
Q
) vs.
Q
plot
can be used to identify the phase nanostructure (crystallographic space group)
and unit cell dimensions. The crystallographic space group is derived from
matching the relative positions (in
Q
) between measured Bragg peaks with
the allowable Miller indices (
h
,
k
,
l
) for different phases. Typically, three or
more such peaks are required for unequivocal identifi cation. This relationship
between relative
Q
spacing and Miller index is summarized for a number of
λ
being the wavelength of the radiation and 2
θ
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