Order and Disorder in Liquid-Crystalline Elastomers - Liquid Crystal Elastomers: Materials and Applications - page 206

The various contributions to the X-ray intensity distribution in the vicinity of the
Bragg position in smectics have been outlined by Kaganer et al. [ 102 ] . Additionally
to the finite size of the system, the effects of the mosaic distribution (distribution
of the layer normals within the illuminated area) have to be taken into account. At
large deviations from q n we can find the power-law due to the algebraic decay of
positional correlations leading to ðq z q n Þ 2 þ n . At smaller distances from q n

the effect of the mosaic distribution gradually takes over, approximating finally to
behavior like ðq z q n Þ 1 þ n . The central part, including the full-width-at-half-

maximum (FWHM), is determined by the residual Bragg peak due to finite-size
domains of the sample [ 99 , 108 ]. The intensity measured in the X-ray experiment
can be represented by the convolution of the various factors mentioned [ 102 ]:

Ið q Þ¼Sð q ÞFð q ÞHð q ÞRð q Þ: (12)

F (q) and H (q) stand for the broadening due to the mosaic distribution and due
the finite size, respectively, while R (q) describes the resolution function of the
setup. Deconvolution of the experimental data provides the required determination
of the structure factor S (q).

4.2 Random Disorder

In condensed matter physics, the effects of disorder, defects, and impurities are
relevant for many materials properties; hence their understanding is of utmost
importance. The effects of randomness and disorder can be dramatic and have
been investigated for a variety of systems covering a wide field of complex
phenomena [ 109 ] . Examples include the pinning of an Abrikosov flux vortex lattice
by impurities in superconductors [ 110 ], disorder in Ising magnets [ 111 ], superfluid
transitions of He 3 in a porous medium [ 112 ] , and phase transitions in randomly
confined smectic liquid crystals [ 113 , 114 ] .
Liquid crystals provide beautiful possibilities to study the structural and
dynamic effects of quenched disorder. Their algebraic decay of positional correla-
tions gives an interesting starting point, they are experimentally easily accessible,
and can be confined within appropriate random porous media. Liquid crystals have
been incorporated into the connected void space of an aerogel, which is a highly
porous (up to 98% void) fractal-like network of multiply connected filaments of
aggregated 3-5-nm diameter silica spheres. Alternatively, quenched disorder can
be introduced in a liquid crystal by dispersing a hydrophilic aerosil (nanosize silica
particles forming a hydrogen-bonded thiotropic gel). Both methods allow the study
of the effects of weak random point forces and torques on the LC order, an idealized
disordering mechanism that affects molecular location and orientation in random
ways but occupies little physical space. Even at very low density of aerogel
or aerosil (about 1-3%), the 1D smectic order is destroyed, in agreement with
general theoretical predictions that generic quenched disorder should do so, no