Order and Disorder in Liquid-Crystalline Elastomers - Liquid Crystal Elastomers: Materials and Applications - page 216

case an exponent of 2.45, independent of temperature, has been predicted theoreti-
cally and is in good agreement with the present experiment. However, no further
details are available to confirm this hypothesis.

5.2.2 Road to Disorder

We return to the siloxane elastomer of Fig. 11 and consider the situation for higher
crosslink concentrations (Fig. 12b ). For c ¼ 15% crosslink, the domain size is

already considerable smaller than for the homopolymer, indicating the end of the
range of increased domain sizes due to crosslinking. With increasing c , the trans-
parency of the samples decreases, which is also expressed by a larger mosaic
distribution and fewer higher harmonics. Whereas for c ¼

10% three harmonics

are observed (Fig. 9b ), for c ¼ 15% only two orders of diffraction occur. Algebraic

decay of the positional correlations is still preserved with 0.01 but is

partly masked by a substantial broadening of the peak along q z and by the increased
mosaic spread. In Fig. 12b , the remaining first harmonic for c ¼

' 0.15

20% is compared
with those at other crosslink densities. It is strongly broadened both along q z
(domain size about 100 nm) and along q x (mosaic distribution of the smectic
layer normal). In this figure, an additional result is included for 15% of the stiff
crosslink V8, which behaves as anticipated for a concentration of the flexible
crosslink V1 appreciably larger than 20%. The results for various concentrations
of both types of crosslink are summarized in Fig. 14 . With increasing concentration
of crosslinks above 10%, the disorder gradually takes over, as indicated by
(i) broadening of the X-ray peak along the layer normal ( Dq z ) and (ii) a crossover
of the lineshape from Gaussian to Lorentzian. Though this behavior is consistent
with the general predictions for random quenched disorder, it is remarkable that the
algebraic decay survives up to rather large crosslink densities of 15%.
To become more quantitative, we note that various factors contribute to the
structure factor in smectics. These include in particular the finite size of the sample
and the effects of the mosaic distribution [see ( 12 ) in Sect. 4.1 ]. For the present
discussion we shall simplify things somewhat and emphasize (a) the broadening of
the central part of the X-ray peak due to the finite size, H (q), and (b) the possible
power-law behavior in the tails of the peak . Let us start with the finite-size term. As
discussed in some detail by Obraztsov et al. [ 7 ], a suitable distribution function to
describe the central part of the X-ray peaks is given by:

" #

2b

ð s b zÞ HðzÞ¼ exp : (16)

2b

This expression gives a Gaussian function for b ¼ 1 and a simple exponential

for b ¼ 0.5, leading to a Gaussian and a Lorentzian lineshape, respectively. Equa-

tion ( 16 ) allows a smooth transition between these cases. The situation 0.5

1
can be described as a stretched Gaussian or equivalently as a compressed

< b <