Order and Disorder in Liquid-Crystalline Elastomers - Liquid Crystal Elastomers: Materials and Applications - page 205

in which;

n ¼ n 2 ; (11)

with given by ( 11 ) . This type of lineshape was first reported for low-molecular-mass

smectics by Als-Nielsen et al. [ 100 ] and then confirmed for various thermotropic [ 101 ,
102 ] and lyotropic lamellar phases [ 103 - 105 ] , smectic polymers [ 106 ], and lamellar
block copolymers [ 107 ]. There are a finite number of power-law peaks of the type of
( 10a ): when 2 the exponent changes sign and the singularities are replaced by

cusp-like peaks. For thermotropic low-molecular-mass smectics,

n >

is small and
positive, typically 0.05-0.1 deep in the smectic-A phase. Equation ( 9 ) indicates that
for less-compressible materials ( B large), such as lyotropic smectics and some poly-
mers, can be even smaller. On the other hand, close to a smectic-nematic transition B

can decrease strongly and might be an order of magnitude larger. When several

higher harmonics are present, the quasi-long-range order can be established unambig-
uously from the scaling relation

n ¼ n 2

.
Algebraic decaying order is demonstrated in Fig. 9 for a typical smectic elasto-
mer. In a double-logarithmic plot with q z q n still on the x -axis, the characteristic
features are a central plateau-like region at small deviations from q n due to the finite
size of the smectic domains, and a power-law behavior in the tails. The latter
regions fulfill the scaling law n / n 2 ¼ ¼ 0.16 0.02, providing a rigorous

proof of algebraic decay.

10 0

10 1

n = 3

10 -1

10 0

n = 2

10 -2

10 -1

10 -3

10 -2

n = 1

10 -4

10 -3

10 -3 10 -2 10 -1 -0.08 -0.04 0.00 0.04 0.08

q-q n (nm -1 ) q-q n (nm -1 )

Fig. 9 Three orders of lineshape for the elastomer depicted in Fig. 11 with 10% crosslinks. The
wings of the peaks (shown logarithmically for emphasis on the right ) indicate algebraic decay
following

ðq q n Þ ð 2 n Þ ( straight lines )[ 7 ]