Order and Disorder in Liquid-Crystalline Elastomers - Liquid Crystal Elastomers: Materials and Applications - page 209

4.3 Fluctuations and Disorder in Smectic Elastomers

In Fig. 1 we summarized the various ways in which LC order and polymer proper-
ties can be combined by attaching mesogenic molecules to, or incorporating in, a
polymer backbone. Once the backbone polymer is weakly crosslinked to form an
elastomer, the resulting macroscopic rubber elasticity interacts with the LC order-
ing field [ 4 ] . In smectic LC elastomers the layers cannot move easily across the
crosslinking points where the polymer backbone is attached. Consequently, layer
displacement fluctuations are suppressed, which effectively stabilizes the 1D peri-
odic layer structure and could under certain assumptions reinstate true long-range
order [ 128 , 129 ]. On the other hand, the crosslinks provide a random network of
defects that could destroy the smectic order [ 130 - 132 ] . Thus, in smectic-A elasto-
mers two opposing tendencies exist: the suppression of layer displacement fluctua-
tions that enhances translational order, and the effect of random disorder that leads
to a highly frustrated equilibrium state.
Let us look at the physical origin of the predicted behavior in some more
detail. On the continuum level, the coupling between the layer fluctuations and
the elastic matrix can be considered as layer pinning by crosslinks, which
constitutes a penalty for local relative displacements. This coupling is additive
to the ordinary smectic elastic energy of deformations and to the elastic energy of
anisotropic rubber network as a whole. The latter contains essentially the five
terms expected for a uniaxial solid on the basis of its symmetry. This includes the
deformation energy related to the shear elastic moduli perpendicular and parallel
to n, C 4 and C 5 respectively, that do not come into play for the liquid smectic
layers. This is essentially the physical reason for a possible solid-like elastic
response in weakly crosslinked smectic elastomers. The rubber elastic constants
are renormalized by the smectic fluctuations and acquire effective values for
two bulk (compression) and three shear moduli. The renormalization is deter-
mined solely by the rubber elastic parameters: shear modulus and coupling
constants. This leads to a combination of four small and one large elastic constant
( C i / C 3 1, 2, 4, 5), which is very different from conventional solids in

which all elastic moduli have about the same large magnitude. A similar situation
occurs in the crystal-B phase of liquid crystals (highly anisotropic molecular
crystal) in which the Landau-Peierls instability is eliminated due to the presence
of a term C 4 q 2

1, i ¼

in the elastic energy. Nevertheless, large layer fluctuations are
still found because of the small value of this elastic modulus compared to the
other modulii [ 133 , 134 ].
The expression for the free energy of a smectic elastomer as a function of layer
displacements is rather complicated. However, it is relatively easy to study its
implications for the two limiting cases, q z !

?

0 and q ⊥ ! 0, leading to the following

dispersion law for the elastomer phonon modes [ 129 ]:

k B T

u 2

ð q Þ

? ¼

; (14)

2 C ef f
5

2 C 5 q 2 ðq z =q 2 B q z þ

? þ ? Þ