Order and Disorder in Liquid-Crystalline Elastomers - Liquid Crystal Elastomers: Materials and Applications - page 189

where l ij is the effective step lengths tensor and L the contour length of the chain. For
conventional nematic or smectic LC polymers of uniaxial symmetry this expression
reduces to the main components of the radii of gyration and step lengths tensor with
respect to n: R g// , R g ⊥ and l // , l ⊥ , respectively. The average value of the contour length
of the chain is given by L ¼ Na ,inwhich N is the average number of monomers in
the chain and a the monomer size. Knowing these values, the main components of the
step length tensor l // and l ⊥ can be determined. In main-chain polymers, the measured
anisotropy l // / l ⊥ ¼ ( R g// / R g ⊥ ) 2 is generally very large. The anisotropy induced in the

backbones of side-chain polymers is much smaller and often is oblate, l // / l ⊥ <

1.
Many macroscopic properties, for example the optical and dielectric anisotropy,
follow the order of the mesogenic rods. However, for polymer networks the backbone
anisotropy is of primary importance because it causes the dramatic elastic response.
In the next section, we give a brief overview of the essential results obtained so far for
chain anisotropies of the various classes of LC polymers.

2.2 Chain Conformation of “End-On” Side-Chain Polymers

For “end-on” side-chain LC polymers, the coupling of the backbones with the
ordering field of the mesogenic rod-like fragments varies over a wide range
depending on the flexibility of the backbone, the spacer, and the rod-rod interac-
tions. Possibly this explains why these mesogenic polymers exhibit practically the
same wealth of LC polymorphism as their low-molar-mass counterparts, including
smectic, hexatic, and crystalline phases [ 10 , 17 - 19 ] .
SANS results on several nematic polyacrylates indicate that the backbone
preferably adopts a weakly prolate shape with R g// / R g ⊥ approximately equal to
1.2-1.5, i.e., the average direction of the backbone is parallel to n [ 10 , 20 ] and is
imposed by the alignment of the mesogenic side groups (Fig. 3a ). These observa-
tions have been confirmed by NMR studies of LC polyacrylates [ 23 ] . This type of
prolate conformation of the backbone is also typical for nematic polysiloxane-based
end-on polymers, especially when the spacer is relatively short. However, less
flexible LC polymethacrylates with the same side-chain and spacer length tend to
coil up in the nematic phase in a subtle oblate configuration (side-chains preferably
perpendicular to the backbone) [ 11 , 12 ] . In the smectic phase, both acrylates and
methacrylates have an oblate configuration and the anisotropy becomes even more
pronounced: R g// / R g ⊥ is approximately 0.3-0.5 [ 12 , 24 ] (Fig. 3b ). The backbones
are to some extent confined in 2D between the smectic sublayers of the mesogenic
cores [ 22 ] . Furthermore, the backbone statistics differ in the directions parallel and
perpendicular to the director. In the perpendicular direction, the mean square of the
radius of gyration

DE is proportional to the degree of polymerization, indicating
a trend towards a Gaussian walk in the plane of layers. Parallel to the director, the
chains show a rod-like behavior, which corresponds to crossing defects, i.e., back-
bones hopping from one layer to another [ 24 ] (Fig. 3b ) . Such a behavior has already

R g ?