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The solid nematic equivalent of the Fredericks transition is found to
depend on a critical field rather than a critical voltage as in the classical
case. This arises because director anchoring is principally to the
solid rubbery matrix of the nematic gel rather than to the sample surfaces.
Moreover, above the threshold field, we find a competition between quartic
(soft) and conventional harmonic elasticity which dictates the director
response. By including a small degree of initial director misorientation,
the calculated field variation of optical anisotropy agrees well with the
conoscopy measurements of Chang et al (Phys.Rev.E56, 595 (1997)
of the electro-optical response of nematic gels.
A molecular model of freely jointed chains of chiral monomers is developed
to describe the piezoelectric effect in chiral nematic elastomers. The
model, an extension of the neo-classical theory of nematic polymer networks,
takes into account a chiral biasing of molecular alignment under shear
which leads to induced polarisation if the monomers contain a transverse
dipole moment. The resulting theory is fully non-linear in elastic deformations,
in the spirit of ordinary rubber elasticity. The expansion to the highest
order in small strains gives the three linear piezoelectric coefficients
predicted by phenomenological models.
We present an experimental and theoretical investigation of the critical
formation of stripe domains in monodomain nematic elastomers. Domains with
alternating sense of director rotation are formed when the material is
stretched perpendicular to the initial director alignment. A wide range
of differing samples are shown to have a singular onset to director rotation
at a threshold deformation and a second singular point at the end of the
stripe domain region. All the data collapses onto a master plot revealing
a universal behaviour. We analyse theoretically the threshold properties
of the stripe phase. The analysis of free energy yields a first order transition
into a fully-coarsened texture without any intermediate state of sinusoidal
modulation.
We consider the elastic and orientational response of a uniform nematic
elastomer subjected to an extension perpendicular to its director. By allowing
a possibility of local shear in the material, we show that the effect of
`soft elasticity' leads to a new regime of director re-orientation, through
a highly non-uniform stripe domain state (in contrast to earlier predictions
and observations of a discontinuous uniform director jump). The molecular
theory developed here gives predictions on two levels: of the general texture
of the stripe state plus the interval of strains in which it occurs, and
of topological properties of the director rotation that are very general
and depend only on chain anisotropy of elastomer but not on details of
the specific material. On the other hand, parameters like the threshold
strain for the domain formation depend on the chemical composition and
on the model used to describe its effect. We discuss and explain experimental
observations of stripe domains both in the perpendicular geometry and when
the stretching direction is at an oblique angle to the director, leading
to asymmetric stripes and different topology.
The result of this work is the general expression for the free energy
of deformations, which combines the effects of large non-symmetric affine
strains in the rubbery network and gradients of curvature deformations
of the director field, $F \sim \lambda^T \{ \nabla {\bf n} \}^2 \lambda
$. We derive the molecular expressions for the elastic constants governing
non-uniform directors in the presence of elastic strains. These constants
depend on the polymer step length anisotropy and - most strikingly - have
an overall
negative sense. We therefore predict that in some circumstances,
especially at large elastic deformations \lambda, these new terms may overpower
the usual, positive Frank elastic moduli of the underlying nematic structure,
as well as the coupling in nematic elastomers of uniform relative rotations
of the director and the elastic matrix. In this event highly distorted
polydomain textures n(r) would be favoured.
Monodomain nematic networks, formed by crosslinking polymer liquid crystals
in ordered states, retain a memory of their anisotropic crosslinking conditions
and thereby show novel elasticity and strain induced nematic transitions.
Orientational transitions can result and these are investigated using a
simple model of nematic elastomers that allows both the the direction and
the magnitude of order to respond to applied strains. We find two temperature
dependent regimes. At low temperatures where nematic effects are strong,
the director rotates and switches with the nematic order largely intact.
At higher temperatures near the thermodynamic phase transition elastic
effects dominate and applied strain destroys the nematic order before it
reforms in the new direction.
Electric fields acting on anisotropic chains induce orientational torques,
which compete with rubber elastic effects. Outcome structures crucially
depend on the mechanical constraints applied to the sample. In set-ups
with no or few constraints, an electric field rotates the nematic director
without resistance, inducing also a spontaneous shape change of a rubber
or gel matrix. When certain strains are prevented in the sample by external
constraints, the magnitude of the elastic barrier is much higher than the
electric contribution and a very high electric field is required to create
an observable director rotation. In weakly anisotropic elastomers, for
instance conventional rubbers which have been strained during crosslinking,
the characteristic field will be considerably lower.
Solid liquid crystals, formed by crosslinking polymeric nematics into
elastomers are shown to display novel and complex elasticity. The internal
(nematic) direction can experience a barrier to its rotation which couples
to standard elasticity. We investigate this elasticity by considering imposed
strains and demonstrate several new orientational phase transitions, caused
by the interaction between applied stress fields and bulk barriers to rotation.
We predict a new phenomena unique to anisotropic rubber -- a `soft
elastic response'; uniaxial strain is developed without resistance below
a critical deformation $\lambda^*$ due to the relaxation of related shear
strains and reorientation of the nematic director.
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In a crosslinked polymer network, which has a lamellar or liquid crystalline
smectic-A order, layer deformations are penalized by an underlying rubber
elasticity. By integrating out phonon modes of the network, we obtain an
effective smectic-A elastic free energy in which there is an energy cost
for rotating smectic layers relative to the elastic network. As a result,
there is long-range translational smectic order with true Bragg peaks in
the X-ray scattering intensity with Debye-Waller factors depending on the
crosslink density of the network. There is also a very anisotropic diffuse
scattering.
The general continuum free energy is derived for a liquid crystalline
(smectic) elastomer under small strain and orientational distortions. Using
group representations theory we obtain all invariants describing the coupling
of translational and orientational degrees of freedom. Possibilities of
uniform rotation of the layer system under shear deformation are outlined.
It is shown that in centrosymmetric materials (smectic A) the only polarizational
response is flexoelectric, that is in the response to non-uniform orientational
deformations. Chiral materials (rubbers of smectic A*) also have a uniform
piezoelectric effect in response to a shear deformation, or an extension
at an angle with the layer normal.
The continuum free energy is derived for a crosslinked network of smectic
C and chiral smectic C* liquid crystal polymers under small strain and
orientational distortions. The coupling between elastic strain, the smectic
C order parameter and the polarization (induced or spontaneous in C* phase)
is examined and several new orientational and polarizational effects are
predicted. It is shown, on the other hand, that some effects in conventional
smectic C liquid crystals, for example the bistability and switching in
ferroelectric C*, no longer exist in corresponding elastomers.
The general phenomenological free energy is derived for a liquid
crystalline elastomer under arbitrary strain and orientational distortions.
Using the group representations method I obtaine all invariants, describing
the coupling of translational and orientational deformations and/or external
electric field. It is shown that in centrosymmetric materials (nematic
rubbers) the only contribution to the free energy which is linear in (small)
gradients of the director, is the coupling with electric field and the
strain tensor (16 independent terms analogous to the flexoelectric effect).
In chiral materials (cholesterics) the electric field couples with the
strain tensor for uniform nematic director (piezoelectric effect, 3 invariants)
and, in the absence of an electric field, translational and linear orientational
distortions interact with each other (couple-stress effects, 6 invariants).
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