## Molecular models of nematic elastomers  and various effects:

skip down to the abstracts on continuum theories

• E.M. Terentjev, M. Warner, R.B. Meyer and J. Yamamoto

• Electromechanical Fredericks Effects in Nematic Gels,

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.

• E.M. Terentjev and M. Warner,

• Piezoelectricity of chiral nematic elastomers,

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.

• H. Finkelmann, I. Kundler, E.M. Terentjev and M. Warner,

• Critical Stripe-Domain Instability of Nematic Elastomers,

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.

• M. Warner and E.M. Terentjev,

• Nematic Elastomers - A New State of Matter? (big review)
Prog. Polym. Sci. v.21, pp.853-891 (1996).

• G.C. Verwey, M. Warner and E.M. Terentjev,

• Elastic Instability and Stripe Domains in Liquid Crystalline Elastomers,
J. Physique II v.6, p.1273 (1996).

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.

• E.M. Terentjev, M. Warner and G.C. Verwey

• Non-uniform Deformations in Liquid Crystalline Elastomers,
J. Physique II v.6, p.1049 (1996).

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.

• P. Bladon, M. Warner and E.M. Terentjev,

• Orientational Order in Strained Nematic Networks,
Macromolecules v.27, 7067, 1994.

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.

• Orientation of Liquid Crystal Elastomers and Gels by an Electric Field,
J. de Physique II v.4, 667, 1994.

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.

• Deformation-induced Orientational Transitions in Liquid Crystal Elastomers, J. de Physique II, v.4, 75, 1994.

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.

• M.Warner, E.M.Terentjev and P.Bladon, Soft Elasticity' -- Deformations without Resistance in Liquid Crystal Elastomers, J. de Physique II, v.4, 93, 1994.
• 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.

• P.Bladon, E.M.Terentjev and M.Warner, Transitions and Instabilities in Liquid Crystalline Elastomers, Phys. Rev. E, v.47, R3838, 1993.

## Continuum theory of nematic and smectic elastomers and more effects...

• E.M. Terentjev,

• THERMODYNAMIC ORDER AND RANDOM DISORDER IN LIQUID CRYSTAL ELASTOMERS,
Makromol. Symp. v.117, 79 (1997).
See the abstract on the page related to random disorder

• P.D. Olmsted and E.M. Terentjev, Mean Field Nematic-Smectic A Transition in a Random Polymer Network,

• Phys. Rev. E53, p.2444 (1996).
See the abstract on the page related to random disorder

• E.M. Terentjev, M. Warner and T.C. Lubensky,

• Fluctuations and Long- Range Order in Smectic Elastomers,
Europhys. Lett. 30, 343, 1995.

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.

• E.M.Terentjev and M.Warner,

• Continuum Theory of Elasticity and Piezoelectric Effects in Smectic A Elastomers,
J. de Physique II, v.4, 111, 1994.

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.

• E.M.Terentjev and M.Warner,

• Continuum Theory of Ferroelectric Smectic C* Elastomers,
J. de Physique II v.4, 849, 1994.

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.

• T.C.Lubensky, E.M.Terentjev and M. Warner, Layer-Network Coupling in Smectic Elastomers, J. de Physique II v.4, 1457, 1994.

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• E.M.Terentjev, Phenomenological Theory of Non-uniform Nematic Elastomers: Free Energy of Deformations and Electric Field Effects, Europhys. Letters, v.23, 27, 1993.

• 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).