Deformation of cholesteric elastomers by uniaxial stress along the helix axis

The deformation of cholesteric elastomers by mechanical stress applied parallel to the helix axis is studied by calculation of the free-energy density. The Frank-elasticity contribution is taken into account. A chiral solvent, present at cross-linking time, is in general considered to be replaced after cross-linking by a solvent with different chirality. Two special cases considered are zero and unchanged solvent chirality, the first known as that of imprinted cholesteric elastomers, the latter equivalent to intrinsic cholesteric elastomers with chemically attached chiral groups. Depending on material parameters and imposed strain, the director can show a tilt towards the helix axis up to the maximum tilt, corresponding to a nematic state. In case of intrinsic elastomers with low conformation anisotropy, direct transitions from untilted to nematic states can be induced by straining. The helix structure of the director field is coarsened with an average wave number different to that of the information inscribed in the network at cross-linking time, if this lowers the average free-energy density. Switching between different states can be achieved with electric fields of reasonable values applied parallel to the helix axis. Spectra of the reflection of polarized light are calculated.     

Thermal diffusion and bending kinetics in nematic elastomer cantilever

Vertically aligned monodomain nematic liquid-crystal elastomers contract when heated. If a temperature gradient is applied across the width of such a cantilever, inhomogeneous strain distribution leads to bending motion. We modelled the kinetics of thermally induced bending in the limit of a long thin strip and the predicted time variation of curvature agreed quantitatively with experimental data from samples with a range of critical indices and nematic-isotropic transition temperatures. We also deduced a value for the thermal diffusion coefficient of the elastomer.     

Semisoft nematic elastomers and nematics in crossed electric and magnetic fields

Nematic elastomers with a locked-in anisotropy direction exhibit semisoft elastic response characterized by a plateau in the stress-strain curve in which stress does not change with strain. We calculate the global phase diagram for a minimal model, which is equivalent to one describing a nematic in crossed electric and magnetic fields, and show that semisoft behavior is associated with a broken symmetry biaxial phase and that it persists well into the supercritical regime. We also consider generalizations beyond the minimal model and find similar results.     

Phase transition in liquid crystal elastomers—A Monte Carlo study employing non-Boltzmann sampling

We investigate nematic-isotropic transition in liquid crystal elastomers employing a variant of Wang-Landau sampling. This technique facilitates calculation of the density of states from which other thermodynamic properties can be obtained. We consider a lattice model of a liquid crystal elastomer and a Hamiltonian which accounts for interactions among liquid crystalline units and interaction of local nematics with global strain. We investigate the effect of varying the strength of coupling between nematic and orientational degrees of freedom. When the local director is coupled strongly to the global strain, the transition is strongly first order. When the strength of the coupling decreases the transition becomes weakly first order. The transition temperature decreases when the coupling becomes weaker. We also report for the first time results on variation of free energy as a function of average energy at different temperatures and coupling constants.     

Anomalous viscoelastic response of nematic elastomers

We report a combined theoretical and experimental study of linear viscoelastic response in oriented monodomain nematic elastomers. The model predicts a dramatic decrease in the dynamic modulus in certain deformation geometries in an elastic medium with an independently mobile internal degree of freedom, the nematic director with its own relaxation dynamics. Dynamic mechanical measurements on monodomain nematic elastomers confirm our predictions of dependence on shear geometry and on nematic order, and also show a very substantial mechanical loss clearly associated with director relaxation.     

Elastic energies for nematic elastomers

We discuss several elastic energies for nematic elastomers and their small strain expansions both in the regime of large director rotations, and in the case that director changes are small. We propose two fully non-linear model anisotropic energies and compare the behavior they predict with the currently available experimental evidence.     

Strongly anisotropic elastic moduli of nematic elastomers: Analytical expressions and nonlinear temperature dependence

Exact formulae for the elastic moduli of the nematic elastomers are obtained by the implicit function method based on somewhat general energy functions. The formulae indicate that both the moduli parallel and perpendicular to the director of the nematic elastomers are smaller than the modulus of the classical elastomers because of the mechanical-nematic coupling. Moreover, the moduli are generally anisotropic due to the biaxiality induced by stretching the nematic elastomers perpendicular to the director. Then we get the explicit analytical expressions of the parallel and perpendicular moduli by making use of the Landau-de Gennes free energy and the neo-classical elastic energy. Very different from the classical elastomers, they are both strongly nonlinear functions of the temperature in the nematic phase. Furthermore, their ratio, the degree of anisotropy, changes with the temperature as well. The results agree qualitatively with some experiments. Better quantitative agreement is obtained by some modifications of the constitutive relation of the elastic energy.     

Nonlinear photoresponse of disordered elastomers

We model the polarization-dependent photocontractions of polydomain nematic elastomers. Contraction initially arises by light-induced director rotation. At higher light intensity, strain recovers because the local order parameter diminishes. Thus we predict photostrains to be nonmonotonic with light intensity and predict the induction of a negative global order parameter for the system of polydomains. Nonmonotonic strains would give curvature reversal in thick samples. Our model also predicts an elongational strain response to nonpolarized light.     

The elastic anisotropy of nematic elastomers

We examine the robustness of order in nematic elastomers under mechanical strains imposed along and perpendicularly to the director when director rotation is prohibited. In contrast to electric and magnetic fields applied to conventional nematics, mechanical fields are shown theoretically and experimentally to greatly affect the degree of nematic order and related quantities. Unlike in liquid nematics, one can impose fields perpendicular to the director, thereby inducing biaxial order which should be susceptible to experimental detection. Nematic elastomers with unchanging director and degree of order should theoretically have the same elastic moduli for longitudinal and transverse extensions. This is violated when nematic order is permitted to relax in response to strains. Near the transition we predict the longitudinal modulus to be smaller than the transverse modulus; at lower temperatures the converse is true, with a crossover a few degrees below the transition. The differences are ascribed to the different temperature dependence of the stiffness of uniaxial and biaxial order. We synthesised side chain single-crystal nematic polymer networks, performed DSC, X-ray, birefringence, and thermo-mechanical characterisations, and then obtained linear moduli from stress-strain measurements. Received 29 September 2000     

Hairpin rubber elasticity

We model the elastic properties of main chain liquid crystalline elastomers, formed by cross linking chains in a strongly nematic state, when they have hairpin defects. We study the response of the elastomer to imposed uniaxial extension along the nematic direction, and employ a microscopic model of how the deformation is distributed non-affinely amongst the hairpin and straight chain populations. The rubber shows a plateau in the stress as a function of the elongation imposed along the director. It is a consequence of the depletion of the actively stretching population of hairpin chains and should not be confused with soft elasticity effects associated with director rotation.     

Elastic effects in disordered nematic networks

Elastic effects in a model of disordered nematic elastomers are numerically investigated in two dimensions. Networks crosslinked in the isotropic phase exhibit an unusual soft mechanical response against stretching. It arises from a gradual alignment of orientationally correlated regions that are elongated along the director. A sharp crossover to a macroscopically aligned state is obtained on further stretching. The effect of random internal stress is also discussed.     

Dynamic soft elasticity in monodomain nematic elastomers

We study the linear dynamic-mechanical response of monodomain nematic liquid crystalline elastomers under shear in the geometry that allows the director rotation. The aspects of time-temperature superposition are discussed at some length and Master Curves are obtained between the glassy state and the nematic transition temperature Tni. However, the time-temperature superposition did not work through the clearing point Tni, due to the transition from the "soft-elasticity" nematic regime to the ordinary isotropic rubber response. We focus on the low-frequency region of the Master Curves and establish the power law dependence of the modulus G' alpha w(a). This law agrees very well with the results of the static stress relaxation, where each relaxation curve obeys the analogous power law G' alpha t(-a) in the corresponding region of long times and temperatures.     

A Generalized High-Elasticity Model to Describe the Stress-Strain Dependence for Polyurethane Elastomers When Stretched at a Constant Rate

The stress versus strain relationships for cross-linked polyetherurethane and polybutadieneurethane elastomers were investigated on samples when simply stretched at a constant rate. Use of the generalized approach earlier developed for equilibrium strain, with related improvements taking into account the relaxation nature of the elastomers, permitted successfully describing the investigated dependences.     

The role of Frank elasticity in cholesteric elastomers

The effect of Frank elasticity on deformations of cholesteric elastomers by mechanical stress applied perpendicular to the helix axis is studied by numerical minimization of the free energy. Above a critical strain, a solution with an only oscillating director is found to be stable in comparison to a distorted helix. At the critical strain, the contractions perpendicular to stress change discontinuously. The critical strain is found to increase with increasing Frank elasticity contribution to the free energy density, and to diverge when the conformation anisotropy of the polymer backbone vanishes. The results are compared with recent experiments which indicated that, in case of weak conformation anisotropies, the Frank elasticity contribution to the free energy cannot be neglected.     

Stress–Strain Dependence of Cross-Linked Single-Phase Polyether Urethane

Stress–strain dependencies of cross-linked elastomers under stretching are considered. The capabilities to describe stress–strain plots of different types for elastomers, including curves with an inflection, by high-elasticity theories are examined. Limitations in the use of common high-elasticity models in the theoretical calculations of stress–strain plots are revealed. The analysis shows that the stress–strain plots with an inflection can be obtained with appropriate adjustment of the concept of finite chain extensibility. We propose a combined two-parameter model based on this concept and on the theory of high elasticity, taking into account the spatial constraints at the initial stage of stretching of the polymer network. Abilities of the new approach to describe the stress–strain dependencies in an adequate manner are demonstrated for five samples of cross-linked polyurethane over the entire range of strain under stretching.     

Calorimetric study of the Paranematic-to-Nematic transition of polydomain side-chain liquid-crystalline elastomers with different mesogen composition

The phase transition behaviour of various nematic side-chain liquid-crystalline elastomers with different mesogen composition has been explored by means of high-resolution ac calorimetry. Polydomain samples of the same crosslinking density and different type of mesogens have been investigated. The results show a strong dependence of the phase transition features upon the composition of the mesogen. The distance from the critical point, reflected in the sharpness of the heat capacity anomalies, increases when adding a shorter-length mesogen. The results provide new insight for the impact of mesogens on the thermodynamic behaviour and, thus, on the thermomechanical response of nematic liquid-crystalline elastomers.     

Algebraic Theory of Linear Viscoelastic Nematodynamics

This paper consists of two parts. The first one develops algebraic theory of linear anisotropic nematic “N-operators” build up on the additive group of traceless second rank 3D tensors. These operators have been implicitly used in continual theories of nematic liquid crystals and weakly elastic nematic elastomers. It is shown that there exists a non-commutative, multiplicative group N₆ of N-operators build up on a manifold in 6D space of parameters. Positive N-operators, which in physical applications hold thermodynamic stability constraints, do not generally form a subgroup of group N₆. A three-parametric, commutative transversal-isotropic subgroup of positive symmetric nematic operators is also briefly discussed. The special case of singular, non-negative symmetric N-operators reveals the algebraic structure of nematic soft deformation modes. The second part of the paper develops a theory of linear viscoelastic nematodynamics applicable to liquid crystalline polymer. The viscous and elastic nematic components in theory are described by using the Leslie–Ericksen–Parodi (LEP) approach for viscous nematics and de Gennes free energy for weakly elastic nematic elastomers. The case of applied external magnetic field exemplifies the occurrence of non-symmetric stresses. In spite of multi-(10) parametric character of the theory, the use of nematic operators presents it in a transparent form. When the magnetic field is absent, the theory is simplified for symmetric case with six parameters, and takes an extremely simple, two-parametric form for viscoelastic nematodynamics with possible soft deformation modes. It is shown that the linear nematodynamics is always reducible to the LEP-like equations where the coefficients are changed for linear memory functionals whose parameters are calculated from original viscosities and moduli.