Symmetries and elasticity of nematic gels

Recent Trends in Continuum Modeling of Liquid Crystal Networks: A Mini-Review

This work aims to provide a comprehensive review of the continuum models of the phase behaviors of liquid crystal networks (LCNs), novel materials with various engineering applications thanks to their unique composition of polymer and liquid crystal. Two distinct behaviors are primarily considered: soft elasticity and spontaneous deformation found in the material. First, we revisit these characteristic phase behaviors, followed by an introduction of various constitutive models with diverse techniques and fidelities in describing the phase behaviors. We also present finite element models that predict these behaviors, emphasizing the importance of such models in predicting the material's behavior. By disseminating various models essential to understanding the underlying physics of the behavior, we hope to help researchers and engineers harness the material's full potential. Finally, we discuss future research directions necessary to advance our understanding of LCNs further and enable more sophisticated and precise control of their properties. Overall, this review provides a comprehensive understanding of the state-of-the-art techniques and models used to analyze the behavior of LCNs and their potential for various engineering applications.

Topological Defects in Solids with Odd Elasticity

Crystallography typically studies collections of point particles whose interaction forces are the gradient of a potential. Lifting this assumption generically gives rise in the continuum limit to a form of elasticity with additional moduli known as odd elasticity. We show that such odd elastic moduli modify the strain induced by topological defects and their interactions, even reversing the stability of, otherwise, bound dislocation pairs. Beyond continuum theory, isolated dislocations can self propel via microscopic work cycles active at their cores that compete with conventional Peach-Koehler forces caused, for example, by an ambient torque density. We perform molecular dynamics simulations isolating active plastic processes and discuss their experimental relevance to solids composed of spinning particles, vortexlike objects, and robotic metamaterials.

Symmetries and Dualities in the Theory of Elasticity

Microscopic symmetries impose strong constraints on the elasticity of a crystalline solid. In addition to the usual spatial symmetries captured by the tensorial character of the elastic tensor, hidden nonspatial symmetries can occur microscopically in special classes of mechanical structures. Examples of such nonspatial symmetries occur in families of mechanical metamaterials where a duality transformation relates pairs of different configurations. We show on general grounds how the existence of nonspatial symmetries further constrains the elastic tensor, reducing the number of independent moduli. In systems exhibiting a duality transformation, the resulting constraints on the number of moduli are particularly stringent at the self-dual point but persist even away from it, in a way reminiscent of critical phenomena.

A theoretical model of collective cell polarization and alignment

Collective cell polarization and alignment play important roles in tissue morphogenesis, wound healing and cancer metastasis. How cells sense the direction and position in these processes, however, has not been fully understood. Here we construct a theoretical model based on describing cell layer as a nemato-elastic medium, by which the cell polarization, cell alignment and cell active contraction are explicitly expressed as functions of components of the nematic order parameter. To determine the order parameter we derive two sets of governing equations, one for the force equilibrium of the system, and the other for the minimization of the system's free energy including the energy of cell polarization and alignment. By solving these coupled governing equations, we can predict the effects of substrate stiffness, geometries of cell layers, external forces and myosin activity on the direction- and position-dependent cell aspect ratio and cell orientation. Moreover, the axisymmetric problem with cells on a ring-like pattern is solved analytically, and the analytical solution for cell aspect ratio are governed by parameter groups which include the stiffness of the cell and the substrate, the strength of myosin activity and the external forces. Our predictions of the cell aspect ratio and orientation are generally comparable to experimental observations. These results show that the pattern of cell polarization is determined by the anisotropic degree of active contractile stress, and suggest a stress-driven polarization mechanism that enables cells to sense their spatial positions to develop direction- and position-dependent behavior. This, in turn, sheds light on the ways to control pattern formation in tissue engineering for potential biomedical applications.

Oriented Active Solids

We present a complete analysis of the linearized dynamics of active solids with uniaxial orientational order, taking into account a hitherto overlooked consequence of rotation invariance. Our predictions include a purely active response of two-dimensional orientationally ordered solids to shear, the possibility of stable active solids with quasi-long-range order in two dimensions and long-range order in three dimensions, generic instability of the solid for one sign of active forcing, and the instability of the uniaxially ordered phase in momentum-conserved systems for large active forcing irrespective of its sign.

Tunable wrinkling of thin nematic liquid crystal elastomer sheets

Instabilities in thin elastic sheets, such as wrinkles, are of broad interest both from a fundamental viewpoint and also because of their potential for engineering applications. Nematic liquid crystal elastomers offer a new form of control of these instabilities through direct coupling between microscopic degrees of freedom, resulting from orientational ordering of rodlike molecules, and macroscopic strain. By a standard method of dimensional reduction, we construct a plate theory for thin sheets of nematic elastomer. We then apply this theory to the study of the formation of wrinkles due to compression of a thin sheet of nematic liquid crystal elastomer atop an elastic or fluid substrate. We find the scaling of the wrinkle wavelength in terms of material parameters and the applied compression. The wavelength of the wrinkles is found to be nonmonotonic in the compressive strain due to the presence of the nematic. Finally, due to soft modes, the critical stress for the appearance of wrinkles can be much higher than in an isotropic elastomer and depends nontrivially on the manner in which the elastomer was prepared.

Finsler geometry modeling and Monte Carlo study of liquid crystal elastomers under electric fields

The shape transformation of liquid crystal elastomers (LCEs) under external electric fields is studied through Monte Carlo simulations of models constructed on the basis of Finsler geometry (FG). For polydomain side-chain-type LCEs, it is well known that the external-field-driven alignment of the director is accompanied by an anisotropic shape deformation. However, the mechanism of this deformation is quantitatively still unclear in some part and should be studied further from the microscopic perspective. In this paper, we evaluate whether this shape deformation is successfully simulated, or in other words, quantitatively understood, by the FG model. The FG assumed inside the material is closely connected to the directional degrees of freedom of LC molecules and plays an essential role in the anisotropic transformation. We find that the existing experimental data on the deformations of polydomain LCEs are in good agreement with the Monte Carlo results. It is also found that experimental diagrams of strain versus external voltage of a monodomain LCE in the nematic phase are well described by the FG model.

Topological bounds of bending energy for lipid vesicles

The Helfrich bending energy plays an important role in providing a mechanism for the conformation of a lipid vesicle in theoretical biophysics, which is governed by the principle of energy minimization over configurations of appropriate topological characteristics. We will show that the presence of a quantity called the spontaneous curvature obstructs the existence of a minimizer of the Helfrich energy over the set of embedded ring tori. In addition, despite the well-realized knowledge that lipid vesicles may present themselves in a variety of shapes of complicated topology, there is a lack of topological bounds for the Helfrich energy. To overcome these difficulties, we consider a general scale-invariant anisotropic curvature energy that extends the Canham elastic bending energy developed in modeling a biconcave-shaped red blood cell. We will show that, up to a rescaling of the generating radii, there is a unique minimizer of the energy over the set of embedded ring tori, in the entire parameter regime, which recovers the Willmore minimizer in its Canham isotropic limit. We also show how elevated anisotropy favors energetically a clear transition from spherical-, to ellipsoidal-, and then to biconcave-shaped surfaces, for a lipid vesicle. We then establish some genus-dependent topological lower and upper bounds for the anisotropic energy. Finally, we derive the shape equation of the generalized bending energy, which extends the well-known Helfrich shape equation.

Mathematical Modeling and Simulations for Large-Strain J-Shaped Diagrams of Soft Biological Materials

Herein, we study stress⁻strain diagrams of soft biological materials such as animal skin, muscles, and arteries by Finsler geometry (FG) modeling. The stress⁻strain diagram of these biological materials is always J-shaped and is composed of toe, heel, linear, and failure regions. In the toe region, the stress is almost zero, and the length of this zero-stress region becomes very large (≃150%) in, for example, certain arteries. In this paper, we study long-toe diagrams using two-dimensional (2D) and 3D FG modeling techniques and Monte Carlo (MC) simulations. We find that, except for the failure region, large-strain J-shaped diagrams are successfully reproduced by the FG models. This implies that the complex J-shaped curves originate from the interaction between the directional and positional degrees of freedom of polymeric molecules, as implemented in the FG model.

Liquid crystal elastomer coatings with programmed response of surface profile

Stimuli-responsive liquid crystal elastomers with molecular orientation coupled to rubber-like elasticity show a great potential as elements in soft robotics, sensing, and transport systems. The orientational order defines their mechanical response to external stimuli, such as thermally activated muscle-like contraction. Here we demonstrate a dynamic thermal control of the surface topography of an elastomer prepared as a coating with a pattern of in-plane molecular orientation. The inscribed pattern determines whether the coating develops elevations, depressions, or in-plane deformations when the temperature changes. The deterministic dependence of the out-of-plane dynamic profile on the in-plane orientation is explained by activation forces. These forces are caused by stretching-contraction of the polymer networks and by spatially varying molecular orientation. The activation force concept brings the responsive liquid crystal elastomers into the domain of active matter. The demonstrated relationship can be used to design coatings with functionalities that mimic biological tissues such as skin.

J-shaped stress-strain diagram of collagen fibers: Frame tension of triangulated surfaces with fixed boundaries

We present Monte Carlo data of the stress-strain diagrams obtained using two different triangulated surface models. The first is the canonical surface model of Helfrich and Polyakov (HP), and the second is a Finsler geometry (FG) model. The shape of the experimentally observed stress-strain diagram is called J-shaped. Indeed, the diagram has a plateau for the small strain region and becomes linear in the relatively large strain region. Because of this highly nonlinear behavior, the J-shaped diagram is far beyond the scope of the ordinary theory of elasticity. Therefore, the mechanism behind the J-shaped diagram still remains to be clarified, although it is commonly believed that the collagen degrees of freedom play an essential role. We find that the FG modeling technique provides a coarse-grained picture for the interaction between the collagen and the bulk material. The role of the directional degrees of freedom of collagen molecules or fibers can be understood in the context of FG modeling. We also discuss the reason for why the J-shaped diagram cannot (can) be explained by the HP (FG) model.

Fluidization and Active Thinning by Molecular Kinetics in Active Gels

We derive the constitutive equations of an active polar gel from a model for the dynamics of elastic molecules that link polar elements. Molecular binding kinetics induces the fluidization of the material, giving rise to Maxwell viscoelasticity and, provided that detailed balance is broken, to the generation of active stresses. We give explicit expressions for the transport coefficients of active gels in terms of molecular properties, including nonlinear contributions on the departure from equilibrium. In particular, when activity favors linker unbinding, we predict a decrease of viscosity with activity-active thinning-of kinetic origin, which could explain some experimental results on the cell cortex. By bridging the molecular and hydrodynamic scales, our results could help understand the interplay between molecular perturbations and the mechanics of cells and tissues.

Adaptive rheology and ordering of cell cytoskeleton govern matrix rigidity sensing

Matrix rigidity sensing regulates a large variety of cellular processes and has important implications for tissue development and disease. However, how cells probe matrix rigidity, and hence respond to it, remains unclear. Here, we show that rigidity sensing and adaptation emerge naturally from actin cytoskeleton remodeling. Our in vitro experiments and theoretical modeling demonstrate a bi-phasic rheology of the actin cytoskeleton, which transitions from fluid on soft substrates to solid on stiffer ones. Furthermore, we find that increasing substrate stiffness correlates with the emergence of an orientational order in actin stress fibers, which exhibit an isotropic to nematic transition that we characterize quantitatively in the framework of active matter theory. These findings imply mechanisms mediated by a large-scale reinforcement of actin structures under stress, which could be the mechanical drivers of substrate stiffness dependent cell shape changes and cell polarity.

Alignment and nonlinear elasticity in biopolymer gels

We present a Landau-type theory for the nonlinear elasticity of biopolymer gels with a part of the order parameter describing induced nematic order of fibers in the gel. We attribute the nonlinear elastic behavior of these materials to fiber alignment induced by strain. We suggest an application to contact guidance of cell motility in tissue. We compare our theory to simulation of a disordered lattice model for biopolymers. We treat homogeneous deformations such as simple shear, hydrostatic expansion, and simple extension, and obtain good agreement between theory and simulation. We also consider a localized perturbation which is a simple model for a contracting cell in a medium.

Phase diagram of selectively cross-linked block copolymers shows chemically microstructured gel

We study analytically the intricate phase behavior of cross-linked AB diblock copolymer melts, which can undergo two main phase transitions due to quenched random constraints. Gelation, i.e., spatially random localisation of polymers forming a system-spanning cluster, is driven by increasing the number parameter μ of irreversible, type-selective cross-links between random pairs of A blocks. Self-assembly into a periodic pattern of A/B-rich microdomains (microphase separation) is controlled by the AB incompatibility χ inversely proportional to temperature. Our model aims to capture the system's essential microscopic features, including an ensemble of random networks that reflects spatial correlations at the instant of cross-linking. We identify suitable order parameters and derive a free-energy functional in the spirit of Landau theory that allows us to trace a phase diagram in the plane of μ and χ. Selective cross-links promote microphase separation at higher critical temperatures than in uncross-linked diblock copolymer melts. Microphase separation in the liquid state facilitates gelation, giving rise to a novel gel state whose chemical composition density mirrors the periodic AB pattern.

Alignment localization in nonlinear biological media

Cells imbedded in biopolymer gels are important components of tissue engineering models and cancer tumor microenvironments. In both these cases, contraction of cells attached to the gel is an important phenomenon, and the nonlinear nature of most biopolymers (such as collagen) makes understanding the mechanics of the contraction a challenging problem. Here, we investigate a unique feature of such systems: a point source of contraction leads to substantial deformation of the environment, but large strains and large alignment of the fibers of the gel are confined to a small region surrounding the source. For fibroblasts in collagen-I, we estimate that the radius of this region is of order 90 μ. We investigate this idea using continuum estimates and a finite element code, and we point out experimental manifestations of the effect.

Nematic fluctuations and semisoft elasticity in liquid-crystal elastomers

We give a detailed theory of nematic fluctuations in liquid-crystal elastomers (LCEs) and calculate relaxation rates as obtained by dynamic light scattering (DLS). In ideal LCEs, a nematic state is formed by a spontaneous orientational symmetry breaking of an isotropic state, manifesting itself in an existence of a coupled director-shear soft mode (Goldstone mode). The relaxation rate of the soft mode (a pure bend and a pure splay mode) goes to zero in a long-wavelength limit. In a real, nonideal sample with a locked-in anisotropy, on the other hand, the relaxation rates of these modes become finite. Nonideal elastomers are characterized by a plateau in the stress-strain curve, and the soft mode can be detected only upon stretching to the point of elastic instability at which the director starts to rotate. We use the semisoft model of Gaussian elasticity to derive relaxation rates as a function of deformation for different scattering geometries. We show that the bend-mode relaxation rate goes to zero at the threshold strain, so it is the soft mode. The splay mode, on the other hand, is not soft because the relaxation rate is finite at the threshold strain. We provide experimental evidence and compare DLS measurements of splay and bend modes of two side-chain LCE samples differing in crosslinking densities. Results of both samples are in complete agreement with the predictions of the semisoft model, which indicates that director relaxation properties are not influenced much by the crosslinking conditions.

Gelation via ion exchange in discotic suspensions

The phase behavior of charged disk suspensions displays a strong dependence on ionic strengths, as the interplay between excluded volume and electrostatic interactions determines the formation of glasses, gels, and liquid crystal states. The various ions in natural soil or brine, however, could present additional effects, especially considering that most platelet structures bear a momentous ion-exchange capacity. Here we observed how ion exchange modulates and controls the interaction between individual disks and leads to unconventional phase transitions from isotropic gel to nematic gel and finally to nematic liquid crystals.

Novel mechanical properties in lamellar phases of liquid-crystalline diblock copolymers

Structural properties of flexible nematic diblock copolymers in the lamellar phase are investigated using a mean-field model. We address two complementary questions on the mechanics of the system: 1) How does the nematic order affect the elasticity of the one-dimensional solid? 2) What effect does the block copolymer microstructure has on the orientation of the nematic director? In the limit when the microstructure does not influence the nematic director orientation we predict a soft lamellar compression mode. When the microstructure does influence the nematic director orientation, small compressions lead to conventional elasticity, until a critical strain is reached, where there is then a transition to a softer response. On the other hand, we show that an identifiable lamellar symmetry provides a direction along which the nematic director prefers to align. Our model provides avenues to explore nonlinear properties of flexible diblock copolymers in which the monomers on both sides have mesogenic side groups.

Response of prestretched nematic elastomers to external fields

We investigate the response of prestretched nematic side-chain liquid single-crystal elastomers to superimposed external shear, electric, and magnetic fields of small amplitude. The prestretching direction is oriented perpendicular to the initial nematic director orientation, which enforces director reorientation. Furthermore, the shear plane contains the direction of prestretch. In this case, we obtain a strongly decreased effective shear modulus in the vicinity of the onset and the completion of the enforced director rotation. For the same regions, we find that it becomes comparatively easy to reorient the director by external electric and magnetic fields. These results were derived using conventional elasticity theory and its coupling to relative director-network rotations.

Soft random solids and their heterogeneous elasticity

Spatial heterogeneity in the elastic properties of soft random solids is examined via vulcanization theory. The spatial heterogeneity in the structure of soft random solids is a result of the fluctuations locked-in at their synthesis, which also brings heterogeneity in their elastic properties. Vulcanization theory studies semimicroscopic models of random-solid-forming systems and applies replica field theory to deal with their quenched disorder and thermal fluctuations. The elastic deformations of soft random solids are argued to be described by the Goldstone sector of fluctuations contained in vulcanization theory, associated with a subtle form of spontaneous symmetry breaking that is associated with the liquid-to-random-solid transition. The resulting free energy of this Goldstone sector can be reinterpreted as arising from a phenomenological description of an elastic medium with quenched disorder. Through this comparison, we arrive at the statistics of the quenched disorder of the elasticity of soft random solids in terms of residual stress and Lamé-coefficient fields. In particular, there are large residual stresses in the equilibrium reference state, and the disorder correlators involving the residual stress are found to be long ranged and governed by a universal parameter that also gives the mean shear modulus.

Polarization dependence of optically driven polydomain elastomer mechanics

We model how polarized and unpolarized light can cause mechanical response in polydomain nematic and related photoelastomers. The reduction of order by heating and the consequential large strains that are known from nematic elastomers can alternatively be caused by light-induced bending of rodlike dye molecules which then equally reduce the order of their nematic hosts. While there is no mechanical response to heating of polydomain elastomers, mechanical response to light is possible by the selective absorption of light according to how domains are aligned with respect to the polarization direction or with respect to the propagation direction in the case of unpolarized light. We find large contractions or elongations, depending on the nature of polarization. The responses are nonmonotonic with light intensity.

Interplay between liquid crystalline and isotropic gels in self-assembled neurofilament networks

Neurofilaments (NFs) are a major constituent of nerve cell axons that assemble from three subunit proteins of low (NF-L), medium (NF-M), and high (NF-H) molecular weight into a 10 nm diameter rod with radiating sidearms to form a bottle-brush-like structure. Here, we reassemble NFs in vitro from varying weight ratios of the subunit proteins, purified from bovine spinal cord, to form homopolymers of NF-L or filaments composed of NF-L and NF-M (NF-LM), NF-L and NF-H (NF-LH), or all three subunits (NF-LMH). At high protein concentrations, NFs align to form a nematic liquid crystalline gel with a well-defined spacing determined with synchrotron small angle x-ray scattering. Near physiological conditions (86 mM monovalent salt and pH 6.8), NF-LM networks with a high NF-M grafting density favor nematic ordering whereas filaments composed of NF-LH transition to an isotropic gel at low protein concentrations as a function of increasing mole fraction of NF-H subunits. The interfilament distance decreases with NF-M grafting density, opposite the trend seen with NF-LH networks. This suggests a competition between the more attractive NF-M sidearms, forming a compact aligned nematic gel, and the repulsive NF-H sidearms, favoring a more expansive isotropic gel, at 86 mM monovalent salt. These interactions are highly salt dependent and the nematic gel phase is stabilized with increasing monovalent salt.

Nematic elastomers: from a microscopic model to macroscopic elasticity theory

A Landau theory is constructed for the gelation transition in cross-linked polymer systems possessing spontaneous nematic ordering, based on symmetry principles and the concept of an order parameter for the amorphous solid state. This theory is substantiated with help of a simple microscopic model of cross-linked dimers. Minimization of the Landau free energy in the presence of nematic order yields the neoclassical theory of the elasticity of nematic elastomers and, in the isotropic limit, the classical theory of isotropic elasticity. These phenomenological theories of elasticity are thereby derived from a microscopic model, and it is furthermore demonstrated that they are universal mean-field descriptions of the elasticity for all chemical gels and vulcanized media.

Nonlinear relative rotations in liquid crystalline elastomers

Relative rotations between the coupled subsystems of a complex material can become crucial in continuum modeling. In this paper the authors focus on the macroscopic description of side-chain liquid crystalline elastomers, where relative rotations between the polymer network and the director orientation associated with the liquid crystalline component are decisive. They extend the known expression for relative rotations to the nonlinear regime, within the framework of a continuum characterization of the materials. This allows the investigation of qualitatively different nonlinear effects determined by relative rotations, and they give an illustrative example. The formalism can easily be transferred to the macroscopic description of magnetic gels and will certainly be helpful in the characterization of other complex systems.

Density functional theory and molecular dynamics investigations on substituted banana-shaped compounds

Density functional theory (DFT) calculations and molecular dynamics (MD) simulations on the atomic level were performed on three different substituted banana-shaped compounds derived from 1,3-phenylene bis[4-(4-n-hexyloxyphenyliminomethyl)benzoate] (P-6-O-PIMB). The DFT studies were carried out on the isolated molecules, and in the MD simulations clusters were treated with up to 64 monomers. The effect of polar substituents, such as chlorine and the nitro group, on the central 1,3-phenylene unit of banana-shaped compounds was investigated. In particular, flexibility, polarity, electrostatic potential (ESP) group charge distributions, B-factors, bending angles and molecular lengths were considered. The MD results were analysed by trajectories of significant torsion angles as well as order parameters such as radial atom pair distribution functions g(r), orientational correlation functions g(o), diffusion coefficients (D) and root mean square deviations (RMSD) values. The g(r) and g(o) values show that a certain long range order is generated by the introduction of a NO(2) group in the 2-position of the central 1,3-phenylene ring. In contrast, the chlorination at the 4 and 6 positions of the central 1,3-phenylene unit decreases the long range order tendency by its perturbation effect on the conformations in such molecules. Moreover, g(r) and g(o) values, as well as diffusion coefficients, show that in the NO(2) substituted compound the formation of microphase areas is preferred. Finally, the aggregation effect in such compounds was studied in a systematic way by a comparison of the conformational properties of the isolated molecules and the monomers in the clusters.

Smectic elastomer membranes

We present a model for smectic elastomer membranes which includes elastic and liquid-crystalline degrees of freedom. Based on our model, we determined the qualitative phase diagram of a smectic elastomer membrane using mean-field theory. This phase diagram is found to comprise five phases, viz., smectic- A -flat, smectic- A -crumpled, smectic- C -flat, smectic- C -crumpled, and smectic- C -tubule phases, where in the latter phase, the membrane is flat in the direction of mesogenic tilt and crumpled in the perpendicular direction. The transitions between adjacent phases are second-order phase transitions. We study in some detail the elasticity of the smectic- C -flat and the smectic- C -tubule phases which are associated with a spontaneous breaking of in-plane rotational symmetry. As a consequence of the Goldstone theorem, these phases exhibit soft elasticity characterized by the vanishing of in-plane shear moduli.

Dynamics of smectic elastomers

We study the low-frequency, long-wavelength dynamics of liquid crystal elastomers, crosslinked in the smectic-A phase, in their smectic-A, biaxial smectic and smectic-C phases. Two different yet related formulations are employed. One formulation describes the pure hydrodynamics and does not explicitly involve the Frank director, which relaxes to its local equilibrium value in a nonhydrodynamic time. The other formulation explicitly treats the director and applies beyond the hydrodynamic limit. We compare the low-frequency, long-wavelength dynamics of smectic-A elastomers to that of nematics and show that the two are closely related. For the biaxial smectic and the smectic-C phases, we calculate sound velocities and the mode structure in certain symmetry directions. For the smectic-C elastomers, in addition, we discuss in some detail their possible behavior in rheology experiments.

Tricritical point induced by smectic ordering of a nematic gel

The authors study volume phase transitions of a nematic gel immersed in a liquid crystal (LC) solvent, which shows a second-order nematic-smectic A phase transition (NST). Combining Flory's elastic energy [Principles of Polymer Chemistry (Cornell University Press, Ithaca, 1953)] for a swelling of the gel with the McMillan model [Phys. Rev. A 4, 1238 (1971)] for smectic ordering, the authors calculate the equilibrium swelling of the gel and smectic order parameters as a function of temperature. The authors take into account an attractive interaction parameter c between the gel and LC solvents. On increasing the value of the coupling constant c, a second-order NST of the gel is changed to a first-order one and a continuous volume phase transition of the gel is changed to a discontinuous one. The authors find a tricritical point of the gel induced by smectic ordering.

Soft elasticity in biaxial smectic and smectic-C elastomers

Ideal (monodomain) smectic-A elastomers cross-linked in the smectic-A phase are simply uniaxial rubbers, provided deformations are small. From these materials smectic-C elastomers are produced by a cooling through the smectic-A to smectic-C phase transition. At least in principle, biaxial smectic elastomers could also be produced via cooling from the smectic-A to a biaxial smectic phase. These phase transitions, respectively, from Dinfinityh to C2h and from Dinfinityh to D2h symmetry, spontaneously break the rotational symmetry in the smectic planes. We study the above transitions and the elasticity of the smectic-C and biaxial phases in three different but related models: Landau-like phenomenological models as functions of the Cauchy-Saint-Laurent strain tensor for both the biaxial and the smectic-C phases and a detailed model, including contributions from the elastic network, smectic layer compression, and smectic-C tilt for the smectic-C phase as a function of both strain and the c-director. We show that the emergent phases exhibit soft elasticity characterized by the vanishing of certain elastic moduli. We analyze in some detail the role of spontaneous symmetry breaking as the origin of soft elasticity and we discuss different manifestations of softness like the absence of restoring forces under certain shears and extensional strains.

Shear mechanical anisotropy of side chain liquid-crystal elastomers: influence of sample preparation

We study the mechanical anisotropy of a series of uniaxial side chain nematic elastomers prepared with the same chemical composition but with different preparation protocols. For all the compounds, the experiments performed as a function of temperature show no discontinuity in both G' (//) and G' ( perpendicular) (the labels // and perpendicular stand for the director parallel, respectively perpendicular to the shear displacement) around the nematic-isotropic (N-I) phase transition temperature determined by DSC. They also all show a small decrease in G' (//) starting at temperatures well above this temperature (from approximately 4( degrees ) C to approximately 20( degrees ) C depending on the compound studied) and leading to a small hydrodynamic value of the G' ( perpendicular)/G' (//) ratio. The measurements taken as a function of frequency show that the second plateau in G' (//) and the associated dip in G (//)" expected from dynamic semi-soft elasticity are not observed. These results can be described by the de Gennes model, which predicts small elastic anisotropy in the hydrodynamic and linear regimes. They correspond to the behavior expected for compounds beyond the mechanical critical point, which is consistent with the NMR and specific heat measurements taken on similar compounds. We also show that a reduction in the cross-linking density does not change the non-soft character of the mechanical response. From the measurements taken as a function of frequency at several temperatures we deduce that the time-temperature superposition method does not apply. From these measurements, we also determine the temperature dependence of the longest relaxation time tau(E) of the network for the situations where the director is either parallel or perpendicular to the shear velocity. Finally, we discuss the influence on the measurements of the mechanical constraint associated with the fact that the samples cannot change their shape around the pseudo phase transition, because of their strong adherence on the sample-bearing glass slides.

Selected macroscopic properties of liquid crystalline elastomers

In this short review we give an overview of selected macroscopic properties of sidechain liquid crystalline elastomers (LCEs) focusing on three closely related topics (a) the influence of relative rotations between the director and the strain field on various reorientation instabilities, (b) the nonlinear stress-strain curves for the polydomain-monodomain transition and for the reorientation transition in LCE monodomains and (c) the shear mechanical response of LCEs in the linear regime. We consider only already existing real materials and do not discuss hypothetical "ideal" systems. We conclude that all observations reported to date can be accounted for without invoking the concept of soft elasticity, but instead relying on macroscopic dynamics in the linear and the nonlinear domain.

Structure of nematic liquid crystalline elastomers under uniaxial deformation

We have used in situ x-ray diffraction and calorimetry to study liquid crystalline elastomers prepared using a one-step photopolymerization method. We used suspended weights to stretch free-standing crystalline elastomer films. With the mechanical stress parallel to the initial director, we observed a gradual nematic to isotropic transition with increasing temperature. The thermal evolution of the nematic order parameter on cooling, together with the observation of isotropic-nematic coexistence over a broad temperature range, suggests that the heterogeneity in the samples introduces a distribution of transition temperatures. With the mechanical stress perpendicular to the initial director, we observed both uniform director rotation and stripe formation, depending on the details of sample preparation.

Phases and transitions in phantom nematic elastomer membranes

Motivated by recently discovered unusual properties of bulk nematic elastomers, we study a phase diagram of liquid-crystalline polymerized phantom membranes, focusing on in-plane nematic order. We predict that such membranes should generically exhibit five phases, distinguished by their conformational and in-plane orientational properties: namely, isotropic-crumpled, nematic-crumpled, isotropic-flat, nematic-flat, and nematic-tubule phases. In the nematic-tubule phase, the membrane is extended along the direction of spontaneous nematic order and is crumpled in the other. The associated spontaneous symmetries breaking guarantees that the nematic tubule is characterized by a conformational-orientational soft (Goldstone) mode and the concomitant vanishing of the in-plane shear modulus. We show that long-range orientational order of the nematic tubule is maintained even in the presence of harmonic thermal fluctuations. However, it is likely that tubule's elastic properties are qualitatively modified by these fluctuations, which can be studied using a nonlinear elastic theory for the nematic tubule phase that we derive at the end of this paper.

Spontaneous flux lattices in ferromagnetic spin-triplet superconductors

A theory is developed for the spontaneous vortex lattice that is expected to occur in the ferromagnetic superconductors ZrZn2, UGe2, and URhGe, where the superconductivity is likely of the spin-triplet nature. The long-wavelength fluctuations of this spontaneous flux lattice are predicted to be huge compared to those of a conventional flux lattice, and to be the same as those for spin-singlet ferromagnetic superconductors. It is shown that these fluctuations lead to unambiguous experimental signatures which may provide the easiest way to observe the spontaneous flux lattice.

Commentary on "Mechanical properties of monodomain side chain nematic elastomers" by P. Martinoty et al

We discuss the rheology experiments on nematic elastomers by Martinoty et al. in the light of theoretical models for the long-wavelength low-frequency dynamics of these materials. We review these theories and discuss how they can be modified to provide a phenomenological description of the non-hydrodynamic frequency regime probed in the experiments. Moreover, we review the concepts of soft and semi-soft elasticity and comment on their implications for the experiments.

Commentary on "Mechanical properties of monodomain side-chain nematic elastomers" by P. Martinoty, P. Stein, H. Finkelmann, H. Pleiner and H.R. Brand

We discuss the background to static and dynamic soft elasticity. The evidence in the static case and the symmetry basis for soft and semi-soft elasticity is well understood. By contrast the dynamic analogy is less clear. Lack of clean time scale separation clouds the interpretation of director relaxation keeping up, or not, with imposed strains. However, the reduction in modulus between geometries obtaining at low frequencies and being lost at high frequencies confirms that director reaction indeed determines dynamical semi-softness.

Landau-de Gennes model for nonuniform configurations in nematic liquid crystalline elastomers

Through a mean field theory, an elastic free energy describing the nonuniform elastic textures observed in nematic elastomers, is proposed. To construct it, an order parameter that describes the nematic-isotropic phase transition through the change of the elastic properties of the strain tensor at the transition point is introduced. The resulting elastic free energy can be written in a form that resembles the Frank free energy of the usual nematic liquid crystals, becoming equivalent to it when the size of the elastic nematic domains is a fixed constant along the whole sample. Using this approach, a model for nonhomogeneous deformations found by Macromolecules 33, 7675 (2000)] in elastomeric thin films of urethane/urea is proposed.

Anomalous elasticity of nematic and critically soft elastomers

Uniaxial elastomers are characterized by five elastic constants. If their elastic modulus C5 describing the energy of shear strains in planes containing the anisotropy axis vanishes, they are said to be soft. In spatial dimensions d less than or equal to 3, soft elastomers exhibit anomalous elasticity with certain length-scale-dependent bending moduli that diverge and shear moduli that vanish at large length scales. Using renormalized field theory at d=3 and to first order in epsilon=3-d, we calculate critical exponents and other properties characterizing the anomalous elasticity of two soft systems: (i) nematic elastomers in which softness is a manifestation of a Goldstone mode induced by the spontaneous symmetry breaking associated with a transition from an isotropic state to a nematic state, and (ii) a particular version of what we call a critically soft elastomer in which C(5)=0 corresponds to a critical point terminating the stability regime of a uniaxial elastomer with C5>0.

Fluctuating nematic elastomer membranes

We study the flat phase of nematic elastomer membranes with rotational symmetry spontaneously broken by an in-plane nematic order. Such a state is characterized by a vanishing elastic modulus for simple shear and soft transverse phonons. At harmonic level, the in-plane orientational (nematic) order is stable to thermal fluctuations that lead to short-range in-plane translational (phonon) correlations. To treat thermal fluctuations and relevant elastic nonlinearities, we introduce two generalizations of two-dimensional membranes in a three-dimensional space to arbitrary D-dimensional membranes embedded in a d-dimensional space and analyze their anomalous elasticities in an expansion about D=4. We find a stable fixed point that controls long-scale properties of nematic elastomer membranes. It is characterized by singular in-plane elastic moduli that vanish as a power law eta(lambda)=4-D of a relevant inverse length scale (e.g., wave vector) and a finite bending rigidity. Our predictions are asymptotically exact near four dimensions.

Universal elasticity and fluctuations of nematic gels

We study elasticity of spontaneously orientationally ordered amorphous solids, characterized by a vanishing transverse shear modulus, as realized by nematic elastomers and gels. We show that local heterogeneities and elastic nonlinearities conspire to lead to anomalous nonlocal universal elasticity controlled by a nontrivial infrared fixed point. Namely, such solids are characterized by universal shear and bending moduli that, respectively, vanish and diverge at long scales, are universally incompressible, and exhibit a universal negative Poisson ratio and a non-Hookean elasticity down to arbitrarily low strains. Based on expansion about five dimensions, we argue that the nematic order is stable to thermal fluctuation and local heterogeneities down to d(lc)<3.

Semisoft elasticity and director reorientation in stretched sheets of nematic elastomers

A two-dimensional effective model for the semisoft elastic behavior of nematic elastomers is derived in the thin film limit. The model is used to investigate numerically the force-stretch curves and the deformed shape, and to resolve the local patterns in the director orientation in a stretching experiment. From the force-stretch curves we recover the two critical stretches which mark the transition from hard to soft and back to hard response. We present an analytical model for their dependence on the aspect ratio of the sample, and compare it with numerical results.

Isotropic-cholesteric transition in liquid-crystalline gels

In a nematic gel, the appearance of nematic order is accompanied by a spontaneous elongation of the gel parallel to the nematic director. If such a gel is made chiral, it has a tendency to form a cholesteric helical texture, in which local elongation of the gel parallel to the nematic director is suppressed due to the requirement of elastic compatibility. We show that a conical helix in which the director makes an oblique angle with respect to the helix axis serves as an energy minimizing compromise between the competing tendencies for elongation and twisting. We find the dependence of the helical cone angle and pitch on the strength of the chirality, and determine the change in sample shape at the isotropic to cholesteric phase transition.