Experimentally it is possible to manipulate the director in a (chiral) smectic-$A$ elastomer using an electric field. This suggests that the director is not necessarily locked to the layer normal, as described in earlier papers that extended rubber elasticity theory to smectics. Here, we consider the case that the director is weakly anchored to the layer normal assuming that there is a free energy penalty associated with relative tilt between the two. We use a recently developed weak-anchoring generalization of rubber elastic approaches to smectic elastomers and study shearing in the plane of the layers, stretching in the plane of the layers, and compression and elongation parallel to the layer normal. We calculate, inter alia, the engineering stress and the tilt angle between director and layer normal as functions of the applied deformation. For the latter three deformations, our results predict the existence of an instability towards the development of shear accompanied by smectic-$C$-like order.
Smectic-C elastomers can be prepared by crosslinking, e.g., liquid crystal polymers, in the smectic-A phase followed by a cooling through the smectic-A to smectic-C phase transition. This transition from $D_{\infty h}$ to $C_{2h}$ symmetry spontaneously breaks rotational symmetry in the smectic plane as does the transition from a smectic-A to a biaxial smectic phase with $D_{2h}$ symmetry. We study these transitions and the emergent elasticity of the smectic-C and biaxial phases in three related models and show that these phases exhibit soft elasticity analogous to that of nematic elastomers.
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, 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…
Stenull and Lubensky [Phys. Rev. E {\bf 76}, 011706 (2007)] have argued that shear strain and tilt of the director relative to the layer normal are coupled in smectic elastomers and that the imposition of one necessarily leads to the development of the other. This means, in particular, that a Smectic-A elastomer subjected to a simple shear will develop Smectic-C-like tilt of the director. Recently, Kramer and Finkelmann [arXiv:0708.2024, Phys. Rev. E {\bf 78}, 021704 (2008)] performed shear experiments on Smectic-A elastomers using two different shear geometries. One of the experiments, which …
We study two aspects of the elasticity of smectic-$A$ elastomers that make these materials genuinely and qualitatively different from conventional uniaxial rubbers. Under strain applied parallel to the layer normal, monodomain smectic-$A$ elastomers exhibit a drastic change in Young's modulus above a threshold strain value of about 3%, as has been measured in experiments by Nishikawa and Finkelmann [Macromol. Chem. Phys. {\bf 200}, 312 (1999)]. Our theory predicts that such strains induce a transition to a smectic-$C$-like state and that it is this transition that causes the change in elastic …
We present a theory for the low-frequency, long-wavelength dynamics of soft smectic-C elastomers with locked-in smectic layers. Our theory, which goes beyond pure hydrodynamics, predicts a dynamic soft elasticity of these elastomers and allows us to calculate the storage and loss moduli relevant for rheology experiments as well as the mode structure.
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 non-hydrodynamic 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 ne…
We present a detailed analysis of the molecular packing of a strained liquid crystal elastomer composed of chiral mesogens in the smectic A phase. X-ray diffraction patterns of the elastomer collected over a range of orientations with respect to the X-ray beam were used to reconstruct the three-dimensional scattering intensity as a function of tensile strain. For the first time, we show that the smectic domain order is preserved in these strained elastomers. Changes in the intensity within a given scattering plane are due to reorientation, and not loss, of the molecular order in directions ort…
The reorientation behaviour of a smectic-A (SmA) elastomer deformed parallel to the smectic layer normal has been interpreted as a Sm-C like transition by Stenull and Lubensky. Experiments, however, prove that such a transition does not occur.
Ideal (monodomain) smectic-$A$ elastomers crosslinked 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 $D_{\infty h}$ to $C_{2h}$ and from $D_{\infty h}$ to $D_{2h}$ symmetry, spontaneously break the rotational symmetry in the smectic planes. We study the ab…
Models of smectic-C liquid crystal elastomers predict that it can display soft elasticity, in which the shape of the elastomer changes at no energy cost. The amplitude of the soft mode and the accompanying shears are dependent on the orientation of the layer normal and the director with respect to the stretch axis. We demonstrate that in some geometries the director is forced to rotate perpendicular to the stretch axis, causing lateral expansion of the sample; a negative Poisson's ratio. Current models do not include the effect of imperfections that must be present in the physical sample. We i…
We study anomalous elasticity in the tubule phases of nematic and smectic elastomer membranes, which are flat in one direction and crumpled in another. These phases share the same macroscopic symmetry properties including spontaneously-broken in-plane isotropy and hence belong to the same universality class. Below an upper critical value $D_c =3$ of the membranes' intrinsic dimension D, thermal fluctuations renormalize the elasticity with respect to elastic displacements along the tubule axis so that elastic moduli for compression along the tubule axis and for bending the tubule axis become le…
Elastomers are artificial polymeric materials created for industrial and commercial applications. Depending on their purpose, they are performing in different species and structure modifications. Our studies focus on the systems of elastomers randomly standing-distributed in a smectic $A$ (Sm$A$) liquid crystal. Basing on the suggestion following from the experiment, that at a phase transition from Sm$A$ to nematic phase caused by an increase of a crosslink concentration, such a system survives a percolation transition at low crossilink concentrations, we propose a modeling explaining this phe…
We present a numerical study of stretching monodomain smectic-A elastomer sheets, computed using the finite element method. When stretched parallel to the layer normal the microscopic layers in smectic elastomers are unstable to a transition to a buckled state. We account for the layer buckling by replacing the microscopic energy with a coarse grained effective free energy that accounts for the fine scale deformation of the layers. We augment this model with a term to describe the energy of deforming buckled layers, which is necessary to reproduce the experimentally observed Poisson's ratios p…
We analytically derive the effective layer elastic energy of smectic-A and smectic-C liquid crystals by adiabatic elimination of the orientational degree of freedom from the generalized Chen-Lubensky model. In the smectic-A phase, the effective layer bending elastic modulus is calculated as a function of the wavelength of the layer undulation mode. It turns out that an unlocking of the layer normal and the director reduces the layer bending rigidity for wavelengths smaller than the director penetration length. In the achiral smectic-C phase, an anisotropic bending elasticity appears due to the…
Structural studies by synchrotron x-ray diffraction on two main-chain smectic-C elastomers reveal the presence of two different relaxation mechanisms in these systems at low and high strains. At low strains, the smectic layers are reoriented with layer-normals distributed in a plane perpendicular to the stretch direction. The system relaxes relatively slowly (time-constant ~ 45 minutes) which is attributed to the flow properties of the LC layers embedded in the elastomer network. At high strains, the equilibration time (~ 4 - 8 minutes) conforms to a faster relaxation and appears to have its o…
In this paper, the two-dimensional pure bending of a hyperelastic substrate coated by a nematic liquid crystal elastomer (abbreviated as NLCE) is studied within the framework of nonlinear elasticity. The governing system, arising from the deformational momentum balance, the orientational momentum balance and the mechanical constraint, is formulated, and the corresponding exact solution is derived for a given constitutive model. It is found that there exist two different bending solutions. In order to determine which the preferred one is, we compare the total potential energy for both solutions…
On the emergence of molecular tilt in a ferroelectric smectic liquid crystal with broken director-inversion symmetry
University of the Basque Country EHU（巴斯克大学）；Universidad de Zaragoza（萨拉戈萨大学）；Instituto de Nanociencia y Materiales de Aragón (INMA), CSIC-Universidad de Zaragoza（阿拉贡纳米科学与材料研究所（INMA），西班牙国家研究委员会-萨拉戈萨大学）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
We study the low-frequency, long-wavelength dynamics of soft and semi-soft nematic elastomers using two different but related dynamic theories. Our first formulation describes the pure hydrodynamic behavior of nematic elastomers in which the nematic director has relaxed to its equilibrium value in the presence of strain. We find that the sound-modes structure for soft elastomers is identical to that of columnar liquid crystals. Our second formulation generalizes the derivation of the equations of nematohydrodynamics by the Harvard group to nematic elastomers. It treats the director explicitly …
A mechanical strain field applied to a monodomain cholesteric elastomer will unwind the helical director distribution. There is an analogy with the classical problem of an electric field applied to a cholesteric liquid crystal, but with important differences. Frank elasticity is of minor importance unless the gel is very weak. The interplay is between director anchoring to the rubber elastic matrix and the external mechanical field. Stretching perpendicular to the helix axis induces the uniform unwound state via the elimination of sharp, pinned twist walls above a critical strain. Unwinding th…
We explore a novel strategy of patterning nematic elastomers that does not require inscribing the texture directly. It is based on varying the dopant concentration that, beside shifting the phase transition point, affects the nematic director field via coupling between the gradients of concentration and nematic order parameter. Rotation of the director around a point dopant source causes topological modification manifesting itself in a change of the number of defects. A variety of shapes, dependent on the dopant distribution, are obtained by anisotropic deformation following the nematic--isotr…
Curvature-induced smectic-C order of tangentially anchored hard spherocylinders on a sphere with a rigidly locked director field
Recognition Physics Institute（Recognition物理研究所）；Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf（杜塞尔多夫海因里希·海涅大学软物质理论物理第二研究所）；Department of Physics, Case Western Reserve University（凯斯西储大学物理系）；Theoretical Department, Joint Institute for High Temperatures, RAS（俄罗斯科学院高温联合研究所理论部）
In liquid crystal elastomers, the orientational order of liquid crystals is coupled with elastic distortions of crosslinked polymer networks. Previous theoretical research has described these materials through two different approaches: a neoclassical theory based on the liquid crystal director and the deformation tensor, and a geometric elasticity theory based on the difference between the actual metric tensor and a reference metric. Here, we connect those two approaches using a formalism based on differential geometry. Through this connection, we determine how both the director and the geomet…
We study the layer-to-layer diffusion in smectic-A liquid crystals of colloidal hard rods with different length-to-diameter ratios using computer simulations. The layered arrangement of the smectic phase yields a hopping-type diffusion due to the presence of permanent barriers and transient cages. Remarkably, we detect stringlike clusters composed of inter-layer rods moving cooperatively along the nematic director. Furthermore, we find that the structural relaxation in equilibrium smectic phases shows interesting similarities with that of out-of-equilibrium supercooled liquids, although there …
We report on and analyze the textural transformations in islands, thicker circular domains, floating in very thin free standing chiral Smectic C* liquid crystal films. As an island is growing, an initial pure bend texture of the c-director changes into a reversing spiral at a critical size. Another distinct spiral texture is induced by changing the boundary condition at the central point defect in the island. To understand these transformations from a pure bend island, a linear stability analysis of the c-director free energy is developed, which predicts a state diagram for the island. Our obs…
Nematic elastomers contract along their director when heated or illuminated (in the case of photoelastomers). We present a conceptual design for an elastomer-based engine to extract mechanical work from heat or light. The material parameters and the geometry of such an engine are explored, and it is shown that its efficiency can go up to 20%.
Particle tracking passive microrheology in 8CB liquid crystals is used to redefine the precessional motion of the orientation of nematic director in liquid crystals. Physical origin of tumbling director in presence of presmectic clusters under zero shear conditions is discussed. Different structural properties (pure nematic phase, presmectic(smectic C and smectic A clusters)) were differentiated with characteristic dependence of $G'$ on $ω$ in the nematic phase of 8CB liquid crystals. Also, dynamic viscosity is observed with a cross over between parallel and perpendicular components as the sme…
We identify problems with the standard complex order parameter formalism for smectic-A (SmA) liquid crystals, and discuss possible alternative descriptions of smectic order. In particular, we suggest an approach based on the real smectic density variation rather than a complex order parameter. This approach gives reasonable numerical results for the smectic layer configuration and director field in sample geometries, and can be used to model smectic liquid crystals under nanoscale confinement for technological applications.
We report the smectic $A_F$, a new liquid crystal phase of the ferroelectric nematic realm. The smectic $A_F$ is a phase of small polar, rod-shaped molecules which form two-dimensional fluid layers spaced by approximately the mean molecular length. The phase is uniaxial, with the molecular director, the local average long-axis orientation, normal to the layer planes, and ferroelectric, with a spontaneous electric polarization parallel to the director. Polarization measurements indicate almost complete polar ordering of the $\sim 10$ Debye longitudinal molecular dipoles, and hysteretic polariza…
Chiral rodlike colloids exposed to strong depletion attraction may self-assemble into chiral membranes whose twisted director field differs from that of a 3D bulk chiral nematic. We formulate a simple microscopic variational theory to determine the elastic moduli of rods assembled into a bi-dimensional smectic membrane. The approach is based on a simple Onsager-Straley theory for a non-uniform director field that we apply to describe rod twist within the membrane. A microscopic approach enables a detailed estimate of the individual Frank elastic moduli (splay, twist and bend) as well as the tw…
We analyse the elastic energy of an intercalated smectic where orientationally ordered polymers with an average orientation varying from layer to layer are intercalated between smectic planes. The lowest order terms in the coupling between polymer director and smectic layer curvature are added to the smectic elastic energy. Integration over the smectic degrees of freedom leaves an effective polymer twist energy that has to be included into the total polymer elastic energy leading to a fluctuational renormalization of the intercalated polymer twist modulus. If the polymers are chiral this in it…
The known ground state of ultrathin smectic films confined to the surface of a sphere is described by four +1/2 defects assembled on a great circle and a director which follows geodesic lines. Using a simple perturbative approach we show that for thick smectic films on a sphere with planar anchoring this solution breaks down, distorting the smectic layers. The instability happens when the bend elastic constant exceeds the anchoring strength times the radius of the inner sphere. Above this threshold, the formation of a periodic chevron-like structure, observed experimentally as well, relieves g…
Liquid crystal elastomers represent a novel class of programmable shape-transforming materials whose shape change trajectory is encoded in the material's nematic director field. Using three-dimensional nonlinear finite element elastodynamics simulation, we model a variety of different actuation geometries and device designs: thin films containing topological defects, patterns that induce formation of folds and twists, and a bas-relief structure. The inclusion of finite bending energy in the simulation model reveals features of actuation trajectory that may be absent when bending energy is negl…
Chiral twisting of the molecular orientation within the layer of a smectic-A liquid crystal has been investigated using circular dichroism spectroscopy. The results indicate that a rotation of the layers away from the alignment direction is induced by the surface electroclinic effect. This leads to an interfacial region where the molecular director twists from the alignment direction until it reaches the layer normal direction. A theory is presented to explain the observed field and temperature dependence of the circular dichroism.
Motivated by the experimentally observed shear-induced destabilization and reorientation of smectic A like systems, we consider an extended formulation of smectic A hydrodynamics. We include both, the smectic layering (via the layer displacement u and the layer normal p) and the director n of the underlying nematic order in our macroscopic hydrodynamic description and allow both directions to differ in non equilibrium situations. In an homeotropically aligned sample the nematic director does couple to an applied simple shear, whereas the smectic layering stays unchanged. This difference leads …
At the A to C transition, smectic elastomers have recently been observed to undergo $\sim$35% spontaneous shear strains. We first explicitly describe how strains of up to twice this value could be mechanically or electrically induced in Sm-$C$ elastomers by rotation of the director on a cone around the layer normal at various elastic costs depending on constraints. Secondly, for typical sample geometries, we give the various microstructures in Sm-$C$ akin to those seen in nematic elastomers under distortions with constraints. It is possible to give explicit results for the nature of the textur…
We have structurally characterized the liquid crystal phase that appears as an intermediate state when a dielectric nematic, having polar disorder of its molecular dipoles, transitions to the almost perfectly polar-ordered ferroelectric nematic. This intermediate phase, which fills a 100-year-old void in the taxonomy of smectics and which we term the "smectic $Z_A$", is antiferroelectric, with the nematic director and polarization oriented parallel to smectic layer planes, and the polarization alternating in sign from layer to layer. The period of this polarization wave ($\sim$180 A) is mesosc…
Liquid crystal elastomers and glasses can have significant shape change determined by their director patterns. Cones deformed from circular director patterns have non-trivial Gaussian curvature localised at tips, curved interfaces, and intersections of interfaces. We employ a generalised metric compatibility condition to characterize two families of interfaces between circular director patterns -- hyperbolic and elliptical interfaces, and find that the deformed interfaces are geometrically compatible. We focus on hyperbolic interfaces to design complex topographies and non-isometric origami, i…
Liquid crystal elastomers are rubbers with liquid crystal order. They contract along their nematic director when heated or illuminated. The shape changes are large and occur in a relatively narrow temperature interval, or at low illumination, around the nematic-isotropic transition. We present a conceptual design of a mechanical, turbine-based engine using photo-active liquid crystal elastomers to extract mechanical work from light. Its efficiency is estimated to be 40%.
Chiral liquid crystals often exhibit periodic modulations in the molecular director; in particular, thin films of the smectic-C* phase show a chiral striped texture. Here, we investigate whether similar chiral modulations can occur in the induced molecular tilt of the smectic-A phase under an applied electric field. Using both continuum elastic theory and lattice simulations, we find that the state of uniform induced tilt can become unstable when the system approaches the smectic-A--smectic-C* transition, or when a high electric field is applied. Beyond that instability point, the system devel…
We use a two-dimensional (2D) elastic free energy to calculate the effective interaction between two circular disks immersed in smectic-$C$ films. For strong homeotropic anchoring, the distortion of the director field caused by the disks generates additional topological defects that induce an effective interaction between the disks. We use finite elements, with adaptive meshing, to minimize the 2D elastic free energy. The method is shown to be accurate and efficient for inhomogeneities on the length scales set by the disks and the defects, that differ by up to 3 orders of magnitude. We compute…
We develop a continuum theory of linear viscoelastic response in oriented monodomain nematic elastomers. The expression for dissipation function is analogous to the Leslie-Ericksen version of anisotropic nematic viscosity; we propose the relations between the anisotropic rubber moduli and new viscous coefficients. A new dimensionless number is introduced, which describes the relative magnitude of viscous and rubber-elastic torques. In an elastic medium with an independently mobile internal degree of freedom, the nematic director with its own relaxation dynamics, the model shows a dramatic decr…
The nematic twist-bend (TB) phase, exhibited by certain achiral thermotropic liquid crystalline (LC) dimers, features a nanometer-scale, heliconical rotation of the average molecular long axis (director) with equally probable left- and right-handed domains. On meso to macroscopic scales, the TB phase may be considered as a stack of equivalent slabs or "pseudo-layers", each one helical pitch in thickness. The long wavelength fluctuation modes should then be analogous to those of a smectic-A phase, and in particular the hydrodynamic mode combining "layer" compression and bending ought to be char…
Deformations that conserve the parallelism and the distances --between layers, in smectic phases; between columns, in columnar phases-- are commonplace in liquid crystals. The resulting deformed textures have the same mass density as in the ground state (an expected property in a liquid) and are at the same time isogonic and isometric, which imposes specific geometric features. The corresponding order parameter singularities extend over rather large, macroscopic, distances, e.g., cofocal conics in smectics. This well-known picture is modified when, superimposed to the 1D or 2D periodicities, t…
