Flexoelectrically driven electroclinic effect in the twist-bend nematic phase of achiral molecules with bent shapes

Structural parameters of twist-bend nematics and splay-bend nematics in Dozov's theory

This paper presents the results of numerical calculations revealing how the structural parameters (i.e., the pitch p_{TB}, the spatial period p_{SB}, and the tilt angle θ_{TB} or θ_{SB}) of twist-bend nematics (N_{TB}) and splay-bend nematics (N_{SB}) depend on the values of elastic constants in Dozov's theory [I. Dozov, Europhys. Lett. 56, 247 (2001)10.1209/epl/i2001-00513-x]. Alternative formulas for p_{TB}, θ_{TB}, p_{SB}, and θ_{SB} have been derived and it has been proved that they give more accurate results than the expressions proposed by Dozov. Although the determination of the fourth-order elastic constants C_{1}, C_{2}, and C_{3} is not feasible in a simple way, the order of magnitude of the sum C_{1}+C_{2} has been easily estimated and is equal to 10^{-31}Jm. Moreover, the numerical calculations have shown that twist-bend nematics can exist even when K_{11} is smaller than 2K_{22} and thus Dozov's criterion K_{11}>2K_{22} for the stability of the N_{TB} phase is not strictly satisfied.

Flexoelectric Ionic Liquid-Grafted Triblock Copolymers for Energy Harvesting under Flexural Deformation

The goal of the present article is to develop flexoelectric polyelectrolyte elastomers for energy harvesting based on a poly(ethylene glycol)-block-poly(propylene glycol)-block-poly(ethylene glycol) dimethacrylate (PEG-b-PPG-b-PEG-DMA) triblock grafted with an ionic liquid (IL) such as allylmethylimidazolium bis(trifluoromethane sulfonyl) imide (AMIMTFSI). The IL-grafted triblock copolymer network possesses a balance of reasonably good ionic conductivity and high ion polarization during cantilever bending. Of particular importance is the achievement of high flexoelectric coefficients in some flexoelectric polyelectrolyte elastomer (FPE) compositions reaching 1368 μC/m at ambient temperature during mechanical deformation under intermittent square-wave bending mode. With the addition of a 10 wt % lithium bis(trifluoromethane sulfonyl) imide (LiTFSI) salt, the flexoelectric coefficient further improved to 1737 μC/m, which is the highest among all piezoelectric and flexoelectric materials hitherto reported, and thus it opens a new opportunity for clean energy harvesting from a vibrating natural environment.

Exploring the Impact of Intermolecular Interactions on the Glassy Phase Formation of Twist-Bend Liquid Crystal Dimers: Insights from Dielectric Studies

The formation of the nematic to twist-bend nematic (NTB) phase has emerged as a fascinating phenomenon in the field of supramolecular chemistry, based on complex intermolecular interactions. Through a careful analysis of molecular structures and dynamics, we elucidate how these intermolecular interactions drive the complex twist-bend modulation observed in the NTB. The study employs broadband dielectric spectroscopy spanning frequencies from 10 to 2 × 109 Hz to investigate the molecular orientational dynamics within the glass-forming thioether-linked cyanobiphenyl liquid crystal dimers, namely, CBSC7SCB and CBSC7OCB. The experimental findings align with theoretical expectations, revealing the presence of two distinct relaxation processes contributing to the dielectric permittivity of these dimers. The low-frequency relaxation mode is attributed to an "end-over-end rotation" of the dipolar groups parallel to the director. The high-frequency relaxation mode is associated with precessional motions of the dipolar groups about the director. Various models are employed to describe the temperature-dependent behavior of the relaxation times for both modes. Particularly, the critical-like description via the dynamic scaling model seems to give not only quite good numerical fittings, but also provides a consistent physical picture of the orientational dynamics in accordance with findings from infrared (IR) spectroscopy. Here, as the longitudinal correlations of dipoles intensify, the m1 mode experiences a sudden upsurge in enthalpy, while the m2 mode undergoes continuous changes, displaying critical mode coupling behavior. Interestingly, both types of molecular motion exhibit a strong cooperative interplay within the lower temperature range of the NTB phase, evolving in tandem as the material's temperature approaches the glass transition point. Consequently, both molecular motions converge to determine the glassy dynamics, characterized by a shared glass transition temperature, Tg.

Dielectric Study of Liquid Crystal Dimers: Probing the Orientational Order and Molecular Interactions in Nematic and Twist-Bend Nematic Phases

Dielectric spectroscopy in frequencies that range from 10 Hz to 1 GHz has been used to study the molecular orientational dynamics of the two types of dimers that form the twist-bend nematic phase over a wide range of temperatures for both nematic and twist-bend nematic phases. The symmetrical and asymmetrical liquid crystal dimers with the cyanobiphenyl mesogenic groups were investigated. The results were analyzed within the framework of the molecular theory of dielectric permittivity for nematogens. The two molecular processes can be assigned to the reorientation of the monomeric unit: the high frequency one to the precessional rotation of the longitudinal components of the cyanobiphenyl groups (CN) and the second (low frequency) to the end-over-end rotation of the CN dipole around the short molecular axis. The precession mode, which is determined by the local order and is almost unaffected by the phase transition from the nematic to the twist-bend phase, while the end-over-end rotation clearly slowed down at the transition, as it is affected by the growth of the intermolecular interactions. The latter corresponds well to the fact revealed by IR spectroscopy about the longitudinal correlation of the molecular dipoles.

Theoretical models of modulated nematic phases

Novel modulated nematic phases, such as twist-bend nematics, splay-bend nematics and splay nematics, are an important subject of research in the field of liquid crystals. In this article fundamental information about the discovery, structure and properties of these phases is presented. Various theoretical models of elastic properties are compared, especially the proposed formulae for the free energy density of modulated nematic phases and the conditions for their stability. The emphasis is put on the variety of material parameters and variables in the mathematical description of the structures. The elastic models are classified according to a few criteria. Flexopolarisation is indicated as a main phenomenon responsible for the formation of modulated nematic phases. The elastic models are used for analysing the deformations of the twist-bend nematic structure in external fields. Dielectric, flexoelectric, ferroelectric and magnetic effects are considered. Two types of distortions are distinguished: microscopic (connected with the deformation of the director distribution) and macroscopic (related to the change of the optic axis direction). This review can be a starting point for further studies, for example computer simulations of modulated phases and design of liquid crystalline devices.

Twist-bend nematic drops as colloidal particles: Electric instabilities

The mesogen 1,''7''-bis(4-cyanobiphenyl-4'-yl)heptane (CB7CB), doped with a small quantity of an amphiphilic compound, is examined in its biphasic state in which twist-bend nematic (N_{TB}) drops are dispersed in the isotropic fluid. Various flexoelectric and electrokinetic responses of small drops in their escaped-radial-like (ER) geometry, and also of larger ones with parabolic focal conic defects, are discussed. A pair of confocal parabolas with their axes along the applied low-frequency electric field undergo periodic dimensional changes so as to contribute flexoelectrically to free-energy reduction. In an ER droplet, the same result is achieved by periodic relocations of the hedgehog core. Sine-wave fields of low frequency and high voltage excite patterned states near zero-voltage crossings and homeotropic alignment at peak voltages. ER drops also exhibit electrohydrodynamic effects; in relatively weak fields, they undergo translatory motion with a velocity that is a quadratic in the field strength; the drift, which occurs over a very wide frequency range, extending from dc to MHz region, is enabled by radial symmetry breaking that their off-centered geometry entails; and the drift direction reverses across a critical frequency. In high fields, vortical flows occurring within an ER N_{TB} drop become discernible. The hydrodynamic effects are discussed based on the Taylor-Melcher leaky dielectric model.

Flexoelectric Polarization in Liquid Crystalline Elastomers Prepared by Cross-Linking under Horseshoe-Shaped Deformation

Flexoelectric polarization, which is caused by symmetry breaking in a distortion of material, was investigated in liquid crystalline elastomers composed of wedge-shaped mesogens prepared by cross-linking under horseshoe-shaped deformation. X-ray diffractometry suggested that splay distortion along the depth direction was induced in the pseudo-isotropic phase. While almost no electric charge was observed in the smectic A phase, an electric charge caused by polarization due to the flexoelectric effect appeared and reached −1367 pC/mm2 in the pseudo-isotropic phase. We tentatively conclude that the macroscopic polarization due to the flexoelectric effect emerged and was fixed in the liquid crystalline elastomers by cross-linking under horseshoe-shaped deformation.

Comparative dielectric and thermally stimulated-depolarization-current studies of the liquid crystal dimers 1″,9″-bis(4-cyanobiphenyl-4'-yl) nonane and heptane and a binary mixture between them, close to the glass transition

We have performed dielectric spectroscopy and thermally stimulated-depolarization-current experiments to study the molecular dynamics of the twist-bend nematic phase close to the glass transition of two members of the 1″,7'-bis(4-cyanobiphenyl-4'-yl)alkane homologous series (CBnCB): the liquid crystal (LC) dimers CB9CB and CB7CB, as well as a binary mixture of both. By doping CB9CB with a small quantity of CB7CB, the crystallization is inhibited when cooling the sample down, while the bulk properties of CB9CB are retained and we can investigate the supercooled behavior close to the glass transition. The study reveals that the inter- and intramolecular interactions of the mixture are similar to those of pure CB9CB and confirms that there is a single glass transition in symmetric LC dimers.

How Do Intermolecular Interactions Evolve at the Nematic to Twist–Bent Phase Transition?

Polarized beam infrared (IR) spectroscopy provides valuable information on changes in the orientation of samples in nematic phases, especially on the role of intermolecular interactions in forming the periodically modulated twist–bent phase. Infrared absorbance measurements and quantum chemistry calculations based on the density functional theory (DFT) were performed to investigate the structure and how the molecules interact in the nematic (N) and twist–bend (N_TB) phases of thioether dimers. The nematic twist–bend phase observed significant changes in the mean IR absorbance. On cooling, the transition from the N phase to the N_TB phase was found to be accompanied by a marked decrease in absorbance for longitudinal dipoles. Then, with further cooling, the absorbance of the transverse dipoles increased, indicating that transverse dipoles became correlated in parallel. To investigate the influence of the closest neighbors, DFT calculations were performed. As a result of the optimization of the molecular cores system, we observed changes in the square of the transition dipoles, which well corresponds to absorbance changes observed in the IR spectra. Interactions of molecules dominated by pairing were observed, as well as the axial shift of the core to each other.

Nematic twist-bend phase of a bent liquid crystal dimer: field-induced deformations of the helical structure and macroscopic polarization

The twist-bend nematic (N_tb) phase is a recent addition to the family of nematic (N) phases of liquid crystals (LCs). A net polar order in the N_tbphase under an external electric field is interesting and it was predicted in several recent theoretical studies. We investigated the field-induced polarization behaviour, dielectric, and electro-optic properties of a bent LC dimer CB7CB in the N and N_tbphases. A threshold-dependent polarization current response was obtained in both the phases under triangular and square-wave input electric fields, existing till frequencies as high as 150 Hz. The polarization switching times were found in ∼1 ms region, especially in the N phase. In the N_tbphase, electric field-induced deformation of the helical structure was observed, like ferroelectric LCs. Dielectric measurements revealed the presence of cybotactic clusters via collective relaxations. The dielectric anisotropy (Δϵ) is negative at the frequencies of polarization measurements. The net polarization resulted from field-induced reorientation of cybotactic clusters and additionally from the field-induced deformation of helical structures in the N_tbphase. We explored the possibility of ionic contributions to the net polarization by synthesizing TiO₂nanoparticles (NPs) dispersed CB7CB LC nanocomposite. Incorporation of the NPs resulted in reduction of the collective order, increase in the ionic impurity content and conductivity, but an extinction of the field-induced polarization response. Our results demonstrate that the net polarization has competing contributions from both ferroelectric-like and ionic origin (up to ∼10 Hz) in the LC phases, but it becomes dominantly ferroelectric-like at higher frequencies.

Structure and Lehmann rotation of drops in a surfactant-doped bent-core liquid crystal

The structure of the nematic (cholesteric) drops that form at the clearing temperature of a mixture of the bent-core molecule CB7CB and the rodlike molecule 8CB doped with a surfactant is optically determined. Using experimental observations and numerical simulations, it is demonstrated that the director field inside these drops is not escaped concentric, as previously proposed, but twisted bipolar. The Lehmann rotation of these drops in the presence of a temperature gradient is described. Their rotation velocity is shown to be proportional to the temperature gradient and to the surface twist angle of the director field and inversely proportional to the drop radius, thus revealing a fundamental scaling law for the Lehmann effect of nematic and cholesteric twisted-bipolar droplets.

Study of the Experimental and Simulated Vibrational Spectra Together with Conformational Analysis of Thioether Cyanobiphenyl-Based Liquid Crystal Dimers

Infrared spectroscopy (IR) and quantum chemistry calculations that are based on the density functional theory (DFT) have been used to study the structure and molecular interactions of the nematic and twist-bend phases of thioether-linked dimers. Infrared absorbance measurements were conducted in a polarized beam for a homogeneously aligned sample in order to obtain more details about the orientation of the vibrational transition dipole moments. The distributions to investigate the structure and conformation of the molecule dihedral angle were calculated. The calculated spectrum was compared with the experimental infrared spectra and as a result, detailed vibrational assignments are reported.

Structure and optical properties of twist-bend nematic liquid crystals doped with chiral dopants

Twist-bend nematic liquid crystals (N_{TB} LCs), although consisting of achiral molecules, possess a spontaneous conic helix. They have been intensively studied and utilized in many applications in recent years. Herein we add chiral molecules to N_{TB} LCs and study their effects on the structure of the conic helix. We observe that the system is in the regular chiral nematic phase at high temperature and is still in the twist-bend nematic phase at low temperature. The addition of the chiral molecules does not induce a twist of the conic helical axis. The main effect of the chiral molecules is increasing the cone angle of the conic helix. We show that the structural chirality parameters in the chiral nematic phase and the twist-bend chiral nematic phase can be calculated from the same intrinsic chirality parameter, which only depends on the molecular structure and concentrations of the chiral molecule. We also observe a pretransitional phenomenon that the helical pitch of the chiral nematic phase increases dramatically when temperature is decreased toward the chiral nematic to twist-bend nematic phase transition temperature.

A Ten-Year Perspective on Twist-Bend Nematic Materials

The discovery of the twist-bend nematic phase (N_TB) is a milestone within the field of liquid crystals. The N_TB phase has a helical structure, with a repeat length of a few nanometres, and is therefore chiral, even when formed by achiral molecules. The discovery and rush to understand the rich physics of the N_TB phase has provided a fresh impetus to the design and characterisation of dimeric and oligomeric liquid crystalline materials. Now, ten years after the discovery of the N_TB phase, we review developments in this area, focusing on how molecular features relate to the incidence of this phase, noting the progression from simple symmetrical dimeric materials towards complex oligomers, non-covalently bonded supramolecular systems.

All-atom simulations of bent liquid crystal dimers: the twist-bend nematic phase and insights into conformational chirality

The liquid crystal dimer 1,7-bis-4-(4'-cyanobiphenyl)heptane (CB7CB) is known to exhibit a nematic-nematic phase transition, with the lower temperature phase identified as the twist-bend nematic (N_TB) phase. Despite the achiral nature of the mesogen, the N_TB phase demonstrates emergent chirality through the spontaneous formation of a helical structure. We present extensive molecular dynamics simulations of CB7CB using an all-atom force field. The N_TB phase is observed in this model and, upon heating, shows phase transitions into the nematic (N) and isotropic phases. The simulated N_TB phase returns a pitch of 8.35 nm and a conical tilt angle of 29°. Analysis of the bend angle between the mesogenic units reveals an average angle of 127°, which is invariant to the simulated phase. We have calculated distributions of the chirality order parameter, χ, for the ensemble of conformers in the N_TB and N phases. These distributions elucidate that CB7CB is statistically achiral but can adopt chiral conformers with no preference for a specific handedness. Furthermore, there is no change in the extent of conformational chirality between the N_TB and N phases. Using single-molecule stochastic dynamics simulations in the gas phase, we study the dimer series CBnCB (where n = 6, 7, 8 or 9) and CBX(CH₂)₅YCB (where X/Y = CH₂, O or S) in terms of the bend angle and conformational chirality. We confirm that the bent molecular shape determines the ability of a dimer to exhibit the N_TB phase rather than its potential to assume chiral conformers; as |χ|_max increases with the spacer length, but the even-membered dimers have a linear shape in contrast to the bent nature of dimers with spacers of odd parity. For CBX(CH₂)₅YCB, it is found that |χ|_max increases as the bend angle of the dimer decreases, while the flexibility of the dimers remains unchanged through the series.

Anatomy of a Discovery: The Twist–Bend Nematic Phase

New fluid states of matter, now known as liquid crystals, were discovered at the end of the 19th century and still provide strong themes in scientific research. The applications of liquid crystals continue to attract attention, and the most successful so far has been to the technology of flat panel displays; this has diversified in recent years and LCDs no longer dominate the industry. Despite this, there is plenty more to be uncovered in the science of liquid crystals, and as well as new applications, novel types of liquid crystal phases continue to be discovered. The simplest liquid crystal phase is the nematic together with its handed or chiral equivalent, named the cholesteric phase. In the latter, the aligned molecules of the nematic twist about an axis perpendicular to their alignment axis, but in the 1970s a heliconical phase with a tilt angle of less than 90° was predicted. The discovery of this phase nearly 40 years later is described in this paper. Robert Meyer proposed that coupling between a vector order parameter in a nematic and a splay or bend elastic distortion could result in spontaneously splayed or bent structures. Later, Ivan Dozov suggested that new nematic phases with splay–bend or twist–bend structures could be stabilised if the appropriate elastic constants became negative. Theoretical speculation on new nematic phases and the experimental identification of nematic–nematic phase transitions are reviewed in the paper, and the serendipitous discovery in 2010 of the nematic twist–bend phase in 1″,7″-bis(4-cyanobiphenyl-4′-yl)heptane (CB7CB) is described.

Photonic Bandgap in Achiral Liquid Crystals-A Twist on a Twist

Achiral mesogenic molecules are shown to be able to spontaneously assemble into liquid crystalline smectic phases having either simple or double-helical structures. At the transition between these phases, the double-helical structure unwinds. As a consequence, in some temperature range, the pitch of the helix becomes comparable to the wavelength of visible light and the selective reflection of light in the visible range is observed. The photonic bandgap phenomenon is reported for achiral liquid crystals.

The Beauty of Twist-Bend Nematic Phase: Fast Switching Domains, First Order Fréedericksz Transition and a Hierarchy of Structures

The twist-bend nematic phase (NTB) exhibits a complicated hierarchy of structures responsible for several intriguing properties presented here. These are: the observation of a fast electrooptic response, the exhibition of a large electroclinic effect, and the observation of an unusual pattern of the temperature dependence of birefringence of bent-shaped bimesogens in parallel-rubbed planar-aligned cells. These unusual effects inspired the use of highly sophisticated techniques that led to the discovery of the twist-bend nematic phase. Results of the optical retardation of a parallel-rubbed planar-aligned cell show that the ‘heliconical angle’ (the angle the local director makes with the optical axis) starts increasing in the high temperature N phase, it exhibits a jump at the N–NTB transition temperature and continues to increase in magnitude with a further reduction in temperature. The liquid crystalline parallel-rubbed planar-aligned and twist-aligned cells in this phase exhibit fascinating phenomena such as a demonstration of the beautiful stripes and dependence of their periodicity on temperature. The Fréedericksz transition in the NTB phase is found to be of the first order both in rubbed planar and homeotropic-aligned cells, in contrast to the second order transition exhibited by a conventional nematic phase. This transition shows a significant hysteresis as well as an abrupt change in the orientation of the director as a function of the applied electric field. Hierarchical structures are revealed using the technique of polymer templating the structure of the liquid crystalline phase of interest, and imaging of the resulting structure by scanning electron microscopy.

The interplay between spatial and heliconical orientational order in twist-bend nematic materials

The helical pitch formed by organic molecules, such as the α-helix of proteins, usually requires hydrogen bonding between chiral units and long-range positional order. It was recently found that certain liquid crystal oligomers can have a twist-bend nematic (NTB) phase with nanoscale heliconical structure without hydrogen bonding, molecular chirality or positional order. To understand the nature of this unique structure, here we present hard and resonant tender X-ray scattering studies of two novel sulfur containing dimer materials. We simultaneously measure the temperature dependences of the helical pitch and the correlation length of both the helical and positional order. In addition to an unexpected strong variation of the pitch with the length of the spacer connecting the monomer units, we find that at the transition to the NTB phase the positional correlation length drops. The helical structure was found not only in the NTB phase but observed even in the upper range of a smectic phase that forms just below the NTB state. The coexistence of smectic layering and the heliconical order indicates a layered (SmATB) phase wherein the rigid units of the dimers are tilted with respect to the smectic layer normal in order to accommodate the bent conformation of the dimers and the tilt direction rotates along the heliconical axis.

Molecular biaxiality determines the helical structure - infrared measurements of the molecular order in the nematic twist-bend phase of difluoro terphenyl dimer

Fourier-transform infrared polarized spectroscopy was employed, to obtain the three components of the infrared absorbance for a series of bent-shaped dimers containing double fluorinated terphenyl core (DTC5Cn, n = 5, 7, 9, 11). The data were used to calculate both uniaxial and biaxial order parameters, for various molecular groups of dimers. The molecule bend was estimated based on the observed differences between the uniaxial order parameters for the terphenyl core and central hydrocarbon linker. The orientational order, distinctly reverses its monotonic trend of increase to decrease at the transition temperature, from the uniaxial nematic to the twist-bend nematic phase as result of the director tilt in latter/(twist-bend) phase. The molecular biaxiality, which is negligible in the nematic phase, starts increasing on entering the twist-bend nematic phase, following a sin-square relationships with the tilt angle. The local director curvature is found to be controlled by the molecular biaxiality parameter.

General liquid-crystal theory for anisotropically shaped molecules: Symmetry, orientational order parameters, and system free energy

A general theory of liquid crystals is presented, starting from the group-theory symmetry analysis of the constituting molecules. A particular attention is paid to the type of elastic free-energies and their relationships with the molecular symmetries. The orientational order-parameter tensors are identified for each molecular symmetry, in a consideration of consistently keeping the leading, characteristic elastic free energies in a model. The order parameters are expressed in terms of symmetric traceless tensors, some of high orders, for all major molecular symmetries, including seven groups of axial symmetries and seven groups of polyhedral symmetries. For spatially inhomogeneous liquid crystals, the couplings of these tensors in the elastic energies are derived by expanding the interaction energies between these molecules. The aim is to provide a general view of the molecular symmetries of individual molecules, orientational order parameters characterizing the orientational distribution functions, and the elastic free energies, all under one single group-theory approach.

Variable pitch hydrodynamic electro-optic gratings utilising bent liquid crystal dimers

Electrohydrodynamic instabilities (EHDI) in liquid crystals form uniform and continuously variable diffractive structures when subject to certain material and geometry determined conditions. A one-dimensional grating is one such diffractive structure, where the refractive index changes periodically in a direction parallel to the initial liquid crystal director. The period of this structure has been shown previously to vary continuously between the values of the cell gap and half-cell gap approximately, allowing continuous angular modulation of optical beams but with a limited angular range. In this work, the lower pitch limit is shown to also be governed in part by the ratio of the splay and bend elastic constants (k₁₁/k₃₃) of the liquid crystal. A host nematic liquid crystal with standard elastic constant ratios (k₁₁/k₃₃ < 1) is doped with odd-alkyl-spaced dimeric liquid crystal CB7CB, to create a liquid crystal mixture with a far higher elastic constant ratio (k₁₁/k₃₃ > 5) than for those previously used in literature EHDI studies. The EHDI gratings formed in this new mixture exhibit pitch lengths significantly below half-cell gap, allowing up to 50% wider angle continuous steering of light. This improves the potential for application in beamsteering and diffractive optical devices.

Chemical-Physical Characterization of a Binary Mixture of a Twist Bend Nematic Liquid Crystal with a Smectogen

Nematic twist-bend phases (NTB) are new types of nematic liquid crystalline phases with attractive properties for future electro-optic applications. However, most of these states are monotropic or are stable only in a narrow high temperature range. They are often destabilized under moderate cooling, and only a few single compounds have shown to give room temperature NTB phases. Mixtures of twist-bend nematic liquid crystals with simple nematogens have shown to strongly lower the nematic to NTB phase transition temperature. Here, we examined the behaviour of new types of mixtures with the dimeric liquid crystal [4′,4′-(heptane-1,7-diyl)bis(([1′,1″-biphenyl]4″-carbo-nitrile))] (CB7CB). This now well-known twist-bend nematic liquid crystal presents a nematic twist-bend phase below T ≈ 104 °C. Mixtures with other monomeric alkyl or alkoxy -biphenylcarbonitriles liquid crystals that display a smectic A (SmA) phase also strongly reduce this temperature. The most interesting smectogen is 4′-Octyl-4-biphenylcarbonitrile (8CB), for which a long-term metastable NTB phase is found at room and lower temperatures. This paper presents the complete phase diagram of the corresponding binary system and a detailed investigation of its thermal, optical, dielectric, and elastic properties.

Nonuniform localized distortions in generalized elasticity for liquid crystals

We analyze a recent generalized free-energy for liquid crystals posited by Virga and falling in the class of quartic functionals in the spatial gradients of the nematic director. We review some known interesting solutions, i.e., uniform heliconical structures, and we find new liquid crystal configurations, which closely resemble some novel, experimentally detected, structures called Skyrmion tubes. These new configurations are characterized by a localized pattern given by the variation of the conical angle. We study the equilibrium differential equations and find numerical solutions and analytical approximations.

Nature-Inspired Emerging Chiral Liquid Crystal Nanostructures: From Molecular Self-Assembly to DNA Mesophase and Nanocolloids

Liquid crystals (LCs) are omnipresent in living matter, whose chirality is an elegant and distinct feature in certain plant tissues, the cuticles of crabs, beetles, arthropods, and beyond. Taking inspiration from nature, researchers have recently devoted extensive efforts toward developing chiral liquid crystalline materials with self-organized nanostructures and exploring their potential applications in diverse fields ranging from dynamic photonics to energy and safety issues. In this review, an account on the state of the art of emerging chiral liquid crystalline nanostructured materials and their technological applications is provided. First, an overview on the significance of chiral liquid crystalline architectures in various living systems is given. Then, the recent significant progress in different chiral liquid crystalline systems including thermotropic LCs (cholesteric LCs, cubic blue phases, achiral bent-core LCs, etc.) and lyotropic LCs (DNA LCs, nanocellulose LCs, and graphene oxide LCs) is showcased. The review concludes with a perspective on the future scope, opportunities, and challenges in these truly advanced functional soft materials and their promising applications.

Photoprogrammable Mesogenic Soft Helical Architectures: A Promising Avenue toward Future Chiro-Optics

Mesogenic soft materials, having single or multiple mesogen moieties per molecule, commonly exhibit typical self-organization characteristics, which promotes the formation of elegant helical superstructures or supramolecular assemblies in chiral environments. Such helical superstructures play key roles in the propagation of circularly polarized light and display optical properties with prominent handedness, that is, chiro-optical properties. The leveraging of light to program the chiro-optical properties of such mesogenic helical soft materials by homogeneously dispersing photosensitive chiral material into an achiral soft system or covalently connecting photochromic moieties to the molecules has attracted considerable attention in terms of materials, properties, and potential applications and has been a thriving topic in both fundamental science and application engineering. State-of-the-art technologies are described in terms of the material design, synthesis, properties, and modulation of photoprogrammable chiro-optical mesogenic soft helical architectures. Additionally, the scientific issues and technical problems that hinder further development of these materials for use in various fields are outlined and discussed. Such photoprogrammable mesogenic soft helical materials are competitive candidates for use in stimulus-controllable chiro-optical devices with high optical efficiency, stable optical properties, and easy miniaturization, facilitating the future integration and systemization of chiro-optical chips in photonics, photochemistry, biomedical engineering, chemical engineering, and beyond.

Structure and grating efficiency of thin cells filled by a twist-bend nematic liquid crystal

A twist-bend nematic (N_{TB}) liquid crystalline phase spontaneously forms modulated structures on a microscale level when confined in thin planar cells. Preliminary studies showed that these cells can be used as polarization gratings. Here we present a theoretical description of the formation of a two-dimensionally modulated structure. By considering the N_{TB} phase as a pseudolayer medium, a threshold condition for the onset of a modulated structure is calculated for weak and strong boundary conditions in the case of initially bookshelf or pretilt alignment of pseudolayers. Based on the modeled structure we determine spatial variation of the optic axis and calculate properties of the transmitted diffracted light. Results of the beam propagation method (BPM) and transfer matrix method are compared and it is shown that a more complex BPM gives better agreement with experimental results, meaning that even in thin cells the diffraction of light inside the grating should not be neglected.

Biaxiality-driven twist-bend to splay-bend nematic phase transition induced by an electric field

Although the existence of the twist-bend (N_TB) and splay-bend (N_SB) nematic phases was predicted long ago, only the former has as yet been observed experimentally, whereas the latter remains elusive. This is especially disappointing because the N_SB nematic is promising for applications in electro-optic devices. By applying an electric field to a planar cell filled with the compound CB7CB, we have found an N_TB-N_SB phase transition using birefringence measurements. This field-induced transition to the biaxial N_SB occurred, although the field was applied along the symmetry axis of the macroscopically uniaxial N_TB Therefore, this transition is a counterintuitive example of breaking of the macroscopic uniaxial symmetry. We show by theoretical modeling that the transition cannot be explained without considering explicitly the biaxiality of both phases at the microscopic scale. This strongly suggests that molecular biaxiality should be a key factor favoring the stability of the N_SB phase.

Theory of the splay nematic phase: Single versus double splay

Recent experiments have reported a novel splay nematic phase, which has alternating domains of positive and negative splay. To model this phase, previous studies have considered a one-dimensional (1D) splay modulation of the director field, accompanied by a 1D modulation of polar order. When the flexoelectric coupling between splay and polar order becomes sufficiently strong, the uniform nematic state becomes unstable to the formation of a modulated phase. Here we reexamine this theory in terms of a recent approach to liquid crystal elasticity, which shows that pure splay deformation is double splay rather than planar single splay. Following that reasoning, we propose a structure with a two-dimensional (2D) splay modulation of the director field, accompanied by a 2D modulation of polar order, and show that the 2D structure generally has a lower free energy than the 1D structure.

Coarse-grained model of the nematic twist-bend phase from a stable state elastic energy

The twist-bend nematic (N_{TB}) phase is a doubly degenerated heliconical structure with nanometric pitch and spontaneous bend and twist deformations. It is favored by symmetry-breaking molecular structures, such as bent dimers and bent-core molecules, and it is currently one of the burgeoning fields of liquid-crystal research. Although tremendous advances have been reported in the past five years, especially in molecular synthesis, most of its potential applications are held back by the lack of a proper and definitive elastic model to describe its behavior under various situations such as confinement and applied field. In this work we use a recently proposed stable state elastic model and the fact that the mesophase behaves as a lamellar structure to propose a mesoscopic or coarse-grained model for the N_{TB} phase. By means of standard procedures used for smectic and cholesteric liquid crystals, we arrive at a closed-form energy for the phase and apply it to a few situations of interest. The predicted compressibility for several values of the cone angle and the critical field for field-induced deformation agree well with recent experimental data.

Ether- and Thioether-Linked Naphthalene-Based Liquid-Crystal Dimers: Influence of Chalcogen Linkage and Mesogenic-Arm Symmetry on the Incidence and Stability of the Twist-Bend Nematic Phase

The twist-bend nematic (N_TB ) phase with a heliconical nanostructure of the local director generating symmetry breaking by achiral bent-shaped molecules is a hot topic of current liquid-crystal science. As opposed to the most common methylene-linked dimers, this study demonstrates chalcogen ether- and/or thioether-linked 6-(4-cyanophenyl)-2-naphthyl-based liquid-crystal dimers with symmetric and asymmetric π-conjugated mesogenic-arm structures that exhibit the N_TB phase. Although the symmetric bis(ether)-linked dimer exhibits only the conventional nematic (N) phase, the asymmetric bis(ether)-linked dimer can form the N_TB phase. All thioether-linked dimers form the N_TB phase, wherein the dimers with asymmetric arms vitrify in the N_TB phase on cooling to room temperature. The phase transitions are discussed in terms of the chalcogen linkage combination, mesogenic-arm symmetry, and spacer length. It is revealed that thioether-linked dimers based on asymmetric π-conjugated mesogenic arms with terminal cyano groups are highly beneficial for the realization of materials that form a wide range of N_TB phases and glassy N_TB states at room temperature.

Gnomonious projections for bend-free textures: thoughts on the splay-twist phase

The Hopf fibration has inspired any number of geometric structures in physical systems, in particular, in chiral liquid crystalline materials. Because the Hopf fibration lives on the three sphere,S 3 , some method of projection or distortion must be employed to realize textures in flat space. Here, we explore the geodesic-preserving gnomonic projection of the Hopf fibration, and show that this could be the basis for a new liquid crystalline texture with only splay and twist. We outline the structure and show that it is defined by the tangent vectors along the straight line rulings on a series of hyperboloids. The phase is defined by a lack of bend deformations in the texture, and is reminiscent of the splay-bend and twist-bend nematic phases. We show that domains of this phase may be stabilized through anchoring and saddle-splay.

Effect of gelation on the Frank elastic constants in a liquid crystalline mixture exhibiting a twist bend nematic phase

We report studies on the Frank elastic constant behaviour of a liquid crystal gel system exhibiting the twist bend nematic (Ntb) phase. Physical gelation is observed to ease the splay and stabilize the twist deformations in the nematic phase preceding the Ntb. More importantly, the ultra-low bend elastic constant (K33) of the system is enhanced by an order of magnitude on gelation. The magnitude of K33 remains high even in the vicinity of the Ntb phase, which otherwise is susceptible to bend deformations. This phenomenon is explained from the point of view of polar interactions in the Ntb system. XRD and dynamic rheology along with the elastic constant data validate this argument. Another salient feature of the system is that gel fibers grown in the direction orthogonal to the helical axis vanish in the Ntb phase as observed from polarizing optical microscopy. A possible reason for this is discussed on the basis of ordering developed in the surrounding medium. This feature gives the possibility of using the Ntb phase as a tool to imprint directional microstructures with a gel network.

Dielectric response of electric-field distortions of the twist-bend nematic phase for LC dimers

Wide band dielectric spectroscopy of bent-shaped achiral liquid-crystal dimers 1″-n″-bis(4-cyanobiphenyl-4'-yl) n-alkanes (CBnCB n = 7, 9, 11) has been investigated in a frequency range 0.1 Hz-100 MHz using planar-aligned cells of sample thicknesses ranging from 2 to 10 (μm) over a temperature range that covers both nematic and twist bend nematic phases. Two peaks in the dielectric spectrum in the higher frequency range are assigned to the molecular relaxation processes. The peak at the highest frequency, ∼40 to 80 MHz, is assigned to an internal precessional rotation of a single unit of the dimer around the director. The mode in the next lower frequency range of 2-10 MHz is assigned to the spinning rotation of the dimer around its long axis. This involves fluctuations of the dipole moment of the bent-shaped conformation that is directed along its arrow direction of the bow shape formed by the dimer. The peak in the frequency range 100 kHz-1 MHz can be assigned to the collective fluctuations of the local director with reference to the helical axis of the N_TB structure. The dependence of its frequency on temperature is reminiscent of the soft mode observed at the SmA^* to SmC^* phase transition. This result clearly corresponds to the electro-clinic effect-the response of the director to the applied electric field in an electro-optic experiment. The lowest frequency mode, observed in the frequency range of 0.1 Hz-100 Hz, is identified with the Goldstone mode. This mode is concerned with the long range azimuthal angle fluctuations of the local director. This leads to an alternating compression and expansion of the periodic structure of the N_TB phase.

Mechanisms to splay-bend nematic phases

While twist-bend nematic phases have been extensively studied, the experimental observation of two dimensional, oscillating splay-bend phases is recent. We consider two theoretical models that have been used to explain the formation of twist-bend phases-flexoelectricity and bond orientational order-as mechanisms to induce splay-bend phases. Flexoelectricity is a viable mechanism, and splay and bend flexoelectric couplings can lead to splay-bend phases with different modulations. We show that while bond orientational order circumvents the need for higher order terms in the free energy, the important role of nematic symmetry and phase chirality rules it out as a basic mechanism.

Metastable room-temperature twist-bend nematic phases via photopolymerization

The heliconical twist-bend nematic (N_{TB}) phase is a promising candidate for novel electro-optic and photonic applications. However, the phase generally exists at elevated temperatures and across a narrow temperature interval, limiting its implementation in device fabrication, which would ideally require the liquid crystal phase to be stable at room temperature. Here we report the formation of room-temperature N_{TB} phases by in situ photopolymerization. A complete phase diagram of the liquid crystal and monomer mixtures is presented and the nature of the polymerized samples is discussed in detail. In contrast to samples before polymerization-where the N_{TB} phases exist at elevated temperatures and across temperature intervals of width <10 °C-all photopolymerized N_{TB} samples are found to be stable at room temperature and exist over a temperature interval of up to 80 °C. Scanning electron microscopy of the polymerized N_{TB} phase shows that the polymer strands assemble at an angle with respect to the direction of the helical axis. This suggests that photopolymerized N_{TB} phases could be used to facilitate the tilt angle measurements in the twist-bend nematic phase.

Multi-level chirality in liquid crystals formed by achiral molecules

Complex materials often exhibit a hierarchical structure with an intriguing mechanism responsible for the ‘propagation’ of order from the molecular to the nano- or micro-scale level. In particular, the chirality of biological molecules such as nucleic acids and amino acids is responsible for the helical structure of DNA and proteins, which in turn leads to the lack of mirror symmetry of macro-bio-objects. To fully understand mechanisms of cross-level order transfer there is an intensive search for simpler artificial structures exhibiting hierarchical arrangement. Here we present complex systems built of achiral molecules that show four levels of structural chirality: layer chirality, helicity of a basic repeating unit, mesoscopic helix and helical filaments. The structures are identified by a combination of hard and soft x-ray diffraction measurements, optical studies and theoretical modelling. Similarly to many biological systems, the studied materials exhibit a coupling between chirality at different levels.

It was previously shown that chiral structures can be formed from achiral bent-shaped mesogens. Here the authors observe hierarchical chiral structures with coupling of chirality at different levels in a system with achiral constituents.

Indication of a twist-grain-boundary-twist-bend phase of flexible core bent-shape chiral dimers

The effect of the molecular chirality of chiral additives on the nanostructure of the twist-bend nematic (NTB) liquid crystal phase with ambidextrous chirality and nanoscale pitch due to spontaneous symmetry breaking is studied. It is found that the ambidextrous nanoscale pitch of the NTB phase increases by 50% due to 3% chiral additive, and the chiral transfer among the biphenyl groups disappears in the NTB* phase. Most significantly, a twist-grain boundary (TGB) type phase is found at c > 1.5 wt% chiral additive concentrations below the usual N* phase and above the non-CD active NTB* phase. In such a TGB type phase, the adjacent blocks of pseudo-layers of the nanoscale pitch rotate across the grain boundaries.

Spherical-cap droplets of a photo-responsive bent liquid crystal dimer

Using a photo-responsive dimer exhibiting the transition between nematic (N) and twist-bend nematic (NTB) phases, we prepared spherical cap-shaped droplets on solid substrates exposed to air. The internal director structures of these droplets vary depending on the phase and on the imposed boundary conditions. The structural switching between the N and NTB phases was successfully performed either by temperature control or by UV light-irradiation. The N phase is characterized by an extremely small bend elastic constant K3, and surprisingly, we found that the droplet-air interface induces a planar alignment, in contrast to that seen for typical calamitic liquid crystals. As a consequence, the director configuration was stabilized in a structure substantially different from that normally found in conventional nematic liquid crystalline droplets. In the twist-bend nematic droplets characteristic structures with macroscopic length scales were formed, and they were well controlled by the droplet size. These results indicated that a continuum theory is effective in describing the stabilization mechanism of the macroscopic structure even in the twist-bend nematic liquid crystal droplets exhibiting director modulations on a scale of several molecular lengths.

Soft modes of the dielectric response in the twist–bend nematic phase and identification of the transition to a nematic splay bend phase in the CBC7CB dimer

Two collective processes resulting from distortion of the heliconical structure of the twist–bend nematic phase of an achiral dimer: one tilt mode due to distortions of the conical angle and second related to long range fluctuation of the cone phase.

Pretransitional behavior of viscoelastic parameters at the nematic to twist-bend nematic phase transition in flexible n-mers

We report dynamic light scattering measurements of the orientational (Frank) elastic constants and associated viscosities among a homologous series of a liquid crystalline dimer, trimer, and tetramer exhibiting a uniaxial nematic (N) to twist-bend nematic (N_TB) phase transition.

Twist-bend nematic liquid crystals based on thioether linkage

“Thioether”-based twist-bend nematogens.

Supramolecular organization of liquid-crystal dimers - bis-cyanobiphenyl alkanes on HOPG by scanning tunneling microscopy

2D supramolecular organization of a series of six cyanobiphenyls bimesogens deposited on highly oriented pyrolytic graphite (HOPG) is studied by scanning tunneling microscopy (STM). The adsorbates are 1,ω-bis(4-cyanobiphenyl-4'-yl)alkanes (CBnCB) with different lengths of their flexible alkyl spacer (containing from 7 to 12 methylene groups). Microscopic investigations at the molecular resolution allow for detailed analysis of the effect of the alkyl spacer length on the type and the extent of the resulting 2D organization. It was demonstrated that bimesogens with shorter spacers (7 and 8 methylene units) organize in a similar manner characterized by the formation of two types of differently ordered monolayers: dense packed, wherein the molecules are oriented in one direction and ordered into parallel rows (layer structure), or less densely packed where they are organized into a chiral windmill-like structure. For derivatives with longer spacers (ranging from 9 to 12 methylene units) the additional effect of parity of carbon atoms in the spacer (even versus odd) is observed. In this range of the spacer lengths even membered bimesogens are also organized in a typical layer structure. However, odd-membered dimers exhibit a much more complex 2D supramolecular organization with a larger unit cell and a helical arrangement of the molecules. Careful comparison of this structure with the 3D structural data derived from the X-ray diffraction investigations of single crystals indicates that for these bimesogens a clear correlation exists between the observed complex 2D supramolecular organization in the monolayer and the organization in one of the crystallographic planes of the 3D nematic twist-bent phase.

Distortions in structures of the twist bend nematic phase of a bent-core liquid crystal by the electric field

The dielectric spectra of the twist bend nematic phase (N_{TB}) of an achiral asymmetric bent-core liquid crystalline compound are studied for determining the various relaxation modes. Dielectric measurements are also carried out under the bias field E up to 8 V/µm. Two molecular and two collective relaxation processes are observed. The orientational order parameters with respect to the local and the main directors determined using molecular modes are used to find the heliconical angle. The results also show that the order parameter with reference to the main director reverses its trend from increasing to decreasing at temperatures of a few degrees above the N_{TB} to N transition. The collective relaxation modes are assigned to (a) distortions of the local director by the electric field at a frequency of ∼100kHz while the periodic helical structure remains unaltered (mode attributed to flexoelectricity); (b) changes in the periodic structure arising from a coupling of the dielectric anisotropy with the electric field at the lowest frequency in the range of 100 Hz-10 kHz. Frequency of the higher frequency collective mode (∼100kHz) depends primarily on the heliconical angle and has anomalous softeninglike behavior at the N-N_{TB} transition. The lowest frequency mode is studied under the bias field E; the modulus of the wave vector gradually vanishes on increasing E (except for an initial behavior, E^{2}<0.1V^{2}/μm^{2}, which is just the opposite). The transition from the twist bend to splay bend structure is observed by a sudden drop in the frequency of this mode, followed by a linear decrease in frequency by increasing E. The results agree with the predictions made from the currently proposed models for a periodically distorted N_{TB} phase.