Flexoelectricity in chiral nematic liquid crystals as a driving mechanism for the twist-bend and splay-bend modulated phases

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.

Stability conditions of a twist-bend nematic phase according to Barbero's theory

The stability conditions of twist-bend nematics given by Barbero et al. [Phys. Rev. E 92, 030501(R) (2015)10.1103/PhysRevE.92.030501] lead to doubtful conclusions when the influence of the magnetic field on the N_{TB} structure is analyzed. For this reason, the stability criteria have been redetermined in the present paper. The calculations have revealed that some of the conditions presented by Barbero and his co-workers are incorrect. It has been shown that the parameters b_{o}K_{33} and η must be positive to induce the formation of a stable twist-bend nematic phase. Furthermore, some additional criteria concerning the value of η have been derived. The phenomenon of the shift of the stability interval in the magnetic field has been analyzed in detail.

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.

Conformational degrees of freedom and stability of splay-bend ordering in the limit of a very strong planar anchoring

We study the self-organization in a monolayer (a two-dimensional system) of flexible planar trimer particles. The molecules are made up of two mesogenic units linked by a spacer, all of which are modeled as hard needles of the same length. Each molecule can dynamically adopt two conformational states: an achiral bent-shaped (cis-) and a chiral zigzag (trans-) one. Using constant pressure Monte Carlo simulations and Onsager-type density functional theory (DFT), we show that the system consisting of these molecules exhibits a rich spectrum of liquid crystalline phases. The most interesting observation is the identification of stable smectic splay-bend (S_{SB}) and chiral smectic-A (S_{A}^{*}) phases. The S_{SB} phase is also stable in the limit, where only cis- conformers are allowed. The second phase that occupies a considerable portion of the phase diagram is S_{A}^{*} with chiral layers, where the chirality of the neighboring layers is of opposite sign. The study of the average fractions of the trans- and cis- conformers in various phases shows that while in the isotropic phase all fractions are equally populated, the S_{A}^{*} phase is dominated by chiral conformers (zigzag), but the achiral conformers win in the smectic splay-bend phase. To clarify the possibility of stabilization of the nematic splay-bend (N_{SB}) phase for trimers, the free energy of the N_{SB} and S_{SB} phases is calculated within DFT for the cis- conformers, for densities where simulations show stable S_{SB}. It turns out that the N_{SB} phase is unstable away from the phase transition to the nematic phase, and its free energy is always higher than that of S_{SB}, down to the transition to the nematic phase, although the difference in free energies becomes extremely small when approaching the transition.

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.

Experimental and Computational Study of a Liquid Crystalline Dimesogen Exhibiting Nematic, Twist-Bend Nematic, Intercalated Smectic, and Soft Crystalline Mesophases

Liquid crystalline dimers and dimesogens have attracted significant attention due to their tendency to exhibit twist-bend modulated nematic (NTB) phases. While the features that give rise to NTB phase formation are now somewhat understood, a comparable structure-property relationship governing the formation of layered (smectic) phases from the NTB phase is absent. In this present work, we find that by selecting mesogenic units with differing polarities and aspect ratios and selecting an appropriately bent central spacer we obtain a material that exhibits both NTB and intercalated smectic phases. The higher temperature smectic phase is assigned as SmCA based on its optical textures and X-ray scattering patterns. A detailed study of the lower temperature smectic ''X'' phase by optical microscopy and SAXS/WAXS demonstrates this phase to be smectic, with an in-plane orthorhombic or monoclinic packing and long (>100 nm) out of plane correlation lengths. This phase, which has been observed in a handful of materials to date, is a soft-crystal phase with an anticlinic layer organisation. We suggest that mismatching the polarities, conjugation and aspect ratios of mesogenic units is a useful method for generating smectic forming dimesogens.

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.

Molecular Flexibility and Bend in Semi-Rigid Liquid Crystals: Implications for the Heliconical Nematic Ground State

The NTB phase phases possess a local helical structure with a pitch length of a few nanometers and is typically exhibited by materials consisting of two rigid mesogenic units linked by a flexible oligomethylene spacer of odd parity, giving a bent shape. We report the synthesis and characterisation of two novel dimeric liquid crystals, and perform a computational study on 10 cyanobiphenyl dimers with varying linking groups, generating a large library of conformers for each compound; this allows us to present molecular bend angles as probability weighted averages of many conformers, rather than use a single conformer. We validate conformer libraries by comparison of interproton distances with those obtained from solution-based 1D 1 H NOESY NMR, finding good agreement between experiment and computational work. Conversely, we find that using any single conformer fails to reproduce experimental interproton distances. We find the use of a single conformer significantly overestimates the molecular bend angle while also ignoring flexibility; in addition, we show that the average bend angle and flexibility are both linked to the relative stability of the NTB phase.

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.

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.

A novel nematic-like mesophase induced in dimers, trimers and tetramers doped with a high helical twisting power additive

From the observation of a previously undiscovered nematic-like mesophase (NX) by Archbold et al., we report on several new binary liquid-crystalline mixtures between the high helical twisting power dopant RM1041 and a selection of dimers with varying average bend angles and conformational landscapes. We also report on mixtures between RM1041 and oligomeric LC materials. We find that dimers and oligomers exhibit not only chiral nematic and twist-bend modulated phases, but also the same NX phase reported by Archbold, indicating that this state of matter (the structure of which is yet to be definitively characterised) is exhibited by a wide range of materials. Mixtures of the dimer CB9CB with a selection of different chiral dopants suggest that it is the helical twisting power of the chiral additive that is responsible incidence of the NX phase.

Structure of nanoscale-pitch helical phases: blue phase and twist-bend nematic phase resolved by resonant soft X-ray scattering

Periodic structures of phases with orientational order of molecules but homogenous electron density distribution: a short pitch cholesteric phase, blue phase and twist-bend nematic phase, were probed by resonant soft X-ray scattering (RSoXS) at the carbon K-edge. The theoretical model shows that in the case of a simple heliconical nematic structure, two resonant signals corresponding to the full and half pitch band should be present, while only the full pitch band is observed experimentally. This suggests that the twist-bend nematic phase has a complex structure with a double-helix built of two interlocked, shifted helices. We confirm that the helical pitch in the twist-bend nematic phase is in a 10 nm range for both the chiral and achiral materials. We also show that the symmetry of the blue phase can be unambiguously determined through a resonant enhancement of the X-ray diffraction signals, by including polarization effects, which are found to be an important indicator in phase structure determination.

Twist-bend nematic phases of bent-shaped biaxial molecules

How can a change in molecular structure affect the relative stability and structural properties of the twist-bend nematic phase (NTB)? Here we extend the mean-field model(1) (C. Greco, G. R. Luckhurst and A. Ferrarini, Soft Matter, 2014, 10, 9318) for bent-shaped achiral molecules, to study the influence of arm molecular biaxiality and the value of the molecule's bend angle on the relative stability of NTB. In particular we show that by controlling the biaxiality of the molecule's arms, up to four ordered phases can become stable. They involve local uniaxial and biaxial variants of NTB, together with uniaxial and biaxial nematic phases. However, a V-shaped molecule shows a stronger ability to form stable NTB than a biaxial nematic phase, where the latter phase appears in the phase diagram only for bend angles greater than 140° and for large biaxiality of the two arms.

Effect of polar intermolecular interactions on the elastic constants of bent-core nematics and the origin of the twist-bend phase

A molecular theory of both elastic constants and the flexoelectric coefficients of bent-core nematic liquid crystals has been developed taking into account dipole-dipole interactions as well as polar interactions determined by the bent molecular shape. It has been shown that if polar interactions are neglected, the elastic constants are increasing monotonically with the decreasing temperature. On the other hand, dipolar interactions between bent-core molecules may result in a dramatic increase of the bend flexocoefficient. As a result, the flexoelectric contribution to the bend elastic constant increases significantly, and the bend elastic constant appears to be very small throughout the nematic range and may vanish at a certain temperature. This temperature may then be identified as a temperature of the elastic instability of the bent-core nematic phase which induces a transition into the modulated phases with bend deformations like recently reported twist-bend phase. The temperature variation of the elastic constants is qualitatively similar to the typical experimental data for bent-core nematics.

Modulated nematic structures induced by chirality and steric polarization

What kind of one-dimensional modulated nematic structures (ODMNS) can form nonchiral and chiral bent-core and dimeric materials? Here, using the Landau-de Gennes theory of nematics, extended to account for molecular steric polarization, we study a possibility of formation of ODMNS, both in nonchiral and intrinsically chiral liquid crystalline materials. Besides nematic and cholesteric phases, we find four bulk ODMNS for nonchiral materials, two of which, to the best of our knowledge, have not been reported so far. These two structures are longitudinal (N_{LP}) and transverse (N_{TP}) periodic waves where the polarization field being periodic in one dimension stays parallel and perpendicular, respectively, to the wave vector. The other two phases are the twist-bend nematic phase (N_{TB}) and the splay-bend nematic phase (N_{SB}), but their fine structure appears more complex than that considered so far. The presence of molecular chirality converts nonchiral N_{TP} and N_{SB} into new N_{TB} phases. Surprisingly, the nonchiral N_{LP} phase can stay stable even in the presence of intrinsic chirality.

Short-range smectic fluctuations and the flexoelectric model of modulated nematic liquid crystals

We show that the flexoelectric model of chiral and achiral modulated nematics predicts the compression modulus that is by orders of magnitude lower than the measured values. The discrepancy is much larger in the chiral modulated nematic phase, in which the measured value of the compression modulus is of the same order of magnitude as in achiral modulated nematics, even though the heliconical pitch is by an order of magnitude larger. The relaxation of a one-constant approximation in the biaxial elastic model used for chiral modulated nematics does not solve the problem. Therefore, we propose a structural model of the modulated nematic phase, which is consistent with the current experimental evidence and can also explain large compression modulus: the structure consists of short-range smectic clusters with a fourfold symmetry and periodicity of two molecular distances. In chiral systems, chiral interactions lead to a helicoidal structure of such clusters.

A Twist-Bend Nematic (NTB ) Phase of Chiral Materials

New chiral dimers consisting of a rod-like and cholesterol mesogenic units are reported to form a chiral twist-bend nematic phase (NTB *) with heliconical structure. The compressibility of the NTB phase made of bent dimers was found to be as large as in smectic phases, which is consistent with the nanoperiodic structure of the NTB phase. The atomic force microscopy observations in chiral bent dimers revealed a periodicity of about 50 nm, which is significantly larger than the one reported previously for non-chiral compounds (ca. 10 nm).

A fibre forming smectic twist–bent liquid crystalline phase

We demonstrate the nanostructure and filament formation of a novel liquid crystal phase of a dimeric mesogen below the twist–bend nematic phase.

Molecular geometry, twist-bend nematic phase and unconventional elasticity: a generalised Maier-Saupe theory

It has been found that bent-shaped achiral molecules can form a liquid crystal phase, called the Twist-Bend Nematic (NTB), which is locally polar and spontaneously twisted having a tilted director, with a conglomerate of degenerate chiral domains with opposite handedness and pitch of a few molecular lengths. Here, using a major extension of the Maier-Saupe molecular field theory, we can describe the transition from the nematic (N) to the NTB phase. We provide a consistent picture of the structural and elastic properties in the two phases, as a function of the molecular bend angle, and show that on approaching the transition there is a gradual softening of the bend mode in the N phase. This points to the crucial role of the molecular shape for the formation of modulated nematic phases and their behaviour.