Liquid Crystal Dimers and the Twist-Bend Phases: Non-Symmetric Dimers Consisting of Mesogenic Units of Differing Lengths

The syntheses and characterisation of the 4-[{[4-({n-[4-(4-cyanophenyl)phenyl]-n-yl}oxy)phenyl]-methylidene}amino]phenyl-4-alkoxybenzoates (CBnOIBeOm) are reported with n=8 and 10 and m=1-10. The two series display fascinating liquid crystal polymorphism. All twenty reported homologues display an enantiotropic nematic (N) phase at high temperature. When the length of the spacer (n) is greater than that of the terminal chain (m), the twist-bend nematic (NTB) phase is observed at temperatures below the N phase. As the length of the terminal chain is increased and extends beyond the length of the spacer up to three smectic phases are observed on cooling the N phase. One of these smectic phases has been assigned as the rare twist-bend smectic C subphase, the SmCTB-α phase. In all the smectic phases, a monolayer packing arrangement is seen, and this is attributed to the anti-parallel associations of the like mesogenic units.

The Interplay of Spacer Chirality and Parity in Mesogenic Dimers

Introducing chirality into soft materials, including liquid crystals (LCs), profoundly impacts their self-organization and physical properties. In this study, we synthesized a novel series of LC dimers with a chiral center as part of their flexible spacer. The dimers were prepared in racemic and enantiomerically pure forms. Their spacer length and parity were varied to investigate the effect of spacer chirality and parity on mesomorphic behavior and on chiral induction in the nematic phase of achiral mesogens. Our results show that the even-membered chiral dimers only have chiral nematic phases. In contrast, the odd-membered dimers display rich mesomorphism, including the intriguing blue phase (BP) and chiral form of the twist-bend nematic phase (N*TB). The observed significant difference in the 3D surface morphology between the racemic and chiral forms of the N*TB phase suggests that the chiral moiety in the spacer promotes a chiral hierarchy. Furthermore, the chiral dimers show a prominent odd-even effect in the helical twisting power in nematic hosts. These findings highlight the importance of the position of the chiral group within the dimeric molecule and provide new insights into how intrinsic chirality in the spacer affects the overall structural chirality.

Mixing twist-bend and ferroelectric nematic liquid crystals

Twist-bend (N_{tb}) and ferroelectric (N_{F}) nematic liquid crystals exhibit several novel effects and new physical properties. Here, we report experimental studies on the phase diagram and some physical properties of binary mixtures of CB9CB and RM734 mesogens. Both N-N_{tb} and N-N_{F} phase transition temperatures and the corresponding enthalpies decrease significantly and, eventually, these transitions disappear at some intermediate compositions, stabilizing wide nematic phase (N). Temperature-dependent birefringence several degrees above the N-N_{tb} phase transition shows strong director tilt fluctuations. The critical range of the fluctuations increases with the nematic range and the critical exponent is consistent with the mean field. The spontaneous polarization of RM734 decreases drastically with the addition of CB9CB mesogen. The temperature dependence of the splay elastic constant of the mixtures' high-temperature nematic (N) phase strikingly differs from that of the pristine CB9CB and RM734 mesogens. The study shows that a small inclusion of either compound has a substantial effect on the phase diagram and physical properties.

DNMR measurements of an asymmetric odd liquid crystal dimer: determination of the intramolecular angle and the degree of order of the two rigid cores

In this work, we present a Deuteron Nuclear Magnetic Resonance (DNMR) study of the non-symmetric odd liquid crystal dimer α-(4-cyanobiphenyl-4'-yloxy)-ω-(1-pyrenimine-benzylidene-4'-oxy) heptane (CBO7O.Py), formed by a pro-mesogenic cyanobiphenyl unit and a bulky pyrene-containing unit, linked via alkoxy flexible chain. We have synthesized two partially deuterated samples: one with the deuterium atoms in the cyanobiphenyl moiety (dCBO7O.Py) and the other one with the deuterium atoms in the pyrenimine-benzylidene unit (CBO7O.dPy). We have performed angular distribution analysis in the SmA glassy state, obtaining the degree of order of both rigid cores and an estimation of the internal molecular angle between both structures. With the results from the angular study, we have been able to determine the degree of order of both rigid units in either the N phase and the SmA phase, far enough from the glass transition. Both rigid cores have the same degree of order close to the nematic-isotropic phase transition, but as the compound is cooled down, the degree of order of the cyanobiphenyl moiety is clearly higher than that of the pyrene-containing unit. The critical behaviour of the order parameter of the pyrene-containing moiety is consistent with the fact that, for CBO7O.Py, the N-I phase transition is tricritical, which seems to indicate that the uniaxial order parameter of the dimer is dominated by the degree of order of the pyrene-containing core.

Spontaneous Twisting of Achiral Hard Rod Nematics

Since Onsager's seminal work, hard rods have been taken as a prototype of nematic liquid crystals, characterized by uniaxial order and a uniform director field as a ground state. Here, using Onsager theory to calculate the free energy in the presence of arbitrary deformations, we find that hard rod nematics have an intrinsic tendency to twist around their ordering axis (double twist), driven by a mechanism in which the orientational fluctuations of particles play a key role. The anisotropic hard core potential used here is arguably the simplest form of interaction able to originate spontaneous breaking of mirror symmetry in a 3D fluid. Our results are discussed in relation to the recent discovery of a double twisted ground state in cylindrically confined lyotropic chromonic liquid crystals.

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.

Microscopic modelling of nematic elastic constants beyond Straley theory

Recent findings on various classes of nematics, whose microscopic structure differs from the prototypical rod-like shape, evidence unusual elastic properties, which challenge existing theories. Here we develop a theoretical and numerical methodology for the calculation of Frank elastic constants, accounting for the coupling between the molecular shape and each specific deformation mode. This is done in the framework of Onsager-Straley's second-virial theory, using a non-local form of the orientational distribution function. The comparison between two benchmark systems, a straight and a bent rod, allows us to illustrate the distinct features of this approach, which include additional order parameters induced by the deformation and, related to this, an ideal contribution to the deformation free energy. Then, using a simple system that can be seen as a minimalist model of liquid crystal trimers, we discuss the impact of different molecular conformations on elastic constants.

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.

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.

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.

A Landau-de Gennes theory for twist-bend and splay-bend nematic phases of colloidal suspensions of bent rods

We develop a phenomenological Landau-de Gennes (LdG) theory for lyotropic colloidal suspensions of bent rods using a Q-tensor expansion of the chemical-potential dependent grand potential. In addition, we introduce a bend flexoelectric term, coupling the polarization and the divergence of the Q-tensor, to study the stability of uniaxial (N), twist-bend (N_TB), and splay-bend (N_SB) nematic phases of colloidal bent rods. We first show that a mapping can be found between the LdG theory and the Oseen-Frank theory. By breaking the degeneracy between the splay and bend elastic constants, we find that the LdG theory predicts either an N-N_TB-N_SB or an N-N_SB-N_TB phase sequence upon increasing the particle concentration. Finally, we employ our theory to study the first-order N-N_TB phase transition, for which we find that K₃₃ as well as its renormalized version K₃₃ ^eff remain positive at the transition, whereas K₃₃ ^eff vanishes at the nematic spinodal. We connect these findings to recent simulation results.

Q-tensor model of twist-bend and splay nematic phases

The twist-bend nematic phase (N_{TB}) is characterized by a conically twisting director and by a dramatic softening of the bend elastic constant before the transition to the N_{TB} phase. In the recently found splay nematic phase (N_{S}) the splay elastic constant tends to zero, resulting in a splay modulation perpendicular to the director. We model both phases with a single Q-tensor free energy including a term that breaks the degeneracy between the splay and bend elastic constant, and a flexoelectric term coupling the divergence of the Q-tensor with polarization. The N_{TB} or N_{S} phase is obtained by a change of sign of one elastic parameter. Measured elastic constants show that the N-N_{TB} transition is mainly driven by the increase of the nematic order, while the N_{S} transition is due to flexoelectric coupling.

The Chiral Twist-Bend Nematic Phase (N*TB )

The twist-bend nematic, N_TB , phase has been observed for chiral materials in which chirality is introduced through a branched 2-methylbutyl terminal tail. The chiral twist-bend nematic phase, N*_TB , is completely miscible with the N_TB phase of the standard achiral material, CB6OCB. The N*_TB phase exhibits optical textures with lower birefringence than those observed for the achiral N_TB phase, suggesting an additional mechanism of averaging molecular orientations. The N*-N*_TB transition temperatures for the chiral materials are higher than the N_TB -N transition temperatures seen for the corresponding racemic materials. This suggests the double degeneracy of helical twist sense in the N T B * phase is removed by the intrinsic molecular chirality. A square lattice pattern is observed in the N* phase over a temperature range of several degrees above the N*_TB -N phase transition, which may be attributed to a non-monotonic dependence of the bend elastic constant.

Fine-tuning the effect of π-π interactions on the stability of the NTB phase

The synthesis and liquid-crystalline properties are reported for novel bent-shaped dimers in which a naphthyl group has been incorporated into the mesogenic cores. In addition to the nematic and twist-bend nematic phase, a new liquid-crystalline phase was observed. The combined experimental and computational study demonstrated how the interplay between the molecular geometry and π-π interactions affects the thermal stability of the twist-bend nematic and nematic phases. Correlation between mesomorphic properties and molecular geometry revealed that a greater conformational diversity leads to a broader distribution of bend-angles and destabilization of the NTB phase. Qualitative correlation between the thermal behaviour and electronic structure of the molecules of a similar geometry suggested that the transition temperatures of both nematic phases depend on the relative contribution of dispersion and electrostatic energies which determines the strength of the π-π interactions. These results provide an insight into how subtle changes in chemical structure can be exploited to tune the intermolecular interactions and influence the thermal stability of the liquid crystalline phase.

Molecular organization in the twist-bend nematic phase by resonant X-ray scattering at the Se K-edge and by SAXS, WAXS and GIXRD

Using a magnetically aligned liquid crystal mixture containing a novel Se-labelled dimer and the difluoroterphenyl dimer DTC5C7, the twist-bend nematic phase (N_tb) was studied by the resonant scattering of hard X-rays and by conventional small and wide-angle X-ray scattering (SAXS, WAXS). Resonant diffraction spots indicated a helix with a 9-12 nm pitch in the N_tb phase and an unprecedentedly high helix orientation. This enabled deconvolution of global and local order parameters. These findings, combined with the simultaneously recorded resonant and non-resonant SAXS and WAXS data, allowed us to construct a locally layered molecular model of the N_tb phase, where the average twisted conformation of each molecule was idealised as a helical segment, matching the local heliconical director field. The dimers were found to be less bent in the N_tb phase than in their minimum energy conformation, and straightening further with increasing temperature. It is proposed that on further heating their low bend angle allows the transition to the normal nematic phase, where the molecules can freely move longitudinally, without the need to perform screw-like motion as in the N_tb phase. At the low-temperature end, the increasing molecular twist becomes unsustainable, leading to a transition to a smectic phase, where no twist is required.

Heliconical smectic phases formed by achiral molecules

Chiral symmetry breaking in soft matter is a hot topic of current research. Recently, such a phenomenon was found in a fluidic phase showing orientational order of molecules—the nematic phase; although built of achiral molecules, the phase can exhibit structural chirality—average molecular direction follows a short-pitch helix. Here, we report a series of achiral asymmetric dimers with an odd number of atoms in the spacer, which form twisted structures in nematic as well as in lamellar phases. The tight pitch heliconical nematic (N_TB) phase and heliconical tilted smectic C (SmC_TB) phase are formed. The formation of a variety of helical structures is accompanied by a gradual freezing of molecular rotation. In the lowest temperature smectic phase, HexI, the twist is expressed through the formation of hierarchical structure: nanoscale helices and mesoscopic helical filaments. The short-pitch helical structure in the smectic phases is confirmed by resonant X-ray measurements.

Systems that form chiral structures from achiral molecules are not common. Here, the authors synthesise a compound consisting of asymmetric and achiral bent-shaped mesogens, which exhibit a variety of liquid crystal phases including one in which chiral structures form from achiral constituent molecules.

^{13}C NMR study of the director distribution adopted by the modulated nematic phases formed by liquid-crystal dimers with odd numbers of atoms in their spacers

The orientational order of the molecules in the bent mesogen CB6OCB has been studied throughout the range of temperature stability of both the N_{U} and N_{TB} liquid-crystal phases by ^{13}C NMR spectroscopy. These spectra provide local order parameters for the para axes of both of the nonequivalent cyanobiphenyl groups and show how they change on entering the twist-bend nematic phase. A key feature of the order parameters is a weak, but clear maximum in the temperature variation of the order parameter prior to the N_{TB} phase. This suggests that the directors in both the N_{U} and N_{TB} phases are tilted with respect to the magnetic field of the spectrometer. Significantly the conformational states of the spacer are comparable in both phases, although the low temperature nematic is chiral but not that at high temperature. It is proposed that the higher temperature, tilted phase could be the splay-bend nematic phase.

The Dependency of Nematic and Twist-bend Mesophase Formation on Bend Angle

We have prepared and studied a family of cyanobiphenyl dimers with varying linking groups with a view to exploring how molecular structure dictates the stability of the nematic and twist-bend nematic mesophases. Using molecular modelling and 1D ¹H NOESY NMR spectroscopy, we determine the angle between the two aromatic core units for each dimer and find a strong dependency of the stability of both the nematic and twist-bend mesophases upon this angle, thereby satisfying earlier theoretical models.

The dependency of twist-bend nematic liquid crystals on molecular structure: a progression from dimers to trimers, oligomers and polymers

This article gives an overview on recent developments concerning the twist-bend nematic phase. The twist-bend nematic phase has been discussed as the missing link between the uniaxial nematic mesophase (N) and the helical chiral nematic phase (N*). After an introduction discussing the key physical properties of the N_TB phase and the methods used to identify the twist-bend nematic mesophase this review focuses on structure property relationships and molecular features that govern the incidence of this phase.

Density functional theory of nematic elasticity: softening from the polar order

Recent experiments have evidenced some unconventional features in the elasticity of nematics, which cannot be explained by standard microscopic theories. Here, in the framework of a second-virial density functional theory, we have developed a general approach, relaxing the usual assumption that the angular distribution of particles with respect to their local director is unaffected by the deformation. We show that, for particles with polar symmetry, a new contribution to the splay and bend deformation free energy arises, associated with the onset of polar order. Calculations for conical and bent-shaped particles reveal dramatic softening of the splay and the bend mode, respectively, which eventually may lead to spontaneous deformation.

Binary System Exhibiting the Nematic to Twist-Bend Nematic Transition: Behavior of Permittivity and Elastic Constants

We describe measurements of the permittivity and Frank elastic constant in the nematic phase of a binary system displaying a transition between the nematic (N) and the recently discovered twist-bend nematic (NTB) phase. Among the salient features observed are (i) the existence of the NTB phase even when the system is loaded with a high concentration (∼64 mol %) of a rodlike component; (ii) a clear signature in permittivity of the N-NTB transition; and (iii) a lower value of the bend elastic constant compared to the splay over a large phase space, with the difference between the two becoming a maximum for an intermediate mixture. These studies further support the surprising idea that the elastic features associated with bent molecules can be further augmented by suitable rodlike additives.

Molecular dynamics of a binary mixture of twist-bend nematic liquid crystal dimers studied by dielectric spectroscopy

We report a comprehensive dielectric characterization of a liquid crystalline binary mixture composed of the symmetric mesogenic dimer CB7CB and the nonsymmetric mesogenic dimer FFO9OCB. In addition to the high-temperature nematic phase, such a binary mixture shows a twist-bend nematic phase at room temperature which readily vitrifies on slow cooling. Changes in the conformational distribution of the dimers are reflected in the dielectric permittivity and successfully analyzed by means of an appropriate theoretical model. It is shown that the dielectric spectra of the mixture reflect the different molecular dipole properties of the components, resembling in the present case the characteristic dielectric spectra of nonsymmetric dimers. Comparison of the nematic and twist-bend nematic phases reveals that molecular dynamics are similar despite the difference in the molecular environment.

Reversible Isothermal Twist-Bend Nematic-Nematic Phase Transition Driven by the Photoisomerization of an Azobenzene-Based Nonsymmetric Liquid Crystal Dimer

The liquid crystal nonsymmetric dimer, 1-(4-butoxyazobenzene-4'-yloxy)-6-(4-cyanobiphenyl-4'-yl) hexane (CB6OABOBu), shows enantiotropic twist-bend nematic, NTB, and nematic, N, phases. The NTB phase has been confirmed using polarized light microscopy, freeze fracture transmission electron microscopy, and X-ray diffraction. The helicoidal pitch in the NTB phase is 18 nm. The NTB-N (TNTBN) and N-I (TNI) transition temperatures are reduced upon UV light irradiation, with the reduction in TNTBN being much larger than that in TNI. An isothermal, reversible NTB-N transition may be driven photochemically. These observations are attributed to a trans-cis photoisomerization of the azobenzene fragment on UV irradiation, with the cis isomers stabilizing the standard nematic phase and the trans isomers stabilizing the NTB phase. The dramatic changes in TNTBN provide evidence that the transition between the normal nematic and twist-bend nematic with spontaneous breaking of chiral symmetry is crucially dependent on the shape of molecular dimers, which changes greatly during the trans-cis isomerization.

Resonant Carbon K-Edge Soft X-Ray Scattering from Lattice-Free Heliconical Molecular Ordering: Soft Dilative Elasticity of the Twist-Bend Liquid Crystal Phase

Resonant x-ray scattering shows that the bulk structure of the twist-bend liquid crystal phase, recently discovered in bent molecular dimers, has spatial periodicity without electron density modulation, indicating a lattice-free heliconical nematic precession of orientation that has helical glide symmetry. In situ study of the bulk helix texture of the dimer CB7CB shows an elastically confined temperature-dependent minimum helix pitch, but a remarkable elastic softness of pitch in response to dilative stresses. Scattering from the helix is not detectable in the higher temperature nematic phase.

A Liquid Crystalline Oligomer Exhibiting Nematic and Twist-Bend Nematic Mesophases

The twist-bend nematic phase (NTB ) has been described as the structural link between the untilted uniaxial nematic phase (N) and the helical chiral nematic phase (N*). The NTB phase exhibits phenomena of fundamental importance to science, that is, 1) the spontaneous formation of a helical pitch on the nanometer scale in a fluid and 2) the spontaneous breaking of mirror symmetry, leading to the emergence of chiral domains in an achiral system. In this Communication, we present a study on T49 [bis(4-(9-(4-((4-cyanobenzoyl)oxy)phenyl)nonyl)phenyl) 4,4'-(nonane-1,9-diyl)dibenzoate], a liquid-crystalline oligomer exhibiting the twist-bend nematic phase, which has a molecular length that is of comparable dimensions to the sub-10 nm pitch determined for CB9CB, and provide new insights into the differentiation between the nano- and macro-science for self-assembling supermolecular systems.

Local distortion energy and coarse-grained elasticity of the twist-bend nematic phase

The recently discovered twist-bend nematic phase of achiral bent-shaped molecules, NTB, has a doubly degenerate ground-state with a periodically modulated heliconical structure and unusual distortion elasticity, the theoretical description of which is still debated. We show that the NTB phase has the same macroscopic symmetry as another periodic mesophase, the chiral smectic-A, SmA*. Based on this NTB/SmA* analogy, we develop a coarse-grained elastic model for the NTB phase. Adopting one of the existing microscopic NTB elastic models, we calculate the coarse-grained elastic constants, coherence and penetration lengths in terms of a few Frank-like nematic elastic coefficients that can be measured in macroscopic experiments. The same coarse-grained approach, applied to different local elastic models, may provide an efficient experimental test of their validity. We show that the anisotropy of the NTB coarse-grained elasticity is opposite to that of the SmA*, leading probably to different configurations of some of the defects of the "layered" NTB structure. Moreover, we argue that the intrinsic chiral frustration of the NTB phase may be resolved by penetration of the twist field into the bulk through a network of screw dislocations of the NTB pseudo-layers, resulting in a twist-bend analogue of the twist grain boundary phase TGBA.

Miscibility studies of two twist-bend nematic liquid crystal dimers with different average molecular curvatures. A comparison between experimental data and predictions of a Landau mean-field theory for the NTB–N phase transition

We have developed a Landau model that predicts a first order twist-bend nematic–nematic phase transition.

Mesophase structure and behaviour in bulk and restricted geometry of a dimeric compound exhibiting a nematic–nematic transition

We demonstrate a liquid crystal system exhibiting a variety of modulated structures on different length-scales: from helicoidal nematic to modulated smectic.

Twist, tilt, and orientational order at the nematic to twist-bend nematic phase transition of 1″,9″-bis(4-cyanobiphenyl-4'-yl) nonane: A dielectric, (2)H NMR, and calorimetric study

The nature of the nematic-nematic phase transition in the liquid crystal dimer 1″,9″-bis(4-cyanobiphenyl-4'-yl) nonane (CB9CB) has been investigated using techniques of calorimetry, dynamic dielectric response measurements, and (2)H NMR spectroscopy. The experimental results for CB9CB show that, like the shorter homologue CB7CB, the studied material exhibits a normal nematic phase, which on cooling undergoes a transition to the twist-bend nematic phase (N(TB)), a uniaxial nematic phase, promoted by the average bent molecular shape, in which the director tilts and precesses describing a conical helix. Modulated differential scanning calorimetry has been used to analyze the nature of the N(TB)-N phase transition, which is found to be weakly first order, but close to tricritical. Additionally broadband dielectric spectroscopy and (2)H magnetic resonance studies have revealed information on the structural characteristics of the recently discovered twist-bend nematic phase. Analysis of the dynamic dielectric response in both nematic phases has provided an estimate of the conical angle of the heliconical structure for the N(TB) phase. Capacitance measurements of the electric-field realignment of the director in initially planar aligned cells have yielded values for the splay and bend elastic constants in the high temperature nematic phase. The bend elastic constant is small and decreases with decreasing temperature as the twist-bend phase is approached. This behavior is expected theoretically and has been observed in materials that form the twist-bend nematic phase. (2)H NMR measurements characterize the chiral helical twist identified in the twist-bend nematic phase and also allow the determination of the temperature dependence of the conical angle and the orientational order parameter with respect to the director.

Elastic continuum theory: towards understanding of the twist-bend nematic phases

The twist-bend nematic phase, N_{TB}, may be viewed as a heliconical molecular arrangement in which the director n precesses uniformly about an extra director field, t. It corresponds to a nematic ground state exhibiting nanoscale periodic modulation. To demonstrate the stability of this phase from the elastic point of view, a natural extension of the Frank elastic energy density is proposed. The elastic energy density is built in terms of the elements of symmetry of the new phase in which intervene the components of these director fields together with the usual Cartesian tensors. It is shown that the ground state corresponds to a deformed state for which K_{22}>K_{33}. In the framework of the model, the phase transition between the usual and the twist-bend nematic phase is of second order with a finite wave vector. The model does not require a negative K_{33} in agreement with recent experimental data that yield K_{33}>0. A threshold is predicted for the molecular twist power below which no transition to a twist-bend nematic may occur.

Apolar bimesogens and the incidence of the twist-bend nematic phase

The nematic twist-bend phase (NTB) was, until recently, only observed for polar mesogenic dimers, trimers or bent-core compounds. In this article, we report a comprehensive study on novel apolar materials that also exhibit NTB phases. The NTB phase was observed for materials containing phenyl, cyclohexyl or bicyclooctyl rings in their rigid-core units. However, for materials with long (>C7) terminal chains or mesogenic core units comprising three ring units, the NTB phase was not observed and instead the materials exhibited smectic phases. One compound was found to exhibit a transition from the NTB phase to an anticlinic smectic C phase; this is the first example of this polymorphism. Incorporation of lateral substitution with respect to the central core unit led to reductions in transition temperatures; however, the NTB phase was still found to occur. Conversely, utilising branched terminal groups rendered the materials non-mesogenic. Overall, it appears that it is the gross molecular topology that drives the incidence of the NTB phase rather than simple dipolar considerations. Furthermore, dimers lacking any polar groups, which were prepared to test this hypothesis, were found to be non mesogenic, indicating that at the extremes of polarity these effects can dominate over topology.

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.

Nematic twist-bend phase with nanoscale modulation of molecular orientation

A state of matter in which molecules show a long-range orientational order and no positional order is called a nematic liquid crystal. The best known and most widely used (for example, in modern displays) is the uniaxial nematic, with the rod-like molecules aligned along a single axis, called the director. When the molecules are chiral, the director twists in space, drawing a right-angle helicoid and remaining perpendicular to the helix axis; the structure is called a chiral nematic. Here using transmission electron and optical microscopy, we experimentally demonstrate a new nematic order, formed by achiral molecules, in which the director follows an oblique helicoid, maintaining a constant oblique angle with the helix axis and experiencing twist and bend. The oblique helicoids have a nanoscale pitch. The new twist-bend nematic represents a structural link between the uniaxial nematic (no tilt) and a chiral nematic (helicoids with right-angle tilt).

Theories predict the existence of a nematic liquid crystal phase with a local twist-bend structure, but no experimental proof is available over the past 40 years. Borshch et al. identify this phase for the first time in two different materials containing dimeric molecules.

Double-well elastic theory for twist-bend nematic phases

The ground state of twist-bend nematic liquid crystals is a heliconical molecular arrangement in which the nematic director precesses uniformly about an axis, making a fixed angle with it. Both precession senses are allowed in the ground state of these phases. When one of the two helicities is prescribed, a single helical nematic phase emerges. A quadratic elastic theory is proposed here for each of these phases which features the same elastic constants as the classical theory of the nematic phase, requiring all of them to be positive. To describe the helix axis, it introduces an extra director field which becomes redundant for ordinary nematics. Putting together helical nematics with opposite helicities, we reconstruct a twist-bend nematic, for which the quadratic elastic energies of the two helical variants are combined in a nonconvex energy. States with minimal energy and opposite helicities may come in contact along an interface belonging to a special family of developable surfaces, including a peculiar cone, which can be constructed through a general method.

Measuring liquid crystal elastic constants with free energy perturbations

A first principles method is proposed to calculate the Frank elastic constants of nematic liquid crystals. These include the constants corresponding to standard splay, twist and bend deformations, and an often-ignored surface-like contribution known as saddle-splay. The proposed approach is implemented on the widely studied Gay-Berne (3, 5, 2, 1) model [J. G. Gay and B. J. Berne, J. Chem. Phys., 1981, 74, 3316], and the effects of temperature and system size on the elastic constants are examined in the nematic phase. The results of simulations for splay, twist, and bend elastic constants are consistent with those from previous literature reports. The method is subsequently applied to the saddle-splay elastic constant k24 which is found to exist at the limits of the Ericksen inequalities governing positive definite free energy. Finally, extensions of the method are discussed that present a new paradigm for in silico measurements of elastic constants.

Twist-bend heliconical chiral nematic liquid crystal phase of an achiral rigid bent-core mesogen

The chiral, heliconical (twist-bend) nematic ground state is reported in an achiral, rigid, bent-core mesogen (UD68). Similar to the nematic twist-bend (N(TB)) phase observed in bent molecular dimers, the N(TB) phase of UD68 forms macroscopic, smecticlike focal-conic textures and exhibits nanoscale, periodic modulation with no associated modulation of the electron density, i.e., without a detectable lamellar x-ray reflection peak. The N(TB) helical pitch is p(TB) ∼ 14 nm. When an electric field is applied normal to the helix axis, a weak electroclinic effect is observed, revealing 50-μm-scale left- and right-handed domains in a chiral conglomerate.

Chiral heliconical ground state of nanoscale pitch in a nematic liquid crystal of achiral molecular dimers

Freeze-fracture transmission electron microscopy study of the nanoscale structure of the so-called "twist-bend" nematic phase of the cyanobiphenyl (CB) dimer molecule CB(CH2)7CB reveals stripe-textured fracture planes that indicate fluid layers periodically arrayed in the bulk with a spacing of d ~ 8.3 nm. Fluidity and a rigorously maintained spacing result in long-range-ordered 3D focal conic domains. Absence of a lamellar X-ray reflection at wavevector q ~ 2π/d or its harmonics in synchrotron-based scattering experiments indicates that this periodic structure is achieved with no detectable associated modulation of the electron density, and thus has nematic rather than smectic molecular ordering. A search for periodic ordering with d ~ in CB(CH2)7CB using atomistic molecular dynamic computer simulation yields an equilibrium heliconical ground state, exhibiting nematic twist and bend, of the sort first proposed by Meyer, and envisioned in systems of bent molecules by Dozov and Memmer. We measure the director cone angle to be θ(TB) ~ 25° and the full pitch of the director helix to be p(TB) ~ 8.3 nm, a very small value indicating the strong coupling of molecular bend to director bend.

Effect of cybotactic clusters on the elastic and flexoelectric properties of bent-core liquid crystals belonging to the same homologous series

We report results of the splay (K_{11}) and bend (K_{33}) elastic constants and the effective flexoelectric coefficient of three bent-core liquid crystals belonging to a homologous series of 4-cyanoresorcinol bisbenzoates with varying chain lengths. Based on the results of x-ray scattering studies, one of the three compounds with a shorter chain length (C4) has few, if any, clusters present in its nematic phase and behaves quite normally, whereas the others two with longer chain lengths (C6 and C7) show the presence of cybotactic nematic phase with smectic C type clusters. These grow in size with a reduction in temperature. K_{33} is found to be the least for C7, whereas it is weakly dependent on temperature. K_{33} is somewhat higher for C4 and C6 and is almost independent of temperature. K_{11} for C6 and C7 is higher by 20% to 50% than C4 depending on the temperature. K_{11} increases linearly with a reduction in temperature for the three compounds. For C6 K_{11}>K_{33} by a factor up to ∼2 depending on the temperature, for C4 it is greater by a factor up to 1.3, and for C7 it is greater by a factor of ∼2.5. These results suggest that the clusters do not have any effect on K_{11}. The magnitude of the effective flexoelectric coefficient e=(|e_{1}-e_{3}|) is measured by creating a uniform lying helix (ULH) configuration in a planar cell. By doping the bent-core system with a small wt% of a chiral dopant, the ULH is obtained by cooling planar cells to the cholesteric phase under weak electric field. The effective flexoelectric coefficient is greater for the bent-core systems than for calamatics but it is much lower than would otherwise have been expected for such systems. |e_{1}-e_{3}| for C4 > C6 ≈ C7 is greater by 20% to 25% than C6 and C7 at the same reduced temperature. These differences in the effective flexoelectric coefficient can easily arise from a difference in the chain lengths among the members of the series but if the presence of clusters were to have an influence on |e_{1}-e_{3}|, then these would reduce it, contrary to the expectations for the bent-core systems.

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

We extend the twist-bend nematic (N(TB)) model to describe the electro-optics of this novel phase. We predict an electroclinic effect (ECE) subject to a dc electric field E applied perpendicular to the helix axis or wave vector q, with rotation of the N(TB) optic axis around E. This linear effect, with its flexoelectric origin, is a close analog to the electro-optic effects observed for chiral liquid crystals. However, in nematics composed of achiral molecules having a bent shape, it is the electro-optic signature of the N(TB) phase. We test our model experimentally in the low-temperature nematic phase of the odd liquid crystal dimer, CB7CB, with its molecules having, on average, a bent shape. The ECE measurements confirm the previously proposed twist-bend nematic structure of this phase, with its broken chiral symmetry, extremely short (<10  nm) doubly degenerate pitch and ultrafast, submicrosecond response times.