Tensorial x-ray structure factor in smectic liquid crystals

Unmasking the structure of a chiral cubic thermotropic liquid crystal phase by a combination of soft and tender resonant X-ray scattering

A resonant X-ray scattering response for two structural models of a chiral cubic phase with a giant unit cell, one composed of a continuous grid and micelles and the other with three continuous grids, is studied theoretically and compared to experimental measurements. For both structural models resonant enhancement of all the symmetry-allowed diffraction peaks is predicted, as well as the existance of several symmetry forbidden peaks (pure resonant peaks). Experimental measurements were performed at the carbon and sulphur absorption edge. Only one pure resonant peak was observed, which is predicted by both models. Two low-angle symmetry allowed peaks, not observed in non-resonant scattering, were resonantly enhanced and their intensity angular dependence can distinguish between the two structural models.

Understanding and Manipulating Helical Nanofilaments in Binary Systems with Achiral Dopants

Here, we report the relationship between helical pitch of the helical nanofilament (HNF) phase formed by bent-core molecule NOBOW and the concentration of achiral dopants 5CB and octane, using linearly polarized resonant soft X-ray scattering (RSoXS). Utilizing theory-based simulation, which fits well with the experiments, the molecular helices in the filament were probed and the superstructure of helical 5CB directed by groove of HNFs was observed. Quantitative pitch determination with RSoXS reveals that helical pitch variation is related to 5CB concentration with no temperature dependence. Doping rodlike immiscible 5CB led to a pitch shortening of up to 30%, which was attributed to a change in interfacial tension. By shedding light not only on phase behavior of binary systems but also enabling control over pitch length, our work may benefit various applications of HNF-containing binary systems, including optical rotation devices, circularly polarized light emitters, and chirality transfer agents.

Ultra-short helix pitch and spiral ordering in cholesteric liquid crystal revealed by resonant soft X-ray scattering

The spontaneous formation of chiral structures offers a variety of liquid crystals (LC) phases that could be further tailored for practical applications. In our work, the characteristic features of spiral ordering in the cholesteric phase of EZL10/10 LC were evaluated. To disclose resonant reflections related to a nanoscale helix pitch, resonant soft X-ray scattering at the carbon K edge was employed. The angular positions of the observed element-specific scattering peaks reveal a half-pitch of the spiral ordering p/2 ≈ 52 nm indicating the full pitch of about 104 nm at room temperature. The broadening of the peaks points to a presence of coherently scattering finite-size domains formed by cholesteric spirals with lengths of about five pitches. No scattering peaks were detectable in the EZL10/10 isotropic phase at higher temperatures. The characteristic lengths extracted from the resonant soft X-ray scattering experiment agree well with the periodicity of the surface "fingerprint" pattern observed in the EZL10/10 cholesteric phase by means of atomic force microscopy. The stability of LC molecules under the incident beam was proven by X-ray absorption spectroscopy in transmission geometry.

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.

Variety of subphase emerging sequences, the frustration of three main phases, SmC_{A}^{*}, SmC^{*}, and SmA, and the long-range interlayer interactions

Prompted by the existence of biaxial subphases 1/4, 2/5, and 3/7 [Phys. Rev. E 96, 012701 (2017)2470-004510.1103/PhysRevE.96.012701], we reconsidered the three-phase frustration and the resulting degeneracy lifting by combining the phase diagram of SmC_{A}^{*}, SmC^{*}, and SmA with the discrete flexoelectric effect. We systematically calculated the phase diagrams and tried to understand the overall picture of the phenomena by means of a simple and intuitively clear way in terms of minimal number of parameters. The treatment naturally explains the highly distorted helical structures of the biaxial subphases as well as the microscopic helical short-pitch of SmC_{α}^{*} which increases or decreases accordingly with rising temperature. The regular subphase emerging sequence is SmC_{A}^{*}(SmC_{α}^{*})-1/4-1/3-2/5-3/7-1/2-SmC^{*}(SmC_{α}^{*}), where the subphases other than 1/3 and 1/2 may or may not emerge. At the same time, we can see a variety of irregular sequences; in particular, any one of the biaxial subphases may singly emerge between SmC_{A}^{*}(SmC_{α}^{*}) and (SmC^{*})SmC_{α}^{*}. Moreover, the experimentally confirmed extraordinary subphase emerging sequence SmC^{*}-1/2-SmC_{α}^{*} appears for particular parameter values. Contrastingly to these affirmative aspects, some calculated results are contradictory to the previously reported experimental results: the change from SmC_{A}^{*} and SmC^{*} to SmC_{α}^{*} is always continuous, the 6-layer 2/3 subphase is not stabilized, and the subphase emerging sequence SmC_{A}^{*}-1/3-SmC^{*} does not appear. The causes of inconsistency and how to resolve them were discussed in comparisons with experimental findings.

Chiral Symmetry Breaking in Liquid Crystals: Appearance of Ferroelectricity and Antiferroelectricity

The study of chiral symmetry breaking in liquid crystals and the consequent emergence of ferroelectric and antiferroelectric phases is described. Furthermore, we show that the frustration between two phases induces a variety of structural phases called subphases and that resonant X-ray scattering is a powerful tool for the structural analysis of these complicated subphases. Finally, we discuss the future prospects for clarifying the origin of such successive phase transition.

Chirality of Liquid Crystals Formed from Achiral Molecules Revealed by Resonant X-Ray Scattering

Intensive research on chiral liquid crystals (LCs) has been fueled by their actively tunable physicochemical properties and structural complexity, comparable to those of sophisticated natural materials. Herein, recent progress in the discovery of new classes of chiral LCs, enabled by a combination of nano- and macroscale investigations is reviewed. First, an overview is provided of liquid crystalline phases, made of chiral and achiral low-weight molecules, that exhibit chiral structure and/or chiral morphology. Then, recent progress in the discovery of new classes of chiral LCs, particularly enabled by the application of resonant X-ray scattering is described. It is shown that the method is sensitive to modulations of molecular orientation and therefore provides information hardly accessible by means of other techniques, such as the sense of helical structures or chirality transfer across length scales. Finally, a perspective is presented on the future scope, opportunities, and challenges in the field of chiral LCs, in particular related to nanocomposites.

Dielectric study of a subphase stabilized in an exceptionally wide temperature range by a delicate balance of interlayer interactions and thermal fluctuations

The chiral smectic phases of calamitic liquid crystals, SmC^{*} and SmC_{A}^{*}, are characterized by the synclinic ferroelectric F ordering and the anticlinic antiferroelectric A ordering in adjacent layers. Various states with mixed A and F orderings are degenerate at the frustrated phase-transition point. The degeneracy lifting is commonly caused by the long-range interlayer interactions (LRILIs), producing a series of biaxial subphases specified by a relative ratio of both orderings, q_{T}=[F]/([A]+[F]). Sandhya et al. [Phys. Rev. E 87, 012502 (2013)PLEEE81539-375510.1103/PhysRevE.87.012502] established, however, the importance of thermal fluctuations in the degeneracy lifting in some binary mixtures of MC881 and MC452. They observed the most intriguing interplay of thermal fluctuations and LRILIs in the stabilization of an apparently single subphase. Since no other detailed experimental study of the subphase has so far been made, we carry out its dielectric investigations and clarify the following five points: (1) the subphase is surely a single phase from ≈80^{∘}C down to room temperature; (2) the imaginary part of complex permittivity ε^{″} shows the weak antiphase mode and hence it must be antiferroelectric q_{T}=1/2; (3) ε^{″} becomes much stronger above ≈80^{∘}C, indicating the emergence of ferroelectric and/or ferrielectric states; (4) the dielectric amplitude gradually increases at least just above the 1/2 subphase, suggesting it be due to a continuous increase of q_{T}; and (5) at low temperatures the antiphase relaxation mode shows irregularities that indicate the important role played by the cooperative motion of the layer undulation as well as of the director tilting.

Molecular Packing in Double Gyroid Cubic Phases Revealed via Resonant Soft X-Ray Scattering

The bicontinuous double gyroid phase is one of the nature's most symmetric and complex structures, the electron density map of which was established long ago. By utilizing small-angle x-ray scattering, resonant soft x-ray scattering at the carbon K edge and model-dependent tensor-based scattering theory, we have not only elucidated morphology but also identified molecular packing in the double gyroid phases formed by molecules with different shapes, i.e., rodlike vs taper shaped, thus validating some of the hypothetical packing models and disproving others. The spatial variation of molecular orientation through the channel junctions in the double gyroid phase can be either continuous in the case of anisotropic channels or discontinuous in the case of isotropic channels depending on the molecular structure and shape.

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.

Chiral Incommensurate Helical Phase in a Smectic of Achiral Bent-Core Mesogens

An achiral, bent-core mesogen forms several tilted smectic liquid crystal phases, including a nonpolar, achiral de Vries smectic A which transitions to a chiral, ferroelectric state in applied electric fields above a threshold. At lower temperature, a chiral, ferrielectric phase with a periodic, supermolecular modulation of the tilt azimuth, indicated by a Bragg peak in carbon-edge resonant soft x-ray scattering, is observed. The absence of a corresponding resonant umklapp peak identifies the superlayer structure as a twist-bend-like helix that is only weakly modulated by the smectic layering.

Resonant x-ray scattering observation of transitional subphases during the electric-field-induced phase transition in a mixture of Se-containing chiral smectic liquid crystals

Using resonant x-ray scattering techniques, transitional subphases during the electric-field-induced phase transition of a mixture of Se-containing chiral liquid crystals, 80% AS657 and 20% AS620, in a planar-aligned cell geometry were investigated, where the prototypical phase sequence SmC_{A}^{*}-SmC_{γ}^{*}-AF-SmC^{*} was observed; the transitional subphases were formed during the transition from the three-layer periodicity phase to the ferroelectric phase. In the lower-temperature range where the three-layer SmCγ^{*} phase appeared under the low electric field, nine- and six-layer subphases and a "streak" pattern appeared in sequence after the transition from the SmCγ^{*} phase with increasing applied electric field; the ferroelectric phase was realized. In the higher-temperature range where the four-layer AF phase appeared under a low electric field, the AF phase changed to a three-layer phase at the medium electric field. The twelve-, nine-, and six-layer subphases subsequently appeared in sequence, and finally the ferroelectric phase was generated with increasing electric field. The molecular arrangements of the field-induced subphases, especially the newly found nine-layer periodicity phase, was analyzed. The successive field-induced phase transition of the present results was compared with that of our previous results for pure Se-containing and Br-containing liquid crystals, and the relation to the three-layer ferrielectric phase was discussed.

Definite existence of subphases with eight- and ten-layer unit cells as studied by complementary methods, electric-field-induced birefringence and microbeam resonant x-ray scattering

A mixture of two selenium-containing compounds, 80 wt. % AS657 and 20 wt. % AS620, are studied with two complementary methods, electric-field-induced birefringence (EFIB) and microbeam resonant x-ray scattering (μRXS). The mixture shows the typical phase sequence of Sm-C_{A}^{*}-1/3-1/2-Sm-C^{*}-Sm-C_{α}^{*}-Sm-A, where 1/3 and 1/2 are two prototypal ferrielectric and antiferroelectric subphases with three- and four-layer unit cells, respectively. Here we designate the subphase as its q_{T} number defined by the ratio of [F]/([F]+[A]), where [F] and [A] are the numbers of synclinic ferroelectric and anticlinic antiferroelectric orderings in the unit cell, respectively. The electric field vs temperature phase diagram with EFIB contours indicates the emergence of three additional subphases, an antiferroelectric one between Sm-C_{A}^{*} and 1/3 and antiferroelectric and apparently ferrielectric ones between 1/3 and 1/2. The simplest probable q_{T}'s for these additional subphases are 1/4, 2/5, and 3/7, respectively, in the order of increasing temperature. The μRXS profiles indicate that antiferroelectric 1/4 and 2/5 approximately have the eight-layer (FAAAFAAA) and ten-layer (FAFAAFAFAA) Ising unit cells, respectively. The remaining subphase may be ferrielectric 3/7 with a seven-layer unit cell, although the evidence is partial. These experimental results are compared with the phenomenological Landau model [P. V. Dolganov and E. I. Kats, Liq. Cryst. Rev. 1, 127 (2014)2168-039610.1080/21680396.2013.869667] and the quasimolecular model [A. V. Emelyanenko and M. A. Osipov, Phys. Rev. E 68, 051703 (2003)1063-651X10.1103/PhysRevE.68.051703].

Transitional subphases near the electric-field-induced phase transition to the ferroelectric phase in Se-containing chiral smectic liquid crystals observed by resonant x-ray scattering

Resonant x-ray scattering experiments revealed transitional subphases near the electric-field-induced phase transition of a Se-containing chiral liquid crystal in a planar aligned cell geometry. In the lower-temperature range (Sm-C_{A}^{*} and three-layer periodicity Sm-C_{γ}^{*} phases), the six-layer periodicity subphase appeared with increasing electric field during the field-induced transition from Sm-C_{γ}^{*} to Sm-C^{*}. In the higher-temperature range [four-layer periodicity antiferroelectric (AF) phase], the peak positions of the three-layer satellites shifted to those of the four-layer satellites and then the satellites corresponding to the five- through seven-layer periodicity appeared in sequence. Near the AF to Sm-C_{α}^{*} phase transition temperature, the layer periodicity increased with applied field. The molecular configurations of the subphases near the field-induced transition are discussed based on the Ising, distorted clock, and perfect clock models.

Effective long-range interlayer interactions and electric-field-induced subphases in ferrielectric liquid crystals

Microbeam resonant x-ray scattering experiments recently revealed the sequential emergence of electric-field-induced subphases (stable states) with exceptionally large unit cells consisting of 12 and 15 smectic layers. We explain the emergence of the field-induced subphases by the quasimolecular model based on the Emelyanenko-Osipov long-range interlayer interactions (LRILIs) together with our primitive way of understanding the frustration in clinicity using the q_{E} number defined as q_{E}=|[R]-[L]|/([R]+[L]); here [R] and [L] refer to the numbers of smectic layers with directors tilted to the right and to the left, respectively, in the unit cell of a field-induced subphase. We show that the model actually stabilizes the field-induced subphases with characteristic composite unit cells consisting of several blocks, each of which is originally a ferrielectric three-layer unit cell stabilized by the LRILIs, but some of which would be modified to become ferroelectric by an applied electric field. In a similar line of thought, we also try to understand the puzzling electric-field-induced birefringence data in terms of the LRILIs.

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.

Probing and controlling liquid crystal helical nanofilaments

We report the first in situ measurement of the helical pitch of the helical nanofilament B4 phase of bent-core liquid crystals using linearly polarized, resonant soft X-ray scattering at the carbon K-edge. A strong, anisotropic scattering peak corresponding to the half-pitch of the twisted smectic layer structure was observed. The equilibrium helical half-pitch of NOBOW is found to be 120 nm, essentially independent of temperature. However, the helical pitch can be tuned by mixing guest organic molecules with the bent-core host, followed by thermal annealing.

Resonant x-ray diffraction spectrum for possible structures of the smectic liquid crystal phase with a six-layer periodicity

With the discovery of the smectic-C(d6)(*) (SmC(d6)(*)) phase showing six-layer periodicity [S. Wang et al., Phys. Rev. Lett. 104, 027801 (2010)] and a recent report of the observation of a possible alternative structure, the need for a reliable and accurate method for distinguishing different possible structures is more urgent than ever. Through simulations using the tensorial structure factor method, we present the resonant x-ray diffraction (RXRD) spectra for different possible structures as proposed in several theoretical studies. Subtle distinctions between models are shown. The ability and limitations of RXRD as a technique for determining the structure of this particular phase is discussed.

Systems with competing interlayer interactions and modulations in one direction: finding their structures

Complex structures in polar smectic systems can be studied within framework of discrete phenomenological models. Considered interactions are usually described by nonlinear trigonometric functions that do not allow for a straightforward search for solutions. The review of three methods reported in the literature are presented and their appropriateness, advantages and disadvantages are discussed. Examples are given as an illustration for each method.

Chiral smectic transition phases appearing near the electric-field-induced phase transition observed by resonant microbeam x-ray scattering

The electric-field-induced phase transition of a chiral liquid crystal containing Br revealed a transition phase between the three-layer periodicity ferrielectric phase and the synclinic ferroelectric phase in the electric field versus temperature phase diagram. Resonant x-ray scattering from the transition phase showed a diffuse streak or spotty weak reflections, which were composed of strong m/3-order (where m = 1 and 2) reflections and other weak peaks. The spotty reflections were found to be related to a 12-layer periodicity phase with a weak contribution from the 15-layer periodicity. An x-ray intensity analysis based on the Ising model suggested that the 12-layer periodicity phase was composed of two three-layer ferrielectric blocks and six synclinic layers. This model indicated that, in the transition phase, the three-layer ferrielectric molecular configuration gradually changed to the synclinic configuration. The diffuse streak appearing around m/3-order reflections near the field-induced transition from the four-layer periodicity phase to the synclinic ferroelectric phase is also discussed.

Superlattice structures observed in the extraordinary phase sequence and analyzed by the phenomenological Landau model and the partially molecular model

We draw several electric-field-temperature (E-T) phase diagrams with electric-field-induced birefringence contours in the nOHFBBB1M7 (n=10) and nOTBBB1M7 (n=11) (C11) mixture system by changing the C11 concentration carefully; some of the mixtures show the unusual extraordinary phase sequence where subphases with the four-, five-, and six-layer superlattice structures emerge above the smectic-C(*) main phase. We try to understand the results in terms of two complementary models that have so far been proposed: the phenomenological Landau model of phase transitions by Dolganov et al. [P. V. Dolganov et al., Phys. Rev. E 86, 020701(R) (2012)] and the partially molecular Emelyanenko-Osipov model [A. V. Emelyanenko and M. A. Osipov, Phys. Rev. E 68, 051703 (2003)]. The observed E-T phase diagram can be well reproduced by the phenomenological model. An emergence of the subphase with the four-layer superlattice structure above smectic-C(*) is also understandable in terms of the partially molecular model. We discuss the pros and cons of the two models as well.

Degeneracy lifting due to thermal fluctuations around the frustration point between anticlinic antiferroelectric SmC(A)* and synclinic ferroelectric SmC*

In the binary mixture phase diagram of MC881 and MC452, the borderline between anticlinic antiferroelectric SmC(A)(*) and synclinic ferroelectric SmC(*) becomes apparently parallel to the temperature ordinate axis at the critical concentration r(c). The free energy difference between SmC(A)(*) and SmC^{*} is extremely small in a wide temperature range near r(c). In such circumstances, by observing Bragg reflection spectra due to the director helical structure and electric-field-induced birefringence, we have observed the continuous change from SmC(A)(*) to SmC(*) for r</~r(c) and the gradual growth of the full-pitch band over 5 °C or more in the apparent SmC(*) temperature region for r>/~r(c). These intriguing phenomena have been explained, successfully at least in the high-temperature region, by a thermal equilibrium between the synclinic and anticlinic orderings and the resulting Boltzmann distribution for the ratio between them; the thermal equilibrium is considered to be attained in a nonuniform defect-assisted way through solitary waves moving around dynamically. We have also discussed qualitatively an important role played by the effective long-range interlayer interactions in the low-temperature region.

Resonant x-ray diffraction study of an unusually large phase coexistence in smectic liquid-crystal films

The recent discovery of the new smectic-C(d6)(*) (SmC(d6)(*)) phase [S. Wang et al., Phys. Rev. Lett. 104, 027801 (2010)] also revealed the existence of a noisy region in the temperature window between the SmC(d6)(*) phase and the smectic-C(d4)(*) (SmC(d4)(*)) phase. Characterized by multiple resonant peaks spanning a wide region in Q(Z), the corresponding structure of this temperature window has been a mystery. In this Letter, through a careful resonant x-ray diffraction study and simulations of the diffraction spectra, we show that this region is in fact an unusually large coexistence region of the SmC(d6)(*) phase and the SmC(d4)(*) phase. The structure of the noisy region is found to be a heterogeneous mixture of local SmC(d6)(*) and SmC(d4)(*) orders on the sub-μm scale.

Polarization periodicity in the B(1) columnar phase determined by resonant x-ray scattering

We report structural results that evidence the polarization distribution of the blocks in the columnar phase of an achiral bent-core liquid crystal. The study was performed using resonant x-ray diffraction at the sulfur K edge on oriented samples aligned on substrates. The extra periodicity is revealed through the violation of the systematic extinction rule of the structural symmetry group along the experimentally accessible diffraction direction. Further data obtained from the polarization analysis of a resonant reflection give information concerning the transition mechanism between B(1) and B(2) phases.

Theory for the evolution of ferroelectric, antiferroelectric, and ferrielectric smectic phases in the electric field

An evolution of Sm-C(A)∗, biaxial intermediate phases, and Sm-C∗ in the electric field is investigated in a framework of molecular-statistical approach [A. V. Emelyanenko et al., Phys. Rev. E 74, 011705 (2006); A. V. Emelyanenko, Eur. Phys. J. E 28, 441 (2009)]. The "electric field-temperature" phase diagrams including the possibility of existence of various tilted smectic phases are plotted and compared with the experimental ones. Permanent transverse molecular dipole moments (without electric field participating only in the spontaneous polarization) were also found to generate the induced polarization in the presence of electric field and to produce very strong dielectriclike effect. This effect is positive in Sm-C∗ (tilt planes have a tendency of orienting along or against the electric field) and is negative in Sm-C(A)∗ and in biaxial intermediate phases (tilt planes have a tendency of orienting perpendicular to the electric field). In the ferrielectric intermediate phases both spontaneous and induced polarizations favor similar tendencies and provide the helix unwinding at very low electric field. At the same time, the tendencies provided by spontaneous and induced polarizations are opposite in Sm-C∗, and therefore the unwinding threshold is larger. It was shown that interplay between spontaneous and induced polarizations can lead to the formation of complex bidomain smectic structures. A single parameter regulating an evolution of structure of Sm-C∗, Sm-C(A)∗, and biaxial intermediate phases in the electric field was found. We suppose that bidomain helical structure is the same as additional ferrielectric phase FiLC existing in some materials just below Sm-C∗. The numerical calculations are done with help of AFLC phase diagram plotter software developed by the author and available at his webpage.

Discovery of a novel smectic-C{*} liquid-crystal phase with six-layer periodicity

We report the discovery of a new smectic-C{*} liquid-crystal phase with six-layer periodicity by resonant x-ray diffraction. Upon cooling, the new phase appears between the SmC{alpha}{*} phase having a helical structure and the SmC{d4}{*} phase with four-layer periodicity. This SmC{d6}{*} phase was identified in two mixtures which have an unusual reversed SmC{d4}{*}-SmC{*} phase sequence. The SmC{d6}{*} phase shows a distorted clock structure. Three theoretical models have predicted the existence of a six-layer phase. However, our experimental findings are not consistent with the theories.

Microbeam resonant x-ray scattering from bromine-substituted bent-core liquid crystals

We studied the local layer structure in the B2 phase of bromine-substituted bent-core liquid crystals in the cell geometry using microbeam resonant x-ray scattering. In the homochiral state of B2 phase, the 3/2 order satellite peak was observed only when the incident x-ray energy is at the K absorption edge of bromine. This result clearly indicates that the B2 homochiral domain forms two-layer superlattice in adjacent layers, the same as in the rodlike Sm-C(A) phase. The work reports on microbeam resonant x-ray scattering experiment from the local layer of the bent-core liquid crystal in the device geometry. Moreover, we can say that bromine is also useful for the analysis of the superstructure of soft materials using resonant x-ray scattering.

Effects of doping on an unusual smectic- C*alpha-smectic-C*FI2-smectic-C* phase sequence

The compound 10OHF has a partially inverted phase sequence, unique among the series of nOHF homologous compounds and all other known liquid crystals, with the smectic-C*FI2 (SmC*FI2) phase occurring at higher temperature than the smectic-C* (SmC*) phase. We present ellipsometric data to identify the phase sequences of 9OHF, 10OHF, 11OHF, and 12OHF. Binary mixtures of 10OHF with C11, a compound with the typical phase sequence among the smectic phases, show that the unusual phase sequence of 10OHF stabilizes upon mixing and that SmC*FI2 predominates over SmC* throughout the entire mixing phase diagram. In thin films of some mixtures, surface interactions induce a reentrant SmC*FI2-SmC*-SmC*FI2 transition in the rest of the film.

Unexpected field-induced phase transitions between ferrielectric and antiferroelectric liquid crystal structures

Liquid crystals are intriguing electrically responsive soft matter systems. We report previously unexplored field-induced changes in the structures of some frustrated liquid crystal phases and describe them theoretically. Specifically, we have discovered using resonant x-ray scattering that the four-layer intermediate smectic phase can undergo either a transition to the ferrielectric (three-layer) phase or to the ferroelectric phase, depending on temperature. Our studies of intermediate phases using electric fields offer a way to test theories that describe ferroelectricity in self-assembling fluids.

Polarization studies of resonant forbidden reflections in liquid crystals

We report the results of resonant x-ray diffraction experiments performed on thick films of a biaxial liquid crystal made of achiral bent-core molecules. Polarization properties of forbidden reflections are observed as a function of the sample rotation angle phi about the scattering vector Q for the first time on a fluid material. The experimental data are successfully analyzed within a tensor structure factor model by taking the nonperfect alignment of the liquid crystal into account. The local structure of the B2 mesophase is hence determined to be SmC_{S}P_{A}.

Deduction of the temperature-dependent structure of the four-layer intermediate smectic phase using resonant X-ray scattering

A binary mixture of an antiferroelectric liquid-crystal material containing a selenium atom and a highly chiral dopant is investigated using resonant X-ray scattering. This mixture exhibits a remarkably wide four-layer intermediate smectic phase, the structure of which is investigated over a temperature range of 16K. Analysis of the resonant X-ray scattering data allows accurate measurement of both the helicoidal pitch and the distortion angle as a function of temperature. The former decreases rapidly as the SmC* phase is approached, whilst the latter remains constant over the temperature range studied at 8 degrees +/-3 degrees. We also observe that the senses of the helicoidal pitch and the unit cell of the repeating four-layer structure are opposite in this mixture and that there is no pitch inversion over the temperature range studied.

Smectic-C*alpha phase with two coexistent helical pitch values and a first-order smectic-C*alpha to smectic-C* transition

Previous results from Kundu using dielectric relaxation have suggested a reentrant antiferroelectric-ferroelectric-antiferroelectric transition in the compound LN36. Our comprehensive studies of this compound using differential optical reflectivity, nonadiabatic scanning calorimetry, null transmission ellipsometry, and resonant x-ray diffraction show that in fact LN36 exhibits the usual phase sequence for chiral smectic liquid crystals: SmA*-SmC*alpha-SmC*-SmC*FI1-SmC*A . Moreover, the SmC*alpha-SmC* transition is a first-order transition, characterized by a discontinuous change in the helical pitch. At temperatures just above the SmC*alpha-SmC* transition, two different values for the helical pitch are simultaneously observed for the first time.

Screwlike order, macroscopic chirality, and elastic distortions in high-density DNA mesophases

We investigate a new screwlike liquid-crystalline ordering in solutions of helical biopolymers and its influence on the state of individual molecules. In the resulting mesophase translational and rotational motions of molecules are coupled in screw fluctuations. We show that in contrast to the case of conventional chiral liquid crystals the elastic distortion does not twist the screw order but leads to overwinding of individual helical molecules. This explains the peculiarities of high-density DNA mesophases.

Extension of the resonant scattering technique to liquid crystals without resonant element

We report X-ray resonant scattering experiments performed on the prototype liquid-crystalline compound MHPOBC doped with a chemical probe containing a resonant atom (selenium). We determined directly for the first time the microscopic 3- and 4-layer structure of the ferrielectric subphases (SmC(FI1)* and SmC(FI2)*) present in MHPOBC. Despite the low fraction of the selenium probe, the resonant signal is strong enough to allow an unambiguous determination of the basic structure of the ferrielectric subphases. These experiments demonstrate that the resonant scattering technique can be extended to liquid crystalline materials without resonant element and may stimulate new studies. A non-resonant Bragg reflection was also found in the SmC(FI1)* phase in pure MHPOBC, consistent with the 3-layer distorted model, but never detected before.

Optical and resonant X-ray diffraction studies confirm a SmC*FI2-SmC* liquid crystal phase sequence reversal

Employing both null transmission ellipsometry and resonant x-ray diffraction, we confirmed the SmC*FI2-SmC* phase sequence reversal in one liquid crystal compound and specially prepared binary mixtures. This phase sequence reversal was predicted by two recent theoretical advances. Moreover, the temperature range for the SmC*FI2 phase increases significantly in the mixture suggesting that such a phase sequence may exist in other compounds.

Resonant X-ray Scattering: A Tool for Structure Elucidation in Liquid Crystals

The use of resonant X-ray scattering to determine structures in liquid crystal systems is surveyed. This powerful experimental technique utilises "forbidden reflections" to determine the subtle differences in interlayer orientation that differentiate several smectic systems. The technique relies on the materials containing an atom to which the X-ray energy can be tuned, usually sulphur or selenium. Experiments are often carried out on free-standing films that provide a highly monodomain structure that allows high-resolution measurements to be made, and, hence, structural details to be determined. Alternatively, resonant scattering has been demonstrated for materials contained in glass devices that permit the application of electric fields to the system, in a manner analogous to that used in liquid crystal devices. The resonant scattering technique provides unequivocal descriptions of the packing in smectic systems, and the way in which the packing is distorted in electric fields. This Minireview describes the principles behind resonant X-ray scattering, its application to liquid crystals and some of the potential for the future.

Discrete flexoelectric polarizations and biaxial subphases with periodicities other than three and four layers in chiral smectic liquid crystals frustrated between ferroelectricity and antiferroelectricity

The subphase for the temperature range that lies in between Sm-C(A)* and the three-layer Sm-C(A)* (1/3)subphase has been confirmed to exist using the measurements of electric-field-induced birefringence, optical rotation, and the characteristic reflection bands in the antiferroelectric liquid-crystalline compound, 1-trifluoromethylundecyl-4-(4'-dodecyloxybiphenyl-4-yl-carbonyloxy)-3-fluorobenzoate (12BIMF10). The measurements of electric-field-induced birefringence and optical rotatory power are made on -thick homeotropic cells, and the characteristic reflection bands are observed in free-standing films of thicknesses ranging from 30 to 50 microm. Several binary mixtures have been prepared by mixing (S)-12BIMF10 with (S)-4-(1-methylheptyloxycarbonyl)-phenyl-4'-octylbiphenyl-4-carboxylate (MHPBC), and the effect of racemization of the compound on the character of the biaxial subphase (other than three and four layers) is also discussed. The results are interpreted in terms of the Emelyanenko-Osipov model [Phys. Rev. E 68, 051703 (2003)]; the effective long-range couplings between the director orientations in separated smectic layers emerge after the minimization of free energy with respect to the total (ordinary spontaneous and discrete flexoelectric) polarizations, lifting the degeneracy and producing the nonplanar structures of the subphases.

Two kinds of smectic-C(alpha)* subphases in a liquid crystal and their relative stability dependent on the enantiomeric excess as elucidated by electric-field-induced birefringence experiment

The electric-field-induced birefringence has been investigated by using a photoelastic modulator, with a view to obtaining a molecular model for the subphases produced by the frustration between ferroelectricity and antiferroelectricity in the chiral smectic liquid crystals. It has been found that even in the bulk, there exist two subphases in the smectic-C(alpha)* (Sm-C(alpha)*) temperature range. By extending the Emelyanenko-Osipov model [Phys. Rev. E 68, 051703 (2003)] to include the temperature dependence of the tilt angle, we have alluded to a possible lifting of the degeneracy at the frustration point P(alpha) , where Sm-C(A)*, Sm-C*, and Sm-A have the same free energy. This leads to the appearance of uniaxial Sm-C(alpha)* characterized by short-pitch helical structures and consequently with a pitch much lower than the optical wavelength. The numerical calculations indicate that the short pitch may generally increase or decrease monotonically with temperature. Depending on the parameter value that represents the relative strength of ferroelectricity and antiferroelectricity, the short-pitch temperature variation may abruptly change from increase to decrease at a temperature; this can be assigned to the observed phase transition between the two Sm-C(alpha)* subphases.

New investigation on the tetragonal liquid-crystalline phase or SmQ

Chiral liquid crystals, for which twist and smectic order strongly compete, can exhibit complex architectures. This is the case of the SmQ phase which shows a high degree of 3D order (tetragonal or hexagonal) as well as a liquid signature. We have combined X-ray scattering and optical observations on a new compound exhibiting a SmQ phase to discriminate between the two models proposed by A.M. Levelut et al. (J. Phys. II 7, 981 (1997)) at the molecular level for the phase with the I4(1)22 space group.

Self-assembled uniaxial and biaxial multilayer structures in chiral smectic liquid crystals frustrated between ferro- and antiferroelectricity

With a view to obtain a molecular model for the subphases produced by the frustration between ferro- and antiferroelectricity in chiral smectic liquid crystals, we report results on two compounds and observe (i) the staircase character of uniaxial Sm C(*)(alpha) itself in the bulk and (ii) the multipeaked characteristic reflection bands due to the modulated helical structures just above the Sm C(*)(A) temperature range. We suggest the emergence of several uniaxial and biaxial subphases. The results show that both types of subphases can be specified by q(T) = [F] / ( [A] + [F] ) in the zero-order approximation; [A] and [F] are the numbers of antiferroelectric and ferroelectric orderings in the unit cell. We consider the basis of both types of subphases, particularly the description of the short-pitch helical structure of Sm C(*)(alpha), in terms of the molecular models so far proposed and emphasize the important role played by the discrete flexoelectric polarization.

Optical rotatory power, biaxiality, and models of chiral tilted smectic phases

Among the chiral tilted smectics, the stable existence has been confirmed in numerous investigations of SmC(*)(A), (antiferroelectric smectic-C(A)) SmC(*)(F11) (SmC(*)(gamma)), SmC(*)(F12) (antiferroelectric, AF) and SmC* phases. The structures of the ferrielectric SmC(*)(F11) and SmC(*)(F12) phases suggested by different models are essentially different although all the models use the three-layer and four-layer periodicity for them. The structures of the phases were investigated using the optical rotatory power (ORP) measurements technique. The ORP was simulated using Berreman's 4 x 4-matrix method. The compound under investigation (S)-1-methylheptyl 4-(4(')-n-undecyloxy-biphenyl-4-yl-carbonyloxy) [acronym (S)-11OF1M7] clearly provides SmC(*)(F11) and SmC(*)(F12) phases, the temperature range for the existence of these phases is about 5 degrees C each. This had not been achieved for the earlier investigated antiferroelectric liquid crystal (AFLC) samples. The results obtained confirm that the unit cell of the molecular structure of these subphases is highly biaxial. Due to the biaxiality the texture of the homeotropic cell under a polarizing microscope appears nonuniform. This requires a special approach to the measurements and a simulation of the ORP, which is discussed in detail. A technique has been designed where the transmitted intensity through a polarizing microscope is measured as a function of the angle of polarization of the incident light. From the observed output, which is a biased sine wave, the ORP is being determined. In the same scan, the wavelength of light is also being automatically altered. Comparing the simulated and measured data, we can conclude that in the SmC(*)(F12) phase the distortion angle of the directors in the Ising model is lower than 10 degrees. Using the Ising model, the pitch in SmC(*)(F11) has been determined and this is found to have a strong temperature dependence.

Interlayer structures of the chiral smectic liquid crystal phases revealed by resonant X-ray scattering

The structures of the liquid crystalline chiral subphases exhibited by several materials containing either a selenium or sulphur atom have been investigated using a resonant x-ray scattering technique. This technique provides a unique structural probe for the ferroelectric, ferrielectric, antiferroelectric, and SmC(*)(alpha) phases. An analysis of the scattering features allows the structural models of the different subphases to be distinguished, in addition to providing a measurement of the helical pitch. This paper reports resonant scattering features in the antiferroelectric hexatic phase, the three- and four-layer intermediate phases, the antiferroelectric and ferroelectric phases and the SmC(*)(alpha) phase. The helicoidal pitch has been measured from the scattering peaks in the four-layer intermediate phase as well as in the antiferroelectric and ferroelectric phases. In the SmC(*)(alpha) phase, an investigation into the helical structure has revealed a pitch ranging from 5 to 54 layers in different materials. Further, a strong resonant scattering signal has been observed in mixtures of a selenium containing material with as much as 90% nonresonant material.

Orientational ordering in the chiral smectic-C*FI2 liquid crystal phase determined by resonant polarized x-ray diffraction

High-resolution resonant polarized x-ray diffraction experiments near the sulfur K edge have been performed on free-standing liquid crystal films exhibiting the chiral smectic-C*FI2 phase. It is widely accepted that this phase has a four-layer repeat unit, but the internal structure of the repeat unit remains controversial. We report different resolved features of the resonant x-ray diffraction peaks associated with the smectic-C*FI2 phase that unambiguously demonstrate that the four-layer repeat unit is locally biaxial about the layer normal and that the measured angle, describing the biaxiality, is in good agreement with optical measurements.

Optical rotation and structure of ferrielectric smectic phases

We analyze the measured optical rotation in three- and four-layer smectic ferrielectric phases within the matrix approach to the light propagation. We show that "perfect" three- and four-layer structures with 120 degrees and 90 degrees phase rotation of the director in neighboring layers give negligible optical rotation of polarized light travelling along the normal to the smectic layers. Significant optical rotation is obtained in deformed three- and four-layer smectic phases. The analysis of the measured optical rotatory power clearly shows that three-layer ferrielectric phases of [4-(1-methylheptyloxycarbonylphenyl)-octylbiphenyl-4-carboxylate] and 4-[(4-[[1(*)-methyl]heptycarboxy]phenyl)carboxy]phenyl-4'-decyloxy-1-benzencarbatioate (10OTBBB1M7) are deformed with the deformation angle of 35 degrees-45 degrees. The deformation angle in the four-layer smectic phase of 10OTBBB1M7 is 70 degrees-90 degrees. This is in reasonable agreement with other experiments and suggests the validity of the "deformed clock model."

Resonant x-ray scattering study of the antiferroelectric and ferrielectric phases in liquid crystal devices

Resonant x-ray scattering has been used to investigate the interlayer ordering of the antiferroelectric and ferrielectric smectic C* subphases in a device geometry. The liquid crystalline materials studied contain a selenium atom and the experiments were carried out at the selenium K edge allowing x-ray transmission through glass. The resonant scattering peaks associated with the antiferroelectric phase were observed in two devices containing different materials. It was observed that the electric-field-induced antiferroelectric to ferroelectric transition coincides with the chevron to bookshelf transition in one of the devices. Observation of the splitting of the antiferroelectric resonant peaks as a function of applied field also confirmed that no helical unwinding occurs at fields lower than the chevron to bookshelf threshold. Resonant features associated with the four-layer ferrielectric liquid crystal phase were observed in a device geometry. Monitoring the electric field dependence of these ferrielectric resonant peaks showed that the chevron to bookshelf transition occurs at a lower applied field than the ferrielectric to ferroelectric switching transition.

Optical rotatory power of different phases of an antiferroelectric liquid crystal and implications for models of structure

The antiferroelectric liquid crystal (AFLC) under investigation possesses different helical smectic phases. The various phases have been identified through a texture under cross-polarizers with a homeotropic alignment of the AFLC. Measurements of the optical rotatory power (ORP) of these phases have elucidated the ability of this method for finding phase transitions between several phases and for investigating the helical structure of the antiferroelectric phases. The optical rotatory power as a function of temperature at a fixed wavelength of light was measured for different phases of the investigated AFLC material. The values of the pitch for some of the phases have been calculated from the ORP data. The results of the ORP rule out the simple "clock" model or a clock model with a long pitch superimposed on to it. The results can be explained only in terms of biaxial models, either Ising-type models or a highly distorted "clock" model. It is also found that in the SmC*A phase the sense of the helix in the investigated material is left handed, and is opposite to that observed in the SmC* phase. The reversal of the helix from left to right handed occurs during the phase transition from the SmC*FI1(SmC*gamma) to SmC*FI2(AF) phase. This fact also allows for SmC*A and AF phases to be distinguished from each other.