INTRODUCTION

Frustration is the cause of formation of a large number of unusual structures. Chiral nematics are the richest system where frustration-induced structures can form [1–3]. Stable and metastable configurations are observed in particular in systems of finite size due to competition between chirality, elasticity, and surface anchoring. Their studies are important both from the fundamental point of view and with regard to many technical applications. Investigations in this direction are of general physical interest in particular in such areas of physics as phase transitions in media with many-component order parameter, topological defects and frustration. The simplest chiral structure is the cholesteric liquid crystal which has helicoidal ordering formed by the rotation of average orientation of long molecular axis (so-called n-director). The period of the structure (helical pitch p) is much larger than characteristic molecular scale. More complex structures can be obtained from the cholesteric phase at high temperatures. Although the jump of the orientational order parameter at these transitions is small, the local orientational order and macroscopic structure can be essentially different. Most well-known structures formed from cholesteric are so-called Blue Phases (BP) [4–7]. Their structure remained a mystery for many decades and turned out to be quite interesting and complex. Blue Phases are formed by an ordered structure of defects (disclinations) due to spatial frustration of local molecular ordering (so-called double twist which cannot fill continuously the three-dimensional space). The cubic cells of Blue Phases contain several millions molecules. Modulated structures with double twist formed by periodically ordered defects can be also observed in other systems, in particular, in cubic helical magnetics [8]. Note that chirality of magnetics is determined by the value and sign of Dzyaloshinskii–Moriya interaction [8, 9].

This paper is devoted to chiral structures with ordering of different dimensions which appear from cholesteric near the transition to isotropic liquid. A sequence of three-dimensional (3D), two-dimensional (2D) and one-dimensional (1D) structures forms on increasing the pitch of cholesteric. Investigations were performed on liquid crystal mixtures of achiral nematic and chiral compound. Samples were prepared from liquid crystal E7 with a wide temperature range of the nematic phase and chiral compound CB15 (Synthon Chemicals). In the cholesteric phase the photonic stop band exists for light of right circular polarization. Cholesteric liquid crystal was capillary introduced into a cell with planar anchoring on the surfaces. We used commercial cells produced by Instec Inc. with the alignment layer favoring homogeneous planar orientation of molecules at the surface, homemade planar and wedge cells. Thickness of the cells varied from 9 to 50 μm. In wedge cells, in particular, cholesteric helical pitch was determined by Cano–Grandjean method [1]. We employed Olympus BX51 polarizing optical microscope equipped by a video camera and Avantes fiber spectrometer. Observations were performed in transmitted and reflected light, with linear and circular light polarization.

We investigated the formation of different structures depending on the cholesteric pitch. All investigated mixtures form the cholesteric phase. Varying the contents of the chiral compound, we changed the value of the cholesteric pitch p and the spectral position of the photonic stop band \({{\lambda }_{B}} = pn\), where n is the average refractive index [3]. Figure 1a shows the dependence of the cholesteric pitch p and wavenumber \(q = 2\pi {\text{/}}p\) on the concentration of the chiral dopant X. Closed symbols are the results obtained by the Cano–Grandjean wedge method for cholesteric with pitch \(p > 0.3{\kern 1pt} \) μm (Fig. 1b). The open symbols are the data obtained from measurements of the selective reflection spectra for cholesteric with short pitch \(p < \) 0.7 μm (Fig. 1c). The data obtained by the two methods on the same mixture well agree with each other (Fig. 1a). Linear increase in q with increase in X is typical for mixtures of nematic with chiral additive [10]. Three structures with 3D, 2D, and 1D ordering formed from cholesteric on heating near the transition to the isotropic phase. In the work we describe their optical and structure peculiarities.

Fig. 1.
figure 1

(Color online) (a) Dependence of the helical pitch p and wavenumber of cholesteric \(q = 2\pi {\text{/}}p\) on contents of the chiral additive X in the nematic liquid crystal. Solid symbols are the data obtained by Cano–Grandjean method (Fig. 1b). Open symbols are the data obtained from measurement of selective reflection spectra (Fig. 1c). In all mixtures planar cholesteric exists at low temperature (yellow stripe). Different structures with three-dimensional (3D), two-dimensional (2D), and one-dimensional (1D) ordering are formed near the transition to the isotropic phase at change of the cholesteric helix: (blue stripe) three-dimensional Blue Phases, (red stripe) two-dimensional structure, and (gray stripe) one-dimensional stripe structure.

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THREE-DIMENSIONAL STRUCTURES

In chiral materials with small cholesteric pitch (p is less than approximately 0.32 μm, right part of the diagram in Fig. 1a) near the transition to the isotropic liquid the cholesteric transforms to three-dimensional structures: Blue Phase I (BPI) and Blue Phase II (BPII) [4–7]. Figures 2a, 2b shows photographs of a polycrystalline sample of BPI and a single-crystal sample of BPII. Local molecular and macrostructure ordering in Blue Phases essentially differ from cholesteric. The local minimum of energy corresponds to double twist—rotation of the orientation of long molecular axes in two perpendicular directions. Formation of macroscopic cubic structure results from geometrical frustration [3]. Cylinders with double twist cannot fill up three-dimensional space. One of the ways to relieve such frustration is to form topological defects (disclinations with strength –1/2), located between double-twist regions. As a result, regular cubic cells are formed. Macroscopic cubic phases may be also considered as structures formed by a set of Fourier harmonics with different period and orientation. At present time it is established that BPI has body centered cubic cell (space group O8), BPII has simple cubic cell (space group O2), the size of elementary cells is several hundred nanometers. “Fog” phase (BPIII) is formed in compounds with short helical pitch and is macroscopically isotropic [7]. In contrast to cholesteric, where a single selective reflection band exits, in three-dimensional Blue Phases BPI and BPII reflections from different crystalline planes can be observed. Reflection spectra of BPI and BPII (Figs. 2a, 2b) are shown in Fig. 3a. In the polydomain BPI structure (Fig. 2a) different colors correspond to crystalline planes (110) and (200), oriented parallel to the film plane. Diffraction is allowed from planes shown in Fig. 3a. The monodomain BPII sample possesses a single [100] reflection peak.

Fig. 2.
figure 2

(Color online) Photographs of three-dimensional (3D), two-dimensional (2D) and one-dimensional (1D) structures. (a) Polycrystalline texture of BPI; (b) single-crystal film of BPII. Ordered (c) 2D and (d) 1D structures. The cholesteric pitch is p = 0.29 μm (3D), 0.352 μm (2D), and 4 μm (1D). The photographs were taken in (a–c) reflection and (d) transmission. The horizontal size of the photographs is (a–c) 60 and (d) 23 μm.

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Fig. 3.
figure 3

(Color online) (a) Reflection spectra of polycrystalline film of BPI with [110], [200], and [211] reflections and single-crystal film of BPII with the [100] reflection. (b) Reflection spectra of the two-dimensional structure in light of (solid line) right circular polarization and (dashed line) left circular polarization. In each spectrum two bands are visible (1 and 2). The ratio of the wavelengths of the maxima of the two bands in each spectrum is about 1.4. The cholesteric pitch p = (a) 0.29 and (b) 0.352 μm.

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With increase in temperature the sequence of transitions cholesteric–BPI–BPII–BPIII is observed. On the temperature—chiral dopant concentration phase diagram with decrease in chirality (increase in cholesteric pitch) near the transition to the isotropic liquid Blue phases appear in sequence BPIII–BPII–BPI, i.e., macroscopically isotropic structure—simple cubic—body centered cubic [7]. A sequence of such type corresponds to the generic phase diagram and sequence of phases following from the Landau weak crystallization theory [4, 11]. We extended and generalized this sequence of transitions with decrease in chirality, including not only three-dimensional 3D phases but also structures with reduced dimensions (2D and 1D).

TWO-DIMENSIONAL STRUCTURE

The situation with structures forming at larger helical pitch (p greater than about 0.32 μm), is more complex. Earlier in literature formation of different unusual textures was described [12–26], including in free standing films [27–29]. However, it is not always clear whether these textures correspond to the same or different structures. Partially the reason is that measurements were made using disordered samples. We succeeded in obtaining the two-dimensional ordered structure near the transition to the isotropic phase [30]. Slow heating of the cholesteric (as a rule at a rate about 0.1 °/min) enabled to obtain regions with regularly ordered system of domains (Fig. 2c). Reflection spectra of the two-dimensional structure in right and left circular polarization are shown in Fig. 3b. Elongated green domains (Fig. 2c) retain the orientation of planar cholesteric near the surface and have the same reflection wavelength as cholesteric (about 575 nm in our case, Fig. 3b). The wide band at smaller wavelength is attributed to double reflections from regions where the cholesteric helix makes an angle approximately 45° with respect to the sample plane [30]. It was found that ordered two-dimensional regions form more easily when the cholesteric phase has planar orientation near the surface. The two-dimensional structure is stable at high temperature (in the temperature range about 1°C near the cholesteric–isotropic liquid transition) and could be preserved as a metastable structure at low temperature (far from the transition). Cholesteric pitch is substantially smaller than the period of the square lattice and film thickness. Sometimes we observed formation of another texture formed by large domains with curved boundaries (Fig. 4). Inside the large domain a point peculiarity is visible in reflection (Fig. 4a). Its color and wavelength of reflection correspond to planar cholesteric. In a number of cases large domains form spatially ordered groups (Fig. 4c). The peculiarities of this structure are rather interesting and resemble the texture of focal conic domains in smectic liquid crystals [1]. The explanation of these textures (related to the circumstance that on scale greater than the helical pitch cholesteric in a number of cases can be regarded as equivalent to smectic, see, for example, [3]) can be a subject of further theoretical studies.

Fig. 4.
figure 4

(Color online) Texture formed from cholesteric with pitch \(p = 0.352{\kern 1pt} \)μm. The pattern resembles the texture of focal conic domains in smectic liquid crystals. (a) Reflection, (b–d) transmission, polarizers are crossed. The domains can form spatially ordered groups (c). (d) Two domains are shown on enlarged scale.

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ONE-DIMENSIONAL STRUCTURE

At large cholesteric pitch (p greater than 1 μm) one-dimensional stripe structure is formed (Fig. 2d). The period of this grating is about half of the cholesteric pitch. Bent striped and more complex structures are formed from cholesteric droplets [31]. In our experiments we also observed similar structures. There are several explanations for the formation of stripe structures in cholesteric [31–34].

Let us now discuss the cause of formation of modulated structures focusing on the two-dimensional square ordering. Some optical peculiarities of our two-dimensional structure resemble confocal domains [12–15]. Confocal domains were first found in smectic liquid crystals and their structure was explained by Friedel, Grandjean [35, 36], and Bragg [37] as formed by complementary (conjugated) defects in the form of an ellipse and a hyperbole. Rosenblatt et al. [13] proposed a model of the confocal structure with two conjugated parabolas for explanation of the textures observed in smectic and cholesteric phases. Significant contribution to understanding of the confocal cholesteric domains was made in particular by Bouligand [12, 38] and Yada, Yamamoto, and Yokoyama [39, 40]. One of the main results of Bouligand [12] was that structures on the opposite sides of liquid-crystal film are shifted by half of the period of two-dimensional ordering. We also observed this peculiarity at the two surfaces in our two-dimensional structure.

Two-dimensional and especially one-dimensional structures are often formed in electric field. In particular their formation is connected to Helfrich–Hurault effect [1, 3]. Detailed investigations of field-induced two- and one-dimensional structures were performed by Senyuk, Smalyukh, and Lavrentovich [17] using fluorescence confocal polarizing microscopy. Two-dimensional structures had square lattice in the plane of the cholesteric film. Domain formation depended on the surface anchoring at the film boundaries. In the one-dimensional structure, periodically located parabolic defects were observed in vertical cross-section of the film with the centers of curvature in the focus of parabola (\({{\lambda }^{{ + 1/2}}}\) disclination) and near the surface (\({{\lambda }^{{ - 1/2}}}\) disclination) [17]. However, in these experiments, two- and one-dimensional structures were induced by the electric field with a transition from two- to one-dimensional at increasing field. In our case the transition between two- and one-dimensional structures occurs spontaneously at increasing cholesteric pitch without application of external forces.

Selinger [41] and Long and Selinger [42] recently proposed a new approach to explain the multiplicity of different modulated structures in chiral nematics and the role of frustration in their formation. The main result of the works [41, 42] is that chiral nematic has a tendency to form local double-twisted configuration due to so-called saddle-splay term in the elastic energy (Frank modulus K24). Because this structure cannot fill up the three-dimensional space, it must be frustrated. That is why usual cholesteric forms as one of the options of relieving frustrations. As shown in [41, 42], very weak orientational anchoring at the boundary or free surfaces also allow to escape from frustrations. It is possible that a similar mechanism can be realized in our experiments near the phase transition to the isotropic phase. As a result, the two-dimensional structure replaces three-dimensional Blue Phases at large cholesteric pitch.

In conclusion, near the transition to the isotropic liquid three different structures can be formed from the cholesteric phase: three-dimensional (3D) Blue Phases at short helical pitch, two-dimensional (2D) structure in the plane of the film at intermediate pitch and one-dimensional (1D) structure at large pitch. It is worth noting that all these structures and their sequence can be obtained using the same nematic material with different quantities of chiral additive in cells of the same thickness, using commercial cells, homemade cells and even cells without special treatment of the surface. This allows to suggest that the observed phenomena are related to universal properties of chiral materials. At present time ordering of confocal domains is the most suitable model for explanation of the peculiarities of observed two-dimensional structures, and frustration can be regarded as the reason of their formation.