Chiral Structures from Achiral Micellar Lyotropic Liquid Crystals under

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Chiral structures from achiral micellar lyotropic liquid crystals under capillary confinement Clarissa F. Dietrich, Per Rudquist, Kristin Lorenz, and Frank Giesselmann Langmuir, Just Accepted Manuscript • Publication Date (Web): 16 May 2017 Downloaded from http://pubs.acs.org on May 16, 2017

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Chiral structures from achiral micellar lyotropic liquid crystals under capillary confinement Clarissa F. Dietrich,† Per Rudquist,‡ Kristin Lorenz, † Frank Giesselmann*, † †

Institute of Physical Chemistry, University of Stuttgart, Pfaffenwaldring 55, 70569 Stuttgart, Germany ‡

Microelectronics and Nanoscience, Chalmers University of Technology, 41296 Göteborg, Sweden

ACS Paragon Plus Environment

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ABSTRACT:

Recently the emergence

of

spontaneous

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reflection

symmetry broken

configurations in achiral chromonic liquid crystals confined in cylindrical capillaries with homeotropic anchoring at the cylinder walls was reported, namely the so-called twisted-escaped radial (TER) and twisted planar polar (TPP) configurations. This new example of spontaneous reflection symmetry breaking in liquid crystals was attributed to the twist elastic modulus which is known to be unusually small in comparison to the splay and bend moduli in the case of chromonic liquid crystals. We now report the experimental observation of reflection symmetry breaking in cylindrical capillaries in the case of a classical, achiral and non-chromonic lyotropic liquid crystal forming a nematic phase of disc-like micelles orienting homeotropically at the capillary walls. We observe the same chiral TER configuration as well as a non-planar twisted polar (TP) configuration. The TP is characterized by two half unit so-called twist disclinations, where the director twist around the line defects drives the formation of a double helix of the disclinations along the axis of the capillary. Additionally, there is a transverse twist between the two disclination lines with the opposite sign of the axial twist. Similarities and differences to the case of chromonic liquid crystals are discussed, in particular we examine the conditions under which spontaneous reflection symmetry breaking occurs in the non-chromonic system. It seems that the chiral TER configuration can be stabilized by the presence of point defects.


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INTRODUCTION The spontaneous appearance of macroscopic chiral domains in a molecular system of achiral components gives rise to fundamental questions such as the molecular mechanisms of reflection symmetry breaking1–4 and the origin of homochirality5 in biomolecules. While the spontaneous resolution of enantiomeric molecules into a conglomerate of crystals with opposite handedness is well known since the groundbreaking work of Pasteur6, chiral segregation in fluid states was widely unknown until the late 1990ties when fluid conglomerates were discovered in certain liquid-crystalline phases formed by elongated bent-shaped achiral molecules. These bent core mesogens densely pack into 2D-fluid smectic layers in which the direction of molecular tilt c (with respect to the smectic layer normal k) and the direction of molecular bent p become longrange ordered. The three vectors – k, p and c – define either a right-handed or a left-handed Cartesian coordinate system and thus give rise to what is called “layer chirality”. As a result, the liquid crystalline phase segregates into a conglomerate of homochiral fluid domains with layers of the same layer chirality as well as racemic domains with alternating layer chirality.7 The recognition of spontaneous reflection symmetry breaking in liquid crystals stimulated intensive research and further examples were found, even in more common liquid crystal phases such as in smectic B8, cubic9, 10 and nematic11-15 phases of achiral molecules. Even in short range ordered isotropic fluids of achiral mesogens the spontaneous formation of (chiral) fluid conglomerates in the isotropic phases of achiral mesogens was reported.16–18 The current status and understanding of reflection symmetry breaking in liquid crystals was summarized in a number of reviews.2–4, 19 While these amazing examples of reflection symmetry breaking in bulk fluids are all inherent to the interplay between molecular conformations and liquid crystalline packing, chiral structures of liquid crystals can also be induced by extrinsic confinement effects, even if the action is


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achiral by symmetry. The oldest and probably most prominent example is the twisted nematic cell (TN cell) which dates back to the early experiments of Mauguin in 191120: A nematic liquid crystal is confined between two rubbed glass plates each of which aligns the liquid crystal director n (the average direction of the long molecular axes of the e.g. rod-like mesogens) parallel to its rubbing direction. If now the rubbing directions of the two glass plates are adjusted at exactly 90 degrees to each other, the nematic director adopts a continuously twisted configuration in order to meet the boundary conditions at both glass plates. The direction of twist however can be left- or right-handed and thus the nematic phase formed in the TN cell after cooling from the isotropic melt segregates into equal amounts of macroscopic chiral domains with opposite twist sense. Neither the 90°-twisted arrangement of glass plates nor the nematic liquid crystal are chiral in themselves, it is the combination of both which leads to reflection symmetry breaking and the formation of macroscopic chiral domains. Another example of confinement induced reflection symmetry breaking can occur in nematic droplets where bipolar twisted structures were observed in the 1980ties.21 Very recently new examples of confinement induced reflection symmetry breaking in the nematic phase of achiral lyotropic chromonic liquid crystals (LCLCs) were reported by four research teams.22–27 The building blocks of LCLCs are columnar stacks of flat organic dye molecules which are dispersed in a solvent, typically water.28, 29 At higher concentrations, the long axes of the columnar stacks become long range orientationally ordered and thereby a uniaxial nematic liquid crystal phase is formed. In comparison to the common nematic phases of rod-like molecules nematic LCLCs are known to have an exceptionally low twist-elastic modulus.30, 31 Under confinement it is the relative magnitudes of the elastic moduli (splay, bend and twist) that makes twist favorable and is the reason for reflection symmetry breaking.22, 26, 32,


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In other words, nematic LCLCs are supposed to release external stresses as far as possible into

twist deformations and indeed several new chiral director configurations (see below) were now observed when these low-twist-modulus phases were confined in achiral cylindrical capillaries. Before we discuss these new chiral configurations, we will first introduce the well-known achiral director configurations of nematics in cylindrical capillaries.34–37 If a nematic liquid crystal is placed in a cylindrical capillary with normal boundary conditions (i.e. aligning the director normal to the capillary wall) we find a situation of frustration between the uniform director field of an undeformed nematic liquid crystal and the radial director field imposed by the boundary conditions. The naive solution of this frustration is the planar radial configuration (PR) shown in Figure 1a. The director is radial everywhere and confined to the cross-sectional plane of the capillary. The director field has a pure splay and a disclination line of topological strength m = +1 along the axis of the capillary. However, as pointed out by Cladis, Kléman and Williams38, 39 the distortions around a defect line of integral strength such as the +1 disclination line in the PR configuration “escape into the third dimension” 34 along the capillary axis and can thus continuously transform into the escaped radial (ER) configuration shown in Fig. 1c. The escape direction can be either to the left or to the right and can even change sign through point defects. The ER director field involves splay and bend deformations but the +1 disclination line has vanished. The ER configuration is thus more stable than the PR configuration provided that the radius of the capillary is much larger than the molecular dimensions. Since the energy per unit length (the line tension) associated with a disclination line of strength m increases as m2, the +1 disclination line in the PR configuration is also unstable towards a splitting into two disclination lines of half-integral strength +1/2. This leads to the planar polar


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(PP) configuration shown in Figure 1b where we find two +1/2 disclination lines close to the capillary walls and an essentially undistorted director field in the center area of the capillary. The PP configuration becomes more stable than the ER configuration if the capillary radius decreases.39

Figure 1. Achiral director configurations of a nematic liquid crystal confined in a cylindrical capillary with homeotropic (normal) boundary conditions. (a) Planar radial (PR), (b) planar polar (PP) and (c) escaped radial (ER) configurations. While the ER configuration is continuous, the PR and PP configurations exhibit one +1 disclination line and two +1/2 disclination lines, respectively. Disclinations are indicated in red.

It is worth to notice that all three configurations – PR, PP and ER – exhibit mirror planes and are thus achiral. Exactly this mirror symmetry is broken in the by Jeong et al. recently discovered


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chiral configurations22, namely the twisted escaped radial (TER) and the twisted planar polar (TPP) configurations shown in Figure 2. A third chiral configuration, the escaped twist (ET) configuration, has been observed in capillaries with tangential (instead of normal) boundary conditions.23,24 Since this configuration is not relevant in the context of this paper, the ET configuration will not be discussed further.

Figure 2. Chiral symmetry broken configurations of nematics in capillaries with normal boundary conditions.22 (a) Twisted escaped radial (TER) and (b) twisted planar polar (TPP). In (a) the cases of weak (left) and strong (middle) surface anchoring are shown. The twist axis is normal and perpendicular to the cylinder axis in TER and TPP, respectively, as illustrated by the nematic director field along the twist axes. Note that the TPP configuration reported in Ref.

22

differs from the twisted polar (TP) state observed in our study.

In the TER configuration (Figure 2a) the splay-bend director field of the ER configuration (Figure 1c) is superimposed by a twist of the director field along all directions normal to the capillary axis.22 This configuration lacks mirror symmetry and, since the twist sense can be leftor right-handed, chiral macroscopic domains with opposite twist sense are observed. Since the


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TER configuration was so far only observed in nematic LCLCs its formation was attributed to the peculiarities of chromonics. Thanks to their exceptionally small twist modulus the total energy decreases if the splay-bent distortion of the ER director field relaxes into the splay-bendtwist distortion of the TER director field. In the TPP configuration (Figure 2b) the director field of the PP configuration (Figure 1b) with its two straight +1/2 disclination lines is twisted along the capillary axis such that now a double helix of the two disclination lines is observed.22 Again, this structure lacks any mirror symmetry and macroscopic chiral domains with opposite twist sense of the double helix are found. The energetic reason for the formation of the proposed TPP configuration is not obvious and still under debate. In this paper we report experimental observations of the same chiral TER configuration and what we call a twisted polar (TP) configuration. TP is related to, but different from, the TPP configuration discussed in Ref. 22. In TPP the director should be confined to planes normal to the cylinder axis. We argue here that in TP the director instead spontaneously twists out of such planes (to reduce bend and splay close to the disclinations) when going around the disclination line. The system used is a classical, non-chromonic and achiral lyotropic liquid crystal forming a nematic phase of disc-like micelles orienting normal to the capillary walls. After cooling from the isotropic phase, we find TER configurations which however slowly relax over several weeks into the non-chiral ER. Cooling down from the isotropic phase under the action of a magnetic field along the capillary axis results in TP configurations, which in the absence of magnetic field transform into a chiral TER configuration where domains of opposite escape direction are separated (and possibly stabilized) by alternating +1 and -1 point defects (TERPD = twisted escaped radial configuration with point defects).


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Due to the fact that micellar systems are closely related to vesicles and biomembranes, the observations presented in this work add to the discussion about chiral symmetry breaking in biological systems. Similarities and differences to the case of chromonic liquid crystals are discussed, and we examine the conditions under which spontaneous reflection symmetry breaking occurs in the non-chromonic lyotropic system.

Experimental Section The lyotropic liquid crystal (LLC) used in this study is a ternary system containing N,Ndimethyl-N-ethyl-hexadecylammoniumbromide (CDEAB, Figure 3a) as surfactant, decan-1-ol (DOH) as co-surfactant and double distilled water as solvent. Above a certain concentration, the CDEAB surfactant molecules (Figure 3a) assemble into micelles. After adding the co-surfactant DOH the micelles become disk-like in shape (Figure 3b) and form a nematic Nd phase. In the Nd phase the principal axes of the micelles are long-range ordered along the director n (Figure 3c), the optic axis of the uniaxial Nd phase, having positive optical anisotropy ∆n = n|| – n⊥ > 0 and negative magnetic anisotropy ∆߯ = ߯ || – ߯⊥.41-43

Figure 3. (a) Molecular structure of the surfactant N,N-dimethyl-N-ethyl-hexadecylammoniumbromide (CDEAB) and corresponding schematic drawing of the surfactant with the polar head group (blue) and the alkyl chain (grey). (b) Aggregation of the surfactant molecules to discotic


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micelles. (c) Scheme of the nematic phase formed by discotic micelles, indicating the refractive indices parallel (n||) and perpendicular (n⊥) to the director as well as the magnetic susceptibilities parallel (߯||) and perpendicular (߯⊥) to the director. (d) Schematic illustration of the homeotropic anchoring at the capillary glass walls. CDEAB and DOH were purchased from Merck KGaA and used without any further purification. First, the CDEAB was dissolved in double distilled water and then the cosurfactant DOH added. The glass vial with screw plug which was used for storing the mixture was sealed with parafilm to prevent from solvent evaporation. The sample was stirred in a thermoshaker (Biosan, PST-60HL) at 40 °C and put on a roller (Phoenix Instrument, RS-TR05) in order to homogenize the mixture. The homogenization took 5 days with alternating use of the thermoshaker and the roller. In this study we used the Nd phase formed at a composition of 32.0 wt% CDEAB, 4.8 wt% DOH and 63.2 wt% water. In the polarizing optical microscope (Leica, DMLP) we observed the Nd phase between room temperature and its clearing temperature at 48°C. The micelle dimensions at this composition were studied with light scattering by C. Görgens giving a height of 32 Å and a diameter of 69 Å, excluding the solvation sphere.44 Mark-capillaries made of glass no. 14 from Hilgenberg with an outer diameter of 0.7 mm and a wall thickness of 0.01 mm were used as capillaries. In supplementary studies borosilicate glass capillaries from Vitrocom having an inner diameter of 0.15 mm were used, the same as in Ref. 22

, and led to similar results. It is well-known that water-based LLCs align homeotropically at the

glass capillary walls (Figure 3d).44,

45

The lyotropic mixture was sucked into the capillary by

using a water pump jet. After filling, both ends of the capillary were sealed by melting the glass with a lighter. The seal was checked via centrifugation.


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Filled capillaries were studied in the polarized optical microscope (POM) with or without an index matching fluid surrounding the capillary (Figure 4). In most cases, we found that the use of the matching fluid does not provide significant additional information (see Figure 4). Photographs (Nikon D40) are thus taken without the index matching fluid for reasons of simplicity unless otherwise stated.

Figure 4. (a) POM image of the same LLC filled capillary (Ø = 700 µm) surrounded by an index matching fluid (rapeseed oil). (b) not surrounded by an index matching fluid.

A Bruker B-E 25v electromagnet with maximum magnetic field strength of 1 Tesla was used for the magnetic field experiments. After filling, the capillaries were heated in a homemade thermostated sample holder into the isotropic phase (≈ 50 °C) which should also rule out possible effects of shear-alignment during filling. Then capillaries were cooled down slowly (0.2 K/h) with the magnetic field along the capillary axis from the isotropic phase into the nematic phase at room temperature. Roughly speaking the capillaries were placed for around one week in the magnetic field.


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Results and Discussion Director configurations in the absence of a magnetic field. The time-dependent development of the director field in the absence of magnetic field treatment is summarized in Figure 5. Capillaries filled in the nematic phase first displayed a Schlieren-like texture (Figure 5a) with no macroscopic average director alignment. After 2 to 3 days at room temperature the virgin texture had been replaced by a defect free configuration with domains of twisted escaped radial (TER) configuration of alternating twist sense. The transition in space between right and left-handed TER regions along the capillary occurred via twist-free escaped radial (ER) regions (Figure 5b). After additional 2 weeks at room temperature, the TER regions had disappeared and the whole capillary exhibited the achiral ER configuration (Figure 5c). A similar TER – ER scenario was observed also when the capillaries were first heated to the isotropic phase and slowly cooled down to room temperature. The TER director twists along the diameter of the capillary (Figure 2a) which gives – in the case of a Mauguin number bigger than one – a rotation of the plane of polarization in addition to retardation of travelling light and makes the center part of the capillary always appear bright between crossed polarizers. Since the birefringence of lyotropic liquid crystals is in general two orders of magnitude smaller (≈ 10-3) than in thermotropics46, a rough estimation reveals a Mauguin number ≥ 1. In capillaries with a diameter of ~700 µm, aligned along one of the crossed polarizers, the TER center part appears orange, compared to the twist-free regions where the TER changes handedness which gives extinction in the center cf. Figure 6a. As the liquid crystal material used is inherently achiral, TER domains with both signs of chirality should be formed with the same probability under spontaneous reflection symmetry breaking, and this also seems to be the case. The local twist sense of the TER can easily be revealed by a slight de-crossing of the polarizers,


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cf. Figures 6b, c, e, and f. Importantly, the fact that the ER configuration appears after long time indicates that the ground state, or lowest energy state, is not TER but in fact ER, and therefore has reflection symmetry. There are several possible reasons for the very long relaxation times. The characteristic lengths are very large – the diameter of the capillary is several hundred micrometers and the wavelength of deformations along the axis may be several millimeters – which in itself should lead to inherently slow dynamics. Furthermore, the initial schlieren texture contains a lot of topological defects that disappear slowly.

Figure 5. Director field evolution in the absence of magnetic field. About 2-3 days after filling or after cooling from the isotropic phase (a) a twisted escaped radial (TER) configuration with regions of alternating twist is obtained (b). After additional two weeks the capillary has adopted a unidirectional escaped radial (ER) configuration (c) which constitutes the ground state.


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The remarkably slow relaxation suggests that the energy difference between the intermediate TER and the final ER configuration is very small and even if the achiral ER is the ground state the relative magnitudes of the elastic constants could still be the key to the formation of the longlived chiral TER states. One could further speculate that the energy balance between ER and TER could be shifted by temperature, composition of the ternary system and/or flow effects. If, for instance, we are close to a lamellar phase the bend and twist elastic constants should be large compared to the splay constant and twist would therefore be suppressed, stabilizing the achiral ER configuration. On the other hand, at lower surfactant concentrations, away from the lamellar phase the twist elastic constant is relatively small, allowing for the relieve of splay and bend into twist, resulting in the reflection symmetry broken TER configuration. Furthermore, a flow along the capillary axis could stabilize ER against TER. For a brief comparison of the relaxation times in thermotropics to those in lyotropics, see 47.

Figure 6. Confirmation and determination of the twist sense in twisted escaped radial domains. (a) Crossed polarizers where TER regions appear bright and orange. A slight de-crossing of the polarizers clockwise (b) and anti-clockwise (c) brightens or darkens regions with opposite twist, respectively. (d) – (e) the same as (a) – (c) using green light.


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Magnetic field induced director configurations. When the capillaries were subjected to an axial magnetic field during the very slow cooling from the isotropic phase to room temperature, we could observe a more complex scenario. This is manifested in Figure 7. Figure 7a depicts a 700 µm thick capillary after initial cooling 0.2 K/h from the isotropic phase under an axial magnetic field of 1 Tesla and then leaving the capillary in the field for additional 2-3 days at room temperature. Starting from the left in Figure 7a we have a state characterized by three extinction bands parallel to the capillary axis, more clearly seen in the left part of the magnification shown in Figure 7b. The quite black extinction region at the center is very narrow while the symmetrically positioned neighbouring bands which are rather grayish are much broader and do therefore not provide perfect extinction. Then, there appears a transition to a configuration with two disclination lines emanating from a point on the cylinder axis. When we continue to the right we see that the two disclination lines form a double helix with a period (pitch) of about 3.5 mm. Looking from the left to the right in the photo of the capillary, the helix changes sign after ~1.5 turns, and after one full turn, it changes handedness again. Although only a few full turns can be observed, it seems that the pitch is the same for both handedness. Figures 7c and d show magnifications of twisted polar (TP) sections with opposite helix sense, determined by watching the movement of the cross-over points of the two disclination arms when rotating the capillary about its axis as illustrated in Figure 7i and j. See also the videos in Supplementary info, SV1 showing a region of the same handedness, whereas SV2 shows a region where the twist sense is changing. Before we interpret the features of the capillary in Figure 7 let us first consider in Figure 8 the case of applying an axial magnetic field H in an ER capillary with ∆߯ < 0. When increasing the field amplitude from zero we might expect a continuous distortion of the ER configuration,


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Figure 7. (a) Photograph of a ~23 mm long section of a magnetic field-treated capillary with diameter 700 µm. (b) Transition point between distorted escaped radial and twisted polar (TP) configurations. (c) and (d) Magnifications of sections with opposite handedness of the TP helix. (e) – (h) In the encircled cross-over points the transverse twist is normal to the plane of the paper. In these regions a de-crossing of the polarizers reveals an opposite sign of twist for left-,


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and right-handed axial twist states reflected by the color shift (orange to bluish or the other way round) for clockwise- and anticlockwise- de-crossing of the polarizers. (i) and (j) Determination of handedness of the axial twist from the movement of the cross-over points of the double helix arms when rotating the cylinder about its axis.


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Figure 8. (a) Schematic illustration of the structure and the optical features (left) and a corresponding photograph (right) of the quiescent escaped radial (ER) configuration between crossed polarizers (indicated by arrows). A magnetic field distorts the ER configuration (b) and after the field is removed a distorted escaped radial state with three dark bands regions (c) is observed for several hours until full relaxation to ER is completed. The blue discs represent the micelles of the lyotropic system. during which the escaped configuration is squeezed into a more planar configuration, cf. Figure 8b. Above some critical field Hc, the total energy of a planar-radial (PR) or -polar (PP) configuration with disclination lines should become smaller than the one of the field-distorted ER configuration. As a result, we expect a field-induced transition from squeezed ER to PR or PP. After removal of the field the system should retain with time the ER ground state. However, we have not observed the PR configuration in our capillaries and there can be several reasons for this. First, with the available equipment we have no possibility to observe the capillary while applying the magnetic field H. Thus, if the PR configuration is indeed obtained at any time, e.g. during cooling from the isotropic phase at H > Hc, it will likely relax to ER as soon as the field is removed and, hence before the capillary is inserted in the microscope. Second, the PP configuration with two m = 1/2 disclinations should be favored against the PR configuration with one m = 1 disclination and a field-induced PR configuration should likely immediately transform into the lower energy PP configuration. The branching of one +1 defect line (PR) into two +1/2 defect lines (PP), is theoretically treated in detail by Shams et al.48 It should be pointed out, however, that in Ref. 48 the singularity-free ER configuration is not discussed. From the above reasoning, we might expect that ER should be destabilized by an axial magnetic field with sufficient amplitude, and consequently, we could have a field-induced


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transition between squeezed ER and PP and a transition from PP to ER when the field is removed. In the case that PP is metastable we should even be able to observe coexistence of PP to ER and the defect (de-)branching under the microscope. We attribute the three-band structure to the far left in the capillary in Figure 7a and b to a fielddistorted version of the ER configuration, explained in Figure 8. The magnetic field squeezes the director field towards a planar director configuration (Figure 8b) and when the field is removed, a transient distorted escaped radial state is obtained (Figure 8c). In a narrow region around the capillary axis the director is parallel to it and forms a “stabilized kink” which gives a narrow extinction region between crossed polarizers, as indicated in Figure 8c. Away from the center the director field splays and bends towards the cylinder wall. In two small regions just symmetrically above and beneath the black narrow extinction band, the director differs from a parallel (or perpendicular) orientation along the capillary axis. These regions where the director is at its maximum 45° to the capillary axis (and also to the polarizer) appear bright. Passing these regions, at some radial distance the director gradually orients normal to the cylinder axis. This gives dark but not fully black bands. At the capillary walls the director meets the boundary conditions, orienting homeotropically and appearing black between crossed polarizers. This transient director field scenario is further supported by the fact that within about 6 hours it had completely relaxed to the quiescent escaped radial (ER) configuration. Furthermore, we have only seen this transient director field in capillaries that have been subjected to magnetic field treatment, which suggests that it is indeed facilitated only by the action of the axial magnetic field. The abrupt transition from the field distorted escaped radial to a metastable polar configuration, stabilized by the magnetic field, is analogous with the disclination branching


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analyzed by Shams et al.48, (see Figure 7b right) as we have a point defect and a planar radial (PR) configuration in the plane containing the transition point. The double helix configuration of the disclination lines reminds of the twisted planar polar (TPP) structure observed by Jeong et al.22 However, the fact that the structure does not appear black between crossed polarizers along and normal to the cylinder, see Figures 7c and d, immediately reveals that it is in fact not a planar structure. Furthermore, a slight de-crossing of the polarizers reveals that this structure is not only twisted along the axis of the capillary, but also perpendicular to the capillary axis, see Figures 7e – h. In the middle between crossover points we indeed get extinction between crossed polarizers, see Figures 7c and d, which shows that in these dark areas the transverse twist axis is horizontal (parallel to the plane of the paper). In other words, in addition to the axial twist revealed by the double helix of line defects there is also a transverse director twist parallel to any diameter joining the arms of the double helix. We have coined this structure the twisted polar (TP) configuration, as it is not perfectly planar anymore. The TP configuration is closely related to the TPP configuration of Jeong et al.22 in the sense that both have a non-escaped (thus more or less planar) director field with two +1/2 disclinations twisted along the capillary axis. However, the TP configuration is distinguished from TPP by its additional transverse director twist between the arms of the double helix formed by the two +1/2 disclinations. We could observe that in the absence of magnetic field such transition points move very slowly (at a speed of ~10 µm/h) to enlarge the ER regions, which shows that the ER configuration is the ground state. We never observed TP regions in capillaries that had not been subjected to axial magnetic fields.


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Breaking of reflection symmetry. We have two reflection symmetry broken configurations in our system, twisted escaped radial (TER) and twisted polar (TP). From the appearance of the TER structure the conclusion can be drawn that, just as in the chromonic case reported by Jeong et al.22, the twist elastic constant in our micellar lyotropic LC is relatively small compared to the splay and bend elastic constants, and if possible, the system will minimize the elastic energy through breaking of reflection symmetry. The planar polar (PP) – twisted polar (TP) reflection symmetry breaking is much less trivial. We can understand, and qualitatively motivate, the field-assisted formation of a polar configuration from the escaped radial. However, we have yet to understand the reasons for the breaking of reflection symmetry, e.g. why we get a TP rather than a PP state. It is unlikely that the extremely weak axial twist with a period of several millimeters would lower the total energy compared to a non-twisted achiral PP configuration. On the contrary, we would rather expect a prevention of a double helix formation to simply minimize the length of the energetically costly disclination lines. We instead suggest that the reflection symmetry breaking occurs at, and due to, the disclination lines, where the director field must be strongly distorted. The symmetry breaking is illustrated in Figure 9, where the cylinders represent the director field. Figure 9a shows a +1/2 disclination (indicated in red) surrounded by a director field (represented by cylinders) which is confined to the plane normal to the disclination line. This is a so-called wedge disclination49, 50 which contains only splay and bend and would correspond to the director field around the two disclination lines in the PP configuration. But as discussed by Frank51, a half unit disclination line can as well be realized by letting the director instead rotate 180° out of the plane of the paper, as shown in Figure 9b. This is called a twist disclination.49, 50, 52 In this case the disclination line follows instead the plane in which the molecules lie.49 This corresponds to


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putting the director on a Möbius strip encircling the disclination. The introduction of twist of course breaks the reflection symmetry and we propose, using the same arguments as stated above concerning the relative magnitudes of the elastic constants, that the system stabilizes the two disclination lines by adopting the chiral configuration in Figure 9b rather than the achiral, planar configuration in Figure 9a. As mentioned by Ranganath53 and discussed by Anisimov and Dzyaloshinskii54, a twist disclination is in general energetically unfavored compared to a wedge disclination in nematics, except in the case of an abnormally small twist elastic constant K22 with K22 < (K11+K33)/2 where K11 and K33 denote the splay and the bend elastic constant, respectively.

Figure 9. Illustration of the reflection symmetry breaking in the vicinity of the half unit disclination line (indicated as red dot). The achiral director field (here represented by cylinders) in (a) lies in a plane normal to the disclination line and contains only splay and bends (wedge disclination). In the reflection symmetry broken state in (b) the director instead twists around the disclination line and follows the plane in which the molecules lie (twist disclination). The chiral


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configuration in (b) is obtained through a rotation of the director in (a) by the angle of 90° about a vertical axis in the plane of the paper, this axis is indicated as orange arrow in (b). (c) Oblique view of (b), along the direction perpendicular to the disclination line (viewing angle indicated above the green arrow), transferred to the cylindrical capillary. The director twist around the disclination has the opposite handedness from the transverse twist. (d) Schematic director field in a cross section of the TP configuration. The reflection symmetry breaking transition from the planar (Figure 9a) to the twisted case (Figure 9b) is realized by a α = 90° rotation of the director about an axis vertical in the plane of the paper, which is indicated as orange arrow in Figure 9b. The twist disclination in Figure 9b appears to be m = +1/2 or m = -1/2 depending on the (oblique) viewing direction. If we observe this configuration, transferred to the cylindrical capillary, from the viewing angle indicated above the green arrow it appears to be m = +1/2, cf. Figure 9c.51 Introducing the twist disclination in Figure 9b, we have in Figure 9c and 9d sketched the director field in a cross section of a TP capillary. It is now twisted parallel to the diameter of the capillary (transverse twist) connecting the disclination lines consistent with our observations. This twist also makes the disclination lines non-parallel which results in the double helix which we observed along the capillary axis (axial twist). An interesting question still to be answered is at what stage the reflection symmetry breaking occurs. It might occur immediately in the branching process, which could mean that the two disclination lines in fact emerge in a rotating fashion, like the two water beams in a garden sprinkler, while at the same time the sprinkler, or rather the branching point, is moving along the axis of the capillary. When the lines have reached the equilibrium position more or less close to the cylinder walls they form the double helix. The rotation direction could be either left or right


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handed, with the same probability. A disturbance, like a dirt particle, or point defect might change the chirality (and the rotation direction of the sprinkler) which will result in a change in handedness in the forming twisted polar (TP) structure. Jeong et al. found that the twisted planar polar (TPP) structure in fact emerged from a moving point22 and maybe we have a similar scenario in forming the TP configuration under the magnetic field. However, the reflection symmetry breaking could also occur from an already existing field-induced planar polar (PP) configuration. It is worth pointing out that while the formation of the non-planar TP configuration can be explained through reflection symmetry breaking at the defect lines (Figure 9), the mechanisms behind the formation of a planar TPP configuration are still not clear.22 In the formation of the twisted polar (TP) structure the reflection symmetry broken state can have right or left with the same probability. By the addition of a tiny amount of chiral dopant, about 0.03 Mol-% of (R)-mandelic acid the formed TP double helix always had the same sign, i.e the whole sample has become homochiral, as shown in Figure 10. Roughly speaking, just one out of 3000 molecules is chiral in this case. The effect of the tiny amount of chiral dopant is to govern the sign of chirality, while the pitch magnitude is not changed drastically (it´s still in the millimeter range). By increasing the dopant concentration up to 0.1 Mol-% a transition to the well-known Fingerprint-Texture could be observed. Therefore, the extreme sensitivity of the sense of chiral induction in the observed TP configuration could constitute the basis for a new type of ultra-sensitive chiral detectors.


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Figure 10. Left handed homochiral TP configuration obtained when the lyotropic liquid crystal is doped with 0.03 Mol-% of (R)-mandelic acid.

Defect stabilization of TER (TERPD). Interestingly, when a large section of the twisted polar (TP) configuration relaxes to the escaped configuration (by means of not being in the magnetic field) a large number of point defects form along the cylinder axis which separate regions of opposite escape directions (Figure 11). Now the configuration between the defects is twisted escaped radial (TER) and as long as the defects are present the chiral structure is stable and the achiral escaped radial (ER) configuration is not formed.


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Figure 11. Defect stabilized twisted escaped radial state (TERPD) obtained after relaxation from the twisted polar (TP) configuration by not being in a magnetic field. (a) The bright center region indicates the twisted state, which is confirmed by de-crossing the polarizers in (b) and (c). The handedness of the TER twist does not change at the point defects. In (d) the pink arrow indicates the slow axis of the optical compensator inserted at 45° to the polarizers. (e) Crossed polarizers rotated 45° around the cylinder axis. The TERPD director field with an alternating row of +1 and -1 point defects is illustrated in (f). We cannot at this stage say if the handedness of the TER configuration is directly related to the handedness of the former TP structure. But if this is the case, the reflection symmetry breaking occurring at the formation of the TP structure would be preserved also in the defect-stabilized


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TER configuration, the so-called TERPD (twisted escaped radial with point defects) configuration. This configuration corresponds to a locked metastable state, because the defectfree ER configuration would be the ground state, but as discussed by Allender et al. the boundary conditions at the ends of the cylindrical capillary prohibit its relaxation to the ER.55 In the regions of the capillary that did not have the TP but the distorted ER structure after removal of the magnetic field we did not see defect-stabilized chiral TER state, but here the structure always relaxed to achiral ER. The possible role of the defects in stabilizing the reflection symmetry broken TER state is not clear. One could speculate that due to the same reasons as described above the defects form chiral seeds due to the strong elastic deformations in their vicinity, i.e. which spontaneously leads to twist, and that the sign of these seeds is set by the chirality of the TP state, from which it the TER state appeared.

Conclusions The experimental observation of chiral director configurations formed by a standard micellar lyotropic nematic in cylindrical capillaries suggests that the recently reported spontaneous reflection symmetry breaking under capillary confinement is not specifically limited to the lyotropic nematic phase of chromonic systems. In fact, it seems to be a more general phenomenon in nematic phases of large supramolecular aggregates such as the disk-like micelles in our micellar nematic or the rod-like stacks of dye-molecules in chromonic nematics. Since in general the elastic constants of nematic liquid crystals are decreasing with increasing size of the mesogenic building blocks56, both micellar and chromonic nematics are thus expected to exhibit anomalies in the size and anisotropy of their elastic moduli in comparison to their thermotropic nematic counterparts which are formed by small, single molecular entities. Actually, as a matter of fact, there were no chiral configurations reported in similar studies using thermotropic liquid


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crystals like MBBA in the same type of capillaries with homeotropic boundary conditions.

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Anomalies in the elastic moduli originating from the large supramolecular building blocks of lyotropics might thus be the common reason behind the ability of chromonic and micellar nematics to escape into chiral twisted structures even under achiral confinement conditions. Since at least in our case of a micellar nematic the time scales of transformations between the various structures observed under capillary confinement (ER, TER, TERPD, TP) are extremely slow, we suppose that energy differences between these configurations are tiny. The global stability of a certain configuration thus critically depends on weak external or internal stimuli such as the presence of a magnetic field or the formation of point defects. For example, the TP configuration is stable under the action of magnetic field of sufficient strength. After removing the magnetic field the TP director field slowly escapes into a TERPD configuration with many +1 point defects, separating sections of opposite escape direction. This TERPD configuration seems to be a locked state which cannot escape anymore into the achiral ER ground state. If on the other hand a tiny amount of chiral dopant is added the TP configuration becomes homochiral and remains stable even in the absence of a magnetic field. In conclusion, the capillary confinement of achiral micellar lyotropic nematics can lead to new chiral director configurations (TER, TERPD and TP including twist disclination lines), unknown in achiral thermotropic nematics but very similar to the chiral director configurations recently reported for chromonic nematic liquid crystals. The emergence of these chiral director configurations as well as the observation of twist disclination lines strongly indicate that micellar lyotropic nematics exhibit the same anomalously small twist elastic moduli K22 as found in chromonic nematics.


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ASSOCIATED CONTENT Supporting Information. SV1 – Region of right-handed twisted polar configuration. The cross-over points of the double helix arms move all to the right when rotating the capillary about its axis. SV2 – Region where the handedness of the twisted polar configuration changes sign. The two double helix arms move towards each other, indicating a change in twist sense.

AUTHOR INFORMATION Corresponding Author *Email: [email protected] Notes The authors declare no competing financial interest.

ACKNOWLEDGMENT This

work

was

to

a

part

supported

by

the

Deutsche

Forschungsgemeinschaft

(DFG Gi 243/4-2).


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(53) Ranganath, G. S. Energetics of Disclinations in Liquid Crystals. Mol. Cryst. Liq. Cryst. 1983, 97, 77-94. (54) Anisimov, S. I.; Dzyaloshinskii, I. E. A new type of disclination in liquid crystals and the stability of disclinations of various types. Soviet. Phys. JETP 1973, 36 (4), 774-779. (55) Allender, D. W.; Crawford, G. P.; Doane J. W. Determination of the Liquid-Crystal Surface Elastic Constant K24. Phys. Rev. Lett. 1991, 67 (11), 1442-1445. (56) De Gennes, P. G.; Prost, J. The Physics of Liquid Crystals. Clarendon Press, Oxford, 1993, 103. (57) Melzer, D.; Nabarro, F. R. N. Cols and noeuls in a nematic liquid crystal with homeotropic cylindrical boundary. Philosophical Magazine 1977, 35 (4), 907-915.

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Capillary confinement of an achiral micellar lyotropic nematic phase leads to novel chiral director configurations such as the twisted polar configuration signified by a left- or a righthanded double helix of half-unit twist disclination lines.


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Chiral Structures from Achiral Micellar Lyotropic Liquid Crystals under

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