Extremely small twist elastic constants in lyotropic nematic liquid crystals

Clarissa F. Dietricha, Peter J. Collingsb,c, Thomas Sottmanna, Per Rudquistd, and Frank Giesselmanna,1

aInstitute of Physical Chemistry, University of Stuttgart, 70569 Stuttgart, Germany; bDepartment of Physics and Astronomy, Swarthmore College, Swarthmore, PA 19081; cDepartment of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104; and dMicrotechnology and Nanoscience, Chalmers University of Technology, 41296 Göteborg, Sweden

Edited by Noel A. Clark, University of Colorado, Boulder, CO, and approved August 26, 2020 (received for review December 18, 2019) Recent measurements of the elastic constants in lyotropic chro- However, if the twist elastic constant is considerably smaller monic liquid crystals (LCLCs) have revealed an anomalously small than the other two, K22 ≪ K11, K33, twist deformations are least twist elastic constant compared to the splay and bend constants. energetically costly. Thus reflection symmetry breaking might Interestingly, measurements of the elastic constants in the micellar occur even in a nonchiral nematic under nonchiral confinement. lyotropic liquid crystals (LLCs) that are formed by surfactants, by A striking example was recently observed in a lyotropic chro- far the most ubiquitous and studied class of LLCs, are extremely monic nematic LC under cylindrical as well as under droplet rare and report only the ratios of elastic constants and do not confinement (3–5). Indeed, in this particular case K22 has been include the twist elastic constant. By means of light scattering, this measured to be one order of magnitude smaller than K11 and K33 study presents absolute values of the elastic constants and their (6, 7). corresponding viscosities for the nematic phase of a standard LLC In a chromonic nematic LC, the orientationally ordered composed of disk-shaped . Very different elastic moduli building blocks are rodlike stacks of dye molecules, suspended in are found. While the splay elastic constant is in the typical range a suitable , usually water (8). They thus constitute a rather of 1.5 pN as is true in general for thermotropic nematics, the twist special class of lyotropic LCs. However, the most ubiquitous elastic constant is found to be one order of magnitude smaller class of lyotropic LCs is formed by surfactants that in water as- (0.30 pN) and almost two orders of magnitude smaller than the semble into rod- or disk-shaped micellar building blocks. In fact, bend elastic constant (21 pN). These results demonstrate that a these micellar or surfactant-based lyotropic LCs have been small twist elastic constant is not restricted to the special case of known for a long time and are abundant in biological structures LCLCs, but is true for LLCs in general. The reason for this extremely (e.g., membranes) and in our daily lives (e.g., de- small twist elastic constant very likely originates with the flexibil- tergents and soaps) (9, 10). ity of the assemblies that are the building blocks of both micellar Our recent observation of confinement-induced reflection and chromonic lyotropic liquid crystals. symmetry breaking in a lyotropic micellar nematic phase (Fig. 1D) suggests that K might also be very small in this case lyotropic liquid crystals | nematic liquid crystals | elasticity | chirality | 22 light scattering (11). Searching the literature we recognized that very little is known about the elasticity of lyotropic nematic phases of disklike =χ =χ (ND) as well as rodlike (NC) micelles: A few K11 a and K33 a nematic is a three-dimensional (3D) fluid (with χ being the diamagnetic anisotropy) measured via mag- phase of matter in which the anisometric building blocks a A netic Frederiks transition have been reported for two lyotropic (e.g., molecules, molecular assemblies, micelles) making up the material maintain a certain amount of long-range orientational Significance order as they diffuse in a fashion similar to that of liquids. At any point in a nematic liquid crystal phase, the anisometric building blocks tend to orient their principal axes along a specific direc- Some soft matter systems behave similarly and some differ- tion called the director and given the symbol n^. If the building ently. Understanding this is one of the significant challenges in blocks are rodlike, the long axes are ordered along the director the field. For example, while the splay and bend elastic con- and if the blocks are disklike, the short axes are ordered. stants for thermotropic and lyotropic nematic liquid crystals are Liquid crystals (LCs) are prime examples of soft matter that similar, the twist elastic constant is nearly an order of magni- are distinguished by having large response functions to even tude smaller in lyotropic nematics. Unfortunately, complete weak external perturbations. The elastic response of the nematic elastic data on lyotropic nematics are restricted to one polymer LC director field n^(~r) is governed by three elastic constants as- and two chromonic systems, omitting the most common and highly studied lyotropic surfactant systems. This article reports sociated with the three principal deformations possible in ne- that a surfactant system of disk-like micelles also possesses an matics (K splay, K twist, and K bend; Fig. 1 A–C) with twist 11 22 33 extremely small twist elastic constant, demonstrating that this being the only chiral deformation of these three. The way a property is general for lyotropic liquid crystals. The reason nematic liquid crystal elastically deforms under the action of likely stems from the higher flexibility of molecular assemblies external forces or under external confinement critically depends compared to molecules. on the relative magnitudes of K11, K22, and K33. In common thermotropic nematic LCs consisting of rodlike molecules, all Author contributions: C.F.D., P.J.C., and F.G. designed research; C.F.D., P.J.C., and T.S. three constants are on the same order of magnitude (typically performed research; C.F.D., P.J.C., T.S., P.R., and F.G. analyzed data; and C.F.D., P.J.C., − − 10 12 to 10 11 N) which in most cases allows for the so-called T.S., P.R., and F.G. wrote the paper. “one-constant approximation” frequently used in liquid crystal The authors declare no competing interest. theory (1). An exception from the one-constant approximation is This article is a PNAS Direct Submission. probably the bent-core nematic phases (BCNs) formed by bent- Published under the PNAS license. core mesogens, the bowlike shape of which considerably deviates 1To whom correspondence may be addressed. Email: [email protected]. from the rodlike shape of common nematic mesogens. Here, This article contains supporting information online at https://www.pnas.org/lookup/suppl/ values of K22 and K33 being one order of magnitude smaller than doi:10.1073/pnas.1922275117/-/DCSupplemental. K11 were reported (2). First published October 16, 2020.

27238–27244 | PNAS | November 3, 2020 | vol. 117 | no. 44 www.pnas.org/cgi/doi/10.1073/pnas.1922275117 Downloaded by guest on September 25, 2021 Here we use the same light-scattering technique of Zhou et al. (7) to determine a complete set of elastic constants and viscos- ities for a surfactant-based lyotropic nematic LC, namely the ternary system N,N-dimethyl-N-ethylhexadecyl-ammonium bro- mide (CDEAB), decan-1-ol, and water. This system forms the ND phase of disklike micelles for which we previously observed reflection symmetry breaking under capillary confinement (11). Very different elastic moduli are found: The twist elastic con- stant K22 = (0.300 ± 0.018) pN is one order of magnitude smaller than K11 = (1.54 ± 0.08) pN and almost two orders of magnitude smaller than the bend elastic constant K33 = (20.8 ± 1.3) pN. We further present a simple model based on the deformation modes of disk-shaped micelles, which makes a plausible argument for the relative magnitudes of the three elastic constants. In view of the finding of a low K22 value for a micellar lyotropic nematic, a low twist modulus is likely a generic property in all classes of lyotropic nematics, which im- plies that the one-constant approximation does not hold for lyotropic nematic systems. Likewise, the viscous behavior of lyotropic nematics does not seem to follow the predictions of the – η Ericksen Leslie model in which twist has the highest value. Furthermore, the very small K22 value makes these phases ex- tremely responsive to chiral perturbations. The quest to under- stand reflection symmetry breaking in a fluid system (and in an aqueous solution in particular) is a long-standing scientific goal, with implications as far ranging as the origin of handedness Fig. 1. (A–C) The three bulk elastic deformations of a nematic liquid crystal in biology. phase: splay (A), twist (B), and bend (C). The blue disks represent the disk- shaped micelles of the investigated LLC, and the director n^ is indicated in Results PHYSICS red. (D) Polarized optical micrograph (crossed polarizers) of the twisted polar The theory of light scattering in nematics was elaborated by de (TP) director configuration of the lyotropic micellar ND phase under the Gennes and Prost (19) and a more recent description is found in confinement of a capillary (diameter: 700 μm). The TP director configura- ref. 20. Following the procedure of ref. 7, this study presents tion breaks reflection symmetry and is characterized by two half-unit twist measurements of all elastic moduli and their corresponding vis- disclination lines forming a double helix. For more information see ref. 11. cosities for a lyotropic nematic micellar LC consisting of disk- (E) In the scattering geometry measuring splay and twist fluctuations, the director n^ is perpendicular to the scattering plane. (F) In the scatter- shaped building blocks. For theoretical and experimental details ing geometry measuring twist and bend fluctuations, the director n^ is par- on this technique, see Materials and Methods and SI Appendix, SI allel to the scattering plane. The incident light is perpendicular to the Text and Fig. S1. Fig. 1 E and F shows the two scattering ge- front and back planes representing the two glass substrates of the LC cell. ometries used, in which the director is either perpendicular ~ → The wave vectors ki and ks indicate incident and scattered light; i (vertical) (splay–twist geometry) or parallel (twist–bend geometry) to the θ and f (horizontal) correspond to their polarizations, respectively. is the scattering plane. The amplitude correlation function g1 is shown scattering angle. versus the delay time τ for θ = 45° and 50° in the case of the splay-twist geometry (Fig. 2 A and B) and for θ = 19° for the twist–bend geometry (Fig. 2C). The yellow dashed line indicating surfactant systems with N phases (12, 13) and one with an N a single stretched exponential fit clearly misses the data in D C τ phase (13). Only one study investigating the N phase of the Fig. 2 A and B especially for larger values of , whereas it fits the D τ cesium perfluorooctanoate (CsPFO)/H O system by conductivity data in Fig. 2C perfectly over the entire regime. This means 2 – measurements reported absolute values for K and K (14). that only one relaxation process is measured in the twist bend 11 33 – Importantly, to the best of our knowledge neither a single geometry. In the splay twist geometry, however, two over- damped relaxation modes can be resolved due to the significant complete set of elastic constants nor a single value of K22 has been reported for surfactant-based lyotropic nematics. difference in the time domain. As the relaxation rate is pro- However, in other classes of lyotropic LCs, complete sets of portional to the ratio of the elastic constant and respective vis- cosity (known as the orientational diffusivity D), this clearly elastic constants including K have been measured by means of 22 indicates (assuming no large difference in the viscosities) that the depolarized dynamic light scattering. The first example in 1985 splay and twist elastic constants are not of the same order of was the lyotropic polymeric nematic phase of racemic poly- γ magnitude. This is in contrast to, for example, the case of the -benzyl-glutamate (PBG) (15). Together with an independent thermotropic calibrating substance 5CB for which no kink in the measurement of K33 via the magnetic Freedericksz transition correlation function can be observed because K11 and K22 are of using the diamagnetic anisotropy of PBG (16), absolute values the same order of magnitude (SI Appendix, Fig. S2A). The as-

for the elastic constants K11, K22, and K33 as well as the corre- signment of the fast or slow mode in Fig. 2 A and B to either η η η sponding viscosities 11, 22, and 33 were reported (15, 17). In splay or twist fluctuations is done by studying the angular de- 2014 Zhou et al. (7) enhanced the analysis of light-scattering pendence of the scattered light intensities. The mode increasing data by introducing a calibration step using the common ther- with scattering angle θ corresponds to scattering from twist motropic nematic LC 4′-n-pentyl-4-cyanobiphenyl (5CB) for fluctuations, whereas the one decreasing with θ is from splay which all viscoelastic properties are well known. This calibration fluctuations (for more details see SI Appendix, SI Text). Without allowed absolute measurements of the elastic constants and their calibrating the data, relative values of the elastic moduli can corresponding viscosities in the chromonic nematic LC disodium easily be obtained by plotting the different scattering amplitude cromoglycate (DSCG)/H2O (7). Since then, this technique has ratios versus the respective angular functions (Fig. 2 D–F). The also been used for thermotropic nematics (18). analysis of the slope shows large elastic anisotropies, i.e., that K22

Dietrich et al. PNAS | November 3, 2020 | vol. 117 | no. 44 | 27239 Downloaded by guest on September 25, 2021 has the smallest value and that K11 and K33 are nearly one to two DOH/H2O (indicated in red). Since the experiments under orders of magnitude higher. By calibrating the lyotropic liquid capillary confinement (11) are performed at room temperature, crystal (LLC) data with measurements from the thermotropic the light-scattering measurements are done at room temperature LC 5CB, absolute values for the elastic constants and the cor- as well. For the sake of comparison, the literature data shown in responding viscosities of the micellar LLC are extracted. Details Table 1, if available, are also measured at room temperature. on the calibration step are explained in SI Appendix, SI Text. The These results represent a comprehensive study of the visco- three Frank elastic constants of the investigated lyotropic mi- elastic properties of a micellar LLC, including a complete set of cellar N phase are (1.54 ± 0.08) pN for K , (0.300 ± 0.018) pN values for the elastic constants and viscosities of a nematic phase D 11 of disk-shaped micelles. for K22, and (20.8 ± 1.3) pN for K33. With that, the corre- sponding viscosities of the splay, twist, and bend deformations Even for the case of thermotropic nematics, few literature data η η η are available on the viscoelastic properties of the ND phase. splay, twist, and bend are calculated (SI Appendix, Fig. S3). A − − Some ratios of an elastic constant to a diamagnetic anisotropy as value of (1.65 ± 0.09) kg·m 1·s 1 is found for η , (5.28 ± 0.32) splay well as the ratio of K33 to K11 have been reported, but only one · −1· −1 η ± · −1· −1 η kg m s for twist, and (87 6) kg m s for bend. Note that absolute measurement of K33 and K11 has been published (21). all experiments are performed at room temperature. Combining the results from experimental data (21–23) with those from theory and simulations (24, 25), it is proposed that in Discussion discotic thermotropic LCs with the normal phase sequence Table 1 summarizes the elastic moduli and viscosities of different (crystalline–columnar-ND–isotropic), K33 < K11 K K22,which thermotropic and lyotropic nematics from the literature along is opposite to thermotropic NC phases where K22 < K11 < K33 with the data for the micellar lyotropic ND phase of CDEAB/ (24–26). However, in the ND phases of the LLCs investigated so far,

Fig. 2. (A–C) Amplitude pseudo–cross-correlation function g1 graphed versus the delay time τ at θ = 45 and 50° in the splay–twist geometry (A and B) and at θ = 19° in the twist–bend geometry (C) at room temperature. The green dots represent the measured data, the yellow dashed line indicates a single stretched exponential fit, and the red line corresponds to a biexponential stretched fit function consisting of a slow and a fast relaxation mode (gray dashed lines). Because the fraction of the slow mode increases with increasing θ, this relaxation process originates from twist fluctuations, and the other (fast) mode, which decreases with θ, originates from splay fluctuations. Note that the single stretched exponential fit in C provides an excellent fit to the data measured in the twist–bend geometry, indicating that bend deformation dominates as expected. For comparison, a single stretched exponential fit is graphed as well (yellow dashed line) in A and B, but here the fit clearly misses the data at higher delay times (B, Inset). (D–F) Plots of the ratios of the splay, twist, and bend amplitudes Ai versus various angular functions. The amplitude Ai is the measured intensity Ii divided by the incident laser power. The respective slopes (gray triangles) of 2 the linear functions give the ratios of the elastic constants. (D) Ratio of the twist amplitude A2 to the splay amplitude A1 graphed versus tan (θst =2).(E) Ratio 2 of the twist amplitude A2 to the bend amplitude A3 graphed versus tan (θtb).(F) Ratio of the splay amplitude A1 to the bend amplitude A3 graphed versus 2 2 tan (θtb)=tan (θtb=2). θtb and θst are the scattering angles in the twist–bend and splay–twist geometry, respectively. The error bars represent the SD of the mean for each individual θ. The straight lines are linear fits constrained to go through the origin.

27240 | www.pnas.org/cgi/doi/10.1073/pnas.1922275117 Dietrich et al. Downloaded by guest on September 25, 2021 Table 1. Overview of the measured elastic constants and the corresponding viscosities of the ND phase of the micellar LLC CDEAB/ DOH/H2O (indicated in red) compared to what is known from the literature for splay (K11), twist (K22), and bend (K33) elastic constants and viscosities (ηsplay , ηtwist, and ηbend ) for thermotropic (gray) and other lyotropic (blue) nematic phases PHYSICS

Values listed for the BCN phase of the bent-core mesogen 4-chloro-1,3-phenylene bis 4-[4′- (9-decenyloxy)benzoyloxy] benzoate (ClPbis10BB) are taken

from ref. 2. Values listed for the thermotropic ND phase refer to hexakis(p-hexloxybenzoyloxy)triphenylene (21).The volume fraction of the lyotropic poly- meric racemic PBG system is 0.2 and the polymer chains have a length-to-diameter ratio of 36 (15, 17). The concentration of the lyotropic chromonic DSCGis

c = 16 wt%, whereas c = 29 wt% for sunset yellow (SSY) (6, 7). The ND phase of the lyotropic micellar CsPFO/H2O contains 49 wt% CsPFO and 51 wt% water (14). The two investigated mixtures of the ND phase of the lyotropic micellar decylammonium chloride (DACl)/H2O/NH4Cl are either 7 mol% DACl, 2.5 mol% NH4Cl, and 90.5 mol% H2O or 7.57 mol% DACl, 2.73 mol% NH4Cl, and 89.70 mol% H2O (12).

this trend is not observed. In fact, K33 has the highest value when constitute the building blocks; and in micellar LLCs the building the nearby more ordered phase is a lamellar phase that allows only blocks are micelles formed mainly due to hydrophobic effects. In splay deformation such that K33 and K22 tend to diverge on short, for all LLCs for which K22 has been measured, chromonic approaching the transition to the lamellar phase (14). As a result, a (4, 29), polymeric (30), and micellar (11), reflection symmetry- high K33=K11 ratio was found for several ND phases of LLCs breaking director configurations are reported. In other words, (12–14)(Table1).Onetheoreticalstudy attributes this behavior to the confinement-induced chiral structures point to a small value of the micellar shape anisotropy dependence on temperature as the K22. On the other hand, although there is evidence of chiral lamellar phase is approached (27). Note that elastic constants structures for achiral thermotropic nematics in spheres (31–36) and typically increase with decreasing temperature and literature data curved cylinders (37, 38), there is little evidence of chiral structures closest to room temperature and close to a lamellar phase have for achiral thermotropic nematics in straight cylinders (39). In been selected. The relatively high K33 (20.8 pN) in the CDEAB/ other words, the induced chiral structures in straight cylindrical DOH/H2O system can be explained with the same reasoning, since confinement are a clear indication of a small K22. This demon- a lamellar phase is nearby (phase diagram in SI Appendix,Fig.S4). strates that an order of magnitude smaller K22 value in lyotropic In chromonic LLCs, the nearby more ordered phase is a columnar nematic phases is the rule rather than the exception, constituting a phase that excludes splay and twist deformations, which might explain unique difference between lyotropic and thermotropic nematics. η η η why the K33=K11 ratiosinTable1aremuchsmaller for these systems. The viscosities splay, twist,and bend are of the same order of Theoretically, twist deformations should also be inhibited by a magnitude for thermotropic 5CB, whereas all lyotropic nematics nearby lamellar or columnar phase. But surprisingly, in all lyo- investigated so far (chromonic, polymeric, and micellar) have dif- tropic nematic phases, independent of whether the building blocks ferent viscosities that span a range of two to three orders of mag- η are rod or disk shaped, K22 is around one order of magnitude nitude. In the case of the NC phase of DSCG, splay appears to be η smaller than the other two elastic moduli. In the micellar CDEAB/ the largest, whereas in the micellar ND phase bend is the largest. DOH/H2Osystem,K22 is almost two orders of magnitude smaller The reverse behavior of NC and ND phases with respect to η η – than K33 and is in the range of the K22 value found in a lyotropic splay vs. bend can be explained according to the Ericksen Leslie DNA nematic (28). These results not only show that a small K22 is model (40–44) with the backflow mechanism (SI Appendix,section independent of the shape of the building block, but also demon- 2 and Fig. S4 and refs. 45–47). Calculating the bend to twist and strate that the composition of the building block and therefore the splay to twist ratios of the orientational diffusivities D3=D2 (= 4.2) intermolecular forces seem to have little effect on this phenome- and D1=D2 (= 16.4), our results are in agreement with the ones non. In chromonic LLCs the building blocks are columns of found for another lyotropic ND phase (48), for which the latter ratio π-stacked dye molecules; in polymeric LLCs long polymer chains is much larger than the former ratio. It seems that for the lyotropic

Dietrich et al. PNAS | November 3, 2020 | vol. 117 | no. 44 | 27241 Downloaded by guest on September 25, 2021 Fig. 3. (A–C) Elastic deformations of a micellar lyotropic ND phase and (D and F) the corresponding distortions of the disk-shaped micelles. All three kinds of deformations are associated with close approaches between micelles, the repulsive interactions between which increase the elastic energy. As a result, flexible micelles may distort their disklike shape to accommodate the specific deformation and lower the elastic energy. To accommodate the splay deformationinA, the micelles distort into a bent shape, as indicated by the red arrows at the center of the . The schematic cross-section of a bent micelle (D) illustrates the corresponding change in the bilayer structure, the nonzero curvature of which requires energy to vary the distances between the surfactant headgroups from their equilibrium distance in the undistorted state. To accommodate the bend deformation in B, the micelles splay into a wedge-like shape. The schematic cross-section of a splayed micelle (E) shows that the splay of the micelle is associated with variations in the bilayer thickness, an energetically expensive distortion which, like for thermotropic smectics, requires a good deal of energy to change the density of tails inside the bilayer. To

accommodate the twist deformation in C, the micelle twists along its long axis parallel to the local twist axis of the ND phase. In F the bilayer structure of a twisted micelle is approximated by a twisted stack of thin bilayer slices. Since in each slice neither the thickness nor the curvature of the bilayer is changed, this is a soft deformation which requires less energy than the distortions in D and E. Twist deformations are thus the least energy-costly fluctuations in a lyotropic

ND phase while bend deformations are the most energy-costly fluctuations.

η η ND phase, backflow lowers splay but not bend which is consistent dispersed in water repel each other on the nanometer length with our results. Furthermore, the high value observed for the bend scale via effective solvent-mediated interactions, so-called hy- – viscosity indicates that bend deformations of the ND phase are dration interactions (51 53), which decay exponentially with the highly dissipative. The reason behind this may be that the bend intermicellar distance and which might take over the role of deformation cannot relax via an internal elastic mode of the micelle steric interactions between hard rods in the nematic Onsager and instead requires the rotational diffusion of the entire assembly. fluid. Disregarding their rim, disk-shaped micelles are essentially In addition, the ratios of elastic constants are much larger in a single, flat, finite, and fluid amphiphile bilayer, the surface area lyotropic nematics than in their thermotropic counterparts. While in of which is a few square nanometers. Let us thus assume that thermotropic 5CB the largest elastic constant K is only less than disk-shaped micelles distort like a single bilayer distorts: 1) Any 33 distortion that requires a change in the bilayer thickness is ener- three times larger than its smallest elastic constant K22,theratio K =K is greater than 60 in the case of the CDEAB/DOH/H O getically very expensive and can practically be neglected (54) and 33 22 2 2) bend distortions that increase the essentially zero curvature of lyotropic N phase (Table 1). How can we explain these remarkable D the undistorted flat bilayer require less energy. The bending en- differences between thermotropic and lyotropic nematics? 2 ergy per surface area, Kb/2R (54) with a typical bend modulus of In comparison to the single-molecule building blocks of ther- − K ∼ 10 20 J (55), is less than or equal to the order of magnitude motropic nematics, the building blocks of lyotropic nematics are b of the thermal energy for reasonably large bend radii R > 10 nm. larger supramolecular assemblies, the nanoscale dimensions of Finally, a (small) twist of the bilayer directed parallel to its surface which allow long wavelength fluctuations and lend these assem- should be the least energy-costly deformation since it changes blies a certain flexibility. Even in the Onsager case of a hard-rod neither the thickness nor the mean curvature of the bilayer. nematic fluid, a distinct difference of elastic constants depending The interplay between the fundamental elastic deformations on the aspect ratio of the hard rods and the steric repulsion be- of an ND phase and the corresponding distortions of its disk- tween them is predicted by theory (49). In the case of flexible shaped micelles is illustrated in Fig. 3. On one hand, the bend assemblies, however, the assembly shape can (at least to a certain elastic energy is reduced very little since the corresponding dis- extent) accommodate a specific elastic deformation of the phase tortion of the micelle requires a change in bilayer thickness, an in such a way that the steric repulsion between the assemblies is energetically expensive deformation that can be neglected. On lowered and thus the elastic energy of that deformation is reduced the other hand, the reduction of twist elastic energy is the most (50). In other words, the actual values of the elastic constants very significant since it is the least energy-costly distortion of the much depend on how the flexible building blocks can distort under micelles. The reduction of splay elastic energy is somewhere the action of a specific elastic deformation of the phase. between these two cases, depending on the actual values of kb These general considerations can now be applied to the case and R. These arguments, essentially based on the distortion of a micellar ND phase. It is well established that micelles modes of disk-shaped micelles, explain at least qualitatively why

27242 | www.pnas.org/cgi/doi/10.1073/pnas.1922275117 Dietrich et al. Downloaded by guest on September 25, 2021 the bend modulus K33 of a micellar lyotropic ND phase is ex- capillary forces at room temperature, and the edges were sealed with UV glue (Norland Optical Adhesive 71) to prevent solvent evaporation. To tremely large and why its twist modulus K22 is anomalously small in comparison to thermotropic nematics. eliminate unwanted alignment effects from the filling process, the samples To conclude, these measurements represent a significant step were heated into the isotropic phase and cooled down to the nematic phase. Due to the relatively large cell thickness, the alignment took 1 to 2 d to in understanding the viscoelastic properties of lyotropic nematic become uniform; applying a magnetic field sped up the process. An example liquid crystals because a full set of elastic constant and viscosity of a homogenously planar aligned LLC cell is shown in the SI Appendix, Fig. values for a discotic micellar lyotropic system is reported. The S6. Being able to align such a thick sample of nematic lyotropic liquid crystal extremely low value for K22 explains the prior observation of is not trivial, since good alignment in such systems has been achieved so far reflection symmetry breaking when this system is confined in only for a rather thin sample thickness of less than 10 μm (58, 59). cylinders and further suggests that such large ratios of elastic constants are a universal feature of lyotropic nematic phases, Dynamic Light Scattering. In the dynamic light-scattering (DLS) setup, the independent of the type and shape of the building blocks. Fi- perpendicular alignment of the incoming laser beam with the cell was nally, considerations of the flexible nature of the building blocks achieved by adjusting the back reflection of the incident laser beam from of lyotropic phases provide a qualitative hypothesis about the the ITO coating. The orientation of the director parallel and perpendicular to the incident light polarization was fixed by a machined sample cell holder origin of the small twist elastic constant. inserted into the scattering compartment of the setup. The samples were measured with a 3D dynamic light-scattering apparatus from LS Instruments Materials and Methods (operation mode: 2D pseudo–cross-correlation) with a high-performance Materials. The LLC investigated in this study is a ternary micellar system DPL Laser (Cobolt) with a wavelength of 561 nm and a maximum power containing N,N-dimethyl-N-ethylhexadecyl-ammonium bromide (CDEAB) as of 400 mW. The polarization of the incident laser light was vertical to the the surfactant, decan-1-ol (DOH) as the cosurfactant, and doubly distilled scattering plane, whereas the scattered light passing through a horizontal water as the solvent. To minimize the amount of dust, filtered DOH and analyzer was detected; this is called depolarized light scattering. The water were used (pore size of filter = 0.2 μm). At a composition of 32.0 wt% depolarized DLS intensity was measured in steps of 1 or 2°, in the twist–bend CDEAB, 4.8 wt% DOH, and 63.2 wt% H2O, a broad nematic phase with geometry typically from θ = 15 to 25° and in the splay–twist geometry disklike micelles (ND) is formed around room temperature (see the phase typically from θ = 20 to 60°, where θ is the scattering angle. For the lyotropic diagram reported in ref. 56 and shown in SI Appendix, Fig. S5). CDEAB and samples the acquisition time for each θ was 2 to 3 h with a laser power of DOH were purchased from Merck KGaA and were used without any further about 20 mW. Light-scattering data were obtained for up to three to four purification. For measurements of the birefringence and the refractive in- individual sample cells of 5CB and the lyotropic CDEAB system, sometimes at dexes, commercially available nylon rubbed LC cells with a thickness of multiple sample positions within the cell. For each single measurement, the 10 μm were used. The birefringence was determined by measuring the op- error in scattering intensity depends on the scattering angle θ. For small and Δ = ± tical retardation with an LC-PolScope giving a value of n 0.0040 0.0002. large angles in the twist–bend geometry (θ = 20 to 30° and 50 to 60°) the PHYSICS This value is consistent with subtracting the individually measured refractive error estimate is around 7 to 10%, since the fraction of either the twist mode = = indexes (nII 1.396 and n⊥ 1.392) of the LLC. The measurement of the two or the splay mode is relatively small in those cases. In the angular regime in refractive indexes nII and n⊥ was done by interferometry with an Ocean between, the error estimate in the measured scattered intensities was Optics Spectrometer according to ref. 57. around 3 to 5%. The acquisition time for the 5CB samples for each θ was 15 min with a laser power of about 0.2 mW. The experiments were per- Sample Alignment. A unidirectional planar alignment of the nematic sample is formed at 298 ± 0.1 K. Decahydronaphthalene with a refractive index of n = crucial for the light-scattering experiments. LC cells were produced in the 1.481 was used as the index matching fluid surrounding the sample cell. MC2 cleanroom facility of the Chalmers University of Technology. Indium tin Refraction was taken into account by changing the laboratory scattering oxide (ITO)-coated glass plates were spin coated with polyimide PI-2610 angle to the scattering angle in the liquid crystal using values for the re- (DuPont) and after curing at 300 °C for 3 h the polyimide was rubbed with a fractive indexes of the LC and the index matching fluid. A detailed de- velvet cloth in a commercial rubbing machine (LC-Tec Automation). Sub- scription of the data analysis and 5CB calibration is found in SI Appendix. strates of size 75 × 75 mm were glued together with parallel rubbing di- rections using a substrate-assembling machine (Ciposa). The cell gap was Data Availability. All data supporting the findings of this study are available μ secured by means of 30- and 100- m diameter spherical silica spacers dis- within this article and/or SI Appendix, with corresponding DLS raw data and persed in the ultraviolet light-activated glue lines. The curing was performed evaluation deposited at DaRUS, the University of Stuttgart data repository under pressure using a vacuum packaging machine after which 6 × (https://doi.org/10.18419/darus-746) (60). 6-mm–sized identical cells were obtained by scribing and breaking, with the machine ensuring that the scribing direction was parallel to the rubbing ACKNOWLEDGMENTS. We thank Tobias Steinle, Zoey S. Davidson, Nadine μ direction. The 30- m cells were used for 5CB to minimize possible multiple Schnabel, Kristina Schneider, and Samuel Sprunt for helpful discussions and scattering, whereas due to the low birefringence of the LLC, 100-μm cells gratefully acknowledge financial support from the Alexander von Humboldt were used for the lyotropic samples. The cells were filled with LLC by Foundation.

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