**Contents**


## **About the Editors**

**Ant ˆonio Martins Figueiredo Neto** is a professor of Physics at the University of Sao Paulo, ˜ Sao Paulo, Brazil, and is Head of the National Institute of Science and Technology on Complex Fluids, ˜ Brazil. He is a member of the organizing committees of more than 20 international conferences and workshops in the field of liquid crystals, magnetic colloids and fluids of biological interest. He has written more than 220 papers in international journals, 1 book published by the Oxford University Press, 5 book chapters, and has given more than 350 presentations at international conferences. He has supervised 22 defended PhD students and 18 Master dissertations. He is a member of the Brazilian Academy of Science and The Academy of Science of the State of Sao Paulo. ˜

**Ingo Dierking** is a senior lecturer/associate professor at the Department of Physics and Astronomy of The University of Manchester, UK. His research is focused on soft matter physics, especially liquid crystals and liquid crystal-based composites, including both thermotropic and lyotropic materials. He is the author and editor of several books and book chapters, and has published more than 140 papers on the topic of liquid crystals in high impact peer-reviewed journals. Dierking is the 2009 recipient of the Hilsum medal of the British Liquid Crystal Society (BLCS) and the 2016 recipient of the Samsung Mid-Career Award for Research Excellence awarded by the International Liquid Crystal Society (ILCS). He is the former Chair of the BLCS and the Secretary of the ILCS, and is the current President of the International Liquid Crystal Society.

## **Preface to "New Trends in Lyotropic Liquid Crystals"**

Lyotropic liquid crystals have long led a shadowy existence in LC research, dominated by its much bigger thermotropic LC brother, who is accountable for the multi-billion dollar industry of flat panel screens and information display devices. Nevertheless, it appears that the equilibrium has shifted slightly, as the research community finds increasing interest in lyotropic systems, such as soft matter nanomaterials, biological materials and systems or active liquid crystals. We have thus decided that it is a good idea to collect some overview articles as well as original research papers on these soft matter systems that have gained much increased interest in recent years, in a Special Issue of the journal Crystals. This Special Issue is now also available in printed book form. The first paper of this collection provides an admittedly subjective and personal view from the editors on the current state of lyotropic liquid crystals. The authors of the more detailed and in-depth reviews were chosen such that a large variety of topics would be discussed, authored by the leading figures in their specialty subjects.

We start with an overview about the novel trends in lyotropic liquid crystals, touching on a number of different aspects, which will be presented in a more elaborate form later on. These are, for example, some new observations by the Giesselmann group on amphiphilic molecules that form phases equivalent to the thermotropic ferroelectric liquid crystal SmC\* phase. Cellulose nanocrystals, as nanomaterials and equally as biological materials, have gained much interest in recent years, especially with respect to their lyotropic behavior, which is discussed in detail in the papers of the Lagerwall and Godinho groups. Furthermore, chromonic liquid crystals, formed by board-like dye molecules, have been shown to offer new insights, as reported by the Lavrentovich group. Similar board-like systems, albeit on a much larger, macromolecular scale, are represented by the lyotropic nematic liquid crystal phase of graphene oxide, discussed by the Dierking group. This is more generalized in the review by the Davis group about the effects of size and shape dispersity on the phase diagram of lyotropics. This is a topic which may also be related to the observation of lyotropic biaxial nematic systems, as discussed experimentally by the group of Neto, and theoretically explored by Salinas et al. At last, a theoretical approach by Izzo investigates the ordering of rods near surfaces.

We hope that the present collection of papers provides a good basis for the further exploration of modern aspects of lyotropic liquid crystals, both from an experimental and a theoretical point of view. Last, but not least, we would like to thank all of our colleagues who have given up their time and contributed to this Special Issue, as well as the editorial staff of "Crystals" for their professional support.

> **Ant ˆonio Martins Figueiredo Neto, Ingo Dierking** *Editors*

## *Review* **Novel Trends in Lyotropic Liquid Crystals**

## **Ingo Dierking 1,\* and Antônio Martins Figueiredo Neto 2,\***


Received: 25 June 2020; Accepted: 10 July 2020; Published: 12 July 2020

**Abstract:** We introduce and shortly summarize a variety of more recent aspects of lyotropic liquid crystals (LLCs), which have drawn the attention of the liquid crystal and soft matter community and have recently led to an increasing number of groups studying this fascinating class of materials, alongside their normal activities in thermotopic LCs. The diversity of topics ranges from amphiphilic to inorganic liquid crystals, clays and biological liquid crystals, such as viruses, cellulose or DNA, to strongly anisotropic materials such as nanotubes, nanowires or graphene oxide dispersed in isotropic solvents. We conclude our admittedly somewhat subjective overview with materials exhibiting some fascinating properties, such as chromonics, ferroelectric lyotropics and active liquid crystals and living lyotropics, before we point out some possible and emerging applications of a class of materials that has long been standing in the shadow of the well-known applications of thermotropic liquid crystals, namely displays and electro-optic devices.

**Keywords:** liquid crystal; lyotropic; chromonic; amphiphilic; colloidal; application

## **1. Introduction**

Lyotropic liquid crystals (LLCs) [1,2] are known from before the time of the discovery of thermotropics by Reinitzer in 1888 [3], which is generally (and rightly) taken as the birth date of liquid crystal research. In the work before this time, for example by Virchow [4], Mettenheimer [5], Planer [6], Loebisch [7] or Rayman [8], the liquid crystalline properties were described, but without the explicit realization that this constituted a novel state of matter. The latter was the significant contribution made by Reinitzer [3] in 1888 and Lehmann [9], who coined the term "*liquid crystal*" in 1889 when studying thermotropic phases. Nevertheless, lyotropic liquid crystal research has been present ever since, albeit on a lower quantitative output than that of thermotropic systems, also because of their less obvious potential in applications, being overshadowed by displays and electro-optic devices. But, as thermotropic liquid crystal research surged in the 1970's to the 2000's, so did that of lyotropic liquid crystals (Figure 1), due to the realization of their importance for biological systems and in colloid science.

Over the last two decades, more and more LC researchers have widened the scope of their work to also include lyotropic phases, and to explore systems of both thermotropic and lyotropic behavior. This paper will try and summarize some of the fascinating recent developments, as lyotropics find their way into an increasing number of liquid crystal laboratories. This will by no means be an exhaustive treatment, but will hopefully provide an overview of the current new trends in lyotropic liquid crystals.

**Figure 1.** Number of publications in the field of lyotropic liquid crystals (LLCs) shown as bars over a time period of 5 years each.

## **2. Lyotropic Liquid Crystals**

## *2.1. Classic Lyotropics from Amphiphiles and Polymers*

Amphiphilic molecules have the striking property of presenting two antagonistic characteristics within the same molecule, i.e., hydrophobicity and hydrophilicity. In contact with polar and/or nonpolar liquids, under proper temperature and relative concentration conditions, they form lyotropic liquid crystalline phases [1]. Nanoscale molecular segregation (self-assembly) gives rise to different molecular aggregates, from micelles to bicontinuous structures. There are different types of amphiphilic molecules, such as anionic, cationic, zwitterionic and non-ionic amphiphiles, detergents and anelydes [10]. Moreover, other more complex molecules belong to this category, the gemini surfactants [11], the rigid spiro-tensides, phospholipids [12], the facial amphiphiles [13] and the bolaamphiphiles [14]. The lyotropic polymorphism encountered in mixtures with amphiphilic molecules is very rich. Figure 2 shows a sketch of a typical lyotropic liquid crystal phase diagram of a mixture composed by an amphiphile and water. The Krafft line separates the crystalline phase region from the liquid region. The critical micelle concentration line separates the single amphiphilic molecule region from the molecular self-assembled region. By increasing the amphiphile relative concentration, the mixture can present the micellar, hexagonal, lamellar and inverted phase structures. A well-known biological example for lyotropic lamellar structures is the lipid bilayers of cell membranes, as shown in Figure 3a. Other examples are Myelin figures.

**Figure 2.** Sketch of a phase diagram of a mixture of an amphiphile and water. In the horizontal axis, the amphiphile concentration is represented. Cubic phases may occur at different areas in the phase diagram.

**Figure 3.** (**a**) Schematic representation of the lyotropic liquid crystalline structure of a cell membrane (reproduced from Wikimedia Commons from OpenStax Anatomy and Physiology). (**b**) Chemical structure of Kevlar®.

Despite the fact that these liquid crystalline phases were extensively studied for many decades, interesting questions still remain to be answered nowadays. Two of these mesophases deserve to be particularly highlighted, the biaxial nematic, NB [15,16], and the chiral [17,18] mesophases.

Controversies have appeared in the literature over the years about the existence of the NB phase and its chemical stability. Theoretical [16,19] and experimental [16,20,21] papers were published about the stability of the NB phase. In this special issue, an extensive discussion about the experimental conditions to stabilize the NB phase is presented.

Chiral lyotropics is, still, a challenge to be fully understood by physicists and chemists. An intriguing question is how the information about chirality passes from one element of the structure (micelle or lamellae) to the next, since there are water molecules between them. Recently, evidences of an equivalent to the thermotropic (chiral smectic C), SmC\*, in lyotropics were reported [18]. These findings further broaden the boundaries of the physical chemistry of lyotropic liquid crystals.

Another class of lyotropic liquid crystals involve polymers in an isotropic, liquid solvent. One classic example found in nature is spider silk, which consists of protein fibers formed from a micellar solution when pushed through a valve at the spiders back under loss of water. This forms oriented crystalline regions of beta-sheets cross-linked via amorphous regions. Another industrially produced high modulus fiber is Kevlar® (Figure 3b), which is produced from the lyotropic liquid crystalline state of the aramide polymer in highly concentrated sulfuric acid. It is used in a wide range of applications, from bullet proof vests to climbing gear.

## *2.2. Inorganic Liquid Crystals*

The most classic example of an inorganic or mineral liquid crystal is vanadium pentoxide, V2O5. Its needle-like nanocrystals form a nematic lyotropic liquid crystal, which was first investigated by Diesselhorst and Freundlich [22,23] in the beginning of the 20th century. They reported the occurrence of birefringence of anisotropic crystallites of vanadium pentoxide when subjected to flow or to an electric field. It was concluded that both mechanisms had the same origin, application of a force to orient the elongated, needle-like crystallites that then exhibited a macroscopic birefringence. After the removal of the force, the system relaxed to yield an isotropic appearance. It has often been reported that a newly prepared V2O5 sol is initially isotropic, while it takes time to observe the development of nematic tactoids [24], as seen in Figure 4.

**Figure 4.** Formation of nematic tactoids in a preparation of vanadium pentoxide, V2O5 (reproduced with permission from Reference [24]), and corresponding needle crystallite orientation within some tactoids.

The phase formation largely depends on the preparation conditions and history of V2O5. This is related to the aging of freshly prepared sol and depends on concentration, temperature and possible electrolyte addition. The formation of the nematic phase is enhanced for large V2O5 concentrations and higher temperatures. The length of the colloidal vanadium pentoxide particle increases at a constant width of approximately 10 nm from the nanometer range to a few micrometers. This is accompanied by a sol-gel transition [25]. Electric field application in the nematic phase indicates that the system has a negative dielectric anisotropy, Δε < 0. This implies that in contrast to standard calamitic nematics with positive anisotropy, the director would switch from homeotropic to planar under electric field application. The phase sequence as a function of increasing concentration of V2O5 is isotropic–biphasic–uniaxial nematic for the fluid suspensions. For further concentration increase, a gel transition is observed and the uniaxial nematic gel is transformed into a biaxial nematic gel [26].

Similarly, aluminum oxyhydroxide, AlOOH, forms nematic tactoids, which on merging exhibit a typical nematic Schlieren-texture [27]. M2Mo6X6, with the metal from the alkalimetal group M = Li, Na, K and X = Se, Te, from the chalcogens group, also represents a general group of inorganic LCs with nematic phases. Crystallites exhibit lengths of approximately several micrometers, and Schlieren-textures, as well as sometimes thread-like textures [28], can be observed when these are dispersed in an isotropic solvent, for example methylformamide. The formation of smectic phases has also been observed. This was shown for FeOOH and for tungstic acid H2WO4 (WO3 • H2O) by the observation of steps in the textures of droplets, indicating smectic layering [29]. A more detailed overview about the structures of anisotropic crystallites, their preparation and the conditions employed to form liquid crystalline phases was published in a review by Sonin [30]. It appears that besides phase diagrams, textures and structures, there has been little work on such systems with respect to modern experimental techniques or applications by self-assembly for nanotechnology. Maybe it can be fruitful to revisit these systems from the different perspectives available today.

## *2.3. Clays*

These natural soil materials of micro- and nano-meter dimensions present high shape anisotropy and contain hydrous aluminum phyllosilicates. Typically, the particles are plate-like, with the plate thickness in the nanoscale. In 1995, Mourchid and co-workers [31] reported an interesting study of aqueous suspensions of clay particles. However, they did not clearly identify a liquid crystalline phase. In 2009, Paineau and co-workers [32] published a paper about a highly diluted (5% volume fraction) aqueous suspension of disk-shaped natural beidellite clay (a phyllosilicate), where a first-order isotropic to nematic phase transition was identified. The optical birefringence of these suspensions (~10−4) is smaller than that from usual micellar lyotropics. The nematic phase may be aligned by electric and magnetic fields and also by shear. The stabilization of a lyotropic structure is assured in an aqueous medium by the existence of electrical charges on the surface of the particles and the hydration of the particles.

More types of clays in aqueous suspensions showed liquid crystalline behavior: bentonite, an aluminum phyllosilicate clay consisting mostly of montmorillonite [33] (Figure 5a), laponite, a synthetic layered magnesium silicate [34] (Figure 5b), and imogolite, an aluminum silicate [35,36]. The imogolite particle has a hollowed cylindrical shape with diameter of about 2 nm and length of the order of hundreds of nm. An interesting, and until now not fully understood behavior of imogolite suspensions, is the presence of a regular streaked texture, observed in the polarizing optical microscope, that resembles textures from cholesteric ordering. The imogolite particles do not have any chiral component and the chiral arrangement (if demonstrated by other experiments) should be due to a particular packing of the cylinders [36]. This type of texture may also originate by a nematic ordering in gels [37]. Aqueous suspensions of nontronite clay also showed a nematic to isotropic phase transition at low (below 10−<sup>3</sup> M/L) ionic strengths [38].

From the theoretical point of view, Onsager's approach [39] qualitatively explains the tendency of the plate-like clay particles to align in an aqueous suspension. However, the liquid crystalline behavior is not observed in concentrated suspensions, where phase segregation occurs. Polydispersity is also an issue that must be addressed when a suspension of clay particles in a solvent with liquid crystalline behavior is aimed at. Usually, a size separation procedure is needed before the preparation.

**Figure 5.** Nematic texture of different clays, (**a**) bentonite and (**b**) laponite (reproduced with permission from Reference [34]). Nematic textures for these materials are generally much less pronounced and typical than those for calamitic thermotropic liquid crystals. White bars correspond to 0.2 mm in (**a**) and 100 μm in (**b**).

## *2.4. Tobacco Mosaic Virus (TMV) and Other Viruses*

The tobacco mosaic virus can be seen as the prototype of a rigid rod system. It is very straight, with a constant length of 18 nm and an often close to monodisperse length distribution round 300 nm. The aspect ratio is thus about 15, and the system is ideally suited to test the Onsager theory [40], see Figure 6. The TMV is a right-handed single-stranded RNA virus which infests the leaves of tobacco, but also other plants, which is clearly visible through a pronounced and characteristic discoloration. Discovered toward the end of the 19th century by Mayer [41], it was first thought to be bacterial, but was later independently shown by Iwanowski [42] and Beijerinck [43] to be of different origin, for which the latter coined the term "virus". It was not until the 1930s that electron microscopic evidence was produced [44], and in 1936, Bawden et al. [45] had already reported the lyotropic liquid crystalline behavior of the tobacco mosaic virus.

**Figure 6.** Electron microscopy image of tobacco mosaic viruses (TMV), showing an aspect ratio of approximately 15. The scale bar indicates 0.2 μm (reproduced from Wikimedia Commons, with no author name supplied).

From the magnetic field aligned nematic phase, Oldenbourg et al. [46] produced small-angle diffraction patterns which allowed the determination of the order parameter S. The X-ray scattering for samples of increasing TMV concentration showed transitions from the isotropic phase at low concentrations, passing through a typical isotropic/nematic two-phase region to a pure nematic phase at high concentrations. The order parameter S within the nematic phase changed from about S = 0.7 at the transition to close to perfect order of the long axis of the TMV S = 1 at rather high concentrations. This is indeed in accordance with predictions by the Onsager model. A very detailed study of the liquid crystalline behavior and physical properties of the TMV lyotropic nematic phase was carried out by Fraden et al. [47]. They measured the sample birefringence not only as a function of concentration and temperature, but also for varying ionic strength and different polydispersity. From these investigations, it was concluded that the stability of the nematic phase of tobacco mosaic virus suspensions is predominantly determined by electrostatic repulsion. Attractive van derWaals forces between the TMV rods supposedly play a much less important role. This provides an indication that the transition from isotropic to nematic is practically based on excluded volume effects. This in turn explains why the predictions of Onsager theory work very well for the TMV liquid crystal, because the theory is based on repulsive steric interactions, ignoring attractive forces between the colloidal particles. Graf and Löwen [48] later predicted the detailed phase diagram of the tobacco mosaic virus from theory and the use of computer simulations. They also described a further transition into smectic phases and colloidal crystals. Different virus suspensions, for example rod-like or semiflexible filamentous bacteriophage fd, have been reported to also exhibit chiral nematic or cholesteric order [49], as well as smectic layering [50], respectively. An overview can be found in the reviews of References [51,52].

We shall see below that novel trends in TMV lyotropic phase research have applicational potential in the production of silica nanostructures through templating. Another more fundamental aspect can be the experimental study of the phase behavior of mixtures, for example of rods and spheres [53], but also other systems like rods and plates, and even rod-rod systems with very different aspect ratios. This would be especially of interest in combination with computer simulations. Other novel aspects may be found in biological and chemical sensing or directed drug delivery.

## *2.5. Lyotropic Phases from DNA*

The DNA macromolecule is a charged anionic polyelectrolyte formed by a right-handed double helix. Small fragments of DNA have a cylindrical shape of about 2 nm of diameter and variable lengths (typically ~50 nm). These fragments can be dispersed in water and present lyotropic liquid crystalline phases [54]. Increasing the DNA concentration in aqueous solutions (depending on the ionic strength and DNA persistence length), the phase sequence experimentally observed is: isotropic, blue phase, cholesteric, columnar hexagonal and crystalline (Figure 7). Disclinations and dislocations were observed in textures of aqueous DNA solutions, identifying the cholesteric phase [55]. Besides the texture inspection, measurements of the circular dichroism were performed to identify this mesophase.

More difficult to be identified is the blue phase, since it exists in a narrow range of temperature and DNA concentration, being optically isotropic. The electron microscopy of freeze-fracture replicas was used to identify the macromolecular arrangement in a double-twist ordering within small cylindrical domains. The optically anisotropic columnar phase was identified by different experimental techniques, mainly X-ray diffraction. The transition from the cholesteric to the columnar phase was shown to be of first-order or continuous. The same DNA solution may show both types of phase transition and, until now, the conditions defining one or the other type of transition are not known. Solutions with long DNA fragments (~70% w/v—comparable to that of in vivo systems) showed a cholesteric phase, with concentration-dependent pitch, and another two-dimensional (2D) phase that resembles the smectic thermotropic phase [56].

**Figure 7.** Transition region of the cholesteric (left) and the high-density region phases (right) in solutions of rod-like DNA molecules (reproduced with permission from Molecular Expressions at the Florida State University Research Foundation. The image can be found at the website https: //micro.magnet.fsu.edu/dna/pages/transition3.html). White bar corresponds to about 300 μm.

Not only long DNA fragments were shown to present liquid crystalline phases. Short fragments (about 8 base pairs) in aqueous solutions with RNA presented columnar and cholesteric phases [57]. The local structure is stabilized due to base stacking forces that promote the end-to-end aggregation of duplexes. An interesting behavior was observed in drying droplets of DNA (persistence length of ~50 nm, 48 k pb) aqueous solutions, where "coffee rings" are formed [58]. The DNA macromolecules accumulate in the droplet edge, forming a lyotropic liquid crystal with concentric-chain orientations.

The atomic arrangement and charge distribution present in DNA fragments open many possibilities of liquid crystalline structures with these building blocks. Salamonczyk and co-workers [59] reported an interesting result about the presence of the smectic-A phase in an aqueous suspension of double-stranded DNA fragments. To achieve this, they increased the DNA flexibility by introducing a spacer in the middle of each duplex. Storm and co-workers discussed the formation of a columnar liquid crystalline structure of self-assembled DNA bottlebrushes [60]. The building block of this structure is made of DNA as the backbone molecule and C4K12 protein polymers as the side chains.

Recently, Brach and co-workers reported a study where important differences in the DNA spatial structure were observed between free DNA and DNA organized in a lyotropic liquid crystalline arrangement [61]. The relations between the liquid crystalline structure and the functionality of living processes involving DNA still challenges researchers and opens a fascinating field of investigation. This last aspect inspires researchers to explore the relations between the liquid crystalline structure and the functionality of living processes involving DNA.

## *2.6. Lyotropic Cholesteric Cellulose Derivatives and Cellulose Nanocrystals*

Cellulose is composed of β-D-glucopyranose units covalently linked with (1–4) glycosidic bonds. Cellulose nanocrystals (CNCs) are obtained from natural cellulose fibers. They are hydrophilic but can be surface functionalized to change their properties in the presence of different solvents [62]. CNCs are stiff, lath-like nanoparticles, with a typical diameter as small as ~6 nm, depending on the preparation method [63], and a length of about 100 nm.

Aqueous suspensions of cellulose nanocrystal particles, chemically prepared to avoid electrostatic stabilization and favoring the steric interaction [64], gave rise to a cholesteric mesophase (see Figure 6), with the typical fingerprint texture [65]. The cholesteric liquid crystalline phase occurs at a volume concentration of nanoparticles of about 10% [66]. One interesting application of the CNCs solution

showing the cholesteric phase is that the mixture can be dried, maintaining the chiral structure, to make films that acquire photonic band gap properties [67].

Cholesteric properties of suspensions of cellulose nanocrystals can be modified by decorating the nanoparticles with polymers [68]. The surface chemistry of the nanoparticles and interacting forces modifies the phase diagrams and the pitch of the suspensions. Long-pitch chiral mesophases were obtained with a decrease in the surface charge of the particles, decreasing the particle–particle interaction [69]. This mesophase is highly viscous and is located in the vicinity of a biphasic region.

Cellulose-based lyotropic mixtures may also stabilize mesophases [70]. Solutions of cellulose tricarbanilate in methyl acrylate and methyl methacrylate were shown to stabilize nematic and cholesteric mesophases at specific relative component concentrations and temperatures [71,72]. Lyotropic mesophases were also obtained in cellulose derivatives (with hydroxypropylcellulose (Figure 8a–d) and ethyl-cellulose) in inorganic solvents [73]. Cellulose acetate phthalate/hydroxypropyl cellulose blends in *N*,*N*-dimethylacetamide showed lyotropic polyphormism under proper temperature and relative concentration conditions [74].

**Figure 8.** Lyotropic textures from (Hydroxypropyl) cellulose (HPC)/water in a polarized microscope: (**a**) ~45% HPC, planar and focal conic textures, (**b**) 55% HPC, focal conic texture, (**c**) 55% HPC, oily streak texture, (**d**) ~65% HPC, planar and focal conic textures (reproduced with permission from Reference [65]).

## *2.7. Nanotubes, Nanorods and Nanowires*

Most of the systems relating to liquid crystalline behavior and nanotubes, nanorods or nanowires are composites, where the nanomaterial is dispersed in a thermotropic liquid crystal. This is often the nematic phase [75–78], occasionally also a smectic phase, often already with an additional functionality available, such as ferroelectric liquid crystals [79]. Such nanomaterials have been dispersed in lyotropic liquid crystals to a much lesser extent [80–83], often in the hexagonal phase for compatibility reasons. Thermotropic liquid crystals are used with carbon nanotubes to directionally orient the nanotubes or nanorods to exploit their extraordinary properties in a predetermined way as an addition to properties provided by the liquid crystal itself. On the other hand, lyotropic liquid crystals may be used as templates for materials in nanotechnology, often washing the liquid crystal out after the templating process. For example, nanowires and nanorods have been produced by synthesis in the lyotropic liquid crystalline state of TiO2 [84] and ZnO [85].

In addition to the dispersions of nanotubes, nanorods or nanowires in thermotropic or lyotropic liquid crystal phases, these materials can in fact also form lyotropic liquid crystals by themselves through dispersion in an isotropic solvent. The behavior is often very similar to that observed for needle-like inorganic liquid crystals, or also the tobacco mosaic virus, and largely follows the description by Onsager's theory. At low concentrations of nanomaterials, an isotropic dispersion is observed, that changes to a biphasic region for increasing concentration, eventually forming a nematic lyotropic phase. For nanotubes, this was first theoretically predicted by Somoza et al. [86]. Experimental evidence followed soon for functionalized multi-wall carbon nanotubes (MWNT) in water [87,88], showing a nematic phase above 4 vol% MWNTs. Instead of covalent functionalization, systems of nanotube-adsorbed DNA were also used, providing the electrostatic repulsion favorable for LC formation [89,90]. Electrostatic repulsion for better dispersion was also used by Davis et al. [91] and Rai et al. [92] when choosing strong acids as isotropic solvents, which led to a protonation of the tube walls, instead of nanotube functionalization or decoration.

ZnO is a wide bandgap semi-conductor, which in nanowire form can exhibit liquid crystalline behavior as a lyotropic nematic [93,94]. Similarly, TiO2 nanowires can assemble into liquid crystal phases [95]. Semiconductor rods of cadmium selenide, CdSe, can be produced with excellent monodispersity and a ratio of length to width of generally 40 to 6 nm, respectively. These are thus ideal candidates to exhibit not only nematic (Figure 9), but also smectic/lamellar ordering of lyotropic liquid crystals, as demonstrated in References [96,97]. The general potential of using nanomaterials in liquid crystals, either to tune the LC properties, to add functionality or to transfer liquid crystal order onto nanomaterials during synthesis or self-assembly in nanotechnology, is enormous. It can be expected that a whole new range of fundamental insights as well as technical applications are still to come in the future.

**Figure 9.** Nematic liquid crystal droplets forming on evaporation of the solvent from a cadmium selenide (CdSe) nano-rod solution (reproduced with permission from Reference [96]).

## *2.8. Graphene Oxide and Other 2D Materials*

Similar to the rod-like and disc-like shapes of calamitic and discotic thermotropic liquid crystals, the analogue to lyotropic nanotubes would be graphene. Graphene itself has been shown to exhibit a nematic liquid crystal phase only in the protonated environment of strong chlorosulphonic acid [98]. A material more similar to the COOH-functionalized nanotubes is graphene oxide (GO), which readily shows liquid crystalline behavior over large concentration ranges [99,100] in a multitude of isotropic solvents, including water. In Figure 10, a well-aligned sample is shown, which exhibits optical properties like a standard calamitic nematic liquid crystal, but with the director in the direction of the sheet normal. The general phase behavior is as discussed for other lyotopic systems: an isotropic phase is followed by a two-phase region, ending in a fully developed nematic phase for increasing graphene oxide concentration. The details of the phase diagram are, on the other hand, dependent on the average size of the GO flakes [101,102], their polydispersity, the polarity of the solvent [101,103] and confinement conditions [101]. It has been shown that graphene oxide liquid crystals respond to applied

electric fields [104,105] and that an electro-optic response can be achieved, which is based on a large Kerr effect [106,107], i.e., an induced birefringence proportional to the square of the applied electric field. The Kerr response times are at present longer than those of the well-discussed thermotropic Blue Phases, due to the higher viscosity of GO-LCs. So far, graphene oxide liquid crystals are by far the best studied of the lyotropic liquid crystals made from 2D materials, and their properties have been summarized in a number of review articles [108–110], although it should be mentioned that here, there are also still many open questions. One of these, which has, for example, been addressed via texture and dielectric studies, is the observation of mixtures of thermotropic nematics with GO [111]. There is some evidence [111,112] that on heating such a dispersion into the isotropic phase of the thermotropic LC, this acts as an isotropic solvent to facilitate the formation of a lyotropic GO liquid crystal phase. This is a very interesting topic, as it implies a thermotropic nematic to lyotropic nematic phase transition, which is not accompanied by any transition enthalpy. This should also be attractive for theoretical interpretation.

**Figure 10.** A lyotropic nematic graphene oxide phase can be oriented by a narrow, untreated glass channel. The quality of orientation is demonstrated by rotation of the sample between crossed polarizers. The director is indicated by *n* (reproduced with permission from Reference [37]).

Lastly, it should also be mentioned that other 2D materials can exhibit liquid crystalline phases in isotropic solvents. One of these is reduced graphene oxide (rGO) [113,114], which regains a certain amount of the conductive behavior observed for graphene, which is absent from graphene oxide. Reports have also been published for molybdenum disulphide, MnO2 [115] and Mxenes [116]. A number of other possible candidates have yet to be investigated further [117]. The topic of graphene oxide liquid crystals, their properties, and dispersions with other lyotropic liquid crystal classes, will certainly be an exciting one over the next years to come. Also, with respect to possible applications, for example as fibers [118,119], in tuneable photonics [120], nanofiltration [121] or reflective displays [122].

## *2.9. Chromonics*

Chromonics are rigid aromatic molecules, with hydrophilic ionic and hydrogen-bond groups located in the peripheries of the molecule [123,124]. π-stacking interactions between these flat molecules favor packing. In the presence of a polar solvent (e.g., water), they stack face to face in columns of different aggregation numbers. The (anisometric) columns consist of the building blocks of lyotropic liquid crystals (Figure 11a–c). Dyes, drugs and even nucleic acids are examples of this type of molecule [125]. Different mesophases were identified in mixtures with this type of molecule: uniaxial nematic [126], hexagonal [127] and lamellar-type structure (proposed for the diethyl ammonium flufenate) [128]. The aggregation of chromonic molecules is isodesmic, where the energy between molecules in a stack is independent of the number of molecules [129]. The addition of salts (e.g., NaCl) to the lyotropic nematic phase of cromolyn aqueous solutions shifted the nematic to isotropic phase boundaries upwards [130].

**Figure 11.** Typical structures of chromonic liquid crystal mesophase (reproduced with permission from https://core.ac.uk/download/pdf/54848113.pdf). (**a**) Isotropic, (**b**) nematic and (**c**) columnar phase.

From the theoretical point of view, besides the classical Onsager's approach, a model considering a competition of charge-like, long-range repulsion and anisotropic short-range attraction among molecules was recently proposed [131]. The nematic and hexagonal phases were obtained in the framework of this model, even at a small volume fraction of molecules, and correlations between the elastic response and stack growth were obtained. The elasticity of chromonic nematics was theoretically investigated (Monte Carlo simulations and Onsager-like model), inspecting the behavior of the Frank elastic constants, K11, K22 and K33, as a function of temperature and molecules' volume fraction [132]. The dependence of the elastic constants with temperature and molecules' volume fraction agree with the experiments. The elastic characteristics of chromonics is evidenced in experiments where the liquid crystal is confined in small droplets (typically from 1 to 100 μm) [133]. A very small twist elastic modulus in the nematic phase seems to be responsible for the appearance of a chiral-twisted bipolar configuration in confined conditions. Chromonic nematics are interesting systems to study topological defect cores in disclinations of +1/2 and −1/2 strengths, that extend to micrometric dimensions [134].

Recently, molecules of the chromonic Sunset Yellow (SSY) were added to classical amphiphilic lyotropic mixtures presenting the biaxial nematic phase [135]. It was shown that SSY exhibits a chaotropic character. Moreover, SSY causes an increase of the micellar shape anisotropy and, consequently, an increase of the biaxial nematic phase domain, with respect to the phase domain of the undoped mixture.

## *2.10. Polar Lyotropic Lamellar Phases*

It took about half a century from the discovery of the first ferroelectric crystal, Rochelle salt [136], for the first fluid material, the chiral smectic C\* phase, to be discovered [137]. It is already remarkable in itself that a fluid material can exhibit a spontaneous polarization, although, due to the SmC\* helix, in its bulk state, it may rather be called helielectric, as true ferroelectricity requires this polarization to be switchable between two stable states. This was demonstrated shortly afterwards with the surface stabilized ferroelectric state [138] and has initiated one of the most active topics in liquid crystal research in the last century. So far, this is all related to thermotropic materials. On the other hand, it was long known that thermotropic and lyotropic phases do show a certain amount of analogy. Both exhibit orientationally ordered nematic phases. The lamellar Lα phase is the lyotropic analogue of the

thermotropic SmA phase. So, why was there no tilted lamellar phase, a lyotropic equivalent of the rather common SmC phase? This was observed for the first time by Schaheutle and Finkelmann [139] and was demonstrated clearly by X-ray diffraction. A second report followed quite a number of years later [140]. These appear to have been the only confirmed cases of a tilted fluid lamellar phase. It was not until 2013 when a chiral amphiphilic molecule was shown to exhibit a lyotropic analogue of the ferroelectric SmC\* phase [18] when added to an isotropic solvent, water or formamide, over a certain range of concentrations.

The fluid tilted lamellar phase was verified via X-ray diffraction and typical textures could be observed, such as a smectic Schlieren-texture (Figure 12a), broken fan-shaped textures (Figure 12b), and in the surface stabilized geometry, a domain texture, which could mutually be brought to extinction when rotated by twice the tilt angle between crossed polarizers (Figure 12c,d). Chirality could be demonstrated by a typical striped texture due to the helical superstructure, just as observed for the thermotropic SmC\* phase. The pitch increased strongly when approaching the orthogonal phase (the SmA\* analogue) on heating at a fixed concentration, which is also generally observed for the thermotropic case. But, the most striking evidence can be found in the ferroelectric electro-optic switching of the surface stabilized state. However, a direct measurement of the spontaneous polarization is not possible due to the overall ionic conductivity of the samples. Lastly, it should be mentioned that the electroclinic effect could also be verified for the orthogonal chiral Lα\* phase [141], which very much resembles that of the SmA\* phase of thermotropics. Other amphiphilic molecules with a similar behavior have been synthesized since [142], and a detailed account of this topic can be found in the review of Reference [143]. The aspect of a conceptual transfer relating to physical properties and structures between thermotropic and lyotropic liquid crystals is certainly a very novel approach and opens up a whole field of future research in lyotropic liquid crystals.

**Figure 12.** Different textures of the lyotropic SmC\* phase. (**a**) SmC\* Schlieren-texture, (**b**) broken fan-shaped chevron texture with typical zig-zag lines, and (**c**) and (**d**) surface stabilized SmC\* domain texture, rotated by twice the tilt angle respectively, leading to mutually dark and bright domains (reproduced with permission from Reference [18]).

## *2.11. Active and Living Lyotropic Nematics*

An emerging topic in liquid crystal research and especially lyotropic liquid crystals is active matter or living liquid crystals. Active matter [144,145] in general are systems that are composed of a large number of constituents, each consuming or transforming energy for the reason of propulsion. These are thus intrinsic non-equilibrium systems. There is a wide variety of such systems found in soft and biological matter, for example a school of fish, bacteria or microtubules. They all have one property in common, they are self-organizing and exhibit collective, self-propelled motion. Liquid crystal-based active matter has recently attracted much increasing interest [146], due to fascinating phenomena observed that are absent in passive liquid crystals [147].

Active liquid crystal systems that are often studied include bacterial suspensions [148,149], microtubule-motor protein systems [150,151] and actin-motor protein systems [152–154].

A quite different system has been proposed, called "living liquid crystals" [155–158]. These are of particular interest to lyotropic liquid crystals, as they represent swimming, live bacteria in a lyotropic nematic phase (Figure 13). The latter have been shown to be supporting bacteria life [159], which is not the case for thermotropic nematic liquid crystals. Lavrentovich and co-workers [155] demonstrated experimentally that bacteria can sense director field deformations. They showed that for pure splay and pure bend deformations, the motion of the bacteria was equally probable in either direction of the director field. For regions with splay-bend deformations, like in the vicinity of topological defects, the motion was directed towards the positive s = 1/2 defect and avoiding the negative s = −1/2 one. By the use of predetermined director patterns, they directed the motion of bacteria and exerted a directing influence on the otherwise chaotic bacterial motion. Active and living liquid crystals exhibit a plethora of fascinating phenomena, spatio-temporal patterns and prospects for biological and biomedical applications. It is thus very likely that this field of lyotropic liquid crystal research will thrive in the future, not only in experiment but also in theory, simulations and applications.

**Figure 13.** Texture of a living liquid crystal with disclination pairs. The bacteria are aligned along the local nematic director, seen by the lines in the textures (reproduced with permission from Reference [155]).

## *2.12. Applications*

To the general public, liquid crystals are often primarily known through their electro-optic applications in displays, light shutters, or optical light modulators. Lyotropics are generally quite unheard of, despite of the fact that we use them on a daily basis. Obviously, a short paragraph on the applications of lyotropic liquid crystals cannot be all encompassing. Besides the various applications in the food and cosmetics industries, as well as detergents [160–164], we will here give an indicative overview of some other, possibly more modern applications.

One of these topics is drug delivery, which is somewhat related to some aspects encountered in the food and cosmetics industry in terms of the targeted and controlled release of an active ingredient. For this, often lyotropic phases can be exploited to trigger the release, for example through a change

of the pH [165], or even by light irradiation [166], where the release is started by molecular switches. For further information, we refer to some relevant reviews [167,168] and citations therein.

Since the seminal work of Abbott and co-workers [169], liquid crystal sensors generally employ a molecularly triggered texture transition of a thermotropic liquid crystal from a dark to bright state (or vice versa) that indicates the absorption of a number of liquid, gas, or biological molecules within the liquid crystal [170]. But, there have also been some reports of lyotropic LC being used for sensing, for example chromonics for biological sensing applications [171], or lyotropic phases of DNA for enzymes [172] and other for antigens [173] and pathogens [174]. In general, particularly the fields of biologically relevant systems like biotechnology, biosensors, drug delivery and biomimetics [175] are of highlighted interest for lyotropic liquid crystals.

Further, the lyotropic phases have found their way into the templated synthesis and self-assembly for functional nanoparticles and nanotechnology. The use of nematic tobacco mosaic viruses has been suggested for the design of silica mesostructures. Exploiting the TMV as a template in the synthesis of inorganic frameworks with ordered porosity was demonstrated by Fowler et al. [176]. They described a method where ordered viruses were first silicated and then thermally removed via biodegradation. This produced silica structures with ordered nanochannels of 20 nm diameter. The method of using a lyotropic structure for the synthesis of nanoparticles [177] and nanostructured materials [178] was quickly picked up and used for a variety of different systems [179,180] (Figure 14a–c). For further information, one may refer to a review article by Hegmann et al. [181] and references therein.

**Figure 14.** Transmission electron microscopy images of silica rods synthesized by templating a lyotropic phase. The scale bars are 2 μm. Different aspect ratios, L/D can be obtained, for example (**a**) L/D ≈ 5, (**b**) L/D ≈ 3, (**c**) L/D ≈ 8, by running the synthesis for different times. The anisotropy is ascribed to the anisotropic supply of reactants, leading to rod-like growth (reproduced with permission from Reference [182]).

Lastly, one may want to mention some applications of dried cellulose nanocrystals, which firstly transfer liquid crystalline order onto dispersed functional nanomaterials and are subsequently dried to produce thin solid films. These can then be employed as chromonic films [183] due to the residual selective reflection of the cholesteric structure, or as plasmonic films with dispersed gold [184,185] or silver [186] nanorods or nanowires, respectively. From the discussed selected examples, it becomes clear that lyotropic liquid crystals have significant potential for future applications, likely to be most intensified in the areas of biotechnology and biomedicine.

## **3. Conclusions**

We hope that we could provide the reader with a short, yet hopefully interesting overview of some of the recent developments and novel trends in lyotropic liquid crystal research. There are certainly still a lot of unanswered fundamental questions in this field, materials developments to be explored and applications to be found, optimized and introduced to the market. This review of new trends in the field has hopefully sparked the interest of readers to explore some of the topics discussed in much more detail in the papers of this special issue, and maybe this issue will help in further bringing together the two fundamental fields of liquid crystal research, thermotropic and lyotropic systems, in the quest to develop an overarching general understanding of this fascinating aspect of soft matter.

**Author Contributions:** This review was written in equal parts by I.D. and A.M.F.N. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Review* **The Lyotropic Analog of the Polar SmC\* Phase**

## **Johanna R. Bruckner \* and Frank Giesselmann \***

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

**\*** Correspondence: johanna.bruckner@ipc.uni-stuttgart.de (J.R.B.); frank.giesselmann@ipc.uni-stuttgart.de (F.G.); Tel.: +49-711-685-64136 (J.R.B.); Tel.: +49-711-685-64460 (F.G.)

Received: 10 October 2019; Accepted: 25 October 2019; Published: 29 October 2019

**Abstract:** Only six years ago, the first clear-cut example of a ferroelectric, lyotropic liquid crystal was discovered. Since then, ongoing investigations in this new research field provided numerous instances of the missing pieces to complete the formerly blank picture of the lyotropic smectic C\* (SmC\*) phase. In this review we wanted to combine these new results and put them into a wider historical and scientific context. We start by giving an introduction about characteristic features of the well-known thermotropic SmC\* phase and why it is so difficult to find a lyotropic equivalent of this fascinating phase. After discussing early examples of achiral lyotropic and swollen SmC phases, we recap the discovery of the first lyotropic SmC\* phase. The molecular features necessary for its formation and its properties are analyzed. We place special emphasis on discussing the long-range orientational order of the tilt direction and the corresponding chirality effects. By comparing these exceptional features with thermotropic and swollen SmC\* phases, we aim to improve not only the understanding of the lyotropic SmC\* phase, but also of the relationship between thermotropic and lyotropic systems in general.

**Keywords:** lyotropic liquid crystals; SmC\* phase; chirality; ferroelectricity; hydrogen bonds; hydration forces

## **1. Introduction**

In the present article, the discovery of the lyotropic analogue of the thermotropic polar smectic C\* phase (lyo-SmC\*), its properties and the prerequisites for its formation are reviewed. We cover the very first examples of fluid and tilted lyotropic lamellar phases and the latest developments in the newly evolved research field. In order to illuminate the special status of the lyo-SmC\* phase among the lyotropic liquid crystal phases, we begin this review with a short recapitulation of the unique features of the thermotropic polar and ferroelectric SmC\* phase, and discuss the obstacles to overcome on the way to finding lyotropic analogues of the former.

## *1.1. Ferroelectricity in Liquid Crystals*

The term ferroelectricity is used in analogy to ferromagnetism and describes the property of certain dielectric materials to have a spontaneous electric polarization, the direction of which can be changed—in most cases reversed—by the action of an external electric field [1]. After the first discovery of ferroelectricity in Rochelle salt by the physicist Joseph Valasek in 1920 [2], it was long believed that ferroelectricity could only be found in solid crystals of low symmetry; namely, those crystals which belong to the ten polar crystallographic point groups (crystal classes) C1, Cs, C*<sup>n</sup>* and C*n*<sup>v</sup> with *n* = 2, 3, 4 and 6. In the 1970s, however, the Harvard physicist Robert B. Meyer realized that a thermotropic smectic C liquid crystal which is composed of chiral molecules has polar C2 symmetry, and might, thus, be the first example of a fluid medium with a spontaneous electric polarization [3,4]. Meyer's discovery initiated one of the most active fields in soft matter research ranging from ferro- and antiferroelectric liquid crystals to the plethora of polar liquid crystal structures formed by bent-core mesogens [5–7].

The structures of the thermotropic fluid smectic liquid crystals, smectic A (SmA) and smectic C (SmC), can be considered as 1D-periodic stacks of 2D-fluid layers formed by elongated molecules, the long axes of which are orientationally ordered along the liquid crystal director **n** (Figure 1). While **n** is parallel to the smectic layer normal **k** (the stacking direction) in SmA (Figure 1a), the director **n** is tilted in SmC by the tilt angle θ with respect to **k** (Figure 1b). The tilt of the director breaks the full rotational symmetry of the uniaxial SmA phase around **k** and makes the SmC phase biaxial. In SmC, the principal symmetry axis is the C2-axis normal to the tilt plane (the plane spanned by **k** and **n**; see Figure 2). In addition, the tilt plane itself is a mirror plane. The non-chiral SmC structure, thus, belongs to the non-polar point group C2h. Chirality, however, excludes the presence of any mirror planes, such that the symmetry of the chiral SmC phase (SmC\*) is reduced to the polar point group C2 [5].

**Figure 1.** Structures of the fluid smectic phases (**a**) smectic A (SmA) and (**b**) smectic C (SmC). Both phases are 1D stacks of 2D fluid layers formed by orientationally ordered mesogenic molecules. In SmA, the director **n** (the mean direction of the long molecular axes) is parallel to the smectic layer normal **k** (the stacking direction of the layers). In SmC, **n** is tilted with respect to **k** by the director tilt angle θ.

The absence of a mirror plane normal to the C2 axis makes this axis a polar axis, in the direction of which the presence of vectorial properties, such as a spontaneous electric polarization **Ps**, is allowed by symmetry. Since the polar C2 axis is normal to both the director **n** and the layer normal **k**, the direction of **Ps** is expressed by [8]:

$$\mathbf{P\_{\theta}} \propto \mathbf{k} \times \mathbf{n},\tag{1}$$

which implies that the magnitude of spontaneous polarization *P*<sup>s</sup> = |**Ps**| increases with the director tilt θ as *P*<sup>s</sup> ∝ sinθ ≈ θ in the first approximation. In conclusion, both the direction and the magnitude of **Ps** depend on the direction and magnitude of tilt. This is known as the polarization-tilt coupling in chiral smectic liquid crystals.

The point group symmetries D∞<sup>h</sup> of non-chiral SmA and D<sup>∞</sup> of chiral SmA\* are too high to allow ferroelectricity and a spontaneous polarization in any direction. In SmA\*, however, an electric polarization induced by an electric field **E** along the smectic layers is linearly coupled to an induced director tilt δθ(*E*) via the polarization-tilt coupling. This so-called electroclinic effect increases in amplitude towards the transition temperature *T*<sup>c</sup> from the SmA\* into the SmC\* phase in a Curie–Weiss-like anomaly [9–11].

**Figure 2.** Symmetries of (**a**) smectic C (SmC) and (**b**) chiral smectic C (SmC\*) liquid crystals. The director **n** is tilted with respect to the smectic layer normal **k** by the tilt angle θ. The tilt direction is indicated by the **c** director and specified by the azimuth angle ϕ. The directions of **k** and **n** define the tilt plane. Normal to the tilt plane, we find a twofold symmetry axis C2, since the directions of +**n** and +**k** are physically equivalent to the directions of −**n** and −**k**, respectively. In addition, the tilt plane is a mirror plane in the case of non-chiral SmC. Both symmetry elements combine to the non-polar point group C2h. In the presence of chiral molecules, however, mirror symmetry is expelled and the C2 axis remains the only symmetry element. Thus, the chiral SmC\* phase has polar C2 symmetry, and a spontaneous polarization vector **Ps** along the C2 axis is allowed by symmetry.

In addition to the spontaneous electric polarization of each smectic layer, chiral smectic C forms a helical superstructure in such a way that the tilt direction—and thus the direction of **Ps** as well—slightly twists from layer to layer along the layer normal **k** (Figure 3). The pitch *p* of the SmC\* helix is typically in the order of several microns and is, therefore, several orders of magnitude larger than the thickness of the smectic layers *d* which are typically in the range of a few nanometers. As a result of this helical superstructure, the spontaneous polarization is macroscopically cancelled out over a full pitch length.

**Figure 3.** Helical superstructure of a chiral smectic C (SmC\*) phase depicted with hard rods (top) and vectors (bottom). The director **n** (black) and the spontaneous polarization **Ps** (red) are twisted along the direction of the layer normal **k** (green) from layer to layer with a helical pitch *p* (blue). The latter is typically in the micron range, and thus, several orders of magnitude larger than the layer spacing *d*.

In 1980, Noel A. Clark and Sven T. Lagerwall discovered that the SmC\* helix formation can be easily suppressed if the phase is confined between two planarly aligning glass plates, the distance between which is less or similar to the SmC\* pitch *p* (Figure 4a) [12]. In this so-called surface stabilized ferroelectric liquid crystal (SSFLC) configuration, the SmC\* phase behaves very much like a classic ferroelectric material. The SSFLC film forms two kinds of domains with opposite tilt directions parallel to the glass surfaces (Figure 4). As a result of the polarization-tilt coupling, these two kinds of tilt domains also have opposite directions of **Ps** and are, therefore, the liquid crystal equivalents to the ferroelectric domains in solid ferroelelectrics. An electric field **E** is applied across the cell switches' tilt directions such that **Ps** is parallel to **E** (Figure 4b). Observed between crossed polarizers, the field-induced reversal of the SmC\* tilt direction gives rise to a very fast and bistable electro-optic effect which attracted tremendous interest for future display applications and initiated a boost of research in the field of ferroelectrics (FLCs), and later, the antiferroelectric (AFLC) liquid crystal field as well [8]. Today, hundreds of thermotropic FLC and AFLC materials are known.

**Figure 4.** Surface stabilized ferroelectric liquid crystals (SSFLCs). (**a**) SmC\* phase confined between two glass plates (with conductive indium tin oxide (ITO) electrodes) of a liquid crystal cell which aligns the SmC\* director parallel to the glass surfaces. All possible tilt directions of the SmC\* director are represented by the so-called tilt cone. Only the two tilt directions parallel to the glass surfaces meet the surface anchoring condition, and thus, the helical SmC\* structure is suppressed if the gap between the glass plates is sufficiently small. The two possible tilt directions (±y) are coupled to opposite directions of the spontaneous polarization **Ps** (up and down). (**b**) The application of an electric field **E** across the cell reorients the **Ps** vectors of all domains into the direction of **E**. The field-induced reversal of **Ps** also reverses the tilt direction of the SmC\* director, which gives rise to a fast electro-optic effect if seen between crossed polarizers. (**c**) In the polarizing optical microscope, the ferroelectric domain structure of the virgin SSFLC configuration is seen between crossed polarizers as an array of bright and dark domains with opposite tilt directions, and thus opposite **Ps** directions. (Reprinted with permission (**a**) from Springer Nature: [13], redrawn after [12]; (**b**,**c**) from A. Bogner: [14].)

Two examples of thermotropic FLC materials are shown in Figure 5. The chiral Schiff base (*S*)-*p*-(*n*-decyloxybenzylidene)-*p*-amino-(2-methylbutyl) cinnamate, code named DOBAMBC, shown in Figure 5a, was actually the first thermotropic SmC\* material which was recognized as a ferroelectric liquid crystal [3] and much of the pioneering work was done with this material. DOBAMBC undergoes a second-order transition from the paraelectric SmA\* to the ferroelectric SmC\* phase below *T*AC = 95 ◦C. Below *T*AC the director tilt angle θ and the spontaneous polarization **Ps** continuously increase from zero and reach far below *T*AC values of 28◦ and 6 nC cm−2, respectively (Figure 5a) [15]. Even though many FLCs have second-order SmA\*–SmC\* transitions like DOBAMBC, materials with first-order ferroelectric transitions are also known. The first example was the chiral biphenyl (*S*,*S*)-4-(3-methyl-2-chloropentanoyloxy)-4 -heptyloxybiphenyl, code named C7 (Figure 5b) [16]. At the transition temperature *T*AC = 55◦C the paraelectric SmA\* phase coexists with the SmC\* phase which has 15◦ tilt and 100 nC cm−<sup>2</sup> polarization at the transition point. In the SmC\* phase below, *T*AC tilt and polarization reach values of up to 23◦ and 300 nC cm−2, respectively (Figure 5b). In the (*T*,*E*)-phase diagram, the first-order SmA\*–SmC\* transition terminates under the action of a strong electric field *E* at a critical point [17].

**Figure 5.** Two examples of ferroelectric SmC\* materials. (**a**) The chiral Schiff base DOMBAMBC has a second-order SmA\*–SmC\* transition at the transition temperature *T*AC = 95 ◦C. At increasing temperatures *T* < *T*AC, the director tilt angle θ and the spontaneous electric polarization *P*<sup>s</sup> continuously decrease to zero values at *T* = *T*AC. At that critical point, the SmC\* phase is identical to SmA\*. (Data taken from [16].) (**b**) The chiral biphenyl C7 has a first-order SmA\*–SmC\* transition. At TAC, the paraelectric SmA\* phase coexists with a ferroelectric SmC\* phase with non-zero θ and *P*s. Values of θ and *P*<sup>s</sup> further increase in the SmC\* phase at decreasing temperatures below *T*AC. (Data taken from [17].)

## *1.2. The Challenge of a Lyotropic SmC\* Phase*

Most thermotropic liquid crystal phases find a counterpart with equivalent structure and same symmetry in the world of lyotropic liquid crystals. The lyotropic counterpart of the thermotropic SmA phase, for instance, is the well-known lamellar α-phase (Lα) which consists of 2D-fluid surfactant bilayers which are separated from each other by fluid solvent layers (Figure 6a). An important exception, however, is the family of tilted fluid smectics. Even though SmC is among the most common phases found in thermotropics, the lyotropic equivalent to the SmC phase (Figure 6b) is almost unknown. To the best of our knowledge, only two examples of lyo-SmC phases and not a single example of a chiral lyo-SmC\* phase were clearly confirmed, e.g., by 2D X-ray diffraction in the literature until 2013.

**Figure 6.** Lyotropic equivalents to the thermotropic fluid smectic phases SmA and SmC. (**a**) The lamellar α-phase (Lα) is a 1D-stack of 2D-fluid surfactant bilayers (lamellae) which are separated from each other by interlamellar solvent layers. Since the director **n** is parallel to the bilayer normal **k**, Lα is the lyotropic equivalent to thermotropic SmA. (**b**) The lyotropic equivalent to the thermotropic SmC phase should have the same fundamental structure as Lα except that the surfactant molecules are tilted by the tilt angle θ into the same direction in all bilayers (synclinic tilt correlation). (Reprinted by permission from Springer Nature: [13].)

We believe that this obvious dissymmetry between thermotropic and lyotropic phases is mainly related to the issue of long-range correlations of tilt directions in a lyotropic medium on different levels:


In view of these general requirements it seems understandable that the counterparts of the thermotropic SmC and SmC\* phases are very rare in lyotropic liquid crystals.

## **2. Examples of Swollen Thermotropic and Lyotropic SmC Phases**

In general, SmC phases containing solvent can be distinguished into one of three categories: the swollen, the hyper-swollen and the true lyotropic SmC phases. In the following, examples for each type and their properties are reviewed, briefly. Regrettably, there are not a lot of examples for hyper-swollen or lyotropic SmC phases, and frequently, they are not investigated in detail. This leaves a big gap of knowledge about the properties of the hyper-swollen, and especially, lyotropic SmC phase and its correlation with the conventional thermotropic SmC phase.

The simplest way to obtain a lyotropic SmC phase, is by mixing an amphiphilic mesogen, which exhibits a SmC phase in the neat state already, with a protic solvent. However, in most cases, the tilted phase will only be stable up to a few weight percent of solvent [19–21]. Instead it is replaced by the Lα phase in which there is no macroscopic tilt angle. These phases are called swollen SmC phases. Typical examples for such swollen SmC phases are shown in Figure 7. Due to the strongly destabilizing impact of the solvent, they are normally not investigated further.

**Figure 7.** (**a**) Molecular constitution of a calamitic mesogen which incorporates a rigid biphenyl core and a diol moiety, the latter providing solubility in protic solvents. For (**b**) n = 12 and m = 3 and for (**c**) n = 10 and m = 1, the amphiphiles show a swollen SmC phase, denoted with SC in the phase diagram. In neither case, is the phase stable with mass fractions of formamide larger than approximately 15 wt%. (Phase diagrams reprinted with permission from Spie [19], and [20]. Copyright 1998 American Chemical Society.)

Thermotropic SmC phases, which can take up considerable amounts of solvent, are the so-called hyper-swollen phases. Kanie et al. [22] discovered that combining a phospholipid with an aromatic cyano azobenzene unit produces an SmC phase, which tolerates a water amount of up to 50 wt% before transforming into the orthogonal Lα phase. This observation was, however, solely based on the detection of characteristic textures; i.e., oily streaks and Schlieren textures.

Tan et al. [23] reported in 2015 that another azobenzene mesogen, this time combined with a hydrophilic polyethylene glycol tail, showed a thermotropic and a lyotropic SmC phase in mixtures with water, formamide and ethylene glycol. They investigated the system by both POM and XRD. For ethylene glycol, the maximum solvent uptake before phase separation was found to be about 50 wt%. The phase diagram shows, that the SmC phase is even stabilized by the addition of ethylene glycol, increasing the thermal phase width from 13 to 18 K. While the molecule behaves similar in mixtures with formamide, the SmC phase is reduced to a thermal width of only 2 K in a water saturated mixture. Investigations of the layer spacing at different concentrations of ethylene glycol show the typical behavior known from thermotropic SmA to SmC phase transitions with an increase of the layer spacing in Lα phase, due to the decreasing orientational order at increasing temperature, and a decrease of the layer spacing in the SmC phase which is connected to the increasing tilt angle. Most remarkable, is that the layer spacing at the phase transition temperature is increased from roughly 5.4 nm in the neat state to 11.7 nm in the ethylene glycol saturated state, giving evidence that a considerable solvent layer is formed between the amphiphile bilayers.

In the same year, an interesting report about hyper-swollen SmA and SmC phases was published by Murase et al. [24]. Once again, the mesogen included a rigid aromatic core and showed amphiphilic behavior. However, contrary to the examples discussed before, the mesogen had no hydrophilic, but instead, a fluorophilic tail, as shown in Figure 8a. Thus, mixtures of the amphiphile with three different perflourinated solvents (e.g., Figure 8b) were investigated. Phase diagrams, such as the one in Figure 8c, were recorded by observation of characteristic textures; i.e., the dark homeotropic texture of the SmA or Lα phase with a few defects and the Schlieren texture of the SmC phase. In all cases, the SmC phase got destabilized by the addition of the perfluorinated solvent, but still appeared in a narrow temperature range at solvent mass fractions higher than 50 wt%. They investigated the temperature-dependent layer spacing for different concentrations and found striking differences for the three solvents. Mixtures of the mesogen with elongated solvents, as shown in Figure 8b, exhibit an almost monotonic increase of the layer spacing with increasing solvent concentration. Within the concentration range of the SmC phase, the layer spacing is roughly tripled in dimension. In contrast to this, mixtures of the more spherically shaped solvent perfluorodecalin did not show a substantial increase of the layer spacing with increasing concentration of the solvent. The authors showed by careful processing of their X-ray data, that the elongated solvent molecules were incorporated in between the partial bilayers of the amphiphiles, while in the perfluorodecalin molecules were localized within the amphiphile bilayers. This case does not meet the common understanding of a lamellar phase, while the others do. Furthermore, the authors speculated about possible mechanisms for the long-range correlation of the tilt across the solvent layers, but did not find a conclusive explanation.

**Figure 8.** (**a**) Chemical structure of the mesogenic perflourinated cyano biphenyl. (**b**) Examples of one of the perflourinated solvents used, (**c**) the corresponding phase diagram and (**d**) the temperature-dependent layer spacing *d*(*T*). The mesogen to solvent ration in (**d**) gradually changes from 1:2 (filled triangles) to 2:1 (filled circles). The layer spacing of the neat mesogen in depicted with open circles. (Reprinted by permission from RCS Publishing: [24].)

Finally, "true" lyotropic SmC-phases, which are formed only in the presence of a solvent, and thus, do not exist in the neat state of the amphiphile, are the least common case. In this case, the presence of the solvent is a necessary condition for the formation of the tilted lamellar mesophase and not just a

destabilizing factor of a formerly thermotropic SmC phase, as in the cases of swollen and hyper-swollen smectics. Therefore, these truly lyotropic SmC phases are the most interesting to study and are in the focus of this review.

Until 2013, there were only two proven reports of truly lyotropic SmC phases. The first publication was by Schaheutle and Finkelmann and was published in 1988 [25]. They investigated a series of amphiphiles with rigid hydrophobic moieties and varying lengths of hydrophilic ethylene glycol units (see Figure 9a). Originally interested in the effects of packing constrains on the shape of micelles, they found that the amphiphiles formed lamellar mesophases only. In all three cases, SmC phases were observed in the presence of water, while there were no liquid crystalline phases in the neat states. They verified the structural analogy of this lyotropic phase with the known thermotropic SmC phase by X-ray diffraction measurements. In Figure 9b, a reprint of this measurement is shown. The two-dimensional diffraction pattern shows the typical features of a fluid, lamellar and tilted phase, i.e., diffuse scattering maxima in the wide angle regime, sharp layer peaks in the small angle region and an azimuth angle differing from 90◦ between the two of them. Even though the tilt angle measurable from this diffraction pattern is only a couple of degrees, it is clearly observable. Comparing amphiphiles with varying hydrophilic-hydrophobic balances, the authors found that the molecule with the shortest polyethylene glycol chain forms the most stable lyotropic SmC phase in a concentration regime between roughly 30 and 70 wt% of surfactant. By increasing the chain length, the concentration range, in which the tilted phase occurs, is diminished and shifted to higher mass fractions of surfactant. In Figure 9c the phase diagram of the investigated amphiphile with an intermediate chain length and water is shown as an example.

**Figure 9.** (**a**) A rigid aromatic core is combined with two polyethylene glycol units on each end, varying from n = 5 to n = 7. (**b**) Two-dimensional X-ray diffraction pattern of a magnetic-field aligned sample with 70 wt% percent of the amphiphile with n = 6 at 26 ◦C. The arrow indicates the direction of the magnetic field (H). The phase diagram corresponding to this system is shown in (**c**). The lyotropic SmC phase is denoted as 'SC' in this diagram; 'S' and 'D' stand either for a supposedly higher ordered smectic or dystetic (i.e., Lα) phase, respectively. (Reprinted by permission form Taylor and Francis Ltd.: [25].)

In spite of Schafheutle's and Finkelmann's outstanding discovery, it took slightly more than a decade until a further example of a truly lyotropic SmC phase was published. Ujiie and Yano [26] reported the occurrence of a lyotropic SmC phase in mixtures of an ionic amphiphile and water. The amphiphile incorporates a rigid hydrophobic azobenzene unit which is connected to a polyethylene

imine chain by a hexamethylene linker. To every imine moiety, a 2-hydroxy ethane group is attached, providing the necessary hydrophilicity. The molecule exhibits a thermotropic SmA phase at elevated temperatures. By adding water, the melting point of the system decreases significantly, but the SmA phase, which turns into a Lα phase, stays stable up to a mass fraction of surfactant as low as 20 wt%. At lower temperatures, in the concentration range between roughly 15 and 75 wt%, the lyotropic SmC phase occurs. The authors provide evidence of the lamellar and fluid nature of this phase by a one-dimensional X-ray diffraction pattern, and for the director tilt by a picture of the typical Schlieren texture.

In addition to those two cases, a further amphiphile was reported to show a true lyotropic SmC phase [27]. However, later investigations revealed that the supposed lyotropic SmC phase was in fact an oblique columnar phase [28].

In view of the very small number of lyotropic SmC phases reported, which only form in the presence of a solvent, one can easily understand why the search for a chiral variant of this rare phase took so long. On the one hand, the missing experience with the design of lyotropic SmC phases, and on the other hand, the more demanding synthesis of chiral components, made it quite difficult to find promising amphiphiles. Furthermore, the knowledge from the often separated research fields of lyotropic and thermotropic liquid crystals had to be combined, to handle the more complicated sample preparation of lyotropic systems and notice the characteristic features of a chiral lyotropic SmC\* phase.

## **3. The Recognition of a First Lyotropic SmC\* Phase**

In 2013, the discovery of a chiral, lamellar tilted and fluid phase—a true lyotropic SmC\* phase—was reported for the first time [29]. The phase is formed by the chiral amphiphile shown in Figure 10a. Without solvent, the amphiphile does not form any stable liquid crystal phase, but after the addition of water or formamide, several lyotropic mesophases appear (Figure 10b).

The two-dimensional X-ray diffraction pattern of a magnetically aligned sample with 64 wt% water which is depicted in Figure 10c, proves without doubt that this is truly a structural equivalent to the thermotropic SmC or SmC\* phase. The sharp, Bragg-like peaks of 1st and 2nd order in the small angle region clearly show that the phase is lamellar; the diffuse maxima in the wide angle region attest that it is fluid; and the azimuth angle between the two of them deviates by 90◦, confirming that the surfactant molecules within the fluid bilayers are tilted with respect to the layer normal **k**. Evidently, the tilt directions are long-range correlated, at least over the macroscopic length scale of the scattering volume. A distinction between the chiral and achiral SmC phase is not possible from X-ray diffraction alone.

Furthermore, the lyotropic SmC\* phase exhibits the same characteristic textures as those known from its thermotropic counterpart. In Figure 11, striking examples of this similarity in terms of textures are given. The Schlieren texture in Figure 11a underlines that the phase is biaxial and the zigzag defects [30] in Figure 11b indicate a substantial layer shrinkage at the Lα to lyo-SmC\* phase transition. Most remarkable is that the lyo-SmC\* phase can be surface stabilized (Figure 11c,d), just as the thermotropic SmC\* phase (cf. Figure 4c). The rotation at about two times the optical tilt angle, exchanges the brightness of the tilt domains between crossed polarizers.

**Figure 10.** The chiral diol depicted in (**a**) forms true lyotropic SmC\* phases in mixtures with water and formamide. (**b**) The phase diagram with water exhibits a cholesteric (N\*), two different columnar phases (Col1 and Col2), a L<sup>α</sup> phase and the lyotropic SmC\* phase. (**c**) A two-dimensional X-ray diffraction pattern of the aligned phase provides clear evidence that the probed lyotropic phase possesses a structure analogous to the thermotropic SmC\* phase. (Phase diagram reprinted by permission from John Wiley and Sons: [29]; X-ray pattern reprinted by permission from Springer Nature: [13].)

**Figure 11.** Typical textures of the lyotropic SmC\* phase observed by polarized optical microscopy. In thicker samples (**a**) Schlieren textures and (**b**) broken fans with zigzag defects appear. (**c**) In the surface stabilized state, bright and dark domains in the range of several hundred micrometers with opposite tilt directions can be observed. (**d**) By rotating the sample between crossed polarizers (P and A), the brightness of the tilt domains reverses. (Reprinted by permission from John Wiley and Sons: [29].)

Even more impressive than the long-range correlation of the director tilt, is that even the subtler precession of the tilt direction along the layer normal **k** is transmitted across the interlamellar solvent layers, resulting in a macroscopically observable chirality. This helical director modulation causes a striped pattern, as exemplarily shown in Figure 12a for a sample with 32 wt% of formamide. The distance between two of these so-called pitch lines is equal to the full pitch, the magnitude of which is in the same order as in thermotropic SmC\* phases. By plotting the pitch versus the temperature,

(Figure 12b) an exponential increase of the values is observed when approaching the Lα phase. This is again, a typical behavior known from the thermotropic SmC\* phase [31–33].

**Figure 12.** The striped texture shown in (**a**) originates from a helical precession of the tilt direction along the layer normal **k** which is depicted in the inset of (**b**). The distance necessary for a full rotation of the director corresponds to the helical pitch *p*, which is plotted versus the temperature *T* in (**b**). (Reprinted by permission from John Wiley and Sons: [29].)

Measurements of the helical twist in dependence of the formamide mass fraction show a trend which is at the first glance counterintuitive, as the twist (the inverse pitch) increases with increasing solvent concentration (Figure 13). Osipov et al. [34] presented a possible explanation, as follows: The twisting is promoted by the chiral centers and counteracted by an elastic force, the effective elastic constant of which is known to be proportional to sin2θ. If we add solvent, the volume density of chiral centers decreases linearly, while the tilt angle decreases at least linearly with the solvent concentration (cf. Figure 19). In this case, the restoring force decays more strongly than the twisting power, and thus, an increasing twist is observed. Nevertheless, this explanation has still to be verified by further experiments.

**Figure 13.** Helical twist *p*−<sup>1</sup> of the lyo-SmC\* phase for different mass fractions of formamide measured by the Cano method [34]. While a strong increase of the twist was measured with increasing solvent content, no temperature dependence was observed. Most likely, the latter is due to a pinning effect between liquid crystalline phase and the glass surfaces of the measurement set up. (Reprinted by permission from Springer Nature: [13].)

The most prominent chirality effect in thermotropic SmC\* is the spontaneous electric polarization of its smectic layers (cf. Figure 3). So far it was not possible to directly measure this spontaneous polarization in a lyotropic SmC\* phase, since its current response to an electric field is dominated by its high electric conductivity. The relatively high conductivity of the lyotropic phase originates from residual ions which cannot be fully avoided in the solvent layers containing water (autoprotolysis) or formamide (hydrolysis to NH4 <sup>+</sup> and HCOO<sup>−</sup> [35,36]). Instead, the presence of a spontaneous polarization in lyo-SmC\* phase was indirectly confirmed by its polar electro-optic switching in the SSFLC state between crossed polarizers. Just as in the thermotropic case, the transmittance of the sample depends on the sign of the electric field (Figure 14), while any dielectric effect is proportional to *E*2, and thus, insensitive to the sign of *E* [29].

**Figure 14.** Polar electro-optic effect of a lyotropic SmC\* phase in the SSFLC geometry (**4** with 20 wt% formamide at 28 ◦C in a 1.6 μm thick cell). The directions of the spontaneous polarization and director tilt depend on the sign of the applied electric field. Therefore, the transmittance of the sample depends on the sign of *E*. Outside the electrode area (upper left corner) the tilt domain texture of the virgin sample can be seen.

## **4. Prerequisites and Properties of Lyotropic SmC\* Phases**

## *4.1. Lyotropic SmC\* Systems—A Delicate Balance*

Since and even before the discovery of the first example of a lyotropic SmC\* phase, there were constant research efforts—especially in and around the soft matter group in Stuttgart—to find a lyotropic analog of the thermotropic SmC\* phase. As discussed in Section 1.2, the first and fundamental challenge was to identify the structural features necessary on a molecular basis for the formation of the lyo-SmC phase. For this, the early examples—namely, the lyotropic systems from Schafheutel and Finkelmann [25], Ujiie and Yano [26] and Pietschmann et al. [27]—were used as templates, although the last example turned out not to form a lyotropic SmC phase later on [28]. All of these model systems share certain common features, which are:


Based on those observations, the amphiphiles listed in Table 1 were synthesized by Porada [28,29] and the Lemieux group [37–39]. All of the molecules incorporate a diol head group, a rigid aromatic core and a hydrophobic alkyl chain. Furthermore, the 1,2-diol head group introduces chirality to the molecules. Often, the amphiphiles only differ in the linker length between the diol group and the aromatic core, the number of oxygen atoms or the nature of the aromatic core. These slight variations allow a systematic analysis of the factors necessary for the formation of the lyotropic SmC\* phase.

The molecules **1** to **8** all possess a phenyl pyrimidine core which is a promoter for the formation of tilted phases in thermotropic liquid crystals [40,41], a hydrophobic heptyl chain or a diol head group. Variations were only made in the linker between the head group and the aromatic core. Amphiphile **4** is the one presented in Section 3 already. Next to water (cf. Figure 10b) it forms a lyotropic SmC\* phase with formamide, too. The phase diagram depicted in Figure 15, reveals that the same mesophases occur as found in mixtures with water. However, the N\*, the Col1, the Col2 and the lyo-SmC\* phases appear in a much more confined concentration range for the benefit of the Lα phase. While the lyo-SmC\* phase is stabilized between roughly 25 and 70 wt% of water, it appears in mixtures with formamide only between 7 and 25 wt% of formamide. This finding underlines the importance of the solvent for the stability of the tilted phase.

**Figure 15.** Phase diagram of the system **4**/formamide measured in heating. Next to the lyotropic SmC\* phase (SmC\* analog), a cholesteric (N\*), two columnar (Col1, Col2) and a rather pronounced lamellar Lα phase occur. (Reprinted by with permission from Springer Nature: [13].)

Elongating the linker by one ethylene glycol unit leads to **5**. Instead of a monotoropic N\* phase, this amphiphile forms a monotropic SmA\* phase in the neat state which is substantially stabilized by the addition of water (Figure 16(a-a)). Moreover, a lyotropic SmC\* phase forms, which was identified by the typical broken fan and schlieren textures (Figure 16(a-b–a-d)). No columnar or cholesteric phases occur in the phase diagram shown in Figure 16b. Another difference to the parent amphiphile **4** is that the lyotropic SmC\* phase is only stable in mixtures with water, but not with formamide. Furthermore, the phase occurs only between 10 to 20 wt% of water. Overall, the lyotropic SmC\* phase is destabilized in comparison to the parent systems with amphiphile **4**.



**Figure 16.** Mixtures of the amphiphile **5** with 19 wt% water show characteristic textures, as is known from thermotropic systems while cooling: (**a-a**) fan texture of the Lα\* phase, (**a-b**) broken fan texture of the lyo-SmC\* phase, (**a-c**) well defined tilt domains of the lyo-SmC\* phase at lower temperatures and (**a-d**) Schlieren texture of the lyo-SmC\* phase surrounded by the broken fan texture. (**b**) Phase diagram of **5** with water. (Reprinted by permission from the Royal Society of Chemistry: [37].)

If the linker consists of three ethylene glycol units (**6**), no lyotropic SmC\* phase is detected at all. Instead an enatiotropic mesophase—SmA\* phase—is formed in the neat state for the first time. Comparing the amphiphiles **4, 5** and **6** suggests that the length of the linker is essential for the formation of the lyotropic SmC\* phase. The linker in **4** seems to have the optimum length for stabilizing the lyo-SmC\* phase.

For the amphiphiles **1**, **2** and **3**, the ethylene glycol unit is replaced by a simple alkyl chain which is attached to the aromatic core by a single ether unit. Even though all three amphiphiles form a variety of lyotropic liquid crystal phases, they neither form any thermotropic phases nor a lyotropic SmC\* phase. Especially for **3**, this is remarkable, considering that the only difference to **4** is the exchange of an oxygen atom in the linker by a CH2-moiety. The change of the molecular length and the flexibility of the linker due to this alteration are negligible. Thus, the most reasonable explanation is, that the modification in the hydrophilicity—namely the ability of the oxygen atom in the linker to accept hydrogen bonds—has a major impact on the formation of the lyotropic SmC\* phase.

The amphiphile **8** is composed of the same alkyl tail, diol head group and linker as the amphiphile **4**. However, the aromatic core is inverted compared to the original molecule. This simple switch in the position of the pyrimidine and the phenyl rings has a major impact on the mesophase behavior of the amphiphile. In the solvent free state, **8** exhibits a SmA\* and a SmC\* phase. Both mesophases stay stable upon the addition of water or formamide up to roughly 50 wt%. No further lyotropic liquid crystal phases were detected.

The amphiphiles **9**–**12** all incorporate the same diol head group and linker as the amphiphile **5**. The phenyl pyrimidine core, however, is substituted by a 2,7-fluorenone core which is known to be an even stronger SmC-promoter in thermotropic smectics [42,43]. From **9** to **11**, the hydrophobic alkoxy tail is elongated. With the shortest hydrophobic chain—a butoxy chain (**9**), neither a thermotropic mesophase nor a lyotropic SmC\* phase forms. Elongating the chain by one methylene unit (**10**) leads to the appearance of a SmA\* phase in the neat state, but only the elongation by two carbon atoms (**11**) allows the formation of lyotropic SmC\* phase with water and formamide. The latter amphiphile already exhibits a SmC\* and a SmA\* phase in the solvent free state. The phase diagram of the system **11**/formamide is presented in Figure 17a and reveals that the thermal stability of the SmC\* phase is gradually decreased by the addition of formamide.

**Figure 17.** (**a**) Already in the neat state, the amphiphile **11** exhibits a SmA\* and a SmC\* phase. In mixtures with formamide, the orthogonal phase is stabilized while the tilted phase is destabilized, but remains present up to 25 wt% of solvent. (**b**) The amphiphile **12** shows only a monotropic lyo-SmC\* phase. Thus, the phase diagram measured while cooling is presented here. During heating, the temperature and concentration range in which the phase occurs is superposed by a two-phase region of crystalline and liquid crystalline phases. (Reprinted by permission from the Royal Society of Chemistry: [37].)

In the amphiphile **12**, one terminal hydrogen atom of the alkyl tail in **11** is replaced by a chlorine atom. The terminal chloro substituent, which favors orthogonal versus tilted phases [44], suppresses the thermotropic SmC\* phase and increases the thermal stability of the SmA\* phase. The lyotropic SmC\* phase is reduced to a monotropic phase. Thus, the phase diagram shown in Figure 17b was measured during cooling.

Concluding this section, it is obvious that every structural subunit of the amphiphiles plays an important role in the formation of the lyotropic SmC(\*) phase. Minor changes in the hydrophilic linker, the hydrophobic chain or the aromatic core easily lead to the appearance or disappearance of the tilted lamellar phase.

## *4.2. Structure and Phase Transitions of Lyotropic SmC\* Phases*

In Section 3 of this review, a two-dimensional X-ray pattern of the lyotropic SmC\* phase (cf. Figure 10c) proved that the structure of the lyotropic SmC\* phase is an analog to its thermotropic equivalent. To address the question of how the amount of solvent affects the structural parameters of the phase, temperature and concentration-dependent measurements of the system **4**/formamide are shown in Figure 18 [13]. For all mixtures investigated a temperature-dependent *d*(*T*) characteristic of thermotropic SmA(\*) to SmC(\*), phase transitions were found: In the Lα phase the lamellar repeat unit *d* increases with increasing temperatures due to the decreasing orientational order. The maximum is reached at the transition from the lamellar Lα to the lyotropic SmC\* phase. By further heating the lamellar repeat unit decreases again, which can be explained by an increasing value of the tilt angle.

A second observation is, that the addition of solvent shifts the *d*(*T*) curves to higher values but does not change their principal shape. To describe these observations in a more quantitative way, we assume that the lamellar repeat unit *d* is composed of two additive contributions:


The total repeat unit *d*(*T*,*w*) can, thus, be written as:

$$d(T, w) = d\_{\mathbb{S}}(w) + d\_{bl}(T),\tag{2}$$

with

$$d\_{\mathbb{S}}(w) = mw\_{\prime} \tag{3}$$

where the slope *m* denotes the increase in solvent layer thickness per wt% of solvent.

**Figure 18.** (**a**) Temperature and concentration-dependent measurements of the lamellar repeat unit *d* of the system **4**/formamide. The weight fraction *w* of formamide is given in the graph. Measured data points are depicted as symbols, while the lines represent fits according to the Equations (2) and (3). (**b**) Calculated thickness of the amphiphile bilayer *dbl(T)* of a hypothetic SmC\* phase in the neat state. (**c**) Calculated solvent layer thickness *ds(w)* for the different weight fractions *w* of formamide. (Reprinted by permission from Springer Nature: [13].)

The measured data points were fitted to these equations resulting in the continuous lines in Figure 18a. From that, the thickness of the pure bilayer *dbl*(*T*) (Figure 18b) and the thickness of the solvent layers *ds*(*T*) for the different mass fractions (Figure 18c) were separated from each other. The reasonable agreement of the measured data and the fits in Figure 18a imply that the solvent molecules form a separate layer between the amphiphile bilayers and do not penetrate them. Furthermore, it allows an estimation of the solvent layer thickness. For example, in a sample of **4** with 25 wt% of formamide, the solvent layer is roughly 2.3 nm thick, while the amphiphile bilayer ranges between 3.3 and 3.6 nm.

The foremost difference between Lα and the lyotropic SmC\* phase, is the collective tilt of the elongated molecules relative to the layer normal. The presence of the tilt was already verified by X-ray diffraction (cf. Figure 10c) and polarizing optical microscopy (cf. Figure 11c,d). In Figure 19 the optical tilt angle θopt of the system **4**/formamide is plotted against the relative temperature for different mass fraction of formamide [45]. For all samples investigated, the tilt angle seems to saturate at lower temperatures. This saturation value decreases with increasing mass fractions of formamide, from about 27◦ at 13 wt% to 12.5◦ at 25 wt% of formamide. Moreover, the course of the measured values in dependence of the temperature changes with the concentration. At high solvent concentrations, the tilt discontinuously drops to zero in a first order phase transition to the Lα phase, while the

transition is continuously second order at low concentrations [13]. This is a remarkable example of a solvent-induced change in the nature of a phase transition.

**Figure 19.** Tilt angle θ*opt* of the system **4**/formamide measured by rotating a surface stabilized sample between crossed polarizers. The values were plotted relative to the temperature *Ttr* of the phase transition from the high temperature phase to the lyotropic SmC\* phase. (Reprinted by permission from John Wiley and Sons: [45].)

A more detailed investigation of the phase transition from the SmC\* analog phase to the high temperature phase was done by differential scanning calorimetry (DSC) [13]. The heat flows measured while cooling and heating are displayed in Figure 20; the discussed transition enthalpies are highlighted in yellow. For the lowest solvent concentration measured in the system 4/formamide, the mesophase above the lyotropic SmC\* phase is a columnar phase. As already required by symmetry arguments, the heat flow measured confirms a 1st order phase transition. At formamide mass fractions between 12 and 22 wt%, the high temperature phase is the lamellar Lα phase. At 12 wt%, the peak is less pronounced than in the former case, but still suggests a phase transition close to 1st order. By increasing the formamide content further, the peak starts to flatten out more and more, only showing a step in the curves for 18 and 22 wt%. This suggests that the order of phase transition is shifted from 1st to 2nd with an increasing amount of solvent. With 27 wt% of formamide, only a non-tilted phase is formed.

From the measured tilt angles and lamellar repeat units, and the estimated thickness of the amphiphile bilayer and the solvent layer a structural model of the lyotropic SmC\* phase was sketched [13]. In Figure 21, a true to scale model of the lyotropic SmC\* phase of the system **4**/formamide with 19 wt% of formamide 10 K below the phase transition from the Lα phase is shown. As depicted, the amphiphile bilayers consist of partial bilayers with a significant interdigitation of the hydrophobic parts. The formamide is not mixed within the amphiphile bilayers but forms separate, well-defined solvent layers. Thus, the amphiphile molecules from different bilayers are not in direct contact with each other, which raises the question once again, of how the long-range correlation of the director tilt across the solvent layers takes place.

**Figure 20.** DSC curves obtained while heating (top) and cooling (bottom) the system **4**/formamide. The weight fraction of formamide is indicated in the graphs. The transition from the high temperature phase to the lyotropic SmC\* phase is highlighted in yellow. In the sample with the smallest amount of formamide the high temperature phase is a columnar (Col1) phase. For the samples with 12 to 22 wt% of formamide, it is a Lα phase. In case of the sample with the highest formamide concentration, the only mesophase occurring is a Lα phase (cf. Figure 15). (Reprinted by permission from Springer Nature: [13].)

**Figure 21.** Model of the lyotropic SmC\* phase for the system **4**/formamide. The model suggested by Bruckner et al. is based on their measurements shown in Figures 18 and 19 and the length *Lcalc.* of the amphiphile calculated by molecular modeling. The sketch corresponds to a sample with 19 wt% of formamide (•) at *T* − *Ttr* = −10 K. (Reprinted by permission from Springer Nature: [13].)

1

ρ

ε

## *4.3. Origin of the Director Tilt*

To learn more about the origin of the director tilt and its long-range correlation, the orientation of the C–N pyrimidine vibration which is directed along the long molecular axis of **4** was selectively probed by polarized micro-Raman spectroscopy [45]. For this a mixture of amphiphile **4**, 15 wt% of formamide was used. The mean direction of the C–N vibration is found to be tilted with respect to **k** and its tilt angle is in good agreement with the director tilt angle optically measured under the same conditions. Thus, the authors concluded that the origin of the tilt in the lyotropic SmC\* phase is a tilt of the aromatic 2-phenylpyrimidine core relative to the layer normal.

However, this still does not explain how the tilt direction is transmitted between adjacent bilayers. To clarify the long-range correlation of the director tilt, the authors made use of an observation mentioned in the Dection 4.1 already: the appearance or disappearance of the lyotropic SmC\* phase by simply changing the solvent. Thus, different solvents were tested for their ability to stabilize the lyotropic SmC\* phase. The solvents and some of their physical data are listed in Table 2.


**Table 2.** Physical data of the solvents which were tested for the formation of a lyotropic SmC\* with the amphiphile **4**. (Adapted from [13].)

<sup>1</sup> Calculated according to n<sup>ρ</sup> = ρ·*N*A/*M*.

As discussed before, the amphiphile **4** forms a lyotropic SmC\* phase in mixtures with water and formamide. The corresponding phase diagrams are depicted in Figure 22a,b once again, for comparability. By changing the solvent from water to formamide, the concentration and temperature range in which the tilted phase occurs is reduced significantly. Exchanging the solvent to *N*-methyl formamide (NMF), leads to the disappearance of the lyotropic SmC\* phase and the two columnar phases (Figure 22c). Moreover, the stability of the lamellar Lα phase is reduced with increasing mass fraction of NMF. When the solvent is replaced by *N*,*N*-dimethyl formamide (DMF), only a N\* phase appears (Figure 22d).

Neither the dielectric permittivities, nor the dipole moments listed in Table 2 may explain this trend. Therefore, pure electrostatic interactions between adjacent amphiphile bilayers cannot explain the long-range correlation of the director tilt in an obvious way. A quantity which follows the trend observed experimentally is the number of hydrogen bond donor atoms. Since all four solvents can accept up to two hydrogen bonds, water and formamide are known to form extended hydrogen bond networks in the liquid state, while NMF only forms chains of hydrogen bonds and DMF no hydrogen bonds with itself. Thus, the authors conclude, that anisotropic hydration interactions which are mediated by a dense three dimensional hydrogen bond network might be responsible for the correlation of the tilt direction in adjacent bilayers [45]. In general, hydration forces have already been known for phospholipids forming lyotropic lamellar phases [48–54], but have never been observed in combination with lamellar, fluid and tilted phases. Several studies show that the orientational ordering of water molecules by the phospholipid head groups can bridge a distance of up to 1 nm [55–60].

*Crystals* **2019**, *9*, 568

**Figure 22.** Phase diagrams of the amphiphile **4** with (**a**) water, (**b**) formamide, (**c**) *N*-methylformamide and (**d**) *N*,*N*,-dimethylformamide. The ability of the solvent to stabilize lyotropic mesophases in general—and the lyotropic SmC\* phase in particular—is diminished in the order of presentation. For the system **4**/DMF, only a schematic phase diagram is presented, which was derived by investigation of a contact sample. (Reprinted by permission from John Wiley and Sons: [45].)

Solvents other than water and formamide which hold two hydrogen bond donor atoms are ethylene glycol and its polymers (PEG). Consequently, these solvents were investigated too [13]. In mixtures of amphiphile **4** and ethylene glycol, two columnar mesophases a N\* and a Lα phase appear. Yet, a lyotropic SmC\* phase is not stabilized. In mixtures with PEG 200 or PEG 300, the only stable mesophase is a cholesteric phase next to a monotropic Lα phase. In conclusion, the number of hydrogen bonds per solvent molecule cannot be the sole determining factor for the stability of the lyotropic SmC\* phase. A second important factor is the density of the hydrogen bond network which is directly related to the number densities of the solvent molecules (Table 2) and the average number of hydrogen bonds per molecule.

To underline the importance of the hydrogen bond network further, the tilt angle in mixtures with ordinary formamide and deuterated formamide were compared [45], since deuteration selectively modifies the strength and dynamics of the hydrogen bond network. The result is shown in Figure 23. Even though the investigated mole fractions of formamide were the same, the saturation value of the sample with deuterated solvent was roughly 15% lower than in the equivalent sample with ordinary formamide. This is indeed, a quite substantial isotope effect which clearly confirms that the correlation mechanism is sensitive to the structure and dynamics of the hydrogen bond network in the solvent layers.

**Figure 23.** Optically measured tilt angle of the system **4**/formamide (circles) and **4**/deuterated formamide (squares) at the same mole fraction relative to the phase transition temperature *T*tr from the Lα phase. (Reprinted by permission from John Wiley and Sons: [45].)

An example which seems to contradict the importance of a strong hydrogen bond network for the long-range correlation of the tilt direction across the solvent layers, is the case of the hyper-swollen perfluorinated smectics [24] (cf. Section 2). The perfluorinated oils which are used for swelling do not form any hydrogen bonds at all. Nonetheless, the hyper-swollen SmC phase stays stable up to an oil content of 60 wt%. We suggest that the mechanism for the long-range tilt correlation with perfluorinated solvents is completely different. Due to the size of the fluorine atoms, a free rotation of the perfluorinated carbon chains is hindered [61,62]. Therefore, perfluorinated polymers are rather stiff and possess a significantly higher persistence length than unfluorinated polymers [63–65]. Therefore, we propose, that the two elongated perfluorinated polymeric oils (cf. Figure 8b) might possess enough rigidity to transmit the tilt direction sterically. This of course, has to be confirmed by experiments and/or simulations.

## *4.4. Electroclinic E*ff*ect*

As we previously discussed in Section 1.1, the D∞-symmetry of a chiral SmA\* phase excludes polar properties, such as a spontaneous polarization. The polarization-tilt coupling, however—present in all chiral smectics—gives rise to the so-called electroclinic effect [9,10], the properties of which resemble, in some aspects, the (inverse) piezoelectricity in non-centrosymmetric solid crystals. An electric field **E** along the SmA\* layers induces an electric polarization **P** in a non-parallel direction to the layer normal **k**. In a chiral medium, the plane spanned by **k** and **P** is no mirror plane. As a result of **E**, the free energy and the distribution of molecular tilt directions are no longer symmetric about the **k**,**P**-plane and the macroscopic director **n** is, therefore, observed to tilt away from **k** in a direction normal to the **k**,**P**-plane. Similar to the field-induced mechanical strain in the inverse piezoelectric effect, the field-induced tilt angle δθ is linear in *E* and the direction of tilt reverses upon reversal of the field direction [9–11].

According to these symmetry arguments, the electroclinic effect must be present in all SmA\* phases even though the effect might be very small. Indeed, a measurable electroclinic effect was only observed in the pretransitional regime of a second-order (or weak first-order) transition from

the SmA\* into the tilted SmC\* phase. In that temperature regime, the electroclinically induced tilt δθ grows hyperbolically towards the critical temperature *T*<sup>c</sup> ≤ *T*AC according to a Curie–Weiss-like behavior [10,11,66].

Since the symmetry arguments leading to the electroclinic effect in SmA\* apply in the very same way to the lyotropic case of a chiral lamellar α-phase, there were several attempts to detect an electroclinic response in Lα\*. A first indication was the observation by Jakli et al. that the chirality of phospholipids makes fluid lamellar phases piezoelectric [67,68]. The discovery of the lyo-SmC\* phase however sheds new light on this issue: in 2017 Harjung et al. observed an electroclinic response in the Lα\* phase of the amphiphile 4 with 23 wt% formamide at temperatures close to its transition into the newly discovered lyo-SmC\* phase [69]. The samples were poured into a liquid crystal cell and a square-wave electric field was applied to its transparent ITO electrodes (Figure 24a). The field *E*(*t*) electroclinically induced an alternating tilt δθ(*t*), which gave rise to an essentially square-wave modulation *I*(*t*) of the light intensity transmitted through the cell between crossed polarizers (Figure 24b). The electroclinic electro-optic response completely vanished in the case of the corresponding non-chiral Lα phase composed of the racemic version of the amphiphile **4**. At least beyond a certain threshold field in the order of 1 V/μm, the electroclinic tilt increased linearly with the field amplitude (Figure 25a) and grew hyperbolically if the temperature was lowered towards the transition temperature into the tilted ferroelectric lyo-SmC\* phase (Figure 25b).

**Figure 24.** Electroclinic electro-optic effect of a chiral lamellar α-phase in the vicinity to its transition into a tilted lyo-SmC\* phase. (**a**) Square-wave electric field E(*t*) applied to lyotropic Lα( \*) samples in a direction normal to the layer normal **k**. (**b**) Corresponding electro-optic response of the lyotropic samples between crossed polarizers measured by the transmitted light intensity *I*(*t*) at a temperature of 0.2 K above the transition into the tilted phase; black line: chiral Lα\* phase of **4** with 23 wt% of formamide; red line: nonchiral Lα phase of racemic **4** with 23 wt% of formamide. (Reprinted by permission from [69] Copyright 2018 by the American Physical Society.)

The electroclinic effect in the Lα\* phase, thus shows all signatures of the electroclinic effect in thermotropic SmA\* phases, namely the effect (i) is chiral in nature, (ii) is essentially linear in the sign and magnitude of the electric field, and (iii) shows Curie–Weiss-like behavior in the pretransitional regime of a tilting transition into lyo-SmC\* [69]. Beyond these striking similarities between the electroclinic effects in SmA\* and Lα\* there are also specific deviations, namely, the slow decay of the electro-optic response in Figure 24b, after switching and the non-linearity at low field strength in Figure 24a. The experiments in [69] indicate that these deviations are related to the comparatively high electric conductivity of lamellar phases which originates from the inevitable presence of ionic impurities in the highly polar and protic solvent layers. Under the action of an electric field, these ionic impurities accumulate at the interfaces between the solid electrodes and the liquid-crystalline electrolyte and form electric double layers which screen the external electric field, and thus reduce the effective field inside the liquid crystal layer. Since the electroclinic effect probes the effective field inside the lyotropic phase,

the screening effect of electric double layers considerably complicates the dynamics of the electroclinic effect in Lα\* phases.

**Figure 25.** Signatures of the electroclinic effect in the Lα\* phase of the chiral amphiphile **4** with 23 wt.% of formamide. (**a**) Electroclinically induced tilt angle δθ versus amplitude *E* of a 400 Hz square-wave electric field at a fixed temperature of 0.5 K above the transition temperature into the lyo-SmC\* phase. The straight line shows the linear regime of δθ(*E*). (**b**) δθ versus temperature *T* at a fixed amplitude *E* = 1.9 V/μm of the 400 Hz square-wave electric field. The solid line shows the Curie–Weiss-like hyperbolic divergence of δθ(*T*). (Reprinted by permission from [69] Copyright 2018 by the American Physical Society.)

All in all, 40 years after the discovery of the electroclinic effect in thermotropic SmA\* it has now become clear that the same effect exists in chiral lyotropic Lα\* phases as well. Since phospholipid cell membranes are in a fluid Lα\* state, this observation might also have certain implications in biophysics.

## **5. Conclusions and Outlook**

After the discovery of ferroelectricity in thermotropic liquid crystals in the 1970s, it has become evidently clear, in recent years, that the lyotropic equivalent to the thermotropic ferroelectric smectic C\* phase indeed exists. The new lyo-SmC\* phase shares many characteristic properties of its thermotropic counterpart:


Beyond these convincing similarities, the new lyotropic SmC\* phase is clearly distinguished from its thermotropic counterpart by the presence of water or water-like solvent layers between the bilayers of amphiphilic mesogens. This raises a lot of new questions which all address the specific role of the solvent. In comparison to thermotropic SmC\*, the solvent concentration is an additional thermodynamic degree of freedom, and thus, an additional "control variable" in lyo-SmC\*. Exploring how the solvent concetration changes properties, such as order parameters, elasticity, viscosity, pitch and polarization, will certainly involve new physics and lead to an improved understanding of interactions in lyotropic systems. In view of the surprising long-range tilt correlation in lyo-SmC\*, the most challenging question is how not only the elastic interactions (tilt) but also the much weaker chiral interactions (twist) are transmitted through the interlamellar solvent layers. Evidently these interactions do transform the disordered solvent into partially ordered solvent. The further investigation of this subtle order is a most interesting subject and key to the detailed understanding of lyotropic SmC\* phases. Even though there are already strong experimental indications that hydrogen bond networks in the solvent layers play an important role in the interlamellar correlation of tilt directions, more experimental and theoretical work is needed to really understand how elastic and chiral interactions are "communicated" across nanoscopic solvent layers. This might also be helpful in our further understanding of chirality effects in other soft matter and biological systems.

The tailored design of new lyo-SmC\* phases remains a non-trivial task, since it relies on a detailed balance between hydrophilic, hydrophobic and tilt-promoting molecular interactions. As far as we know today, the formation of lyo-SmC\* phases requires, on the one hand, protic water-like solvents which are able to form extended hydrogen bond networks, and on the other hand, rod-shaped amphiphiles terminated by a hydrogen-bonding hydrophilic head group that is able to connect to the hydrogen bond network of the solvent. The molecular structure of the amphiphile further contains a rigid tilt-promoting core which is linked to the head group by a slightly hydrophilic spacer chain. The further proof and extension of this rather specific design concept is a future challenge for synthetic liquid crystal chemistry.

Last but not least, all attempts to measure the actual values of spontaneous polarization in lyo-SmC\* phases by field-reversal, dielectric or pyroelectric techniques have failed so far, due to the high electrolytic conductivity of lyo-SmC\* phases originating from the mobile ionic impurities located in the solvent layers. Since *P*s is the key parameter of any ferroelectric material, this situation is highly unsatisfactory and it remains another future challenge to directly measure the spontaneous polarization in an electrolytically conductive medium.

In terms of possible applications, the use of the lyo-SmC\* phases in displays—as known from its thermotropic counterpart—namely, in SSFLC devices [70], is not very promissing, considering the high electrolytic conductivity and the potential leakage of the solvents. At the same time, the solvent opens new perspectives; e.g., in chirality sensing. For instance, the enantiomeric excess of physiologically-important, watersoluble drug molecules might be probed by their effect on the electo-optic switching time. In addition, the discovery of the elctroclinic effect in Lα\* has certain implications in the biophysics of cell membranes, since these are in a lamellar α state and contain substantial amounts of chiral inclusions; e.g., cholesterols and membrane proteins.

All in all, the recent discovery of the lyotropic equivalent to the thermotropic ferroelectric SmC\* phase has not only bridged a long-standing gap between the worlds of thermotropic and lyotropic liquid crystals, it also opened new research directions to the understanding of solvent-mediated chirality effects in lyotropic and biological systems.

**Author Contributions:** Both authors contributed equally.

**Funding:** Parts of our research were funded by *Deutsche Forschungsgemeinschaft*, DFG Gi 243/4, and *Landesgraduiertenförderung* of the state Baden-Württemberg.

**Acknowledgments:** Mikhail Osipov, Friederike Knecht and Marc Harjung are gratefully acknowledged for valuable discussions and input.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**


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