#
The Ever Elusive, Yet-to-Be-Discovered Twist-Bend Nematic Phase^{ †}

^{†}

## Abstract

**:**

## 1. Are the N_{X} and N_{TB} Phases Equivalent?

_{X}phase, so named because its molecular organization was unfamiliar. In the N

_{X}phase, nonlinear dimer liquid crystals exhibit a very tight 1D twist modulation of the orientational ordering with a nanoscale pitch of ~10 nm. Meyer envisioned the twist-bend nematic phase (N

_{TB}) specifically as a different kind of hierarchical meta-structure, a nematic having the apolar, uniaxial nematic director

**n**(

**n**$\leftrightarrow $ −

**n**, D

_{∞h}symmetry) spontaneously bending on the macroscale (~1 µm). Nevertheless, Meyer’s N

_{TB}phase has been used by some [2,3,4] to explain the nanoscale roto-translation modulation exhibited by bent or V-shaped dimer mesogens (e.g., CB-7-CB) in the N

_{X}phase, a second, lower-temperature nematic phase discovered in 1991 [5]. This explanation suggests that Meyer’s twist-bend nematic phase has already been discovered. A growing body of literature has accumulated around the putative N

_{X}= N

_{TB}equivalence without questioning it [3,4,6].

_{X}phase, published in 2016, provides a closer match to the experimental values than the original 2011 article which claimed that N

_{X}is the sought-after N

_{TB}phase [2]. This new theory proposed by Vanakaras and Photinos (VP) suggests that the N

_{X}phase is actually a novel liquid crystal (LC) phase—a new type of nematic that they call the polar-twisted nematic (N

_{PT}). Initially described within a mean field approximation, the VP theory predicts spontaneous chiral symmetry breaking and the formation of chiral domains of opposite handedness [7]. Their predictions were corroborated by simulations using detailed modeling of dimer LCs [8].

_{X}= N

_{TB}supposition [9,10,11]. Its sole reference to alternative views is to a critique of a critique [12], one that obscures Meyer’s coherent conjecture, relegating it to a genealogical affiliation with an ill-defined “family” of twist-bend nematics. Dunmur cites only research based on the assumption that N

_{X}and N

_{TB}are one and the same, and, in this regard, it is one-sided, not a comprehensive historical account. Much of the cited research is solid and has advanced the field, but the work of VP demonstrates that its underlying equation of N

_{X}= N

_{TB}is erroneous. For that reason, the mass of papers (more than 600) citing the same 2011 base publication by Dunmur et al. [2] has had a twofold effect on the science of LCs: it has suppressed not only any investigation of the new polar-twisted nematic but also the search for Meyer’s twist-bend nematic via the claim that such a phase has already been found. While there have always been researchers who have been more circumspect about the N

_{X}= N

_{TB}supposition (e.g., Chen et al., 2013 [13]), publications after 2016 should reference and fully address the advances in understanding made by the VP theory. Given that the VP theory is not mere speculation but an established methodology, a failure to address the way it differentiates the N

_{X}and N

_{TB}phases in future will retard advances in the understanding of the nematic phases exhibited by nonlinear, achiral mesogens.

## 2. Achiral Mesogens with Chiral Nematic Phases

_{TB}phase has not been observed to date, little can be said about prerequisite mesogen chemical structures other than that the molecular symmetry should be C

_{2V}or lower.) Meyer’s so-called twist-bend nematic N

_{TB}, derives from the interplay between the bend elasticity of the nematic director field

**n**(

**r**) and its associated flexoelectric polarization

**P**(

**r**) [1]. A half century later, Vanakaras’ and Photinos’ analysis of the nematic phase of nonlinear dimer molecules predicted spontaneous chiral symmetry breaking, the so-called polar-twisted nematic N

_{PT}[7]. Some have claimed that Vanakaras’ and Photinos’ new understanding of the N

_{X}phase only differs from prior interpretations semantically [12]; this is not the case, as was explained in 2020 [11].

_{X}phase requires an understanding of the limitations of Meyer’s theory, namely, its foundation in continuum elasticity theory in nematics; my colleagues and I have published articles explaining the constraints imposed on Meyer’s conjecture by elasticity theory in 2020 [9] and 2021 [10].

_{TB}theory and the N

_{X}phase, which Vanakaras’ and Photinos’ N

_{PT}theory illuminates: (1) The pitch of the form chirality is measured in micrometers in N

_{TB}theory versus nanometers in the N

_{X}phase. (2) The twisting entity is qualitatively different in the two theories, a nematic director

**n**in N

_{TB}theory versus a polar director

**m**in the N

_{X}phase, as delineated by N

_{PT}theory.

#### 2.1. The Polar-Twisted Nematic Phase

_{PT}is an orientationally ordered fluid phase without density modulations (i.e., it is nematic), and, like Marvin Freiser’s 1970 predicted biaxial nematic phase (N

_{B}) [14], it is also derived from theory utilizing minimal molecular modeling [7]. The most striking property of the new N

_{PT}nematic is its nanoscale modulation of the local polar orientational order (Figure 1 Left), a prediction of the VP model that is in remarkable quantitative agreement with experimental observations in the N

_{X}phase (Figure 1 Center)—the pitch of the nanoscale orientational modulation is ~3L (three dimer lengths) [13,15]. As Figure 1 emphasizes, the nominal pitch in the N

_{TB}phase (Right) is 100$\times $ larger than the observed pitch in the N

_{X}phase (Center); the difference is so large that the figure cannot accommodate it without the dotted blue lines leaving the page (Right).

^{PT}~10 nm) is much lower because it derives from the local, polar, molecular packing of dimer LCs. The dimers displaying the N

_{X}phase have a bent or V-shaped contour with associated polarity (electrostatic and/or shape) and a C

_{2v}average molecular symmetry. In the VP simulations, the dimer C

_{2}symmetry axes locally align (Figure 1 Left magnified inset), generating a polar phase director

**m**that roto-translates about a local axis h, yielding a 1D modulation of orientational order with a pitch that is in agreement with experimental observations in the N

_{X}phase, i.e., L

^{PT}= L

^{X}. The polar molecular packing in the polar-twisted phase defining

**m**is unconstrained by elasticity theoretical considerations, and

**m**is free to tightly spiral spontaneously about h to circumvent low-entropy, ferroelectric polarity [7,8]. Polarity is averaged out over one pitch length in the direction of the modulation, and since both left- and right-handed chiral packing (spirals) are equally probable, in macroscopic regions, the N

_{PT}phase is predicted to be uniaxial with balanced chirality as indeed observed experimentally in the N

_{X}phase.

#### 2.2. The Twist-Bend Nematic Phase

**n**tilts with a constant cone angle θ

_{c}and spirals about a macroscopic direction z in the N

_{TB}phase (Figure 1 Right). (The resulting N

_{TB}structure is asymptotically related to that of a traditional cholesteric or chiral nematic N* phase, where θ

_{c}= 90°.) Since the twist-bend structure must conform to the physics of Frank–Oseen elastic theory, continuum elasticity restricts the magnitude of such deformations. As a result, the lower bound on the pitch L

^{TB}of the spiraling nematic director in the twist-bend phase is in the order of microns, i.e., in order for the 1D modulations of the orientational order in the N

_{TB}phase to be consonant with Frank–Oseen theory, the L

^{TB}pitch must be $\gtrsim $100L

^{X}(Figure 1 Center).

_{TB}phase conforms to the limitations of Frank–Oseen theory. De Gennes carefully explained those limitations in his canonical text of 1974 [17], updated in 1993 [18]. Meyer certainly understood those limitations when he conjectured that (flexoelectric) polarization could spontaneously drive the formation of a space filling director topology that is labeled the twist-bend nematic:

“Although a state of uniform torsion is possible, a state of constant splay is not possible in a continuous three-dimensional object. A state of pure constant bend is also not possible, although a state of finite torsion and bend is possible. The latter is a modified helix in which the director has a component parallel to the helix axis. In laboratory coordinates,

_{z}= cosφ, n

_{x}= sinφcost

_{0}z, n

_{y}= sinφsint

_{0}z.

**n**; t

_{0}is the helical wavenumber and in Figure 1 Right, θ

_{c}= φ. In fact, he prefaces his description of the N

_{TB}phase with language that parallels that of de Gennes [17] pp. 58, 61; [18] p. 100:

“Changes in the magnitude of the order parameters in a nematic phase are high energy local processes. However gradual changes in the orientation of the director are low energy processes capable of being induced by small external perturbations. A continuum elasticity theory has been developed to describe these curvature structures” [1] p. 291.

## 3. The Second Nematic Phase in Dimer LCs

**n**. Such restrictions apply to all continuum elastic theories of nematics, even those with extreme elastic constants [19]. Those limitations effectively exclude Dunmur and collaborators’ original assumption that N

_{X}= N

_{TB}[2]; that equivalence requires the director field

**n**(

**r**) to twist through an angle of π over a distance of approximately three molecular lengths (~9 nm), a distance scale over which

**n**itself is undefined. In other words, there are not enough molecules in a volume element around

**r**to specify

**n**(

**r**). As emphasized before, “The issue is not the mere definition of the director in some volume v, but the deformation of

**n**(

**r**), described by the curvatures of the director field, i.e., it has to do with lengths. In order to describe the curvatures of

**n**(

**r**), the director has to be definable over a small volume v around

**r**, and of course such a description is meaningful only if the length scale of the curvature of

**n**(

**r**) is much larger than the dimensions of v.” [10]. Such constraints do not apply to the N

_{PT}phase, where the ferroelectric polarization associated with the molecular packing of V-shaped molecules is alleviated by nanoscale torsion—roto-translation of the polar director

**m**on the scale of a few molecular dimensions [7,8].

_{X}and N

_{TB}phases, recent citation practices continue to assume they are identical. A 2023 report reviews a variety of dimer LCs exhibiting the N

_{X}phase, calls it the N

_{TB}phase, and appears to be exhaustively documented (100 references) [20]; yet, it does not cite Meyer’s original work. Instead, it relies on a 2001 elasticity-based model by Dozov [19], one sourced in visually inspiring simulations [21] that are, however, technically flawed [8]. Its claim that the Dozov model supports the N

_{X}= N

_{TB}proposition stretches the limits of continuum elasticity; that model computes a twist pitch “$\le $100L $\cong $ 300 nm, rather small but still macroscopic” [19], a value 30$\times $ larger than the measured pitch in the N

_{X}phase. The VP model of the N

_{PT}phase requires no such gymnastics.

## 4. A New Heuristic

_{X}= N

_{TB}assumption can be rectified with a new heuristic, one that clearly differentiates the possible representations of nematics. To that end, we might consider a classification scheme delineating two length scales:

**I.**- Local nematic ordering: local organization dictated directly by intermolecular attractive dispersion forces regulated by excluded volume considerations [22], e.g., the uniaxial, biaxial, polar-twisted (N
_{U}, N_{B}, N_{PT}) nematic phases. **II.**- Topological nematic ordering: defined on a larger length scale by soft trajectories, reflecting analogously “soft residual” molecular interactions, of the (typically uniaxial) nematic director
**n**, e.g., chiral, splay-bend, twist-bend (N*, N_{SB}, N_{TB}) nematic phases.

_{K}) phase comprising a contiguous (chiral) director field; it is displayed below for pedagogical purposes. It shows soft/slow changes in the trajectory of the uniaxial director

**n**(

**r**); its associated flexoelectric polarization

**P**(

**r**) is aligned along the bend vector

**n**x(∇x

**n**). It is devised to communicate the unique, hierarchical, macroscale meta-structure in the types of phases Meyer predicted a half century ago (i.e., twist-bend and splay-bend nematics). A failure to differentiate these two types of nematic organizations, local molecular (

**I**) and topological (

**II**), will continue to obfuscate distinctions between nematic phases and to confound interpretations of new experimental findings (e.g., NMR [24] and simulations [25]) in bent-core and dimer liquid crystals.

## 5. Concluding Remarks

_{PT}phase is similar to Freiser’s predicted biaxial nematic N

_{B}phase. The latter reaffirmed the notion that biased, intermolecular interactions (correlated azimuthal angles among board-shaped mesogens) could manifest on a macroscale, and its alleged discovery was greatly acclaimed [26,27]. The organization in N

_{PT}similarly derives from unique intermolecular interactions (correlated polar ordering), but its predictions continue to be ignored [4,20] even though the N

_{PT}nematic phase perfectly describes the molecular organization in the N

_{X}phase and may even account for incompletely understood behavior in other bent-core mesogens [24]. The V-shape of the CB-7-CB dimers are inherently biaxial, and, in its N

_{X}phase, the dimers assume ferroelectric packing arrangements. The low entropy of such configurations is averted by winding them into tight, right- and left-handed helices having a nanoscale pitch.

_{X}phase was discovered [5], some thought that it represented the long sought-after twist-bend nematic phase postulated by Meyer [2]. Then, Vanakaras and Photinos showed that, on the contrary, the N

_{X}phase was a new phase, which they christened the polar-twisted nematic phase. The nanoscale modulation of the orientational order in dimer LCs unequivocally precludes the second, lower temperature N

_{X}phase from being Meyer’s (or Dozov’s) twist-bend nematic. Those topological nematic models diverge from the observations in the N

_{X}nematic; the N

_{PT}model reveals this divergence.

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**

**Molecular organization in nematic phases.**

**Left**: Schematic of the locally polar supramolecular structure of the polar-twisted nematic phase. The local polarization

**m**spirals about a helix axis h and generates a 1D modulation of the polar orientational order with pitch L

^{PT}~10 nm.

**Center**: Freeze-fracture transmission electron microscopy image of the N

_{X}phase of CB-7-CB, illustrating a 1D modulation pitch L

^{X}= 8 nm, excerpted from Figure 4a in ref. [15].

**Right**: The apolar director

**n**(

**r**) has a heliconical trajectory about z in the twist-bend nematic phase with an anticipated pitch L

^{TB}of ~1000 nm. The magnified view in the insets showing that the microscopic structural organizations of average mesogen shapes are approximately to scale.

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Samulski, E.T.
The Ever Elusive, Yet-to-Be-Discovered Twist-Bend Nematic Phase. *Crystals* **2023**, *13*, 1648.
https://doi.org/10.3390/cryst13121648

**AMA Style**

Samulski ET.
The Ever Elusive, Yet-to-Be-Discovered Twist-Bend Nematic Phase. *Crystals*. 2023; 13(12):1648.
https://doi.org/10.3390/cryst13121648

**Chicago/Turabian Style**

Samulski, Edward T.
2023. "The Ever Elusive, Yet-to-Be-Discovered Twist-Bend Nematic Phase" *Crystals* 13, no. 12: 1648.
https://doi.org/10.3390/cryst13121648