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# Nematic twist–bend phase in an external field

Edited by Ivan Dozov, Université Paris-Sud, Orsay, France, and accepted by Editorial Board Member John D. Weeks September 10, 2018 (received for review December 15, 2017)

## Significance

The twist–bend nematic liquid crystalline phase is the fifth nematic structure recognized in nature [Chen D, et al. (2013) *Proc Natl Acad Sci USA* 110:15931–15936], and its stabilization is explained by assuming a coupling between polar and nematic orderings. It exhibits macroscopically chiral heliconical orientational order on the 10-nm scale and represents a unique example of spontaneous chiral symmetry breaking for a system of achiral molecules. Understanding how an external field affects the stability of this phase can shed further light on the origin of this induced twist and is of relevance to potential applications. Within the Landau–de Gennes theory we find that for compounds with positive anisotropy the helix unwinds to a polar nematic, however negative material anisotropy gives rise to a rich sequence of new nematic phases obtained via a mechanism of flattening the conical spiral.

## Abstract

The response of the nematic twist–bend (

The twist–bend nematic (

The first possibility is recognized as the chiral

The

While phenomenologically the spontaneous distortion of the

A mesoscopic-level explanation of how molecular polarity of bent-core molecules can generate modulated polar nematic phases and, hence, effectively lower the bend elastic constant has been proposed to be due to the flexopolarization couplings, where derivatives of the alignment tensor (or of the director field) induce a net polarization (17, 18, 26, 29) (see supplemental material for ref. 37).

## Model

We regard

The minimal coupling model in

As always in Landau theory, the temperatures **4**–**6** contain all symmetry-allowed invariants up to fourth order in **3**) predicts the existence of two further 1D modulated nonchiral, polar nematic phases with transverse and longitudinal polarization being modulated along just one direction (18). But it is important to observe that the flexopolarization term alone (

Alternative mesoscopic scenarios pertaining to the stability of

In the case of modulated nematics, their response to an external field can become highly nontrivial (11, 49). In cholesterics, for example, it is possible to unwind the orientational spiral through an intermediate heliconical structure (50, 51), both for bulk sample (52) and in confined geometry (53, 54). A more comprehensible, field-induced modification of cholesterics involves reorientation of the helical axis (11) or changing the pitch (55). Similar effects can be expected for

The purpose of this paper is to study, in a systematic way, a response of the bulk **3**–**6**). As **3**) supplemented by the dielectric (diamagnetic) term, **8** any coupling with the field that is independent of orientation. We think that the dielectric (diamagnetic) term should dominate, at least for sufficiently strong fields, and disregard a possible direct interaction between the dipole moments and the field.

It is now useful to rewrite Eq. **3** in terms of reduced (dimensionless) quantities, which reveals the redundancy of four parameters in the expressions (Eqs. **4**–**6**) and allows us to set them to one from the start (17, 18, 39, 61). We introduce the reduced quantities F, f, r, Q, P, **3**–**6** and **8** then become**14**–**18**) by limiting to a family of all One-Dimensional Modulated Nematic Structures (ODMNS), periodic at most in one spatial direction (18). Starting with the

All possible ODMNS structures can be parameterized with the aid of the plane wave expansions of **18**, giving

Our starting point is the identification of homogeneous structures of wave vector k that can be constructed out of Q and P, among which should be the **22** represent the reference N phase with the director along

Setting **22** gives the cholesteric phase of the conical angle **1**), while the simplest parameterization of the **22**. In this simplified case, the conical angle is given by**26** is k-independent, expressing the fact that z dependence of Q in Eq. **22** is generated by a rotation. This means that **22****—**namely, the biaxial nematic.

Each of the structures identified so far can be polar (Eq. **23**). For **22** and **23** with

## Results

In seeking a globally stable structure among ODMNS, we need to take into account both homogeneous and inhomogeneous trial states given by the plane wave expansions of Q and P. The complexity of the ODMNS minimization depends on the number of amplitudes used in this expansion, which is controlled by the maximal value

More specifically, we consider the bulk

The relaxation method for the discretized free energy determines **29**) was solved iteratively until self-consistency with the required accuracy was achieved. We used very large bulk samples of

Results of the numerical analysis are illustrated for the case when **18**, the

To gain an insight into fine structure of stable phases, we plot characteristic observables for each of them in Figs. 6–10. First, we present the behavior of eigenvalues for **25**). In Fig. 6, we present

## Discussion

Scientists have long sought to understand how chiral states can be generated in a liquid state from nonchiral matter. Now strong evidence is found that a new class of nematics, called nematic twist–bend, provides such an example. This entropically induced state is realized because the underlying molecules have a specific shape. Here we presented possible transformations of the

For materials with positive anisotropy, the unwinding of the helix to the uniaxial (prolate) nematic structure is obtained, however here for sufficiently strong fields, a polar nematic appears more stable. Interestingly, in the

An interesting question that is left is whether currently available applied fields allow for potential experimental verification of our predictions. A connection between the physical and reduced fields can be estimated by a direct replacement of the LdeG parameters with their director field equivalents, akin to ref. 67. Since **30** and taking

If the nematic twist–bend phase is separated from the isotropic phase by the uniaxial nematic, we can set the scale for the field from properties of this intermediate nematic phase (Eq. **4**). A connection between the physical and reduced fields requires in this case an estimate of b and c parameters (Eq. **13**). One may also find it useful to know **13**). More specifically, for **2**, **33**–**35**, in Eq. **13** one can obtain a different relation between the dimensionless and physical fields, expressed in terms of experimentally accessible quantities at the nematic–isotropic phase transition. More precisely,**37** gives a somewhat higher value of 0.1 K]; mass density ^{3}; and molar mass

The final remark is that fields analyzed in this paper can also be of surface Rapini–Papoular form (11): **31**, has a similar mathematical structure as Eq. **18**, where the role of external field

### Closing Note.

While our paper was subject to the reviewing procedure, we have learned that Merkel et al. (72) have confirmed experimentally the electric field-stabilized

## Acknowledgments

We thank the referees and the editor for valuable comments. This work was supported by National Science Center in Poland Grant DEC-2013/11/B/ST3/04247. G.P. acknowledges the support of Cracowian Consortium “Materia-Energia-Przyszłość im. Mariana Smoluchowskiego” within the “Krajowy Naukowy Ośrodek Wiodący” (KNOW) grant.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: grzegorz{at}th.if.uj.edu.pl, lech.longa{at}uj.edu.pl, or achrzano{at}usk.pk.edu.pl.

Author contributions: G.P., L.L., and A.C. performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. I.D. is a guest editor invited by the Editorial Board.

Published under the PNAS license.

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