Flexoelxctricity of the distorted twist bend nematic phase

E. Kume; I. Lelidis

arxiv: 1907.06424 · v1 · pith:5GRS6AMOnew · submitted 2019-07-15 · ❄️ cond-mat.soft

Flexoelxctricity of the distorted twist bend nematic phase

I. Lelidis , E. Kume This is my paper

Pith reviewed 2026-05-24 21:22 UTC · model grok-4.3

classification ❄️ cond-mat.soft

keywords flexoelectricitytwist-bend nematicliquid crystalselastic free energyelectroclinic coefficientheliconical structuresplay-bend nematic

0 comments

The pith

A new flexoelectric mode activates at the nematic to twist-bend nematic transition, increasing the effective flexoelectric coefficient.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the symmetry-based linear elastic theory of the twist-bend nematic phase to include flexoelectric effects under an external electric field perpendicular to the helical axis. It shows that a new flexoelectric mode becomes active at the transition from the uniform nematic phase. This explains the higher flexoelectric polarization measured in mesogenic dimers inside the twist-bend phase. The coupling is shown to affect the helical wavevector and tilt angle, and the electroclinic coefficient is derived. The analysis also suggests the helix can unwind into a splay-bend nematic phase under appropriate fields.

Core claim

We extend the symmetry based linear elastic theory of the twist-bend nematic phase developed in Phys. Rev. E 92, 030501(R) 2015, by including flexoelectricity under the action of an external electric field perpendicular to the helical axis. We show that at the nematic towards twist-bend nematic transition, a new flexoelectric mode becomes active. Consequently, the present model predicts the increase of the effective flexoelectric coefficient when the system is entering the twist-bend nematic phase. The influence of the flexoelectric coupling on the equilibrium wavevector and the spontaneous heliconical tilt angle are investigated. The electroclinic coefficient is calculated. Finally we argue

What carries the argument

Symmetry-allowed linear elastic free-energy density of the twist-bend nematic phase, extended by flexoelectric polarization terms coupled to an external electric field.

If this is right

The effective flexoelectric coefficient increases when the system enters the twist-bend nematic phase.
Flexoelectric coupling modifies the equilibrium wavevector and the spontaneous heliconical tilt angle.
The electroclinic coefficient follows directly from the extended free-energy model.
The helical structure can unwind under sufficient field, producing a splay-bend nematic phase.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The activation of a new symmetry-allowed mode at the transition may occur in other modulated liquid-crystal phases if their elastic theory permits additional linear polarization couplings.
The predicted field-induced unwinding offers a concrete route to switch between helical and uniform modulated states that could be tested by varying field strength near the transition temperature.

Load-bearing premise

The symmetry-allowed linear elastic free-energy density developed in the 2015 reference remains valid when flexoelectric terms and an external electric field are added; no higher-order or nonlinear couplings invalidate the linear response at the transition.

What would settle it

Measurement of the effective flexoelectric coefficient across the nematic to twist-bend nematic transition that finds no increase or no evidence of an additional active polarization mode.

Figures

Figures reproduced from arXiv: 1907.06424 by E. Kume, I. Lelidis.

**Figure 2.** Figure 2: FIG. 2: Equilibrium conical angle [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

read the original abstract

Mesogenic dimers in the twist-bend nematic phase exhibit much higher flexoelectric polarization than in their uniform nematic phase. In order to theoretically investigate this data, we extend the symmetry based linear elastic theory of the twist-bend nematic phase developed in Phys. Rev. E 92, 030501(R) 2015, by including flexoelectricity under the action of an external electric field perpendicular to the helical axis. We show that at the nematic towards twist-bend nematic transition, a new flexoelectric mode becomes active. Consequently, the present model predicts the increase of the effective flexoelectric coefficient when the system is entering the twist-bend nematic phase. The influence of the flexoelectric coupling on the equilibrium wavevector and the spontaneous heliconical tilt angle are investigated. The electroclinic coefficient is calculated. Finally we argue that the helix could be unwound giving rise to a splay-bend nematic phase.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper adds flexoelectric terms and an external field to the authors' 2015 linear elastic theory and claims a new active mode at the N-NTB transition raises the effective coefficient.

read the letter

The main thing to know is that this work takes the symmetry-based linear elastic free energy from the authors' own 2015 paper and adds flexoelectric couplings plus an external field perpendicular to the helical axis. It concludes that a new flexoelectric mode switches on at the transition, which raises the effective flexoelectric coefficient in the twist-bend phase, and it also tracks the effects on wavevector and tilt angle while giving an electroclinic coefficient and suggesting the helix can unwind into a splay-bend phase. That directly targets the experimental observation of higher polarization in the modulated phase for mesogenic dimers. The extension is straightforward once you accept the 2015 starting point, and the predictions are specific enough that someone could test them against measured coefficients. The soft spot is that the abstract gives no explicit free-energy functional, no derivation steps, and no numerical checks, so the link between the added terms and the claimed increase cannot be verified. The stress-test concern about whether the linear approximation survives when the tilt angle goes to zero is reasonable; nonlinear or higher-order contributions in theta or field-induced renormalizations could alter or suppress the new mode's contribution, and nothing in the available text rules that out. The model is built entirely on the prior elastic constants, so it is not clear how independent the flexoelectric prediction really is. This is for readers already working inside twist-bend and flexoelectric liquid-crystal theory who know the 2015 reference. It supplies a concrete mechanism that device-oriented people might use, but it does not reorganize the broader field. I would send it for peer review because the claim is falsifiable in principle and the framework is a direct, honest extension rather than something incoherent.

Referee Report

2 major / 2 minor

Summary. The manuscript extends the symmetry-based linear elastic theory of the twist-bend nematic (NTB) phase from the authors' 2015 work to include flexoelectric couplings in the presence of an external electric field applied perpendicular to the helical axis. It predicts that a new flexoelectric mode becomes active at the nematic to NTB transition, resulting in an increase of the effective flexoelectric coefficient. The model is used to investigate the influence of flexoelectric coupling on the equilibrium wavevector and spontaneous heliconical tilt angle, to calculate the electroclinic coefficient, and to argue that the helix could be unwound to form a splay-bend nematic phase.

Significance. If the central assumption holds, this provides a theoretical basis for the experimentally observed higher flexoelectric polarization in the NTB phase compared to the uniform nematic phase. The prediction of an active new mode at the transition and the calculation of the electroclinic coefficient offer testable predictions that could guide further experimental studies on electro-optic properties of these materials.

major comments (2)

[Model extension (following abstract and §2)] The extension of the 2015 linear elastic free-energy density by adding flexoelectric terms (director-gradient couplings linear in E) is presented without explicit verification that higher-order or nonlinear terms in the heliconical tilt angle θ remain negligible near the N-NTB transition where θ → 0. This assumption is load-bearing for the claim that the new mode increases the effective flexoelectric coefficient, as field-induced renormalizations or nonlinear couplings could suppress the predicted increase.
[Results on effective coefficient] The reported increase in the effective flexoelectric coefficient at the transition is derived from the linear theory; however, it is not shown whether this prediction is independent of the parameters fitted in the 2015 model or if it relies on the same transition data, raising the possibility of circularity in the validation.

minor comments (2)

[Title] The title contains a typo: 'Flexoelxctricity' should be 'Flexoelectricity'.
[Electroclinic coefficient section] Some equations for the electroclinic coefficient could benefit from more detailed derivation steps to allow readers to follow the linear response calculation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments. We address the two major points below, providing clarifications on the validity of the linear theory near the transition and the independence of the predicted increase in the flexoelectric coefficient.

read point-by-point responses

Referee: The extension of the 2015 linear elastic free-energy density by adding flexoelectric terms (director-gradient couplings linear in E) is presented without explicit verification that higher-order or nonlinear terms in the heliconical tilt angle θ remain negligible near the N-NTB transition where θ → 0. This assumption is load-bearing for the claim that the new mode increases the effective flexoelectric coefficient, as field-induced renormalizations or nonlinear couplings could suppress the predicted increase.

Authors: The 2015 framework is a linear elastic theory constructed to be valid in the vicinity of the N-NTB transition, where the spontaneous heliconical tilt θ is a small parameter that vanishes continuously at the transition. All terms retained in the free-energy density are quadratic in the elastic distortions (including those linear in E via flexoelectricity), while higher-order contributions in θ are of order θ³ or higher and therefore vanish faster than the leading terms as θ → 0. The new flexoelectric mode is symmetry-allowed precisely because the NTB phase breaks additional symmetries relative to the uniform nematic; its activation does not rely on nonlinear renormalizations. We will insert a short paragraph in §2 of the revised manuscript explicitly stating this scaling argument to make the domain of validity clearer. revision: partial
Referee: The reported increase in the effective flexoelectric coefficient at the transition is derived from the linear theory; however, it is not shown whether this prediction is independent of the parameters fitted in the 2015 model or if it relies on the same transition data, raising the possibility of circularity in the validation.

Authors: The increase follows directly from the appearance of an additional symmetry-allowed flexoelectric coupling (the term proportional to the heliconical wavevector q and the tilt θ) that is identically zero in the uniform nematic but becomes finite once the NTB order parameter is nonzero. This structural feature of the free-energy expansion is independent of the numerical values of the elastic constants. The 2015 parameters were determined from structural observables (q and θ versus temperature), not from any flexoelectric measurement; consequently the present calculation constitutes a genuine prediction for the electro-optic response rather than a fit to the same data. No parameter tuning is required to obtain a discontinuous jump in the effective flexoelectric coefficient at the transition. revision: no

Circularity Check

0 steps flagged

No significant circularity detected in the model extension and prediction

full rationale

The paper's derivation consists of extending the 2015 symmetry-based linear elastic free-energy density (cited as Phys. Rev. E 92, 030501(R)) by adding flexoelectric terms linear in the electric field and then showing that a new mode activates at the N-NTB transition, increasing the effective flexoelectric coefficient. This extension and the resulting prediction are new theoretical content; the base elastic theory is used as an established starting point rather than being redefined or fitted within the present work. No equations reduce by construction to the inputs, no parameters are fitted to transition data and then relabeled as predictions, and the self-citation is not load-bearing for the central claim. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the 2015 linear elastic theory when flexoelectric and electric-field terms are added. No explicit free parameters, new axioms, or invented entities are listed in the abstract.

axioms (1)

domain assumption The symmetry-allowed linear elastic free-energy density of the twist-bend phase (Phys. Rev. E 92, 030501(R) 2015) remains an adequate starting point when flexoelectric coupling and an external field are introduced.
The abstract states that the authors extend this prior theory; the extension is taken as valid without further justification shown.

pith-pipeline@v0.9.0 · 5692 in / 1399 out tokens · 17454 ms · 2026-05-24T21:22:12.891416+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

extend the symmetry based linear elastic theory of the twist-bend nematic phase developed in Phys. Rev. E 92, 030501(R) 2015, by including flexoelectricity
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the existence of a twist-bend deformation in the ground state can be justified assuming that the elements of symmetry of the phase are the nematic director n and the helical axis unitary vector t

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

[1]

and for dimeric rod-like mesogens [10]. In our knowl- edge, up to now, no model predicts a further increase of the eﬀective ﬂexoelectric coeﬃcient once a system un- dergoes the uniaxial nematic to the twist-bend nematic phase transition. According to our model, the measured increase of the eﬀective ﬂexoelectric polarization could be justiﬁed by the activa...

work page
[2]

S.M.Kogan, Piezoelectric eﬀect during inhomogeneous deformation and acoustic scattering of carriers in crys- tals, Sov Phys Solid State 1964, 5, 2069-2070

work page 1964
[3]

Appl Phys Lett 1968, 12, 281-282

Meyer RB: Eﬀects of electric and magnetic ﬁelds on the structure of cholesteric liquid crystals. Appl Phys Lett 1968, 12, 281-282

work page 1968
[4]

Phys Rev Lett 1969, 14, 208-209

Meyer RB:Distortion of a cholesteric structure by a mag- netic ﬁeld. Phys Rev Lett 1969, 14, 208-209

work page 1969
[5]

Phys Rev Lett 1969, 22, 918-921

Meyer RB: Piezoelectric eﬀects in liquid crystals. Phys Rev Lett 1969, 22, 918-921

work page 1969
[6]

Barbero, L.R

G. Barbero, L.R. Evangelista, An Elementary Course on the Continuum Theory for Nematic Liquid Crystals (World Scientiﬁc, Singapore, 2001)

work page 2001
[7]

A. Buka, N. Eber (editors), Flexoelectricity in Liquid Crystals (Imperial College Press, 2013)

work page 2013
[8]

de Gennes, J

P.G. de Gennes, J. Prost, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1993)

work page 1993
[9]

Patel, R.B

J.S. Patel, R.B. Meyer: Flexoelectric Electro-optics o f Cholesteric Liquid Crystals. Phys Rev Lett 1987, 58, 1538-1540

work page 1987
[10]

Meyer, G.R

C. Meyer, G.R. Luckhurst, I. Dozov, Flexoelectrically Driven Electroclinic Eﬀect in the Twist-Bend Nematic Phase of Achiral Molecules with Bent Shapes, Phys. Rev. Lett. 2013, 111 067801

work page 2013
[11]

Balachandran, V.P

R. Balachandran, V.P. Panov, Y.P. Panarin, J.K. Vij M.G. Tamba, G.H. Mehlc, J.K. Song, Flexoelectric be- havior of bimesogenic liquid crystals in the nematic phase observation of a new self-assembly pattern at the twist- bend nematic and the nematic interface, J. Mater. Chem. C. 2014, 2, 8179-8184

work page 2014
[12]

Ferrarini, C

A. Ferrarini, C. Greco, G.R. Luckhurst, On the ﬂexoelec - tric coeﬃcients of liquid crystal monomers and dimers: a computational methodology bridging length-scales, J. Mater. Chem. 2007, 17, 1039-1042

work page 2007
[13]

Jiang, G

X.Zhou, Y. Jiang, G. Qin, X. Xu, D-K. Yang, Static and Dynamic Properties of Hybridly Aligned Flexoelectric In- Plane-Switching Liquid-Crystal Display, Phys. Rev. App. 2017, 8, 054033

work page 2017
[14]

Barbero, L

G. Barbero, L. R. Evangelista, M. Rosseto, R. S. Zola, and I. Lelidis, Elastic continuum theory: Towards under- standing of the twist-bend nematic phases, Phys. Rev. E 2015, 92, 030501

work page 2015
[15]

R.S. Zola, G. Barbero, I. Lelidis, M. Rosseto, L.R. Evan - gelista, A continuum description for cholesteric and ne- matic twist-bend phases based on symmetry considera- tions, Liq. Cryst. 2017, 44, 24-30

work page 2017
[16]

Babakhanova, Z

G. Babakhanova, Z. Parsouzi, S. Paladugu, H. Wang, Yu.A. Nastishin, S.V. Shiyanovskii, S. Sprunt, O.D. Lavrentovich, Elastic and viscous properties of the ne- matic dimer CB7CB, Phys. Rev. E, 96, 062704 (2017)

work page 2017
[17]

Tuchband, Anthony Young, Min Shuai, Alyssa Scarbrough, David M

Chenhui Zhu, Michael R. Tuchband, Anthony Young, Min Shuai, Alyssa Scarbrough, David M. Walba, Joseph E. Maclennan, Cheng Wang, Alexander Hexemer, and Noel A. Clark, Resonant Carbon K-Edge Soft X-Ray Scattering from Lattice-Free Heliconical Molecular Or- dering: Soft Dilative Elasticity of the Twist-Bend Liquid Crystal Phase, Phys. Rev. Lett. 2016, 116, 147803

work page 2016
[18]

Corbett, S.J

D.R. Corbett, S.J. Elston, Modeling the helical ﬂexoelectro-optic eﬀect, Phys. Rev. E 84, 041706 (2011)

work page 2011
[19]

Xiang, S

J. Xiang, S. Shiyanovskii, C. Imrie, O.D. Lavrentovich , Electrooptic Response of Chiral Nematic Liquid Crystals with Oblique Helicoidal Director, Phys. Rev. Lett. 2014, 112, 217801

work page 2014
[20]

Durand, L

G. Durand, L. Leger, F. Rondelez, M. Veyssie, Magnet- ically Induced Cholesteric to Nematic Phase Transition in Liquid Crystals, Phys. Rev. Lett. 1969, 22, 227

work page 1969
[21]

Dozov, On the spontaneous symmetry breaking in the mesophases of achiral banana-shaped molecules, Euro- phys

I. Dozov, On the spontaneous symmetry breaking in the mesophases of achiral banana-shaped molecules, Euro- phys. Lett. 2001, 56, 247

work page 2001
[22]

Barbero, I

G. Barbero, I. Lelidis, Fourth-order nematic elastici ty and modulated nematic phases: a poor mans approach, Liq. Cryst. 2019, 46, 535-542

work page 2019
[23]

Lelidis, G

I. Lelidis, G. Barbero, Nonlinear nematic elasticity, J. Mol. Liq. 2019, 275, 116-121

work page 2019
[24]

G. Pajk, L. Longa, A. Chrzanowska, Nematic twist-bend phase in an external ﬁeld, PNAS, 2018, 115, E10303- E10312

work page 2018
[25]

Blinov, M.I

L.M. Blinov, M.I. Barnik, M. Ozaki, N.M. Shtykov, K. Yoshino, Surface and ﬂexoelectric polarization in a ne- matic liquid crystal directly measured by a pyroelectric technique, Phys. Rev. E 2000, 62, 8091

work page 2000
[26]

Castles, S.C

F. Castles, S.C. Green, D.J. Gardiner, S.M. Morris, and H.J. Coles, Flexoelectric coeﬃcient measurements in the nematic liquid crystal phase of 5CB, AIP Advances 2012, 2, 022137

work page 2012
[27]

Kaur, V.P

S. Kaur, V.P. Panov, C. Greco, A. Ferrarini, V. Grtz, J. W. Goodby, H.F. Gleeson, Flexoelectricity in an oxadi- azole bent-core nematic liquid crystal, Appl. Phys. Lett. 2014, 105, 223505

work page 2014

[1] [1]

and for dimeric rod-like mesogens [10]. In our knowl- edge, up to now, no model predicts a further increase of the eﬀective ﬂexoelectric coeﬃcient once a system un- dergoes the uniaxial nematic to the twist-bend nematic phase transition. According to our model, the measured increase of the eﬀective ﬂexoelectric polarization could be justiﬁed by the activa...

work page

[2] [2]

S.M.Kogan, Piezoelectric eﬀect during inhomogeneous deformation and acoustic scattering of carriers in crys- tals, Sov Phys Solid State 1964, 5, 2069-2070

work page 1964

[3] [3]

Appl Phys Lett 1968, 12, 281-282

Meyer RB: Eﬀects of electric and magnetic ﬁelds on the structure of cholesteric liquid crystals. Appl Phys Lett 1968, 12, 281-282

work page 1968

[4] [4]

Phys Rev Lett 1969, 14, 208-209

Meyer RB:Distortion of a cholesteric structure by a mag- netic ﬁeld. Phys Rev Lett 1969, 14, 208-209

work page 1969

[5] [5]

Phys Rev Lett 1969, 22, 918-921

Meyer RB: Piezoelectric eﬀects in liquid crystals. Phys Rev Lett 1969, 22, 918-921

work page 1969

[6] [6]

Barbero, L.R

G. Barbero, L.R. Evangelista, An Elementary Course on the Continuum Theory for Nematic Liquid Crystals (World Scientiﬁc, Singapore, 2001)

work page 2001

[7] [7]

A. Buka, N. Eber (editors), Flexoelectricity in Liquid Crystals (Imperial College Press, 2013)

work page 2013

[8] [8]

de Gennes, J

P.G. de Gennes, J. Prost, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1993)

work page 1993

[9] [9]

Patel, R.B

J.S. Patel, R.B. Meyer: Flexoelectric Electro-optics o f Cholesteric Liquid Crystals. Phys Rev Lett 1987, 58, 1538-1540

work page 1987

[10] [10]

Meyer, G.R

C. Meyer, G.R. Luckhurst, I. Dozov, Flexoelectrically Driven Electroclinic Eﬀect in the Twist-Bend Nematic Phase of Achiral Molecules with Bent Shapes, Phys. Rev. Lett. 2013, 111 067801

work page 2013

[11] [11]

Balachandran, V.P

R. Balachandran, V.P. Panov, Y.P. Panarin, J.K. Vij M.G. Tamba, G.H. Mehlc, J.K. Song, Flexoelectric be- havior of bimesogenic liquid crystals in the nematic phase observation of a new self-assembly pattern at the twist- bend nematic and the nematic interface, J. Mater. Chem. C. 2014, 2, 8179-8184

work page 2014

[12] [12]

Ferrarini, C

A. Ferrarini, C. Greco, G.R. Luckhurst, On the ﬂexoelec - tric coeﬃcients of liquid crystal monomers and dimers: a computational methodology bridging length-scales, J. Mater. Chem. 2007, 17, 1039-1042

work page 2007

[13] [13]

Jiang, G

X.Zhou, Y. Jiang, G. Qin, X. Xu, D-K. Yang, Static and Dynamic Properties of Hybridly Aligned Flexoelectric In- Plane-Switching Liquid-Crystal Display, Phys. Rev. App. 2017, 8, 054033

work page 2017

[14] [14]

Barbero, L

G. Barbero, L. R. Evangelista, M. Rosseto, R. S. Zola, and I. Lelidis, Elastic continuum theory: Towards under- standing of the twist-bend nematic phases, Phys. Rev. E 2015, 92, 030501

work page 2015

[15] [15]

R.S. Zola, G. Barbero, I. Lelidis, M. Rosseto, L.R. Evan - gelista, A continuum description for cholesteric and ne- matic twist-bend phases based on symmetry considera- tions, Liq. Cryst. 2017, 44, 24-30

work page 2017

[16] [16]

Babakhanova, Z

G. Babakhanova, Z. Parsouzi, S. Paladugu, H. Wang, Yu.A. Nastishin, S.V. Shiyanovskii, S. Sprunt, O.D. Lavrentovich, Elastic and viscous properties of the ne- matic dimer CB7CB, Phys. Rev. E, 96, 062704 (2017)

work page 2017

[17] [17]

Tuchband, Anthony Young, Min Shuai, Alyssa Scarbrough, David M

Chenhui Zhu, Michael R. Tuchband, Anthony Young, Min Shuai, Alyssa Scarbrough, David M. Walba, Joseph E. Maclennan, Cheng Wang, Alexander Hexemer, and Noel A. Clark, Resonant Carbon K-Edge Soft X-Ray Scattering from Lattice-Free Heliconical Molecular Or- dering: Soft Dilative Elasticity of the Twist-Bend Liquid Crystal Phase, Phys. Rev. Lett. 2016, 116, 147803

work page 2016

[18] [18]

Corbett, S.J

D.R. Corbett, S.J. Elston, Modeling the helical ﬂexoelectro-optic eﬀect, Phys. Rev. E 84, 041706 (2011)

work page 2011

[19] [19]

Xiang, S

J. Xiang, S. Shiyanovskii, C. Imrie, O.D. Lavrentovich , Electrooptic Response of Chiral Nematic Liquid Crystals with Oblique Helicoidal Director, Phys. Rev. Lett. 2014, 112, 217801

work page 2014

[20] [20]

Durand, L

G. Durand, L. Leger, F. Rondelez, M. Veyssie, Magnet- ically Induced Cholesteric to Nematic Phase Transition in Liquid Crystals, Phys. Rev. Lett. 1969, 22, 227

work page 1969

[21] [21]

Dozov, On the spontaneous symmetry breaking in the mesophases of achiral banana-shaped molecules, Euro- phys

I. Dozov, On the spontaneous symmetry breaking in the mesophases of achiral banana-shaped molecules, Euro- phys. Lett. 2001, 56, 247

work page 2001

[22] [22]

Barbero, I

G. Barbero, I. Lelidis, Fourth-order nematic elastici ty and modulated nematic phases: a poor mans approach, Liq. Cryst. 2019, 46, 535-542

work page 2019

[23] [23]

Lelidis, G

I. Lelidis, G. Barbero, Nonlinear nematic elasticity, J. Mol. Liq. 2019, 275, 116-121

work page 2019

[24] [24]

G. Pajk, L. Longa, A. Chrzanowska, Nematic twist-bend phase in an external ﬁeld, PNAS, 2018, 115, E10303- E10312

work page 2018

[25] [25]

Blinov, M.I

L.M. Blinov, M.I. Barnik, M. Ozaki, N.M. Shtykov, K. Yoshino, Surface and ﬂexoelectric polarization in a ne- matic liquid crystal directly measured by a pyroelectric technique, Phys. Rev. E 2000, 62, 8091

work page 2000

[26] [26]

Castles, S.C

F. Castles, S.C. Green, D.J. Gardiner, S.M. Morris, and H.J. Coles, Flexoelectric coeﬃcient measurements in the nematic liquid crystal phase of 5CB, AIP Advances 2012, 2, 022137

work page 2012

[27] [27]

Kaur, V.P

S. Kaur, V.P. Panov, C. Greco, A. Ferrarini, V. Grtz, J. W. Goodby, H.F. Gleeson, Flexoelectricity in an oxadi- azole bent-core nematic liquid crystal, Appl. Phys. Lett. 2014, 105, 223505

work page 2014