# Role of Lifshitz Invariants in Liquid Crystals

## Abstract

**:**

## 1. Introduction

## 2. The Dzyaloshinskii-Moriya Coupling

_{3}[16]. More recently, the coupling of spin waves with the optical phonons has been discussed in the framework of Lifshitz invariant, for the same material [17]. An antiferromagnetic vector $\overrightarrow{L}$ characterises the BiFeO

_{3}spin structure. The Landau-Ginzburg energy density [3] of the spin structure is the following sum of four terms:

_{3}, as shown in Ref.16. The second term ${f}_{exch}$ in (2) is the inhomogeneous exchange energy, where $A$ is a stiffness constant. In the third term, K

_{u}is the uniaxial anisotropy. ${f}_{ME}$ is the coupling of an external electric field $\overrightarrow{E}$ with a spatial uniform inner field ${\overrightarrow{H}}_{DM}=\overrightarrow{d}\times \overrightarrow{L}$, where $\overrightarrow{d}=\left(0,0,{P}_{z}\right)$, and β the homogeneous magneto-electric constant. This term is originated from a magneto-electric-like DM interaction.

## 3. The Flexoelectricity in Liquid Crystals

## 4. Periodic Distortions in Nematics

**Figure 1.**The frame of reference and the angles used to describe the director, represented by the rod-like molecule.

**Figure 2.**Comparison of the free energy behaviours in the case of the uniform configuration and for the distorted one.

## 5. The Chiral Nematic and the Smectic Phase

## 6. The Saddle-Splay Elasticity at Surfaces

## 7. The Hybrid Cell and the Flexoelectricity

**Figure 3.**Frame of reference for the hybrid cell in the upper part of the figure. In the lower part, the free energies as a function of the electric field, in the case of planar and homeotropic configurations. Note the presence of a threshold.

**Figure 4.**(a) Behaviour of the free energies of planar, hybrid (HAN) and homeotropic configurations, as functions of the electric field. Note the existence of two thresholds for the transition between the planar and the HAN configuration and between the HAN and the homeotropic configuration. (b) The two curves in grey show how the energy of the HAN configuration changes for the presence of flexoelectricity. According the sign of the flexoelectric parameter, the threshold field is raised or lowered.

## 8. The Phase Diagrams of the Hybrid Cell

**Figure 5.**Phase diagrams of the HAN cell, for a fixed choice of the surface parameter ${b}_{o}=1$. We change the value of parameter ${b}_{1}$ and find the value of the thresholds of the electric dimensionless field $\widehat{\mathsf{\xi}}$. There are three regions in the diagrams where planar, homeotropic and hybrid alignments are allowed. The phase diagram is depending on the values of flexoelectric parameter $\mathsf{\Pi}$. Diagram (d) shows the behaviour of a cell when the flexoelectric parameter $\mathsf{\Pi}$ changes. Note that the hybrid configuration disappears when flexoelectric parameter is higher than 1.3.

**Figure 6.**Behaviour of $cos\theta $ as a function of the reduced cell thickness $\widehat{y}=y/d$ in the case of positive and negative flexoelectric coefficients, for different values of the dimensionless electric field (some values are reported on the curves). Note that, as the field increases, the role of surface is suppressed and the angle at the planar surface increases. As the electric field is higher that the threshold value, the cell becomes homeotropic and then $cos\theta =0$.

## 9. Nematics in Cylindrical Geometry

**Figure 7.**Cylindrical cell and frame of references on the left and on the right the angles of director chosen for calculations.

**Figure 8.**Behaviour of $\theta $ as a function of the reduced radial coordinate $r/R$ in the case of flexoelectric coefficient $\mathsf{\Pi}$ equal to 0 and 1, for different values of the anchoring parameter b and dimensionless electric field (some values are reported on the curves). As the electric field is higher that a threshold value, angle $\mathsf{\theta}$ goes to zero, that is the director field is parallel to the cylinder axis.

**Figure 9.**Behaviour of $cos\theta $ as a function of the reduced radial coordinate $r/R$, for different values of the flexoelectric coefficient $\mathsf{\Pi}$ and of the dimensionless electric field (some values of $\widehat{\mathsf{\xi}}$ are reported on the curves). The value of the anchoring strength is fixed in all the figures. Note that a negative value of the flexoelectric parameter is strongly favouring the alignment of director parallel to cylinder axis, and then the threshold electric field is very low. If the flexoelectric parameter is positive and large, the distorted configuration is favoured, and the threshold field, needed for suppressing this configuration, is increased. As shown in the lower part of the figure, when $\mathsf{\Pi}$ is very large, $cos\theta $ is oscillating as the electric field increases. A very large field is required to suppress the distortion and have $\theta =0$.

**Figure 10.**Phase diagram of the cylindrical confinement, when the anchoring parameter b is fixed and equal to 6. The three regions are denoted by U for the uniform alignment of director parallel to the cylinder axis, D when the director has a deformed configuration, and O if director is oscillating and cosine becomes negative too.

## 10. The Saddle-Splay Contribution and the PHAN Cell

_{0}= 0, where z is the axis perpendicular to the cell plane. The planar wall is at z

_{1}= d, where d is the thickness of the cell. The easy-axis of the planar alignment is chosen coincident with the x-axis. The director $\overrightarrow{n}$ is described as:

## 11. Conclusions

## References

- Landau, L.D.; Lifshitz, E.M. Statistical Physics; Pergamon Press: Oxford, UK, 1980. [Google Scholar]
- Braginsky, A.Y. Phenomenological theory of phase transitions to a state with irregular phase of the order parameter. Phys. Rev. B
**2002**, 66, 054202:1–054202:8. [Google Scholar] - Dzyaloshinskii, I.E. Theory of helicoidal structures in antiferromagnets. Sov. Phys. JETP
**1964**, 19, 960–971. [Google Scholar] - Bogdanov, A.N.; Rössler, U.K.; Pfleiderer, C. Modulated and localized structures in cubic helimagnets. Physica B
**2005**, 259-361, 1162–1164. [Google Scholar] - Kadomtseva, A.M.; Popov, Yu.F.; Vorob’ev, G.P.; Ivanov, V.Yu.; Mukhin, A.A.; Balbashov, A.M. Specific features of the magnetic field-induced orientational transition in EuMnO
_{3}. JETP Letters**2005**, 81, 590–593. [Google Scholar] [CrossRef] - Dozov, I. On the spontaneous symmetry breaking in the mesophases of achiral banana-shaped molecules. Europhys. Lett.
**2001**, 56, 247–253. [Google Scholar] [CrossRef] - Sarkissian, H.; Park, J.B.; Zeldovich, B.Ya.; Tabirian, N.V. Liquid crystal structure with transverse periodic alignment. In Quantum Electronics and Laser Science Conference, QELS '05, Baltimore, Maryland, USA, May 22, 2005; Volume 3, pp. 1597–1599.
- Marinova, Y.; Kosmopoulos, J.; Weissflog, W.; Petrov, A.G.; Photinos, D.J. Flexoelectricity of wedge-like molecules in nematic mixtures. Mol. Cryst. Liquid Cryst.
**2001**, 357, 221–228. [Google Scholar] [CrossRef] - Pikin, S.A. Structural Transformations in Liquid Crystals; Taylor & Francis: New York, NY, USA, 1991. [Google Scholar]
- Lavreontovich, O.D.; Pergamenshchik, V.M. Periodic structures in nematic thin-layers. Pisma V. Zhur. Tekh. Fiz.
**1989**, 15, 73–78. [Google Scholar] - Lavreontovich, O.D.; Pergamenshchik, V.M. Periodic domain-structures in thin hybrid nematic layers. Mol. Cryst. Liquid Cryst.
**1990**, 179, 125–132. [Google Scholar] - Moriya, T. Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev.
**1960**, 120, 91–98. [Google Scholar] - Coffey, D.; Rice, T.M.; Zhang, F.C. Dzyaloshinskii-Moriya interaction in the cuprates. Phys. Rev. B.
**1991**, 44, 10112–10116. [Google Scholar] [CrossRef] - Bogdanov, A.N.; Rössler, U.K.; Wolf, M.; Müller, K.-H. Magnetic structures and reorientation transitions in noncentrosymmetric uniaxial antiferromagnets. [arXiv:cond-mat/0206291, 2002].
- Lifschitz, E.M. On the theory of second-order phase transitions. Zh. Eksp. Teor. Fiz.
**1941**, 11, 255–269. [Google Scholar] - Sparavigna, A.; Strigazzi, A.; Zvezdin, A.K. Electric-field effects on the spin-density wave in magnetic ferroelectrics. Phys. Rev. B
**1994**, 50, 2953–2957. [Google Scholar] [CrossRef] - de Sousa, R.; Moore, J.E. Optical coupling to spin waves in the cycloidal multiferroic BiFeO
_{3}. [arXiv:0706.1260v2, 2007]. - Bobylev, Y.P.; Chigrinov, V.G.; Pikin, S.A. Threshold flexoelectric effect in nematic liquid crystals. J. Phys.
**1979**, 40, 331. [Google Scholar] - Meyer, R.B. Piezoelectric effects in liquid crystals. Phys. Rev. Lett.
**1969**, 22, 918–921. [Google Scholar] [CrossRef] - Maranganti, R.; Sharma, N.D.; Sharma, P. Electromechanical coupling in non piezoelectric materials due to nanoscale nonlocal size effects: Green's function solutions and embedded inclusions. Phys. Rev. B
**2006**, 74, 014110:1–014110:14. [Google Scholar] [CrossRef] - Lagerwall, S.T. Ferroelectrics: ferroelectric and antiferroelectric. Liq. Cryst.
**2004**, 301, 15–45. [Google Scholar] - Clark, N.A.; Lagerwall, S.T. Submicrosecond bistable electro-optic switching in liquid crystals. Appl. Phys. Lett.
**1980**, 36, 899–901. [Google Scholar] [CrossRef] - Weissflog, W.; Lischka, C.; Diele, S.; Pelzl, G.; Wirth, I.; Grande, S.; Kresse, H.; Schmalfuss, H.; Hartung, H.; Stettler, A. Banana-shaped or rod-like mesogens? Molecular structure, crystal structure and mesophase behaviour of 4,6-dichloro-1,3-phenylene. Mol. Cryst. Liq. Cryst.
**1999**, 333, 203–235. [Google Scholar] [CrossRef] - Benguigui, L.; Jacobs, A.E. Reentrant smectic-C and smectic-C* phases in liquid crystals under an electric field. Phys. Rev. E
**1994**, 49, 4221–4227. [Google Scholar] [CrossRef] - Sparavigna, A.; Lavrentovich, O.D.; Strigazzi, A. Periodic stripe domains and hybrid-alignment regime in nematic liquid crystals: Threshold analysis. Phys. Rev. E
**1994**, 49, 1344–1352. [Google Scholar] [CrossRef] - Barbero, G.; Sparavigna, A.; Strigazzi, A. The structure of the distortion free-energy density in nematics: Second-order elasticity and surface terms. Nuovo Cim. D
**1990**, 12, 1259–1272. [Google Scholar] [CrossRef] - Frank, F.C. On the theory of liquid crystals. Discuss. Faraday Soc.
**1958**, 25, 19–28. [Google Scholar] [CrossRef] - Lubensky, T.C. Molecular description of nematic liquid crystals. Phys. Rev. A
**1970**, 2, 2497–2514. [Google Scholar] [CrossRef] - Nehring, J.; Saupe, A. On the elastic theory of uniaxial liquid crystals. J. Chem. Phys.
**1971**, 54, 337–343. [Google Scholar] [CrossRef] - Pergamenshchik, V.M. Surface-like-elasticity-induced spontaneous twist deformations and long-wavelength stripe domains in a hybrid nematic layer. Phys. Rev. E
**1993**, 47, 1881–1892. [Google Scholar] [CrossRef] - Lelidis, I.; Barbero, G. Modulated structures in nematic monolayers formed by symmetric molecules. Phys. Rev. E
**2005**, 71, 022701:1–022701:4. [Google Scholar] [CrossRef] - Helfrich, W. Z. Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforschung
**1973**, 28c, 693–703. [Google Scholar] - Parthasarathy, R.; Groves, J.T. Curvature and spatial organization in biological membranes. Soft Matter
**2007**, 3, 24–33. [Google Scholar] [CrossRef] - Barbero, G.; Evangelista, L.R. An Elementary Course on the Continuum Theory for Nematic Liquid Crystals; World Scientific: Singapore, 2001. [Google Scholar]
- Cheung, D.L.G. Structures and Properties of Liquid Crystals and Related Molecules from Computer Simulation . Ph.D Thesis, University of Durham, Durham, UK, 2002. [Google Scholar]
- Schmidt, D.; Schadt, M.; Helfrich, W.Z. Liquid-crystalline curvature electricity: The bending mode of MBBA. Z. Naturforschung
**1972**, 27a, 277–280. [Google Scholar] - Barbero, G.; Valabrega, P.T.; Bartolino, R.; Valenti, B. Flexoelectricity in the hybrid aligned nematic cell. Liq. Cryst.
**1986**, 1, 483–490. [Google Scholar] [CrossRef] - Dozov, I.; Martinot-Lagarde, P.; Durand, G. Flexoelectrically controlled twist of texture in a nematic liquid crystal. J. Phys. Lett.
**1982**, 43, L365–L370. [Google Scholar] [CrossRef] - Warrier, S.; Madhusudana, N.V. An AC electrooptic technique for measuring the flexoelectric coefficient and anchoring energies of nematics. J. Phys. II
**1997**, 7, 1789–1803. [Google Scholar] - Blinov, L.M.; Barnik, M.I.; Ohoka, H.; Ozaki, M.; Yoshino, K. Separate measurements of the flexoelectric and surface polarization in a model nematic liquid crystal p-methoxybenzylidene-p
^{′}-butylaniline: Validity of the quadrupolar approach. Phys. Rev. E**2001**, 64, 031707:1–031707:4. [Google Scholar] [CrossRef] - Harden, J.; Mbanga, B.; Ėber, N.; Fodor-Csorba, K.; Sprunt, S.; Gleeson, J.T.; Jákli, A. Giant flexoelectricity of bent-core nematic liquid crystals. Phys. Rev. Lett.
**2006**, 97, 157802:1–157802:4. [Google Scholar] [CrossRef] - Barberi, R.; Barbero, G.; Gabbasova, Z.; Zvezdin, A. Flexoelectricity and alignment phase transitions in nematic liquid crystals. J. Phys. II
**1993**, 3, 147–164. [Google Scholar] - Sparavigna, A.; Wolf, R.A. Electron and ion densities in corona plasma. Czech. J. Phys.
**2006**, 56, B1062–B1067. [Google Scholar] [CrossRef] - Barbero, G.; Strigazzi, A. Influence of the surface-like volume elasticity on the critical thickness of a hybrid aligned nematic cell. J. Phys. Lett.
**1984**, 45, L857–L862. [Google Scholar] [CrossRef] - Palto, S.P.; Mottram, N.J.; Osipov, M.A. Flexoelectric instability and a spontaneous chiral-symmetry breaking in a nematic liquid crystal cell with asymmetric boundary conditions. Phys. Rev. E
**2007**, 75, 061707:1–061707:8. [Google Scholar] [CrossRef] - Belavin, A.A.; Polyakov, A.M. Metastable states of 2-dimensional isotropic ferromagnets. JETP Lett.
**1975**, 22, 245–247. [Google Scholar] - Abdullaev, F.K.; Galimzyanov, R.M.; Kirakosyan, A.S. Local magnon modes and resonances for dynamical skyrmions in Heisenberg two-dimensional ferromagnets. Phys. Rev. B
**1999**, 60, 6552–6557. [Google Scholar] [CrossRef]

© 2009 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Sparavigna, A.
Role of Lifshitz Invariants in Liquid Crystals. *Materials* **2009**, *2*, 674-698.
https://doi.org/10.3390/ma2020674

**AMA Style**

Sparavigna A.
Role of Lifshitz Invariants in Liquid Crystals. *Materials*. 2009; 2(2):674-698.
https://doi.org/10.3390/ma2020674

**Chicago/Turabian Style**

Sparavigna, Amelia.
2009. "Role of Lifshitz Invariants in Liquid Crystals" *Materials* 2, no. 2: 674-698.
https://doi.org/10.3390/ma2020674