# Molecular–Statistical Theory for the Description of Re-Entrant Ferroelectric Phase

## Abstract

**:**

## 1. Introduction

## 2. Molecular Model

## 3. Free Energy of Smectic Molecules with Transverse Dipoles in Rigid Cores and in Flexible Tails

## 4. Spontaneous Polarization in the Absence of Piezoelectricity and Flexoelectricity

## 5. Polarization in the Presence of Electric Field: Re-Entrant Ferroelectric Phase

## 6. Temperature Induced Transition Between Synclinic And Anticlinic Smectic Phases

## 7. Helical Rotation, Elasticity and Deformation of Sm-${\mathbf{C}}^{*}$, Sm-${\mathbf{C}}_{\mathit{A}}^{*}$ and Sm-${\mathbf{C}}_{\mathrm{Re}}^{*}$ in the Electric Field

## 8. Dielectric Response

## 9. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Average Multiple of Various Projections of Dipole Moments $\mathbf{\mu}$ and ${\mathbf{\mu}}^{*}$

## Appendix B. Free Energy Expansion in Taylor Series with Respect to the Local Field

## References

- Meyer, R.B. Ferroelectric Liquid Crystals. Mol. Cryst. Liq. Cryst.
**1977**, 36, 69–71. [Google Scholar] [CrossRef] - Chandani, A.D.L.; Gorecka, E.; Ouchi, Y.; Takezoe, H.; Fukuda, A. Antiferroelectric Chiral Smectic Phases Responsible for the Tristable Switching in MHPOBC. Jpn. J. Appl. Phys.
**1989**, 28, L1265. [Google Scholar] [CrossRef] - Novotna, V.; Hamplova, V.; Kaspar, M.; Glogarova, M.; Bubnov, A.; Lhotakova, Y. Phase diagrams of binary mixtures of antiferroelectric and ferroelectric compounds with lactate units in the mesogenic core. Ferroelectrics
**2004**, 309, 103–109. [Google Scholar] [CrossRef] - Bubnov, A.; Novotna, V.; Hamplova, V.; Kaspar, M.; Glogarova, M. Effect of multilactate chiral part of the liquid crystalline molecule on mesomorphic behaviour. J. Mol. Struct.
**2008**, 892, 151–157. [Google Scholar] [CrossRef] - Novotna, V.; Glogarova, M.; Hamplova, V.; Kaspar, M. Re-entrant ferroelectric phases in binary mixtures of ferroelectric and antiferroelectric homologues of a series with three chiral centers. J. Chem. Phys.
**2001**, 115, 9036–9041. [Google Scholar] [CrossRef] - Bubnov, A.; Kaspar, M.; Hamplova, V.; Glogarova, M.; Samaritani, S.; Galli, G.; Andersson, G.; Komitov, L. New polar liquid crystalline monomers with two and three lactate groups for preparation of side chain polysiloxanes. Liq. Cryst.
**2006**, 33, 559–566. [Google Scholar] [CrossRef] - Kaspar, M.; Hamplova, V.; Novotna, V.; Glogarova, M.; Pociecha, D.; Vanek, P. New series of ferroelectric liquid crystals with two or three chiral centres exhibiting antiferroelectric and hexatic phases. Liq. Cryst.
**2001**, 28, 1203–1207. [Google Scholar] - Na, Y.; Naruse, Y.; Fukuda, N.; Orihara, H.; Fajar, A.; Hamplova, V.; Kaspar, M.; Glogarova, M. E-T Phase Diagrams of an Antiferroelectric Liquid Crystal with Re-Entrant Smectic C* Phase. Ferroelectrics
**2008**, 364, 13–19. [Google Scholar] [CrossRef] - Catalano, D.; Domenici, V.; Marini, A.; Veracini, C.A.; Bubnov, A.; Glogarova, M. Structural and orientational properties of the ferro, antiferroelectric, and re-entrant smectic C* phases of ZLL7* by Deuterium NMR and other experimental techniques. J. Phys. Chem. B
**2006**, 110, 16459–16470. [Google Scholar] [CrossRef] [PubMed] - Domenici, V.; Marini, A.; Menicagli, R.; Veracini, C.A.; Bubnov, A.M.; Glogarova, M. Dynamic behaviour of a ferroelectric liquid crystal by means of Nuclear Magnetic Resonance and Dielectric Spectroscopy. In Proceedings of the SPIE 6587, Liquid Crystals and Applications in Optics, Prague, Czech Republic, 16–19 April 2007; Volume 6587, p. 65871F1. [Google Scholar]
- Domenici, V.; Bubnov, A.; Marini, A.; Hamplova, V.; Kaspar, M.; Glogarova, M.; Veracini, C.A. The ferroelectric SmC* phase studied by means of 2H and 13C NMR: structural and orientational features. In Proceedings of the 37th Topical Meeting of the German Liquid Crystal Society, Stuttgart, Germany, 1–3 April 2009; pp. 115–116. [Google Scholar]
- Emelyanenko, A.V. Molecular-ststistical approach to a behaviour of ferroelectric, antiferroelectric and ferrielectric smectic phases in the electric field. Eur. Phys. J. E
**2009**, 28, 441–455. [Google Scholar] [CrossRef] [PubMed] - Emelyanenko, A.V. Theory for the evolution of ferroelectric, antiferroelectric, and ferrielectric smectic phases in the electric field. Phys. Rev. E
**2010**, 82, 031710. [Google Scholar] [CrossRef] [PubMed] - Emelyanenko, A.V.; Ishikawa, K. Smooth transitions between biaxial intermediate smectic phases. Soft Matter
**2013**, 9, 3497–3508. [Google Scholar] [CrossRef] - Emelyanenko, A.V. Induction of new ferrielectric smectic phases in the electric field. Ferroelectrics
**2016**, 495, 129–142. [Google Scholar] [CrossRef] - Emelyanenko, A.V.; Osipov, M.A. Theoretical model for the discrete flexoelectric effect and a description for the sequence of intermediate smectic phases with increasing periodicity. Phys. Rev. E
**2003**, 68, 0517033. [Google Scholar] [CrossRef] - Emelyanenko, A.V.; Fukuda, A.; Vij, J.K. Theory of the intermediate tilted smectic phases and their helical rotation. Phys. Rev. E
**2006**, 74, 011705. [Google Scholar] [CrossRef][Green Version] - Emelyanenko, A.V.; Filimonova, E.S. Molecular-statistical approach to the description of tilted smectic phases. Phase Transit.
**2018**, 91, 984–993. [Google Scholar] [CrossRef]

**Figure 2.**(Color online) Molecular model: $\{\mathbf{x},\mathbf{y},\mathbf{z}\}$ is the laboratory frame; $\{\mathbf{w},{\mathbf{m}}^{\prime},\mathbf{n}\}$ is the local frame related to molecular core, where axis $\mathbf{w}$ is perpendicular to the tilt plane of director $\mathbf{n}$; $\{\mathbf{b},\mathbf{c},\mathbf{a}\}$ is the local frame related to molecular tail, where axis $\mathbf{b}$ is perpendicular to the tilt plane of tail axis $\mathbf{a}$. Inset (

**a**) shows the projection on the director tilt plane, and inset (

**b**) shows the projection on the tilt plane of tail axis.

**Figure 3.**(Color online) Positional correlation of terminal molecular dipoles. Dipoles located in the cores are essentially larger, but they do not correlate.

**Figure 4.**(Color online) Spontaneous polarization along axis $\mathbf{k}$ (

**a**) and along axis $\mathbf{m}$ (

**b**) as a function of reduced temperature at ${\mu}_{\mathrm{ef}}=1$ and ${S}_{2}={S}_{4}=0.01$ (red circles for the synclinic phase and blue up triangles for the anticlinic phase); $0.05$ (red rectangles and blue down triangles, respectively); $0.08$ (red diamonds and blue stars, respectively). Here ${T}^{*}$ is the transition temperature into Sm-${A}^{*}$; ${B}_{0}/{B}_{1}=0.04$; ${B}_{1}^{-1}=0.25$.

**Figure 5.**Transition tilt angle, at which the proper spontaneous polarization ${\mathbf{p}}_{0}$ arises, as a function of nematic order parameter of flexible tails at ${\mu}_{\mathrm{ef}}=1$ in the synclinic phase (1) and in the anticlinic phase (2).

**Figure 6.**(Color online) (

**a**) Electric field–Temperature phase diagram at ${\mu}_{\mathrm{ef}}=1$, ${c}_{p}=-0.05$, ${c}_{f}=0.02$, ${v}_{1}=-1.05$, ${v}_{3}=2.77$, ${v}_{5}=0.025$, ${S}_{2}={S}_{4}=0$, ${S}_{1}=0.5$, ${S}_{3}=0.25$, ${B}_{0}/{B}_{1}=0.04$; ${B}_{1}^{-1}=0.25$. Here ${T}^{*}$ is the phase transition temperature into Sm-${A}^{*}$, solid black thin lines detach Sm-${C}_{\mathrm{re}}^{*}$, Sm-${C}_{A}^{*}$ and Sm-${C}^{*}$, dash blue thick line detaches the above phases from the bidomain synclinic smectic phase with tilt plane projections either along or against the electric field, and dash dot red thick line detaches helical phases from the unwound ones. Black arrows inside molecules show the direction of polarization ${\mathbf{p}}^{*}$; (

**b**) Experimental phase diagram–reproduced with permission from Ref. [8]. Copyright Taylor and Francis, 2008.

**Figure 7.**Equilibrium helical pitch at $E=0$ in Sm-${C}_{\mathrm{re}}^{*}$, Sm-${C}_{A}^{*}$ and Sm-${C}^{*}$ at ${\mu}_{\mathrm{ef}}=1$, ${c}_{p}=-0.05$, ${c}_{f}=0.02$, ${v}_{1}=-1.05$, ${v}_{3}=2.77$, ${v}_{5}=0.025$, ${S}_{2}={S}_{4}=0$, ${S}_{1}=0.5$, ${S}_{3}=0.25$, ${B}_{0}/{B}_{1}=0.04$; ${B}_{1}^{-1}=0.25$. Here ${T}^{*}$ is the phase transition temperature into Sm-${A}^{*}$.

**Figure 8.**Deformation of smectic layers in electric field $\mathbf{E}$ in the presence of spontaneous polarization along smectic layer normal $\mathbf{k}$.

**Figure 9.**Dielectric permittivity related to reorientation of spontaneous polarization in the electric field in Sm-${C}_{\mathrm{re}}^{*}$, Sm-${C}_{A}^{*}$ and Sm-${C}^{*}$ at ${\mu}_{\mathrm{ef}}=1$, ${c}_{p}=-0.05$, ${c}_{f}=0.02$, ${v}_{1}=-1.05$, ${v}_{3}=2.77$, ${v}_{5}=0.025$, ${S}_{2}={S}_{4}=0$, ${S}_{1}=0.5$, ${S}_{3}=0.25$, ${B}_{0}/{B}_{1}=0.04$; ${B}_{1}^{-1}=0.25$, ${\mu}^{*}E/\left({K}_{\gamma}{k}_{B}{T}^{*}\right)=4.55$. Here ${T}^{*}$ is the phase transition temperature into Sm-${A}^{*}$.

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Emelyanenko, A.V. Molecular–Statistical Theory for the Description of Re-Entrant Ferroelectric Phase. *Crystals* **2019**, *9*, 583.
https://doi.org/10.3390/cryst9110583

**AMA Style**

Emelyanenko AV. Molecular–Statistical Theory for the Description of Re-Entrant Ferroelectric Phase. *Crystals*. 2019; 9(11):583.
https://doi.org/10.3390/cryst9110583

**Chicago/Turabian Style**

Emelyanenko, Alexander V. 2019. "Molecular–Statistical Theory for the Description of Re-Entrant Ferroelectric Phase" *Crystals* 9, no. 11: 583.
https://doi.org/10.3390/cryst9110583