Antiferroelectric liquid crystals: Interplay of simplicity and complexity

Antiferroelectric liquid crystals: Interplay of simplicity and complexity

Hideo Takezoe*

Department of Organic and Polymeric Materials, Tokyo Institute of Technology,O-okayama, Meguro-ku, Tokyo 152-8552, Japan

Ewa Gorecka†

Department of Chemistry, Warsaw University, Al. Zwirki i Wigury 101, 02-089 Warsaw,Poland

Mojca Cepič‡

Jozef Stefan Institute, Jamova 39, 1000 Ljubljana, Sloveniaand Faculty of Education, University of Ljubljana, Kardeljeva pl 16, 1000 Ljubljana,Slovenia

�Published 23 March 2010�

This paper reviews nearly 20 years of research related to antiferroelectric liquid crystals and gives ashort overview of possible applications. “Antiferroelectric liquid crystals” is the common name forsmectic liquid crystals formed of chiral elongated molecules that exhibit a number of smectic �Sm�tilted structures with variation of the strong-tilt azimuthal direction from layer to layer �i.e.,nonsynclinic structures�. The phases have varying crystallographic unit periodicity from a few �SmC

�*�,

four �SmCFI2* �, three �SmC

FI1* �, and two �SmC

A* � smectic layers and all of the phases possess liquidlike

order inside the layer. The review describes the discovery of these phases and various methods usedfor their identification and to determine their structures and their properties. A theoretical descriptionof these systems is also given; one of the models—the discrete phenomenological model—ofantiferroelectric liquid crystals is discussed in detail as this model allows for an explanation of phasestructures and observed phase sequences under changes of temperature or external fields that is mostconsistent with experimental results.

DOI: 10.1103/RevModPhys.82.897 PACS number�s�: 61.30.Dk, 42.70.Df, 42.79.Kr

CONTENTS

I. Introduction 897

II. Discovery of Antiferroelectric and Ferrielectric

Liquid Crystal Phases 899

A. Toward the discovery of the antiferroelectric phase 899

B. Identification of the antiferroelectric phase structure 900

C. Discovery of the SmC�* and intermediate phases 903

III. The Search for Internal Structures of Multilayer

Phases 905

A. The crystallographic unit cell 905

B. Secondary helical modulation of the phases 908

C. Polarity of multilayer phases 909

D. Temperature sequence of phases 910

E. Present understanding of phase structures 911

IV. Theoretical Modeling of Phases and Phase Sequences 911

A. Continuous model 912

1. The nontilted SmA phase 914

2. The simply modulated SmC* phase 914

3. The simply modulated SmCA* phase 914

4. The two simply modulated ferrielectric

SmCSM* phases 914

5. The incommensurate SmCIC* phase 915

B. Simple clock model 916C. Origin of interactions 919D. Distorted clock model 920

1. Lock in to commensurate periods 9222. The four-layer SmC

FI2* phase 923

3. The three-layer SmCFI1* phase 925

4. Some additional interesting results ofdiscrete modeling 925

5. Discrete model with long-range interactions 9266. Ising-like models 927

E. Other theoretical approaches 9281. Molecular statistical approach 9282. Computer simulations 929

F. Discussion 930V. Applications 930

VI. Conclusions and Open Questions 932Glossary 933Acknowledgments 934References 934

I. INTRODUCTION

Liquid crystals are now indispensable materials forflat panel displays. However, the discovery of this classof materials was made only 120 years ago �Kelker, 1988�.Since then, a variety of phases has been discovered, andeven recently new phases have been found and charac-

*[email protected]†[email protected]‡[email protected]

REVIEWS OF MODERN PHYSICS, VOLUME 82, JANUARY–MARCH 2010

0034-6861/2010/82�1�/897�41� ©2010 The American Physical Society897

http://dx.doi.org/10.1103/RevModPhys.82.897

terized. Thus, liquid crystals are not only useful for prac-tical application but also a rich source for the study ofphysics and chemistry of soft matter.

The formation of liquid crystalline structures, particu-larly the nematic phase, is naively explained by an elec-trostatic attractive force, i.e., the van der Waals interac-tion, and a repulsive force, i.e., the excluded volumeeffect �Chandrasekhar, 1977; de Gennes and Prost,1993�. For rodlike or disklike molecules, these interac-tions lead to the nematic phase, where the long axes ofthe rodlike molecules and the disk normal of the disk-like molecules orient on average toward a particular di-rection called the director. However, one cannot de-scribe the antiferroelectric �AF� and ferrielectric phases,in which molecules form layered structures with com-plex director changes from layer to layer, by these basicinteractions only. The structures that we discuss areshown in Fig. 1. As the temperature is lowered, thephases usually observed are �a� the SmC

�* phase which

has a short pitch, �b� the conventional ferroelectricSmC* phase, �c� the SmCFI2

* and �d� the SmCFI1* phases

with unit cells of four and three layers, respectively, and�e� the antiferroelectric SmC

A* phase. The purpose of

this review is to summarize the current knowledge aboutthe antiferroelectric and ferrielectric phases, describinghow these structures are experimentally identified andhow the formation of and transitions between thesestructures are theoretically explained.

For the system to be ferroelectric it is necessary tohave sufficiently low symmetry at least Cn or Cnv �seeFig. 2�. Why? If the symmetry is C�h as for the nonpolarSmA phase, the uniaxial molecules, described as simplerods �Fig. 2, upper right�, are free to rotate around theirlong axes. This rotation does not allow for polar order ofdipoles in the smectic plane �transverse polarization�.Moreover, the horizontal mirror plane symmetry, if itexists in the system, does not allow for polar order ofdipoles in the direction perpendicular to the smecticplane �longitudinal polarization�. Therefore such sys-tems cannot be polar. Polar properties cannot appeareven in a system of biaxial lathlike molecules, which re-duces the symmetry to C2h �Fig. 2, top middle�, because

the orientation of transverse molecular dipole momentsis symmetrically distributed around the C2 axis. How-ever, the phase structure can be macroscopically polar ifit has C2v symmetry. This can be realized, for example,in a smectic formed of nontilted “half rods” �Fig. 2, left�or in a smectic with C2 symmetry if half rods are addi-tionally tilted from the layer normal. In both cases theC2 axis prevents longitudinal polarization but transverse�in-plane� polarization can exist. In liquid crystals, re-duction of the symmetry is necessary to obtain polarproperties—C2v symmetry is obtained in a nontiltedsmectic made of bent-core molecules �Niori et al., 1996�and C2 symmetry is realized if a SmC phase is formed ofchiral molecules since chirality removes the mirror planein the system �Meyer et al., 1975�. The actual value ofthe electric polarization depends on the hindrance ofrotation around the long molecular axis.

Polar properties can be found in other systems as well.Even in diskotic columnar phases, the introduction ofchirality works in a similar way as in the tilted chiralsmectic phases. Scherowsky and Chen �1994� and Bockand Helfrich �1995� obtained a helical diskotic columnarphase and found a ferroelectric switching between twostates with uniformly tilted disks with respect to the di-rection of the column. The second possibility of intro-ducing polar properties by changing the shape of themolecules from disklike to conical was also studied�Gorecka et al., 2004; Kishikawa et al., 2005; Takezoe etal., 2006�. Bent-core liquid crystals are also typical ex-amples of systems with low symmetry. In many bent-core smectic phases, the bent core lowers the symmetryof the system to C2v by preventing free rotation aboutthe long axis �Niori et al., 1996�. It is worth mentioningthat for these systems formed of molecules with bent

SmC SmC SmC SmC SmCor or

SmC SmC

* * * * *� FI2 FI1 A

(a) (b) (c) (d) (e)

* *� �

12

3

5

1

54321

3

3

3 4

4

4

51

1

12

22

FIG. 1. Schematic illustration of molecular ordering in struc-tures of �a� SmC

�* , �b� SmC*, �c� SmC

FI2* , �d� SmC

FI1* , and �e�

SmCA* phases, which emerge as temperature decreases.

C

CC

C

2 2h

2h2v

C axis2

C h

C axis

FIG. 2. Simple illustration of realization of Cn and Cnv sym-metries necessary for polar systems in nontilted �top� and tilted�bottom� smectic phases.

898 Takezoe, Gorecka, and Čepič: Antiferroelectric liquid crystals: Interplay …

Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010

cores, phases with C1 symmetry are also possible; forthese, polarization is not constrained to any specific di-rection �Jákli et al., 2001; Bailey and Jákli, 2007;Gorecka et al., 2008�.

In the above examples of smectic phases, the polariza-tion direction is perpendicular to the molecular longaxis, except in the phases with local C1 symmetry. In theuniaxial nematic phase, however, the polar directionmust be parallel to the director because of the free ro-tation of molecules about their long axis. Therefore, it ismuch more difficult to realize polar order in the nematicphase. Two examples have been reported so far, i.e.,polypeptides �Yen et al., 2006� and aromatic polyesters�Koike et al., 2007�. In both cases, the dipole-dipole in-teraction between rigid organizations of sequential di-poles seems to be the origin of the polar order �Ter-entjev et al., 1994�. In this sense, these ferronematicphases are proper ferroelectrics, for which spontaneouspolarization appears as a consequence of dipole-dipoleinteractions and which have never been realized in otherliquid crystalline phases.

II. DISCOVERY OF ANTIFERROELECTRIC ANDFERRIELECTRIC LIQUID CRYSTAL PHASES

A. Toward the discovery of the antiferroelectric phase

The ferroelectric SmC* phase was discovered byMeyer �Meyer et al., 1975; Meyer, 1977� on the basis ofelegant symmetry considerations. Because of the pos-sible application to fast displays, extensive studies havebeen conducted from the physics, chemistry, and engi-neering viewpoints. During this course of studies, atleast three groups noticed peculiar behaviors in their in-dividual materials. However, it has taken a long time tounderstand that these peculiarities can be attributed toin the antiferroelectric phase. This was probably becauseof a common wisdom for liquid crystals, namely, “mol-ecules in liquid crystal tend to align parallel to eachother.”

The materials in which the three groups foundthe antiferroelectric phase are listed in Fig. 3: 4-�1-methylheptyloxycalbonyl�phenyl 4�-octyloxybiphenyl-4-carboxylate �MHPOBC� �Inukai et al., 1985�, 1-methyl-heptyl-terephthalidene-bis-aminocinnamate �MHTAC��Levelut et al., 1983�, and �R�- and �S�-1-methylpentyl

4�-�4�-n-decyloxybenzoyloxy� biphenyl-4-carboxylates�Goodby and Chin, 1988�. All these compounds weresynthesized rather a long time ago before the identifica-tion of the antiferroelectric phase structure and wererecognized as new ferroelectric liquid crystal �FLC� ma-terials. The MHPOBC compound was first reported in1985 by Inukai et al. �1985�. In the first internationalconference on ferroelectric liquid crystals �FLC87� inArcachon, France, two groups pointed out unusual be-haviors in this compound. Hiji et al. �1988� reported athird stable state exhibiting a dark view between crossedpolarizers when one of the polarizers is parallel to thesmectic layer. Furukawa et al. �1988� reported a verysmall dielectric constant and a threshold behavior in theelectro-optic response in the lower-temperature regionof SmC*, suggesting a new phase SmY*. It is now clearthat these experimental facts unambiguously indicatedthe structure called the herringbone or antiferroelectricstructure, where molecules in neighboring layers aretilted in opposite directions. In this phase the antiparal-lel or anticlinic molecular tilt in adjacent layers results inthe structure showing light extinction between crossedpolarizers along the layer normal and an electric-field-induced transition to the ferroelectric structure with athreshold and extinction direction inclined from thelayer normal, as shown in Fig. 4.

An MHTAC analog was first synthesized in 1976, andMHTAC first appeared in the literature in 1983 �Levelutet al., 1983�. Phase identification was attempted based ontexture observation �mainly�, miscibility tests, and x-rayanalysis. Detailed experiments gave many hints of theantiferroelectric structure. For instance, some observedmany 1/2 disclination defects, which were later recog-nized as characteristic for an antiferroelectric structure.Unfortunately, they could not establish the phase struc-ture. The antiferroelectric structure of this material wasfirst reported at the Second International Conference onFLCs �1989, Göteborg� �Galerne and Liebert, 1989�. Atthe same conference, Takezoe et al. also reported theantiferroelectricity of MHPOBC �Takezoe, Chandani,Gorecka, et al., 1989; Takezoe, Chandani, Lee, et al.,�1989�.

COOn-C8H17O COO-CH(CH3)-n-C6H13*

COOn-C8H17On-C8H17O COO-CH(CH3)-n-C6H13)-n-C6H13*

CH N= CH CH=NCH= CHCOOCH(CH3)C6H13*

6 3)OCOCH=H13C CH(CH*

CH N= CH CH=NCH= CHCOOCH(CH3)C6H13*

CHCOOCH(CH3)C6H13)C6H13*

6 3)OCOCH=H13C CH(CH*

6 3)OCOCH=H13CH13C CH(CH*

COOn-C10H21O COO-CH(CH3)-n-C4H9*

COOCOOn-C10H21O COO-CH(CH3)-n-C4H9)-n-C4H9*

1: MHPOBC

2: MHTAC

3: 10B1M5

FIG. 3. Chemical structures of important compounds in whichthe antiferroelectric phase was identified early.

E=0E EE=0E E

(a) (b) (c)

FIG. 4. The local molecular arrangements of �b� the antiferro-electric phase and �a� the positive and �c� negative electric-field-induced ferroelectric phases.

899Takezoe, Gorecka, and Čepič: Antiferroelectric liquid crystals: Interplay …


Goodby and Chin �1988� synthesized �R�- and �S�-1-methylpentyl 4�-�4�-n-decyloxybenzoyloxy�biphenyl-4-carboxylates in 1988 and reported the phase behaviorwithout noticing the existence of the antiferroelectricphase in these compounds. They actually found two dis-tinct phases below the ferroelectric SmC* phase andnoted the influence of racemization on the phase se-quence. They suggested several phase structures such asSmI*-like or SmF*-like, depending on optical purity, andsimply SmC* phases with different senses of the helicalmodulation. Finally they reported that these two phasesare actually ferrielectric and antiferroelectric �SmC

A* �

phases �Goodby et al., 1992�.The proposal for the application of antiferroelectric

liquid crystals in display devices was made in 1988 be-fore the phase identification, because of their tristableswitching with a sharp threshold and double hysteresis�Chandani et al., 1988�. The tristable switching is observ-able by two methods: the electro-optic effect and switch-ing current measurements. These results are shown inFig. 5 �Chandani et al., 1988�. The transmittance showedthree stable states under the application of a triangular-wave voltage, and two switching current peaks appearedwhen sharp transmittance changes occurred between thetristable states. Figure 6 shows the apparent tilt angle asa function of a dc electric field �Chandani et al., 1988�.The apparent tilt angle is zero in the absence of the fieldand shows sharp changes to finite positive and negativeangles with thresholds and hysteresis, as reported at theFirst International FLC Conference in 1987 by Hiji et al.�1988� and Furukawa et al. �1988�. Clearly, a switchingdevice could be made by utilizing the threshold and the

hysteresis. Actually, Chandani et al. demonstrated theelectro-optic performance: transmittance changes on ap-plication of positive and negative pulses superposed on abiased voltage, as shown in Fig. 7 �Chandani et al., 1988�.This is the essential principle of the tristable antiferro-electric display �Yamamoto et al., 1992�. Despite the ob-servation of the double hysteresis loop �Fig. 6� charac-teristic of the antiferroelectric phase the identification ofthe antiferroelectric phase was not made at this stage. Itwas another year before it was clarified that the electro-optic switching appeared due to the electric-field-induced antiferroelectric-ferroelectric transition.

B. Identification of the antiferroelectric phase structure

As mentioned, two groups presented the antiferro-electric structure shown in Figs. 1�e� and 4�b� at the

0V

oltag

e(V

)T

ransm

itta

nce

0

Curr

ent(a

.u.)

(a)

(b)

(c)

+10

-10

0V

oltag

e(V

)T

ransm

itta

nce

0

Curr

ent(a

.u.)

0 100 200 300 400 500

Time (ms)

(a)

(b)

(c)

+10

-10

FIG. 5. Transmittance change and switching current observedby applying a triangular-wave voltage to MHPOBC. Note thattwo changes in the transmittance and two current peaks areobserved at the same voltages. From Chandani et al., 1988.

+20

+10

0

-10

-20

-5-10

+5 +10

Applied Voltage (V)

Appare

nt

Tilt

Angle

(deg)

+20

+10

0

-10

-20

-5-10

+5 +10

Applied Voltage (V)

Appare

nt

Tilt

Angle

(deg)

FIG. 6. Apparent tilt angle in MHPOBC under the applicationof dc electric voltages. Double hysteresis loop is observed.From Chandani et al., 1988.

0 200 400

Time (ms)

0

6

0

1

2

Tra

nsm

itta

nce

(a.u

.)V

oltag

e(v

)

0 200 400

Time (ms)

0

6

0

1

2

Tra

nsm

itta

nce

(a.u

.)V

oltag

e(v

)

FIG. 7. Electro-optic response of MHPOBC obtained by ap-plication of positive and negative electric pulses in addition toa bias voltage of 6 V. From Chandani et al., 1988.



same conference in 1989 �Galerne and Liebert, 1989;Takezoe, Chandani, Gorecka, et al., 1989; Takezoe,Chandani, Lee, et al., 1989�. They used the different ma-terials and methods to identify the phase structure. Wenow introduce four methods used to identify the antifer-roelectric phase, some of which are still the most conve-nient and decisive methods for identification even now.

Chandani, Gorecka, et al. �1989� proposed a doublehelical structure shown in Fig. 8 for the new phase anddesignated it SmC

A* , where A stands for antiferroelectric

with respect to polar organization or anticlinic with re-spect to the tilt interlayer structure. They used selectivereflection at oblique incidence to a thick sample cell, inwhich the helical structure shown in Fig. 8 is formedwith the helical axis perpendicular to the substrate sur-faces. Imagine a cholesteric �chiral nematic, N*� liquidcrystal with helical modulation of the director. Thelength in which the local director �optical indicatrix�turns by 2� is one pitch. However, the actual periodicityis only half of the pitch because of the head-tail equiva-lence of the director �Fig. 9�. Therefore, the first-orderBragg reflection �selective reflection� emerges at thewavelength corresponding to one optical pitch �the ac-tual periodicity multiplied by the average refractive in-dex�. The situation is the same for the SmC* and SmC

A*

phases for normal incidence of light. If the incident lightis oblique with respect to the axis of the helical modula-tion, a periodical structure with one full pitch must beobserved in the SmC* structure �Fig. 9�. If one measuresthe selective reflection using obliquely incident light, thereflected wavelength still indicates half of the pitchlength in the N* and SmC

A* structures. This fact makes

the obliquely incident reflection spectra in the SmC* andSmC

A* structures different; i.e., a full-pitch band emerges

in the SmC* structure �Hori, 1983; Ouchi et al., 1984� butnot in the SmC

A* structures, as shown in Fig. 10. The first

identification of the SmCA* helical structure by Chan-

dani, Gorecka, et al. �1989� was thus made. They con-firmed that the tilted smectic phase has a helical struc-ture similar to that of the choleresteric N* phase. Strictlyspeaking, however, they did not experimentally show thelayer-by-layer anticlinic structure.

A compatible result with the selective reflection ex-periment was obtained by Muševi~ et al. �1993� usingquasielastic backward light scattering. Gapless phasondispersion was obtained in the SmC

A* phase of

MHPOBC, showing a minimum at q=2qc, where qc isthe wave vector of the unperturbed helical structure, asshown in Fig. 11. This result is consistent with analternating-tilt molecular orientational structure. Itshould be noted that the molecular fluctuation, the am-plitude �soft� mode, and the phase �sometimes incor-

x

z

y�

half pitch

molecule dipolelayer plane

x

z

y�

half pitch

molecule dipolelayer plane

FIG. 8. Helical structure of the SmCA* phase.

N SmC SmCA* * *

actu

alpit

ch

actu

alper

iod

actu

alpit

ch actu

alper

iod

FIG. 9. Schematic structure and corresponding variation ofindicatrix along the helical axis in the N*, SmC

A* , and SmC*

phases. Note that the helices in the N* and SmCA* phases are

optically the same, but that in the SmC* phase is different.

0.5 1.0 1.5 2.0 2.5

Wavelength (mm)

Tra

nsm

itta

nce

0.6

1.0

SmC*

SmCFI1*

SmCA*

0.5 1.0 1.5 2.0 2.5

Wavelength (mm)

Tra

nsm

itta

nce

0.6

1.0

SmC*

SmCFI1*

SmCA*

FIG. 10. Transmittance spectra at oblique incidence in theSmC*, SmC

FI1* �or SmC

�*�, and SmC

A* phases of MHPOBC.

Note that no transmittance loss is observed at the wavelengthindicated by an arrow in the SmC

A* phase, as explained. No

transmittance loss is observed in SmCFI1* because of the long

pitch. From Chandani, Gorecka, et al., 1989.

FIG. 11. Gapless antiferroelectric phason in the SmCA* phase

measured by the photocorrelation technique. From Muševi~ etal., 1993.



rectly named the Goldstone� mode are not easy to de-tect by dielectric measurements because themacroscopic dipole moment nearly vanishes. In thissense, the photon correlation technique is useful for de-tecting such fluctuation modes in the SmC

A* phase

�Muševi~ et al., 1993�.Galerne and Liebert �1990� first showed the layer-by-

layer anticlinic structure. They called the novel phasethe SmO* phase, which was later identified to be identi-cal to the SmC

A* phase �Cladis and Brand, 1993; Heppke

et al., 1993; Takanishi et al., 1993�. They prepared theSmO* film on the free surface of the isotropic MHTACdroplets by precise control of the temperature. Underdefinite layer numbers, they applied an in-plane electricfield to measure the polarization direction and foundthat the polarization direction alters from layer to layer,as shown in Fig. 4�b�. However, it should be noted thatin thin films of the SmC

A* phase, in order to properly

describe the electric-field response we must take intoaccount not only the polarization perpendicular to thetilt plane but also the layer-boundary polarization paral-lel to the tilt plane, which inevitably exists because theup-down symmetry of molecular distribution is brokenat surface layers. Therefore the surface layers must havepolarization along the long molecular axes as well �Linket al., 1996, 2001�. The former is zero for even-layeredstructures, whereas the latter is zero for odd-layeredstructures.

Another experiment to confirm the layer-by-layer al-ternation of polarization and tilt was conducted by Bahrand Fliegner �1993a, 1993b� using free-standing films.They used transmission ellipsometry with obliquely inci-dent light. They measured the retardation �+ and �−under opposite electric fields perpendicular to the inci-dence plane. Figure 12 shows the results for �a� three-and �b� four-layer free-standing films. A large differencein the retardations in the SmC* phase is observed inboth films because of the tilt of the optical axis eitheraway from or toward the incoming laser beam. However,the SmC

A* phase is distinctly different in odd- and even-

layered films. In the four-layer film, the retardation doesnot depend on the field direction, as shown in Fig. 12�b�.In the three-layered film, a smaller retardation differ-ence for opposite field directions is shown than in theSmC* phase, as shown in Fig. 12�a�. These measure-ments were performed using two-, three-, and four-layerfilms and verified the layer-by-layer alternation of thepolarization and tilt. They also performed the same ex-periments in the SmC�* phase and showed the three-layered repeating unit structure of this phase �Bahr etal., 1994�.

A very simple method for the identification of theSmC

A* phase was proposed by Takanishi, Takezoe,

Fukuda, Komura, and Watanabe �1992� and Takanishi,Takezoe, Fukuda, and Watanabe �1992�. The method isbased on schlieren texture observation. It is known thatnematic cells exhibit two- and four-brush disclination.Analogous four-brush disclination is possible for SmC*,although two-brush disclination is prohibited because adefect plane with opposite tilts is inevitably introduced

�Figs. 13�a�, 13�b�, and 14�a��. Takanishi et al. suggestedthat the plane defect could be removed in the SmC

A*

phase if screw dislocations exist �Fig. 13�c��. Actuallythey showed the existence of two-brush defects �Fig.14�b�� and attributed this structure to dispiration, whichhad been theoretically predicted �Harris, 1970� withoutclear experimental observations. Since this method issimple and applicable even to the racemic SmCA phase,it is widely used for the identification of the SmC

A*

phase.The SmC

A* phase can also be easily identified by direct

birefringence measurements in the helix-unwound state.Because of the anticlinic structure, the birefringence be-comes small. The difference of the birefringence be-tween the unwound SmC* and the unwound SmC

A*

phases is usually easily seen when planar textures areobserved. In the particular case where the tilt angle is45°, the birefringence becomes zero. This optical charac-teristic provides an attractive display device even with-out the necessity of rubbing treatment and good align-ment �D’Have, Dahlgren, et al., 2000; D’Have, Rudquist,et al., 2000�.

Another important technique for identifying theSmC

A* and ferrielectric phases, resonant x-ray diffrac-

tion, is discussed later.

three layers

100 120 140

Temperature ( C)

0.8

1.0

1.2

1.4

1.6

0.8

1.0

1.2

1.4

0.6

four

layers

three

layers

Reta

rdation

(deg)

Reta

rdation

(deg)

SmASmC*

SmC *A

SmC *A

SmC*SmA

(a)

(b)

E

�-

E

�+three layers

100 120 140

Temperature ( C)

0.8

1.0

1.2

1.4

1.6

0.8

1.0

1.2

1.4

0.6

four

layers

three

layers

Reta

rdation

(deg)

Reta

rdation

(deg)

SmASmC*

SmC *ASmC *A

SmC *ASmC *A

SmC*SmA

(a)

(b)

EE

�-

�-

EE

�+

�+

FIG. 12. Phase retardation of �a� three- and �b� four-layer free-standing films of MHPOBC by the oblique incidence of lightunder the application of positive and negative fields perpen-dicular to the optical plane. From Bahr et al., 1994.



Today we know hundreds of materials having theSmCA phase; most of them are chiral compounds.

C. Discovery of the SmC�* and intermediate phases

Unambiguously identified subphases so far are theSmC

�* , the SmCFI1

* �or SmC�*�, the SmC

FI2* �or AF or

SmCAF* � in addition to SmC* �or the SmC

�*�, and the

SmCA* phases, the structures of which are summarized in

Fig. 15. Here MHPOBC-family compounds togetherwith some other important compounds are drawn in theorder of stronger ferroelectricity or weaker antiferro-electricity. The material at the top �CF278� has the ferro-

electric SmC* phase only �strong ferroelectric nature�and the material at the bottom, TFMHPOCBC, has onlythe antiferroelectric SmC

A* �strong antiferroelectric na-

ture�. From the top to the bottom, additional phasessuch as SmC

�* , SmCFI1

* , and SmCFI2* emerge and disap-

pear one by one. In MHPOBC, the SmC* and SmCA*

phases coexist and, in MHPBC, the SmC* phase disap-pears. Fukui et al. �1989� first found some distinct phasesin a small temperature range of MHPOBC. Chandani,Ouchi, et al. �1989� also found the same phase sequenceand designated the phases as the SmC

�* phase, the SmC

�*

phase, and the SmC�* phase in this order from high tem-

perature to the SmCA* phase as the lowest-temperature

phase. They assigned the SmC�* as the SmC* phase �Tak-

ezoe, Lee, Chandani, et al., 1991�. Later, however, somereported that the SmC

�* phase has antiferroelectric na-

ture �Li et al., 1996; Sako et al., 1996; Jakli, 1999� and itwas finally shown that it is actually the SmCFI2

* phase�Gorecka et al., 2002� in optically very pure materials,but the SmC* phase emerges when the optical purityis slightly decreased �Gorecka et al., 2002�. HenceMHPOBC, the first antiferroelectric liquid crystal, hasall subphases of the tilted SmC* family.

As shown in Fig. 1, the repeating units of the SmC*,SmC

A* , SmCFI1

* , and SmCFI2* phases have one, two, three,

and four layers, respectively. All of these phases are chi-ral, and every subphase has the helical structure. TheSmC

�* phase was initially mysterious but was finally as-

signed as a tilted phase with extremely short pitch.

dislocation

screw

edge disclination

smectic layer

c-director

(a)

(c)

(b)

dislocation

screw

edge disclination

smectic layer

c-director

dislocation

screw

edge disclination

smectic layer

c-director

(a)

(c)

(b)

FIG. 13. C-director maps �a�, �b� in the SmC* phase and �c� theSmC

A* phase with screw dislocation lines. Note that four-brush

defects are possible in the SmC* phase but two-brush defectscannot exist because of a defect plane with discontinuousC-director change �solid line in �a��. Two-brush defects canpossibly be introduced in the SmC

A* phase by screw disloca-

tions. From Takanishi, Takezoe, Fukuda, and Watanabe, 1992.

FIG. 14. �Color online� Schlieren textures in �a� the SmC and�b� SmCA phases. Note that only four-brush defects are ob-served in the SmC phase, but two-brush defects are alsopresent in the SmCA phase, as indicated by arrows.

-(40.0 C)-Cry.

SmA-(120.2 C)-SmC�*-(118.7 C)-SmC*FI2-(117.2 C)-SmC*FI1

SmA-(105.5 C)-SmC �*-(99.5 C)-SmCA*-(73.3 C)-SmIA*-(66.1 C)-Cry.

MHPOCBC C 8H17COO

MHPOOCBC C8H

17 OCO COO COOCH(CH3)C6 H13*

SmA-(93.3 C)-SmCA*-(66.9 C)-Cry.

EHPOCBC C 8H17COO

MHPOBC C 8H17 O

A*

MHPBC

TFMHPBC C8H

17

SmA-(75.0 C)-SmC �*-(74.3 C)-SmC A*

TFMHPOBC

TFMHPOCBC

C 8H17 O

SmA-(115.5 C)-SmCA*

SmA-(108.9 C)-SmCA*

C8H

17COO

COO COOCH(CH3)C6 H13*


C8H

17COO COOCH(CH3)C6 H13

*C

8H

17COO COOCH(CH3)C6 H13

*COO COOCH(CH3)C6 H13

*



COO COOCH(CF3)C6 H13*


COO COOCH(C H5 )C6 H13)C6 H13*

2





SmA-(80 C)-SmC*-(28 C)-Cry.

CF278 C11H23 O COO COOCH(CF3)(CH )2 2*

F

4 OC H5C11H23 OC11H23 O COO COOCH(CF3)(CH )2 2*

F

4 OC H5

Incre

asin

gA

Fo

rder

Incre

asin

gF

ord

er

oo

SmA-(87.2 C)-SmC*-(41.9 C)-SmC*FI1o o

SmA-(87.2 C)-SmC*-(41.9 C)-SmC*FI1o o o

o o o-(116.3 C)-SmCA*o-(116.3 C)-SmCA*o

SmA-(76.3 C)-SmC �o

SmA-(76.3 C)-SmC �o

*-(72.1 C)-SmC*FI2-(66.4 C)-SmC*FI1 -(64.9 C)-SmCoo o*-(72.1 C)-SmC*FI2-(66.4 C)-SmC*FI1 -(64.9 C)-SmCoo

*-(72.1 C)-SmC*FI2-(66.4 C)-SmC*FI1 -(64.9 C)-SmCoo o

oooo

oo

o o

o

o

FIG. 15. Compounds that show SmCA* and subphases.

MHPOBC-family compounds are listed in the order of stron-ger ferroelectricity or weaker antiferroelectricity.



As mentioned, the SmC�* phase was found early. As

temperature decreases the SmC�* phase usually appears

between the nontilted SmA phase and one of the sub-phases, the SmC* or the SmC

A* phase. Soon after the

discovery of this phase in 1989, a detailed investigationwas launched. The SmC

�* phase cannot be distinguished

from the SmA phase by texture observation. Other mea-surements, however, can easily detect the SmA-SmC

�*

phase transition. �1� The smectic layer starts to shrink atthe transition, suggesting that the SmC

�* phase is a tilted

phase �Takanishi et al., 1991; Isozaki, Hiraoka, et al.,1992�. Isozaki, Hiraoka, et al. �1992� used MHPOCBCwith a phase sequence of Iso-SmA-SmC

�*-SmC

A* and a

wide temperature range �6 °C� of the SmC�* phase to

observe the transition from or to SmA and SmC�* by the

temperature dependence of the chevron angle �layer-tiltangle� together with dielectric measurements, as shownin Fig. 16. �2� Switching current peaks start to appear onapplication of a triangular-wave field to a MHPOBCcell. Their number, shape, and position change within anarrow temperature range of the phase, and finally asingle switching current peak is observed when the tran-sition to SmC* occurs �Takanishi et al., 1991�. These be-haviors suggest complicated structural change withchange in applied field and temperature. �3� Pretransi-tional effects are observed in the SmA phase: the dielec-tric constant �Fukui et al., 1990; Hiraoka, Chandani, etal., 1990; Hiraoka, Taguchi, et al., 1990; Hiraoka et al.,1992; Isozaki, Hiraoka, et al., 1992� and induced circulardichroism �Lee, Ouchi, et al., 1990� increase with de-creasing temperature, and inflection points are clearlyseen at the transition point. The above observations sug-gest that SmC

�* is a SmC*-like phase. Because it is a

chiral phase, a helical structure was expected. After ex-tensive effort to measure the period of this structure, thepitch was found to be extremely short, depending ontemperature and sample thermal history, i.e., 60–250 nm�Laux et al., 1996�, as theoretically predicted by Cepi~and Žekš �1995� and Roy and Madhusudana �1996�.More detailed pitch measurements in the SmC

�* phase

were made later by Cady et al. �2002� and Hirst et al.�2002� using resonant x-ray diffraction �see Sec. III.B�.

The three- and four-layer periodicities in SmCFI1* and

SmCFI2* , respectively, were suggested in the early stage of

their discovery. For the phase determination, conoscopywas adequate �Gorecka et al., 1990�. We now summarizethe conoscopic data in the SmC

�* , SmC*, SmCFI2

* ,SmCFI1

* , and SmCA* phases, some of them presented in

Fig. 17. In all cases, the applied field is along the verticaldirection. In a sufficiently strong electric field the cono-scopic figures will be the same as for the unwound SmC*

phase for all phases, although a strong enough field isnot always accessible experimentally. In Figs.17�a�–17�d� the field dependence of the conoscopic fig-ures in SmC* is shown �Gorecka et al., 1990�. In theabsence of an electric field, a uniaxial figure almost thesame as the one in SmA and SmC

�* is observed, although

the center of the isogyre is not dark. This is because ofthe optical rotatory power due to the helical structure.With increasing field, the center shifts in a direction per-pendicular to the electric field and biaxiality emerges,indicating unwinding of the helix and a consequent ap-parent tilt, i.e., of the average molecular direction overthe layers �Figs. 17�c� and 17�d��. The optical plane isdefined by a plane containing two optical axes, the onewith the largest and the one with the smallest refractiveindices, respectively. The optical plane is perpendicularto the field direction, indicating that the axis of the mini-mal refractive index is perpendicular to the field and themolecular long axis.

The SmC�* phase looks only slightly distorted under

an electric field, as shown in Fig. 17�e�, which means thatunder applied field the helix is not completely unwound

SmASmC *ASmC *�

SmASmC *A

SmC *�

10

0

-10

10

0

La

ye

rT

iltA

ngle

(de

g)

�’

Temperature ( C)

88 93 98 103 108

SmASmC *ASmC *ASmC *�SmC *�

SmASmC *ASmC *A

SmC *�SmC *�

10

0

-10

10

0

La

ye

rT

iltA

ngle

(de

g)

�’

Temperature ( C)

88 93 98 103 108

FIG. 16. Simultaneous measurements of dielectric constantand chevron angles. Two measurements clearly identify thephase transitions SmA-SmC

�*-SmC

A* and indicate that SmC

�* is

a tilted phase. From Isozaki, Hiraoka, et al., 1992.

230 V/mm8 V/mm170 V/mm 170 V/mm

0 V/mm 7 V/mm 9 V/mm 13 V/mm

(a)

(f)(e) (g) (h)

(b) (c) (d)

230 V/mm8 V/mm170 V/mm 170 V/mm

0 V/mm 7 V/mm 9 V/mm 13 V/mm

(a)

(f)(e) (g) (h)

(b) (c) (d)

FIG. 17. Conoscopic figures in the �a�–�d� SmC*, �e� SmC�* , �f�

SmCA* , �g� SmC

FI2* , and �h� SmC

FI1* phases. The SmC

FI2* and

SmCA* phases under the same field are shown. They show es-

sentially the same pattern, suggesting that SmCFI2* has more or

less an antiferroelectric nature, but the threshold fields for themolecular reorientation are totally different. Note also thecharacteristic conoscopic image in the SmC

FI1* phase.



�Isozaki, Hiraoka, et al., 1992; Okabe et al., 1992�. In theSmC

A* phase subjected to an electric field the optical

plane is perpendicular to the field, as shown in Fig. 17�f�.No tilt is visible, which indicates that the structure is inan unwound anticlinic state. The SmCFI2

* phase wasfound to exhibit the same behavior as the SmC

A* phase

with varying field, although the helix unwinding occursat a much lower electric field compared to the field inthe SmC

A* phase, as shown in Fig. 17�g� �Okabe et al.,

1992�. The antiferroelectric behavior was the reason thatthe SmCFI2

* phase was at first named the AF or SmCAF*

phase �Okabe et al., 1992�. For this phase Isozaki,Fujikawa, et al. �1992� suggested an antiferroelectricstructure based on the Ising model, where the tilt direc-tion changes every two layers and the repeating unitconsists of four layers.

The structure of the SmCFI1* phase, first called the

SmC�* phase, was recognized in 1990 from its novel

conoscopic figures under an applied electric field, asshown in Fig. 17�h� �Gorecka et al., 1990�. The cono-scopic figure of this phase is quite unusual: a large tilt inthe major axis of the optical indicatrix occurs perpen-dicular to the field direction, and biaxiality with the op-tical plane parallel to the field direction arises. If weconfine ourselves to the Ising model for simplicity, aplausible model is as follows: The ratio of the moleculartilts in opposite senses is far from unity, so that finitepolarization remains uncanceled and the molecule re-sponds by reorientation toward the field direction�Gorecka et al., 1990�. Then what is the ratio of mol-ecules tilting in opposite senses? To solve this problem,Takezoe, Lee, Ouchi, Fukuda, et al. �1991� consideredthe twisting power in the SmCFI1

* phase �Lee, Ouchi, etal., 1990�. It was known that helix handedness in theSmC

A* phase is always opposite to that in the SmC*

phase �Lee, Ouchi, et al., 1990; Li et al., 1991; Yamamotoet al., 1992�. Since the suggested Ising SmCFI1

* structureconsists of synclinic �ferroelectric� and anticlinic �antifer-roelectric� neighboring layers, the twisting power in theSmCFI1

* can be determined by the ratio of the twistingpowers at the SmC*-like and SmC

A* -like interfaces. This

assumption leads to the following simple relation:

1

p�SmCFI1* �

=F

p�SmC*�+

1 − F

p�SmCA* �

, �1�

where p is the helical pitch or wavelength of the selec-tive reflection band in the respective phases and F is thefraction of the parallel-tilted, i.e., synclinic, layers. Usingthe value of p in each phase, F was estimated to be 0.3.Therefore, the ratio of the synclinic and anticlinic layerswas concluded to be 3:7. Since SmCFI1

* is a thermallystable phase, the structure should have a periodic regu-lar structure with a rather short length. The simplest ra-tio satisfying this condition is 1:2 with a repeating unit ofthree layers �Takezoe, Lee, Ouchi, Fukuda, et al., 1991�.In this way, the three-layered model of SmCFI1

* ap-peared. The triple hysteresis loop �Lee, Chandani, et al.,1990�, the five-step electro-optic response �Hiraoka,Taguchi, et al., 1990�, the intermediate apparent tilt

angle �Hiraoka, Chandani, et al., 1990�, and the dielectricconstant �Fukui et al., 1990; Hiraoka, Taguchi, et al.,1990� between SmC* and SmC

A* also supported this

structural model. Although later considerations did notsupport the intuitive path to it, the three-layered struc-ture of the ferrielectric SmCFI1

* phase really exists andwas experimentally confirmed �Mach et al., 1999�.

As mentioned, the repeating units of one �SmC*�, two�SmC

A* �, three �SmCFI1

* �, and four �SmCFI2* � layers were

conclusively reported for these four phases at least onthe basis of the Ising model. The theoretical basis ofthese model structures was confirmed by an axially nextnearest-neighbor interaction model including thirdneighbor interaction �Yamashita and Miyazima, 1993�.In contrast, Cepi~ and Žekš �1995� and Roy andMadhusudana �1996� developed a discrete phenomeno-logical model, leading to essentially a clock model, i.e.,the tilt azimuthal angle makes one turn in three and fourlayers for SmCFI1

* and SmCFI2* with a slight incommensu-

rability due to chirality. This model was supported byresonant x-ray diffraction measurements by Mach et al.�1998�. However, the intermediate model structures, de-formed Ising or deformed clock models, were finally ex-perimentally evidenced by optical properties �Akizuki etal., 1999�, ellipsometry �Johnson et al., 2000; Fera et al.,2001�, detailed analysis of the resonant x-ray data�Johnson et al., 2000; Matkin et al., 2001�, dynamic lightscattering �Konovalov et al., 2001�, optical rotatorypower �Muševi~ and Škarabot, 2001; Shtykov et al.,2001�, specular x-ray reflection �Fera et al., 2001�, and soon. The details will be discussed in the following.

III. THE SEARCH FOR INTERNAL STRUCTURES OFMULTILAYER PHASES

A. The crystallographic unit cell

After the investigation of the chiral polar phases,around the end of the 1990s, it was widely accepted thatapart from the synclinic-ferroelectric SmC* phase andthe anticlinic-antiferroelectric SmC

A* phase, three more

distinct phases—the SmCFI1* , the SmCFI2

* , and the SmC�*

phases—exist, as mentioned previously. However, thesituation was far from being clear. The basic repeatingunit cell of these phases was still a subject of speculationbased on indirect evidence. The structure of the unit cellin the intermediate phases, i.e., the two phases whichappear between the SmC* and the SmC

A* phases, was

still unsolved; some insisted that they could be describedby an Ising model and others that the structures aremore consistent with the clock model. Moreover, someresearchers claimed that in their systems a larger num-ber of intermediate phases could be identified �Fukudaet al., 1994�. Even for the most studied prototype anti-ferroelectric material MHPOBC, the ferroelectric na-ture of the phase identified as SmC* by the Tokyo groupwas still questioned.

The first unambiguous proof that the SmCFI2* , SmCFI1

* ,and SmC

A* phases have unit cells of four, three, and two



layers, respectively, came in 1998 from a resonant x-rayscattering experiment �Mach et al., 1998, 1999�. Conven-tional x-ray measurements allow for the studies of den-sity modulation along the layer-normal direction onlyand are not sensitive to the orientation of the tilt direc-tion in the layers. The transition between intermediatephases is generally not detectable by this method unlesschanges in the layer thickness are seen at the phase tran-sition temperature. This can be related to small discon-tinuity of the tilt angle or to a change in the molecularinterdigitation between layers �Hamplova et al., 2003�.The difference between phases could also be detected ifelectron densities at layer boundaries �in the Ising modelsynclinic and anticlinic� are appreciably different andthus signals of a superlattice become visible. Such a sig-nal, related to the three-layer structure, was reported forthe SmCFI1

* phase by careful x-ray studies using a strongsynchrotron source �Fernandes et al., 2006�.

Unlike the conventional x-ray method, resonant x-raydetection, when the beam energy is tuned to the absorp-tion edge of one of the atoms present in molecular struc-ture, usually sulfur, selenium, or bromine, is sensitive tothe distribution of this particular atom in the crystalstructure �Dmitrienko, 1983�. In other words, the scat-tered beams from consecutive layers do not interferedestructively and the superlattice structure becomes vis-ible in the x-ray diffraction �Fig. 18�. The additional sig-nals appear as satellites of the main reflection, with wavevector

q = q0�l ±m

n� , �2�

where m=1 �first-order peak� or 2 �second-order peak�,n is the number of layers forming the unit cell of thephase, q0=2� /d is the reciprocal vector of main reflec-tion, d is the layer thickness, and l takes the values 0, ±1,±2.

For the preliminary resonant x-ray studies the me-sogenic material 10TBBB1M7, containing a sulfur atomin the mesogenic core, was chosen. For this materialMach et al. demonstrated that indeed in the SmC

A*

phase, apart from the nonresonant signal, correspondingto the layer spacing, additional peaks related to double-layer periodicity exist �Figs. 18�d� and 18�e��. They alsoreported that in the temperature range of the SmCFI1

*

and SmCFI2* phases, signals related to three- and four-

layer periodicities were found, respectively �Figs. 18�b�and 18�c��. On the other hand, in the SmC

�* phase modu-

lation of several layers was detected, incommensuratewith the layer thickness �Fig. 18�a��. The polarizationsand intensities of the superlattice signals seemed toagree with a clock model. For the clocklike structure inan experiment in which the incident x-ray beam is �polarized �polarized in the plane perpendicular to theplane containing the incident and the scattered beam�,the first-order peaks �m=1� should be � polarized �po-larized parallel to the plane containing the incident andthe scattered beam� and the second-order peaks �m=2�should be � polarized, as indeed was found in experi-

ments. The Ising model predicts �-polarized signals forthe first-order reflections. The clocklike structure of thetwo intermediate phases was further supported by care-ful nonresonant x-ray measurements, where no subhar-monics of the main periodicity in any of the studiedcompounds were observed �Fera et al., 2001�, suggestingthat all interlayer boundaries in SmCFI2

* and SmCFI1*

phases are indeed identical, as predicted by the model.However, it soon turned out that although resonantx-ray studies showed correctly the superlattice periodic-ity of the intermediate phases, the arguments about theirinternal structure were premature.

The suggested clock model was inconsistent with theresults of optical studies. Short helical structures shouldbe optically uniaxial with the optical axis along the layernormal, as the dielectric tensor components describingproperties of the structure in the direction perpendicularto the layer normal �in-plane components� are equal.Thus, the samples observed along the layer-normal di-rection �for example, free-standing films� should havezero birefringence. The optical rotatory power should

FIG. 18. X-ray intensity scans in �a�–�d� the indicated phases ofthe �R� enantiomer and �e� the racemic 10OTBBB1M7. Addi-tional signals related to the layer superstructure appear, apartfrom the second harmonic of the signal related to the layerstructure, qz /q0=0. From Mach et al., 1998.



also be negligible, as the helical modulation of the tiltdirection with the period of the modulation extendingover only a few smectic layers is much shorter than thewavelength of the incident light. In contrast to these pre-dictions, films drawn in the SmCFI1

* and SmCFI2* phases

show pronounced birefringence �Fig. 21�; thus opticalstudies strongly suggested that the structure of interme-diate phases—the SmCFI1

* and the SmCFI2* phases—is ei-

ther Ising-like �although this was excluded by formerx-ray studies� or should be described by a new, differentmodel. The best candidates were hybrid structures be-tween the Ising-like and clock structures—called dis-torted clock structures. The question was how one coulddetermine the distortion � from the clock or Ising struc-ture �Figs. 19 and 20�.

The biaxiality of the sample resulting from the distor-tion from the clock structure can be measured quantita-tively by optical ellipsometry. Using this techniqueJohnson et al. �2000� concluded that the structures of theSmCFI1

* and SmCFI2* phases are actually much closer to

the Ising than to the clock model. For the MHPBC ma-terial the distortion � from the uniplanar Ising structurein the SmCFI2

* phase is only about 5° �instead of 45° forthe four-layer clock structure� �Fig. 19� and in theSmCFI1

* phase it is only about 30° �instead of 60° for thethree-layer clock structure� �Fig. 20�.

Measurements of optical rotatory power were alsoproved to be useful for the detection of clock distortion�Akizuki et al., 1999; Muševi~ and Škarabot, 2001;Shtykov et al., 2001; Cepi~ et al., 2002�. According to themodified de Vries’ equation �de Vries, 1951� the opticalrotatory power depends on the anisotropy of the in-plane components of the dielectric tensor, calculated fora single crystallographic unit cell, and the effects of thelonger optical helix due to the rotation of this unit cell inspace �Fig. 21, top inset�. In the SmC*, SmC

A* , SmCFI1

* ,and SmCFI2

* phases the optical helix can be detected

from the selective reflection of the incident light in somecases if the optical helix is in the range of the visible orthe near infrared light wavelengths. If the period of theoptical helix, the pitch, is in the range of the visible orinfrared light wavelengths, it can also be measured inplanar samples as the distance between dechiralizationlines �Brunet and Williams, 1978; Glogarova et al., 1983�.From information about the pitch and the optical rota-tory power, the in-plane components of the dielectrictensor and thus the distortion � from the Ising structurecan be estimated. Because the optical rotatory power isrelated to the anisotropy of the dielectric tensor of thesingle crystallographic unit cell, it should be maximal forthe uniplanar Ising structure and zero for the ideal clockmodel. Using optical rotatory power measurements itwas deduced that for 10OTBBB1M7, i.e., the compoundused for the first resonant x-ray measurements �Mach et

1

2

4

3

��1

2

4

3

(a) (b)

23

14

(c)

FIG. 19. Considered director projections on the smectic layerfor the SmC

FI2* phase: �a� Ising structure, �b� distorted struc-

ture, and �c� clock structure. The distortion � is measured withrespect to the tilt direction in the Ising structure.

1

2

3 ��1

23

(a) (b)

1

2

3

(c)

FIG. 20. Considered director projections on the smectic layerfor the SmC

FI1* phase: �a� Ising structure �b�, distorted struc-

ture, and �c� clock structure. The distortion � is measured withrespect to the tilt direction in the Ising structure

FIG. 21. �Color online� Optical rotary power and selective re-flection for HHBBMz compound. The temperature depen-dence of the optical rotatory power and selective reflectionwavelength �inset�; note that within the temperature range ofthe SmC

FI1* phase helix-twist inversion takes place �top� for the

material 12 HHBBMz �Dzik et al., 2005�. Textures of SmCFI1*

and SmC* phases in free suspended film samples were ob-served between crossed polarizers, subject to a temperaturegradient �the arrow indicates the direction of increasing tem-perature�. In SmC* the color of selective reflection is visible�dark lower right corner, green online�, in SmC

FI1* at the tem-

perature at which the optical helix becomes unwound clearbirefringence colors appear �bright upper left region, blue on-line� �bottom�.



al., 1999�, the distortion from the uniplanar structure isnot larger than 10° in both SmCFI1

* and SmCFI2* ; distor-

tion less than 10° in the SmCFI1* phase was also found for

MHPOBC �Muševi~ and Škarabot, 2001�. Some materi-als might even have an ideal planar structure �Dzik et al.,2005�. Moreover, it was shown that the distortion angleis nearly temperature independent �Brimicombe et al.,2007�.

In summary, taking into account results of the ellip-sometry and the optical rotatory power measurements,it became evident that neither the SmCFI1

* nor theSmCFI2

* phase could possibly have the simple clock struc-ture. Only the measurements of the SmC

�* phase seemed

to be consistent with the clock structure as in this phasenegligible optically rotatory power was found �Muševi~and Škarabot, 2001� and optical uniaxiality was con-firmed.

Reports about optical studies stimulated further x-rayinvestigations. During the years 2001 and 2002 some re-ports appeared that provided additional evidence formore detailed description of the intermediate phasestructures by high resolution resonant x-ray method�Cady et al., 2001; Hirst et al., 2002�. The calculation ofthe tensorial form factors by Levelut and Pansu showedthat in phases with an additional optical helix, superim-posed on the basic few-layered structure, both first- andsecond-order resonant signals should be additionallysplit and observed at

q = q0�l ± m� 1

n+

d

p�� , �3�

where p is the pitch length �Levelut and Pansu, 1999�.The splitting is larger for shorter pitches. While the in-tensities of the second-order signal m=2, split due to thehelix, should always be equal, the intensities of the splitfirst order m=1 signals are more complex. The intensityratio for these signals departs quickly from 1, as the ba-sic crystallographic unit becomes distorted from theclock structure and for the ideal Ising structure oneof the signals disappears. For the SmCFI2

* phase inMHDDOPTCOB, by comparing the intensities of peakspositioned near 1.25q0 and 1.75q0, Cady et al. �2001� de-duced that the structure is almost Ising-like and the dis-tortion is only about 10° from the Ising uniplanarity andnearly temperature independent. Similar results wereobtained soon after for a few other materials havingSmCFI2

* and SmCFI1* phases �Wang et al., 2006�.

Finally, after some years of intensive research, consis-tency between the optical and x-ray results was ob-tained. The SmCFI2

* and SmCFI1* phases have a basic

crystallographic unit cell of four and three layers, re-spectively, in which the tilt directions in smectic layersform a strongly distorted clock structure �Figs. 19 and20�. Additionally, a secondary long-wavelength periodic-ity, the optical helix, is formed. Only SmC

�* has a few-

layered clock structure �Fig. 22�. In the following yearsother properties of the phases also became more under-standable.

B. Secondary helical modulation of the phases

For nearly all compounds the twist sense of the opticalhelix changes in the SmC*-SmC

A* and also in the

SmCFI2* -SmCFI1

* phase transitions �Cepi~ et al., 2002; La-gerwall et al., 2006�. This is consistent with constant chi-ral nearest-neighbor interlayer interactions in the system�see Sec. IV�. However, in a number of materials thesense of the helical-modulation sign reverses within oneof the phases. The helical-modulation inversion occursin SmC

A* in MHPOBC but in many other materials it

happens in one of the intermediate phases, although thetemperature range of these phases is usually very nar-row. Due to the strong temperature dependence of theoptical helix length in the vicinity of the helix sense in-version the texture changes are rather violent. The be-havior was sometimes misinterpreted as an additionalphase transition �Nguyen et al., 1994�. On the otherhand, the helical modulation does not reverse at thetransition from the SmC* to the SmC

�* phase. The two

phases seem to have similar structure except that thepitch in SmC

�* is very short, below the resolution of mi-

croscopic observations. Its length can be determinedonly by the resonant x-ray method. Gleeson and Hirst�2006� measured the helical pitch in SmC

�* in different

materials and found a pitch ranging from 5 to 54 layers�Mach et al., 1999�. With a few assumptions the length ofthe helical modulation can also be estimated from ellip-sometric data. The pitch is usually incommensurate withthe smectic layer and can be as tight as 3.5 layers �Cadyet al., 2002; Wang et al., 2006; Liu et al., 2007�. The shortpitch in the SmC

�* phase can also be estimated by the

comparison of Friedel fringes which appear due to thechanges of the propagating light ellipticity and thewedge fringes in free surface drops of liquid crystals. Forthe four-layer structure of the SmC

�* phase, the phase

0.000 0.005 0.010 0.0152

3

4

5

6

7

8

9

10

Cn H2n+1O C6H13C S

O

C O

O

C O

O

HC*

CH3

Cn H2n+1OCn H2n+1O C6H13C6H13C S

O

C S

O

C O

O

C O

O

C O

O

C O

O

HC*C*

CH3CH3

C10.25 C11.30C10.80 C12.00C10.90C11.00C11.04

ξξ ξξ

(TAC

-T)/TAC

0.00 0.02 0.04 0.06 0.080

2

4

6

(nC-n)/n

C

Dataβ=0.6

∆∆ ∆∆ξξ ξξ

FIG. 22. Evolution of the pitch at the SmC*-SmC�* phase tran-

sition for mixtures of compounds with n=10 and 11. The jumpin the pitch at the phase transition �inset� decreases with de-creasing concentration of the n=11 homolog in the mixture.The studied system shows the evolution from first-order toovercritical behavior. From Liu et al., 2006.



difference between tilts in neighboring layers is exactly� /2 and phase retardation for the two polarizations oflight does not appear. Therefore the structure seen bylight has no chiral properties and the continuous changeof the structure through the � /2 phase difference givesrise to a peculiar behavior of the period of the Friedelfringes �Laux et al., 1999, 2000�.

As the temperature is lowered and the system ap-proaches the transition to the SmC* phase the pitchgreatly increases. The evolution between SmC

�* and

SmC* phases can be either discontinuous �Ema et al.,2000; Cady et al., 2002� or continuous �Liu et al., 2006�.Since both phases have the same symmetry the systemcan be considered as an analog to the gas-liquid system.The order parameter for this transition is the helicalpitch, which can evolve continuously beyond the criticalend point. The existence of such a critical end point forthe SmC*-SmC

�* phase transition was shown by Liu et al.

�2006�. In the vicinity of the critical point the depen-dence of the jump in the pitch is described by a mean-field critical exponent close to 0.5.

C. Polarity of multilayer phases

The tilted phases have more or less distinct polarproperties, which can be explained by a distorted clockmodel. Both the two-layered SmC

A* phase and the dis-

torted four-layered SmCFI2* phase have compensated

electric polarization in the crystallographic unit cell andthus they have antiferroelectric properties. The SmC*

and smectic SmC�* are ferroelectric, as their crystallo-

graphic unit cell is just a single electrically polarizedlayer; the SmCFI1

* phase with its three-layered structureand only partially compensated polarization is ferrielec-tric.

The polar properties of the phases are easily distin-guished by the dielectric response using dielectric spec-troscopy. Fluctuations of polarization with the wave vec-tor q=0 �thus homogeneous in space� can be followed bythis method �Merino et al., 1996; Panarin et al., 1998�.

The dielectric response in the SmC* and SmC�* phases

is dominated by the Goldstone mode, which should bemore properly called the helix-distortion mode �Fajar etal., 2002; Douali et al., 2004�. This mode is related tocollective movements of the molecules on the tilt cone�azimuthal mode� which also affect the polarization vec-tor’s direction. The amplitude of these fluctuations issmaller and the relaxation time is shorter for shorterhelical pitches. This is the reason why the dielectric re-sponse in the SmC* phase is much stronger and hasmuch lower relaxation frequency than the mode ob-served in the SmC

�* phase.

In the antiferroelectric phases, the SmCA* and SmCFI2

*

phases, the dielectric response is at least one order ofmagnitude weaker than in the ferroelectric SmC* andSmC

�* phases. There are two main collective fluctuations

which contribute to the dielectric response �Parry-Jonesand Elston, 2002�. A mode with higher relaxation fre-quency appears due to antiphase azimuthal tilt angle

fluctuations �Fig. 23� in which the antiferroelectric struc-ture of the crystallographic unit cell is distorted. Thelower-frequency mode is usually attributed to helix dis-tortion. This mode is visible due to the small noncom-pensated polarization in the crystallographic unit cell�Lagerwall, 2005; Dolganov and Zhilin, 2008�.

The dielectric response in the SmCFI1* phase is much

stronger than in the antiferroelectric SmCA* phase but

weaker than in the ferroelectric SmC* phase �Figs. 24and 25�. Its characteristic feature is a broad distributionof mode relaxation frequencies usually in the range of10 Hz–1 kHz. The polydisperse character of the dielec-tric response suggests that, apart from antiphase azi-muthal tilt angle changes disturbing the three-layeredequilibrium structure and the helix-distortion mode,some other mechanism might also contribute strongly tothe dielectric response in the same frequency window.The candidate could be movements of boundaries be-tween domains of differently oriented polarizations ofthe three-layered unit cell �Miyachi et al., 1994� �see Fig.40 in Sec. IV�. Such an origin of the dielectric response issupported by the strong influence of surface interactionson the dielectric response and the history of electric fieldapplication to the sample �Miyachi et al., 1994�. Gener-ally, in all phases, in addition to the described azimuthalmodes related to the changes of the polarization direc-tion in the smectic layers, amplitude modes, related tothe changes of the tilt angle magnitude and thus polar-ization �due to the tilt-polarization coupling� in the lay-ers, also contribute to the dielectric response. However,this contribution is weak except in the vicinity of thephase transition to the nontilted SmA phase �Bourny etal., 2000�.

The difference in the polarity of phases is also shownin their electro-optical response �Panarin et al., 1997�.While in ferroelectric phases application of an electricfield reorients polarizations along the helix continuouslyand the threshold for the switching is nearly negligible,in antiferroelectric phases the switching usually takesplace in two steps: initially the unwinding of the helicalstructure occurs and at higher electric fields the antifer-roelectric order transforms to ferroelectric order withthe polarization oriented parallel to the electric field�Parry-Jones and Elston, 2001�. The threshold electricfield necessary to break antiferroelectric interactionsand impose the ferroelectric structure is of the same or-der of magnitude for the SmCFI2

* and SmCA* phases and

1 2

1 4

2 3

1 4

2 3

FIG. 23. Polar azimuthal antiphase modes in the antiferroelec-tric �a� SmC

A* and �b� SmC

FI2* phases. Note that polarization is

perpendicular to the tilt �marked by small arrows�. When thestructure is deformed due to the fluctuations �dashed arrows�the polarization in the unit cell does not cancel out.



decreases on approach to the ferroelectric SmC* phase.In the SmC

A* phase, the induction of the synclinic struc-

ture by application of an electric field �or temperature� isvisible in thin cells as characteristic stripes propagatingin the sample in the direction parallel to the smecticlayers. The velocity at which the stripes propagate isproportional to the strength of the applied field. Thestripes were described as solitary waves resulting fromstrong coupling of the layers �Li et al., 1995; Bahatt et al.,2001�. In the SmCFI1

* phase the structural changes arealso stepwise when an electric field is applied. At lowerfields the helical modulation unwinds; as the field in-creases the structural changes from the three-layeredferrielectric structure to the ferroelectric structure occur.

Generally the helical pitch in the SmC�* phase is so

short that its switching often mimics the antiferroelectricbehavior—the helically modulated structure with veryshort pitch rapidly unwinds and rewinds if the polariza-

tion is measured using triangular voltage dependence�Lagerwall, 2005�. The same behavior, i.e., tristableswitching, was observed even in the SmC* phase with anultrashort pitch �Itoh et al., 1997�.

D. Temperature sequence of phases

Unlike to the SmC�*-SmC* phase transition, which can

be continuous or discontinuous, the other phases appearby first-order transitions �Ema et al., 2000�, except forsome unusual systems with a critical end point for the

(b)

(a)

FIG. 24. Temperature-frequency dependence of �left� real and�right� imaginary parts of dielectric permittivity for MHPOBCwith high optical purity �showing SmA, SmC

�* , SmC

FI2* ,

SmCFI1* , and SmC

A* �. Note that the antiferroelectric phases

SmCA* and SmC

FI2* have much weaker dielectric response than

SmC�* and SmC

FI1* . From Gorecka et al., 2002.

T[C

]

T[ C](b)

(a)

FIG. 25. �Color online� Phase diagram and corresponding dif-ferential calorimetric scans. �Left� Phase diagram for MH-POBC showing the influence of optical purity on the subphasesequence. �Right� Differential Scanning calorimetry scans forthe mixtures of different optical purity. From Gorecka et al.,2002.



SmC*-SmCA* phase transition �Song et al., 2008�. As the

temperature is lowered the most general sequence isSmC

�*-SmC*-SmCFI2

* -SmCFI1* -SmC

A* . In some systems

some phases might be missing or be metastable; how-ever, as the temperature is lowered it is generally ob-served that the crystallographic unit cell of phase struc-tures shortens from many layers in the SmC

�* phase to

four layers in the SmCFI2* phase, to three layers in the

SmCFI1* phase, and finally to two layers in the SmC

A*

phase. While the SmC and SmCA phases are observed inracemic mixtures or in nonchiral materials, SmC

�* ,

SmCFI1* , and SmCFI2

* are found only in systems with highoptical purity �Fukuda et al., 1994�. In most systems theenantiomeric excess has a dramatic influence on the ob-served phase sequence. For example, in the prototypematerial MHPOBC it was found that in samples ofhigh optical purity the phase sequenceSmC

�*-SmCFI2

* -SmCFI1* -SmC

A* is observed, while in a

slightly racemized sample the antiferroelectric SmCFI2*

phase disappears and the ferroelectric SmC* phase ap-pears in the same temperature window. The phase se-quence of the slightly racemized sample isSmC

�*-SmC*-SmCFI1

* -SmCA* �Gorecka et al., 2002�, as re-

ported by the first studies of this system �Chandani,Gorecka, et al., 1989�. This solved the long-lasting con-troversy of some reporting antiferroelectric properties�Li et al., 1996; Sako et al., 1996; Jakli, 1999� for thephase below the SmC

�* phase and other ferroelectric

properties for presumably the same phase �Chandani,Gorecka, et al., 1989; Gorecka et al., 1990�.

Although shortening of the crystallographic unit cellseems to be the most common situation as the tempera-ture is lowered, it should be mentioned that in somematerials the SmCFI1

* phase �Laux et al., 2000� or theSmCFI2

* phase �Wang et al., 2006, 2009� interferes be-tween the SmC

�* and SmC* phases. In some systems a

reentrant phenomenon is observed where the phase se-quence SmC*-SmC

A* -SmC* �Kašpar et al., 2001; Pociecha

et al., 2001� appears. Recently a SmC�*-SmCFI1

* -SmC*

phase sequence was also reported �Sandhya et al., 2008�and even SmCFI2

* -SmC*-SmCFI2* reentrancy in some mix-

tures �McCoy et al., 2008� was detected.It should also be mentioned that some claim �Isozaki,

Hiraoka, et al., 1992; Chandani et al., 2005; Emely-anenko et al., 2006; Panov et al., 2007� that they are ableto detect more than five tilted smectic C-type phases�e.g., with the unit cell larger than four smectic layers� insome materials or mixtures. However, these additionalphase transitions are detected indirectly, for example, assmall, nearly experimental-resolution-limited, jumps inthe thickness of the film made of the material. Only veryrecently six-layer structure was confirmed by resonantx-ray experiment in narrow temperature window be-tween SmC� and SmCFI2 phases �Wang et al., 2010�.

The three- and four-layered phases were also found tobe stable in device geometry. It was proved that anelectric-field-induced transition from the structure of theintermediate three- or four-layered phase to the ferro-electric SmC* structure occurs in thin glass cells. One

can imagine that application of a sufficiently high elec-tric field stabilizes the SmC* phase. Extensive studies onthe electric-field-induced phase transition have beenmade by study of the dielectric response under a biasfield �Hiraoka, Taguchi, et al., 1990; Panarin et al., 1997�and the apparent tilt angle �Hiraoka et al., 1991� conos-copy �Isozaki, Hiraoka, et al., 1992; Fukuda et al., 1994�,induced polarization �Hou et al., 1997; Panarin et al.,1997�, pyroelectric �Shtykov et al., 2001�, and resonantx-ray diffraction �Jaradat et al., 2008� measurements. Aparticularly interesting result is the field-induced se-quential transition SmCFI2

* -SmCFI1* -SmC*. This transi-

tion behavior is found in E-T phase diagrams deter-mined by other experimental methods �Isozaki,Hiraoka, et al., 1992; Fukuda et al., 1994; Shtykov et al.,2001�, as shown in Figs. 26�a� and 26�b�. Recently Jara-dat et al. showed that in some materials the four-layeredstructure undergoes a transition to the three-layeredstructure at weaker fields before the SmC* phase occurs�Jaradat et al., 2008�. They used resonant x-ray diffrac-tion and found that a diffraction peak at q /q0=2/3 dis-appeared and instead q /q0=3/4 emerged on applicationof a field. The predicted E-T diagram is similar to thosedetermined by conoscopy and pyroelectric measure-ments.

E. Present understanding of phase structures

After two decades of intensive studies the picture ofall five definitely confirmed phases is finally consistent.The results obtained by a number of complementarytechniques proved that the crystallographic unit cells ofthe SmC*, SmC

A* , SmCFI1

* , and SmCFI2* phases consist of

one, two, three, and four layers. The structure within thebasic unit of the SmCFI1

* and SmCFI2* phases is neither

clocklike nor Ising, but it has properties of both. Mostmeasurement showed that distortion from the clockmodel is rather high; apparently the tendency for a uni-planar structure is pronounced in these systems. The ori-entation of the basic unit cell additionally slowly rotatesalong the layer normal giving an additional long opticalhelix. Moreover, it turns out that the SmC

�* and SmC*

phases are structurally identical, differing only quantita-tively in the length of the helical pitch, which is some-times even smaller than five �Liu et al., 2007� smecticlayers in the SmC

�* phase.

Now the question arises as to the molecular interac-tions at the origin of these structures.

IV. THEORETICAL MODELING OF PHASES AND PHASESEQUENCES

An old joke says: a theorist is a person whom nobodybelieves except himself; an experimentalist is a personwhom everybody believes except himself. Following thisjoke one might think that a theoretical understanding ofnewly discovered phenomena is not of much impor-tance. It could wait for the maturity of the problemwhen an extensive set of experimental results would al-



low for control of the model at the early stage of itsdevelopment. However, theorists do not share this opin-ion. The first theories appear almost always immediatelyafter the discovery of a new phenomenon. This was truefor antiferroelectric liquid crystals. Various groups,many of them having experience with the theoreticalmodeling of ferroelectric liquid crystals, became inter-ested in the new materials immediately. Some of themodels developed from proposed structures, while theothers independently predicted structures that nobody

except the authors believed in. Surprisingly, a few ofthese predictions were later confirmed by experimentalobservations, but some are still in their ghostlike form.

The reader wants to know what we theoretically un-derstand today and what can he or she apply for under-standing of his or her experimental results. Thereforethe theoretical section is organized as follows: We beginwith the two most widely used models. The continuousmodel can be used for understanding of the two phaseswith less complex structure—the SmC* and the SmC

A*

phase. It is especially useful for the analysis of intralayerstructures in displays. The discrete model is more gen-eral as it allows for experimentally consistent structuresof all observed phases as described in Sec. III.E. Thediscrete model makes a bridge between microscopic in-teractions and phenomenological coefficients and givesreasons for the phase sequence dependence on opticalpurity. Therefore this model is discussed in detail. Inaddition, we present an overview of other models andapproaches for modeling of antiferroelectric liquid crys-tals.

A. Continuous model

The first theoretical model of the antiferroelectric liq-uid crystals was proposed almost immediately after theirdiscovery �Orihara and Ishibashi, 1990�. Orihara andIshibashi constructed a theory that was based on thebilayer structure of the antiferroelectric SmC

A* phase.

They assumed that bilayer periodicity is an essentialproperty of these systems and they introduced four two-dimensional order parameters, an approach that hadproven successful in a number of studies of antiferro-electric and antiferromagnetic crystalline systems �Kit-tel, 1951�. They introduced two two-dimensional orderparameters �Fig. 27� �j and P� j, which are the property ofthe layer,

�j = njxnjz,− njynjz ,�4�

P� j = Pjx,Pjy ,

and four two-dimensional order parameters as theircombinations,

(a)

(b)

(c)

FIG. 26. E-T phase diagrams proposed by �a� conoscopy inslightly racemized �R:S=90:10� TFMH-PBC �Fukuda et al.,1994�, �b� pyroelectricity in 12OF1M7 �Shtykov et al., 2001�,and �c� resonant x-ray diffraction �Jaradat et al., 2008�. In �b� ris a parameter expressing polar strength, 1 for SmC* and 0 forSmC

A* .

�j

y

x

z

�

�Pj

FIG. 27. Tilt �j and polarization P� j order parameters presentedschematically for an average molecule in the jth layer.



�f,j = 12 ��2j+1 + �2j� � �f�z� ,

�a,j = 12 ��2j+1 − �2j� � �a�z� ,

�5�P� f,j = 1

2 �P� 2j+1 + P� 2j� � P� f�z� ,

P� a,j = 12 �P� 2j+1 − P� 2j� � P� a�z� .

The order parameters with the subscript f are nonzero inthe ferroelectric SmC* phase, and order parameters withthe subscript a are zero. The opposite is true in the an-tiferroelectric SmC

A* phase, where the f parameters are

zero and the a parameters are nonzero. As they explic-itly describe ferroelectric or antiferroelectric propertiesof the structure, they are called the ferroelectric �f� andthe antiferroelectric �a� tilt and polarization. Althoughlayer order parameters abruptly change from layer tolayer, the sum and the difference of order parameters inneighboring layers vary slowly along the layer normal.This allows for the treatment of the order parameters ascontinuous variables. The free energy was constructedfrom terms allowed by symmetry considerations. It has asimilar form to the Pikin-Indebom free energy for ferro-electric liquid crystals �Pikin and Indebom, 1978�. Thefree energy for the antiferroelectric liquid crystal systemis expressed with both types of order parameters withadditional coupling terms. Here,

G =12

aaa2 +

14

baa4 +

12

aff2 +

14

bff4 +

12

�1a2f

2

+12

�2��a · �f�2 + a��a � P� a� + f��f � P� f�

+1

2�aPa

2 +1

2�fPf

2 +12

a� ��a

�z�2

+12

f� ��f

�z�2

− �a��a ��a

�z�

z− �f��f �

��f

�z�

z

+ �a�P� a ��a

�z�

z+ �f�P� f �

��a

�z�

z. �6�

The first two terms describe the anticlinic ordering ofthe tilt and the next two terms its synclinic ordering.Terms with coefficients �1 and �2 give the couplings be-tween the ferroelectric and antiferroelectric tilt orderparameters. The parameters a and f give “piezoelec-tric” couplings of the ferroelectric polarization with theferroelectric tilt and the antiferroelectric polarizationwith the antiferroelectric tilt, respectively. The next twoterms present the electrostatic contributions of theferroelectric and antiferroelectric dipolar orderings. Theterms with a and f give the energy contributions ofelastic deformations of both tilt order parameters. Chi-ral interactions between molecules in neighboring layersare given by the terms with parameters �a and �f. In adescription of the order parameter within these terms az component with a zero value is formally added to theinitially two-dimensional order parameter, which allows

for the correct description of the chiral term using thecross product. The flexoelectric inductions of the ferro-electric and antiferroelectric polarizations are given bythe terms �f and �a, respectively. The temperature-dependent coefficients are

aa = �a�T − Ta� ,�7�

af = �f�T − Tf�

and Ta and Tf are the temperatures where the anticlinicand synclinic tilts would appear without the presence ofthe piezoelectric, elastic, and other interactions. To de-scribe the phase sequence which is always found in an-tiferroelectric liquid crystals, where the antiferroelectricphase is the phase stable within the lowest temperaturerange, the temperature Ta should be lower than the tem-perature Tf.

In order to obtain stable solutions the free energy hasto be minimized with respect to all four order param-eters. The bilinear coupling of the polarizations with thetilt allows for a straightforward elimination of the polar-izations, and the free energy obtains a much simplerform,

G =12

aaa2 +

14

baa4 +

12

aff2 +

14

bff4 +

12

�1a2f

2

+12

�2��a · �f�2 +12

a� ��a

�z�2

+12

f� ��f

�z�2

− �a��a ��a

�z�

z− �f��f �

��f

�z�

z, �8�

where the parameters are renormalized as

aa = aa − �aa2, af = af − �ff

2,

�a = �a + �aa�a, �f = �f + �ff�f, �9�

a = a − �a�a2, f = f − �f�f

2.

Orihara and Ishibashi �1990� neglected the terms thatconsidered derivatives of the tilt order parameters andtherefore limited the analysis to nonmodulated struc-tures. Žekš and Cepi~ later analyzed possible structuresand their stability if elastic and chiral interactions arenot neglected and found phases that were helicallymodulated �Žekš and Cepi~, 1993�. A year later thesame structures were found by Lorman et al. indepen-dently �Lorman et al., 1994�. Recently, the model wasalso analyzed in regions of coefficient values where suchsimple modulated structures are not stable and an in-commensurate modulated structure was found �Cepi~and Žekš, 2007�.

We now describe the structures allowed by the con-tinuous model. The experimental observations haveshown homogeneous order with helical modulation butno higher periodicities in samples. Therefore, the con-stant amplitude approximation seems reasonable andthe magnitude of both order parameters is considered asconstant, �a and �f. In addition, a simple helical modu-



lation with only one wave vector q is also included in thetrial solution �ansatz� as well as a general but fixed phasedifference �0 between the ferroelectric and antiferro-electric order parameters in structures where both orderparameters exist. We call solutions of this type simplymodulated phases in the following as they are describedby a single wave vector of modulation q:

�a = �acos qz,sin qz ,�10�

�f = �fcos�qz + �0�,sin�qz + �0� .

Using Eq. �10� the free energy adopts a simpler form

G = 12 �aa − 2�aq + aq2��a

2 + 14ba�a

4 + 12 �af − 2�fq

+ fq2��f

2 + 14bf�f

4 + 12�1�a

2�f2 + 1

2�2�a2�f

2�cos �0�2.

�11�

Minimization of the free energy �11� is rather simple.Five different structures are stable depending on the val-ues of the model coefficients: One solution describes thenontilted SmA phase. There are four different solutionsthat describe helically modulated phases with only oneperiod of modulation and one solution with a more com-plex structure where the competition between two dif-ferent favorable periods of modulation results in a soli-tonlike structure not commensurate either with the layerthickness or with either of the two favorable periods. Inthe following, the short description of solutions is given.

1. The nontilted SmA phase

The trivial solution of the free-energy minimizationequations is stable when both �a and �f are zero. It isstable at the highest temperatures and it develops to oneof the two tilted structures, the SmC* or the SmC

A*

phase, below the transition temperature. The dynamicalproperties of the system �Žekš et al., 1991� reveal thecharacter of the phase below SmA. In all nontilted sys-tems having tilted phases at lower temperatures twotypes of tilt fluctuations exist—synclinic and anticlinic.The fluctuation that has the lower frequency and con-denses at higher temperature defines the nature of thetilted phase below the SmA phase. If the synclinic fluc-tuations have lower frequency, the ferroelectric SmC*

phase becomes stable as the temperature decreases andthe antiferroelectric SmC

A* phase becomes stable if the

anticlinic fluctuation is the one with the lower frequency.

2. The simply modulated SmC* phase

For certain sets of model coefficients and a certainrange of temperatures, the stable solution requires that�a is zero and �f is nonzero. The structure is recognizedas the structure of the ferroelectric SmC* phase pre-dicted by Meyer and synthesized by his co-workers al-ready in 1974 �Meyer et al., 1975�. The molecules in eachlayer are tilted by �f, which increases with decreasingtemperature and the structure is helically modulatedwith the wave vector qf=�f / f, which is temperature in-dependent. The temperature independence of the wave

vector qf is actually not consistent with experiments, butinclusion of higher-order chiral terms can account forthis inconsistency. We do not discuss this detail, as ourattempt is to discuss only the main features of the sim-plest possible continuous model.

3. The simply modulated SmCA* phase

The solution where �a is nonzero and �f is zero de-scribes the antiferroelectric phase with the wave vectorof helicoidal modulation qa=�a / a. The temperature re-gion of its stability depends on the choice of the tem-perature Ta and the ratio of the model coefficients ba /bf.As both transition temperatures Tf and Ta and the val-ues of the fourth-order coefficients bf and ba define thegeneral phase sequence, the model also describes simpleferroelectric systems where the antiferroelectric phasedoes not develop.

4. The two simply modulated ferrielectric SmCSM* phases

The model also allows for two phases with one wave-vector modulation and both order parameters differentfrom zero. The phase difference �0 between the two or-der parameters is well defined only in these structures.The minimization of the free energy with respect to �0allows for two different structures. The uniplanar struc-ture is characterized by having the ferroelectric and an-tiferroelectric order parameters in the same plane ��0=0 or ��, so all layers are tilted to the same plane de-fined by the layer normal and the tilt �Fig. 28�a��. It isstable for negative values of the coefficient �2.

For positive values of the coefficient �2 the structureswhere the ferroelectric �f and antiferroelectric �a orderparameters lie in two mutually perpendicular planes��0= ±� /2� are stable. The solution presents the struc-

j+1

j

� j�

� j+1�

� j+1+�

� j=�

� f�

� j+1�

� j=�

� a�

�

(a)

� j+1+�

� j=�

� f�

� j+1�

� j=�

� a��

j+1

j

� j+1�

� j�

(b)

FIG. 28. In the ferrielectric SmCSM* phase both order param-

eters, �f and �a, are nonzero. �a� For �2�0 the order param-eters are parallel, molecules are tilted within the same plane,but the magnitude of the tilt differs in odd and even layers. �b�For �2�0 the order parameters �f and �a are perpendicular.The magnitude of molecular tilts is equal in all layers; however,the directions of tilts in odd and even layers are at an angle.



ture where the tilts in neighboring layers have the samemagnitude but their directions are at an angle �Fig.28�b��. The temperature development of both order pa-rameters has the same form for both structures,

�a2 =

�af − 2�fq + fq2�� − �aa − 2�aq + aq2��f

�a�f − �2 ,

�12�

�f2 =

�aa − 2�aq + aq2�� − �af − 2�fq + fq2��a

�a�f − �2 ,

where �=�1+�2 cos �0 and the wave vector q is the re-sult of the free-energy minimization. The temperaturerange of phase stability lies between the stability rangeof the ferroelectric and antiferroelectric phases, provid-ing the model parameter �2 is large enough to bind thedirections of both order parameters �Figs. 29–31�.

Although these two structures were ruled out as can-didates for the structure of the SmC

�* phase by the first

direct resonant x-ray observations, the structure of the

unwound ferrielectric SmCSM* phase for the coupling �2

�0 develops if an electric field is applied to the antifer-roelectric SmC

A* phase.

5. The incommensurate SmCIC* phase

The phase sequence depends on the choice of modelcoefficients. Comparison to experiments gives the theo-

FIG. 29. Tilt dependence on temperature for �2�0. �Top� Thetemperature dependence of the ferroelectric �f and the antifer-roelectric �a order parameters for �2�0. �Bottom� Corre-sponding tilts in even and odd smectic layers. The negativesign of the tilt magnitude in the odd layers means the tilt is inthe opposite direction to the tilt in even layers.

FIG. 30. Tilt dependence on temperature for �2�0. �Top� Thetemperature dependence of the ferroelectric �f and antiferro-electric �a order parameters for �2�0. �Bottom� Correspond-ing angle between directions of the tilts in even and odd smec-tic layers.

FIG. 31. Temperature dependence of the wave vector q for thesimple modulation is qualitatively equal for both signs of �2.



rist a hint about the magnitudes and signs of the coeffi-cients. Therefore the space of model coefficients is usu-ally analyzed in order to find conditions for whichdifferent phases are stable �Žekš and Cepi~, 1993; Lor-man et al., 1994; Olson et al., 2002; Hamaneh and Taylor,2004�. However, detailed analysis of the model coeffi-cients space revealed another curiosity: only simplymodulated structures cannot be stable for any set ofmodel coefficients.

A consideration can give us a hint. Imagine the systemwhich could be described by the negligible coupling �2=0. This condition means that the ferroelectric order pa-rameter �f does not influence the antiferroelectric orderparameter �a even if both of them exist. Therefore theyform two independent generally different helices, theferroelectric and the antiferroelectric ones, guided bythe competition in chiral and elastic properties ex-pressed by considering the two order parameters inde-pendently. If the coupling in the system is weak, the sys-tem solves the frustration between two different helicesby formation of domains with a single wave vector sepa-rated by domains where the order parameters flip fromone relative orientation to the opposite one. Althoughthe problem was pointed out long ago �Žekš and Cepi~,1993� a more detailed analysis with solutions obtainedwithin the constant amplitude approximation was per-formed only recently �Cepi~ and Žekš, 2007�. However,the structure allowed by the model seems not a candi-date for any real structure found in antiferroelectric liq-uid crystals, and it can be considered as one of the ghost-like solutions, possibly applicable in completely differentsystems �Pociecha et al., 2003�.

The continuous model and its structures have a fewimportant flaws. First, the high-temperature phase, theSmC

�* phase, which often develops through a continuous

transition from the nontilted SmA phase, could not bereproduced within this model. The higher-temperaturetilted phase is always either the SmC* phase or theSmC

A* phase. Other structures become stable only at

lower temperatures where the tilt is already present.Second, the resonant x-ray scattering experiments haveshown that two distinctly different intermediate phasesexist, the SmCFI2

* phase with four-layer periodicity andthe SmCFI1

* phase with three-layer periodicity. Repro-duction of both phases within the continuous modelwould require introduction of many new order param-eters. The required number of independent tilt and po-lar order parameters is the same as the basic periodicityof the phase. The number of coupling terms betweenthese order parameters that enter the free energy is evenhigher, making theoretical equations extremely hard tomanage. Finally, the resonant x-ray method has shownthat of the four different structures allowed by the con-tinuous model only the SmC* and the SmC

A* phases are

consistent with experiments. Therefore this model can-not satisfy expectations.

The model is still useful for consideration of variousphenomena within these two phases: the structuresformed in the cell, under the application of an electric

field �Parry-Jones and Elston, 2001�, where the electricfield induces the structure of the unwound bilayer ferri-electric phase, the structure of defects, and more. When-ever one faces a situation where it is expected that theorder parameters �ferroelectric, antiferroelectric, orboth� do not change significantly at distances compa-rable to the molecular length or the layer thickness, thecontinuous approach is the most appropriate.

B. Simple clock model

Limitations of the continuous model having the form�6� are quoted at the end of the previous section. So, is itpossible to construct free energy which allows for addi-tional structures without introducing new order param-eters?

In the following we shall present an alternative ap-proach to the continuous modeling of antiferroelectricliquid crystals—the discrete model. We discuss two vari-ants of discrete modeling: the simple clock and the dis-torted clock models.

The basic idea of the discrete approach is as follows:

• The interlayer and intralayer interactions are consid-ered separately.

• Instead of elastic deformations which allow forgradual changes of order parameters only, abruptand significant changes of order parameters fromlayer to layer are allowed.

• Elastic deformations are not written in derivativesbut as couplings of order parameters in neighboringlayers.

• Interactions with more distant layers than neighbor-ing layers are taken into account.

Orihara and Ishibashi �1990� had sown the seed forthe discrete approach using the continuous model. Soonafter publication of the continuous model for antiferro-electric liquid crystals Sun et al. published the analysis ofinterlayer interactions �Sun et al., 1993� that led to theproposed form of the model �6�. They wrote expressionsfor elastic and chiral terms in the form of differencesbetween tilt order parameters in neighboring layers upto the distance to the next-nearest neighbors. These ex-pressions were the basis of terms that phenomenologi-cally express interlayer interactions within the discretedescription of the free energy. Their analysis showedthat the continuous description is allowed only in sys-tems in which next-nearest layers prefer parallel �syn-clinic� tilt directions. However, they did not consider theother possibility: the favorable orientation of tilts in thenext-nearest layers as anticlinic.

Favorable anticlinicity in next nearest layers intro-duces frustration into the system. Favorable ferroelectricor antiferroelectric order, even a combination of both ofthem, as present in solutions of the continuous model, isalways in contradiction with anticlinicity in the next-nearest layers. In all solutions of the continuous modeltilts in next-nearest layers are parallel �Figs. 32�a�–32�c��.



The system can suppress frustration by modulation ofthe order parameters from layer to layer. The tilt orderparameter can vary in magnitude, in direction, or inboth. If variation of the magnitude were important, thex-ray measurements should have shown variations of thesmectic layer thickness. No such observations have beenreported so far. Therefore we conclude that structureswith amplitude variation are not present at least not inbulk. The next possibility is variation of the tilt direc-tion. Ising models, which will be discussed later, assumedthat the tilt is bound to one plane defined by the layernormal and the tilt direction. Such structures are said tobe uniplanar �Rovšek et al., 2000�. This assumptionmeans that tilts in neighboring layers can be either par-allel �the difference in tilt directions in neighboring lay-ers, also called the phase difference, is equal to zero� orantiparallel �the phase difference is equal to ��.

In addition, there are two questions that need to beanswered if one wants to pursue the uniplanar structuresfor the missing SmC

�* phase:

• What is the origin of the uniplanar constraint? Ob-servations of properties of ferroelectric and antifer-roelectric phases did not reveal any reasons for theconstraint of the tilt to one plane only.

• How can the transition from the nontilted SmAphase to SmC

�* be continuous, as experimentally ob-

served? If the structure of the phase leads to differ-ent interactions between neighboring layers becauseinterfaces can be both parallel, both antiparallel, orcombined and therefore the magnitude of the tilt inthese layers should differ, the phase could developonly through a discontinuous phase transition fromthe nontilted SmA phase or at lower temperatures inbulk samples.

As the two questions cannot be answered satisfacto-rily one should discuss a last possibility: how to avoid orat least lessen the competition between favorable inter-actions with nearest and next-nearest layers: the out-of-plane tilt modulation. The argument is the following. Astructure where the directions of tilts in neighboring lay-ers form a large angle can result in an angle close to � inthe next-nearest layers �Figs. 32�d� and 32�e��. Put sim-ply, if the system “pays” in nearest layers by not havingexactly parallel or antiparallel tilts, it benefits by the tiltsbeing almost antiparallel or at least far from parallel inthe next-nearest layers. As a final effect, the free energyof the structure can be lower than the free energy ofstructures formed in the uniplanar way.

This understanding of structures is essentially differ-ent from that of the continuous approach. The largeangle of a few tenths of degrees formed by tilt directionsin neighboring layers contradicts the concept of elasticenergy connected to small changes of order parameterfrom layer to layer. Although the change of the tilt di-rection from layer to layer is already included in thecontinuous model, it could still be considered as an in-ternal structure within the crystallographic unit cell oftwo layers.

Another question concerns interlayer interactionswhich do not significantly decrease with distance. But ifinteractions between nearest layers originate in differentinteractions that have opposite effects they may cancelout, at least partially. If so, the interactions with next-nearest layers may become comparable to interactionswith nearest layers. This possibility for the structure wasfirst reported by Cepi~ and Žekš �1995�. The microscopicorigin of the interaction will be discussed later in Sec.IV.C when the reader will already be acquainted withthe model coefficients and their effects on the structure.

The free energy in its simplest form that includes theconsiderations above consists of a few terms only and isexpressed in tilt order parameters. Tilt order parameters�Fig. 27� are a layer property and are allowed to varysignificantly from layer to layer. Due to the significantchanges allowed, the classical expression for the elasticenergy is replaced by a phenomenologically expresseddirect interaction between layers in a form that allowslarge changes of direction from layer to layer. The inter-actions are separated to intralayer interactions and inter-layer interactions extending to the next- and next-next-nearest layers. Each layer interacts with the layers aboveand below itself. The free energy is then

Gj = 12�0�T − T0��j

2 + 14b0�j

4 + 14a1��j+1 · �j + �j · �j−1�

+ 14 f1��j � �j+1 + �j � �j−1�z + 1

16a2��j · �j+2

+ �j · �j−2� . �13�

The first two terms, rewritten from the continuousmodel, comprise interactions between molecules fromthe same layer and account for the main tilt dependenceon the temperature. The subscript zero denotes the in-teractions within the same layer. The third term a1 de-scribes the achiral interaction between the nearest-

(a) (c)

2,4

1,3

4

3

2

1

(b)

1,3 2,4

(d)

4

3

1

2

(e)

1

2

3

4

FIG. 32. For favorable anticlinic tilts in neighboring layers allbilayer structures �a�–�c� are disfavored. The short helix allowsfor the reduction of frustration as the neighboring layerslightly deviates from the favorable �d� synclinicity or �e� anti-clinicity; however, the next-nearest layers slightly deviate fromthe favorable anticlinicity.



neighboring layers. The subscript 1 reminds the readerthat the interactions considered are in the first-neighboring layer. In its simplest bilinear form it canonly account for favorable anticlinicity �a1�0� or syncli-nicity �a1�0�. The term f1 describes the chiral interac-tions with nearest-neighboring layers. The sign of thecoefficient depends on the handedness of the sample,i.e., on the enantiomeric excess. It is zero for racemicmixtures. Although chiral interactions are in generalconsidered as very weak, one is not allowed to excludethem, as experiments clearly show the large influence ofoptical purity on structures and phase sequences. Thelast term a2 gives achiral interlayer interactions to thesecond neighboring layers. It might favor synclinicity�a2�0�, which is a required condition for the formationof bilayer primitive cells. If anticlinicity is favored �a2�0�, competition between nearest-layer and next-nearest-layer interactions takes place. For strong enoughcompetition �a2� �a1��, the new structure with largeangles between tilt directions in neighboring layers be-comes stable �Figs. 32�d� and 32�e��.

As in the continuous model, we search for a solutionwith constant magnitude of the tilt in all layers, consis-tent with the uniform layer thickness reported by x-rayobservations �Isozaki, Hiraoka, et al., 1992�. For the sim-plest structure we assume that the angle between tiltdirections in neighboring layers is constant as well,

�j = �cos�j� + �0�,sin�j� + �0� , �14�

where � is the magnitude of the tilt, � is the phase dif-ference, �0 is the angle the tilt in the zeroth layer formswith the arbitrary direction in space chosen as a coordi-nate axis, and j runs over all layers in the sample. Thefree energy �13� adopts the form

Gj = 12�0�T − T0��2 + 1

4b0�4 + 12a1�2 cos � + 1

2 f1�2 sin �

+ 18a2�2 cos 2� . �15�

If one neglects the chiral interaction, the minimizationof the free energy with respect to the phase difference �leads to three different tilted structures. Conditions fortheir stability are simple,

SmC*:a1

a2� − 1, � = 0,

SmCA* :

a1

a2� 1, � = � , �16�

SmC�* : − 1 �

a1

a2� 1 � = arccos�−

a1

a2� .

Two structures present in the continuous model can eas-ily be recognized. The solution with �=0 presents astructure with synclinic neighboring layers of the SmCphase. The solution with �=� presents a structure withanticlinic neighboring layers of the SmCA phase. Bothstructures can be found in racemic mixtures; due to theabsence of chirality, polarization is not present and onlyclinicity can be observed. With inclusion of the weak

chiral term, the phase difference is distorted from 0 or �for an angle �,

� = ±f1

�a1� + a2, �17�

where the positive sign gives the deviation from 0 for theferroelectric structure and for positive f1, while the nega-tive sign gives the deviation from � for the antiferroelec-tric structure and for the positive sign of f1. Thereforethe helical modulations have opposite signs in the chiralferroelectric SmC* and antiferroelectric SmC

A* phases of

the same material, as is consistent with experimental ob-servations.

The third structure is helically modulated even in theabsence of chirality and the period of modulation ex-tends over a few layers only. The modulation is not theresult of the chiral interactions and is, in the absence ofchiral terms, doubly degenerate. If chiral interactionsare present, the degeneracy is lifted and the helicalmodulation has only one sense defined by the chiralityof the molecules. This was suggested as a possible struc-ture for the SmC

�* phase. It is consistent with experimen-

tal observations: the molecules are tilted, due to the sub-light wavelength pitch of the helical modulation theoptical properties do not reveal optical rotatory poweror other indications of the helical modulation such asselective reflection, and the properties are due to aver-aging effects similar to the properties of the SmA phase.The phase can be stable in the temperature window be-low the nontilted SmA phase and can evolve continu-ously from SmA. Direct resonant x-ray scattering ex-periments have confirmed this helicoidally modulatedstructure of the SmC

�* phase with pitch extending over a

few layers only �Mach et al., 1998, 1999�. Later detailedmeasurements in various systems have shown that thepitch in the SmC

�* phase can have any incommensurate

values including very small values close to three layers�Liu et al., 2007� as well as rather large values around 20layers �Johnson et al., 1999�.

Before direct resonant x-ray observations were madethe short-pitch model for the SmC

�* phase was not

widely accepted. However, there were some indirect in-dications that the suggested structure could allow for anexplanation of the oscillating behavior of the differencebetween the ordinary and extraordinary refraction indi-ces as temperature decreases found in the free-standingfilm in the SmC

�* phase �Bahr et al., 1995; Schlauf et al.,

1999� �Fig. 33, top�. The reasoning is the following. Thefree-standing film consisting of a few tenths of layers isthick enough that helices of the short-pitch SmC

�* phase

develop. When the pitch is commensurate with a num-ber of film layers, the average optical indicatrix isuniaxial with the axis along a layer normal. If the pitch isnot commensurate, the remnant part of the helix con-tributes to the biaxiality of the indicatrix and to the dif-ference between the largest and smallest refraction indi-ces. This difference is measured ellipsometrically. Thefilm is oriented using a small electric field perpendicularto the layer normal, which defines the eigendirections of



the optical indicatrix. The oblique light perpendicular tothe electric-field direction is incident on the film and thephase difference between the ordinary and extraordi-nary rays is measured for two opposite directions of theelectric field. The oscillating behavior of the phase dif-ference indicates the increase and the decrease of thebirefringence as the temperature decreases due to thetemperature dependence of the helical pitch length andits consequent remnant part �Rovšek et al., 1996�.

The model was further developed to account for thewhole phase sequence and the microscopic origin of thevalues and signs of the model coefficients was discussed�Žekš and Cepi~, 1998; Škarabot et al., 1998�. To accountfor the whole phase sequence of MHPOBC includingthe SmC

�* and SmC

�* phases within one model and a

single set of model coefficients, the discrete model wasmerged with the continuous one rewritten in a discreteform. The idea was the following: the continuous modelsuccessfully reproduced three of the phases, the SmC*,the SmC

A* , and the SmC

�* between them. What was miss-

ing was the phase between the SmC* and the SmA,which was now occupied with the short-pitch structuresproposed by the discrete model �Cepi~ and Žekš, 1995�.

The discrete form of the continuous model is obtainedby inserting definitions of the ferroelectric and antifer-

roelectric order parameters �5� in the free energy �6� asdiscussed by Sun et al. �1993�. The coupling of the ferro-electric and antiferroelectric order parameters expressedin layer order parameters suggested another term whichdescribes the biquadratic coupling between neighboringlayers,

14bQ�j · j+1�2. �18�

Here the model coefficient bQ can be positive, favoringperpendicular orientation of tilts in neighboring layers,or negative, favoring parallel or antiparallel orientationof tilts in neighboring layers. A positive value of bQleads to a new form of frustration even when a2�0 andinteractions between next-nearest layers favor synclinic-ity. It was shown that electrostatic quadrupolar interac-tions lead to a positive sign of bQ �Cepi~ and Žekš,2007�. If the coefficient a1 changes sign from negative topositive and a2 from positive to negative as temperaturedecreases, the phase sequence found in MHPOBC canbe reproduced. However, the structure proposed for theSmC

�* phase by the continuous model was soon ruled

out by direct structural observations �Mach et al., 1998�;therefore the combinations of the two models was rec-ognized as incorrect.

The structures with short pitch are also called clockstructures, following the well-known model from solid-state physics �Chaikin and Lubensky, 1995�, and alsofrom the pictures of the structures, since the bird’s-eyeview of the set of molecules representing the averagemolecule in the layer are reminiscent of a clock hand atvarious hours �Figs. 19�c� and 20�c��. Therefore, thesame observation that unambiguously proved the exis-tence of large changes in direction from layer to layer;i.e., the short-pitch helical structure for the SmC

�* phase,

ruled out the bilayer model of ferrielectric SmC�* or as it

is called the SmCFI1* phase. The structures suggested for

the intermediate phases by resonant x-ray investigationswere clock structures of commensurate numbers of lay-ers �three and four�, which seemed the most simple andconsistent with the resonant x-ray data.

C. Origin of interactions

In the model presented above we discussed the effectsof different terms and the signs of model coefficients.However, we have not discussed the origin of the termsand their coefficients, why signs are positive or negative,and why they change with temperature. All interactionswere simply described as effective interactions ex-pressed in tilts. Each term that appears in the free en-ergy �13� has its origin in intermolecular interactions.

The first two terms originate in van der Waals attrac-tive interactions and repulsive steric interactions be-tween molecules in neighboring layers. Both coefficientsinclude the essential competition between the entropicdesire for disorder and the best packing to reduce voidsbetween molecules within the same layer. The expres-sion is the same as in the continuous model �Pikin and

FIG. 33. Comparison of theoretical analysis and experimentalobservations of phase retardation. �Top� The measurement ofthe phase retardation between the ordinary and the extraordi-nary ray for a freestanding film consisting of 122 layers in theSmC

�* phase. From Bahr et al., 1995. �Bottom� The calculation

of the phase difference based on the theoretical model of theSmC

�* phase for the same number of layers. From Rovšek et

al., 1996.



Indebom, 1978�, but in continuous models the contribu-tion to the free energy is not separated into intralayerand interlayer interactions.

The achiral interlayer interactions given by the coeffi-cient a1 have three different sources. The molecular dif-fusion or partial interpenetration between neighboringlayers promotes parallel tilts in interacting layers as hardmolecular cores cannot bend and adapt to different di-rections of tilts in neighboring layers �Fig. 34�a��. As dif-fusion is the consequence of entropic effects it is reason-able to expect that this contribution decreases withdecreasing temperature. The contribution is present inchiral samples and racemic mixtures and as it favors par-allel ordering in neighboring layers it contributes nega-tively to the coefficient a1.

Molecules in neighboring layers are attracted by vander Waals forces as well. It was shown �Bruinsma andProst, 1994� that the anisotropic properties of layers donot influence interlayer interactions if layers possess anideal liquid order, i.e., without any positional correla-tions between molecules. This is valid for van der Waalsattraction as well as for electrostatic interactions. There-fore, the tilt orientation should not be affected by thevan der Waals and electrostatic interactions betweenlayers. However, in any material the ideal positionallynoncorrelated disorder is never realized. At least somepositional correlations can be expected even for mol-ecules in neighboring layers, but if this is the case, thenfor long molecules one has to consider that distancesbetween parts of molecules are very different and thecloser parts feel a stronger attraction �Fig. 34�b��. Theantiparallel ordering of positionally correlated mol-ecules becomes more favorable as the effective distancesbetween parts of the interacting molecules becomeshorter. The contribution seems not to change signifi-cantly with decreasing temperature and as it favors an-tiparallel order it contributes positively to the coefficienta1.

The last contribution to a1 has an achiral symmetrybut is present in chiral samples only. The tilt inducespolarization which is proportional to the tilt of chiral

molecules. Therefore the tilted chiral molecules possesseffective dipole moment perpendicular to the tilt andthese dipoles interact electrostatically. Although the di-poles from neighboring layers if they are oriented alongthe layer do not interact in the absence of positionalcorrelations �Bruinsma and Prost, 1994�, it was shownthat if simple positional correlations are assumed anti-parallel dipoles are favored �Cepi~ and Žekš, 1997�. Asthe polarization is perpendicular to the tilt and its direc-tion is defined by the molecular chirality, the favorableantiparallel orientation of dipoles is transferred to thefavorable antiparallelism of the tilt. The interaction isalso not significantly influenced by the temperature. Be-cause of its favored antiparallelism it contributes posi-tively to a1.

Now we can see that as the temperature decreases,the negative contribution to the a1 term decreases, thecoefficient may become very small and even change sign.The reasoning is consistent with the experimental obser-vation that antiferroelectric SmC

A* is always stable at the

lowest temperature in the phase sequence.The molecular interpenetration, i.e., diffusion, does

not contribute to the interactions in the next-nearest lay-ers. As one can assume at least weak positional correla-tions to the next-nearest layers we are left with two con-tributions mentioned before that both favor antiparallelordering. From this it seems that it is natural that thecoefficient a2 is always positive. However, only antifer-roelectric liquid crystals are materials where the interac-tions to nearest layers become weak enough and frus-trating competition between them takes place.

We now spend a few words on the chiral interactiongiven by the f1 term. This term is the discrete form of theusual Lifshitz term in the continuous model �Pikin andIndebom, 1978�. As mentioned, this is not the only in-teraction that depends on chirality. The piezoelectriccoupling between the tilt and the polarization is of chiralorigin as well. However, the origin of the f1 term is thevan der Waals interactions. Various molecular bonds andtheir polarizability contribute to these. If the molecule ischiral, then the van der Waals field around the moleculereflects its chiral symmetry. Another chiral moleculefound in this field therefore adopts an orientation whichdue to chirality is not strictly parallel or antiparallel tothe molecule that is the source of the van der Waalsfield. Since the chiral center is not a large part of themolecule it seems that the direct chiral interaction givenby the f1 term is weak. As the effect of the term favorsnonparallelism of molecules, it influences the magnitudeof the phase difference �. This influence is probablysmall but cannot be separated from the influences ofother interactions, at least not in the SmC

�* phase. How-

ever, the important effect is that chiral interactions liftthe degeneracy between left- and right-handed helicalmodulations.

D. Distorted clock model

The resonant x-ray measurements posed a new ques-tion for theoretical understanding. What is the reason

(a) (b)

FIG. 34. The schematic presentation of microscopic origins forbilinear interactions between nearest layers. �a� Molecules thatdiffuse through layers promote the same direction of the tilt.�b� Distances between parts of the molecules in neighboringlayers are shorter for anticlinic orientation than for synclinic.



for lock in to three and four layers? Are there any inter-actions present in the system that extend over longerrange than next nearest layers? The nature of interac-tions discussed previously unfortunately did not give anyreasons for interactions of longer range. Therefore itseems that one has to consider the effect of polarizationin the free energy explicitly. The free energy has to in-clude the chiral piezoelectric coupling to make the effectof chirality more transparent. Interlayer polar interac-tions should be taken into account as well. As the stud-ies of odd-even number of layers effects in free-standingracemic ferroelectric liquid crystalline films revealed thepossible importance of the flexoelectric interactions�Andreeva et al., 1999�, one should consider them. Theflexoelectric interaction is not of chiral origin and can besignificant for systems formed of any incompletely sym-metric molecules. The flexoelectric term

��P� ·��

�z� �19�

was always considered as unimportant in continuousmodels, but for large changes of the tilt direction it is notnecessarily so. The discrete form of the free energy in-cluding polarization explicitly is �Cepi~ and Žekš, 2001�

Gj = 12�0�T − T0�j

2 + 14�0j

4 + cp��j � �� j�z + 12b0�� j

2

+ 14a1��j · �j+1 + �j · �j−1� + 1

4 f1��j+1 � �j + �j � �j−1�

+ 14b1�� j · �� j+1 + �� j · �� j−1� + 1

2�� j · ��j+1 − �j−1� .

�20�

In this expression the order parameter � was introducedin order to lift the confusion about the polarization ori-gin. The polarization appears due to the ordering ofmolecules that have a shape with geometric polar sym-metry �Cepi~ and Žekš, 2001� given by the order param-eter �. The polarization of the layer is directly propor-tional to the ordering of geometric dipoles,

P� j = P0�� j. �21�

The polarization P0 depends on the molecular proper-ties and is hidden in the piezoelectric coefficient cp.�Cepi~ and Žekš, 2001� were to reduce the confusionabout the origin of the polar ordering: it is not the elec-trostatic interactions but mainly van der Waals and stericinteractions. Because the electrostatic polarizationwithin the layer is directly proportional to the extent ofthe geometric polar ordering, one calls the order param-eter �� simply the polarization. The interactions betweenthe layer polarizations extend to more distant layers, atleast to the next-nearest layers. However, it was shown�Emelyanenko and Osipov, 2003� that flexoelectric inter-actions have similar effects as dipolar interactions to thenext-nearest layers, but they are expected to be moreimportant. Therefore we limit electrostatic interactionsto the same layer and to nearest layers only �b0 and b1�.The coefficients are both positive. b0 is positive becausethe polarization is not the primary order parameter andb1 is positive because dipolar interactions favor antipar-

allel ordering of dipoles. The coefficients of terms ex-pressed in tilts �a1 , f1� include only steric �diffusion� andvan der Waals interactions.

The flexoelectric interaction includes the fact that thedirection of the tilt in the layers above and below theconsidered layer influences hindered molecular rotationand consequently induces polarization in the layer. Theflexoelectric term � is written in a discrete form; herethe difference in tilts above and below is coupled to thepolarization. In the expression for the free energy theinteractions with the next-nearest layers expressed intilts are neglected as well, as it is expected that the vander Waals interactions to the next-nearest layers are ofthe same magnitude as or even smaller than the dipolardirect interactions.

The elimination of the polarization revealed some in-teresting facts. All terms expressed in polarizations �� jcan be written in the matrix form

GP = � · C� · + 12� · B� · � . �22�

Here � and are 2N-dimensional vectors of the form

� = �1x, . . . ,�jx, . . . ,�Nx,�1y, . . . ,�jy, . . . ,�Ny ,�23�

= 1x, . . . ,jx, . . . ,Nx,1y, . . . ,jy, . . . ,Ny .

The matrix B� couples polarization in the same layer andbetween the neighboring layers. It is a 2N-dimensionalthree-diagonal matrix and is composed from two

N-dimensional tridiagonal matrices B� ,

�B� 0

0 B�� , �24�

with elements having the form

Bj,j = b0,�25�

Bj,j±1 = Bj±1,j = 12b1.

The matrix C� is the antisymmetric matrix composed of

two different N-dimensional matrices C� and M� as

� M� C�

− C� M�� , �26�

where the elements of the matrices are

Mj,j±1 = ± 12� ,

�27�Cj,j = cp.

The minimization of Eq. �20� with respect to the polar-ization � leads to the set of linear equations

� = − B� −1C� , �28�

where B� −1 is an inverse matrix of the five-diagonal ma-trix B� . With this solution, the part of the free energy�Eq. �20�� that gives interlayer interactions can be writ-ten in the form



Gint =12

j�

kak��j · �j+k� + fk��j � �j+k�z� . �29�

This expression clearly shows that after elimination onecan find interactions extending over more distant neigh-boring layers. They are indirect, a consequence of in-duced polarizations due to the tilt variation, i.e., theflexoelectric effect. If the interactions decrease signifi-cantly with increasing distance, one can safely keep onlyachiral and chiral coefficients of smaller k. As a firstguess as to how many distant layers should contribute,the experimentally shown basic periods of the primitivecell �three and four layers� were used. k was limited tothree in achiral and to two in chiral terms. These limitswere chosen because the four-layer structure of theSmCFI2

* phase could be the consequence of the interac-tions with next-nearest layers and the three-layer struc-ture of the SmCFI1

* phase seems to require interaction tothird neighbors which favor synclinic orientation,

a1 = a1 + � cp2

b0+

14

�2

b0��b1

b0� ,

a2 =12

�2

b0,

a3 = −18

�2

b0�b1

b0� , �30�

f1 = f1 − 2cp�

b0,

f2 =cp�

b0�b1

b0� .

If interlayer interactions change with temperature, onecan expect many phases, each stabilized within the tem-perature range where one or the other interaction is pre-vailing or the competition is strong. The negative sign ofthe achiral third-nearest-neighbor interaction gives hopefor limiting the period to three layers. The positive signof the effective achiral interactions to the next nearestlayers favors four-layer structures.

Some groups made detailed analyses of the phase dia-grams in dependence of various model coefficients ak

and fk �Cepi~ and Žekš, 2001; Olson et al., 2002�. Theydid not find a strict lock in of the periods but broadeningof the regions where the period was close to three layersor close to four layers with coexistence of both. The

effective interlayer coefficients ak and fk are not free tovary, but they depend on the original interlayer interac-tions, such as piezoelectric and flexoelectric effects andelectrostatic dipolar interactions between nearest-neighboring and next-nearest-neighboring layers. Be-cause they are not free to vary, many sets �values� of the

coefficients ak and fk cannot be obtained using the dis-crete model coefficients cp ,� ,b0, etc., which reasonablydescribe interlayer interactions. For example, as men-

tioned, all b coefficients have to be positive, as electro-static dipolar interactions within the layer are not theorigin of the polar ordered state, and between the layersthe antiferroelectric order of dipoles is always favored�Cepi~ and Žekš, 1997�.

1. Lock in to commensurate periods

Within the discrete model with temperature-dependent model coefficients one is able to reproducesomething similar to the observed phase sequence.Semistabilized structures with periods around three andfour layers can also be reproduced, especially if surfaceinteractions are taken into account. However, consis-tency with experimental results could not be reached inthis way. Clock structures, which are solutions of themodel �20� and which are also suggested by the firstresonant x-ray scattering results, could not explain thelarge optical rotatory power observed in the three- andfour-layer phases. Additionally, it became clear thatwithin the four-layered SmCFI2

* phase a reversal of theoptical rotatory power sign appears very often, whichindicates the possibility of helix reversal �Philip et al.,1995�. Careful studies revealed biaxial properties offilms at the temperature where optical rotation vanishes�Fig. 35�, clearly disallowing the simple symmetric clockstructure �Cepi~ et al., 2002�.

Experimentalists �Akizuki et al., 1999� proposed amodification of the clock model in which phase differ-ences between neighboring layers are not the same. Theclock structure is expected to be flattened �Figs. 19�b�and 20�b�� or distorted. Today these structures are calleddistorted clock structures in the literature. These struc-tures allow for the biaxiality of the crystallographic unitcell and the consequent optical rotatory power of thehelically modulated structures having three- and four-layer basic periods.

The interactions considered within the model �20� didnot give any hints about a favorable “flattening” of thestructure. What kind of interactions could possibly bethe reason for the flat structures and the lock in?

The theoretical explanation followed from simpler al-though also interesting system �Pociecha et al., 2001�.The homologous series of mPIRn shows an interestingreentrant behavior �Fig. 36�. The first phase that is stablebelow SmA upon cooling is the ferroelectric SmC*

phase which develops into the antiferroelectric SmCA*

phase. Surprisingly, for short homologs the ferroelectricphase reenters. The only possible reason for the expla-nation of the phenomenon was a nonmonotonic tem-perature dependence of the interactions between near-est layers. However, the microscopic origin of thereentrance of the favorable synclinicity was not clear. Itturned out that, in spite of the previous understanding,interlayer biquadratic coupling bQ is not of electrostaticorigin only, which requires bQ to be positive �Cepi~ andŽekš, 1997�, but interactions of geometric quadrupolarorigin as well �Neal et al., 1997� contribute negatively tothe sign of the coefficient bQ. The importance of geo-metric quadrupolar interactions is typical for lathlike



molecules where an additional ordering within the layertakes place �Žekš, 1984�. If molecules order within thelayer with short axes parallel �Fig. 37�a��, the system fa-vors both the diffusion of molecules between the layers,which favors parallel ferroelectric ordering in neighbor-ing layers �Fig. 37�b��, and shorter distances betweenparts of molecules, which favor antiparallel tilt ordering,i.e., the antiferroelectric structure �Fig. 37�c�� at thesame time. The interaction can be described by

14bQ�j · j+1�2, �31�

with a negative coefficient bQ.Addition of the term to the discrete model allowed for

solutions whose periods and distortions were consistentwith experimental observations; even better, as a result

of the minimization the helix reversal within the four-layered SmCFI2

* phase came out as an intrinsic propertyof the model �Cepi~ et al., 2002� �Fig. 38�.

Why are four- and three-layer structures stabilizedwhen the geometric quadrupolar term is present? Thefavorable synclinicity or anticlinicity of the interactionstrengthens the next-nearest achiral interactions whichfavor only one type of ordering, synclinic or anticlinic.

2. The four-layer SmCFI2* phase

As a hand-waving argument consider the formation ofthe four-layered structure from the point of energeticcosts and benefits. For the special situation where a1 iszero, a2 and the quadrupolar bQ interactions define the

FIG. 35. �Color online� The one surface free film of the �R�enantiomer of 10TBBB1M7, the material studied by resonantx-ray scattering �Mach et al., 1998�. �a� At the temperature ofthe helix reversal in the SmC

FI2* phase the film is clearly bire-

fringent and �b� oriented in the electric field. �c� The film ob-servation using a plate reveals the orientation of the largerand the smaller refraction indices. From Cepi~ et al., 2002.

T’[

C]

FIG. 36. Phase diagram of the homologous series 8PIRn inwhich the length of the alkyl chain attached to the chiral car-bon atom is varied. Inset: Phase diagram for the homologousseries mPIR4 in which the length of the achiral alkoxy chain ischanged. In both cases a reentrant synclinic phase appears.From Pociecha et al., 2001.

(a)

(b) (c)

FIG. 37. The schematic presentation of microscopic origins forbiquadratic interactions between nearest layers. �a� Lathlikemolecules ordering within the layer. �b� Partial interlayer dif-fusion �gray molecules� favors synclinic ordering. �c� Intermo-lecular distances are on average shorter for the anticlinic or-dering �solid line in �c�� as favored by van der Waals attractionthan those for the synclinic ordering �dotted line in �b��.



structure, and the rest of the interactions we consider asnegligible for a moment. The favorable structure isshown in Fig. 39�a� and consists of alternating synclinicand anticlinic neighboring layers. All layers are “happy”as the next-nearest neighbors are strictly anticlinic,which is favored by interactions with the next-nearestlayers �a2 positive�. Nearest neighbors are happy as well.The relative structure is either synclinic or anticlinic,both structures favored by the quadrupolar couplingwith bQ negative. Now, switch on the chiral interactiongiven by f1, which might be rather significant �see Eq.�30�� if the flexoelectric coupling � is comparable toother terms. The parallelism of synclinic layers is dis-torted and the direction of the tilt in neighboring layersforms the angle 2� �Fig. 39�b�, left�. The anticlinic pair isalso distorted but in the opposite sense and the direc-tions form an angle �−2� �Fig. 39�b�, right�. The struc-ture has a periodicity of four layers and is strictly com-mensurate �Fig. 39�c��. If interactions with more distantlayers are negligible, the four-layered structure is com-mensurate without any additional modulations. How-ever, chiral interactions with next nearest layers f2 in-duce slight nonantiparallelism of tilts in next-nearest-neighboring layers and give rise to a helical modulationon a longer scale �Fig. 39�d��. It is rather easy to find theset of model coefficients that gives the correct phase se-quence, the optical helix reversal within the SmCFI2

*

phase, opposite helices in the SmC* and SmCA* phases,

as well as experimentally consistent phase differences

between neighboring layers. However, a detailed analy-sis of values requires experimental data on the same ma-terial synthesized within the same procedure in onelaboratory. Such an extensive and elaborate study onone single material is still missing to our knowledge.Therefore, for the time being the model coefficients areonly tentative �Cepi~ et al., 2002�.

The four-layer structure of SmCFI2* differs in symme-

try from the helical structure of the SmC�* , SmC*, and

SmCA* phases, which can all be described by one tilt and

one phase difference. The distorted clock structure witha period of four layers has two different phase differ-ences between neighboring layers, say � and �, whichinterchange. To find the regions where such a structureis stable, one has to consider contribution to the freeenergy of interfaces having both phase differences. Theexpected solution has the following form:

�2j = �cos�j� + j��,sin�j� + j�� ,�32�

�2j+1 = �cos��j + 1�� + j��,sin��j + 1�� + j�� ,

and the free energy has to be averaged over two layers.This solution is generally not commensurate to four lay-ers and can present the structure with any periodicity,even completely incommensurate ones. However, itturns out that the minimum of the free energy is ob-tained only when the sum �+� is close to �. A small

FIG. 38. Theoretical reproduction of the phase sequence ob-served in 10TBBB1M7. Evolution of the distortion ��2=2� inthe SmC

FI2* and �3=2� in the SmC

FI1* phase� �top� and the

phase difference � between the neighboring repeating units independence on the temperature �bottom�. From Cepi~ et al.,2002.

j, j+1j+2, j+3

j

j+1

j+2

j

j+1

j+3

j+2

j

j+1

j+3

j+2 j+1

�

(a)

(b)

(c)

(d)

FIG. 39. Schematic presentation of the four-layer structureconstruction. �a� The four-layered structure is stable in systemswith quadrupolar interactions �bQ�0� and negligible achiralbilinear coupling between nearest layers �a1�0�. �b� If the chi-ral interaction between neighboring layer, f1, is not negligible,the synclinic pair �left� is distorted in the opposite direction tothe anticlinic pair �right�. �c� The consequence is a four-layeredrepeating unit if the chiral interaction f1 is not negligible. �d�For non-negligible chiral interactions to more distant layers, f2,the sum of the phase differences between neighboring layersover the repeating unit is not 2�.



deviation from � results in a long scale modulation ob-served in four-layer systems. At specific conditions, strictlock in exists, where the sum is exactly � and the struc-ture is locked to exactly four layers. It is experimentallyobserved as the temperature where the optical rotatorypower diverges and changes sign, where the free-standing films become biaxial within the temperature re-gion of the SmCFI2

* phase, and where the pitch of theoptical modulation diverges. These changes are not ac-companied by any other evidences of a phase transitionsuch as differential scanning calorimetry peaks or simi-lar. But because of this behavior this “commensurate”temperature was mistaken several times for a phasetransition to a new phase.

3. The three-layer SmCFI1* phase

Similarly, the ansatz that considers three-layeredstructures also leads to a stable solution for certainchoices of the model coefficients. From Fig. 40�a� onecan see that this structure can be described by the se-quence of three phase differences �, �, and �, whosesum locks to approximately 2� in the three-layer struc-ture. It turns out that for symmetry reasons only the

solution where �=� can be stable. The expected solu-tion for the minimization is

�3j = �cos�2j� + j��,sin�2j� + j�� ,

�3j+1 = �cos��2j + 1�� + j��,sin��2j + 1�� + j�� , �33�

�3j+2 = �cos�2�j + 1�� + j��,sin�2�j + 1�� + j�� ,

and the free energy has to be averaged over three layers.The deviation of angles � and � from zero and � for

both three- and four-layer phases is defined by chiralinteractions, mainly by the combination of the flexoelec-tric and piezoelectric interactions that define the magni-

tudes of the f1 and f2 coefficients.It seems that a solitonlike structure �Fig. 40�c��, where

the direction of the polarization in the three-layer crys-tallographic unit cell reverses dynamically within do-main walls, might give consistent explanations of the ex-perimental observations �Cepi~ et al., 2002�.

The three-layer structure is of special interest fortheorists. As pointed out by Dolganov et al. �2002, 2008�,the symmetry of the phase also requires modulation ofthe tilt amplitude. Tilts in layers having phase differ-ences with both neighboring layers closer to � should belarger than in layers having a phase difference with oneof the neighboring layers closer to zero. The reason isthat the phase appears in the region where nearest-layerinteractions favor anticlinicity. The modulation of the tiltamplitude leads to the modulation of density and poten-tial observability even by nonresonant x-ray diffraction.In fact, the predicted satellite peak �Dolganov et al.,2002� was found only in the three-layer phase SmCFI1

*

�Fernandes et al., 2006�.In the systems studied only structures with unit cells

up to four layers have been undoubtedly experimentallyconfirmed, and discrete phenomenological modeling wasable to reproduce them. We searched for minimal solu-tions for structures with a number of combinations hav-ing different phase differences, locking to various short-range periodicities. Unfortunately, we were not able tofind stable structures of longer periodicities unless theperiodicity in layers was already imposed by the periodof the cycling boundary conditions �Emelyanenko andOsipov, 2003; Emelyanenko et al., 2006�. As mentionedbefore, the six-layer sturcture was found only recently�Wang et al., 2010�. The existence of the six-layer phasewithin the region of the SmC

�* without imposture of pe-

riodic boundary conditions is theoretically studied atpresent.

4. Some additional interesting results of discrete modeling

The discrete modeling has some additional advan-tages over the continuous model. The restricted geom-etry in free-standing films that consist of a few layersonly or have a distinctly different smectic order in only afew layers close to the surface than in the interior pre-sents systems where the transition from the structuresstabilized by surface interactions toward structures sta-

FIG. 40. Schematic presentation of the three-layer structureconstruction. �a� The three-layered structure is stable in sys-tems with quadrupolar interactions �bQ�0�. �b� When the chi-ral interaction between nearest layers, f1, is not negligible, thesum of the phase differences over the repeating unit is not 2�.�c� The soliton lattice is formed dynamically. From Cepi~ et al.,2002.



bilized by bulk interactions can be studied. Theoreticalconsideration of free-standing films revealed additionalcuriosities �Rovšek et al., 2000� that were later also ob-served experimentally �Conradi et al., 2004, 2005�. Free-standing films seem ideal systems to study structures ofphases and were used as such �Bahr and Fliegner, 1993a;Johnson et al., 1999; Schlauf et al., 1999�. In free-standingfilms a significant fraction of the volume is exposed tointeractions at the surfaces, which differ from the inter-actions in the interior of the film due to the missinginteractions with neighboring layers or larger orderingdue to the surface tension. This might have an importanteffect on the structure of the film as a whole, resulting inmodulation of the tilt amplitude, not present in bulkphases and the consequent longitudinal flexoelectric ef-fect �Andreeva et al., 1999�, as well as the existence ofuniplanar phases similar to the first Ising-like structuralmodels �Rovšek et al., 2000�.

Similarly, the analysis of the SmC�* structure in an ex-

ternal electric field using the discrete model has givenfew surprising results. The short-pitch structures do notbehave like structures with longer periods of helicalmodulation where soliton type of unwinding �Benguiguiand Jacobs, 1994; Kutnjak-Urbanc and Žekš, 1995� takesplace. The period of the structure already locks to cer-tain commensurate periods at low external fields form-ing devil’s staircase transitions as the electric field in-creases �Fig. 41�. At higher electric field the devil’sstaircase transforms to a harmless staircase where theperiod of helical modulations increases in periods of in-teger numbers of layers. Finally the unwinding that oc-curs at the critical field is always discontinuous in sys-tems with achiral interactions between the next-nearestlayers �Rovšek et al., 2004�.

The results might explain the steps found in apparenttilt angle during initial studies of antiferroelectric liquidcrystals �Hiraoka et al., 1991� and the behavior of phasesin external electric fields, results which lead to Ising-likestructural and theoretical models. Although the analysishas been done for the electric field, which couples polar-ization with the electric field linearly, one might specu-late about similar behavior in the magnetic field, whichcouples with tilt quadratically. Measurements haveshown that even unwinding of the long periods originat-ing in weak interactions requires extremely high mag-netic fields �Škarabot et al., 2000�, so one can hardlyhope for unwinding of the short helix in the SmC

�* phase

by application of an external magnetic field. But the in-fluences of surfaces can be in some circumstances con-sidered as fields which couple with the tilt quadratically.As the period of the modulation changes as the tem-perature is lowered, the surface field can lead to a lockof certain commensurate periods and result in the ob-served steplike behavior of the SmC

�* phase �Isozaki,

Hiraoka, et al., 1992� also as the temperature changes.These problems still wait for more detailed experimentaland theoretical research.

5. Discrete model with long-range interactions

In spite of successful results of discrete modeling hav-ing nearest-neighbor interactions only, Eq. �20� includingthe biquadratic term �31�, theoretical curiosity still ex-ists: is the biquadratic coupling the only possible originof structures locked to commensurate periods or doother, maybe simpler, mechanisms exist, which couldalso allow for commensurate periods of longer ranges?One idea for possible long-range interactions, whichmight give rise to lock ins, was suggested by Bruinsmaand Prost �1994�. They showed that neither ordered di-poles nor van der Waals interactions lead to an inter-layer interaction that favors some particular relative or-dering of the tilt direction, providing that layers can beconsidered as ideal liquids without positional correla-tions of molecules in neighboring layers. However, ther-mally excited polarization waves result in the weak longrange repulsion forces that were postulated within Isingmodels �Yamashita, 1996, 1997�.

The idea of competition between short- and long-range interlayer interactions which result in structureslocked to commensurate periods was considered by Ha-maneh and Taylor �2004, 2005, 2007�. They limited the

FIG. 41. Structural changes of the SmC�* phase in external

electric field. �Top� Distorted helical structure in an externalelectric field. The pitch in the absence of the field extends oversix layers; in the field the commensurate structure of four ro-tations in 25 layers is stable for the considered field. �Bottom�The structures are locked to commensurate ratios of distortedand nondistorted helices p /p0 at lower electric field. At higherelectric field helices lock to integer numbers of layers in thepitch p. From Rovšek et al., 2004.



short-range interaction to neighboring layers only. Thecontribution to the free energy due to short-range inter-actions depends on the relative orientations of the tilt inneighboring layers. The most general form of this inter-action is �Hamaneh and Taylor, 2007�

Vsr = vsr l

��l+1 − �l� . �34�

In their model the preferential angle formed by tilt inneighboring layers was considered,

Vsr = − vsr l

cos��l+1 − �l − �� , �35�

although they did not consider the origin of that pre-ferred angle. In principle the angle can be a conse-quence of competing bilinear short-range interactions�Cepi~ and Žekš, 1995, 2001�, especially when strong chi-ral interactions are present.

They proposed that interactions to more distant layersoriginate in bending fluctuations of smectic layers. Theytook into account the anisotropy of elastic constants forbending of smectic layers, which is a consequence of theanisotropic molecular tilt order in the smectic layer ex-plicitly. The energy associated with bending of the smec-tic layer along the tilt direction is necessarily differentfrom the energy for bending perpendicularly to the tilt.Therefore fluctuation in one direction is more favorablethan in the other. The bending fluctuation contributionto the free energy is

fl =12�kx

�2ul

�xl2 + ky

�2ul

�yl2 � . �36�

Here the x and y directions are considered parallel andperpendicular, respectively, to the tilt direction in theconsidered layer. As the normal modes contributemostly at lower frequencies, the Debye-like approxima-tion was used and integration of the contribution of dif-ferent modes resulted in the simple form of the freeenergy

F = −v

Nd l

cos��l+1 − �l − �� − �J2, �37�

where v gives the mean-field strength of the short-rangeinteractions, N is the number of layers, and d is the layerthickness. The new order parameter J is defined as

J =1

N l

cos 2�l. �38�

Here the direction in which the flattening occurs is cho-sen as direction x. The order parameter is zero for anystructure where all directions are equivalently occupied.This is also valid for commensurate clocklike structures.However, if the phase �l occupies more directions closerto zero or �, this leads to a flattened structure and it isalso more likely that the structure is commensurate. Ingeneral, one can also imagine structures that have anorder parameter defined as in Eq. �38� equal to 1 and arenot commensurate �for instance, a general Ising se-

quence extending over the whole sample without a unitcell�. However, such structures do not present the mini-mum of the free energy. The mean-field interaction ofthe ordered systems given by � depends on the level ofanisotropy �the ratio �kx−ky� / �kx+ky��, the magnitudeof the average elastic constant, temperature, and layerthickness. The expression is rather complicated and de-tails of its derivation are given by Hamaneh and Taylor�2004, 2005�. The commensurate structures are likely toappear if the coefficient � is approximately equal to 1.Starting from the general experimental data for tiltangles, magnitudes of the switching external electricfields, layer polarizations, and layer thicknesses, they de-duced that the value of � could be close to unity in somesystems and therefore the mechanism is worth consider-ing �Hamaneh and Taylor, 2004, 2005�. They also calcu-lated the range and strength of the entropy-induced in-teraction and showed that this interaction depends onthe ratio of the two constants for bending a layer oftilted smectic liquid crystal along the tilt direction andperpendicular to it. They can be of a longer range forsmall ratios but they decay as an inverse third power ofthe interlayer distance. In addition to the structure withthe longest unit cell of four layers found in the distortedclock model, they also found one additional commensu-rate structure with the unit cell extending over six layers.

6. Ising-like models

Another model that requires a potential that boundstilts strongly into one plane formed by the molecular tiltand the layer normal was initiated by the first structuresproposed for intermediate phases and the SmC

�* phase.

In these tentative structures the sequences of synclinicand anticlinic neighboring layers formed the basic re-petitive unit of the structure. Structures with differentperiods that should transform one into another upontemperature changes and should in principle have vari-ous commensurate periods were considered as devil’sstaircase structures. Yamashita and Miyazima �1993� firstattempted to explain devil’s staircase structures, the tem-perature ranges of the stability, and their macroscopicproperties using an Ising-like model. The main assump-tion of the Ising-like model�s� is the limitation of the tiltto one plane only—uniplanarity—consistent with earlyproposed structures �Isozaki, Hiraoka, et al., 1992�. Thetilt plane is defined by the layer normal and the molecu-lar tilt, and directions out of this plane are considered asprohibited or highly energetically disfavored. As thislimitation leads to only two possible directions of the tiltand x-ray studies showed no variations of the layerthickness, a constant tilt amplitude was also assumed.The tilt order within the smectic layer was described bythe Ising-type variable typical for a layer, where +1means the tilt is in one direction and −1 in the oppositedirection. Within these assumptions the free energy waswritten as �Yamashita, 1996, 1997�



G = − J �i,j�

sisj − J1 i

sisi+1 − J2 i

sisi+2 − J3 i

sisi+3.

�39�

Here the first term gives the intermolecular interactionswithin the layer. The coefficient J1 can be either positive,favoring ferroelectric structures, or negative, favoringantiferroelectric structures. The J2 term introduces com-petition and is always considered as negative. The J3term is the most important for the stability of structureswith periodicities of three layers.

The stability of the four phases is drawn in �J1 / �J2�,J3 / �J2�� space in Fig. 42 �Yamashita and Miyazima, 1993�,where q is given by n �number of waves in a period�divided by m �number of layers in a period�. In the re-gion of J1�0, a ferroelectric phase �q=0, SmC*� is sta-bilized. When J1 is negative, two possible orientationsare possible: �1� an antiferroelectric phase �q=1/2,SmC

A* � is stabilized for J3�0 since the third-nearest

neighbors always tilt in opposite directions; SmCFI1* �q

=1/3� is stabilized for J3�0 since the third-nearestneighbors always tilt in the same direction. Near the ori-gin �small J1 and J3 compared with J2� SmCFI2

* �q=1/4� isstabilized since the next nearest neighbors always favorthe opposite tilt. The phase diagram obtained in theform of a devil’s flower �Chaikin and Lubensky, 1995� isshown in Fig. 43 �Yamashita, 1997�. Apart from thestructures with complicated q numbers, the order of thephase sequence, i.e., SmC*-SmCFI2

* -SmCFI1* -SmC

A* was

explained.There was a problem in the construction of Ising-like

models: the relation to the microscopic origins of thephenomenological coefficients. The origin of the restric-tion of the tilt to one plane was not known. In addition,more complicated terms, where many molecules contrib-ute, required sophisticated statistical analysis with anumber of assumptions. Yamashita tried to explain themicroscopic origin �Yamashita, 1996� and to introducethe effect of breaking the tilt restriction �Tanaka andYamashita, 2000� with only a limited success.

E. Other theoretical approaches

Although Hamaneh and Taylor �2004, 2005� pointedout that a minor change in synthesis, like addition of onesingle group, changes the phase diagram of the sampledrastically and therefore it is highly unlikely that intra-layer and interlayer molecular interactions can be under-stood from first principles, theorists have not given upon determining how the model coefficients are related tomolecular properties. Two approaches are known: themolecular statistical approach and computer simula-tions. The molecular statistical approach presents mol-ecules as sources of multipole fields of different origins�Neal et al., 1997; Emelyanenko and Osipov, 2003� andthe free energy of a system is derived using statisticalthermodynamics. In computer simulations interactionsbetween molecules are modeled in more or less detailincluding molecular properties such as shape or thepresence of dipoles, and the results of simulations givesome information about the system behavior or evenabout some macroscopic properties which can be relatedto the phenomenological considerations and experimen-tal results. Both approaches are discussed in the follow-ing; however, for more detail see the overview articles ofWilson �1997, 2007�.

1. Molecular statistical approach

This is another powerful theoretical approach. Its pur-pose is to establish a relationship between the generalmacroscopic description discussed before and the inter-molecular interactions at the microscopic level. Mol-

FIG. 42. �Color online� Possible stable structures allowed bythe Ising model with interactions to the third-nearest-neighboring layers.

0 2J1/ IJ2 I

-2-4

T/

IJ2I

014

13

152

9

314

27

38

25

310

413

411

10

-60

12

q=

(SmC�*) (AF) (SmC*)(SmCA*)

T

q=1/3

q=1/4

SmC * Fi1 Fi2 SmC*A

0 2J1/ IJ2 I

-2-4

T/

IJ2I

014

13

152

9

314

27

38

25

310

413

411

10

-60

12

q=

(SmC�*) (AF) (SmC*)(SmCA*)

T

q=1/3

q=1/4

SmC * Fi1 Fi2 SmC*A

FIG. 43. Devil’s flower, the phase diagram in dependence ofthe temperature allowed by the Ising model. The arrow indi-cates the path of the coefficient ratio which accounts for thefour- and the three-layered structures.



ecules are modeled as rigid biaxial molecules �Neal et al.,1997; Osipov and Fukuda, 2000� with additional dipolesand/or quadrupoles; other properties can also be in-cluded. Usually only pair intermolecular interactions tothe nearest molecules are considered. This approachstarts from an effective potential, which is usually con-structed from van der Waals attractive and repulsive in-teractions, electrostatic forces, and the like. The free en-ergy is calculated as a statistical sum of contributionsdue to the intermolecular interactions. As effective po-tentials often depend on the state of order expressed inorder parameters, the order parameters can be calcu-lated from self-consistent equations, or the expressionfor the free energy can be obtained by expansion of itsdependence for small values of the order parameters. Inthe latter case, the relation to phenomenological modelsof existing theories can be established; the importance ofsome terms in phenomenological theories was consid-ered and the sign and even magnitude of the phenom-enological model coefficients were calculated �Emely-anenko and Osipov, 2003�. On the other hand, thistreatment can also result in the construction of aLandau-type free energy for the system or in the intro-duction of new important terms in existing expressionsfor the free energy �Emelyanenko and Osipov, 2003�. Asthe whole approach starts from basic molecular proper-ties such as shape or dipoles, the results can be helpful inavoiding the trial and error approach during the processof organic synthesis as well as a hint for computer simu-lations.

2. Computer simulations

Molecules forming liquid crystals with tilted polarsmectic phases consist of 100 or even more atoms, insome places bound with flexible rotatable bonds andhaving different polarizabilites, some of the bonds hav-ing nonuniform charge distributions leading to effec-tively electrostatic molecular dipoles or quadrupoles,and some of the connected groups being chiral. Suchmolecules generally possess a number of different con-formations changing over the time span due to fluctua-tions or as a reaction to interactions with neighboringmolecules. The molecules are generally large in com-parison with intermolecular distances giving rise to astrong dependence of the interactions on the relativepositions and orientations of the interacting molecules.All described properties give rise to changes of molecu-lar behaviors on different time and length scales. Forexample, the changes of molecular conformations andvibrations are on the order of femtoseconds, but the for-mation of clusters near the phase transitions takes 106

longer. On the other hand, atomistic simulations thatconsider all atoms in the molecules and interactions be-tween them can consider a few hundreds of molecules atmost using advanced computers and techniques. But, forconsideration of phase transitions one needs to considerhundreds of thousands or even a few millions of inter-acting molecules. It is highly unlikely that a single simu-

lation can consider phenomena on both short and longtime and length scales.

In spite of extensive work and rapidly developingcomputers, researchers have had to be satisfied withsimulations for simpler systems, nematics and smectics,only. They were able to extract experimentally and theo-retically consistent elastic and flexoelectric coefficientsfor nematics �Allen and Masters, 2001� and they mod-eled isotropic nematic and nematic smectic phase tran-sitions �Care and Cleaver, 2005�, but accurate densitiesof materials are still an open question. Molecular simu-lations of complex molecules forming complex phasesare still at the beginning of their evolution �Maiti et al.,2004�. As phenomenological modeling uses hand-wavingarguments for the signs of model coefficients leading tocertain structures, the integration of these guessed inter-actions with computer simulations can lead to an under-standing of the intermolecular interactions as well asserving as a hint to molecular design for better materialproperties for applications.

Computer simulations are of a few types. The simplesttype is a spin model, where a molecule or groups ofmolecules is presented as vectors �spins� that interactwith neighbors through a pair potential. Then randomchanges of orientations are applied on individual vectorsand the energy is calculated as a criterion for the accep-tance or rejection of the change. The degree of order iscalculated with respect to the average vector direction.However, these types of model give only orientationalorder, no translations of molecules are possible, andtherefore the model can only account for phase transi-tions where the orientational order changes.

The second class of models is called single-site coarse-grain molecular models. The interaction potentials arehard anisotropic potentials to which additional attractiveparts can be added. Typical representatives of thesetypes of models are models where molecules are mod-eled as hard spheroids, hard spherocylinders, or hardellipsoids. The temperature does not influence the struc-tures in hard models and therefore the models are moreappropriate for lyotropic liquid crystals. The changes inorder are induced by changes of density. Most modelsuse Monte Carlo simulations where the criteria for hardpotentials are rather simple. All positions that result inoverlapping configurations are rejected. Thermotropicliquid crystals are better modeled by soft potentials, forexample, the Gay-Berne potential �Gay and Berne,1981�. Here,

Uij = 4�0��ui,uj��ui,uj, rij��

�� 0

rij + ��ui,uj, rij� + �0�12

− � �0

rij + ��ui,uj, rij� + �0�6� . �40�

Here repulsion is proportional to 1/r12 and attraction isproportional to 1/r6, where r is the distance between thetwo interacting molecules. The dependence on unit vec-tors u and r describes the dependence of the interactions



on orientations. The properties of the potential are ad-justed by the constants �0 and �0. The model can begeneralized for disklike, biaxial, or noncentrosymmetricmolecules. It is also relatively easy to add dipoles orquadrupoles. For example, the addition of a longitudinaldipole in the molecular center or two off-center dipolesled to the formation of the tilted smectic.

The most demanding are the atomistic models thatconsider intermolecular interactions in detail. In thesemodels the interatomic bonds in molecules are consid-ered as flexible rotatable, bonds, conformations of mol-ecules are allowed, etc. Although these models are themost realistic, the number of molecules that can be con-sidered nowadays is still too small �up to 1000� and thelife span of the simulations still too short to simulatephase transitions and other properties. However, all-atom simulations have shown that the populations ofdifferent complex molecular conformations differs indifferent phases �Cheung et al., 2004�. This result is im-portant as it explains the surprising stability of variousphases. The computing time could be shortened by theapplication of coarse-grain models, where molecules areusually modeled by a series of Gay-Berne potentials.

F. Discussion

All phenomenological models presented are still inuse for consideration in different circumstances. Evenstrongly distorted clocklike structures are present insome systems, as shown by x-ray experiments �Gleesonand Hirst, 2006�.

The continuous model is practical for consideration ofsystems where only the two simplest phases, the SmC*

and SmCA* phases, exist. It allows for the straightforward

relation of model coefficients and macroscopic proper-ties. It also allows for implementation of results fromother solid systems in the liquid crystals due to the simi-larities of the models.

The clock model without biquadratic coupling proveditself effective close to the transition to the SmA phasewhere the biquadratic coupling is still weak. For morequalitative analysis the biquadratic coupling can safelybe neglected. The model was also often used in its renor-malized version where only interactions expressed in tiltvectors were used. The origin and limitations of themodel coefficients describing initial interactions weresometimes overseen. A number of groups have consid-ered their experimental results within this slightlypoorer model and got many results consistent with ex-perimental observations �Douali et al., 2004�. Variousstructural behaviors were found including the recent dis-covery of continuous evolution of the short pitch fromfew-layer pitches toward two layers �Liu et al., 2007� aspredicted by Cepi~ and Žekš �1996�.

The distorted clock model with inclusion of the biqua-dratic term has also experienced a life of its own. Theflexoelectric contribution was derived from microscopicorigins that were an important contribution to the un-derstanding of microscopic interactions �Emelyanenkoand Osipov, 2004�. Sometimes this model is also called

the “Emelyanenko model”; however, it is the same asthe clock model �Chandani et al., 2005; Emelyanenko etal., 2006�.

Discrete modeling has proven its efficiency; howeverthere are still a number of open problems that might beaddressed by the discrete approach. There are still miss-ing answers: how many commensurate structures existwith more than four layers in the unit cell? Do they forma distinct phase with a phase transition between them?Some have not found such stable structures �Jaradat etal., 2008�; others claim that they have, although it seemsthat their periodicities could also be the consequence ofthe cycling boundary conditions of short period usedduring the solution of minimization equations �Emely-anenko and Osipov, 2004; Emelyanenko et al., 2006�.

Another open question concerns the link or interplaybetween the continuous and the discrete approach.Within a layer one has to consider structure continu-ously; from layer to layer changes of the tilt directionscan be large. How can both of the approaches be con-nected? Dynamics and stability have been theoreticallyand experimentally studied for simple helices. Butanalysis of dynamics of structures with three and fourlayers is still lacking. Theoretical analysis of films andcomparison of predictions with experiments can revealthe influence of chiral and quadrupolar interactions onthe evolution of uniplanar structures. Discrete modelinghas been successfully applied for other systems �Po-ciecha et al., 2003, 2006; Nishida et al., 2006� and it isworth mentioning that it might be applied in all systemswhere structural changes can be large at short distances,i.e., on a tenth of a nanoscale.

There are also open questions with respect to othertheoretical approaches. The link between computersimulations, the molecular statistical approach, and thephenomenological modeling will help the organic chem-ists in their search for materials with larger potentials forapplications. As simple phenomenological modeling isless time demanding than some computer simulations,the hints one can get from structures formed by complexmolecules modeled in computer simulations can alsohelp to improve phenomenological models.

V. APPLICATIONS

As mentioned in Sec. II, the history of antiferroelec-tric liquid crystal research started in 1988 as a displayapplication without knowledge of the molecular orienta-tion or structure �Chandani et al., 1988�. At that time,many engineers were developing ferroelectric liquidcrystal displays �FLCDs� using surface-stabilized states�Clark and Lagerwall, 1980�. FLCDs were quite promis-ing because of their many advantages for passive matrixaddressing, such as fast response speed, the memory ef-fect, and threshold behavior �Fig. 44�. On the otherhand, there were some drawbacks such as mechanicaldamage on alignment, low contrast ratio, and charge ac-cumulation. Antiferroelectric liquid crystal displays�AFLCDs� were expected to solve these problems, asmentioned below. Hence extensive efforts toward devel-



oping AFLCDs have been made �Johno et al., 1990; Ya-mada et al., 1990; Yamamoto et al., 1992�.

The fundamental concept of AFLCDs is shown inFigs. 45 and 46. There are two bistable states for theapparent tilt angle in positive- and negative-field re-gions. We can use the two bistable states alternately, sothat charge accumulation problems can be solved. Thedriving scheme is shown in Fig. 45 together with thetransmittance change against an applied voltage �Fig. 46,right�. Here crossed polarizers are located parallel to thesmectic layer. The AF state gives a dark and the twoferroelectric states �F+ and F−� give bright views. Thesimplified driving voltage scheme �left� shows, at the be-ginning of each frame, the device is reset to AF by thetermination of the field for a short period. If the firstpulse �VD� in addition to a bias voltage �V0� is higherthan the higher threshold voltage �Vth

H�, i.e., V0+VD

�VthH, either F+ or F− is realized depending on whether

the field is positive or negative. On the other hand, if thefirst pulse VD is low such that V0+VD�Vth

H, the deviceremains in AF. By use of a bias voltage between thelower and higher threshold voltages Vth

L and VthH, Vth

L

�V0�VthH, we can always use two bistable states alter-

nately. This leads to the advantage of no charge accumu-lation.

The response time is as fast as in FLCDs, in the rangebetween microseconds and milliseconds depending onthe applied field and temperature. The viewing anglecharacteristics are as good as in FLCDs since the switch-ing occurs essentially within the plane parallel to thesubstrates. As for the mechanical shock problems, a self-recovery effect was demonstrated; the deformed layerstructure is recovered during the switching �Itoh et al.,1991�. However, the recovery is not perfect, so that thelow-contrast problem still remains. In addition, there isanother essential problem in the contrast ratio inAFLCDs. As shown in Fig. 45, transmittance leakageoccurs when Vth

H is approached, due to the pretransitionfrom the SmC

A* to the SmC* state. Such problems bring-

ing about low contrast ratio were solved by materialsdevelopment.

DENSO presented a prototype display in 1997, asshown in Fig. 44. However, AFLCDs as well as FLCDshad lots of problems: �1� weakness to mechanical dam-

FIG. 44. �Color online� Antiferroelectric liquid crystal displaydeveloped by DENSO.

F

-

1st frame 2nd frameTime

Transmittance

Voltage

VV

V

V

V

th

0

D

D

0

0

H

VthH

VthL-

-

VthL

F+ F

-

1st frame 2nd frameTime

Transmittance

Voltage

VV

V

V

V

th

0

D

D

0

0

H

VthH

VthL-

-

VthL

F+

FIG. 45. Driving scheme for AFLCDs. On the right-hand side,optical response �transmittance vs applied voltage� is shown.Simplified driving scheme is shown on the left. The short pe-riod of 0 V at the first stage of each frame resets the system tothe AFLC state. The first �high or low� pulse VD in addition tothe bias voltage V0 selects for the antiferroelectric or ferroelec-tric state.

Tra

nsm

itta

nce

Voltage

0 VthHVth

L-VthH- Vth

L

Tra

nsm

itta

nce

Voltage

Tra

nsm

itta

nce

Voltage

cis

trans

trans

light

on

light

off

before

light

irradiation

applied

bias voltage(a)

(c)

(b)

Tra

nsm

itta

nce

Voltage

0 VthHVthHVth

L-VthL-Vth

H-VthH- Vth

LVthL

Tra

nsm

itta

nce

Voltage

Tra

nsm

itta

nce

Voltage

cis

trans

trans

light

on

light

off

before

light

irradiation

applied

bias voltage(a)

(c)

(b)

FIG. 46. Principle of spatial light modulator using antiferro-electric liquid crystals with photoisomerizable molecules. Thethreshold voltage shift is associated with trans-cis isomeriza-tion by light irradiation. Dark and bright �off and on� statescan be switched under a bias voltage.



age due to the existence of smectic layer structures, �2�smectic layer rotation on application of an electric field�Ozaki et al., 1994; Nakayama et al., 1995�, �3� tempera-ture dependence of physical parameters such as tiltangle and viscosity, �4� rather low contrast ratio, and soon. These are not problems in nematic liquid crystal dis-plays �NLCDs�. In addition, the development of NLCDsrapidly solved various problems such as the switchingspeed and viewing angle �Kim and Song, 2009�. Forthese reasons, DENSO could not commercializeAFLCDs.

Antiferroelectric liquid crystal materials with a tiltangle near 45° have a special optical property, i.e., theanticlinic molecular orientation between neighboringlayers gives isotropy at least when viewed from the nor-mal direction. Such a material was first reported in 1976�MHTAC �Figs. 1 and 2��, although it was not identifiedas an antiferroelectric phase at that time. Robinson et al.�1997, 1998� also reported such materials. More recently,Drzewinski et al. �Drzewinski et al., 1999; Dabrowski,2000� synthesized partially fluorinated compounds witha high tilt angle near 45°. D’Have, Dahlgren, et al. �2000�and D’Have, Rudquist, et al. �2000� used these materialsto measure the optical performance associated with theSmC

A* -SmC* field-induced phase transition. They

showed that a homogeneously aligned SmCA* cell be-

comes uniaxial with the optical axis perpendicular to thecell normal when the tilt angle is 45° and actuallyshowed an excellent dark state in the absence of a field.Because of the isotropy around the cell-normal direc-tion, the darkness does not depend on the layer-normaldirection. This means that surface alignment treatmentis not necessary. Although the advantages of using 45°antiferroelectrics are clear, displays using such materialsare still far from being commercial.

The AFLCDs mentioned above are of passive matrixaddressing type and are bistable devices essentially forblack-and-white displays. For full-color images, displayof gray levels is indispensable. To realize a full-color dis-play as in Fig. 44, spatial and/or temporal multiplexing isnecessary. Otherwise, continuous brightness change hasto be achieved by other techniques. For this purpose,polymer-stabilized AFLCDs were examined �Straussand Kitzerow, 1996�. Another display mode that is calledV-shaped switching and is capable of displaying gray lev-els was proposed by Inui et al. �1996�. Although exten-sive effort has been made from basic and applicationapproaches, we do not describe the details since there isstill discussion as to whether the performance is basedon a ferroelectric, ferrielectric, or antiferroelectric phase�Matsumoto et al., 1999; Park et al., 1999; Rudquist et al.,1999�.

Another possible application is as a spatial lightmodulator. Spatial light modulators are key for informa-tion processing. Actually liquid crystal panels behave asspatial light modulators, by which a lot of information istransformed electrically into a two-dimensional image.They can be addressed optically; parallel optical infor-mation processing becomes possible. Applications are towide-image encryption and motive object extraction us-

ing digital image processing, noise removal, pattern rec-ognition using spatial frequency filtering, stereo display,and image reconstruction using holograms �opticalphase conjugation�. Thus, it is a promising technology.The material is composed of an antiferroelectric liquidcrystal doped with photoisomerizable compounds suchas azobenzene. The device performance is based on thechange in the threshold voltage by photoisomerizationof azobenzene, as shown in Fig. 46. When the homoge-neously aligned AFLC cells are irradiated with uv light,the photochromic molecules are excited from trans to cis�from �a� to �b� in Fig. 46�. This photoisomerizationslightly distorts the orientation around the dye mol-ecules and effectively decreases the order parameter, re-sulting in a decrease of the transition temperature�Moriyama et al., 1993�. This transition temperature de-crease brings about a decrease of the threshold voltagefor the field-induced antiferroelectric-ferroelectric phasetransition. If the cell is under dc bias voltage just belowVth

H, the transition occurs from the antiferroelectric tothe ferroelectric state on light irradiation. The ferroelec-tric state is memorized under the bias voltage after thelight irradiation is turned off because of the bistability�Fig. 46�c��. It is easy to return to the antiferroelectricstate by termination of the bias voltage for a short pe-riod.

A response time of about 30 ms was reported on irra-diation of a 10 mW Ar laser without focusing. The trans-mittance can be controlled by either the light power orthe light pulse duration. From a microscope observation,it was found that the gray level recording is based on thearea ratio of the antiferroelectric and ferroelectric do-mains but not an intermediate state. On irradiation,ferroelectric domains grow along the layer direction�Kajita et al., 1993�. Therefore the spatial resolution ofthe image construction is higher along the smectic layer-normal direction, i.e., at least several tens of microme-ters, than the layer direction. Improvements in perfor-mance such as the response time can be made byproperly choosing the liquid crystals and photochromiccompounds.

VI. CONCLUSIONS AND OPEN QUESTIONS

Antiferroelectric liquid crystals are systems formed byelongated chiral molecules ordered in smectic layers,which possess a variety of phases with different macro-scopic properties. The specific systems are the results ofthe interplay of various interactions always presentwhen systems are formed of more or less complex mol-ecules. Three basic intermolecular interactions alwayscompete with the entropic move toward disorder. Thevan der Waals attractive interactions favor the shortestpossible intermolecular distances. As the molecules arelarge �a few nanometers� in comparison with intermo-lecular distances between neighboring centers of masses,the attraction cannot be modeled by simple forms of vander Waals attractions between centers of masses. Theattraction, which is the consequence of the interactionsbetween induced dipoles in different bonds within the



molecules, depends significantly on the distances be-tween parts of different molecules, which can generallybe much larger or much smaller than the distances be-tween molecular centers of masses. Therefore the mo-lecular shape has an important role in the formation ofthe nanostructures present in antiferroelectric liquidcrystals. As close packing competes, on the other hand,with steric or excluded volume interactions and the elec-trostatic interactions because molecules have a signifi-cant dipole moment, the final structure is formed due tothe delicate balance between the interactions and en-tropic tendency. These complex interactions are respon-sible for the fact that antiferroelectric materials are rarerthan ferroelectric ones.

Although the present understanding of these systemsis approaching maturity, many phenomena are still notwell understood. Therefore an overview of the experi-mental results and the problems they cause for under-standing of structures and properties can provide ideasfor the solution of future problems studied in these sys-tems. Even more useful, the polar properties can be cre-ated in other systems where molecules or parts of mol-ecules also have a complicated enough structure to allowthe construction of a polar vector associated with theorientation of certain molecular properties. One ex-ample is application of the knowledge acquired in themodeling of antiferroelectric liquid crystals to explana-tion of DNA structure. The shape of the parts of DNAmolecules can be described by the direction of the dy-adic axis, which has all the properties of polarizationexcept the significant electrostatic polarization associ-ated with the zigzag shape in tilted chiral smectics. Be-cause parts of the molecule are also chiral, they exhibitthe range of phenomena typical for antiferroelectric liq-uid crystals �Lorman and Podgornik, 2005; Manna et al.,2007�. The experimental methods and theoretical rea-soning can also be applied in rather similar achiral polarsystems, where polar packing is granted due to the spe-cific bananalike molecular shape �Roy andMadhusudana, 1997; Cepi~ et al., 1999; Nishida et al.,2006; Niigawa et al., 2007�.

This paper gives an overview of the indications thatlater led to the discovery of antiferroelectric liquid crys-tals. The methods of identification developed at thattime are still crucial for the recognition of phases today.The five different tilted phases discussed in this paperrange from the ferroelectric SmC* phase to the antifer-roelectric SmC

A* phase with the basic repeating unit ex-

tending over two layers, and three rather complexphases which have longer basic repeating units. Two ofthem, which develop directly above the antiferroelectricSmC

A* phase, the three-layered SmCFI1

* phase and thefour-layered SmCFI2

* phase, have distinct ferrielectricand antiferroelectric properties. The SmC

�* phase, which

develops in the temperature region directly below theSmA phase where the tilt is still small, is helically modu-lated with a very short pitch extending over a few smec-tic layers only. Thorough experimental studies accompa-nied by theoretical modeling led to the consistent modelof the system that is accepted nowadays.

A number of details are still open questions worthy ofstudy. We quote a few of them which are more or lessrelated to our work. Some experimental observationsstill await explanation. The softening of the polar modein an external electric field in the antiferroelectric SmC

A*

phase �Dzik et al., 2005� is an interesting question. Thebehavior of structures with more layers in the basic re-peating unit in external fields, such as an electric field orthe influence of surfaces, is also an interesting problem.The predicted devil’s and harmless staircase behaviors ofthe SmC

�* phase �Rovšek et al., 2004� are also a difficult

experimental problem. A detailed study of the polariza-tion in the layer due to the combination of the flexoelec-tric and piezoelectric effects �Cepi~ et al., 2002� and theconsequent nonperpendicularity of the tilt and polariza-tion within the layer in more complex structures is stilllacking. Although the phase diagram is theoreticallyquite well understood, the dependence of the generalphase diagram on interlayer interactions was only par-tially reproduced �Vaupoti~ and Cepi~, 2005�. The influ-ence of the quadrupolar coupling probably significantlyinfluences the formation of the devil’s staircase in anexternal electric field. Another interesting question con-cerns the effects of chirality. Does chirality affect onlypolarization and interlayer chiral interactions or does italso affect other properties such as the flexoelectric in-teraction or even intermolecular interactions within thelayer? How do the positions of chiral groups affect in-termolecular interactions in general �Dierking, 2005�?

The potential of antiferroelectric liquid crystals forapplication is probably not sufficiently appreciatednowadays. As technology has solved a number of prob-lems in the construction and efficiency of displays work-ing on the basis of nematic liquid crystals �Kim andSong, 2009�, development of the displays using new andmore complex materials is not of such importance at themoment. However, complex structures found in antifer-roelectric liquid crystals offer multiple memory states,fast responses, and interesting optical properties thatcan also be useful in the future in different applicationswe are not able to imagine at present.

Last, complex antiferroelectric liquid crystalline sys-tems provide an area where scholars can learn variousexperimental methods, phenomenological modelingbased on description of various types of order; they canstudy the influence of anisotropic and chiral propertiesand the combination of them both. Studies of behaviorin restricted geometries such as Langmuir-Blodget orfree-standing films can provide an understanding of thecrossover behavior from two- to three-dimensional sys-tems. The experiences gained can be efficiently used in anumber of studies in soft matter such as biological sys-tems. The understanding of effects of competing mecha-nisms can be used almost everywhere.

GLOSSARYSmA The nontilted or orthogonal smectic phase.

This describes its basic property that theelongated molecules are oriented perpen-dicular to the layer and parallel to the layer



normal; the molecules are not tilted. Thisphase is the highest temperature smecticphase in the liquid crystalline phase se-quence.

SmC* The usual ferroelectric synclinically tiltedphase. The structure is helicoidally modu-lated and the pitch extends over 100 andmore layers. Due to the polarization in thelayers the structure is susceptible to an exter-nal electric field. Originally, the structure wasnamed SmC

�* as it appeared below the SmC

�*

phase.SmC

�* Phase in which the elongated molecules are

tilted and the layers are polar. The structureis helicoidally modulated but the period ofmodulation is short and can extend from afew layers up to a few tenths of layers. Thephase always appears directly below the non-tilted SmA phase.

SmCA* The antiferroelectric anticlinically tilted

phase. Due to the opposite tilts in neighbor-ing layers, the polarizations have opposite di-rections in neighboring layers as well. Thepolarizations cancel out over two layers andthe system behaves antiferroelectrically. Thestructure is helically modulated, forming adouble helix, which extends over 100 andmore layers. The phase usually appears as thelowest temperature phase in the phase se-quence of the SmC*-type phases.

SmCFI1* Structure with a three-layer periodicity. The

tilt directions in neighboring layers form al-ternatively two different angles � and �, and2�+��2�. The polarizations do not cancelout over three layers and the structure be-haves ferrielectrically. Deviation from thethree-layer basic periodicity results in a longhelicoidal modulation extending over a fewhundred layers. When discovered, the struc-ture was named SmC

�* as it appeared below

the SmC�* phase.

SmCFI2* The structure with a four-layer periodicity.

The tilt directions in neighboring layers formtwo different angles � and �, and 2�+2��2�. Because the polarizations cancel outover four layers the structure is antiferroelec-tric, although the subscript FI2 recalls the his-torical consideration of the phase as ferrielec-tric. Deviation from the four-layer basicperiodicity results in a long helicoidal modu-lation extending over a few hundred layer. Insome papers the phase is called SmC

AF* as it

is antiferroelectric.

ACKNOWLEDGMENTS

The authors acknowledge Damian Pociecha and Na-taša Vaupoti~ for many stimulating discussions and care-ful reading of the paper. The financial support of Slov-

enian Ministry of Higher Education, Science, andTechnology to research Program No. P1-0055 is ac-knowledged.

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Antiferroelectric liquid crystals: Interplay of simplicity and complexity