Systems with Competing Interlayer Interactions and Modulations in One Direction: Finding their Structures

DOI: 10.1002/cphc.201301101

Systems with Competing Interlayer Interactions andModulations in One Direction: Finding their StructuresMojca Cepic*[a]

1. Introduction

Smectic liquid crystals are formed when the free energy of thesystem is lowered if elongated molecules that already have ori-entationally ordered long axes, organize in smectic layers uponlowering the temperature. The free energy is lowered furtherby new types of additional order. For example, elongated mol-ecules tilt, which decreases the effective distances betweenparts of neighboring molecules if they are uneven. If elongatedmolecules are chiral, the tilt results in induced polarization andin variation of the tilt direction from layer to layer. Finally, inantiferroelectric liquid crystals several structures appear whichdiffer in patterns of the tilt direction modulation mainly. Thepatterns are commensurate, the periodicity extends over 2, 3,4 or 6 layers, or a pattern is incommensurate with a periodicityextending over few layers only, but also up to several hun-dreds of layers. As another example one can discuss systemsformed of bent shaped or popularly called banana-like mole-cules, where an order of secondary axes appears and is usuallyassociated with a polarization in the same direction. As inthese systems tilt is also allowed and is actually rathercommon, a whole zoo of structures having various combina-tions of tilt and polarization order appear.

In continuation we limit our discussion to smectic systemsthat are described by constant nematic and smectic order pa-rameters, and where one additional order, tilt or polar order, isneeded to describe the structure of the phase at least. Evenmore, we focus the analysis to various ways how this thirdorder is modulated.

How the systems, studied in this conceptual review paper,were recognized historically? Modulations of the tilt in tiltedsmectic phases appear in systems formed of chiral liquid elon-

gated molecules that form tilted smectic phases. Material withsuch properties was synthesized following Meyer’s suggestionthat tilted smectics composed of chiral molecules should havepolar layers.[1] The polarization of the layer was expected andwas actually observed.[2] The helical modulation of the tilt di-rection appeared and it was shown that the helical modulationof the tilt is always present in chiral tilted smectic phasesexcept in some cases at a single temperature, where the lefthanded helix changes into right handed helix. The phenomen-on is called helix reversal.[3]

Fifteen years later, a new material MHPOBC, which was ex-pected to be a ferroelectric liquid crystal with a large polariza-tion, showed peculiar behaviour. Upon lowering the tempera-ture, in the region where the ferroelectric liquid crystallinephase was expected, the set of four distinct phases was ob-served.[4] The lowest-temperature phase had antiferroelectricproperties and was called an antiferroelectric liquid crystallinephase denoted as SmC*

A . One of the phases at higher tempera-tures was recognized as an ordinary ferroelectric SmC* phase.The phase between the SmC*

A and the SmC* phase has ferri-electric properties and was at that time called the SmC*

g

phase. The phase that appears strictly below the orthogonalsmectic SmA phase and above the ferroelectric SmC* phasewas called SmC*

a . The polar properties of the SmC*a phase vary

upon temperature changes and its optical appearance is simi-lar to the orthogonal SmA phase. A few years later anotherphase with antiferroelectric properties in a different tempera-ture window was recognized.[5] As the two phases, the SmC*

g

and this novel phase share the same temperature window be-tween the antiferroelectric SmC*

A and the ferroelectric SmC*

phase, they were both called intermediate phases and a newname code was introduced for both of them—the SmC*

g wasrenamed to SmC*

Fi1 and the new antiferroelectric phase wascalled SmC*

Fi2. Few years ago another phase in the set of anti-ferroelectric phases was discovered—the SmC*

6d phase that isextremely rare and appears strictly below the SmC*

a phase.[6]

Table 1 gives a few standard materials that form chiral polarsmectics. Phase sequence starting from an orthogonal SmA

Complex structures in polar smectic systems can be studiedwithin framework of discrete phenomenological models.Considered interactions are usually described by nonlinear trig-onometric functions that do not allow for a straightforward

search for solutions. The review of three methods reported inthe literature are presented and their appropriateness, advan-tages and disadvantages are discussed. Examples are given asan illustration for each method.

[a] Prof. Dr. M. CepicFaculty of EducationUniversity of Ljubljana, Kardeljeva pl. 161000 Ljubljana (Slovenia)andJozef Stefan InstituteJamova 39, 1000 Ljubljana (Slovenia)E-mail : [email protected]

� 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim ChemPhysChem 0000, 00, 1 – 14 &1&

These are not the final page numbers! ��

CHEMPHYSCHEMCONCEPTS

phase that appears upon lowering the temperature is givenbelow the molecular structure. In these materials all typicalstructures can be found, providing the enantiomeric excess issufficiently high. The material, in which the most recent phasethe SmC*

6d phase was discovered, is a mixture of more compo-nents, therefore it is not given in the Table 1. There are severalsimilarities with respect to the shape and the internal structurein the presented materials. Its molecular length (approx. 3 nm)is much longer than its width (ca. 0.5 nm). When a liquid ora liquid crystal is formed, distances between parts of differentmolecules can be much shorter then distances between mole-cules measured in distances between centers of masses, givingrise to essentially anisotropic intermolecular interactions. Allmolecules consist of less flexible core from benzene rings andmore flexible alkyl tails. Molecules forming antiferroelectricliquid crystals usually have a longer core than molecules form-ing the ferroelectric phase, resulting in higher smectic order.Molecules forming ferroelectric phases usually have one COgroup. Molecules forming antiferroelectric phases have two COgroups, resulting in a larger molecular dipole. The arrangementof more dipoles often contributes to a significant molecularquadrupole as well. Finally, all molecules have chiral groups at-tached at the end of the molecular core influencing rotationaround long molecular axes.

Although structures of phases found in antiferroelectricliquid crystals are known to researchers studying them explicit-ly, it is worth to describe them in more detail. Schematic repre-sentations of the structures are presented in Figure 1 a. Mole-cules are organized in smectic layers and they are tilted awayfrom a layer normal. The magnitude of the tilt is described by

an angle q that the average long molecular axis forms with thelayer normal, usually corresponding to the z coordinate axis.Another important piece of information for the structure is thetilt direction that is given by an angle f measured as an anglebetween the tilt projection onto the smectic layer and thechosen direction within a smectic layer corresponding to x co-ordinate axis. As molecules in the layer may tilt in any averagedirection, the structure of a single layer is presented as a conewith an apex angle 2 q and the average direction within thelayer is shown as an ellipsoid on this cone (Figure 1 a). Suchpresentations are often stylized further for more complexstructures. The cone representing a layer is given as a circle ina bird’s eye view and the arrow gives the tilt direction (Fig-ure 1 a below). For more complex structures with modulationsover several layers, the projection of cones (circles) on onesmectic plane is given. In order to follow the structure fromlayer to layer, arrows denoting tilt directions are marked with

Table 1. Standard materials exhibiting phases found in antiferroelectricliquid crystals. The phase sequence upon lowering temperature, startingfrom orthogonal SmA phase, including all existing tilted polar phases, isgiven for optically pure materials with an exception of MHPOBC, that isslightly racemized and has more phases in a sequence then a pure one.

Material Chemical formula and the corresponding phasesequence

DOBAMBC[2]

SmA$SmC*

MHPOBC[33]

SmA$SmC*a$SmC*$SmC*

Fi2$SmC*Fi1$SmC*

A

MHPBC[17]

SmA$SmC*a$SmC*

Fi2$SmC*Fi1$SmC*

A

MHPOCBC[17]

SmA$SmC*a$SmC*

Fi2$SmC*Fi1$SmC*

A

TFMHPOBC[17]

SmA$SmC*A

Figure 1. a) Top: Ellipsoid on a cone presenting a molecule tilted and orient-ed as given by order parameters. a) Bottom: Bird’s eye perspective on thecone and the simplified presentation of the layer structure with the tilt andthe polarization order marked as arrows. b) Top: Bent-shape molecule withthe same tilt presented in two possible favourable orientations—gray andwhite. b) Bottom: Symbolic presentation of the molecular side view for bothorientations presented above. The bold line gives the tilt of the bent mole-cules. The arrow notation shows polarization pointing toward or away fromthe reader.



CHEMPHYSCHEMCONCEPTS www.chemphyschem.org

www.chemphyschem.org

a number of the corresponding layer in the elementary periodof modulation.

More detailed description of structures follows the historicalorder of discoveries of their structures. The ferroelectric SmC*

phase is given in Figure 2 c. The system where elongated mole-cules order themselves into smectic layers and tilt away from

the layer normal is formed from chiral molecules. In chiral sys-tems the rotation around the long molecular axis is hindered ifmolecules are tilted and one orientation of the molecules inthe presence of other chiral molecules exists where a moleculesspends more time on average. This is the reason that the com-ponent of the polarization perpendicular to the tilt and thelayer normal do not cancel out. The smectic layer is thereforepolar, which means that such liquid crystals are very sensitiveto an external electric field. The chirality of the sample definesthe triad: the tilt, the polarization and the layer normal. Thepolarization is proportional to the enantiomeric excess and isopposite for opposite handedness of the same material.No prediction of polarization direction with respect to the tiltand the handedness can be done if different materials are con-sidered. The tilt direction is parallel in neighboring layer but inchiral samples the parallelism is not exact and a small angle isformed between tilts in neighboring layers resulting in a helicalmodulation extending over few hundreds of layers.

The structure of the antiferroelectric phase SmC*A was sug-

gested shortly after its discovery already in 1989.[4] The tilts inneighboring layers have opposite directions (Figure 2 f) and sodo polarizations. Therefore the polarization cancels out overtwo layers. The structure is the origin of low response of thephase to weaker external electric fields, and a sudden structur-al change to the structure with a higher polarization at

a threshold electric field. Such behavior is typical for antiferro-electrics and it has given the phase with these properties itsname—the antiferroelectric SmC*

A phase. This phase always ap-pears in these systems as a lower-temperature liquid crystallinephase and has given them a general name—the antiferroelec-tric liquid crystals.

The structure of the phase that appears strictly below theSmA phase, the SmC*

a phase, was predicted theoretically in1995[7] and was confirmed by resonant X-ray scattering in1998.[8] The tilt direction changes from layer to layer for a fixedgeneral phase difference a (Figure 2 a). The phase is thereforehelically modulated in the same way as a regular ferroelectricSmC* phase, only the period of the helical modulation extendsfrom few to few tenths of layers being incommensurate withthe number of layers in general, while the typical period ofmodulation in the SmC* extends over few hundreds of layers.For shorter periods of modulations the structures behave anti-ferroelectrically, for longer periods its behavior is more similarto the behavior of the ferroelectric phase in an external field.Due to the very short pitch of the modulation the structure ofthe SmC*

a phase is optically similar to the SmA phase.For the SmC*

Fi1 and the SmC*Fi2, the resonant X-ray scattering

has shown strictly commensurate periodicities. The modulationof the SmC*

Fi1 phase extends over three smectic layers. The di-rection of tilts in neighboring layers differs for either phase dif-ference a or for the phase difference b, which are not equal.The sequence of phase differences is well defined as phase dif-ference a is always followed twice by phase difference b (seeFigure 1 e). The commensurate period of the elementary unitexists because the sum a+ 2 b�2 p. A slight deviation of thesum from 2 p results in an additional helical modulation on thescale of several hundreds of smectic layers.

The modulation of the SmC*Fi2 phase extends over four smec-

tic layers. Its structure is also defined by a sequence of two dif-ferent phase differences a and b that interchange. The suma+ b�p is shown in Figure 1 d. The slight deviation of thesum of both phase differences from p, again results in a helicalmodulation on much longer scale.

Finally, the elementary modulation of the most recently dis-covered SmC*

6d phase extends over six layers. It appears strictlybelow the SmC*

a phase. The structure is very similar to theSmC*

Fi1 phase as two different phase difference interchange inthe sequence a,b,b, but the sum of the three angles in the se-quences is a+ 2 b�p (Figure 2 b).

In 1995 new systems of polar smectics formed of moleculeswith a bent core were discovered.[9] Their typical chemicalstructure is given in Table 2. The bent core of the moleculeusually consists of five benzene rings. Two CO groups are at-tached close to the benzene ring in the middle and the orien-tation of CO groups is associated with the orientation of thebent core. Both sides of the molecule have flexible alkyl chainsattached. The special shape of molecular core hinders rotationaround the long molecular axes that additionally stimulatesspatial separation of cores from chains and organization ofmolecules into smectic layers. The zoo of phases in these sys-tems is even richer then in antiferroelctric liquid crystals.Besides smectic phases, quasi-columnar phases exist where

Figure 2. Series of possible structures that appear in antiferroelectric liquidcrystals upon decreasing temperature. Cones denote the magnitude of thetilt in the smectic layer ; arrows denote the tilt direction. Bird’s eye view ofthe projection of several layers onto a single smectic plane alows for obser-vation of additional structural details of the complex phases. a) SmC*

a phase,b) SmC*

6d phase, c) SmC* phase, d) SmC*Fi2 phase, e) SmC*

Fi1 phase, f) SmC*A

phase. Alternative names of phases are given as well.





columns are formed from clusters of molecules. Columnar or-ganization is often modulated in different directions,[10] insome systems columns themselves develop from layers[11] orbroken layers in the form of twisted ribbons.[12] Phases werenamed according to the time of their naming as Bx, for exam-ple B1, B2 and so on. In this concept paper we discuss onlystructures where bent-shaped molecules organize in smecticlayers that are polarly ordered within the B2 phase. Within thisphase the whole set of structures is stable and the structureswere given descriptive names.

Several types of structures within B2 are possible.[13] To de-scribe those structures let us first introduce the graphical pre-sentation of order in bent-core systems. As seen in Figure 1 b(top right), the polarization is associated with the bent orienta-tion and is marked as an arrow. The tilt is associated with theorientation of the line that schematically connects ends of thecore. As the polarization can have two directions with respectto the tilt, it is necessary to present the polarization and thetilt in each layer. The tilt is marked as a bold line and the polar-ization is marked with arrows, as described in details in thecaption of the Figure 1 b.

We discuss six types of polar structures that are consideredas subphases of the B2 phase. The bent molecules organize ina layered order bent in one direction only. As CO groups arefound close to the middle of the molecule, the average orien-tation of the bent defines also the direction of the polarization.As the polarization can be influenced by an external field, thepolarization is used for a description of the bent orientation.Two structures have the order equal in all layers. In the recent-ly discovered SmAPF structure phase molecules in layers arenot tilted and polarizations have the same direction (ferroelec-tric order) in all layers. In the SmCSPF structure molecules in alllayers are tilted in the same direction (synclinically tilted) andalso the polarization is the same in all layers. Four remainingstructures have an elementary period extending over two

layers. In the SmAPA structuremolecules are not tilted, but po-larizations in neighboring layershave opposite directions (antifer-roelectric order). In the SmCAPF

structure molecules are tilted inopposite directions (anticlinical-ly) but the polarization order isferroelectric. Two combinationsof tilt and order remain, namelythe synclinically tilted antiferroe-lectrically ordered structure ofthe SmCSPA and the anticlinicallytilted antiferroelectrically or-dered structure of the SmCAPA.All structures are presented inTable 3.

Interlayer organization is im-portant as it defines macroscopicproperties. As said, the modula-tion in bent-shaped systems ex-tends over two layers at most.

Structures with antiferroelectric polar order have antiferroelec-tric properties. All structures are biaxial, but synclinically tiltedstructures have significantly different optical properties thananticlinic and non-tilted structures.

The systems discussed above are generally considered aspolar smectics due to existence of polar smectic layers. Our in-tention is to discuss the methods that allow one to find firstthe structures described above and next to find structures towhich those systems develop under influences of externalfields or in restricted geometries. Some of those problemshave already been discussed in the literature but many re-mained open.

The paper is organized as follows: First we define order pa-rameters necessary for description of complex structures foundin discussed systems. We present the free energy for both sys-tems—chiral and achiral polar smectics, and discuss possibleinteractions that contribute to the free energy and result in

Table 2. Standard materials forming achiral polar systems. Names of the phases are marked as generally ac-cepted. All structures described in this paper correspond to the phase called B2.

Material Chemical formula and the corresponding phase sequence

P-8-O-PIMB[9]

I$B2$B3$Cr

NOBOW[39]

I$B2$B4$Cr

W586[40]

I$SmAd$SmAdPF$Cr

Table 3. Possible ordering of bent-core molecules in smectic layers withcorresponding model coefficients.

a0,P>0 a0,P<0a1,x<0

a0,P<0a1,x>0

a1,P>0

SmAPA W1>0, SmCSPA W1<0, SmCAPA

a1,P<0

SmAPF W1<0, SmCSPF W1>0, SmCAPF





complex structures. The differences between two systems arestressed. Next we present three different methods for obtain-ing structures that minimize the free energy in such systemsand discuss their advantages and disadvantages and resumethe discussion in conclusions.

The construction of the free energy and the methods toobtain structures are not limited to smectic systems or even toliquid crystals. The methods can be used in any situation,where interactions can be written in a discrete form and canbe separated to interactions that determine the basic energeti-cally demanding structural elements, the tilt and the polariza-tion in our case, and to interactions that determine energeti-cally less demanding modulations. The separation of interac-tions to two energetic levels is not compulsory but can simpli-fy numerical efforts. Examples can be found in crystalline polarheterostructures,[14] structures formed at molecular level, inwell-known situations described by the Frenkel–Kontorovamodel and its variations[15] and others.

2. Systems with Competing InterlayerInteractions

For all systems described above it is characteristic that interac-tions, which act between molecules in the same smecticlayer—the intralayer interactions—are stronger than interac-tions between molecules in neighboring layers—interlayer in-teractions. In addition, changes of the tilt magnitude are relat-ed to much larger changes in the free energy than changes inthe tilt direction in general. Significant interactions betweenmolecules occur only if parts of the molecules (at least) arerather close to each other. Which interactions contribute incomplex systems formed from rather large molecules? The re-pulsive part of van der Waals intermolecular interactions isknown as steric interactions, which consider excluded volumeeffects. On the other hand, attractive van der Waals interac-tions tend to decrease intermolecular distances. Both parts ofthe van der Waals interactions are of short range, with a signifi-cant impact to distances much smaller than longer dimensionsof elongated molecules forming liquid crystalline phases.In polar systems electrostatic interactions are important aswell, but as smectic layers have liquid-like positional order ofmolecules, the impact of these interactions is present as longas positional intermolecular correlations exist.[16] For liquid-likesystems this means a typical dimension of a single molecule.Within the layer this dimension is defined by a width of themolecule, through the layers by a length of the molecule.To the free energy of the smectic liquid crystals intralayer inter-actions contribute the most important part. So, the elementarystructure of the layer is determined by stronger intralayer inter-actions. As intralayer interactions in the bulk do not have a pre-ferred direction, the changes in the tilt direction do not requireenergy. Interlayer interactions are important to nearest neigh-boring layers only and they determine interlayer organization,that is, the relative directions of tilts in neighboring layers.Interlayer interactions induce also minor changes of tilts in in-teracting layers consequently.

Let us shortly present the most elaborate free energies forthe two systems described in introduction for the chiral polarsmectics formed of elongated chiral molecules and for theachiral polar smectics where polar ordering is induced by or-dering of the bent-shaped molecules in the layer. Free energiespresented in continuation allow for all experimentally con-firmed phase structures and phase sequences.[17, 18]

For description of the layer order we assume that the nem-atic and the smectic orders are constant and we further limitour studies to two vectorial order parameters—the tilt and thepolarization, which are defined in the same way for elongatedas well as for bent-shaped molecules (Figure 1). The tilt orderparameter for molecules in the jth smectic layer ~xj recapitu-lates the quadrupolar nature of the up-down molecular pack-ing symmetry [Eq. (1)]:

~xj ¼ fnj;x nj;z; nj;y nj;zg ð1Þ

where ~nj ¼ fnj;x; nj;y; nj;zg is a director in the jth smectic layer.The director is related to the direction of average long molecu-lar axis in chiral polar smectics (Figure 1 a) or to the directionof the longest dimension of the bent-shaped molecule(Figure 1 b).

The polarization in the chiral smectic liquid crystals is an im-proper order parameter and it is induced by the tilt. Whena chiral molecule is tilted and encircled by other tilted chiralmolecules, rotation around long molecular axis is hindered andone favorable orientation exists due to the chiral symmetry ofthe molecule. The molecule spends more time in this position,the molecular dipoles do not cancel out on average and thelayer is polar. An isolated tilted layer is polarized perpendicular-ly to the tilt due to symmetry reasons,[1] but in more complextilted structures the polarization can have a general direc-tion.[19] In fact, the hindrance of the rotation is a steric effectand the polarization is the consequence only. Its direction isparallel to the average molecular geometric polar axis which isnot necessary associated to the direction of the moleculardipole. However, the average polarization has the same direc-tion as an average geometric axis, it is a property that is mea-sured easily and we suggest that although by origin geometri-cal, the order parameter is called polarization (Figure 1 a),given by Equation (2):

~Pj ¼ fPj;x; Pj;yg ð2Þ

The reasoning described above is even more evident in sys-tems of bent-shaped molecules. Imagine rotation of the bent-shaped molecule encircled by other bent-shaped molecules.As a molecule rotates around the long molecular axis, two ori-entations are more favorable than others. Out of the two ori-entations, given as a grey and a white molecule in Figure 1 b,one of them is the most favorable, because then moleculescan pack the most tightly. Therefore, ordering of bents is pre-ferred sterically. Molecular dipoles due to the CO groups closeto the central benzene ring is parallel or antiparallel to the





second molecular axis defined by the bent order, and the layerwith ordered bents is polar even without a tilt. As the polariza-tion of systems formed of ordered bent shaped molecules maybe polar even without a tilt the polarization is a proper orderparameter.

The free energy of the system consisting of chiral elongatedmolecules expressed in order parameters up to the fourthorder is [Eq. (3)]:[17, 20]

Gj ¼12

a0 x2j þ

14

b0 x4j þ cp

~xj �~Pj

� �zþ 1

2 e0P2

jþ

14~a1

~xj �~xjþ1 þ~xj �~xj�1

� �þ 1

8bQ

~xj �~xjþ1

� �2þ ~xj �~xj�1

� �2� �þ

14~f1 ~xjþ1 �~xj þ~xj �~xj�1

� �zþ

12

m~Pj � ~xjþ1 �~xj�1

� �þ 1

4 e1

~Pj �~Pjþ1 þ~Pj �~Pj�1

� �

ð3Þ

Here the first line of Equation (3) gives contribution of intralay-er interactions expressed in tilt and polarization order parame-ters to the free energy of the jth smectic layer. The only tem-perature-dependent parameter is a0, which changes sign atthe transition temperature to the tilted phase in an isolatedlayer without any polarization. Additional interactions such asthe piezoelectric term (cp), that couples the tilt and the polari-zation, increase the transition temperature. If the tilt appears,as the tilted phase becomes stable for example, the polariza-tion proportional to the tilt appears as well. On the otherhand, if on a non-tilted chiral smectic an electric field is ap-plied, the layers become polarized, and also the molecules tiltconsequently. Finally, if the layer is polar, the electrostaticenergy of the ordered dipoles has to be taken into account.

The form of interlayer interactions to the free energy allowsalso for large angles between tilt directions in neighboringlayers. Three types of interactions contribute to the interlayerinteractions expressed in tilt, namely steric or van der Waals re-pulsive interactions, van der Waals attractive interactions andelectrostatic interactions between dipoles or quadrupoles.Layers interact with neighboring layers only and interactionsto neighboring layers are marked by a subscript 1. Coefficients~a1, bQ and ~f1 give steric and attractive van der Waals interac-tions. Negative ~a1 favors parallel tilts in neighboring layers, ifsteric interactions due to a migration of molecules betweenlayers prevail. At lower temperatures, attractive van der Waalsinteractions prevail over entropic tendency for molecular mi-gration and antiparallel tilts in neighboring layers become pref-erential. Similar reasoning accounts for the quadrupolar cou-pling if molecules can be considered as a flattened lath-likeobject. The quadrupolar ordering of short molecular axes re-sults in either an enhanced molecular migration through layersstimulating parallel tilts or in an enhanced attraction of mole-cules from neighboring layers stimulating antiparallel tilts. The~f1 coefficient is chiral, and has the opposite sign in samplesformed by molecules of opposite handedness and is zero inracemic mixtures. The van der Waals field of molecules witha chiral symmetry, to which the whole molecular structure con-tribute with all the atoms and the polarizabilities of chemical

bonds between them, is also chiral. The favorable orientationof another chiral molecule with the same handedness posi-tioned in such a field is different from an equal moleculehaving the opposite handedness. Usually a non-parallelism intilts in neighboring layers is favored as a result. As chiral inter-actions are usually much weaker than non-chiral ones, the re-sulting modulation extends over several hundreds of layers orlifts a degeneracy between left- and right-handed structures al-lowed without chiral interactions.

The last two terms consider interactions that are presentdue to the polar nature of molecules. The simplest explanationfor the flexoelectrically induced polarization, contribution isgiven by the flexoelectric m coefficient, is the following. Whenmolecules are tilted, the rotation around the molecular longaxes is hindered. The magnitude of induced polarization de-pends on orientation, which is preferable for a tilted moleculesurrounded by other tilted molecules. But this favorable orien-tation is influenced also by interactions with the molecules inthe layers above and below the considered layer. If the tilt inneighboring layers differ, the hindrance of the rotation is differ-ent and therefore affects the magnitude of the polarization. Insome cases the difference in interactions influences also the fa-vorable molecular orientation along the long axis and conse-quently also the direction of the induced polarization.[19]

Another system, very similar to that described in Equa-tion (3) is a system formed of (very often) achiral, polar mole-cules with a bent core. Both systems differ in few ways, and al-though differences appear in interactions they do not influ-ence the search for structures. Let us consider main differen-ces. Phenomena that are present in bent core systems do notappear due to the chiral molecular shape but due to the bentmolecular shape, Table 2. Next, tilt and polarization order pa-rameter are independent and phases with polarization andwithout a tilt exist. Due to molecular bent shape, the favorabletilt is perpendicular to the polarization, however, tilt is degen-erated as molecules can tilt in two directions that are both per-pendicular to the tilt. When layers are polar and tilted, they arechiral. For these systems a chiral order parameter can be intro-duced as Equation (4):

sj ¼ xj � Pj

� �z ð4Þ

Chiral layers interact among themselves in various ways, theycan prefer equal chirality in neighboring layers, resulting in ho-mochiral samples as well as the opposite chirality in neighbor-ing layers, resulting in antichiral samples.[21] Systems of bent-shaped molecules can be doped by chiral dopants or bentshape molecules can be synthesized in chiral forms as well.Chiral effects in such systems are not yet fully understood. It isknown that doping expands homochiral domains of one signon account of homochiral domains of the opposite signs.[22]

An electroclinic effect for a doped systems was predicted[23]

and several other possible effects remain to be studied.

The free energy of the systems formed of bent-core mole-cules is given by Equation (5):





Gj ¼12

aP;0 P2j þ

14

bP;0 P4j þ

12

ax;0 x2j þ

14

bx;0 x4j þ

12

W0 s2jþ

14

aP;1~Pj �~Pjþ1 þ~Pj �~Pj�1

� �þ 1

4ax;1

~xj �~xjþ1 þ~xj �~xj�1

� �þ

14

W1 sjsjþ1 þ sjsj�1

� �ð5Þ

Let us describe in detail differences between Equation (3) andEquation (5). The form of the first two terms in Equation (5) isequal to the form of the second two terms in the same equa-tion and the same as the first two terms in Equation (3), butthey are given in polarization order parameters. The form ofthe first four terms reflects the fact that both the tilt and thepolarization are proper order parameters. Also here the first co-efficient aP,0 = aP(T�T0) is only temperature dependent andchanges sign when the structure becomes polar. The coeffi-cient bP,0 is positive for continuous transitions to the polarphase. In most of the cases the transition to the first phasebelow the ordinary SmA phase is discontinuous, therefore thecoefficient bP,0 is negative and the expansion of the free energyneeds the sixth-order term in polarization 1

6 cP;0 P6j

� �. As the

nature of the phase transition is not a matter of discussion inthis concept paper, we will not go into further details in thisrespect. The next difference appears in the chiral order param-eter and in the contribution of the interactions related to it.The chiral order parameter can be recognized also in a piezo-electric term in Equation (3), and it is present in all tilted struc-tures. In bent-core systems it is present only when both orderparameters exist and have different directions in the layer,which is usually the case. The coefficient W0 is negative, giventhat it is sterically more favorable that molecules tilt in the di-rection perpendicular to the polarization. As there is no pre-ferred direction with respect to the polarization, the system isdegenerate with respect to the chiral order parameter. It alsoseems that in most of the systems the chiral parameters arecoupled across the layers with W1 positive favoring antichiralstructures, that is, chiral order parameters have opposite signsin neighboring layers.[21] Coefficient aP,1 can be either positive,favoring antiparallel orientations of polarizations in neighbor-ing layers due to prevailing dipolar electrostatic interactionsand diffusion of molecular branches through layers, or nega-tive favored by attractive van der Waals forces betweenbranches of bent-core molecules. Also coefficient ax,1 can beeither negative due to the diffusion of molecular branchesthrough layers or positive due to the attractive van der Waalsforces between branches of molecules in neighboring layers.As interactions are a result of competition between variouscontributions to the effective interaction expressed in a modelcoefficient, the coefficient can change the sign. Interactionsare usually rather well determined and depend mostly on mo-lecular structure.[24] However, they can change sign uponchanging concentration of components, preferring synclinic oranticlinic and antiferroelectric or ferroelectric structures inmixtures.[25]

3. Stable Structures

Systems where interactions to neighbors exist are known fora long time. Heterostructures in ferroelectrics were analyzed indetails, revealing several interesting phenomena like devil’sand harmless staircase in structure evolution, that were foundin one-dimensional systems where the polarization can haveone direction only.[15] As some experimental observations re-vealed steps in temperature or electric-field-dependent proper-ties and the similarity of smectic layers to polar ferroelectriclayers in heterostructures of ferroelectric crystals was recog-nized, the Ising-like structures similar to structures in layeredferroelectrics were suggested for various phases. The tilt in thesmectic layer was believed to be bound to one plane, al-though there were no persuasive arguments given why thatshould be so. The tilt was described as an Ising-like variable,and various structures as sequences of “positive” and “nega-tive” tilts.[5] Few years were needed before the XY character ofthe tilt order parameter was recognized and that variations intilt direction out of plane were considered as more probablethan changes in the tilt magnitude only.[7, 8]

Studies of polar smectic systems have revealed that struc-tures are modulated on a short length scale of a few layersand in chiral polar systems of elongated molecules on a longerscale extending over several hundreds of layers in addition tothe short-range modulation. To find the structures one has tominimize the free energy with respect to all two-dimensionalpolarizations and with respect to all two-dimensional tilts. Dueto interactions with neighboring layers, equations are coupled.As interactions are expressed using trigonometric functions,equations are also highly non-linear. Free energy has usuallyseveral rather close metastable minima that further complicatethe search for stable structures. Another complication is severalmodel coefficients. Although the measurements clearly indi-cate the periodicity of structures and also further details ofstructures, the model coefficient space is not straightforwarddue to several contributing and competing interactions. Forexample, the flexoelectric interaction favors the largest possi-ble difference in direction of tilt above and below the interact-ing layer. Therefore one expects antiparallel tilts for next-near-est-neighbor layers. This interaction competes with other near-est-layer interactions which favor parallel or antiparallel tilts inneighboring layers and ferroelectric or antiferroelectric order-ing of polarizations in neighboring layers. The guess for coeffi-cients which lead to the stable structures with shorter periodsis consideration of structures favored by an isolated interac-tion.[26] For example, if the flexoelectric coefficient is large andother coefficients are small, one should search for a solutionthat has oppositely tilted molecules in next-nearest-neighborlayers. The period of such a modulation, that is, the distancethat includes the whole sequence of tilts as a primitive cell ofa repeating structure, is four layers. Therefore in a search forstructures one is not completely bound to a trial and error.

But let us return to the basic question—how to find stablestructures? Three different methods are used by researchers.Here one of the methods is discussed in more details as it isless demanding for the calculation of structures but more de-





manding in the presentation of the results.[26] The other twoapproaches[27, 28] are described as well. Although the methodgiven in ref. [27] was used for finding stable structures it ismuch more appropriate for studies of structures in externalfields.[29] Similarly, the method proposed by ref. [28] was usedfor studying structures in free-standing films.[30, 31] Reflectingvarious circumstance in which the methods were used, the ad-vantages and disadvantages of the three methods will be dis-cussed and situations in which one of approaches is betterthan other will be considered.

In all three methods, the final structure is described as a setof order parameters characteristic for each layer.

3.1. Relative Orientations of Tilts

The first method considers the relative orientations of orderparameters, that is, the angles between directions of tilts inneighboring layers called phase differences, and operates ina space of phase differences.[7, 18, 20] Solutions, which correspondto commensurate stable structures have in general shorter pe-riodicities than their equivalents in the real space. For example,the four-layer structure of the SmC*

Fi2 phase has periodicity oftwo phase differences instead of four order parameters, whichsignificantly reduces the number of equations. Let us illustratethe search for a solution for one example from structuresfound in antiferroelectric liquid crystals (Figure 2).

In Equation (3) polarizations appear in bilinear terms only.Therefore polarizations can easily be eliminated and the freeenergy is expressed in tilt order parameters only. The freeenergy of the single layer, interacting with its surrounding lim-ited to interlayer interactions only, has after elimination of thepolarization a very simple form [Eq. (6)][20]:

Gint;j ¼X3

k¼1

12

ak~xj �~xjþk

� �þX2

k¼1

12

fk~xj �~xjþk

� �z

!

þ 14

bQ~xj �~xjþ1

� �2:

ð6Þ

In Equation (6) interlayer interactions with neighboring layersare not written as the average of interaction with layers aboveand below the interacting layers because every term appearstwice in the summation over the whole sample and allows fora shorter version of the equation. The flexoelectric interactionsgiven by m couple indirectly more distant than neighboringlayers although direct interactions extend to nearest neighborsonly. The strength of interactions decreases with more distant

layers. Achiral interactions ak decrease as e0

e1

� �k�2and chiral in-

teractions fk decrease as e0

e1

� �k�1. Therefore we consider achiral

interactions to next nearest layers and chiral interactions tonext nearest layers only [Eq. (7)]:

a1 ¼ ~a1 þ e0 c2p þ

14

e0 m2

� �e0

e1

� �

a2 ¼12

e0 m2

a3 ¼ �18

e0 m2 e0

e1

� �

f1 ¼ ~f1 � 2 e0 cp m

f2 ¼ e0 cp me0

e1

� �

ð7Þ

Writing the order parameter~xj as Equation (8):

~xj ¼ qj fcos�j; sin�jg ð8Þ

and naming the angle formed by tilt directions in neighboringlayers as shown in Equation (9):

aj ¼ �jþ1 � �j ð9Þ

one describes the structure by a set of tilt magnitudes qj char-acteristic for layers and the set of phase differences aj forwhich the tilt direction changes from the layer j to the layerj + 1. The search for structures starts with the sequence ofphase difference aj that is finite. The length of the sequence isguessed by the symmetry of the periodical structure in the realworld. Let us illustrate this approach for the search of a three-layer structure, which was suggested by the resonant X-raymeasurements.[32] The suggested structure is seen in Figure 2 e.The theory has to answer the following questions: Does sucha structure present a minimum of the free energy? Which in-teractions define the angle a in Figure 2 d and how large can itbe? Which interactions define the additional helical modula-tion on a longer scale given by d and which periodicities aretypical? The structure is described by a sequence of phase dif-ferences in the form a,b,b. As one layer has different phase dif-ferences above and below the other two, one should alsoexpect a slightly different tilt in one layer out of three. The tiltdifference is small as the changes of the tilt are associatedwith the changes of the layer thickness, which is energeticallydifficult to compensate by changes in the tilt direction as longas the structure studied is not very close to the continuoustransition to the tilted phase. So, the search for the structure isdone in three steps—first the constant tilt approximation isused and the Ansatz [Eq. (10)]:

~x3j ¼ q fcos j aþ 2 bð Þð Þ; sin j aþ 2 bð Þð Þg~x3jþ1 ¼ q fcos j aþ 2 bð Þ þ bð Þ; sin j aþ 2 bð Þ þ bð Þg~x3jþ2 ¼ q fcos j aþ 2 bð Þ þ 2 bð Þ; sin j aþ 2 bð Þ þ 2 bð Þg

ð10Þ

is inserted into the free energy. The resulting average freeenergy per layer is characteristic for an arbitrary set of threeconsecutive layers [Eq. (11)]:





G3 ¼12

a0 q2 þ 14

b0 q4 þ 16

a1 q2 cosaþ 2 cosbð Þþ

16

a2 q2 2 cos aþ bð Þ þ cos2bð Þþ

12

a3 q2cos 2 aþ bð Þþ

112

bQ q4 cos2 aþ 2 cos2 bð Þ þ 16

f1 q2 sinaþ 2 sinbð Þþ

16

f2 q2 2 sin aþ bð Þ þ sin2 bð Þ

ð11Þ

Instead of six independent equations that would be necessaryin the space of tilt magnitudes and directions, one is left withthree rather simple minimization equations only. Here oneshould be aware, that this sequence of phase differences leadsto a commensurate structure only, if the sum of the threephase differences is a fraction of 2 p. For example, if a+ 2 b=

2 p, the basic structural unit consists of three layers. If the sumis equal to p, the structural unit consists of six layers and thesolution, if it is stable, presents the structure of the SmC*

6d ; ifthe sum is equal to 2 p/3, the unit would extend over ninelayers and so on. It is not necessary that considered structureshave any of such limitations. In those cases structures wouldnot be commensurate with the layer thickness although thesolution could be described by two different angles and twodifferent tilt magnitudes only.

The final solution for structures that are commensurate inthe short range and additionally modulated with a long-rangeperiodicity is found in the following way. First : the constantamplitude in the layers and the commensurate periodicity isused as an Ansatz. The sum of the angles in the basic period istherefore set to a fraction of 2 p as described before. This re-duces the free energy to dependence on the magnitude of thetilt and a single phase difference. If the sequence of phase dif-ferences is longer, the symmetry reduces the number of differ-ent phase differences in the sequence and the commensurabil-ity requirement reduces the number of different angles further.All structures that were confirmed by resonant X-ray, havea single unknown phase difference at this step. Next, the mini-mization is done with respect to two different phase differen-ces, using the commensurate solution as a first approximation,for three phase differences, for example a,2 p-a/2,2 p-a/2. Forweak chiral interactions one is usually close to the solutioneven in this approximation, but the next minimization is donefor two different angles a and b where the commensurateangles are used as a first approximation. The difference d=

2 p�a�2 b gives the helical modulation on a longer scale. Forlarge chiral coefficients additional minimization usually con-verges to a different solution, presenting a helically modulatedphase with a single phase difference and only such simplestructures of the SmC*

a phase are stable. Finally, the differencein tilts is calculated using the equal tilts as a first approxima-tion. The procedure is repeated a few times until the desiredaccuracy is obtained. The free energy of the solution has to becompared to free energies of structures with different sequen-ces, for example, different number of angles in the sequence.

This method is the most appropriate for the search of struc-tures in bulk. It has several advantages. The most important isprobably that the method allows for structures having anyphase differences. There is no limitation for deviations of tilt di-rections out of one plane. One should also not neglecta severe reduction of equations that have to be solved. Anoth-er advantage is that the system of equations includes rotation-al invariance for solutions and allows for incommensurable sol-utions of various periodicities and short-range structures ob-tained by the same method. Finally, when one becomes famili-ar with effects of various model coefficients, the search formore specific stable structures or ranges of coefficients, inwhich they might be stable, becomes rather straightforward.

At this point it is worth to discuss another interesting ques-tion that is initiated by experimental results. Up to now, inter-mediate phases as well as the SmC*

a phase were found inchiral samples only. Is the chirality really necessary for the sta-bility of those phases? Let us suggest a hypothesis from thetheoretical point of view. The method used above allowed forthe analysis of a general phase diagram of antiferroelectricliquid crystals.[33–35] It allows for all observed structures havingonly one temperature dependent coefficient a0 = a(T�T0) anda single interlayer interaction ~a1 influenced by the magnitudeof the tilt. All the complexity of modulations having periods ofseveral layers arises due to the competition of various interac-tions that extend to neighboring layers only. Consequent inter-actions have longer range of achiral and chiral characters. Thefree energy Equation (3) has two chiral coefficients only: thepiezoelectric coefficient cp that defines the magnitude of thepolarization in the layer and the interlayer chiral interactionsgiven by f1. The latter is usually considered as small. Equa-tion (8) shows that interactions to neighboring layers a1 are in-fluenced by chirality, but they exist also in achiral samples.So, from this point one can safely conclude that the synclinicSmC phase and the anticlinic SmCA phase exist in racemic mix-tures that were indeed observed. The SmC*

a phase and theSmC*

Fi2 exist in regions where an effective coefficient a2 islarger than a1 that is close to zero. If quadrupolar interactionsgiven by negative bQ are already strong, they grow with in-creasing tilt, the SmC*

Fi2 phase is stable, if not, the SmC*a phase

is stable instead. So, both phases can in principle appear inracemic mixtures as well, where polarization is not present.The precondition is a large flexoelectric effect. Its influencesare represented by m in the free energy. However, interactionswith third-nearest neighboring layers a3 depend on combina-tion of the flexoelectric and the piezoelectric effect. In somecircumstances these interactions in combinations with quadru-polar interactions with negative bq can stabilize the three-layerSmC*

Fi1 phase or the six-layer SmC*6d phase. Therefore these two

phases should not be found in racemic mixtures according tothe theoretical understanding. It seems that existence of polar-ization, that is, a significant enantiomeric excess shifts the can-cellation of entropic forces favoring synclinic tilts and electro-static dipolar interactions favoring anticlinic tilts in neighboringlayers to such temperature ranges where the interplay of otherinteractions results in complex structures. In the absence ofpolarization the competition is not so pronounced anymore





and only simpler phases are stabilized. A hint for finding morecomplex structures in achiral samples is hidden in the flexo-electric effect. Combination of molecules having large flexo-electric interactions, lath-like shape to foster the in-plane tiltsand allowed entropic movement of molecules between layers,might give a hope to find more complex structures in racemicmixtures.

The straightforwardness in the search for stable structures iseasily demonstrated in the case of bent-core systems. Bent-core molecules do not rotate around their long axes as rapidlyas elongated molecules, which is indicated by the magnitudeof the polarization. One also cannot expect a significant flexo-electric effect related to different orientation of tilt or polariza-tion in neighboring layers as in chiral system of elongated mol-ecules. Without an important flexoelectric contribution, the ef-fective interactions to more distant layers do not exist. If so,the free energy given in Equation (5), that includes direct inter-actions to neighboring layers only, leads to the solution ofonly one periodicity, that is, the structures can be described bya single phase difference in the tilt ax and a single phase differ-ence in polarization aP, if it is expressed in the space of phasedifferences. As the polarization is a proper order parameter, itsmagnitude and orientation could be considered separately[Eq. (12)]:

~Pj ¼ Pjfcosyj; sinyjg: ð12Þ

The magnitude of the tilt and the polarization are constant asall layers are found in equal position. The free energy is simpli-fied to Equation (13):

G ¼ 12

aP;0 P2 þ 14

bP;0 P4 þ 12

ax;0 x2 þ 14

bx;0 x4þ

12

W0 P2x2 sin2 yj � �j

� �2þ

14

aP;1 P2 cos yjþ1 � yj

� �þ cos yj � yj�1

� �� þ

14

ax;1 x2 cos �jþ1 � �j

� �þ cos �j � �j�1

� �� þ

14

W1 P2x2 sin yj � �j

� �sin yjþ1 � �jþ1

� �þ sin yj�1 � �j�1

� ��

ð13Þ

A closer look to the free energy shows that there are threephase differences, the already mentioned phase difference be-tween polarizations in neighboring layers, aP,j = yj + 1�yj, thephase difference between tilts in neighboring layers,ax,j =fj + 1�fj, and the new phase difference between the tiltand the polarization in the same layer, Dj =yj�fj. As all threeangles are never coupled, they can be minimized separately.The negative sign of aP,1 stabilizes ferroelectric structures,which is the parallel ordering of polarization in neighboringlayers or aP,j = 0. The positive sign of the same coefficient sta-bilizes antiferroelectric ordering in polarization, which is theantiparallel ordering of polarization in neighboring layers oraP,j = p. The negative sign of W0 allows for tilts to be perpen-dicular to the polarization only. The tilt directions however isnot defined and is doubly degenerated or Dj ¼ � p

2. Molecules

in the layer can tilt in both directions with respect to the polar-ization and actually one meets both types of domains in sam-ples. But relative orientation of tilts is well defined. For nega-tive ax,1 tilts are synclinic or ax,j = 0, for positive coefficient ax,1,tilts in neighboring layers are anticlinic or ax,j = p. Finally, thepositive chirality coupling W1 favors layers with opposite signsof cross products of the tilt and polarization, while the nega-tive sign favors the same sign of chirality in neighboring layers.From this simple consideration, one recognizes that the lon-gest periodicity of the structure could be two layers only.Although there are more order parameters, as long as thesystem is bound to simple smectic layers, structures are lesscomplex than in systems formed of chiral elongated molecules(Table 3). More complex phases appear when layers becomemodulated. However, our discussion of structure ends here.

The method that considers relative phase differences hasalso disadvantages. Combinations of equations containing sev-eral trigonometric functions are usually not solved analytically.But also numerical methods are not always simple. The freeenergy dependence on model coefficients allows for severalmetastable minima, which complicates numerical search forsolutions. Considerations of relative orientations in neighbor-ing layers is also not the most appropriate when systems arefound in external fields or in films. Although it is possible toanalyze evolutions of structures with periodicities of few layerswhen the long range modulation is already unwound or infree standing films with few layers only, this approach is notvery powerful when the sequence of phase differences cannotbe transformed into sequence of a few different angles only,like in weak external fields or thicker free standing films. Alter-native methods discussed in the continuation are more appro-priate in such cases.

3.2. Periodic Boundary Conditions

The method discussed in this subsection was first proposed inref. [27]. It was later adapted for solving different problems.[29]

The method is widely used in solid-state physics in studiesphonons in crystals. Solutions are searched within a predefinedperiodicity having one to several smectic layers. The numberof equations is defined by a number of layers expected toform a commensurate structure in the smectic liquid crystal.Cycling boundary conditions are used. Symmetry reasoningcan further reduce the number of equations. However, one setof equations leads to structures with only one commensurateperiod. In order to verify its stability, several periodicities haveto be considered and the energies of their stable structureshave to be compared. For each periodicity a separate set ofequations has to be derived and solved. As the length of theperiodicity increases, so does the number of equations whichmakes the numerics more and more complicated and less andless reliable. Incommensurate structures cannot be foundusing this approach.

Let us illustrate the method on an example of search fora fictitious structure having a five layer periodicity. One has tosolve the following set of non-linear equations. The freeenergy considering only interlayer interactions has to be re-





written that contains only layers in assumed elementaryperiod, that is, five [Eq. (14)]:

G�;5 ¼12

q2X5

j¼1

a1cos �jþ1 � �j

� �þ a2cos �jþ2 � �j

� �þ

�

a3cos �jþ3 � �j

� �þ 1

2bQ q2cos �jþ1 � �j

� �2þ

f1sin �jþ1 � �j

� �þ f2sin �jþ2 � �j

� ��:

ð14Þ

Here the enumerator for f starts again from 1 if it exceeds 5,the periodicity of the studied structure. To find the stable solu-tion in a constant amplitude approximation the free energyhas to be minimized with respect to all five different tiltorientations [Eq. (15)]:

@G�;5

@�j¼ 1

2a1 q2 sin �jþ1 � �j

� �� sin �j � �j�1

� �� þ

12

a2 q2 sin �jþ2 � �j

� �� sin �j � �j�2

� �� þ

12

a3 q2 sin �jþ3 � �j

� �� sin �j � �j�3

� �� þ

12

bQ q4 cos �jþ1 � �j

� �sin �jþ1 � �j

� �� cos �j � �j�1

� �sin �j � �j�1

� ��

12

f1 q2 cos �j � �j�1

� �� cos �jþ1 � �j

� �� þ

12

f2 q2 cos �j � �j�2

� �� cos �jþ2 � �j

� �� :

ð15Þ

The set of five equations is highly nonlinear and it is relativelyeasy only if one assumes uniplanar structures, that is, directionof all tilts are in one plane in the first approximation. Also devi-ations from the uniplanarity can be considered as long asthese deviations are so small that they allow for expansion oftrigonometric expressions to Taylor series of low order. So, thestructures obtained by these methods[27, 36] are similar to thestructures presented by first Ising-like suggestions for multilay-er structures.[5] One can also consider phase differences only,but this transforms the search for the solution to the methoddiscussed in a previous subsection.

This method has several advantages in slightly differentproblems, in studies of structures in external fields[29, 37] and soforth. In external fields, electric, magnetic or also the influencesof surface treatments, the rotational degeneracy is lost. Eachbasic unit has an orientation that is preferable in an externalfield. The initial structure without the presence of the field isknown, as it was obtained by other methods, for example thestructure and the periodicity of the helically modulated SmC*

a

phase. External fields have surprising effects. The structure firstbecomes commensurate as each repeating part or an approxi-mate period has a favorable orientation in the external field.The commensuration additionally supports the calculation ofthe structure within a finite cyclic period. Further the structuretransforms within commensurate basic units with weak firstorder transitions and at higher fields also the number of smec-tic layers forming a structural unit increases. In general, in anexternal field incommensurate structures become commensu-

rate and the structure evolves upon increasing field throughdevil’s and harmless staircase of commensurate periods to theunwound state.[29]

3.3. Trial Structures

The third method uses a very smart trick to avoid the problemof cycling boundary conditions and its problems with prede-termined periodicities on the one hand and the similar prob-lem with predetermined periodicities in the space of phase dif-ferences on the other. The method makes the number of equa-tions manageable by using a finite set of layers described byorder parameters.[28] The sample for which the structure issearched is considered as a free-standing film thick enoughthat interior of the film is not affected by the missing interac-tions at surface layers. Structure that develops in the interior ofthe film is a good candidate for a structure in a bulk as well.At the beginning the set of tilts describing a certain structureis given. The set is then numerically minimized and the struc-ture that has the lowest free energy is compared to structuresthat result from different initial sets. Structures, for which it isbelieved to present real structures have to be robust with re-spect to the choice of initial sets.

Also this method is appropriate for problems in an alterna-tive way. Instead of making an assumption of the structurethat is numerically driven to its minimum, one considers struc-tures that continuously develop one from another startingfrom a non-tilted SmA phase. One considers in details the sta-bility of phases in the following way.[30, 31] The temporary tiltorder parameter in each layer is written as a sum of an orderparameter that minimizes the free energy and the temporarysmall deviation from this value due to fluctuations [Eq. (16)]:

~xj ¼~xj;0 þ d~x ð16Þ

The free energy of the free standing film is obtained as thesum of the free energies of all layers considering the fact thatsome interactions at surface layers are missing. To consider themissing interactions correctly the tilts are considered as zero inlayers, where the numerator of the layer is smaller than 1 orlarger than N. The numerical consideration is the most simplein usual Cartesian coordinates. The free energy is expanded influctuations to the second order. The contribution of fluctua-tions to the free energy can be written in the matrix form[Eq (17)]:

dG ¼XN

j¼1

¼ 12

c � G2 � c ð17Þ

which, in a system of N layers, is [Eq. (18)]:

c ¼ fdx1;x; :::; dxj;x; :::; dxN;x; dx1;y; :::; dxj;y; :::dxN;yg ð18Þ

and G2 is a matrix that connects fluctuations of various typesin the system. It is obtained as a matrix of second derivativesof the free energy that includes fluctuations with respect to





components of c. Eigenvalues of the matrix G2 are all positiveas long as the SmA phase is stable. When the lowest eigenval-ue becomes zero, the transitions to the tilted phase occur. Thecorresponding eigenvector indicates the structure and thevector of the same symmetry can be used as a first approxima-tion in the search for the structures below the transition tem-perature. The structure in the film is found by an iterative pro-cedure.[30] Stability ranges of various types of structures arefind by the same criterion. The solution is correct if the mini-mal eigenvalue of the G2 is zero (Goldstone mode criterion).The new structure evolves at the temperature, where thesecond eigenvalue becomes zero.

The procedure discussed is nothing special. Why mention itexplicitly? Films are limited and therefore general structurescan be obtained by direct numerical minimization of the freeenergy as suggested by ref. [28]. However, the approachshown above also has well-defined approximate structures,which converge well to the correct structure. By proper choiceof path through the phase diagram, changing either interlayercouplings or the temperature in small steps one can straight-forwardly reach stability limits of different structures and canalso try the convergence and stability of various “forced” ap-proximations, which do not evolve continuously one from theother. Stability limits and energies of such structures, if theyexist, can give suggestions for the bulk structures or allow tostudy phenomena that are limited to restricted geometries,like appearance of uniplanar structures[30] in regions where hel-ically modulated structures are stable only in bulk.

4. Conclusions

This concept paper discusses approaches for finding complexstructures described by two-dimensional order parameterswithin systems with competing interactions having a discreteform and are accompanied by illustrative examples. Three dif-ferent approaches are presented and their appropriateness, ad-vantages and disadvantages are discussed.

The first method, the search for structures in the space ofphase differences has no limitations in numerically accessedangles formed by tilts in neighboring layers and allows for in-commensurate structures in general. After finding the solutionin the phase space it is transformed to polar coordinates, de-scribing structures in tilt magnitudes and directions in eachlayer. The method has two main disadvantages: the studiedstructures have predefined symmetry in the space of phase dif-ferences. Although this space allows for much wider range ofstructures than the search for solutions in the real tilt and itsdirection space, still some solutions could be overseen. Thesecond disadvantage is that the method is not appropriate forstudies of systems in external fields and in free-standing films.

The second method uses periodic boundary conditions inthe search for structures. The number of the layers in the as-sumed structure is therefore fixed and in the search for solu-tions one is bound to commensurate structures having as-sumed periods. For initial hints of the structure, limitation tosmall deviations from uniplanar structure significantly simplifiesnumerical problems, however larger angles were observed ex-

perimentally and they complicate the numerics especially iflonger periods are studied. However, this method is perfectlysuitable when external fields are applied and in fact, severalstudies have been done.[29, 36] External fields define the symme-try of the structures and incommensurate structures alwaystransform to closest commensurate ones. For commensuratestructures the periodic boundary conditions in the study arethe most appropriate. Using this method one could use polarpresentation for the order parameters or they can be ex-pressed in Cartesian coordinates as two dimensional order pa-rameters. Both representations can be used in the search forsolutions depending on the nature of the problem.

The third method considers free-standing films. In thismethod, the structure is again defined by two dimensionalorder parameters given for each layer, however one can alsoexpect large variation of tilts close to the surface layers due tothe missing interactions or due to the enhanced smectic orderin surface layers that are usually not very important in bulksamples. Order parameters are described as two dimensionalvectors in Cartesian coordinates. Thick film studies give goodideas about physics in the bulk, as an interior of the film isa good approximation of a bulk. Phenomena in thin films areeasier to study using this method as also the stability of suchfilms can be studied in details.

These complicated systems were also studied by using stat-istical mechanics approaches.[27, 38] The approach is actuallycomplementary to the phenomenological approach and thetwo approaches support one another. Attempts have beenmade to relate the coefficients used in phenomenological de-scription of the free energy to the intermolecular interactions.As molecules forming liquid crystals are large, consisting ofmore than hundred atoms and corresponding bonds, a strictstatistical approach is difficult and several approximations areusually done. The phenomenological hand waving can givehints for eventual approximations. On the other hand, if a stat-istical approach starting from a molecular level leads to thesame form of a phenomenological coefficient, the phenom-enological reasoning is additionally supported. For a phenom-enological model few details with respect to the model coeffi-cients, which can often be related to macroscopic properties,are important. One should know the magnitudes or at leastthe ratios of various interactions. Here statistical mechanicscan give a valuable insight. Without theoretical considerationstatistical mechanics offer, magnitudes and signs of coefficientsgiving interactions could be guessed from experimental resultsonly. But there are still several problems that remain open, forexample a simple one, namely how the chemical formula re-lates to the magnitude and the sign of the piezoelectric coeffi-cient when a system is formed from molecules. But also a phe-nomenological approach, when it is supported by good-handwaving guesses, is not without explanatory and predictive abil-ity. With phenomenological theories one is left with guesseswith respect to phenomenological coefficients that are sup-ported by experimental evidence. Quoting just one example,although several others exist : the solutions of the puzzle thattwo in the MHPOBC, the SmC* and the SmC*

Fi2 appeared in thesame temperature window if different groups were studying





the system. A phenomenological reasoning was the SmC*Fi2 is

more probable in a material with higher polarization, that iswith a higher enantiomeric excess. In fact, in purer samplesthe SmC*

Fi2 was found in the same temperature window, wherethe SmC* phase was found in a slightly racemized sample.So different groups were working with materials of differentpurity.[33]

Acknowledgements

The author thanks Hideo Takezoe, Ewa Gorecka and Damian Po-ciecha for many stimulating discussions. The financial support ofSlovenian Agency for Research and Development, Research Pro-gram P10055, is acknowledged.

Keywords: antiferroelectric liquid crystals · bent-core liquidcrystals · complex structures · phenomenological models ·polar smectics

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Received: November 22, 2013

Revised: January 16, 2014

Published online on && &&, 2014




http://dx.doi.org/10.1080/15421407708084469

http://dx.doi.org/10.1080/15421407708084469

http://dx.doi.org/10.1080/15421407708084469

http://dx.doi.org/10.1143/JJAP.28.L1265




http://dx.doi.org/10.1039/jm9940400997

http://dx.doi.org/10.1039/jm9940400997

http://dx.doi.org/10.1039/jm9940400997

http://dx.doi.org/10.1039/jm9940400997

http://dx.doi.org/10.1103/PhysRevLett.81.1015



http://dx.doi.org/10.1039/jm9960601231

http://dx.doi.org/10.1039/jm9960601231

http://dx.doi.org/10.1039/jm9960601231

http://dx.doi.org/10.1039/jm9960601231

http://dx.doi.org/10.1126/science.1170027




http://dx.doi.org/10.1080/13583149908047731

http://dx.doi.org/10.1080/13583149908047731

http://dx.doi.org/10.1080/13583149908047731

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



http://dx.doi.org/10.1039/b103506c



http://dx.doi.org/10.1103/PhysRevE.60.6803



http://dx.doi.org/10.1080/15421406.2011.568333

http://dx.doi.org/10.1080/15421406.2011.568333

http://dx.doi.org/10.1080/15421406.2011.568333

http://dx.doi.org/10.1126/science.278.5345.1924




CONCEPTS

M. Cepic*

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Systems with Competing InterlayerInteractions and Modulations in OneDirection: Finding their Structures

One, two, three: Complex structures inpolar smectics with commensurate orincommensurate modulations to layerthickness (see picture, commensurateperiods are marked yellow) can be stud-ied within the framework of a discretephenomenological model. Three meth-ods are presented and their appropri-ateness, advantages and disadvantagesare discussed. Examples are given as anillustration for each method.



Documents

Systems with Competing Interlayer Interactions and Modulations in One Direction: Finding their Structures