Systems with competing interlayer interactions and modulations in one direction: finding their structures.

CHEMPHYSCHEM CONCEPTS DOI: 10.1002/cphc.201301101

Systems with Competing Interlayer Interactions and Modulations in One Direction: Finding their Structures Mojca Cˇepicˇ*[a]

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

search for solutions. The review of three methods reported in the literature are presented and their appropriateness, advantages and disadvantages are discussed. Examples are given as an illustration for each method.

1. Introduction Smectic liquid crystals are formed when the free energy of the system is lowered if elongated molecules that already have orientationally ordered long axes, organize in smectic layers upon lowering the temperature. The free energy is lowered further by new types of additional order. For example, elongated molecules tilt, which decreases the effective distances between parts of neighboring molecules if they are uneven. If elongated molecules are chiral, the tilt results in induced polarization and in variation of the tilt direction from layer to layer. Finally, in antiferroelectric liquid crystals several structures appear which differ in patterns of the tilt direction modulation mainly. The patterns are commensurate, the periodicity extends over 2, 3, 4 or 6 layers, or a pattern is incommensurate with a periodicity extending over few layers only, but also up to several hundreds of layers. As another example one can discuss systems formed of bent shaped or popularly called banana-like molecules, where an order of secondary axes appears and is usually associated with a polarization in the same direction. As in these systems tilt is also allowed and is actually rather common, a whole zoo of structures having various combinations of tilt and polarization order appear. In continuation we limit our discussion to smectic systems that are described by constant nematic and smectic order parameters, and where one additional order, tilt or polar order, is needed to describe the structure of the phase at least. Even more, we focus the analysis to various ways how this third order is modulated. How the systems, studied in this conceptual review paper, were recognized historically? Modulations of the tilt in tilted smectic phases appear in systems formed of chiral liquid elon[a] Prof. Dr. M. Cˇepicˇ Faculty of Education University of Ljubljana, Kardeljeva pl. 16 1000 Ljubljana (Slovenia) and Jozef Stefan Institute Jamova 39, 1000 Ljubljana (Slovenia) E-mail: [email protected]

2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

gated molecules that form tilted smectic phases. Material with such properties was synthesized following Meyer’s suggestion that tilted smectics composed of chiral molecules should have polar layers.[1] The polarization of the layer was expected and was actually observed.[2] The helical modulation of the tilt direction appeared and it was shown that the helical modulation of the tilt is always present in chiral tilted smectic phases except in some cases at a single temperature, where the left handed helix changes into right handed helix. The phenomenon is called helix reversal.[3] Fifteen years later, a new material MHPOBC, which was expected to be a ferroelectric liquid crystal with a large polarization, showed peculiar behaviour. Upon lowering the temperature, in the region where the ferroelectric liquid crystalline phase was expected, the set of four distinct phases was observed.[4] The lowest-temperature phase had antiferroelectric properties and was called an antiferroelectric liquid crystalline phase denoted as SmC*A . One of the phases at higher temperatures was recognized as an ordinary ferroelectric SmC* phase. The phase between the SmCA* and the SmC* phase has ferrielectric properties and was at that time called the SmCg* phase. The phase that appears strictly below the orthogonal smectic SmA phase and above the ferroelectric SmC* phase was called SmCa* . The polar properties of the SmC*a phase vary upon temperature changes and its optical appearance is similar to the orthogonal SmA phase. A few years later another phase with antiferroelectric properties in a different temperature window was recognized.[5] As the two phases, the SmC*g and this novel phase share the same temperature window between the antiferroelectric SmC*A and the ferroelectric SmC* phase, they were both called intermediate phases and a new name code was introduced for both of them—the SmC*g was renamed to SmC*Fi1 and the new antiferroelectric phase was * . Few years ago another phase in the set of anticalled SmCFi2 ferroelectric phases was discovered—the SmC*6d phase that is extremely rare and appears strictly below the SmC*a phase.[6] Table 1 gives a few standard materials that form chiral polar smectics. Phase sequence starting from an orthogonal SmA ChemPhysChem 0000, 00, 1 – 14

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CHEMPHYSCHEM CONCEPTS Table 1. Standard materials exhibiting phases found in antiferroelectric liquid crystals. The phase sequence upon lowering temperature, starting from orthogonal SmA phase, including all existing tilted polar phases, is given for optically pure materials with an exception of MHPOBC, that is slightly racemized and has more phases in a sequence then a pure one. Material

Chemical formula and the corresponding phase sequence

www.chemphyschem.org Although structures of phases found in antiferroelectric liquid crystals are known to researchers studying them explicitly, it is worth to describe them in more detail. Schematic representations of the structures are presented in Figure 1 a. Molecules are organized in smectic layers and they are tilted away from a layer normal. The magnitude of the tilt is described by

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

phase that appears upon lowering the temperature is given below the molecular structure. In these materials all typical structures can be found, providing the enantiomeric excess is sufficiently high. The material, in which the most recent phase the SmC*6d phase was discovered, is a mixture of more components, therefore it is not given in the Table 1. There are several similarities with respect to the shape and the internal structure in the presented materials. Its molecular length (approx. 3 nm) is much longer than its width (ca. 0.5 nm). When a liquid or a liquid crystal is formed, distances between parts of different molecules can be much shorter then distances between molecules measured in distances between centers of masses, giving rise to essentially anisotropic intermolecular interactions. All molecules consist of less flexible core from benzene rings and more flexible alkyl tails. Molecules forming antiferroelectric liquid crystals usually have a longer core than molecules forming the ferroelectric phase, resulting in higher smectic order. Molecules forming ferroelectric phases usually have one CO group. Molecules forming antiferroelectric phases have two CO groups, resulting in a larger molecular dipole. The arrangement of more dipoles often contributes to a significant molecular quadrupole as well. Finally, all molecules have chiral groups attached at the end of the molecular core influencing rotation around long molecular axes. 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

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

an angle q that the average long molecular axis forms with the layer normal, usually corresponding to the z coordinate axis. Another important piece of information for the structure is the tilt direction that is given by an angle f measured as an angle between the tilt projection onto the smectic layer and the chosen direction within a smectic layer corresponding to x coordinate axis. As molecules in the layer may tilt in any average direction, the structure of a single layer is presented as a cone with an apex angle 2 q and the average direction within the layer is shown as an ellipsoid on this cone (Figure 1 a). Such presentations are often stylized further for more complex structures. The cone representing a layer is given as a circle in a bird’s eye view and the arrow gives the tilt direction (Figure 1 a below). For more complex structures with modulations over several layers, the projection of cones (circles) on one smectic plane is given. In order to follow the structure from layer to layer, arrows denoting tilt directions are marked with ChemPhysChem 0000, 00, 1 – 14

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CHEMPHYSCHEM CONCEPTS a number of the corresponding layer in the elementary period of modulation. More detailed description of structures follows the historical order of discoveries of their structures. The ferroelectric SmC* phase is given in Figure 2 c. The system where elongated molecules order themselves into smectic layers and tilt away from

Figure 2. Series of possible structures that appear in antiferroelectric liquid crystals upon decreasing temperature. Cones denote the magnitude of the tilt in the smectic layer; arrows denote the tilt direction. Bird’s eye view of the projection of several layers onto a single smectic plane alows for observation of additional structural details of the complex phases. a) SmC*a phase, * phase, c) SmC* phase, d) SmC*Fi2 phase, e) SmC*Fi1 phase, f) SmC*A b) SmC6d phase. Alternative names of phases are given as well.

the layer normal is formed from chiral molecules. In chiral systems the rotation around the long molecular axis is hindered if molecules are tilted and one orientation of the molecules in the presence of other chiral molecules exists where a molecules spends more time on average. This is the reason that the component of the polarization perpendicular to the tilt and the layer normal do not cancel out. The smectic layer is therefore polar, which means that such liquid crystals are very sensitive to an external electric field. The chirality of the sample defines the triad: the tilt, the polarization and the layer normal. The polarization is proportional to the enantiomeric excess and is opposite for opposite handedness of the same material. No prediction of polarization direction with respect to the tilt and the handedness can be done if different materials are considered. The tilt direction is parallel in neighboring layer but in chiral samples the parallelism is not exact and a small angle is formed between tilts in neighboring layers resulting in a helical modulation extending over few hundreds of layers. The structure of the antiferroelectric phase SmC*A was suggested shortly after its discovery already in 1989.[4] The tilts in neighboring layers have opposite directions (Figure 2 f) and so do polarizations. Therefore the polarization cancels out over two layers. The structure is the origin of low response of the phase to weaker external electric fields, and a sudden structural change to the structure with a higher polarization at 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

www.chemphyschem.org a threshold electric field. Such behavior is typical for antiferroelectrics and it has given the phase with these properties its name—the antiferroelectric SmC*A phase. This phase always appears in these systems as a lower-temperature liquid crystalline phase and has given them a general name—the antiferroelectric liquid crystals. The structure of the phase that appears strictly below the SmA phase, the SmC*a phase, was predicted theoretically in 1995[7] and was confirmed by resonant X-ray scattering in 1998.[8] The tilt direction changes from layer to layer for a fixed general phase difference a (Figure 2 a). The phase is therefore helically modulated in the same way as a regular ferroelectric SmC* phase, only the period of the helical modulation extends from few to few tenths of layers being incommensurate with the number of layers in general, while the typical period of modulation in the SmC* extends over few hundreds of layers. For shorter periods of modulations the structures behave antiferroelectrically, for longer periods its behavior is more similar to the behavior of the ferroelectric phase in an external field. Due to the very short pitch of the modulation the structure of the SmCa* phase is optically similar to the SmA phase. * and the SmC*Fi2 , the resonant X-ray scattering For the SmCFi1 has shown strictly commensurate periodicities. The modulation * phase extends over three smectic layers. The diof the SmCFi1 rection of tilts in neighboring layers differs for either phase difference a or for the phase difference b, which are not equal. The sequence of phase differences is well defined as phase difference a is always followed twice by phase difference b (see Figure 1 e). The commensurate period of the elementary unit exists because the sum a + 2 b 2 p. A slight deviation of the sum from 2 p results in an additional helical modulation on the scale of several hundreds of smectic layers. * phase extends over four smecThe modulation of the SmCFi2 tic layers. Its structure is also defined by a sequence of two different phase differences a and b that interchange. The sum a + b p is shown in Figure 1 d. The slight deviation of the sum of both phase differences from p, again results in a helical modulation on much longer scale. Finally, the elementary modulation of the most recently dis* phase extends over six layers. It appears strictly covered SmC6d below the SmCa* phase. The structure is very similar to the * phase as two different phase difference interchange in SmCFi1 the sequence a,b,b, but the sum of the three angles in the sequences is a + 2 b p (Figure 2 b). In 1995 new systems of polar smectics formed of molecules with a bent core were discovered.[9] Their typical chemical structure is given in Table 2. The bent core of the molecule usually consists of five benzene rings. Two CO groups are attached close to the benzene ring in the middle and the orientation of CO groups is associated with the orientation of the bent core. Both sides of the molecule have flexible alkyl chains attached. The special shape of molecular core hinders rotation around the long molecular axes that additionally stimulates spatial separation of cores from chains and organization of molecules into smectic layers. The zoo of phases in these systems is even richer then in antiferroelctric liquid crystals. Besides smectic phases, quasi-columnar phases exist where ChemPhysChem 0000, 00, 1 – 14

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CHEMPHYSCHEM CONCEPTS

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layers. In the SmAPA structure molecules are not tilted, but polarizations in neighboring layers Material Chemical formula and the corresponding phase sequence have opposite directions (antiferroelectric order). In the SmCAPF structure molecules are tilted in P-8-O-PIMB[9] opposite directions (anticlinically) but the polarization order is ferroelectric. Two combinations I$B2$B3$Cr of tilt and order remain, namely the synclinically tilted antiferroelectrically ordered structure of NOBOW[39] the SmCSPA and the anticlinically tilted antiferroelectrically orI$B2$B4$Cr dered structure of the SmCAPA. All structures are presented in W586[40] Table 3. Interlayer organization is important as it defines macroscopic I$SmAd$SmAdPF$Cr properties. As said, the modulation in bent-shaped systems extends over two layers at most. Structures with antiferroelectric polar order have antiferroeleccolumns are formed from clusters of molecules. Columnar ortric properties. All structures are biaxial, but synclinically tilted ganization is often modulated in different directions,[10] in structures have significantly different optical properties than some systems columns themselves develop from layers[11] or broken layers in the form of twisted ribbons.[12] Phases were anticlinic and non-tilted structures. named according to the time of their naming as Bx, for examThe systems discussed above are generally considered as polar smectics due to existence of polar smectic layers. Our inple B1, B2 and so on. In this concept paper we discuss only tention is to discuss the methods that allow one to find first structures where bent-shaped molecules organize in smectic the structures described above and next to find structures to layers that are polarly ordered within the B2 phase. Within this which those systems develop under influences of external phase the whole set of structures is stable and the structures fields or in restricted geometries. Some of those problems were given descriptive names. have already been discussed in the literature but many reSeveral types of structures within B2 are possible.[13] To demained open. scribe those structures let us first introduce the graphical preThe paper is organized as follows: First we define order pasentation of order in bent-core systems. As seen in Figure 1 b rameters necessary for description of complex structures found (top right), the polarization is associated with the bent orientain discussed systems. We present the free energy for both systion and is marked as an arrow. The tilt is associated with the tems—chiral and achiral polar smectics, and discuss possible orientation of the line that schematically connects ends of the interactions that contribute to the free energy and result in core. As the polarization can have two directions with respect to the tilt, it is necessary to present the polarization and the tilt in each layer. The tilt is marked as a bold line and the polarTable 3. Possible ordering of bent-core molecules in smectic layers with ization is marked with arrows, as described in details in the corresponding model coefficients. caption of the Figure 1 b. We discuss six types of polar structures that are considered a0,P < 0 a0,P < 0 a0,P > 0 as subphases of the B2 phase. The bent molecules organize in a1,x < 0 a1,x > 0 a layered order bent in one direction only. As CO groups are found close to the middle of the molecule, the average oriena1,P > 0 tation of the bent defines also the direction of the polarization. As the polarization can be influenced by an external field, the polarization is used for a description of the bent orientation. SmAPA W1 > 0, SmCSPA W1 < 0, SmCAPA Two structures have the order equal in all layers. In the recently discovered SmAPF structure phase molecules in layers are not tilted and polarizations have the same direction (ferroeleca1,P < 0 tric order) in all layers. In the SmCSPF structure molecules in all layers are tilted in the same direction (synclinically tilted) and also the polarization is the same in all layers. Four remaining SmAPF W1 < 0, SmCSPF W1 > 0, SmCAPF structures have an elementary period extending over two Table 2. Standard materials forming achiral polar systems. Names of the phases are marked as generally accepted. All structures described in this paper correspond to the phase called B2.


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CHEMPHYSCHEM CONCEPTS complex structures. The differences between two systems are stressed. Next we present three different methods for obtaining structures that minimize the free energy in such systems and discuss their advantages and disadvantages and resume the discussion in conclusions. The construction of the free energy and the methods to obtain structures are not limited to smectic systems or even to liquid crystals. The methods can be used in any situation, where interactions can be written in a discrete form and can be separated to interactions that determine the basic energetically demanding structural elements, the tilt and the polarization in our case, and to interactions that determine energetically less demanding modulations. The separation of interactions to two energetic levels is not compulsory but can simplify numerical efforts. Examples can be found in crystalline polar heterostructures,[14] structures formed at molecular level, in well-known situations described by the Frenkel–Kontorova model and its variations[15] and others.

2. Systems with Competing Interlayer Interactions For all systems described above it is characteristic that interactions, which act between molecules in the same smectic layer—the intralayer interactions—are stronger than interactions between molecules in neighboring layers—interlayer interactions. In addition, changes of the tilt magnitude are related to much larger changes in the free energy than changes in the tilt direction in general. Significant interactions between molecules occur only if parts of the molecules (at least) are rather close to each other. Which interactions contribute in complex systems formed from rather large molecules? The repulsive part of van der Waals intermolecular interactions is known as steric interactions, which consider excluded volume effects. On the other hand, attractive van der Waals interactions tend to decrease intermolecular distances. Both parts of the van der Waals interactions are of short range, with a significant impact to distances much smaller than longer dimensions of elongated molecules forming liquid crystalline phases. In polar systems electrostatic interactions are important as well, but as smectic layers have liquid-like positional order of molecules, the impact of these interactions is present as long as positional intermolecular correlations exist.[16] For liquid-like systems this means a typical dimension of a single molecule. Within the layer this dimension is defined by a width of the molecule, through the layers by a length of the molecule. To the free energy of the smectic liquid crystals intralayer interactions contribute the most important part. So, the elementary structure of the layer is determined by stronger intralayer interactions. As intralayer interactions in the bulk do not have a preferred direction, the changes in the tilt direction do not require energy. Interlayer interactions are important to nearest neighboring 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 interacting layers consequently. 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

www.chemphyschem.org Let us shortly present the most elaborate free energies for the two systems described in introduction for the chiral polar smectics formed of elongated chiral molecules and for the achiral polar smectics where polar ordering is induced by ordering of the bent-shaped molecules in the layer. Free energies presented in continuation allow for all experimentally confirmed phase structures and phase sequences.[17, 18] For description of the layer order we assume that the nematic and the smectic orders are constant and we further limit our studies to two vectorial order parameters—the tilt and the polarization, which are defined in the same way for elongated as well as for bent-shaped molecules (Figure 1). The tilt order parameter for molecules in the jth smectic layer ~ xj recapitulates the quadrupolar nature of the up-down molecular packing symmetry [Eq. (1)]: ~ xj ¼ fnj;x nj;z ; nj;y nj;z g

ð1Þ

where ~ nj ¼ fnj;x ; nj;y ; nj;z g is a director in the jth smectic layer. The director is related to the direction of average long molecular axis in chiral polar smectics (Figure 1 a) or to the direction of the longest dimension of the bent-shaped molecule (Figure 1 b). The polarization in the chiral smectic liquid crystals is an improper order parameter and it is induced by the tilt. When a chiral molecule is tilted and encircled by other tilted chiral molecules, rotation around long molecular axis is hindered and one favorable orientation exists due to the chiral symmetry of the molecule. The molecule spends more time in this position, the molecular dipoles do not cancel out on average and the layer is polar. An isolated tilted layer is polarized perpendicularly to the tilt due to symmetry reasons,[1] but in more complex tilted structures the polarization can have a general direction.[19] In fact, the hindrance of the rotation is a steric effect and the polarization is the consequence only. Its direction is parallel to the average molecular geometric polar axis which is not necessary associated to the direction of the molecular dipole. However, the average polarization has the same direction as an average geometric axis, it is a property that is measured easily and we suggest that although by origin geometrical, the order parameter is called polarization (Figure 1 a), given by Equation (2): ~ Pj ¼ fPj;x ; Pj;y g

ð2Þ

The reasoning described above is even more evident in systems of bent-shaped molecules. Imagine rotation of the bentshaped molecule encircled by other bent-shaped molecules. As a molecule rotates around the long molecular axis, two orientations are more favorable than others. Out of the two orientations, given as a grey and a white molecule in Figure 1 b, one of them is the most favorable, because then molecules can pack the most tightly. Therefore, ordering of bents is preferred sterically. Molecular dipoles due to the CO groups close to the central benzene ring is parallel or antiparallel to the ChemPhysChem 0000, 00, 1 – 14

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CHEMPHYSCHEM CONCEPTS second molecular axis defined by the bent order, and the layer with ordered bents is polar even without a tilt. As the polarization of systems formed of ordered bent shaped molecules may be polar even without a tilt the polarization is a proper order parameter. The free energy of the system consisting of chiral elongated molecules expressed in order parameters up to the fourth order is [Eq. (3)]:[17, 20] 1 1 1 2 Pj þ Pþ xj ~ a0 x2j þ b0 x4j þ cp ~ z 2 4 2 e0 j 1 2 2 ~1 ~ xjþ1 þ ~ xj ~ xj1 þ bQ ~ xjþ1 þ ~ xj ~ xj1 xj ~ xj ~ þ a 8 ð3Þ ~f1 ~ xj þ ~ xj ~ xj1 þ xjþ1 ~ z 1 ~ ~ ~ ~ ~ Pj ~ Pj1 Pj Pjþ1 þ ~ m Pj xjþ1 xj1 þ 4 e1

Gj ¼ 1 4 1 4 1 2

Here the first line of Equation (3) gives contribution of intralayer interactions expressed in tilt and polarization order parameters to the free energy of the jth smectic layer. The only temperature-dependent parameter is a0, which changes sign at the transition temperature to the tilted phase in an isolated layer without any polarization. Additional interactions such as the piezoelectric term (cp), that couples the tilt and the polarization, increase the transition temperature. If the tilt appears, as the tilted phase becomes stable for example, the polarization proportional to the tilt appears as well. On the other hand, if on a non-tilted chiral smectic an electric field is applied, the layers become polarized, and also the molecules tilt consequently. Finally, if the layer is polar, the electrostatic energy of the ordered dipoles has to be taken into account. The form of interlayer interactions to the free energy allows also for large angles between tilt directions in neighboring layers. Three types of interactions contribute to the interlayer interactions expressed in tilt, namely steric or van der Waals repulsive interactions, van der Waals attractive interactions and electrostatic interactions between dipoles or quadrupoles. Layers interact with neighboring layers only and interactions to neighboring layers are marked by a subscript 1. Coefficients a~1 , bQ and ~f1 give steric and attractive van der Waals interactions. Negative a~1 favors parallel tilts in neighboring layers, if steric interactions due to a migration of molecules between layers prevail. At lower temperatures, attractive van der Waals interactions prevail over entropic tendency for molecular migration and antiparallel tilts in neighboring layers become preferential. Similar reasoning accounts for the quadrupolar coupling if molecules can be considered as a flattened lath-like object. The quadrupolar ordering of short molecular axes results in either an enhanced molecular migration through layers stimulating parallel tilts or in an enhanced attraction of molecules from neighboring layers stimulating antiparallel tilts. The ~f1 coefficient is chiral, and has the opposite sign in samples formed by molecules of opposite handedness and is zero in racemic mixtures. The van der Waals field of molecules with a chiral symmetry, to which the whole molecular structure contribute with all the atoms and the polarizabilities of chemical 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

www.chemphyschem.org bonds between them, is also chiral. The favorable orientation of another chiral molecule with the same handedness positioned in such a field is different from an equal molecule having the opposite handedness. Usually a non-parallelism in tilts in neighboring layers is favored as a result. As chiral interactions are usually much weaker than non-chiral ones, the resulting modulation extends over several hundreds of layers or lifts a degeneracy between left- and right-handed structures allowed without chiral interactions. The last two terms consider interactions that are present due to the polar nature of molecules. The simplest explanation for the flexoelectrically induced polarization, contribution is given by the flexoelectric m coefficient, is the following. When molecules are tilted, the rotation around the molecular long axes is hindered. The magnitude of induced polarization depends on orientation, which is preferable for a tilted molecule surrounded by other tilted molecules. But this favorable orientation is influenced also by interactions with the molecules in the layers above and below the considered layer. If the tilt in neighboring layers differ, the hindrance of the rotation is different and therefore affects the magnitude of the polarization. In some cases the difference in interactions influences also the favorable molecular orientation along the long axis and consequently also the direction of the induced polarization.[19] Another system, very similar to that described in Equation (3) is a system formed of (very often) achiral, polar molecules with a bent core. Both systems differ in few ways, and although differences appear in interactions they do not influence the search for structures. Let us consider main differences. Phenomena that are present in bent core systems do not appear due to the chiral molecular shape but due to the bent molecular shape, Table 2. Next, tilt and polarization order parameter are independent and phases with polarization and without a tilt exist. Due to molecular bent shape, the favorable tilt is perpendicular to the polarization, however, tilt is degenerated as molecules can tilt in two directions that are both perpendicular to the tilt. When layers are polar and tilted, they are chiral. For these systems a chiral order parameter can be introduced as Equation (4): s j ¼ xj P j z

ð4Þ

Chiral layers interact among themselves in various ways, they can prefer equal chirality in neighboring layers, resulting in homochiral samples as well as the opposite chirality in neighboring layers, resulting in antichiral samples.[21] Systems of bentshaped molecules can be doped by chiral dopants or bent shape molecules can be synthesized in chiral forms as well. Chiral effects in such systems are not yet fully understood. It is known that doping expands homochiral domains of one sign on 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 molecules is given by Equation (5): ChemPhysChem 0000, 00, 1 – 14

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CHEMPHYSCHEM CONCEPTS 1 1 1 1 1 a P 2 þ b P 4 þ a x2 þ b x4 þ W s 2 þ 2 P;0 j 4 P;0 j 2 x;0 j 4 x;0 j 2 0 j 1 1 ð5Þ Pj ~ Pjþ1 þ ~ Pj ~ Pj1 þ ax;1 ~ xjþ1 þ ~ xj ~ xj1 þ aP;1 ~ xj ~ 4 4 1 W s s þ sj s j1 4 1 j jþ1 Gj ¼

Let us describe in detail differences between Equation (3) and Equation (5). The form of the first two terms in Equation (5) is equal to the form of the second two terms in the same equation and the same as the first two terms in Equation (3), but they are given in polarization order parameters. The form of the first four terms reflects the fact that both the tilt and the polarization are proper order parameters. Also here the first coefficient aP,0 = aP(TT0) is only temperature dependent and changes sign when the structure becomes polar. The coefficient bP,0 is positive for continuous transitions to the polar phase. In most of the cases the transition to the first phase below the ordinary SmA phase is discontinuous, therefore the coefficient bP,0 is negative and the expansionof the free energy 1 needs the sixth-order term in polarization 6 cP;0 P6j . As the nature of the phase transition is not a matter of discussion in this concept paper, we will not go into further details in this respect. The next difference appears in the chiral order parameter and in the contribution of the interactions related to it. The chiral order parameter can be recognized also in a piezoelectric term in Equation (3), and it is present in all tilted structures. In bent-core systems it is present only when both order parameters exist and have different directions in the layer, which is usually the case. The coefficient W0 is negative, given that it is sterically more favorable that molecules tilt in the direction perpendicular to the polarization. As there is no preferred direction with respect to the polarization, the system is degenerate with respect to the chiral order parameter. It also seems that in most of the systems the chiral parameters are coupled across the layers with W1 positive favoring antichiral structures, that is, chiral order parameters have opposite signs in neighboring layers.[21] Coefficient aP,1 can be either positive, favoring antiparallel orientations of polarizations in neighboring layers due to prevailing dipolar electrostatic interactions and diffusion of molecular branches through layers, or negative favored by attractive van der Waals forces between branches of bent-core molecules. Also coefficient ax,1 can be either negative due to the diffusion of molecular branches through layers or positive due to the attractive van der Waals forces between branches of molecules in neighboring layers. As interactions are a result of competition between various contributions to the effective interaction expressed in a model coefficient, the coefficient can change the sign. Interactions are usually rather well determined and depend mostly on molecular structure.[24] However, they can change sign upon changing concentration of components, preferring synclinic or anticlinic and antiferroelectric or ferroelectric structures in mixtures.[25]


www.chemphyschem.org 3. Stable Structures Systems where interactions to neighbors exist are known for a long time. Heterostructures in ferroelectrics were analyzed in details, revealing several interesting phenomena like devil’s and harmless staircase in structure evolution, that were found in one-dimensional systems where the polarization can have one direction only.[15] As some experimental observations revealed steps in temperature or electric-field-dependent properties and the similarity of smectic layers to polar ferroelectric layers in heterostructures of ferroelectric crystals was recognized, the Ising-like structures similar to structures in layered ferroelectrics were suggested for various phases. The tilt in the smectic layer was believed to be bound to one plane, although there were no persuasive arguments given why that should be so. The tilt was described as an Ising-like variable, and various structures as sequences of “positive” and “negative” tilts.[5] Few years were needed before the XY character of the tilt order parameter was recognized and that variations in tilt direction out of plane were considered as more probable than changes in the tilt magnitude only.[7, 8] Studies of polar smectic systems have revealed that structures are modulated on a short length scale of a few layers and in chiral polar systems of elongated molecules on a longer scale extending over several hundreds of layers in addition to the short-range modulation. To find the structures one has to minimize the free energy with respect to all two-dimensional polarizations and with respect to all two-dimensional tilts. Due to interactions with neighboring layers, equations are coupled. As interactions are expressed using trigonometric functions, equations are also highly non-linear. Free energy has usually several rather close metastable minima that further complicate the search for stable structures. Another complication is several model coefficients. Although the measurements clearly indicate the periodicity of structures and also further details of structures, the model coefficient space is not straightforward due to several contributing and competing interactions. For example, the flexoelectric interaction favors the largest possible difference in direction of tilt above and below the interacting layer. Therefore one expects antiparallel tilts for next-nearest-neighbor layers. This interaction competes with other nearest-layer interactions which favor parallel or antiparallel tilts in neighboring layers and ferroelectric or antiferroelectric ordering of polarizations in neighboring layers. The guess for coefficients which lead to the stable structures with shorter periods is consideration of structures favored by an isolated interaction.[26] For example, if the flexoelectric coefficient is large and other coefficients are small, one should search for a solution that has oppositely tilted molecules in next-nearest-neighbor layers. The period of such a modulation, that is, the distance that includes the whole sequence of tilts as a primitive cell of a repeating structure, is four layers. Therefore in a search for structures one is not completely bound to a trial and error. But let us return to the basic question—how to find stable structures? Three different methods are used by researchers. Here one of the methods is discussed in more details as it is less demanding for the calculation of structures but more deChemPhysChem 0000, 00, 1 – 14

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manding in the presentation of the results.[26] The other two approaches[27, 28] are described as well. Although the method given in ref. [27] was used for finding stable structures it is much more appropriate for studies of structures in external fields.[29] Similarly, the method proposed by ref. [28] was used for studying structures in free-standing films.[30, 31] Reflecting various circumstance in which the methods were used, the advantages and disadvantages of the three methods will be discussed and situations in which one of approaches is better than other will be considered. In all three methods, the final structure is described as a set of order parameters characteristic for each layer.

1 e0 ~1 þ e0 c2p þ e0 m2 a1 ¼ a 4 e1 1 a2 ¼ e0 m2 2 1 e a3 ¼ e0 m2 0 e1 8

ð7Þ

f1 ¼ ~f1 2 e0 cp m e f2 ¼ e0 cp m 0 e1 Writing the order parameter ~ xj as Equation (8): ~ xj ¼ qj fcosj ; sinj g

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

3.1. Relative Orientations of Tilts The first method considers the relative orientations of order parameters, that is, the angles between directions of tilts in neighboring layers called phase differences, and operates in a space of phase differences.[7, 18, 20] Solutions, which correspond to commensurate stable structures have in general shorter periodicities than their equivalents in the real space. For example, the four-layer structure of the SmC*Fi2 phase has periodicity of two phase differences instead of four order parameters, which significantly reduces the number of equations. Let us illustrate the search for a solution for one example from structures found in antiferroelectric liquid crystals (Figure 2). In Equation (3) polarizations appear in bilinear terms only. Therefore polarizations can easily be eliminated and the free energy is expressed in tilt order parameters only. The free energy of the single layer, interacting with its surrounding limited to interlayer interactions only, has after elimination of the polarization a very simple form [Eq. (6)][20]: 2 X 1 ~ ~ Gint;j ¼ xjþk þ ak ~ f x xjþk xj ~ z 2 2 k j k¼1 k¼1 2 1 xjþ1 : þ bQ ~ xj ~ 4 3 X 1

! ð6Þ

In Equation (6) interlayer interactions with neighboring layers are not written as the average of interaction with layers above and below the interacting layers because every term appears twice in the summation over the whole sample and allows for a shorter version of the equation. The flexoelectric interactions given by m couple indirectly more distant than neighboring layers although direct interactions extend to nearest neighbors only. The strength of interactions decreases with more distant k2 e layers. Achiral interactions ak decrease as e01 and chiral in k1 e0 teractions fk decrease as e1 . Therefore we consider achiral interactions to next nearest layers and chiral interactions to next nearest layers only [Eq. (7)]: 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

ð8Þ

aj ¼ jþ1 j

ð9Þ

one describes the structure by a set of tilt magnitudes qj characteristic for layers and the set of phase differences aj for which the tilt direction changes from the layer j to the layer j + 1. The search for structures starts with the sequence of phase difference aj that is finite. The length of the sequence is guessed by the symmetry of the periodical structure in the real world. Let us illustrate this approach for the search of a threelayer structure, which was suggested by the resonant X-ray measurements.[32] The suggested structure is seen in Figure 2 e. The theory has to answer the following questions: Does such a structure present a minimum of the free energy? Which interactions define the angle a in Figure 2 d and how large can it be? Which interactions define the additional helical modulation on a longer scale given by d and which periodicities are typical? The structure is described by a sequence of phase differences in the form a,b,b. As one layer has different phase differences above and below the other two, one should also expect a slightly different tilt in one layer out of three. The tilt difference is small as the changes of the tilt are associated with the changes of the layer thickness, which is energetically difficult to compensate by changes in the tilt direction as long as the structure studied is not very close to the continuous transition to the tilted phase. So, the search for the structure is done in three steps—first the constant tilt approximation is used and the Ansatz [Eq. (10)]: ~ x3j ¼ q fcosðjða þ 2 bÞÞ; sinðjða þ 2 bÞÞg ~ x3jþ1 ¼ q fcosðjða þ 2 bÞ þ bÞ; sinðjða þ 2 bÞ þ bÞg

ð10Þ

~ x3jþ2 ¼ q fcosðjða þ 2 bÞ þ 2 bÞ; sinðjða þ 2 bÞ þ 2 bÞg is inserted into the free energy. The resulting average free energy per layer is characteristic for an arbitrary set of three consecutive layers [Eq. (11)]: ChemPhysChem 0000, 00, 1 – 14

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CHEMPHYSCHEM CONCEPTS G3 ¼


1 1 1 a q2 þ b0 q4 þ a1 q2 ðcosa þ 2 cosbÞþ 2 0 4 6

1 a q2 ð2 cosða þ bÞ þ cos2bÞþ 6 2 1 a q2 cosð2 a þ bÞþ 2 3 1 1 b q4 ðcos2 a þ 2 cos2 bÞ þ f1 q2 ðsina þ 2 sinbÞþ 12 Q 6 1 2 f q ð2 sinða þ bÞ þ sin2 bÞ 6 2

ð11Þ

Instead of six independent equations that would be necessary in the space of tilt magnitudes and directions, one is left with three rather simple minimization equations only. Here one should be aware, that this sequence of phase differences leads to a commensurate structure only, if the sum of the three phase 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 sum is equal to p, the structural unit consists of six layers and the * ; if solution, if it is stable, presents the structure of the SmC6d the sum is equal to 2 p/3, the unit would extend over nine layers and so on. It is not necessary that considered structures have any of such limitations. In those cases structures would not be commensurate with the layer thickness although the solution could be described by two different angles and two different tilt magnitudes only. The final solution for structures that are commensurate in the short range and additionally modulated with a long-range periodicity is found in the following way. First: the constant amplitude in the layers and the commensurate periodicity is used as an Ansatz. The sum of the angles in the basic period is therefore set to a fraction of 2 p as described before. This reduces the free energy to dependence on the magnitude of the tilt and a single phase difference. If the sequence of phase differences is longer, the symmetry reduces the number of different phase differences in the sequence and the commensurability requirement reduces the number of different angles further. All structures that were confirmed by resonant X-ray, have a single unknown phase difference at this step. Next, the minimization is done with respect to two different phase differences, using the commensurate solution as a first approximation, for three phase differences, for example a,2 p-a/2,2 p-a/2. For weak chiral interactions one is usually close to the solution even in this approximation, but the next minimization is done for two different angles a and b where the commensurate angles are used as a first approximation. The difference d = 2 pa2 b gives the helical modulation on a longer scale. For large chiral coefficients additional minimization usually converges to a different solution, presenting a helically modulated phase with a single phase difference and only such simple structures of the SmCa* phase are stable. Finally, the difference in tilts is calculated using the equal tilts as a first approximation. The procedure is repeated a few times until the desired accuracy is obtained. The free energy of the solution has to be compared to free energies of structures with different sequences, for example, different number of angles in the sequence. 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

This method is the most appropriate for the search of structures in bulk. It has several advantages. The most important is probably that the method allows for structures having any phase differences. There is no limitation for deviations of tilt directions out of one plane. One should also not neglect a severe reduction of equations that have to be solved. Another advantage is that the system of equations includes rotational invariance for solutions and allows for incommensurable solutions of various periodicities and short-range structures obtained by the same method. Finally, when one becomes familiar with effects of various model coefficients, the search for more specific stable structures or ranges of coefficients, in which they might be stable, becomes rather straightforward. At this point it is worth to discuss another interesting question that is initiated by experimental results. Up to now, intermediate phases as well as the SmC*a phase were found in chiral samples only. Is the chirality really necessary for the stability of those phases? Let us suggest a hypothesis from the theoretical point of view. The method used above allowed for the analysis of a general phase diagram of antiferroelectric liquid crystals.[33–35] It allows for all observed structures having only one temperature dependent coefficient a0 = a(TT0) and a single interlayer interaction a~1 influenced by the magnitude of the tilt. All the complexity of modulations having periods of several layers arises due to the competition of various interactions that extend to neighboring layers only. Consequent interactions have longer range of achiral and chiral characters. The free energy Equation (3) has two chiral coefficients only: the piezoelectric coefficient cp that defines the magnitude of the polarization in the layer and the interlayer chiral interactions given by f1. The latter is usually considered as small. Equation (8) shows that interactions to neighboring layers a1 are influenced by chirality, but they exist also in achiral samples. So, from this point one can safely conclude that the synclinic SmC phase and the anticlinic SmCA phase exist in racemic mixtures that were indeed observed. The SmC*a phase and the * exist in regions where an effective coefficient a2 is SmCFi2 larger than a1 that is close to zero. If quadrupolar interactions given by negative bQ are already strong, they grow with increasing tilt, the SmC*Fi2 phase is stable, if not, the SmC*a phase is stable instead. So, both phases can in principle appear in racemic mixtures as well, where polarization is not present. The precondition is a large flexoelectric effect. Its influences are represented by m in the free energy. However, interactions with third-nearest neighboring layers a3 depend on combination of the flexoelectric and the piezoelectric effect. In some circumstances these interactions in combinations with quadrupolar interactions with negative bq can stabilize the three-layer * phase or the six-layer SmC*6d phase. Therefore these two SmCFi1 phases should not be found in racemic mixtures according to the theoretical understanding. It seems that existence of polarization, that is, a significant enantiomeric excess shifts the cancellation of entropic forces favoring synclinic tilts and electrostatic dipolar interactions favoring anticlinic tilts in neighboring layers to such temperature ranges where the interplay of other interactions results in complex structures. In the absence of polarization the competition is not so pronounced anymore ChemPhysChem 0000, 00, 1 – 14

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and only simpler phases are stabilized. A hint for finding more complex structures in achiral samples is hidden in the flexoelectric effect. Combination of molecules having large flexoelectric interactions, lath-like shape to foster the in-plane tilts and allowed entropic movement of molecules between layers, might give a hope to find more complex structures in racemic mixtures. The straightforwardness in the search for stable structures is easily demonstrated in the case of bent-core systems. Bentcore molecules do not rotate around their long axes as rapidly as elongated molecules, which is indicated by the magnitude of the polarization. One also cannot expect a significant flexoelectric effect related to different orientation of tilt or polarization in neighboring layers as in chiral system of elongated molecules. Without an important flexoelectric contribution, the effective interactions to more distant layers do not exist. If so, the free energy given in Equation (5), that includes direct interactions to neighboring layers only, leads to the solution of only one periodicity, that is, the structures can be described by a single phase difference in the tilt ax and a single phase difference in polarization aP, if it is expressed in the space of phase differences. As the polarization is a proper order parameter, its magnitude and orientation could be considered separately [Eq. (12)]: ~ Pj ¼ Pj fcosyj ; sinyj g:

ð12Þ

The magnitude of the tilt and the polarization are constant as all layers are found in equal position. The free energy is simplified to Equation (13): 1 1 1 1 a P2 þ bP;0 P4 þ ax;0 x2 þ bx;0 x4 þ 2 P;0 4 2 4 1 2 W P2 x2 sin2 yj j þ 2 0 1 ð13Þ a P2 cos yjþ1 yj þ cos yj yj1 þ 4 P;1 1 ax;1 x2 cos jþ1 j þ cos j j1 þ 4 1 W P2 x2 sin yj j sin yjþ1 jþ1 þ sin yj1 j1 4 1

in the layer can tilt in both directions with respect to the polarization and actually one meets both types of domains in samples. But relative orientation of tilts is well defined. For negative 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, the positive chirality coupling W1 favors layers with opposite signs of cross products of the tilt and polarization, while the negative sign favors the same sign of chirality in neighboring layers. From this simple consideration, one recognizes that the longest periodicity of the structure could be two layers only. Although there are more order parameters, as long as the system is bound to simple smectic layers, structures are less complex than in systems formed of chiral elongated molecules (Table 3). More complex phases appear when layers become modulated. However, our discussion of structure ends here. The method that considers relative phase differences has also disadvantages. Combinations of equations containing several trigonometric functions are usually not solved analytically. But also numerical methods are not always simple. The free energy dependence on model coefficients allows for several metastable minima, which complicates numerical search for solutions. Considerations of relative orientations in neighboring layers is also not the most appropriate when systems are found in external fields or in films. Although it is possible to analyze evolutions of structures with periodicities of few layers when the long range modulation is already unwound or in free standing films with few layers only, this approach is not very powerful when the sequence of phase differences cannot be transformed into sequence of a few different angles only, like in weak external fields or thicker free standing films. Alternative methods discussed in the continuation are more appropriate in such cases.

G¼

A closer look to the free energy shows that there are three phase differences, the already mentioned phase difference between polarizations in neighboring layers, aP,j = yj + 1yj, the phase difference between tilts in neighboring layers, ax,j = fj + 1fj, and the new phase difference between the tilt and the polarization in the same layer, Dj = yjfj. As all three angles 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 neighboring layers or aP,j = 0. The positive sign of the same coefficient stabilizes antiferroelectric ordering in polarization, which is the antiparallel ordering of polarization in neighboring layers or aP,j = p. The negative sign of W0 allows for tilts to be perpendicular to the polarization only. The tilt directions however is p not defined and is doubly degenerated or Dj ¼ 2. Molecules 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

3.2. Periodic Boundary Conditions The method discussed in this subsection was first proposed in ref. [27]. It was later adapted for solving different problems.[29] The method is widely used in solid-state physics in studies phonons in crystals. Solutions are searched within a predefined periodicity having one to several smectic layers. The number of equations is defined by a number of layers expected to form a commensurate structure in the smectic liquid crystal. Cycling boundary conditions are used. Symmetry reasoning can further reduce the number of equations. However, one set of equations leads to structures with only one commensurate period. In order to verify its stability, several periodicities have to be considered and the energies of their stable structures have to be compared. For each periodicity a separate set of equations has to be derived and solved. As the length of the periodicity increases, so does the number of equations which makes the numerics more and more complicated and less and less reliable. Incommensurate structures cannot be found using this approach. Let us illustrate the method on an example of search for a fictitious structure having a five layer periodicity. One has to solve the following set of non-linear equations. The free energy considering only interlayer interactions has to be reChemPhysChem 0000, 00, 1 – 14

&10&




written that contains only layers in assumed elementary period, that is, five [Eq. (14)]: G;5 ¼

5 1 2X a1 cos jþ1 j þ a2 cos jþ2 j þ q 2 j¼1

1 2 a3 cos jþ3 j þ bQ q2 cos jþ1 j þ 2 f1 sin jþ1 j þ f2 sin jþ2 j :

ð14Þ

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

3.3. Trial Structures

The third method uses a very smart trick to avoid the problem of cycling boundary conditions and its problems with predetermined periodicities on the one hand and the similar problem with predetermined periodicities in the space of phase difHere the enumerator for f starts again from 1 if it exceeds 5, ferences on the other. The method makes the number of equathe periodicity of the studied structure. To find the stable solutions manageable by using a finite set of layers described by tion in a constant amplitude approximation the free energy order parameters.[28] The sample for which the structure is has to be minimized with respect to all five different tilt searched is considered as a free-standing film thick enough orientations [Eq. (15)]: that interior of the film is not affected by the missing interactions at surface layers. Structure that develops in the interior of @G;5 1 2 the film is a good candidate for a structure in a bulk as well. ¼ a1 q sin jþ1 j sin j j1 þ @j 2 At the beginning the set of tilts describing a certain structure 1 is given. The set is then numerically minimized and the struc2 a q sin jþ2 j sin j j2 þ 2 2 ture that has the lowest free energy is compared to structures 1 2 that result from different initial sets. Structures, for which it is a q sin jþ3 j sin j j3 þ 2 3 believed to present real structures have to be robust with reð15Þ 1 4 b q cos jþ1 j sin jþ1 j cos j j1 sin j j1 spect to the choice of initial sets. 2 Q Also this method is appropriate for problems in an alterna 1 tive way. Instead of making an assumption of the structure f1 q2 cos j j1 cos jþ1 j þ 2 that is numerically driven to its minimum, one considers struc 1 tures that continuously develop one from another starting f2 q2 cos j j2 cos jþ2 j : 2 from a non-tilted SmA phase. One considers in details the stability of phases in the following way.[30, 31] The temporary tilt The set of five equations is highly nonlinear and it is relatively order parameter in each layer is written as a sum of an order easy only if one assumes uniplanar structures, that is, direction parameter that minimizes the free energy and the temporary of all tilts are in one plane in the first approximation. Also devismall deviation from this value due to fluctuations [Eq. (16)]: ations from the uniplanarity can be considered as long as these deviations are so small that they allow for expansion of ~ ð16Þ xj;0 þ d~ x xj ¼ ~ trigonometric expressions to Taylor series of low order. So, the structures obtained by these methods[27, 36] are similar to the The free energy of the free standing film is obtained as the structures presented by first Ising-like suggestions for multilaysum of the free energies of all layers considering the fact that [5] er structures. One can also consider phase differences only, some interactions at surface layers are missing. To consider the but this transforms the search for the solution to the method missing interactions correctly the tilts are considered as zero in discussed in a previous subsection. layers, where the numerator of the layer is smaller than 1 or This method has several advantages in slightly different larger than N. The numerical consideration is the most simple problems, in studies of structures in external fields[29, 37] and so in usual Cartesian coordinates. The free energy is expanded in forth. In external fields, electric, magnetic or also the influences fluctuations to the second order. The contribution of fluctuaof surface treatments, the rotational degeneracy is lost. Each tions to the free energy can be written in the matrix form basic unit has an orientation that is preferable in an external [Eq (17)]: field. The initial structure without the presence of the field is known, as it was obtained by other methods, for example the N X 1 dG ¼ ¼ c G2 c structure and the periodicity of the helically modulated SmCa* ð17Þ 2 j¼1 phase. External fields have surprising effects. The structure first becomes commensurate as each repeating part or an approxiwhich, in a system of N layers, is [Eq. (18)]: mate period has a favorable orientation in the external field. The commensuration additionally supports the calculation of c ¼ fdx1;x ; :::; dxj;x ; :::; dxN;x ; dx1;y ; :::; dxj;y ; :::dxN;y g ð18Þ the structure within a finite cyclic period. Further the structure transforms within commensurate basic units with weak first order transitions and at higher fields also the number of smecand G2 is a matrix that connects fluctuations of various types tic layers forming a structural unit increases. In general, in an in the system. It is obtained as a matrix of second derivatives external field incommensurate structures become commensuof the free energy that includes fluctuations with respect to 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim


&11&


CHEMPHYSCHEM CONCEPTS components of c. Eigenvalues of the matrix G2 are all positive as long as the SmA phase is stable. When the lowest eigenvalue becomes zero, the transitions to the tilted phase occur. The corresponding eigenvector indicates the structure and the vector of the same symmetry can be used as a first approximation in the search for the structures below the transition temperature. The structure in the film is found by an iterative procedure.[30] Stability ranges of various types of structures are find by the same criterion. The solution is correct if the minimal eigenvalue of the G2 is zero (Goldstone mode criterion). The new structure evolves at the temperature, where the second eigenvalue becomes zero. The procedure discussed is nothing special. Why mention it explicitly? Films are limited and therefore general structures can be obtained by direct numerical minimization of the free energy as suggested by ref. [28]. However, the approach shown above also has well-defined approximate structures, which converge well to the correct structure. By proper choice of path through the phase diagram, changing either interlayer couplings or the temperature in small steps one can straightforwardly reach stability limits of different structures and can also try the convergence and stability of various “forced” approximations, which do not evolve continuously one from the other. Stability limits and energies of such structures, if they exist, can give suggestions for the bulk structures or allow to study phenomena that are limited to restricted geometries, like appearance of uniplanar structures[30] in regions where helically modulated structures are stable only in bulk.

4. Conclusions This concept paper discusses approaches for finding complex structures described by two-dimensional order parameters within systems with competing interactions having a discrete form and are accompanied by illustrative examples. Three different approaches are presented and their appropriateness, advantages and disadvantages are discussed. The first method, the search for structures in the space of phase differences has no limitations in numerically accessed angles formed by tilts in neighboring layers and allows for incommensurate structures in general. After finding the solution in the phase space it is transformed to polar coordinates, describing structures in tilt magnitudes and directions in each layer. The method has two main disadvantages: the studied structures have predefined symmetry in the space of phase differences. Although this space allows for much wider range of structures than the search for solutions in the real tilt and its direction space, still some solutions could be overseen. The second disadvantage is that the method is not appropriate for studies of systems in external fields and in free-standing films. The second method uses periodic boundary conditions in the search for structures. The number of the layers in the assumed structure is therefore fixed and in the search for solutions one is bound to commensurate structures having assumed periods. For initial hints of the structure, limitation to small deviations from uniplanar structure significantly simplifies numerical problems, however larger angles were observed ex 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

www.chemphyschem.org perimentally and they complicate the numerics especially if longer periods are studied. However, this method is perfectly suitable when external fields are applied and in fact, several studies have been done.[29, 36] External fields define the symmetry of the structures and incommensurate structures always transform to closest commensurate ones. For commensurate structures the periodic boundary conditions in the study are the most appropriate. Using this method one could use polar presentation for the order parameters or they can be expressed in Cartesian coordinates as two dimensional order parameters. Both representations can be used in the search for solutions depending on the nature of the problem. The third method considers free-standing films. In this method, the structure is again defined by two dimensional order parameters given for each layer, however one can also expect large variation of tilts close to the surface layers due to the missing interactions or due to the enhanced smectic order in surface layers that are usually not very important in bulk samples. Order parameters are described as two dimensional vectors in Cartesian coordinates. Thick film studies give good ideas about physics in the bulk, as an interior of the film is a good approximation of a bulk. Phenomena in thin films are easier to study using this method as also the stability of such films can be studied in details. These complicated systems were also studied by using statistical mechanics approaches.[27, 38] The approach is actually complementary to the phenomenological approach and the two approaches support one another. Attempts have been made to relate the coefficients used in phenomenological description of the free energy to the intermolecular interactions. As molecules forming liquid crystals are large, consisting of more than hundred atoms and corresponding bonds, a strict statistical approach is difficult and several approximations are usually done. The phenomenological hand waving can give hints for eventual approximations. On the other hand, if a statistical approach starting from a molecular level leads to the same form of a phenomenological coefficient, the phenomenological reasoning is additionally supported. For a phenomenological model few details with respect to the model coefficients, which can often be related to macroscopic properties, are important. One should know the magnitudes or at least the ratios of various interactions. Here statistical mechanics can give a valuable insight. Without theoretical consideration statistical mechanics offer, magnitudes and signs of coefficients giving interactions could be guessed from experimental results only. But there are still several problems that remain open, for example a simple one, namely how the chemical formula relates to the magnitude and the sign of the piezoelectric coefficient when a system is formed from molecules. But also a phenomenological approach, when it is supported by good-hand waving guesses, is not without explanatory and predictive ability. With phenomenological theories one is left with guesses with respect to phenomenological coefficients that are supported by experimental evidence. Quoting just one example, although several others exist: the solutions of the puzzle that two in the MHPOBC, the SmC* and the SmC*Fi2 appeared in the same temperature window if different groups were studying ChemPhysChem 0000, 00, 1 – 14

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CHEMPHYSCHEM CONCEPTS the system. A phenomenological reasoning was the SmC*Fi2 is more probable in a material with higher polarization, that is with a higher enantiomeric excess. In fact, in purer samples * was found in the same temperature window, where the SmCFi2 the SmC* phase was found in a slightly racemized sample. So different groups were working with materials of different purity.[33]

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Received: November 22, 2013 Revised: January 16, 2014 Published online on && &&, 2014


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CONCEPTS M. Cˇepicˇ* && – && Systems with Competing Interlayer Interactions and Modulations in One Direction: Finding their Structures


One, two, three: Complex structures in polar smectics with commensurate or incommensurate modulations to layer thickness (see picture, commensurate periods are marked yellow) can be studied within the framework of a discrete phenomenological model. Three methods are presented and their appropriateness, advantages and disadvantages are discussed. Examples are given as an illustration for each method.


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Stripe systems with competing interactions on quasi-one dimensional periodic substrates.

Interlayer interactions in graphites.

Design of small MEMS microphone array systems for direction finding of outdoors moving vehicles.

Bio-Inspired Miniature Direction Finding Acoustic Sensor.

Interactions of platinum metals and their complexes in biological systems.

Large Frequency Change with Thickness in Interlayer Breathing Mode--Significant Interlayer Interactions in Few Layer Black Phosphorus.

Competing Interactions of RNA-Binding Proteins, MicroRNAs, and Their Targets Control Neuronal Development and Function.

Direction finding by hornets under gravitational and centrifugal forces.

Proteoglycans: structures and interactions.

Sustained selective attention to competing amplitude-modulations in human auditory cortex.

Structural ordering of trapped colloids with competing interactions.

An efficient algorithm for direction finding against unknown mutual coupling.

Intermolecular interactions in solid-state metalloporphyrins and their impacts on crystal and molecular structures.

ES and TRiC chaperonin components.

A Direct Coarray Interpolation Approach for Direction Finding.

Tunable aggregation by competing biomolecular interactions.

2 one-dimensional random-exchange ferromagnetic zigzag ladder, which exhibits competing interactions in a critical regime.

Self-induced uniaxial strain in MoS2 monolayers with local van der Waals-stacked interlayer interactions.

Ferrocene-based supramolecular structures and their applications in electrochemical responsive systems.

Repeated Sprint Ability in Young Basketball Players: Multi-direction vs. One-Change of Direction (Part 1).

Cluster formation in fluids with competing short-range and long-range interactions.

Physical Studies of P450-P450 Interactions: Predicting Quaternary Structures of P450 Complexes in Membranes from Their X-ray Crystal Structures.

Finding Communities by Their Centers.

Competing ParA structures space bacterial plasmids equally over the nucleoid.