Antiferroelectric liquid crystals: Interplay of simplicity and complexity

This paper reviews nearly $20\text{years}$ of research related to antiferroelectric liquid crystals and gives a short overview of possible applications. “Antiferroelectric liquid crystals” is the common name for smectic liquid crystals formed of chiral elongated molecules that exhibit a number of smectic (Sm) tilted structures with variation of the strong-tilt azimuthal direction from layer to layer (i.e., nonsynclinic structures). The phases have varying crystallographic unit periodicity from a few $\left(\mathrm{Sm}{\mathrm{C}}_{\alpha }^{*}\right)$, four $\left(\mathrm{Sm}{\mathrm{C}}_{FI2}^{*}\right)$, three $\left(\mathrm{Sm}{\mathrm{C}}_{FI1}^{*}\right)$, and two $\left(\mathrm{Sm}{\mathrm{C}}_A^{*}\right)$ smectic layers and all of the phases possess liquidlike order inside the layer. The review describes the discovery of these phases and various methods used for their identification and to determine their structures and their properties. A theoretical description of these systems is also given; one of the models—the discrete phenomenological model—of antiferroelectric liquid crystals is discussed in detail as this model allows for an explanation of phase structures and observed phase sequences under changes of temperature or external fields that is most consistent with experimental results.

Figure 1

Schematic illustration of molecular ordering in structures of (a) $\mathrm{Sm}{\mathrm{C}}_{\alpha }^{*}$, (b) $\mathrm{Sm}{\mathrm{C}}^{*}$, (c) $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}2}^{*}$, (d) $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}1}^{*}$, and (e) $\mathrm{Sm}{\mathrm{C}}_A^{*}$ phases, which emerge as temperature decreases.

Figure 2

Simple illustration of realization of $C_n$ and $C_{nv}$ symmetries necessary for polar systems in nontilted (top) and tilted (bottom) smectic phases.

Figure 3

Chemical structures of important compounds in which the antiferroelectric phase was identified early.

Figure 4

The local molecular arrangements of (b) the antiferroelectric phase and (a) the positive and (c) negative electric-field-induced ferroelectric phases.

Figure 5

Transmittance change and switching current observed by applying a triangular-wave voltage to MHPOBC. Note that two changes in the transmittance and two current peaks are observed at the same voltages. From 28.

Figure 6

Apparent tilt angle in MHPOBC under the application of dc electric voltages. Double hysteresis loop is observed. From 28.

Figure 7

Electro-optic response of MHPOBC obtained by application of positive and negative electric pulses in addition to a bias voltage of $6\mathrm{V}$. From 28.

Figure 8

Helical structure of the $\mathrm{Sm}{\mathrm{C}}_A^{*}$ phase.

Figure 9

Schematic structure and corresponding variation of indicatrix along the helical axis in the ${\mathrm{N}}^{*}$, $\mathrm{Sm}{\mathrm{C}}_A^{*}$, and $\mathrm{Sm}{\mathrm{C}}^{*}$ phases. Note that the helices in the ${\mathrm{N}}^{*}$ and $\mathrm{Sm}{\mathrm{C}}_A^{*}$ phases are optically the same, but that in the $\mathrm{Sm}{\mathrm{C}}^{*}$ phase is different.

Figure 10

Transmittance spectra at oblique incidence in the $\mathrm{Sm}{\mathrm{C}}^{*}$, $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}1}^{*}$ (or $\mathrm{Sm}{\mathrm{C}}_{\gamma }^{*}$), and $\mathrm{Sm}{\mathrm{C}}_A^{*}$ phases of MHPOBC. Note that no transmittance loss is observed at the wavelength indicated by an arrow in the $\mathrm{Sm}{\mathrm{C}}_A^{*}$ phase, as explained. No transmittance loss is observed in $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}1}^{*}$ because of the long pitch. From 27.

Figure 11

Gapless antiferroelectric phason in the $\mathrm{Sm}{\mathrm{C}}_A^{*}$ phase measured by the photocorrelation technique. From 138.

Figure 12

Phase retardation of (a) three- and (b) four-layer free-standing films of MHPOBC by the oblique incidence of light under the application of positive and negative fields perpendicular to the optical plane. From 7.

Figure 13

$C$-director maps (a), (b) in the $\mathrm{Sm}{\mathrm{C}}^{*}$ phase and (c) the $\mathrm{Sm}{\mathrm{C}}_A^{*}$ phase with screw dislocation lines. Note that four-brush defects are possible in the $\mathrm{Sm}{\mathrm{C}}^{*}$ phase but two-brush defects cannot exist because of a defect plane with discontinuous $C$-director change [solid line in (a)]. Two-brush defects can possibly be introduced in the $\mathrm{Sm}{\mathrm{C}}_A^{*}$ phase by screw dislocations. From 182.

Figure 14

(Color online) Schlieren textures in (a) the SmC and (b) $\mathrm{Sm}{\mathrm{C}}_A$ phases. Note that only four-brush defects are observed in the SmC phase, but two-brush defects are also present in the $\mathrm{Sm}{\mathrm{C}}_A$ phase, as indicated by arrows.

Figure 15

Compounds that show $\mathrm{Sm}{\mathrm{C}}_A^{*}$ and subphases. MHPOBC-family compounds are listed in the order of stronger ferroelectricity or weaker antiferroelectricity.

Figure 16

Simultaneous measurements of dielectric constant and chevron angles. Two measurements clearly identify the phase transitions SmA-$\mathrm{Sm}{\mathrm{C}}_{\alpha }^{*}\text{−}\mathrm{Sm}{\mathrm{C}}_A^{*}$ and indicate that $\mathrm{Sm}{\mathrm{C}}_{\alpha }^{*}$ is a tilted phase. From 89.

Figure 17

Conoscopic figures in the (a)–(d) $\mathrm{Sm}{\mathrm{C}}^{*}$, (e) $\mathrm{Sm}{\mathrm{C}}_{\alpha }^{*}$, (f) $\mathrm{Sm}{\mathrm{C}}_A^{*}$, (g) $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}2}^{*}$, and (h) $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}1}^{*}$ phases. The $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}2}^{*}$ and $\mathrm{Sm}{\mathrm{C}}_A^{*}$ phases under the same field are shown. They show essentially the same pattern, suggesting that $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}2}^{*}$ has more or less an antiferroelectric nature, but the threshold fields for the molecular reorientation are totally different. Note also the characteristic conoscopic image in the $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}1}^{*}$ phase.

Figure 18

X-ray intensity scans in (a)–(d) the indicated phases of the (R) enantiomer and (e) the racemic 10OTBBB1M7. Additional signals related to the layer superstructure appear, apart from the second harmonic of the signal related to the layer structure, $q_z∕q_0=0$. From 126.

Figure 19

Considered director projections on the smectic layer for the $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}2}^{*}$ phase: (a) Ising structure, (b) distorted structure, and (c) clock structure. The distortion $\psi$ is measured with respect to the tilt direction in the Ising structure.

Figure 20

Considered director projections on the smectic layer for the $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}1}^{*}$ phase: (a) Ising structure (b), distorted structure, and (c) clock structure. The distortion $\psi$ is measured with respect to the tilt direction in the Ising structure

Figure 21

(Color online) Optical rotary power and selective reflection for HHBBMz compound. The temperature dependence of the optical rotatory power and selective reflection wavelength (inset); note that within the temperature range of the $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}1}^{*}$ phase helix-twist inversion takes place (top) for the material 12 HHBBMz (49). Textures of $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}1}^{*}$ and $\mathrm{Sm}{\mathrm{C}}^{*}$ phases in free suspended film samples were observed between crossed polarizers, subject to a temperature gradient (the arrow indicates the direction of increasing temperature). In $\mathrm{Sm}{\mathrm{C}}^{*}$ the color of selective reflection is visible (dark lower right corner, green online), in $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}1}^{*}$ at the temperature at which the optical helix becomes unwound clear birefringence colors appear (bright upper left region, blue online) (bottom).

Figure 22

Evolution of the pitch at the $\mathrm{Sm}{\mathrm{C}}^{*}\text{−}\mathrm{Sm}{\mathrm{C}}_{\alpha }^{*}$ phase transition for mixtures of compounds with $n=10$ and 11. The jump in the pitch at the phase transition (inset) decreases with decreasing concentration of the $n=11$ homolog in the mixture. The studied system shows the evolution from first-order to overcritical behavior. From 122.

Figure 23

Polar azimuthal antiphase modes in the antiferroelectric (a) $\mathrm{Sm}{\mathrm{C}}_A^{*}$ and (b) $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}2}^{*}$ phases. Note that polarization is perpendicular to the tilt (marked by small arrows). When the structure is deformed due to the fluctuations (dashed arrows) the polarization in the unit cell does not cancel out.

Figure 24

Temperature-frequency dependence of (left) real and (right) imaginary parts of dielectric permittivity for MHPOBC with high optical purity (showing $\mathrm{SmA}$, $\mathrm{Sm}{\mathrm{C}}_{\alpha }^{*}$, $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}2}^{*}$, $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}1}^{*}$, and $\mathrm{Sm}{\mathrm{C}}_A^{*}$). Note that the antiferroelectric phases $\mathrm{Sm}{\mathrm{C}}_A^{*}$ and $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}2}^{*}$ have much weaker dielectric response than $\mathrm{Sm}{\mathrm{C}}_{\alpha }^{*}$ and $\mathrm{Sm}{\mathrm{C}}_{FI1}^{*}$. From 69.

Figure 25

(Color online) Phase diagram and corresponding differential calorimetric scans. (Left) Phase diagram for MHPOBC showing the influence of optical purity on the subphase sequence. (Right) Differential Scanning calorimetry scans for the mixtures of different optical purity. From 69.

Figure 26

$E\text{−}T$ phase diagrams proposed by (a) conoscopy in slightly racemized $(\mathrm{R}:\mathrm{S}=90:10)$ TFMH-PBC (57), (b) pyroelectricity in 12OF1M7 (174), and (c) resonant x-ray diffraction (94). In (b) $r$ is a parameter expressing polar strength, 1 for $\mathrm{Sm}{\mathrm{C}}^{*}$ and 0 for $\mathrm{Sm}{\mathrm{C}}_A^{*}$.

Figure 27

Tilt ${\vec{\xi }}_j$ and polarization ${\vec{P}}_j$ order parameters presented schematically for an average molecule in the $j\text{th}$ layer.

Figure 28

In the ferrielectric $\mathrm{Sm}{\mathrm{C}}_{\mathrm{SM}}^{*}$ phase both order parameters, ${\vec{\xi }}_f$ and ${\vec{\xi }}_a$, are nonzero. (a) For ${\gamma }_2<0$ the order parameters are parallel, molecules are tilted within the same plane, but the magnitude of the tilt differs in odd and even layers. (b) For ${\gamma }_2>0$ the order parameters ${\vec{\xi }}_f$ and ${\vec{\xi }}_a$ are perpendicular. The magnitude of molecular tilts is equal in all layers; however, the directions of tilts in odd and even layers are at an angle.

Figure 29

Tilt dependence on temperature for ${\gamma }_2<0$. (Top) The temperature dependence of the ferroelectric ${\vec{\xi }}_f$ and the antiferroelectric ${\vec{\xi }}_a$ order parameters for ${\gamma }_2<0$. (Bottom) Corresponding tilts in even and odd smectic layers. The negative sign of the tilt magnitude in the odd layers means the tilt is in the opposite direction to the tilt in even layers.

Figure 30

Tilt dependence on temperature for ${\gamma }_2>0$. (Top) The temperature dependence of the ferroelectric ${\vec{\xi }}_f$ and antiferroelectric ${\vec{\xi }}_a$ order parameters for ${\gamma }_2>0$. (Bottom) Corresponding angle between directions of the tilts in even and odd smectic layers.

Figure 31

Temperature dependence of the wave vector $q$ for the simple modulation is qualitatively equal for both signs of ${\gamma }_2$.

Figure 32

For favorable anticlinic tilts in neighboring layers all bilayer structures (a)–(c) are disfavored. The short helix allows for the reduction of frustration as the neighboring layer slightly deviates from the favorable (d) synclinicity or (e) anticlinicity; however, the next-nearest layers slightly deviate from the favorable anticlinicity.

Figure 33

Comparison of theoretical analysis and experimental observations of phase retardation. (Top) The measurement of the phase retardation between the ordinary and the extraordinary ray for a freestanding film consisting of 122 layers in the $\mathrm{Sm}{\mathrm{C}}_{\alpha }^{*}$ phase. From Bahr et al., 1995. (Bottom) The calculation of the phase difference based on the theoretical model of the $\mathrm{Sm}{\mathrm{C}}_{\alpha }^{*}$ phase for the same number of layers. From 164.

Figure 34

The schematic presentation of microscopic origins for bilinear interactions between nearest layers. (a) Molecules that diffuse through layers promote the same direction of the tilt. (b) Distances between parts of the molecules in neighboring layers are shorter for anticlinic orientation than for synclinic.

Figure 35

(Color online) The one surface free film of the (R) enantiomer of 10TBBB1M7, the material studied by resonant x-ray scattering (126). (a) At the temperature of the helix reversal in the $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}2}^{*}$ phase the film is clearly birefringent and (b) oriented in the electric field. (c) The film observation using a $\lambda$ plate reveals the orientation of the larger and the smaller refraction indices. From 19.

Figure 36

Phase diagram of the homologous series 8PIRn in which the length of the alkyl chain attached to the chiral carbon atom is varied. Inset: Phase diagram for the homologous series mPIR4 in which the length of the achiral alkoxy chain is changed. In both cases a reentrant synclinic phase appears. From 161.

Figure 37

The schematic presentation of microscopic origins for biquadratic interactions between nearest layers. (a) Lathlike molecules ordering within the layer. (b) Partial interlayer diffusion (gray molecules) favors synclinic ordering. (c) Intermolecular distances are on average shorter for the anticlinic ordering [solid line in (c)] as favored by van der Waals attraction than those for the synclinic ordering [dotted line in (b)].

Figure 38

Theoretical reproduction of the phase sequence observed in 10TBBB1M7. Evolution of the distortion (${\alpha }_2=2\psi$ in the $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}2}^{*}$ and ${\alpha }_3=2\psi$ in the $\mathrm{Sm}{\mathrm{C}}_{\mathrm{FI}1}^{*}$ phase) (top) and the phase difference $\delta$ between the neighboring repeating units in dependence on the temperature (bottom). From 19.

Figure 39

Schematic presentation of the four-layer structure construction. (a) The four-layered structure is stable in systems with quadrupolar interactions $(b_Q<0)$ and negligible achiral bilinear coupling between nearest layers $({\tilde{a}}_1\approx 0)$. (b) If the chiral interaction between neighboring layer, ${\tilde{f}}_1$, is not negligible, the synclinic pair (left) is distorted in the opposite direction to the anticlinic pair (right). (c) The consequence is a four-layered repeating unit if the chiral interaction ${\tilde{f}}_1$ is not negligible. (d) For non-negligible chiral interactions to more distant layers, ${\tilde{f}}_2$, the sum of the phase differences between neighboring layers over the repeating unit is not $2\pi$.

Figure 40

Schematic presentation of the three-layer structure construction. (a) The three-layered structure is stable in systems with quadrupolar interactions $(b_Q<0)$. (b) When the chiral interaction between nearest layers, ${\tilde{f}}_1$, is not negligible, the sum of the phase differences over the repeating unit is not $2\pi$. (c) The soliton lattice is formed dynamically. From 19.

Figure 41

Structural changes of the $\mathrm{Sm}{\mathrm{C}}_{\alpha }^{*}$ phase in external electric field. (Top) Distorted helical structure in an external electric field. The pitch in the absence of the field extends over six layers; in the field the commensurate structure of four rotations in 25 layers is stable for the considered field. (Bottom) The structures are locked to commensurate ratios of distorted and nondistorted helices $p∕p_0$ at lower electric field. At higher electric field helices lock to integer numbers of layers in the pitch $p$. From 166.

Figure 42

(Color online) Possible stable structures allowed by the Ising model with interactions to the third-nearest-neighboring layers.

Figure 43

Devil’s flower, the phase diagram in dependence of the temperature allowed by the Ising model. The arrow indicates the path of the coefficient ratio which accounts for the four- and the three-layered structures.

Figure 44

(Color online) Antiferroelectric liquid crystal display developed by DENSO.

Figure 45

Driving scheme for AFLCDs. On the right-hand side, optical response (transmittance vs applied voltage) is shown. Simplified driving scheme is shown on the left. The short period of $0\mathrm{V}$ at the first stage of each frame resets the system to the AFLC state. The first (high or low) pulse $V_D$ in addition to the bias voltage $V_0$ selects for the antiferroelectric or ferroelectric state.

Figure 46

Principle of spatial light modulator using antiferroelectric liquid crystals with photoisomerizable molecules. The threshold voltage shift is associated with trans-cis isomerization by light irradiation. Dark and bright (off and on) states can be switched under a bias voltage.