Physics of liquid crystals of bent-shaped molecules

Thermotropic liquid crystals can be formed by various molecular shapes, some discovered over 125 years ago. The simplest and most-studied liquid crystals are made of rod-shaped molecules and led to today’s omnipresent liquid crystal displays. While applied scientists and engineers have been perfecting liquid crystal displays, a large group of liquid crystal scientists have become excited about liquid crystals of bent-shaped (banana-shaped) molecules. These compounds were first reported 20 years ago and since then have taken center stage in current liquid crystal science. The “banana mania” is due to the fact that even a small kink in the molecular shape leads to fundamentally new properties and phases. This review summarizes the large variety of novel structures and physical properties and describes the underlying physics. The dependence of macroscopic properties on both the shape of the molecules and the flexibility of the central core is emphasized. Most rigid bent-core molecules form smectic and sometimes columnar structures; only a minority forms nematic phases. By contrast, most flexible bent-core molecules form nanostructured nematic phases, including the twist-bend nematic phase discovered very recently.

Figure 1

Phases of bent-core liquid crystals. (a) Uniaxial nematic, (b) polar with uniform $\widehat{\mathit{n}}$, (c) polar with bend of $\widehat{\mathit{n}}$, and (d) biaxial.

Figure 2

Phases of pear-shaped liquid crystals. (a) Uniaxial nematic, (b) polar with uniform $\widehat{\mathit{n}}$, and (c) polar with splay of $\widehat{\mathit{n}}$.

Figure 3

Chiral symmetry breaking in the formation of right- and left-handed dimers of bent-core molecules. From [268].

Figure 4

Modulated nematic phases of bent-core molecules. (a) Twist bend and (b) splay bend.

Figure 5

Historic bent-shaped liquid crystal molecules with opening angles (a) about 60°, (b) 120°, and (c) 120°. The compound (a), synthesized by Vorländer and Apel in 1932, has an isotropic—nematic—crystal (I 219 °C N 171 °C Cr) phase sequence on cooling. The compound (b), synthesized by Vorländer in 1929, shows an isotropic—smectic—crystal (I 224°C $Sm$ 186°C Cr) phase sequence on cooling. The compound (c), synthesized by Watanabe in 1996, has an isotropic—smectic 1—smectic 2—crystal (I 158°C $Sm1$ 140°C $Sm2$ 72°C Cr) phase sequence on cooling.

Figure 6

Sketch of hypothetical nematic phase of bent-core liquid crystals illustrating the smectic clusters enclosed in dotted lines.

Figure 7

Experimental proof of the smectic clusters in bent-core nematic liquid crystals. (a) Representative cryo-TEM pattern of a three-ring bent-core liquid crystal material in the nematic phase. (b) Corresponding small-angle x-ray scattering results with the illustration of the smectic layers in clusters.

Figure 8

Molecular structure of bent-core nematics (a) C7, C12 and (b) A131. The length of the C7 molecule is about 3.8 nm, and the width is 0.8 nm.

Figure 9

Conoscopic images of a homeotropic sample in (a) $N_U$ and (b) hypothetical $N_B$ phases. The splitting of isogyres shown in (b) can be observed in the $N_U$ phase when the director varies along the direction of observation.

Figure 10

Biaxiality-mimicking effect of thermal contraction and expansion in homeotropic cells of $N_U$. (a) Cooling of the $N_U$ phase of nonsymmetrically substituted thiadiazole abbreviated J35 or DT6Py6E6 results in transient but long-lived splitting of isogyres that mimic biaxial order ([229]), (b) equilibrium director in the homeotropic cell, and (c) director deformations in a homeotropic cell caused by thermal contraction and expansion that produce split isogyres in conoscopic patterns of a deformed $N_U$.

Figure 11

Bulk point defects: hedgehogs in (a) $N_U$ and (b) $N_B$. Surface point defects: boojums in (c) $N_U$ and (d) $N_B$. Note that the point defects in $N_B$ are always connected to singular disclinations. Point defects provide another opportunity for an experimental test of bulk biaxial order. Schemes courtesy of Bohdan Senyuk.

Figure 12

Colloidal polystyrene spheres in the planar cells ($h=25\mu \mathrm{m}$) of (a) C12, sphere diameter $8\mu \mathrm{m}$; (b) C7, diameter $5\mu \mathrm{m}$; (c) scheme of director configuration with boojums around the sphere in $N_U$; (d) a hypothetical scheme of director configuration around the sphere in $N_B$. The singular lines shown in (d) are not observed experimentally in the studied biaxial candidate materials. From [400].

Figure 13

Rheological properties of several bent-core nematogens. (a) Shear rate dependences of the viscosities in the isotropic phases 10 °C above the $I\text{−}N$ transition. In comparison the viscosities of two rod-shaped materials 5CB and 8CB are also shown. (b) Shear rate dependences of the elastic modulus in the isotropic phases 10 °C above the $I\text{−}N$ transition. The shear moduli for 5CB and 8CB are not shown because they are within the experimental error of $\sim 5\mathrm{Pa}$. From [16].

Figure 14

(a) Variation of splay ($K_{11}$) and bend ($K_{33}$) elastic constants as a function of reduced temperature of a bent-core material with a broad range of nematic phase. The inset shows $K_{11}$ and $K_{33}$ at the function of birefringence ($\mathrm{\Delta }n$). (b) Schematic of the biasing of the kink directions by rotation around the long axis to relieve bend elastic energy. From [388].

Figure 15

Summary of dielectric measurements on a bent-core nematic material. Top: Molecular structure and abbreviated name of the material. Main panel: Frequency dependence of the real part of the dielectric constant parallel to the director at selected temperatures below the $I\text{−}N$ transition. Upper inset: Temperature dependence of the relaxation frequencies in the isotropic and nematic phases. From [376].

Figure 16

Textures of the BPIII phase of a BCN doped with 2 wt % chiral dopant (BDH1281) in $10\mu \mathrm{m}$ film with in-plane electrodes. The texture is black at (a) zero electric fields and becomes birefringent when (b) 11 V in-plane voltage is applied. From [195].

Figure 17

Summary of the EC patterns in a $15\mu \mathrm{m}$ thick BCN film (molecular structure shown in Fig. 15) in planar alignment. PS: parallel stripes; PW: prewavy; ER: empty region. The length scale represents $100\mu \mathrm{m}$ in each picture. The rubbing direction is vertical for PS and horizontal for the others. From [486].

Figure 18

Schematic of the ferroelectric cluster mechanism proposed to explain the “giant” flexoelectricity. From [164].

Figure 19

Smectic-$A$ ($SmA$) phases of bent-core molecules. (a) $SmA$, (b) $SmA{P}_R$, (c) $SmA{P}_F$, and (d) $SmA{P}_A$.

Figure 20

Illustration of the V-shaped electro-optic switching in the $SmA{P}_F$ phase. (Left) Cartoon showing the reorientation of the polar axis in a bookshelf-aligned cell (substrates parallel to the page) in response to applied electric fields. (a) Bookshelf layer structure with $P$ parallel to the substrates at $E=0$; (b), (c) partial reorientation of $P$ with applied field $E_1$ (subsaturation); (d), (e) at $E_{\text{sat}}$ (saturation) with $P$ along $E$. (Right) Electrostatic V-shaped switching with optical latching. $A$: Applied voltage (blue) changing with time; $B$: current response (black) to voltage ramp (constant current); $C$: constant applied voltage results in $D$: no current and essentially no energy dissipation (latching). From [237].

Figure 21

Evidence of the existence of the $SmA{P}_A$ phase. (a) Schlieren texture of the $SmA{P}_A$ phase with smectic layers parallel to the substrates. The existence of two brush defects (shown by yellow circles). (b) X-ray pattern of a sample with director oriented vertically. The small-angle peaks in the vertical direction are proof of the orthogonal smectic structure. From [116].

Figure 22

Tilted, polar smectic phases of bent-core molecules. (a) $Sm{C}_s{P}_F$, (b) $Sm{C}_a{P}_F$, (c) $Sm{C}_s{P}_A$, and (d) $Sm{C}_a{P}_A$.

Figure 23

Tipping (leaning), polar smectic phases of bent-core molecules. (a) $Sm{T}_s{P}_F$, (b) $Sm{T}_a{P}_F$, (c) $Sm{T}_s{P}_A$, and (d) $Sm{T}_a{P}_A$.

Figure 24

Generalized smectic phases of bent-core molecules with tilt, tipping (leaning), and polar order.

Figure 25

Pictorial explanation of the formation and possible structures of layer undulated $SmC{P}_{\mathrm{mod}}$. (a) Illustration of the four possible $SmCP$ structures (for comparison, see also Fig. 22). The $SmCP$ smectic layer structure has spontaneous polar order and tilt of the bows, which makes the layers chiral, with handedness indicated by color (cyan or magenta, dark and light gray), and gives the four possible bilayer phases. Lines with end bars (T’s) indicate the projection of the bent-shaped molecules so that the bar indicating the end is nearest the reader. (b) Preferred structure of a polarization field will be either positive (red, dark gray) or negative (green, light gray) splay. (c) Stripes assuming positive splay can fill space if the required defects (green lines, light gray) have sufficiently low energy. (d) Elemental splay stripe varieties, depending on handedness and orientation of $P$. Although everywhere we draw the polarization splay with the bent-core vertex splayed out, it may be the other way around. (e) Illustration of the layer expansion at the splay defect lines that is the mechanism driving the layer undulation. (f), (g) The layers in the intermediate regions must tilt to establish uniform layer spacing along $s$. This organization of $P$, chirality, and undulation [yellow (orange) displaced up (down)] is inferred from the optical and x-ray experiments. The handedness changes at defect lines shown in blue and does not change at lines in red. This makes the undulation crest curvature different from that of the troughs. (g) Director strain, minimized in the undulated synclinic ($C_S{P}_F$) local structure, tends to suppress the undulation in the anticlinic ($C_A{P}_F$) structure. (h) Layer interdigitation provides effective molecular packing at the polarization splay defects to produce a distinct ($B_1$) family of 2D ordered phases. From [70].

Figure 26

Summary of experimental results on a completely asymmetric bent-core material [molecular structure shown above (b)]. (a) Scattering wave number ($q$) dependence of the normalized intensity of small-angle x-ray scattering. Red: $SmC{P}_{\mathrm{mod}}$ phase; green: $SmA{P}_{\mathrm{mod}}$ phase. The splitting of the (1, $k$) and (1, $-k$) peaks (where $k=2$, 3, 4) in the $SmC{P}_{\mathrm{mod}}$ phase is due to the tilt. (b), (c) Cryo-TEM textures in the $SmC{P}_{\mathrm{mod}}$ and $SmA{P}_{\mathrm{mod}}$ phases, respectively. They show only nonmodulated smectic layering with layer spacing larger than the (1,0) peaks show in bulk x ray. (d), (f) POM textures of a $5\mu \mathrm{m}$ film at the isotropic-$SmC{P}_{\mathrm{mod}}$ transition. Textures are typical to switchable $B_7$ textures characterized by striped ribbon motifs as seen in (d) and in the bottom of (f). (e) Model of the layer structure explaining the formation of the striped ribbon motifs. From [504].

Figure 27

Structural cartoons of three columnar phases formed by bent-core liquid crystals: (a) the $B_1$ phase with $p2mg$ symmetry. (b) Reversed orthogonal phase $B_{1\text{rev}}$, and (c) tilted $B_{1\text{rev}}^{\text{obl}}$ phase. From [481].

Figure 28

Molecular structure of the prototypical $B_7$ material, $n\text{−}\mathrm{OPIMB}\text{−}{\mathrm{NO}}_2$ with $n=7$ (inset in the middle) and typical POM texture of a $10\mu \mathrm{m}$ film between crossed polarizers at the isotropic $B_7$ transition. (a) Banded myelinic texture with director and layer structure corresponding to that shown in Fig. 25; (b) a combination of broken fan texture, myelinic texture, and a hexagonally patterned myelinic texture; (c) combination of spiral texture ([207]) with banded myelinic texture; and (d) a combination of broken fan (right) and banded myelin textures (left).

Figure 29

Summary of TEM profiles and the suggested packing model of $B_7$ (splayed $B_1$) materials. (a) Cryo-TEM profile of ${9\text{−}\mathrm{OPIMB}\text{−}\mathrm{NO}}_2$ showing modulations with $b\sim 8.3\mathrm{nm}$ and $L\sim 125\mathrm{nm}$; (b) freeze-fracture (FF) TEM profile of ${8\text{−}\mathrm{OPIMB}\text{−}\mathrm{NO}}_2$ showing topological features similar to that found by cryo-TEM. From N. A. Clark. (c) Histogram of the height profile and (d) suggested packing model of the molecules. From [503].

Figure 30

Freestanding fibers of bent-core liquid crystals. (a) POM textures at the meniscus of the filaments in different phases. (a1) 8-OPIMB-NO2 in the nonswitchable $B_7$ (splayed $B_1$) phase. (a2) A material in the $SmC{P}_{\mathrm{mod}}$ phase; (a3), (a4) bare and chlorine substituted BCS in $SmCP$; (a5) a BCN in nematic phase. The bar indicates $10\mu \mathrm{m}$. From [203]. (b1) Proposed concentric layer structure of $SmCP$ filaments; (b2), (b3) FF-TEM images of $B_7$ filaments with modulation direction along the filament axis (b2) and with an angle $\gamma$ with respect to the filament axis (b3). (b2), (b3) From [70].

Figure 31

(a) Schematic of the column cross section made of four bent-shaped molecules; (b) in the ${\text{Col}}_h{P}_A$ phase, the molecules form a conelike structure with noncompensated dipole moment illustrated by an arrow; and (c) a schematic of the polar switching in the ${\text{Col}}_h$ phase of polycatenar molecules. (a), (b) From [149]. . (c) From [442].

Figure 32

Illustration of the properties of a BC mesogen W508 (molecular structure with phase sequence at the top) in the DC phase. (a) Freeze-fracture TEM image with the illustration of the formation of the sponge-type domains in the inset. (b) Optical textures of the DC phase between slightly uncrossed (top and bottom) and crossed polarizers (middle). From [183].

Figure 33

Pictorial of the HNF phase. (a) Bent-core mesogenic molecules, which form well-defined smectic layers. (b) Demonstration of the projections of the lattices of the core arms (yellow and violet). (c) Sketch of an individual nanofilament. (d) FF-TEM image of NFs of the molecule shown in (a) at $T=25°\mathrm{C}$. (e), (f) FF-TEM images of nanofilaments growing independently or collectively in packs. (g) FF-TEM image of the bulk HNF phase exhibiting large domains of parallel, coherently twisting NFs. Scale bars, (d) 60 nm, (e), (g) 400 nm, and (f) 370 nm. From [182].

Figure 34

TEM image and models showing double-twist structure. (a) Cryo-TEM image of an area of P8OPIMB with gradually increasing thickness from bottom right to top left. The vectors ${\overrightarrow{h}}_1$, ${\overrightarrow{h}}_2$, and ${\overrightarrow{h}}_3$ illustrate the direction and periodicity of the filaments. Scale bar is 100 nm. (b) Transmitted electron intensity profile along the line $ABCD$, indicating a gradual increase of the film thickness from $A$ to $D$. (c) Equilibrium arrangement of different layers of helical pastas; from bottom to top are the first, second, and third layers of filament, respectively. (d) Sketch of the proposed arrangement of the helical nanofilaments. From [502].

Figure 35

Odd-even effect in the phase transition temperatures and entropy of melting for dimers with the general formula $\mathrm{NC}{({\mathrm{C}}_6{\mathrm{H}}_4)}_2\mathrm{O}{({\mathrm{CH}}_2)}_x\mathrm{O}{({\mathrm{C}}_6{\mathrm{H}}_4)}_2\mathrm{CN}$ and monomers ${\mathrm{CH}}_3{({\mathrm{CH}}_2)}_{x-1}\mathrm{O}{({\mathrm{C}}_6{\mathrm{H}}_4)}_2\mathrm{CN}$ with the same number $x$ of carbon atoms in the alkyl-oxy end chain. Entropy changes are shown in units of the gas constant. Plotted with the data reported by Emsley et al. From [112].

Figure 36

Odd-even effect of the aliphatic linkers with all methylene groups in transconformations on the molecular shapes: (a) 1,7-bis (4-cyanobiphenyl-$4^{\prime }$-yl)hexane (CB6CB), $x=6$, shows a rodlike shape while (b) 1,7-bis (4-cyanobiphenyl-$4^{\prime }$-yl)heptane (CB7CB), $x=7$, is bent. The CB7CB molecular shape closely follows a circular arch of a radius as suggested by [455]. Chemical structure by Greta Babakhanova.

Figure 37

Temperature dependences of physical properties of a uniaxial nematic phase formed by (a) odd dimer DTC5C9 and (b) monomer MCT5: (c) birefringence; (d) anisotropy of dielectric permittivity; (e) elastic constants of splay, twist, and bend; (f) expanded view of the temperature dependence of $K_{33}$ in the $N_U$ phase of DTC5C9; (g) ratio of splay and bend elastic constants $K_{11}/K_{33}$. From [74].

Figure 38

Temperature dependences of material properties of CB7CB: (a) birefringence; (b) dielectric permittivities; (c) dielectric anisotropy; (d) splay $K_{11}$, twist $K_{22}$, and bend $K_{33}$ constants; (e) detailed data for $K_{33}$; and (f) orientational viscosities. From [15].

Figure 39

Temperature dependences of birefringence of CB9CB measured in heating (red squares) and cooling (blue circles) under zero (open symbols) and 22 T magnetic fields (solid symbols). Inset (a) shows the field dependence of birefringence at 130 °C, which is above the $I\text{−}N$ transition temperature in absence of the field. Inset (b) shows the shift of $T_{N\text{−}I}$ as a function of the square of the magnetic induction.

Figure 40

Cholesterics in an external electric field: (a) right-angle helicoidal $Ch$ in absence of the electric field; (b) realignment of the $Ch$ with $K_{33}>K_{22}$ into a “fingerprint” texture by the electric field; (c), (d) field-induced ${Ch}_{\mathrm{OH}}$ state for $K_{33}<K_{22}$; as the electric field increases, the pitch $P_{Ch}$ and the conical angle $\theta$ both decrease. The figures are not to scale as the experimental cell thickness is typically 20–50 times larger than the cholesteric pitch.

Figure 41

Electric field-controlled colors of a cell with the $C{h}_{\mathrm{O}\mathrm{H}}$ structure shown as (a) polarizing optical microscope textures and (b) reflection spectra. Incident light is unpolarized. The rms amplitude of the electric field is indicated on the figures. Mixture $\mathrm{CB}7\mathrm{CB}:\mathrm{CB}11\mathrm{CB}:5\mathrm{CB}:\mathrm{S}811=49.8:30:16:4.2\mathrm{wt}\%$. Temperature $27.5°\mathrm{C}$. Data by Olena Iadlovska.

Figure 42

(a) Magnetic field dependence of the wavelength ${\lambda }_p$ and (b) full width at half maximum (FWHM) $\mathrm{\Delta }\lambda =(n_e-n_0)P$ of the selective reflection peak observed at different temperatures. From [382].

Figure 43

Polarizing microscope textures of periodic structural deformations of a dimeric mixture KA ([5]) doped with (a) 1 wt%, (b) 2 wt%, and (c) 3 wt% of chiral dopant S811, respectively. Planar alignment, the rubbing direction is along the dotted arrow. Film thickness $h=5\mu \mathrm{m}$, approximate values of the cholesteric pitch $P_{Ch}$ are (a) 8, (b) 4, and (c) $2\mu \mathrm{m}$; (d) scheme of periodic undulations with director bend that develops in a planar cell with a thickness slightly larger than the equilibrium period $P_{Ch}/2$ of $Ch$ (shown on the left side). The resulting stripes are parallel to the rubbing direction. The regions of tilt are shadowed in blue, and the regions of layers dilation are shadowed in yellow.

Figure 44

Typical polarizing optical microscope textures of the $N_{\mathrm{TB}}$ phase (a) in a sessile droplet of $N_{\mathrm{TB}}$ phase of CB7CB placed on a rubbed substrate. (b) In comparison the texture in the $SmA$ phase of material ${\mathrm{CHF}}_2\mathrm{O}({\mathrm{C}}_6{\mathrm{H}}_4)\mathrm{CH}=\mathrm{N}({\mathrm{C}}_6{\mathrm{H}}_4){\mathrm{C}}_4{\mathrm{H}}_9$. (c) 3D scheme of layer curvatures within three focal conic domains with singularities in the shape of ellipse and hyperbolae. (d) Stripe patterns in the $N_{\mathrm{TB}}$ phase of CB7CB. (e) Stripe patterns in the $N_{\mathrm{TB}}$ phase of DTC5C9. Textures (b) and (d) from Greta Babakhanova.

Figure 45

Structural transitions triggered by the electric field $\mathit{E}=(0,0,E)$ in homeotropic cells of (a), (b) $N_U$ and (c)–(e) $N_{\mathrm{TB}}$ materials with $\mathrm{\Delta }ϵ<0$. (a) PolScope texture showing the director projection onto the horizontal $x\text{−}y$ plane with predominantly bend deformations, (b) 3D scheme of director deformations, (c) nucleation of a toroidal focal conic domain in $N_{\mathrm{TB}}$, (d) expansion of splay and saddle-splay deformations of the helicoidal axis $\widehat{\mathit{t}}$ in the form of an oily streak, and (e) cross section of the oily streak; continuous lines are equidistant layers of $N_{\mathrm{TB}}$. See [39] for more details on Frederiks transition in $N_{\mathrm{TB}}$.

Figure 46

FF-TEM textures of the $N_{\mathrm{TB}}$ phase of CB7CB (a) FF-TEM textures and corresponding fast Fourier transform (FFT) patterns with pitch $P_{\mathrm{TB}}=8.05\mathrm{nm}$ viewed in the planes parallel to the optic axis. The arrows point toward domain boundaries, which are roughly perpendicular to the $N_{\mathrm{TB}}$ layers. The presence of domains is also revealed by a diffuse intensity pattern in FFT, marked by a white arrow. (b), (c) FF-TEM image of Bouligand arches formed as imprints of the oblique helicoidal structure onto the fracture plane that is (b) almost perpendicular and (c) almost parallel to the helicoid axis of $N_{\mathrm{TB}}$. The insets show the corresponding schemes of Bouligand arches of two types in $N_{\mathrm{TB}}$. All scale bars are 100 nm. (d)–(f) Three types of Bouligand arches predicted by Eq. (31): (d) type I, alternating wide and narrow arches, observed when $\psi <{\theta }_{\mathrm{TB}}$; (e) type II, observed when $\psi >{\theta }_{\mathrm{TB}}$; (f) type III, equidistant arches, observed for heliconical structures only when $\psi ={\theta }_{\mathrm{TB}}$; in $Ch$, the equidistant arches are observed for any tilt angle $0<\psi <\pi /2$. From [39].

Figure 47

Packing of bent-core molecules: (a) in a 2D $x\text{−}y$ plane, bent-core molecules with the fixed angle $\chi$ between the rigid arms can be arranged along a circle of a certain radii $R_{\mathrm{mol}}$ that fits the molecular shape; see Fig. 36. Filling the interior $R<R_{\mathrm{mol}}$ and exterior $R>R_{\mathrm{mol}}$ of the circle with similarly bent molecules $\chi =\text{const}$ is impossible; (b) bent-core molecules escape into the third $z$ dimension along a helicoidal trajectory thus preserving a constant bend in the entire 3D space but suffering twist deformations; (c) helicoidal chain of CB7CB molecules with the pitch $P_{\mathrm{TB}}$, conical angle ${\theta }_{\mathrm{TB}}$, and the contour length $2\pi R_{\mathrm{mol}}$ measured along one full turn of the helicoid. Also shown are two extreme geometries: a flat circular chain ${\theta }_{\mathrm{TB}}=\pi /2$ and a straight chain ${\theta }_{\mathrm{TB}}=0$, both of the same contour length $2\pi R_{\mathrm{mol}}$; (d) scheme of a “duplex helical tilted chain” as a building unit of CB7CB; (e), (f) models of helicoidal flagellae studied by [22], long before $N_{\mathrm{TB}}$ were observed, in out-of-phase (e) and in-phase (f) arrangements. (a), (b), and (d) From V. Borshch. (e), (f) From [22].