Banana and Tetrahedral Phases
by H. Pleiner
(external collaboration with P.E. Cladis and H.R. Brand)
Banana (or bow-) shaped molecules can order spontaneously in smectic superstructures such that the geometrically preferred molecular axes give rise to a macroscopic polarization [C]. Depending on the orientation relative to the smectic layers (untilted, tilted about a non-polar or about the polar axis, tilted about two axes) various phases are possible. The polarization can be 1-, 2-, or 3-dimensional with components in, or across, the layers. Some of the phases possess a handedness (spontaneous twist) that is geometrical in nature and does not result from molecular shapes, thus allowing both types of handedness to occur (ambidextrous phases) [D]. These low symmetry smectic phases still bear a lot of interesting scientific questions, e.g. regarding defect structures and phase transitions [E,F]. Quite recently, dolphin phases came into consideration (also called "peelable banana phases" referring to bananas lacking up-down symmetry, in contrast to the "minimal" ones having this symmetry). There, two polar axes give rise to low symmetry phases even without tilt or with only one tilt.
In principle, biaxial nematic phases made of banana and dolphin molecules [E,F] are a possibility, although up to date, no such phase has been identified experimentally with certainty.
Smectic phases made of achiral biaxial molecules:
phase | local symmetry | polar axes | untilted axes | polarization components |
spontan. splay | spontan. 'bend' | spontan. twist |
CM | D2h | 0 | all | none | no | no | no |
C | C2h | 0 | 1 | none
| no | no | no |
C T | Ci | 0 | 0 |
none |
no | no | no |
|
C P | C2v | 1 | all | 1 (in-plane) | 1 | 2 | no |
C P' |
1 (out-of-plane) |
C B2 | C2 | 1 | 1 (polar) |
1 (in-plane) |
1 | 2 | 1 |
C B1 | C1h | 1 | 1
| 2 |
2 | 4 | no |
C G | C1 | 1 | 0 | 3 |
3 | 6 | 3 |
|
C Q | C1h | 2 | all |
2 (in-plane) |
2 | 4 | no |
C Q' |
2 (out/in-plane) |
C D1 | C1h | 2 | 1
| 2 |
2 | 4 | no |
C DG | C1 | 2 | 1 (polar) | 3 |
3 | 6 | 3 |
|
C R | C1 | 3 | all | 3 |
3 | 6 | 3 |
The symmetry classification (e.g. C B1 and C B2) given above should not be confused with the names B1, B2 etc., which serve as a list to discriminate various banana phases, experimentally.
The low symmetry phases shown above come in several modifications (e.g. eight for CG), differing by the orientation of the polarization and/or by the handedness. Stacking such different modifications one can get a
host of globally ferro- and antiferroelectric phases with and without a helix [C,D]. Thus the global symmetry of those heterogeneously stacked phases can be different from the local one. Helix-free ferroelectric phases seem to be very interesting for fast-switching devices, especially when the rotation of the polarization occurs within a layer plane with only minimal changes of the smectic layer spacing [G].
Meanwhile, columnar banana phases (2D crystal- and 1D liquid-like) are discussed as a possibility to explain the peculiar properties of the so-called "B7" phase. Here, the molecules are not necessarily stacked into columns, but an undulated smectic structure also has the symmetry properties of a columnar phase. Columnar discotic phases made of achiral molecules are [I,K]: hexagonal (Colh) and
rectangular (Colr) non-polar phases,
hexagonal (ColPh, ColPh1 and ColPh2, a, c, b, respectively)
and rectangular (ColPr, ColPr1 and ColPr2) (achiral) polar phases,
phase | local symmetry | polarization | 2D lattice | 1st rank tensor |
spontan. splay | spontan. 'bend' | spontan. twist |
Colh | D6h | none | hex | none | no | no | no |
Colr | D2h | none | rect | none
| no | no | no |
|
ColPh | C6V | along columns (k) | hex (l1,l2) |
1D along polarization |
1 | 2 | no |
ColPh1 | C1h | in the k/l1, k/l2 or l1/l2 plane | hex (l1,l2)
| 2D in the plane |
2 | 4 | no |
ColPh2 | C2V | along l1 or l2 | hex (l1,l2) | 1D along polarization |
1 | 2 | no |
ColPr | C2V | along columns (k) | rect (l1,l2) |
1D along polarization |
1 | 2 | no |
ColPr1 | C1h | in the k/l1, k/l2 or l1/l2 plane | rect (l1,l2)
| 2D in the plane |
2 | 4 | no |
ColPr2 | C2V | along l1 or l2 | rect (l1,l2) | 1D along polarization |
1 | 2 | no |
|
ColPi | C1 | inclined to columns and to lattice directions | any | 3D any direction |
3 | 6 | 3 |
We argue that a columnar phase ColPi composed
of achiral molecules, not previously
considered for classic columnar phases, is sufficient
to account for many of the unusual physical properties of B7.
If the entities that make up the columns have a preferred geometric (non-polar) axis n (e.g. the disk normal), which is different from the polarization P (i.e. the entities are biaxial) and different from the column axis k (i.e. the entities are tilted), then additional cases are possible [K]
phase | local symmetry | polarization | 2D lattice | 1st rank tensor |
spontan. splay | spontan. 'bend' | spontan. twist | tilt of n |
ColPi2 | C2 | along l1 or l2 | any | 1D along polarization |
1 | 2 | 1 | yes |
ColPi | C1 | in the l1/l2 plane or (generally) inclined to k | any | 3D any direction |
3 | 6 | 3 | yes |
Of course, there are various antiferro-, ferri- and mixed ferro-/antiferroelectric modifications possible.
The isotropic phase above the B7 phase has peculiar flow properties in the presence of electric fields, which are hardly possible for a true isotropic phase. This phase could be a (optically isotropic) tetrahedratic phase, which has no center of inversion and whose internal structure is described by a 3rd rank tensor order parameter. As a consequence of the existence of tetrahedratic order an applied electric field or an applied temperature gradient generates flow [J,K]. Reciprocally, a shear flow applied to a tetrahedratic phase leads to an induced electric field and a temperature gradient. In addition, we find [L] that this optically isotropic phase
becomes uniaxial under the influence of an external electric field, E, resulting in a phase with
C3v-symmetry.
For an applied simple
shear flow, the system becomes biaxial and a
time-dependent
state with C1 symmetry arises.
We discuss [L] to what extent deformations induced by
external forces and flows on this optically isotropic
phase, which we call a "deformable
tetrahedratic phase",
are consistent with observations at
the isotropic-B7 transition found recently
in compounds composed of banana-shaped molecules
and suggest a number of experiments to test the
conclusions of this model.
In addition [M], an electric field together with an oblique temperature gradient induces an electrical current perpendicular to both fields. Spatial variations of the tetrahedratic or the nematic order parameter generate reversible stresses inducing flow.
If tetrahedratic order is present in banana liquid crystals, it not only changes the physics of the isotropic phase, but also that of other banana phases [N]. We have investigated effects that arise, when both, tetrahedratic (octupolar) and nematic (quadrupolar) order is present. First, a huge shift of the transition temperature between the (optically) isotropic and the anisotropic phase in an external electric field is found, where the shift is linear in the field in accordance with experiments. This feature is a hallmark of the presence of tetrahedratic order.
Second, there exists a linear gradient term coupling nematic and tetrahedratic order. Thus, although the molecules are achiral, there is the possibility of nematic helices accompanied by counter-rotating tetrahedratic helices, where both hands are equivalent and equally likely, explaining the experimental findings of ambidextrous chirality in banana phases [N]. The same linear gradient term allows for defect-free splay bend textures that are less energetic compared to the uniform state. An unusual feature of these splay bend textures is the fact that they have a non homogeneous, space dependent free energy density [O]. Such textures are believed to play an important role in the interpretation of giant, biaxial moving myelin textures seen close to the B7 phase.
Recent Publications:
H.R. Brand and H. Pleiner, "Hydrodynamic and electrohydrodynamic properties of the smectic CM phase in liquid crystals", [A]
J. Phys. II (France) 1, 1455 (1991)
H.R. Brand and H. Pleiner, "The biaxial nematic-smectic CM transition in polymeric liquid crystals", [B]
Macromol. Chem., Rapid Commun. 12, 539 (1991), [418 kB pdf-file]
H.R. Brand, P.E. Cladis, and H. Pleiner, "Symmetry and Defects in the CM Phase of Polymeric Liquid Crystals", [C]
Macromol. 25, 7223 (1992), [428 kB pdf-file] and figure, [38 kB pdf-file]
- contains the achiral ferroelectric CP phase
H.R. Brand, P.E. Cladis, and H. Pleiner, "Macroscopic Properties of Smectic CG Liquid Crystals", [D]
Eur. Phys. J. B 6, 347 (1998) [429 kB pdf-file]
Erratum: Eur. Phys. J. B 31, 147 (2003), [324 kB pdf-file] DOI: 10.1140/epjb/e2003-00018-6
- contains also the CB1 and the CB2 banana phases
H. Pleiner, "Biaxial Smectic Phases", [E1]
Symposium TU Berlin, unpublished manuscript (1998), [574 kB pdf-file]
Proceedings Freiburger Arbeitstagung Flüssigkristalle 28 , 16 (1999), [591 kB pdf-file]
H.R. Brand, P.E. Cladis, and H. Pleiner, "Polar biaxial liquid crystalline phases with fluidity in two and three spatial dimensions", [E2]
Int. Journal of Engin. Science 38, 1099 (2000) [1,7 MB pdf-file]
- contains examples for complicated stacks
P.E. Cladis, H.R. Brand, and H. Pleiner, "Fluid Biaxial Banana Phases: Symmetry at Work", [G]
Liquid Crystals Today 9, part 3/4, 1 (1999), [490 kB pdf-file] and
Ferroelectrics 243, 221 (2000) [197 kB pdf-file]
H.R. Brand, P.E. Cladis, and H. Pleiner, "Antiferroelectricity in Liquid Crystals", [H]
Phys. Rev. Lett. 86, 4974 (2001, [130 kB pdf-file] DOI: 10.1103/PhysRevLett.86.4974
H.R. Brand, P.E. Cladis, and H. Pleiner, "Symmetries and physical properties of polar columnar phases in materials composed of achiral molecules", [I]
Europhys. Lett. 57, 368 (2002), [671 kB pdf-file]
H.R. Brand, H. Pleiner, and P.E. Cladis, "Flow properties of the optically isotropic tetrahedratic phase", [J]
Eur. Phys. J. E 7, 163 (2002), [769 kB pdf-file] DOI:10.1140/epje/i200201113
H. Pleiner, H.R. Brand, and P.E. Cladis, "Polar Columnar and Tetrahedratic Phases: The B7 Phase and Beyond", [K]
Proceedings Freiburger Arbeitstagung Flüssigkristalle 30, 8 (2002), [357 kB pdf-file]
Mol. Cryst. Liq. Cryst. 396, 169 (2003) [474 kB pdf-file] DOI: 10.1080/15421400390213285
P.E. Cladis, H. Pleiner, and H.R. Brand, "Deformable tetrahedratic phases:
The effects of external fields and flows", [L]
Eur. Phys. J. E 11, 283 (2003) [360 kB pdf-file]
DOI: 10.1140/epje/i2002-10157-2
H.R. Brand, P.E. Cladis, and H. Pleiner, "Selected macroscopic consequences of tetrahedratic order", [M]
Ferroelectrics 315, 165 (2005) [148 kB pdf-file] DOI: 10.1080/00150190490509656
H.R. Brand, H. Pleiner, and P.E. Cladis, "Tetrahedratic Cross-couplings:
Novel Physics for Banana Liquid Crystals", [N]
Physica A 351, 189 (2005) [193 kB pdf-file] DOI: 10.1016/j.physa.2004.12.027
H. Pleiner, P.E. Cladis, and H.R. Brand, "Splay-bend textures involving tetrahedratic order", [O]
Eur. Phys. J. E 20, 257 (2006) [364 kB pdf-file] DOI: 10.1140/epje/i2005-10129-0
H.R. Brand, P.E. Cladis, and H. Pleiner, "Reversible macroscopic dynamics of polar nematics: Reversible currents
and their experimental consequences",
Phys. Rev. E 79, 032701 (2009) [160 kB pdf-file] DOI:
10.1103/PhysRevE.79.032701
H.R. Brand and H. Pleiner, "Macroscopic Behavior of Non-polar Tetrahedratic Nematic Liquid
Crystals",
Eur. Phys. J. E 31, 37 (2010) [1 MB pdf-file] DOI:
10.1140/epje/i2010-10547-9
H. Pleiner and H.R. Brand, "Low Symmetry Tetrahedral Nematic Liquid Crystal Phases:
Ambidextrous Chirality and Ambidextrous Helicity",
Eur. Phys. J. E 37, 11 (2014) [3.3 MB pdf-file] DOI: 10.1140/epje/i2014-14011-8
H. Pleiner and H.R. Brand, "Tetrahedral Order in Liquid Crystals",
Braz. J. Phys. 46, 565 (2016) [2.9 MB pdf-file] DOI: 10.1007/s13538-016-0438-z
H.R. Brand and H. Pleiner, "On the influence of a network on optically isotropic fluid phases with
tetrahedral/octupolar order",
Eur. Phys. J. E 40, 34 (2017) [273 kB pdf-file] DOI:
10.1140/epje/i2017-11523-7
H.R. Brand and H. Pleiner, "Two-fluid model for a fluid with tetrahedral-octupolar order",
Phys. Rev. E 104, 044705 (2021) [278 kB pdf-file] DOI:
10.1103/PhysRevE.104.044705
For tetrahedral order in magnetic systems cf. Magnetic Fluids and Elastomers
preprints of the theory group
Other research topics:
Instabilities in Complex Fluids, Active Soft matter
Rheology of Complex Fluids
Non-magnetic Liquid Crystalline Polymers and Elastomers
Membranes, Films and Surface Waves
General Mesophases
Magnetic Fluids and Elastomers
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