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]:

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.

	  




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Other research topics:
Instabilities in Complex Fluids, Active Soft matter
Rheology of Complex Fluids
Liquid Crystalline Polymers and Elastomers
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General Mesophases
Unconventional Ferrofluids

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HARALD PLEINER
Last modified: June 14th, 2006