11institutetext: Harald Pleiner 22institutetext: Max Planck Institute for
Polymer Research, Mainz, Germany
Tel.: +49-6131-379246
Fax: +49-6131-379340
22email<EMAIL_ADDRESS>33institutetext: Helmut R. Brand
44institutetext: Theoretische Physik III, Universität Bayreuth, 95440
Bayreuth, Germany
44email<EMAIL_ADDRESS>
# Tetrahedral Order in Liquid Crystals

Harald Pleiner  Helmut R. Brand

(Received: 8 June 2016 / Published online: 15 August 2016
© The Authors 2016. This article is published with open access at
Springerlink.com)

###### Abstract

We review the impact of tetrahedral order on the macroscopic dynamics of bent-
core liquid crystals. We discuss tetrahedral order comparing with other types
of orientational order, like nematic, polar nematic, polar smectic, and active
polar order. In particular, we present hydrodynamic equations for phases,
where only tetrahedral order exists or tetrahedral order is combined with
nematic order. Among the latter we discriminate between three cases, where the
nematic director (a) orients along a 4-fold, (b) along a 3-fold symmetry axis
of the tetrahedral structure, or (c) is homogeneously uncorrelated with the
tetrahedron. For the optically isotropic Td phase, which only has tetrahedral
order, we focus on the coupling of flow with e.g. temperature gradients and on
the specific orientation behavior in external electric fields. For the
transition to the nematic phase, electric fields lead to a temperature shift
that is linear in the field strength. Electric fields induce nematic order,
again linear in the field strength. If strong enough, electric fields can
change the tetrahedral structure and symmetry leading to a polar phase. We
briefly deal with the T phase that arises when tetrahedral order occurs in a
system of chiral molecules. To case (a), defined above, belong (i) the non-
polar, achiral, optically uniaxial D2d phase with ambidextrous helicity (due
to a linear gradient free energy contribution) and with orientational
frustration in external fields, (ii) the non-polar tetragonal S4 phase, (iii)
the non-polar, orthorhombic D2 phase that is structurally chiral featuring
ambidextrous chirality, (iv) the polar orthorhombic C2v phase, and (v) the
polar, structurally chiral, monoclinic C2 phase. Case (b) results in a
trigonal C3v phase that behaves like a biaxial polar nematic phase. An example
for case (c) is a splay bend phase, where the ground state is inhomogeneous
due to a linear gradient free energy contribution. Finally we discuss some
experiments that show typical effects related to the existence of tetrahedral
order. A summary and perspective is given.

###### Keywords:

Bent-core liquid crystals Hydrodynamics Phase Behavior Symmetries Macroscopic
Properties Electric Octupolar Order

††journal: Brazilian Journal of Physics

##  1 Introduction

The quantitative macroscopic description of liquid crystals (LC) in terms of
partial differential dynamic equations, free energy functionals, Ginzburg-
Landau energies and the like, has been developed over the last 50 years MPP
\- mbuch . For a long time, it has been sufficient to consider nematic-like
phases with preferred (non-polar) directions or smectic (and columnar) phases
with one-dimensional layer (or two-dimensional lattice) structures. With the
advent of bent-core (or banana) LC CM92 ; Takezoe96  this has changed
considerably. It turned out that, first, polar (vector) order plays an
important role, in particular in smectic-like structures, and second,
tetrahedral order is an additional type of order, necessary to describe the
phases found in these new materials. We will only briefly discuss the role of
polar order, but concentrate on the various aspects of tetrahedral order. To
do that we first recall the traditional types of orientational order, give
some motivations why these are insufficient in the case of bent-core
materials, and then discuss tetrahedral order and and its interplay with other
types of orientational order. In the subsequent Sections 2 \- 4 we give
detailed accounts of the hydrodynamics of the various phases where tetrahedral
order is involved. The experimental situation is discussed in Section 5,
followed by a Summary and Perspective, Section 6.

Liquid crystals generally are anisotropic fluids. This is due to the existence
of one (or more) preferred directions in the fluid, either due to rotational
order (e.g. nematic LCs), or translational order (e.g. smectic and columnar
LCs), or both (e.g. smectic C LCs). As a result, at least some of their
(macroscopic) material properties, like e.g. heat conduction, viscosity, or
sound velocity, depend on the orientation. Thereby, the rotational and/or
translational symmetry of isotropic liquids is spontaneously broken by the
occurrence of ordered structures at an equilibrium phase transition. The
latter can be obtained by changing some control parameters, like temperature,
pressure or concentration (in a mixture) leading to the distinction of
thermotropic, barotropic, and lyotropic LCs, respectively.

###  1.1 Nematic order

Well-known are (uniaxial thermotropic) nematic LC, consisting of rod-like or
plate-like molecules that are disordered in the isotropic state. When cooling
down into the nematic phase they spontaneously align their preferred molecular
axes in the mean forming a macroscopic preferred direction, the director $n$,
with $\mbox{\boldmath$n$}^{2}=1$ rendering the phase anisotropic. In nematic
LC one cannot discriminate between the director and its opposite direction (no
”head” or ”tail”), even if the molecular axes do have this distinction. Thus,
$n$ is not a true vector. The common procedure to circumvent this shortcoming
is to use use $n$ as a true vector with the additional requirement that all
macroscopic equations are invariant under the exchange of $n$ with
$-\mbox{\boldmath$n$}$.

The (uniaxial) nematic order is described by the quadrupolar, traceless
symmetric order parameter tensor, correctly reflecting the $n$ to
$-\mbox{\boldmath$n$}$ invariance,

$Q_{ij}=\frac{1}{2}S(n_{i}n_{j}-\frac{1}{3}\delta_{ij})$ (1)

Due to its quadrupolar structure no vector can be extracted from it. Here,
$S$, the strength of the order, is the second moment of the microscopic
orientations of all molecules with respect to the preferred direction $n$.
There is $S=0$ in the isotropic phase, and the extremum values $S=1$ and
$S=-1/2$ for perfect order in the rod-like and plate-like case, respectively.
A realistic value for many rod-like nematic LC is $S\approx 0.8$ within the
nematic phase, which decreases approaching the nematic to isotropic phase
transition, with a jump of $\Delta S\approx 0.4$ at the transition. It can be
measured e.g. through the dielectric anisotropy PGdG .

The order is spontaneous and the orientation of $n$ is arbitrary, as long as
there are no orienting external fields or boundaries. Therefore, the two
rotations of the director, $\delta\mbox{\boldmath$n$}$ (with
$\mbox{\boldmath$n$}\cdot\delta\mbox{\boldmath$n$}=0$), are the slow
additional hydrodynamic variables (”symmetry variables”) due to the nematic
order MPP . Deviations of the order parameter $S$ from its equilibrium value
relax in a finite time (except near phase transitions) and are often
neglected.

In case the molecules order themselves with respect to two different molecular
directions, a biaxial order results with the order parameter tensor

$Q_{ij}=\frac{1}{2}S(n_{i}n_{j}-\frac{1}{3}\delta_{ij})+\eta(m_{i}m_{j}-l_{i}l_{j})$
(2)

where $\eta$ is a measure for biaxiality. The directions $m$ and $l$ are
directors with a $m$ to $-\mbox{\boldmath$m$}$ and $l$ to
$-\mbox{\boldmath$l$}$ invariance. Depending on the symmetries of the phase
considered there can be additional relations between the directors, e.g. for
orthorhombic phases the three directors have to be mutually orthogonal. The
symmetry variables are the three rotations of the rigid tripod
$\\{\mbox{\boldmath$n$},\mbox{\boldmath$m$},\mbox{\boldmath$l$}\\}$, e.g.
described by
$\mbox{\boldmath$m$}\cdot\delta\mbox{\boldmath$n$}+\mbox{\boldmath$n$}\cdot\delta\mbox{\boldmath$m$}$,
$\mbox{\boldmath$m$}\cdot\delta\mbox{\boldmath$l$}+\mbox{\boldmath$l$}\cdot\delta\mbox{\boldmath$m$}$,
and
$\mbox{\boldmath$l$}\cdot\delta\mbox{\boldmath$n$}+\mbox{\boldmath$n$}\cdot\delta\mbox{\boldmath$l$}$.
For finite rotations, that is in a nonlinear description, these variables do
not commute Nbiax ; biaxnemliu .

###  1.2 Polar order

A different type of orientational order is polar order. In polar nematic LC
the preferred direction is a true vector, with ”head” and ”tail”
distinguishable. The order parameter is the polarization vector polnema

$P_{i}=Pp_{i}$ (3)

with $P$, the value of the spontaneous polarization that characterizes the
strength of the polar order, and $p$ the unit vector that denotes the
(arbitrary) orientation of the polar direction. For rod-like systems,
thermotropic polar nematic LC are rare in nature. One reason might be that a
finite sample of homogeneous ($\mbox{\boldmath$p$}=\textrm{const.}$) polar
nematic LC exhibit opposite surface charges in the planes perpendicular to
$p$, which give rise to destabilizing electrostatic forces. Second, the
homogeneous state is not the energetic ground state, since the existence of
$P$ allows for spontaneous structures with a constant splay texture splay .
However, the latter cannot be space filling and is necessarily connected to
defects.111In chiral nematics the existence of a pseudoscalar allows for the
existence of spontaneous twist, which can fill space without defects.

The hydrodynamic variables are the two rotations of the polarization
direction, $\delta\mbox{\boldmath$p$}$ (with
$\mbox{\boldmath$p$}\cdot\delta\mbox{\boldmath$p$}=0$). The absolute value of
the polarization $P$ is linearly susceptible to an external electric field and
is therefore often kept as (slowly relaxing) variable in a macroscopic
description including, e.g., pyroelectricity.

Figure 1: Model bent-core molecule with nematic ($n$) and polar ($p$)
directions

Figure 2: Orientations in polar smectic LC: (upper left) untilted CP phase,
(upper right) p-tilted CB1 phase, (lower left) n-tilted CB2 phase and (lower
right) twice tilted CG phase; stars and rings mean a vector points out of, or
into, the drawing plane, respectively.

For bent-core molecules the situation is rather different. Typically they have
a non-polar (long) axis, $\mbox{\boldmath$n$}_{mol}$ and perpendicularly a
(short) polar axis $\mbox{\boldmath$p$}_{mol}$, cf. Fig. 1. They can align in
layered structures to form various smectic phases with polar properties (Fig.
2). There are two untilted phases, C${}_{\textrm{P}}$ with the nematic axis
$n$ along the layer normal and the polar axis $p$ within the layer CM92 , and
C${}_{\textrm{P'}}$, vice versa. Both are of $C_{2v}$ symmetry. One can tilt
the C${}_{\textrm{P}}$ structure in two ways. Rotation about the polar axis
results in the C${}_{\textrm{B2}}$ phase of $C_{2}$ symmetry and rotating
about the $p\times n$ direction gives the C${}_{\textrm{B1}}$ phase of
$C_{1h}$ symmetry. If both ways of tilting are combined, the CG phase arises,
where no symmetry element is left ($C_{1}$) D . The C${}_{\textrm{B2}}$ and
C${}_{\textrm{G}}$ phases are chiral due to their structure, even when the
molecules are achiral. This allows for the occurrence of energetically
equivalent left- and right-handed helices G1 ; G2  (called ambidextrous
chirality). If layers of different polarization direction and/or different
tilt direction are stacked, various overall structures with different
properties can be obtained, e.g. ferro-, ferri-, or antiferroelectric, and
helical or non-helicalE1 ; E2 . There is also the possibility of polar
columnar phases I . If bent-core molecules align both their axes in a biaxial
nematic way, the result is a polar biaxial nematic phase (N${}_{\textrm{I}}$)
of $C_{2V}$ symmetry E2 ; F .

A different type of polar order can occur in active systems,222There is also
the possibility of an axial, non-polar active order AG . like schools of fish,
bird flocks, insect swarms, growing bacteria, and biological motors. If the
active part of those systems moves relative to a passive background, the
relative velocity denotes a preferred polar direction AF . This type of order
is dynamic, since it is provided by the motion of the active part and
vanishes, when the motion stops. It does not occur spontaneously by an
equilibrium phase transition from an disordered to an ordered state, but is
due to the (internal) driving typically via chemical reactions (food
consumption, metabolism). The active state is a non-equilibrium one,
constantly dissipating the supplied energy. If the driving stops, the system
falls back into a passive, disordered state AH . The order parameter in this
case is the relative velocity

$F_{i}=Ff_{i}$ (4)

where the amount of the velocity, $F$, is a measure of the strength of the
order. Its non-zero value even in a stationary state, is due to the driving
and indicates the non-equilibrium nature of the system. The unit velocity,
$f$, is a polar vector and characterizes the direction of the polar order.
This direction is not prescribed by the driving and is therefore arbitrary.
Its two rotations are the symmetry variables. In that respect $f$ is similar
to the polar nematic case, $p$. However, a velocity changes sign under time
reversal, while a polarization does not. Therefore, the hydrodynamics of
active polar systems is quite different from that of (passive) polar nematic
LC. In particular, the former systems exhibit new non-trivial couplings among
various hydrodynamic variables, linear advection properties, active stresses,
second sound, and asymmetries between forward and backward traveling sound
excitations, which is a clear indication of non-equilibrium AF ; AI .

###  1.3 Tetrahedral order

During the development of more sophisticated bent-core materials, it became
apparent that an important ingredient in the macroscopic description was still
missing. Compounds were found that showed a phase transition between two
different (optically) isotropic phases. Being one of them the true isotropic
phase without any order, the other must have a type of order that is not
detectable in a microscope, meaning the dielectric tensor has to be isotropic.
That immediately rules out polar or nematic order. Obviously, tetrahedral
order Fel  \- BPC02  does qualify, since it is described by a third rank
tensor $T_{ijk}$, that cannot influence lower rank material tensors, like the
rank-2 dielectric tensor. In analogy to the nematic order that has the
quadrupolar structure of a second moment of an orientational distribution,
tetrahedral order can be called octupolar, since it is related to the third
moment.

The tetrahedral (octupolar) order parameter

$T_{ijk}=N\sum_{\beta=1}^{4}n_{i}^{\beta}n_{j}^{\beta}n_{k}^{\beta}$ (5)

is a fully symmetric rank-3 tensor expressed by an amplitude $N$ and by the 4
tetrahedral unit vectors, $\mathbf{n^{\beta}}$, with $\beta\in\\{1,2,3,4\\}$
defining a tetrahedron, cf. Fig. 3. For actual calculations one can use e.g.
the representation of $T_{ijk}$ Fel ; RadLub02

$T_{ijk}=\frac{N}{\sqrt{3}}\begin{pmatrix}1&1&-1&-1\\\ 1&-1&1&-1\\\
-1&1&1&-1\end{pmatrix}$ (6)

that is used in the left hand sides of Figs. 3 and 4. The tetrahedral
structure has four 3-fold symmetry axes, the tetrahedral vectors
$n_{i}^{\beta}$, and three 2-fold (proper) and 4-fold improper ($\bar{4}$),
symmetry axis, the Cartesian directions x,y,z in Eq. (6). The latter means
that a 900 rotation about such an axis has to be followed by a spatial
inversion, in order to arrive at the initial structure, cf. Fig. 4. Inversion
is an operation, where a structure is either reflected through a point in
space, or mirrored by three mutually orthogonal planes. If the resulting
structure is different from the original one, inversion symmetry is broken.
The existence of improper rotation axes is a sign of broken inversion
symmetry.

Another useful representation of the tetrahedral structure is

$T_{ijk}=\frac{N}{3}\begin{pmatrix}0&-\sqrt{2}&-\sqrt{2}&2\sqrt{2}\\\
0&-\sqrt{6}&\sqrt{6}&0\\\ 3&-1&-1&-1\end{pmatrix}$ (7)

where one tetrahedral vector is along the z-axis, and one of the remaining
three has a lateral projection along the x-axis, only. Note that both
structures, Eqs. (6) and (7), have an inverted counter part (all signs within
the braces changed, shown on the right hand sides of Figs. 3 and 4); those are
different from the original ones, but can be used equivalently to describe the
structures.

A phase that has only tetrahedral order is of $T_{d}$ symmetry, a subgroup of
cubic symmetry. It breaks inversion symmetry, since $T_{ijk}$ changes sign
under inversion due to the odd number of (tetrahedral) vectors involved. It is
non-polar, i.e. one cannot extract a vector from $T_{ijk}$, because of
$T_{ikk}\equiv 0$. It does not imply any nematic order, since
$T_{ikl}T_{jkl}\sim\delta_{ij}$ is isotropic. Because of the two mirror
planes, defined by two non-adjacent tetrahedral vectors (e.g. 1/4 or 2/3 in
Fig. 4), chirality is also excluded. Only if the molecules themselves are
chiral, a phase of (chiral) $T$ symmetry arises, which will be dealt with
below, separately.

Figure 3: The four tetrahedral vectors pointing to the edge of a tetrahedron
and (right) the mirror image - from BP-D2d

Figure 4: A projection of the tetrahedral structure and its mirror image of
Fig. 3 onto the x/y plane; stars and rings mean a vector points out of, or
into, the drawing plane, respectively. The two structures differ by a 900
rotation about the z-axis – note that the actual numbering of the tetrahedral
vectors is irrelevant, since they are all equivalent.

A trivial example for broken inversion symmetry is polar order, since $p$
becomes $-\mbox{\boldmath$p$}$ after inversion, which is different from the
original structure. On the other hand, nematic order $Q_{ij}$ is invariant
under inversion. Inversion symmetry is also broken in chiral systems, where a
pseudoscalar exists, typically called $q_{0}$, that changes sign under
inversion and reflects the two possibilities of left- and right-handed
structures. However, the broken inversion symmetry in tetrahedral order is
quite different, since it does not show polarity, nor chirality.

Among the peculiar features, which we will discuss in detail in Section 2, is
the possibility that an applied external electric field leads to (i) a
temperature shift of the (optically isotropic) tetrahedral to nematic phase
transition and (ii) induces nematic order in the Td phase. Both, the
transition shift and the induced order are proportional to the field strength
N , rather than to the square of the field strength, as it is common when only
ordinary isotropic phases are involved. The experimental detection of such
linear-field strength transition shifts Weissflog  accompanied by flow
phenomena in some bent-core materials is another indication that tetrahedral
order plays an important role for these systems. Another consequence of the
broken inversion symmetry in tetrahedral phases is the reversible coupling of
flow to (vectorial) generalized forces, like electric fields or gradients of
temperature or concentration. This piezo-like dynamic coupling constitutes
ambi-polarity: although polar fluxes are induced, the inverted $T_{ijk}$
structure gives fluxes in the opposite direction and the overall phase is non-
polar.

An external electric field (but not a magnetic one) not only orients the
tetrahedral structure such that one of the tetrahedral vectors, $n_{i}^{1}$,
is parallel or antiparallel to the field Fel3 , it also imposes a torque on
the other three L . If the tetrahedral structure is soft enough and the field
$E$ high enough, a deformed structure is obtained, where the angle between
$n_{i}^{1}$ and the others is reduced, asymptotically ($E\to\infty$) to
$90^{o}$ giving a pyramidal structure, cf. Section 2.3. In addition, there is
an overall rotation about $n_{i}^{1}$ (or $E_{i}$), whose rotation sense is
reversed for the inverted tetrahedral structure or if the electric field is
reversed.

If the molecules are chiral, the chiral tetrahedral phase T can be obtained
Fel ; PB14 . Its $T$ symmetry333We keep the standard notation for this type of
symmetry, confident that confusion with the temperature $T$ does not occur.
lacks any mirror plane and the tetrahedral $\bar{4}$ axes of the Td phase are
reduced to (proper) 2-fold axes. There is a pseudoscalar $q_{0}$ of molecular
origin. Similar to the case of cholesteric LC a helical structure of a given
handedness reduces the free energy, but only if the helical axis is one of the
tetrahedral vectors (the 3-fold symmetry axes). There are appropriate static
as well as dissipative Lehmann-type couplings Lehmann  among rotations of the
tetrahedral structure and e.g. the thermal degree of freedom. In contrast to
the Td phase, there is flow alignment in the chiral T phase, in particular,
for simple shear flow (with the vorticity direction along one of the
tetrahedral vectors) the tetrahedral structure is rotated about this direction
by an angle that depends on $q_{0}$, but not on the shear rate, PB14  and
Section 2.4.

###  1.4 Combined tetrahedral and nematic order

In a system where both, nematic ($Q_{ij}$) and tetrahedral order ($T_{ijk}$)
exist simultaneously, the overall symmetry of the possible phases depend on
the relative orientation of the two structures. This problem is investigated
using a Landau description setting up a free energy in terms of the two order
parameter tensors

$E_{L}=E_{L}^{Q}+E_{L}^{T}+E_{L}^{(mix)}$ (8)

where $E_{L}^{Q}$ and $E_{L}^{T}$ are the well-known Landau energy expressions
for a (pure) nematic PGdG  and a (pure) tetrahedral phase Fel3 ; RadLub02 ,
respectively. They contain terms quadratic, cubic and quartic in the order
parameter in the first case, but only quadratic and quartic ones in the second
case, due to the broken inversion symmetry of $T_{ijk}$. Since the
orientational symmetry is broken spontaneously in a nematic phase as well as
in a tetrahedral phase, $E_{L}^{Q}$ and $E_{L}^{T}$ cannot depend on the
relative orientation of the two structures. Thus, the minimum of the mixed
Landau energy

$\displaystyle E_{L}^{(mix)}$ $\displaystyle=$ $\displaystyle
d_{1}Q_{il}Q_{jm}T_{ilk}T_{jmk}$ (9) $\displaystyle+$
$\displaystyle\frac{d_{2}}{2}(Q_{im}Q_{jl}T_{ilk}T_{jmk}+Q_{ij}Q_{lm}T_{ilk}T_{jmk})$

gives the ground state of the combined system. It is of fourth order in the
order parameters and might therefore be small, in particular close to the
phase transition. We will therefore consider two cases, first the strong
coupling limit, where $E_{L}^{(mix)}$ is large and leads to a rigid relative
orientation of the two structures (”correlated order”) in Section 3, and the
weak coupling limit, where $E_{L}^{(mix)}$ is neglected (”uncorrelated order”)
in Section 4, where other energies become important.

For the case of correlated order there are two well-defined distinct
geometries (restricting ourselves for the moment to uniaxial nematic order):
Either the nematic director is along of one of the 2-fold (or $\bar{4}$)
symmetry axes of the tetrahedral structure, or it is along one of the 3-fold
axes, the tetrahedral vectors. To show that these two possibilities indeed
lead to energetic minima, one can take, without loss of generality, the
director along the z-axis, with the tetrahedral structure given in Eq. (6) and
Eq. (7), for the first and second case respectively. Indeed, the first case is
the ground state, if $d_{1}+d_{2}>0$, while $d_{1}+d_{2}<0$ gives the second
one.

In the first case (the director along a $\bar{4}$ axis) a tetragonal biaxial
structure is obtained that lacks inversion symmetry. It is of $D_{2d}$
symmetry and its hydrodynamics is discussed in detail in Section 3.1. The
hallmark of this phase is the existence of ambidextrous helicity PB14 .
Although this phase is not chiral, the formation of a non-homogeneous ground
state in the form of a helix is possible. The combined tetragonal biaxial
structure rotates about one of the $\bar{4}$ axes that are perpendicular to
the director, when going along this helical axis. The physical origin is the
broken inversion symmetry that allows for a linear gradient energy,
$\varepsilon_{l}=\xi T_{ijk}n_{i}\nabla_{j}n_{k}$, involving linearly
$\nabla_{x}n_{y}+\nabla_{y}n_{y}$ (with the helical axis along the
z-direction). One can discriminate left- and right-handed helices because the
inverted tetragonal biaxial structure is different from the original one, in
particular, what can be called right-handed for the original structure is
left-handed for the inverted one and vice versa. Obviously, the energetic gain
due to the helix does not depend on the handedness and is therefore coined
”ambidextrous”. In that respect it is similar to the ambidextrous chirality in
the smectic CB2 and CG phases, which are however (structurally) chiral, in
contrast to the D2d phase, where we call this phenomenon ambidextrous
helicity, cf. Section 3.1.2.

Another peculiar feature of this phase is the orientational frustration in an
external electric field. A nematic director is oriented in an electric field
due to the dielectric anisotropy either along the field or perpendicular to
it. The orientation energy is proportional to the square of the field
strength. In the Td phase the electric orientation effect is cubic in the
field strength and orients one of the tetrahedral vectors parallel or
antiparallel to the field direction Fel3 . In the D2d phase, where the
director and the tetrahedral vectors are at an oblique angle $\phi_{D}$ with
$\cos\phi_{D}=\pm 1/\sqrt{3}$, the nematic and tetrahedral orientation cannot
be achieved simultaneously. As a result, depending on the values of the
coupling constants one can obtain e.g. for small fields the nematic-type of
orientation that deviates, however, for higher fields towards the tetrahedral
orientation, cf. Section 3.1.3. Similar frustration effects occur for the
orientation by boundaries, but not for magnetic fields, since the latter do
not orient the tetrahedral structure.

We also discuss relative rotations in the D2d phase, where the director and
the tetrahedral structure deviate from their equilibrium orientation. If the
relaxation of this variable is slow enough, it can influence the dynamics of
the D2d phase, cf. Section 3.1.5.

If the uniaxial nematic direction $n$ along one of the $\bar{4}$ axes of the
tetrahedral structure is accompanied by (orthogonal) biaxial nematic
directions $m$ and $l$, phases of even lower symmetry occur. Depending on
whether the nematic structure is tetragonal or orthorhombic, and on how $m$
and $l$ are oriented with respect to the tetrahedral vectors, non-polar phases
of $S_{4}$ (S4) and $D_{2}$ (D2) symmetry occur, cf. Section 3.2, as well as
polar ones with $C_{2v}$ (C2v) and $C_{2}$ (C2) symmetry, cf. Section 3.3.
Phase transitions among various tetrahedral phases are described in RadLub02 .

In the tetragonal S4 phase the nematic directors $m$ and $l$ are equivalent,
but oriented obliquely within the plane perpendicular to $n$, Section 3.2.1.
The $\bar{4}$ symmetry axis (along $n$) is the only symmetry element left. Its
hydrodynamics is rather similar to that of the D2d phase.

The D2 phase is orthorhombic with all nematic directions along one of the
$\bar{4}$ tetrahedral directions, Section 3.2.2. Thereby, the three preferred
directions are reduced to 2-fold symmetry axes, which are the only symmetry
elements left. For the hydrodynamic description the most important additional
feature of the D2 phase (compared to D2d) is its chirality, since it only
contains proper rotation axes and no mirror planes anymore. Chirality is due
to the structure (and not due to the chirality of molecules) described by the
pseudoscalar $q_{0}=n_{i}n_{j}m_{k}m_{p}l_{q}l_{r}\epsilon_{ikq}T_{jpr}$ with
the orthorhombic nematic directors $n$, $m$, $l$. This gives rise to
ambidextrous chirality, since the inverted structure (with the opposite
chirality) is energetically equivalent and comes in addition to the
ambidextrous helicity that is already present in the D2d phase. To make things
even more complicated, both effects favor helices about all three 2-fold axes
generating strong frustration due to the steric incompatibility of helices
about different axes (similar to the case of biaxial cholesteric LC PB90 ).

If one orients the orthorhombic directors $m$ and $l$ within the tetrahedral
planes $n^{1}/n^{4}$ and $n^{2}/n^{3}$ (instead of along the $\bar{4}$ axes as
in the D2 phase) one gets the C2v phase, Section 3.3.2, which is polar, but
achiral, since the planes $n^{1}/n^{4}$ and $n^{2}/n^{3}$ are mirror planes.
The polar axis is the (former) $\bar{4}$ axis along $n$, since a flip of that
axis can no longer be compensated by a $\pi/4$ rotation, since $m$ and $l$ are
not equivalent. If $m$ and $l$ are oriented obliquely within the plane
perpendicular to $n$, the polar C2 phase occurs, Section 3.3.3. It is chiral,
since the mirror planes are removed. It could also be obtained by replacing
the tetragonal biaxial nematic structure of a S4 phase by an orthorhombic one.
The hydrodynamics of these polar phases will be discussed together with that
of the C3v phase, introduced next.

In the second case of correlated nematic and tetrahedral structure (with the
director along one of the tetrahedral axes, instead of a $\bar{4}$ axis) the
resulting trigonal biaxial structure is polar. The preferred polar direction,
$p_{i}\sim T_{ijk}Q_{jk}$, is given by the tetrahedral vector along the
director. The hydrodynamics of this C3v phase is rather similar to that of the
(uniaxial) polar nematic phase polnema , but has one additional hydrodynamic
degree of freedom, the rotation about the polar axis, and a reduced symmetry,
$C_{3v}$ (compared to $C_{\infty v}$ in polar nematics). This gives rise to a
more complicated structure of all rank-3 (and higher) material tensors. A
brief discussion of the hydrodynamics is given in Section 3.3.1.

In the case $E_{L}^{(mix)}=0$ there is no energy that locks a homogeneous
nematic and a homogeneous tetrahedral structure (”homogeneously
uncorrelated”). Therefore we amend the Landau energy (8) by Ginzburg-type
gradient terms

$\displaystyle E_{GL}$ $\displaystyle=$
$\displaystyle\gamma(\nabla_{k}Q_{ij})^{2}+\delta(\nabla_{k}T_{ijl})^{2}+{\cal
D}\,T_{ijk}\nabla_{k}Q_{ij}.$ (10)

The linear gradient term exists because of the broken inversion symmetry of
$T_{ijk}$ and is not related to chirality. It allows for inhomogeneous phases
having a lower energy than the homogeneous ones. In Section 4 we discuss as an
example, splay-bend textures of the nematic director accompanied by those of
the tetrahedral structure O . In the nematic splay-bend texture the
orientation of the director periodically oscillates along an axis in the
plane. Taking this axis as one of the $\bar{4}$ axes (and a second one to
define the splay-bend plane) the tetrahedral splay-bend structure (with the
same periodicity) is rotated independently about the second $\bar{4}$ axes by
a constant angle $\phi$. The total energy of the combined system is negative
due to linear gradient term, despite the energy density not being constant in
space. The state with a constant energy density is energetically slightly less
favorable. This also holds for a generalization of the nematic splay-bend
texture with a constant tilt into the third dimension.

In closing this section we briefly address the work performed in the area of
microscopic and molecular modeling. Macroscopic and molecular symmetries of
unconventional nematic phases have been studied in detail in Ref. mettout .
The analysis presented in mettout  also includes a polar order parameter as
well as third rank tensors of tetrahedral/octupolar type along with a
discussion of possible isotropic - nematic phase transitions. Microscopic
models leading to phase diagrams for liquid crystalline phases formed by bent-
core molecules using a generalized Lebwohl-Lasher lattice model with
quadrupolar and octupolar anisotropic interactions were studied in longa1 ;
longa2 ; longa3 . The techniques used include mean field theory and Monte
Carlo simulations. Among the liquid crystalline phases found are the
tetrahedral phase (Td in our notation), a tetrahedral nematic phase (D2d) with
$D_{2d}$ symmetry, as well as a chiral tetrahedral nematic phase (D2) with
$D_{2}$ symmetry. In addition, the classical phases expected for rod-like
molecules, namely the uniaxial nematic phase and the orthorhombic biaxial one
(with $D_{2h}$ symmetry) were found. In Ref. longa2  an estimate for the pitch
in the chiral tetrahedral nematic phase has been presented and the possibility
to find ambidextrous chirality for the nonchiral nematic phase formed by bent-
core molecules was elucidated. In Ref. longa3  the spontaneous formation of
macroscopic domains of opposite optical activity has been investigated in the
context of bent-core systems and ferrocenomesogens for two spatial dimensions.

##  2 Tetrahedral Hydrodynamics

###  2.1 Hydrodynamics of the Td phase without external fields

Hydrodynamics is a powerful systematic tool to describe the dynamics of
macroscopic systems. It is applicable to situations where most of the many
degrees of freedom have locally relaxed to their equilibrium values, and only
a few ones are slow enough to be dealt with explicitly by partial differential
equations. The latter class comprises the conserved quantities that cannot
relax locally, but can only be transported, like mass, momentum, and energy.
In an Eulerian description their local densities,
$\rho(\mbox{\boldmath$r$},t)$, $g_{i}(\mbox{\boldmath$r$},t)$, and
$\varepsilon(\mbox{\boldmath$r$},t)$, respectively, are space-time fields
obeying the conservation laws

$\displaystyle\frac{\partial}{\partial t}\rho+\nabla_{i}g_{i}$
$\displaystyle=$ $\displaystyle 0$ (11) $\displaystyle\frac{\partial}{\partial
t}g_{i}+\nabla_{j}\sigma_{ij}$ $\displaystyle=$ $\displaystyle 0$ (12)
$\displaystyle\frac{\partial}{\partial
t}\varepsilon+\nabla_{i}j_{i}^{\varepsilon}$ $\displaystyle=$ $\displaystyle
0$ (13)

where the nabla operator denotes partial spatial derivation
$\mbox{\boldmath$\nabla$}={\partial}/{\partial\mbox{\boldmath$r$}}$. The mass
current $g_{i}$ in Eq. (11) is the momentum density, while the stress tensor
$\sigma_{ij}$ and the heat current $j_{i}^{\varepsilon}$ are still to be
determined. Angular momentum conservation does not give rise to an additional
dynamic equation, but to some restrictions for the stress tensor. For an
extended discussion cf. MPP ; mbuch .

For isotropic liquids the conserved quantities are the only hydrodynamic
variables and Eqs. (11)-(13) are the basis for the universal set of Navier-
Stokes equations. For fluids with internal structures, like LC, the symmetry
or Goldstone variables have to be taken into account, additionally. In the Td
phase the tetrahedral structure breaks 3-dimensional rotational symmetry
spontaneously, i.e., the orientation of the structure is arbitrary. Any rigid
rotation of the structure leads to a different state, which however, has the
same internal energy as any of the others. Therefore, there is no restoring
force and an $\omega=0$ excitation (Goldstone mode) results. For an
inhomogeneous distortion with a characteristic, macroscopic wavelength $k$,
the dynamics $\omega(k)$ is slow and vanishes in the hydrodynamic limit
$\omega(k)\to 0$ for $k\to 0$. The same behavior is found for the conserved
variables as is obvious from Eqs. (11)-(13).

From the general changes of the order parameter from its equilibrium value,
$\delta T_{ijk}=T_{ijk}-T_{ijk}^{eq}$, the projection

$\delta\Gamma_{i}\equiv\frac{1}{4\alpha}\epsilon_{ipq}T_{pkl}\,\delta T_{qkl}$
(14)

with the conventional normalization $27\alpha=32\,N^{2}$, describes the
rotations of the tetrahedral structure according to the broken rotational
symmetry. Note that $\delta\Gamma_{i}$ is even under spatial inversion. Since
$\delta\Gamma_{i}$ is not conserved, its dynamic equation has the form of a
balance equation

$\frac{\partial}{\partial t}\Gamma_{i}+Y_{i}=0$ (15)

with a yet undetermined quasi-current $Y_{i}$.

Since (finite) rotations in three dimension do not commute, $\delta\Gamma_{i}$
is not a vector, nor are its components rotation angles (except in linear
approximation). Indeed,

$\displaystyle(\delta_{1}\delta_{2}-\delta_{2}\delta_{1})\Gamma_{i}$
$\displaystyle=\frac{1}{2\alpha}\epsilon_{ipq}(\delta_{1}T_{pjk})(\delta_{2}T_{qjk})$
$\displaystyle=2\epsilon_{ipq}(\delta_{1}\Gamma_{p})(\delta_{2}\Gamma_{q})$
(16)

two subsequent changes cannot be interchanged. This is similar to e.g.
rotations in biaxial nematics Nbiax ; biaxnemliu  or of the preferred
direction in superfluid 3He-A ho . Equation (14) can be inverted to give
$\delta T_{qkl}=2\,\epsilon_{ipq}T_{pkl}\,\delta\Gamma_{i}$. It is easy to
check that this special $\delta T_{ijk}$ fulfills the requirements for the
absence of polar order, $\delta T_{ijj}=0$, and of nematic order, $\delta
T_{ikl}T_{jkl}+T_{ikl}\delta T_{jkl}=0$.

All the relations above are given in terms of Cartesian coordinates, which
however only serve as a proxy for rotationally invariant descriptions using
vectors and tensors and their appropriate products. Therefore, we can use for
actual calculations any representation of $T_{ijk}$ that is suitable. Most of
the time we use the orientations given in Eqs. (6) and (7).

In addition to the hydrodynamic variables discussed above, there are systems
or situations, where a few mesoscopic, fast-relaxing degrees of freedom become
slow enough to be relevant for a macroscopic description. Examples are elastic
strains in viscoelastic fluids or the scalar nematic order parameter $S$ in
nematic LC close to phase transitions and in the vicinity of defect cores. In
the Td phase the strength of tetrahedral order, $N$, is of that character.
Most of the time we will assume that $N$ has already relaxed to its
equilibrium value $N_{0}$, which one can then take as unity. When deviations
$\delta N=N-N_{0}$ are relevant for the macroscopic dynamics, the balance
equation

$\frac{\partial}{\partial t}N+X=0$ (17)

with the quasi-current $X$ will be used. In contrast to the symmetry
variables, however, even homogeneous changes $\delta N$ cost energy and lead
to a finite restoring force. As a consequence, appropriate excitations are
non-hydrodynamic $\omega(k\to 0)=\omega_{0}$ with a finite gap $\omega_{0}\neq
0$. Some aspects of the order dynamics will be discussed at the end of this
section.

In order to set up the complete dynamics of the Td phase we apply
thermodynamics locally to the relevant variables. The first law of
thermodynamics (Gibbs relation), describing energy conservation including
heat,

$d\varepsilon=\mu
d\rho+v_{i}dg_{i}+h_{i}^{\Gamma\prime}d\Gamma_{i}+\Psi^{\Gamma}_{ij}d\nabla_{j}\Gamma_{i}+Td\sigma$
(18)

relates changes of all variables to changes of the entropy density $d\sigma$.
The prefactors are the conjugate quantities, chemical potential $\mu$,
velocity $v_{i}$, molecular tetrahedral fields $h_{i}^{\Gamma\prime}$,
$\Psi^{\Gamma}_{ij}$, and temperature $T$. Like in nematic LC we have added
gradients of the symmetry variable, since without external fields
$h_{i}^{\Gamma\prime}$ has to vanish. The two contributions can be combined to
$h_{i}^{\Gamma}d\Gamma_{i}$ with

$h_{i}^{\Gamma}=h_{i}^{\Gamma\prime}-\nabla_{j}\Psi^{\Gamma}_{ij}-2\epsilon_{ikl}\Psi^{\Gamma}_{kj}\nabla_{j}\Gamma_{l}$
(19)

The last (nonlinear) contributions is due to Eq. (2.1).

Setting up a phenomenological expression for the energy density,

$\varepsilon=\varepsilon_{0}+\frac{1}{2\rho}\mbox{\boldmath$g$}^{2}+\frac{1}{2}K_{ijkl}^{\Gamma}(\nabla_{j}\Gamma_{i})(\nabla_{l}\Gamma_{k})$
(20)

the conjugate quantities follow by partial derivation,
$\mu=\partial\varepsilon/\partial\rho$, $v_{i}=\partial\varepsilon/\partial
g_{i}$, $h_{i}^{\Gamma\prime}=\partial\varepsilon/\partial\Gamma_{i}$,
$\Psi^{\Gamma}_{ij}=\partial\varepsilon/\partial\nabla_{j}\Gamma_{i}$, and
$T=\partial\varepsilon/\partial\sigma$. The first contribution to Eq. (20) is
the free energy density of an isotropic liquid expressed by well-known
susceptibilities, like compressibility, specific heat, and thermal expansion.
The second one is the kinetic energy and the last one is the gradient energy
for inhomogeneous rotations. It contains three (Frank-like) susceptibilities

$\displaystyle K_{ijkl}^{\Gamma}$
$\displaystyle=K_{1}^{\Gamma}(\delta_{ij}\delta_{kl}+\delta_{il}\delta_{jk})+K_{2}^{\Gamma}\delta_{ik}\delta_{jl}$
$\displaystyle+K_{3}^{\Gamma}T_{jlp}T_{ikp}\text{\,.}$ (21)

completing the statics of the Td phase.

The Gibbs relation allows to interchange the entropy density with the energy
density as a dynamic variable

$\frac{\partial}{\partial t}\sigma+\nabla_{i}j_{i}^{\sigma}=R/T$ (22)

making Eq. (13) redundant. The source term, the entropy production, is written
in terms of the dissipation function $R$. With the help of the dynamic Eqs.
(11) - (13), (15) and (22) the Gibbs relation (18) can be written as

$R=-\sigma_{ij}\nabla_{j}v_{i}-j_{i}^{\sigma}\nabla_{i}T+Y_{i}\,h_{i}^{\Gamma}+g_{i}\nabla_{i}\mu\geq
0$ (23)

suppressing an irrelevant surface term (div$j$).

According to the second law of thermodynamics, entropy is conserved, i.e.
$R=0$, only for reversible processes, while irreversible ones are dissipative
with $R>0$. It is possible to write any current or quasi-current as a sum of a
reversible (superscript R) and a dissipative (superscript D) part,
$\sigma_{ij}=\sigma_{ij}^{R}+\sigma_{ij}^{D}$,
$j_{i}^{\sigma}=j_{i}^{\sigma,R}+j_{i}^{\sigma,D}$, and
$Y_{i}=Y_{i}^{R}+Y_{i}^{D}$, while $g_{i}$, the momentum density is reversible
and has no irreversible part. The reversible (dissipative) parts have the same
(opposite) time reversal behavior as the time derivative of the appropriate
variable. All variables (and conjugates) have a definite time reversal
behavior, e.g., $\sigma$, $\rho$, $T$, $\mu$, $\nabla_{j}\Gamma_{i}$,
$\Psi^{\Gamma}_{ij}$, and $\varepsilon$ are invariant, while $g_{i}$ and
$v_{i}$ change sign implying that also $\sigma_{ij}^{R}$, $j_{i}^{\sigma,D}$,
and $Y_{i}^{D}$ are invariant and $\sigma_{ij}^{D}$, $j_{i}^{\sigma,R}$, and
$Y_{i}^{R}$ change sign.

The framework of linear irreversible thermodynamics is used to derive the
irreversible parts of the currents and quasi-currents. It has a solid
microscopic basis in linear response theory guaranteeing compatibility with
general physical principles. Descriptions based on linear irreversible
thermodynamics have successfully applied even to systems driven far from
equilibrium. It is based on a linear relationship between the irreversible
currents and the thermodynamic forces that drive the system out of
equilibrium. In equilibrium, $T^{(eq)}$, $\mu^{(eq)}$, and $v_{i}^{(eq)}$ are
constant (the latter is typically put to zero due to Galilean invariance),
while $h_{i}^{\Gamma(eq)}$ is zero. Thus, $\nabla_{i}T$, $\nabla_{i}\mu$,
$\nabla_{j}v_{i}$, and $h_{i}^{\Gamma}$ are candidates for thermodynamic
forces. However, $\nabla_{i}\mu$ must not enter the dissipation function,
since $g_{i}$ is reversible, and only symmetrized gradients
$A_{ij}=(1/2)(\nabla_{j}v_{i}+\nabla_{i}v_{j})$ are allowed, since solid body
rotations must not change the entropy of the system444A solid body rotation is
equivalent to a change of the point, from which a system is viewed..

Taking into account spatial inversion behavior additionally, we find the
symmetry-allowed contributions

$R=\frac{\kappa}{2}(\mbox{\boldmath$\nabla$}T)^{2}+\frac{1}{2\gamma^{\Gamma}}(\mbox{\boldmath$h$}^{\Gamma})^{2}+\frac{1}{2}\nu_{ijkl}A_{ij}A_{kl}$
(24)

with the (dissipative) transport parameters, heat conduction $T\kappa$, and
tetrahedral rotational viscosity $\gamma^{\Gamma}$, and with the viscosity
tensor $\nu_{ijkl}$. The first two terms are isotropic, while the rank-4
material tensor has a form different from the isotropic case Fel2

$\displaystyle\nu_{ijkl}$ $\displaystyle=$
$\displaystyle\eta_{1}(\delta_{jl}\delta_{ik}+\delta_{il}\delta_{jk}-\frac{2}{3}\delta_{ij}\delta_{kl})+\zeta\delta_{ij}\delta_{lk}$
(25) $\displaystyle+\eta_{2}T_{ijp}T_{klp}$

with an additional deformational viscosity, $\eta_{2}$, due to the tetrahedral
order. It leads to additional stresses in symmetric shear flows, but not in
elongational ones. Positivity of $R$ requires some positivity conditions on
the transport parameters, in particular $\gamma^{\Gamma}>0$, $\eta_{1}>0$,
$\zeta-(2/3)\eta_{1}>0$ and $\eta_{1}+\eta_{2}>0$.

The dissipative currents follow from Eq. (24) by partial derivation

$\displaystyle j_{i}^{\sigma,D}$ $\displaystyle=$
$\displaystyle-\kappa\nabla_{i}T$ (26) $\displaystyle Y_{i}^{D}$
$\displaystyle=$ $\displaystyle\frac{1}{\gamma^{\Gamma}}h_{i}^{\Gamma}$ (27)
$\displaystyle\sigma_{ij}^{D}$ $\displaystyle=$
$\displaystyle\nu_{ijkl}A_{kl}$ (28)

The reversible part of the dynamics has two origins. Either there are
phenomenological reversible contributions to the currents, or the
contributions are due to symmetry and/or other general requirements. The first
kind comes with phenomenological reversible material coefficients, similar to
the case of dissipative ones. However, in the reversible case there is no
potential from which such contributions could be derived, since $R=0$.
Instead, one looks for couplings between reversible currents and forces that
are allowed by time reversal symmetry and inversion symmetry and choose the
phenomenological parameters such that $R=0$. In isotropic liquids no such
phenomenological reversible couplings exist. On the other hand, in nematic LC
the reversible couplings between director rotations and simple shear flow
(flow alignment and back flow) are a well-known example. Its phenomenological
parameter, generally called $\lambda$, determines the director alignment angle
under shear flow. In the Td phase there is no flow alignment effect, since
there is no preferred direction that could be aligned, but there is a
(reversible) coupling between flow and the thermal degree of freedom

$\displaystyle\sigma_{ij}^{ph}$ $\displaystyle=$
$\displaystyle-\Gamma_{2}T_{ijk}\nabla_{k}T$ (29) $\displaystyle
j_{i}^{\sigma,ph}$ $\displaystyle=$ $\displaystyle\Gamma_{2}T_{ijk}A_{jk}$
(30)

In particular, a temperature gradient generates (symmetric) shear stresses,
the geometry of which depends on the orientation of the temperature gradient
with respect to the tetrahedral orientation. Assuming the temperature gradient
(taken as z-axis) is along one of the $\bar{4}$ axis, the stresses induced by
Eq. (30) are $\sigma_{xy}=\sigma_{yx}$ according to the structure Eq. (6). Due
to the viscous stress - strain rate coupling, there is a stationary planar
flow pattern perpendicular to the temperature gradient shown in Fig. 5.

Vice versa, a shear flow, say, in the x/y plane that defines the perpendicular
directions ($z$ or $-z$) produces via Eq. (29) a heat flux in a definite polar
direction ($z$ for $\Gamma_{2}<0$ or $-z$ for $\Gamma_{2}>0$). Of course, this
does nor imply that the Td phase is polar. If the tetrahedral structure is
inverted, $T_{ijk}$ changes sign and the induced currents will point in the
opposite direction, what could be viewed as induced ambi-polarity. If both
variants are present in different parts of the same sample, this ambi-polarity
shows up directly.

Figure 5: Planar symmetric shear flow induced by a temperature gradient
perpendicular to the shear plane - from BPC02 .

The reversible transport parameter $\Gamma_{2}$ is without a rigorous upper
bound in magnitude and can have either sign. It is easy to check that these
contributions cancel each other in the entropy production, Eq. (23), since
$\sigma_{ij}^{ph}A_{ij}=-j_{i}^{\sigma,ph}\nabla_{i}T$.

The non-phenomenological contributions to the reversible currents are mainly
due to transport. In the Eulerian description of hydrodynamics a variable can
change its value at a given point by advection, i.e. by transporting (with
velocity $v_{i}$) material with a different value of that variable to this
point, e.g. $(\partial/\partial t)^{(adv)}\rho=-\nabla_{i}(\rho v_{i})$. For
vectorial quantities also convection (with vorticity
$\mbox{\boldmath$\omega$}=(1/2)\textrm{curl}\,\mbox{\boldmath$v$}$) is
possible. In particular, in a linearized description, where $\delta\Gamma_{i}$
is a vector of rotation angles, one gets $(\partial/\partial
t)^{(trans)}\Gamma_{i}=-v_{j}\nabla_{j}\Gamma_{i}+\omega_{i}$.

All these transport contributions have to add up to zero in the entropy
production. This is obtained by counter terms in the stress tensor, involving
the isotropic pressure $p$ and a nonlinear stress $\sigma_{ij}^{nl}$, which
exists in similar form also in nematic LC, where it is called Ericksen stress.
For an exposition of the method and its application to tetrahedral phases cf.
mbuch ; BP-D2d . The final, nonlinear result for the total reversible currents
is

$\displaystyle j_{i}^{\sigma,R}$ $\displaystyle=$ $\displaystyle\sigma
v_{i}+j_{i}^{\sigma,ph}$ (31) $\displaystyle Y_{i}^{R}$ $\displaystyle=$
$\displaystyle
v_{j}\nabla_{j}\Gamma_{i}-\frac{1}{2}\omega_{i}+\frac{1}{2\alpha}\epsilon_{ipq}\epsilon_{mjl}T_{pjk}T_{qlk}\,\omega_{m}$
(32) $\displaystyle\sigma_{ij}^{R}$ $\displaystyle=$ $\displaystyle
v_{i}g_{j}+\delta_{ij}p+\sigma_{ij}^{nl}+\sigma_{ij}^{ph}$ (33)

with
$4\sigma_{ij}^{nl}=2\Psi^{\Gamma}_{kj}\nabla_{i}\Gamma_{k}+2\Psi^{\Gamma}_{ki}\nabla_{j}\Gamma_{k}-3\epsilon_{ijk}\nabla_{l}\Psi^{\Gamma}_{kl}$.
In Eqs. (29) - (33), as well as in the following, the superscript $ph$ refers
to the reversible parts of the currents that carry phenomenological
coefficients the value of which cannot be simply determined by invariance
arguments.

The last term in Eq. (32) demonstrates that in the nonlinear domain
$\delta\Gamma_{i}$ does not behave like an ordinary vector under finite
rotations. There is no phenomenological reversible coupling to symmetrized
shear flow, with the effect that the tetrahedral structure cannot be oriented
in simple shear flow. In the dynamic momentum Eq. (12), the pressure term
appears as $\nabla_{i}p$, which is given by the Gibbs-Duhem equation

$\nabla_{i}p=\sigma\nabla_{i}T+\rho\nabla_{i}\mu+g_{j}\nabla_{i}v_{j}-h_{j}^{\Gamma}\nabla_{i}\Gamma_{j}$
(34)

where $h_{i}^{\Gamma}$ is defined in Eq. (19) Note that the stress tensor is
either symmetric or the divergence of an antisymmetric tensor, which is the
requirement of angular momentum conservation MPP . This form is obtained by
applying a condition on the conjugate quantities that follows from the
rotational invariance of the energy density $\varepsilon$.

Very often LC are mixtures of several different components. For a binary
mixture, whose components are individually conserved, there are two mass
conservation laws for $\rho_{1}$ and $\rho_{2}$. They can be replaced by the
total mass conservation Eq. (11) and a dynamic equation for the concentration
$c=\rho_{1}/\rho$

$\frac{\partial}{\partial t}c+v_{i}\nabla_{i}c+\frac{1}{\rho}\,j_{i}^{c}=0$
(35)

whose conjugate quantity, the osmotic pressure, $\Pi=\rho\mu_{c}$, is related
to the difference of the chemical potentials $\mu_{c}=\mu_{1}-\mu_{2}$. It
follows from an appropriately extended $\varepsilon_{0}$ in Eq. (20). The
concentration variable is rather similar to the temperature variable, and
$j_{i}^{c}$ has the same structure as $j_{i}^{\sigma}$. In particular, the
dissipative part contains diffusion and thermodiffusion,

$j_{i}^{c,D}=-D\nabla_{i}\mu_{c}-D_{T}\nabla_{i}T$ (36)

where the latter also occurs in Eq. (26) as $-D_{T}\nabla_{i}\mu_{c}$. There
is also the tetrahedral-specific reversible coupling to flow,
$j_{i}^{c,ph}=\Gamma_{3}T_{ijk}A_{jk}$, with the counter term
$-\Gamma_{3}T_{ijk}\nabla_{k}\mu_{c}$ in the stress tensor, Eq. (29).

###  2.2 Orienting, flow- and order-inducing external fields

In nematic LC it is well known that an external static electric fields orients
the director due to the dielectric anisotropy effect. In equilibrium the
director is either along the field or perpendicular to it. Therefore,
rotations away from the field direction in the former case, or out of the
perpendicular plane in the latter case, are no longer Goldstone modes, but
lead to a relaxation towards the equilibrium orientation. Since this is
typically a weak coupling, it is customary to keep director rotations as
variables to describe the macroscopic dynamics, and taking the symmetry
unchanged as D∞h. In the same spirit we will treat the tetrahedral phase in
external fields in this section.

The electrostatic degree of freedom is described by the electric field $E_{i}$
and the displacement field $D_{i}$. In the Gibbs relation (18) this adds the
electric energy change $d\varepsilon^{E}=E_{i}dD_{i}$. To make contact with
the familiar description in nematic LC, we switch to the Legendre transformed
energy $\tilde{\varepsilon}^{E}\equiv\varepsilon^{E}-E_{i}D_{i}$ giving rise
to $d\tilde{\varepsilon}^{E}=-D_{i}dE_{i}$. The variable $D_{i}$ is thereby
given by the thermodynamic conjugate (and generalized force) $E_{i}$ via
$D_{i}=-(\partial\tilde{\varepsilon}^{E}/\partial E_{i})$. The tetrahedral
symmetry allows for a cubic electric field energy Fel

$\tilde{\varepsilon}^{E}=-\epsilon_{1}T_{ijk}E_{i}E_{j}E_{k}$ (37)

that orients the tetrahedral structure such that one of the tetrahedral
vectors, say $n^{1}_{i}$, is parallel or antiparallel to the field $E_{i}$
depending on the sign of the susceptibility $\epsilon_{1}$. Switching from the
tetrahedral structure to its inverse ($T_{ijk}\to-T_{ijk}$ or $N\to-N$) the
role of $\mathrm{sign}(\epsilon_{1})$ is reversed.

In principle, other vectorial external fields, like temperature or
concentration gradients can have a similar orienting effect via an energy like
$\varepsilon^{T}\sim T_{ijk}(\nabla_{i}T)(\nabla_{j}T)(\nabla_{k}T)$.
Boundaries with a polar surface normal act in the same way orienting one
tetrahedral vector perpendicular to the surface. A magnetic field $H_{i}$
cannot orient the tetrahedral structure due to the odd time reversal behavior
of magnetic fields, but there is an additional orienting effect, when both,
electric and magnetic fields are present, since the energy
$\tilde{\varepsilon}^{EH}=-(\epsilon_{2}/3)T_{ijk}(E_{i}H_{j}H_{k}+E_{j}H_{i}H_{k}+E_{k}H_{i}H_{j})$
is possible BPC02 . The electric field energies lead to quadratic
contributions to the displacement field that come in addition to the linear,
isotropic one

$D_{i}=\chi
E_{i}+\epsilon_{1}T_{ijk}E_{j}E_{k}+\epsilon_{2}T_{ijk}H_{j}H_{k}.$ (38)

For the relaxation of the tetrahedral structure in an external field, the
orienting energy Eq. (37) provides the driving force. Since it defines the
equilibrium orientation, it is zero for linear deviations, since
$E_{i}E_{j}E_{k}\delta T_{ijk}=0$ (with $\delta
T_{ijk}=2\,\epsilon_{mpi}T_{pjk}^{eq}\,\delta\Gamma_{m}$), while for quadratic
ones one gets

$\tilde{\varepsilon}^{E}_{2}=\epsilon_{1}E_{i}E_{j}E_{k}\,\delta^{(2)}T_{ijk}=\frac{32}{9}|\epsilon_{1}N|E_{0}^{3}(\delta\mbox{\boldmath$\Gamma$}_{\perp})^{2}$
(39)

with $\delta^{(2)}T_{ijk}\equiv 2\,\epsilon_{mpi}\,\delta
T_{pjk}\,\delta\Gamma_{m}$. It describes the energy related to rotations of
the tetrahedral structure perpendicular to an external field of strength
${E}_{0}$. As a result there is a non-vanishing restoring force, even in the
homogeneous case,
$h_{i}^{\Gamma}=h_{i}^{\Gamma\prime}=\partial\tilde{\varepsilon}^{E}/\partial\delta\Gamma_{i}$
and transverse rotations relax with the inverse relaxation time

$\lambda^{rel}=16\,\frac{|\epsilon_{1}N|}{\gamma^{\Gamma}}\,E_{0}^{3}$ (40)

according to the dissipative dynamics, Eq. (27). The relaxation is cubic in
the field strength, in contrast to the nematic case, where it is quadratic.

There is also the analog of flexoelectricity

$\tilde{\varepsilon}^{flex}_{2}=e_{1}E_{j}E_{k}\nabla_{i}T_{ijk}=\frac{16}{9}e_{1}NE_{0}^{2}\,(\textrm{curl}\mbox{\boldmath$\Gamma$})_{z}$
(41)

involving $\nabla_{x}\Gamma_{y}-\nabla_{y}\Gamma_{x}$ when the field is in
z-direction. Here, the effect is quadratic in the field amplitude, while in
nematic LC it is linear.

The existence of $T_{ijk}$ also allows for piezoelectricity

$\tilde{\varepsilon}^{piezo}_{2}=d_{1}T_{ijk}E_{i}u_{jk}$ (42)

in solid systems, where a strain tensor $u_{ij}$ describes elasticity.

The dynamics is also affected by an external field, in particular if there are
electric charges, $\rho_{e}$, as is often the case for LC. They are related to
the displacement field by $\nabla_{i}D_{i}=\rho_{e}$ and the dynamic balance
equation for $D_{i}$ is the charge conservation law

$\frac{\partial}{\partial t}\rho_{e}+\nabla_{i}j_{i}^{e}=0$ (43)

The reversible part of the electric current BPC02

$j_{i}^{e,R}=v_{i}\nabla_{i}\rho_{e}+\Gamma_{1}T_{ijk}A_{jk}$ (44)

contains a phenomenological coupling to flow, similar to the thermal and
solutal currents in Eq. (30) and after (35). It is balanced to give zero
entropy production by a contribution to the stress tensor

$\sigma_{ij}^{e,ph}=-\Gamma_{1}T_{ijk}E_{k}$ (45)

in analogy to Eq. (29). If the electric fields (along the z-axis) orients the
tetrahedral structure in the way described above, the stresses induced by Eq.
(45) are $\sigma_{xx}=\sigma_{yy}=-(1/2)\sigma_{zz}$ giving rise (via viscous
coupling) to 3-dimensional elongational flow, called uniaxial (biaxial) for
$\Gamma_{1}>0$ ($\Gamma_{1}<0$), cf. Fig. 6. Note that a reversal of the field
direction also changes the velocity directions and interchanges uniaxial with
biaxial elongational flow.

Figure 6: Uniaxial 3-dimensional elongational flow in z-direction for
$\Gamma_{1}>0$; for $\Gamma_{1}<0$ the flow directions are reversed (biaxial
elongational flow in the x/y plane) - from OMueller .

The dissipative part of the electric current BPC02

$\displaystyle j_{i}^{e,D}$ $\displaystyle=$
$\displaystyle\sigma^{E}E_{i}+D^{E}\nabla_{i}\mu_{c}+\kappa^{E}\nabla_{i}T$
(46)

describes the coupled diffusions among the electric, thermal and concentration
degrees of freedom (e.g. Ohm’s law, Peltier effect, etc.) with appropriate
counter terms in the heat and concentration current, Eq. (26) and (36). Here,
the material parameters relating currents with forces are rank-2 material
tensors that, however, are isotropic, since tetrahedral order cannot occur in
such tensors. The dissipative transport parameters have to fulfill certain
positivity relations (e.g. $\sigma^{E}>0$ or
$\kappa\sigma^{E}>[\kappa^{E}]^{2}$) to guarantee $R>0$.

In an isotropic liquid an external electric field can induce nematic order
(Kerr effect PGdG ), which also leads to a shift of the thermodynamic
isotropic to nematic phase transition. The effect is based on the dielectric
anisotropy of the molecules and is quadratic in the field amplitude. In the Td
phase tetrahedral order provides an additional mechanism for inducing nematic
order, as well as for shifting the nematic phase transition. However, both the
strength of induced nematic order and the transition shift are linear in the
field strength. This is due to the interaction energy N

$\tilde{\varepsilon}_{EQT}=\Gamma\,E_{i}Q_{jk}T_{ijk}$ (47)

which is linear in the field amplitude. This energy exists, since with Eq. (7)
$E_{k}T_{ijk}$ has the structure of uniaxial nematic order
$(\delta_{iz}\delta_{jz}-(1/3)\delta_{ij})\sim Q_{ij}$. The coefficient
$\Gamma$ is a true scalar quantity that can have either sign.

Assuming in the Td phase small unstable nematic fluctuations with energy pgdg2

$\varepsilon_{Q}=(A/2)Q_{ij}Q_{ij}$ (48)

with $A>0$, an electric field can stabilize them due to the coupling Eq. (47).
Minimizing $\tilde{\varepsilon}_{EQT}+\varepsilon_{Q}$ the stable induced
nematic order is linear in the field strength

$Q_{ij}^{ind}=\frac{\Gamma}{A}\,E_{k}T_{ijk}$ (49)

and of prolate or oblate geometry depending on the sign of $\Gamma$, Fig. 7.

Figure 7: Electric field induced nematic order, prolate (oblate) for
$\Gamma>0$ ($<0$) - from N .

The onset of the nematic phase transition is also shifted due to the
interaction energy, Eq. (47). The Landau expansion for the onset of nematic
order reads

$\tilde{\varepsilon}=\frac{4}{9}\Gamma
E_{0}NS-\frac{\epsilon_{a}}{6}E_{0}^{2}S+\frac{\alpha}{2}S^{2}+\frac{\beta}{3}S^{3}+\frac{\gamma}{4}S^{4}$
(50)

containing the linear (tetrahedral) and the quadratic (isotropic) electric
contributions in addition to the field-free expansion of the nematic order
parameter $S$; $N_{0}$ is the constant tetrahedral order parameter. We note
that the prefactors of $S^{2}$, $S^{3}$ and $S^{4}$ in Eq. (50) are different
from the prefactors one obtains when starting with $Q_{ij}$ instead of $S$.

Without the external field the first order transition takes place at
$\alpha=\alpha_{c}\equiv(2\beta^{2}/9\gamma)$, where a finite
$S_{c}=-(2\beta)/(3\gamma)$ occurs. With the standard form
$\alpha=a(T-T^{*})$, where $T^{*}$ is the fictitious second order phase
transition temperature, the transition temperature is
$T_{c}=T^{*}+\alpha_{c}/a$. Assuming the changes due to the field to be small,
the transition temperature and the order parameter jump are shifted

$\displaystyle T_{c}(E_{0})$ $\displaystyle=$ $\displaystyle
T_{c}+3\frac{\gamma}{\beta}\frac{L_{E}}{a}$ (51) $\displaystyle S_{c}(E_{0})$
$\displaystyle=$ $\displaystyle S_{c}+\frac{9}{2}\frac{\gamma}{\beta}L_{E}$
(52)

with $L_{E}=(4/3)\Gamma NE_{0}-(1/6)\epsilon_{a}E_{0}^{2}$ demonstrating the
linear field shift in the tetrahedral phase, in addition to the quadratic,
isotropic one.

###  2.3 Strong, structure-changing external fields

In this section we will deal with strong external electric fields that are
able to distort the tetrahedral structure of the Td phase. This might be
possible, when the tetrahedral order is weak, in particular close to the phase
transition or in the vicinity of defects.

An external electric field not only provides the orientation of the (rigid)
tetrahedral structure as described in the preceding section, it also invokes a
torque on the individual tetrahedral vectors. Taking for definiteness
$\mbox{\boldmath$E$}=E_{0}\mbox{\boldmath$n$}^{1}$, with $E_{0}\geq 0$,
$\epsilon_{1}>0$, and $N>0$ the electric free energy Eq. (37)

$\tilde{\varepsilon}^{E}=-{\epsilon_{1}}N\sum^{4}_{\beta=1}(\mbox{\boldmath$E\cdot
n$}^{\beta})^{3}$ (53)

leads to the non-vanishing torques

$\displaystyle\mbox{\boldmath$n$}^{2}\times\frac{\partial\tilde{\varepsilon}^{E}}{\partial\mbox{\boldmath$n$}^{2}}$
$\displaystyle\sim$ $\displaystyle(0,1,1)$ (54)
$\displaystyle\mbox{\boldmath$n$}^{3}\times\frac{\partial\tilde{\varepsilon}^{E}}{\partial\mbox{\boldmath$n$}^{3}}$
$\displaystyle\sim$ $\displaystyle(-1,0,-1)$ (55)
$\displaystyle\mbox{\boldmath$n$}^{4}\times\frac{\partial\tilde{\varepsilon}^{E}}{\partial\mbox{\boldmath$n$}^{4}}$
$\displaystyle\sim$ $\displaystyle(1,-1,0).$ (56)

where we have used the orientation (6) for the tetrahedral vectors. These
torques are perpendicular to $\mbox{\boldmath$n$}^{1}$ and tend to rotate
$\mbox{\boldmath$n$}^{2,3,4}$. Assuming such a rotation with a (yet
undetermined) finite amplitude $b$, the rotated unit vectors are given by

$\displaystyle\mbox{\boldmath$n$}^{2E}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\frac{1}{\sqrt{1+2b^{2}}}(1,-1+\sqrt{3}b,1+\sqrt{3}b)$
(57) $\displaystyle\mbox{\boldmath$n$}^{3E}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\frac{1}{\sqrt{1+2b^{2}}}(-1-\sqrt{3}b,1,1-\sqrt{3}b)$
(58) $\displaystyle\mbox{\boldmath$n$}^{4E}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\frac{1}{\sqrt{1+2b^{2}}}(-1+\sqrt{3}b,-1-\sqrt{3}b,-1).$
(59)

while
$\mbox{\boldmath$n$}^{1E}=\frac{1}{\sqrt{3}}(1,1,-1)=\mbox{\boldmath$n$}^{1}$
is undistorted.

Figure 8: The angle $\phi$ between $\mbox{\boldmath$n$}^{1}$ and
$\mbox{\boldmath$n$}^{\bar{\beta}E}$ as a function of the deformation
amplitude $b$. In the asymptotic limit $b\to\infty$ the latter are
perpendicular to the field direction $n^{1}$ \- from L .

These individual rotations distort the tetrahedral structure as is manifest in
the relative angles, whose cosines are given by

$\mbox{\boldmath$n$}^{1E}\mbox{\boldmath$\cdot
n$}^{\bar{\beta}E}=-\frac{1}{3}\frac{1}{\sqrt{1+2b^{2}}}$ (60)

for $\bar{\beta}=2,3,4$ and by

$\mbox{\boldmath$n$}^{2E}\mbox{\boldmath$\cdot
n$}^{3E}=\mbox{\boldmath$n$}^{2E}\mbox{\boldmath$\cdot
n$}^{4E}=\mbox{\boldmath$n$}^{3E}\mbox{\boldmath$\cdot
n$}^{4E}=-\frac{1}{3}\,\frac{1+3b^{2}}{1+2b^{2}}$ (61)

Equation (60) describes the rotation towards the field direction, where the
tetrahedral angles $\phi(b=0)\equiv\phi_{T}\approx 109.5^{o}$ (or
$\cos\phi_{T}=-1/3$) approach $\phi=90^{o}$ in the asymptotic limit
$b\to\infty$ (Fig. 8). In parallel, the angles among the vectors
$\mbox{\boldmath$n$}^{\bar{\beta}E}$ change from $\phi=\phi_{T}$ in the
tetrahedral case, $b=0$, towards $\phi=120^{o}$ for $b\to\infty$, resulting in
a regular pyramidal structure. For finite $b$ the distorted structure is of
$C_{3v}$ symmetry with only one 3-fold symmetry axis left,
$\mbox{\boldmath$n$}^{1E}$, and three equivalent vertical symmetry planes
containing this preferred axis and any of
$\mbox{\boldmath$n$}^{\bar{\beta}E}$.

Figure 9: a) The deformed tetrahedron viewed along the field direction for
$b\in[0,\infty\\}$ (200 values shown as $\Box$). Not only grow the angles
among the $\mbox{\boldmath$n$}^{\bar{\beta}E}$ from $\phi_{T}$ to $120^{o}$
and shrink the angles between $\mbox{\boldmath$n$}^{1E}$ and any
$\mbox{\boldmath$n$}^{\bar{\beta}E}$ from $\phi_{T}$ to $90^{o}$, there is
also an overall rotation of the structure about $\mbox{\boldmath$n$}^{1E}$ of
$90^{o}$ maximum. b) The transition from the tetrahedron to the pyramid and
the accompanied rotation seen from the side - from L .

In addition, there is an overall rotation of all
$\mbox{\boldmath$n$}^{\bar{\beta}E}$ about the field direction. This is
obvious from

$\mbox{\boldmath$n$}^{2E}\mbox{\boldmath$\cdot
n$}^{2}=\mbox{\boldmath$n$}^{3E}\mbox{\boldmath$\cdot
n$}^{3}=\mbox{\boldmath$n$}^{4E}\mbox{\boldmath$\cdot
n$}^{4}=\frac{1}{\sqrt{1+2b^{2}}}$ (62)

demonstrating that for $b\to\infty$ all three
$\mbox{\boldmath$n$}^{\bar{\beta}E}$ have been rotated by $90^{o}$, Fig. 9.
The rotation sense depends on the sign of $b$, but is irrelevant. What looks
like a clockwise rotation when viewed from above, is a counter-clockwise one
when viewed from below. The rotation sense is also changed, when the
tetrahedral structure is replaced by its inverse. The equivalence of $b$ with
$-b$ is also manifest in the energetics discussed below, where only $b^{2}$
occurs.

If the system follows the torques provided by the external field, its electric
free energy is certainly lowered. Indeed, the free energy of the distortion,
$\Delta\tilde{\varepsilon}^{E}\equiv\tilde{\varepsilon}^{E}(\mbox{\boldmath$n$}^{\beta
E})-\tilde{\varepsilon}^{E}(\mbox{\boldmath$n$}^{\beta})$, is found using Eq.
(53) to be

$\Delta\tilde{\varepsilon}^{E}=-\frac{1}{18}\epsilon_{1}E_{0}^{3}\left(1-(1+2b^{2})^{-3/2}\right)\leq
0.$ (63)

It vanishes by definition for $b=0$ and is a monotonically decreasing function
with increasing $b^{2}$.

On the other hand, the thermodynamic ground state is that of the undistorted
tetrahedral structure. Therefore, any deviation from the tetrahedral angle
$\phi_{T}$ increases the energy, which can be written phenomenologically in a
kind of harmonic approximation as

$\displaystyle\varepsilon_{def}$ $\displaystyle=$
$\displaystyle\frac{B_{1}}{2}\sum_{\gamma,\beta>\gamma}\Bigl{(}\mbox{\boldmath$n$}^{\gamma
E}\mbox{\boldmath$\cdot n$}^{\beta E}+\frac{1}{3}\Bigr{)}^{2}$ (64)
$\displaystyle=$
$\displaystyle\frac{B_{1}}{6}\left[\Bigl{(}1-\frac{1}{\sqrt{1+2b^{2}}}\Bigr{)}^{2}+\Bigl{(}\frac{b^{2}}{1+2b^{2}}\Bigr{)}^{2}\right]\geq
0.$ (65)

It is zero for $b=0$ and positive for finite $b$, increasing monotonically
with increasing $b^{2}$.

Figure 10: The total deformation energy $\varepsilon_{def}^{tot}$ scaled by
$\epsilon_{1}E_{0}^{3}$ and by $V_{0}/V_{E}$, cf. Eq. (67), as a function of
$b_{0}$ (with a vertical offset of $-4/9$) - from L .

The equilibrium value of $b^{2}=b_{0}^{2}$ is given by the minimum of the sum
of the two energies related to tetrahedral distortions due to external
electric fields,
$\varepsilon_{def}^{tot}=\Delta\tilde{\varepsilon}^{E}+\varepsilon_{def}$.
Minimization leads to the condition

$\sqrt{1+2b_{0}^{2}}\,\bigl{(}2+4b_{0}^{2}-A\bigr{)}=2+2b_{0}^{2}$ (66)

with $A=\epsilon_{1}E_{0}^{3}/B_{1}$. Generally, in the physically relevant
limit when the external field only slightly distorts the Td phase, $A$ is
small. In that case555This relation also applies to the opposite limit, when
$A$ is large., $b_{0}^{2}=A/4$, and the total distortion energy
$\varepsilon_{def}^{tot}=-(3/16)A^{2}$ is negative. In Fig. 10 it is shown
that $\varepsilon_{def}^{tot}$ is negative for all values of $b_{0}^{2}$. This
means there is no threshold for the deformations due to an external field and
even a very small field leads to a non-zero deformation. However, the
deformation amplitude $b_{0}$ will be very small and hardly measurable, in
particular for large $B_{1}$ . Of course, in the rigid limit,
$B_{1}\to\infty$, there is $b_{0}^{2}\to 0$ and $\varepsilon_{def}^{tot}\to
0$. This scenario is rather robust, e.g. the small amplitude behavior is the
same, when $\varepsilon_{def}$ is replaced by the more complicated deformation
energy $T_{ijk}^{E}T_{ijk}^{E}-T_{ijk}T_{ijk}$, where
$T_{ijk}^{E}=N\sum_{\beta=1}^{4}n_{i}^{\beta E}n_{j}^{\beta E}n_{k}^{\beta
E}$.

Finally we remark that the deformations induced by the electric torques also
change the volume of the structure given by the four vertices
$\mbox{\boldmath$n$}^{\beta E}$

$\frac{V_{E}}{V_{0}}=\frac{1}{16}\frac{4+9b_{0}^{2}}{(1+2b_{0}^{2})^{3/2}}\,\,\left(1+3\sqrt{1+2b_{0}^{2}}\right)$
(67)

with $V_{0}=8/(9\sqrt{3})$ the volume of the undistorted tetrahedron. In the
limit $b\to\infty$ the volume of the pyramid is $V_{\infty}=\sqrt{3}/4$, which
is by 15,625% smaller than without distortion. Assuming that the volume of the
structure is related to the volume of the molecules involved, it means that
the electric torques lead to an increase of the density of the system.

For a tetrahedral phase that is deformed by a strong external field rotations
of aligned tetrahedral vector $\mbox{\boldmath$n$}^{1}$ are clamped, e.g. they
relax on a time scale faster than the other hydrodynamic variables. Thus,
compared to an isotropic fluid the only additional degree of freedom is the
rotation of the (deformed) tetrahedron about the field. On the other hand,
such a phase is polar and uniaxial, and the material properties become
anisotropic due to the external field. In Sec. 3.3.1 we will give a more
detailed account of the full hydrodynamics of a thermodynamic phase with
$C_{3v}$ symmetry, where the preferred direction exists spontaneously and its
rotations are hydrodynamic excitations.

###  2.4 The chiral T phase

The Td phase considered so far (made up of achiral molecules) is achiral.
However, using chiral molecules one can get its chiral analog, the T phase Fel
. This is similar to conventional nematic LC that become cholesteric (chiral
nematic), when the molecules are replaced by chiral ones or chiral molecules
are added. Bent-core molecules can be chiralized in different ways. To get the
T phase, the simplest way is to assume that the two tails of such molecules
are symmetrically chiralized, Fig. 11 (left). If such a molecules is mirrored
at a plane or inverted, the chirality is changed, Fig. 11 (right), and the two
forms cannot be brought into coincidence by mere rotations.

Figure 11: Model of a symmetrically chiralized bent-core molecule (left) with
its mirror image (right); filled (orange) circles mean e.g. positive
chirality, while the (orange) stars indicate negative chirality - from PB14 .

To get a chiral phase, one has to employ a specific model, where two bent-core
molecules of the same chirality are combined in a steric arrangement
resembling the tetrahedral vectors 1-4 and 2-3 in Fig. 12, What has been a
$\bar{4}$ axis in the Td phase is here reduced to a (proper) 2-fold symmetry
axis, and the planes spanned by vectors 1/4 and 2/3 are no longer mirror
planes, Fig. 13,

Figure 12: Model of two bent-core molecules (1-4 and 2-3) with the same
chirality, arranged to fit into the tetrahedral geometry (left) with its
mirror image (right); not only is the tetrahedral geometry inverted, but also
the molecular chirality has changed - from PB14 . Figure 13: Projection of the
tetrahedral structure of Fig. 12 onto the $x/y$ plane. The $z$ axis is reduced
to a 2-fold axis, since a $\pi/4$ rotation with an additional inversion
preserves the structure, but changes the chirality. For the same reason, the
planes spanned by vectors 1/4 or 2/3 are no mirror planes - from PB14 .

with the result that only three 2-fold and four 3-fold symmetry axes exist.
The former are the $x,y,z$ directions, while the latter are the tetrahedral
axes 1-4, which are equivalent since they have the same chirality. Such an
arrangement of bent-core molecules ensures the compensation of the molecular
polarity and results in the T phase being non-polar.

The hydrodynamics of the achiral Td phase has been given above in Sec. 2.1. We
therefore concentrate on the differences between the hydrodynamics of the T
compared to the Td phase. In both phases the same set of hydrodynamic
equations is used. Differences occur in the static and dynamic couplings due
to the chirality of the T phase, which is manifest by the existence of a
pseudoscalar $q_{0}$. The rotational elastic gradient free energy (cf. Eq.
(20)

$\varepsilon_{g}=\frac{1}{2}K_{ijkl}^{\Gamma}(\nabla_{j}\Gamma_{i})(\nabla_{l}\Gamma_{k})+q_{0}K_{1}^{lin}\nabla_{i}\Gamma_{i}$
(68)

contains a chiral term $\sim q_{0}K_{1}^{lin}$ linear in the gradients of the
rotations $\delta\Gamma_{i}$. Generally, a linear gradient term favors a
spatially inhomogeneous structure. In the present case, a helical rotation of
$T_{ijk}$ about any of the 3-fold axes (the tetragonal vectors) reduces the
free energy by
$\Delta\varepsilon=-\tfrac{1}{2}(q_{0}K_{1}^{lin})^{2}/(2K_{1}^{\Gamma}+K_{2}^{\Gamma})$.
What looks like a linear splay term is physically a linear twist contribution,
quite similar to the familiar case of cholesteric LC. The optimum helical
pitch, $q_{h}=\frac{3}{2}q_{0}K_{1}^{lin}/(2K_{1}^{\Gamma}+K_{2}^{\Gamma})$,
is generally different from the chiral pseudoscalar of the phase, $q_{0}$,
since there is no a priori reason that $K_{1}^{lin}$ is related to
$K_{1,2}^{\Gamma}$. An analogous statement holds for ordinary cholesteric LC
oswaldComm . Helical rotations about the 2-fold axes do not lower the free
energy, since the linear gradient term is zero in that case and the quadratic
term, $\sim K_{3}^{\Gamma}$, increases the free energy.

The similarity to the cholesteric phase also holds for chiral Lehmann-type
contributions, both static (in the free energy),
$\varepsilon_{c}=q_{0}(\xi^{\rho}\delta\rho+\xi^{\sigma}\delta\sigma+\xi^{c}\delta
c)\nabla_{i}\Gamma_{i}$, and dynamic (in the dissipation function),
$R=q_{0}h_{k}^{\Gamma}(\Psi^{E}E_{k}+\Psi^{T}\nabla_{k}T+\Psi^{c}\nabla_{k}c)$.
They relate in the dissipative currents the scalar degrees of freedom
(temperature, concentration, density etc.) with the rotations of the
tetrahedron.

The reversible part of the current of tetrahedral rotations, $Y_{i}^{R}$, Eq.
(32), contains additionally a chiral coupling to the rate of strain tensor

$\displaystyle Y_{i}^{R}=$ $\displaystyle q_{0}\lambda T_{ijk}A_{jk}$ (69)
$\displaystyle\sigma_{ij}=$ $\displaystyle q_{0}\lambda T_{ijk}h_{k}^{\Gamma}$
(70)

with the appropriate counter term in the stress tensor.

Now that there are couplings to both, rotational and symmetric shear flows, a
stationary alignment of $T_{ijk}$ in simple shear is possible that is
independent of the flow rate. This scenario is very much like the flow
alignment in nematics, although there it is an achiral reversible effect. In
particular, if one of the 3-fold tetrahedral axis is in the vorticity
direction, the tetrahedron is rotated about this direction by an angle
$\theta$, Fig. 14,

$\frac{1}{\cos 2\theta}=\frac{16}{27}\,q_{0}\lambda$ (71)

Figure 14: Projection of the tetrahedral structure onto the $x/y$ plane with
one of the tetrahedral axes ($\mbox{\boldmath$n$}^{1}$) along the z-axis
(corners 2,3,4 lie below the x/y plane). This is also the vorticity direction
of the simple shear $\nabla_{y}v_{x}=S$. The structure is rotated in the shear
plane by an angle $\theta$ that is independent of the shear rate $S$ \- from
PB14 .

that only depends on the material parameters $q_{0}\lambda$. Only 3-axes, but
no 2-axes, can be aligned. In the Td phase there is no flow alignment, since
there is no $q_{0}$. The remainder of the dynamics of the T phase is as in the
achiral Td phase.

##  3 Phases with Correlated Tetrahedral and Nematic Order

###  3.1 The non-polar tetrahedral uniaxial nematic D2d phase

####  3.1.1 Statics

When both, tetrahedral and nematic order are present, we have shown in Sec.
1.4 that a possible ground state is the D2d phase. The nematic director $n$ is
fixed to be along one of the $\bar{4}$ axes of the tetrahedron (the z axis in
Fig. 15). The angle between the director and the tetrahedral axes is half the
tetrahedral angle $\phi_{D}=\phi_{T}/2$ or $\phi_{D}=\pi-\phi_{T}/2$ with
$\mbox{\boldmath$n\cdot n$}^{\beta}=\pm 1/\sqrt{3}$. The tetrahedral 3-fold
axes ($\beta=1,2,3,4$ in Fig. 15) are not symmetry axes any longer, but the
z-axis is still a $\bar{4}$ symmetry axis, while the x- and y-axis are reduced
to be 2-fold.. The planes defined by vectors 1/4 and 2/3 are symmetry planes
prohibiting chirality of the D2d phase. This phase is non-polar, because of
the $n$ to $-\mbox{\boldmath$n$}$ equivalence and the absence of a polar
vector $T_{ijk}Q_{jk}=0=T_{ijk}n_{j}n_{k}$. It can be viewed as a uniaxial
nematic LC with a transverse structure that resembles orthorhombic biaxial
nematics, but without inversion symmetry.

Figure 15: The director $n$ along the $\bar{4}$ symmetry axis in the D2d
phase. Vectors 1,2,3,4 denote the four tetrahedral vectors
$\mbox{\boldmath$n$}^{\beta}$ with $\beta=1,2,3,4$ \- from PB14 .

Since the director and the tetrahedron are rigidly coupled, the number of
hydrodynamic variables is the same as in the Td phase. However, instead of
using the three tetrahedral rotations $\delta\mbox{\boldmath$\Gamma$}$, Eq.
(14), we will split them up into (two) rotations of the director
$\delta\mbox{\boldmath$n$}=(\mbox{\boldmath$n$}\times\delta\mbox{\boldmath$\Gamma$})$
and one rotation about the director

$\delta\Omega\,\equiv\,\mbox{\boldmath$n\cdot$}\delta\mbox{\boldmath$\Gamma$}=(1/4\alpha)\,n_{i}\epsilon_{ipq}T_{pjk}\,\delta
T_{qjk}.$ (72)

By construction $\delta\Omega$ is even under parity and time reversal and odd
in $n$, but is not a true scalar (concerning its behavior under rotations -
see below). There is no direct way of detecting this degree of freedom
optically. Only through its (static) coupling to the director rotations (see
below), it might be accessible to experiments.

The hydrodynamic description in terms of $\delta\mbox{\boldmath$n$}$ and
$\delta\Omega$ facilitates the comparison with ordinary nematic LC and is
closer to experiments, where the director and its rotations are accessible by
optical means. We will assume the nematic as well as the tetrahedral order
parameter strength, $S$ and $N$, respectively, to have relaxed on a very short
time scale to their constant equilibrium values, $S_{0}$ and $N_{0}$, which we
will take as unity. We also restrict ourselves to a linear hydrodynamic
description, in particular we disregard the consequences of the nonlinear non-
commutativity relation Eq. (2.1). A recipe how to deal with them and some
results can be found in Sec. 4 of BP-D2d . There, we also discuss other
general aspects of nonlinear hydrodynamics, including the dependence of
material parameters on state variables and material tensors on orientations of
the structure, as well as the occurrence of non-harmonic thermodynamic
potentials, transport derivatives, and nonlinear pressure and stresses.

In this setting the Gibbs relation Eq. (18) takes the form

$\displaystyle d\varepsilon$ $\displaystyle=$ $\displaystyle\mu
d\rho+v_{i}dg_{i}+h_{i}^{\prime}dn_{i}+\Psi_{ij}d\nabla_{j}n_{i}$ (73)
$\displaystyle+h^{\Omega\prime}d\Omega+\Psi_{i}^{\Omega}d\nabla_{j}\Omega+Td\sigma$

with the conjugates $h^{\Omega}=h^{\Omega\prime}-\nabla_{i}\Psi_{i}^{\Omega}$
and $h_{i}=h_{i}^{\prime}-\nabla_{j}\Psi_{ij}$ omitting the nonlinearities
shown in Eq. (19).

The gradient free energy reads

$\displaystyle\varepsilon_{g}$ $\displaystyle=$
$\displaystyle\frac{1}{2}K_{ikjl}(\nabla_{i}n_{k})(\nabla_{j}n_{l})+K_{7}\delta_{ik}^{\perp}\epsilon_{lij}(\nabla_{l}\Omega)(\nabla_{j}n_{k})$
(74)
$\displaystyle+(K_{5}n_{i}n_{j}+K_{6}\delta_{ij}^{\perp})(\nabla_{i}\Omega)(\nabla_{j}\Omega)$

with the transverse Kronecker $\delta_{ij}^{\perp}=\delta_{ij}-n_{i}n_{j}$
projecting onto the plane perpendicular to $n_{i}$. There are four Frank-type
orientational elastic coefficients related to distortions of the director

$\displaystyle K_{ijkl}$
$\displaystyle=K_{3}n_{i}n_{j}\delta_{kl}^{\perp}+(K_{1}-2K_{2})\delta_{ik}^{\perp}\delta_{jl}^{\perp}$
$\displaystyle+K_{2}(\delta_{il}^{\perp}\delta_{jk}^{\perp}+\delta_{ij}^{\perp}\delta_{kl}^{\perp})+K_{4}n_{p}n_{q}T_{ijp}T_{klq}$
(75)

which is one more than in the uniaxial nematic case: The $K_{4}$ term is
related to $(\nabla_{x}n_{x})(\nabla_{y}n_{y})$, and vanishes in the
transversely isotropic case. In addition there are two coefficients related to
distortions of $\Omega$ ($K_{5,6}$) and a mixed one ($K_{7}$). The latter
links inhomogeneous $\Omega$ rotations, $\nabla_{z}\Omega$, to director twist,
$\nabla_{x}n_{y}-\nabla_{y}n_{x}$. Assuming that the tetrahedral structure is
clamped at solid surfaces with $n$ homeotropic, a circular Couette cell with a
fixed plate at the bottom and a rotating one at the top will create a finite
$\nabla_{z}\Omega$. By the $K_{7}$ coupling twist of the director is induced
in the x-y-plane.

The total number of seven Frank coefficients corresponds to the number of such
coefficients in orthorhombic biaxial nematic LC. In particular, The $K_{4}$
term can be written as $K_{4}(m_{i}l_{j}+l_{i}m_{j})(m_{k}l_{l}+l_{k}m_{l})$
using the transverse directors $m$ and $l$ of a orthorhombic biaxial nematic
LC Nbiax . This demonstrates that the lack of inversion symmetry does not
influence the Frank-like (quadratic) gradient energy. On the other hand,
however, a linear gradient energy is possible in the D2d phase

$\varepsilon_{l}=\xi\,T_{ijk}\,n_{i}\nabla_{j}n_{k}$ (76)

which is forbidden in ordinary nematic LC due to the inversion symmetry. It is
related neither to linear splay, $\nabla_{x}n_{x}+\nabla_{y}n_{y}$ (present in
polar nematics), nor to linear twist, $\nabla_{x}n_{y}-\nabla_{y}n_{x}$
(present in chiral nematics), but involves the combination
$\nabla_{x}n_{y}+\nabla_{y}n_{x}$. As it is well known from cholesteric liquid
crystals PGdG  and polar nematics splay ; polnema ; polarchol , the appearance
of linear gradient terms in the deformation energy of a director field allows
for the possibility of an inhomogeneous ground state.

Before we start the discussion of the implications of the linear gradient
energy, we mention related cross-couplings between director deformations of
the $\xi$ -type (Eq.(76)) and all the scalar hydrodynamic variables

$\varepsilon_{c}=T_{ijk}n_{i}\nabla_{j}n_{k}(\xi^{\rho}\delta\rho+\xi^{\sigma}\delta\sigma)$
(77)

unknown for usual nematics. Analogous terms for possible additional scalar
variables, like concentration variations $\delta c$ in mixtures, or variations
of the order parameters $\delta N$ or $\delta S$, can be written down
straightforwardly. These terms resemble the structure of static Lehmann terms
in cholesteric liquid crystals leh , although here they do not involve a
(Lehmann) rotation of the director, but transverse bend deformations.

The statics is completed by adding up all the energy contributions
$\varepsilon=\varepsilon_{0}+\varepsilon_{g}+\varepsilon_{l}+\varepsilon_{c}+\tilde{\varepsilon}_{E}$,
where $\varepsilon_{0}$ is the part of isotropic liquids and
$\tilde{\varepsilon}_{E}$ describes the influence of external fields (see
below). The conjugate quantities then follow from $\varepsilon$ as partial
derivatives according to Eq. (73).

####  3.1.2 Ambidextrous Helicity

The linear gradient energy contribution Eq. (76) allows for a inhomogeneous
ground state. Indeed, it is straightforward to show that a helical state has a
lower free energy than the homogeneous state. In this helical state the
director and the tetrahedral structure rotate together about one of the 2-fold
axes. 666The picture of counter-propagating nematic and tetrahedral helices
suggested in Ref. N  is based on a misinterpretation of the results and is not
possible.  These are the $x$ and $y$ axis in the geometry of Eq.(6), where the
director is along the $z$ axis. Choosing the $x$ axis as helical axis for
definiteness, the director (and the $\bar{4}$ axis) is given by

$n_{i}=\delta_{iz}C+\delta_{iy}S$ (78)

with $C=\cos(q_{0}x)$ and $S=\sin(q_{0}x)$, while for the tetrahedral vectors
one finds

$\frac{\pm 1}{\sqrt{3}}\begin{pmatrix}1&1&-1&-1\\\ C-S&-C+S\ \ \ \ &C+S\ \
&-C-S\\\ -C-S\ \ \ \ &C+S\ \ \ &C-S\ \ \ &-C+S\end{pmatrix}$ (79)

where the minus sign refers to the inverted tetrahedral structure. It is this
possibility to discriminate between original and inverted structure that
allows for different helical rotation senses.

Indeed, the helical state has a free energy, which is by
$\Delta\varepsilon=(1/2)(\xi^{2}/K_{2})$ smaller than that of the homogeneous
state, independent of the sign of Eq. (79). On the contrary, the helical wave
vector, $q_{0}=\mp\,\xi/K_{2}$, depends on that sign and the helical rotation
sense is reversed for the inverted structure. The two different possibilities
are demonstrated in Fig. 16, where the inverted structure on the right has the
opposite rotation sense compared to the original structure on the left. We
call this phenomenon ’ambidextrous helicity’. Ambidextrous means no energetic
preference for the two possibilities, in analogy to ’ambidextrous chirality’
e.g. in the CB2 (B2) phase, where the left and right handed helices are also
energetically equivalent G1 ; G2 . In the latter case the term ’chirality’ is
appropriate, since this phase is structurally chiral, while the D2d phase is
achiral and no pseudoscalar can be constructed from $n$ and $T_{ijk}$.

The helical wave vector $q_{0}$ is proportional to $\xi$, the material
parameter of the linear gradient term, and changes sign with it. For materials
with $\xi<0$ (there is no general principle that fixes this sign) the
situation is as shown in Fig. 16, while for the case $\xi>0$, the roles of
’original’ and ’inverted’ structure are interchanged. However, this is
irrelevant, since what is called ’original’ or ’inverted’ is arbitrary.

Figure 16:  Projection of a D2d structure onto the y/z plane on the left, with
the inverted structure on the right. The director $n$ is along the z axis, the
x axis sticks out of the drawing plane, and circles and asters denote
tetrahedral vectors that point out or into the drawing plane, respectively.
Rotations about the x axis with the sequence $A\rightarrow z\rightarrow B$
constitute an opposite helical sense for the right and left structure. The
sequence $B\rightarrow z\rightarrow A$ reverses the rotating sense in both
cases - from BP-D2d .

The choice of one of the 2-fold axes (x- or y-direction) as the helical axis
leads to the maximum energy gain, while for the rotation axis
$\mbox{\boldmath$b$}=\mbox{\boldmath$e$}_{x}\pm\mbox{\boldmath$e$}_{y}$ no
gain at all is found, since the linear gradient energy is zero in that case.

In a spontaneous formation of the D2d phase, helices of different rotation
sense and about different orthogonal axes might occur randomly at different
places of the sample, since all the possibilities discussed above are equally
likely. The D2d symmetry would be present only locally in the domains, and,
when averaged, an almost isotropic behavior can be expected. Even, if in a D2d
phase a pure helical state (with a single helicity and single helix
orientation) has been formed, averaging such a structure over a length scale
large compared to the pitch, the resulting symmetry of the system is
isomorphic to the situation arising when a cholesteric phase is averaged over
length scales large compared to the cholesteric pitch.

Therefore, we will use the local description in the following. This means, we
assume locally D2d symmetry, but with the $\xi$-term in the free energy,
Eq.(76), which reflects the lack of inversion symmetry. This procedure is
frequently used in cholesterics, which are locally described as nematics with
the additional linear twist energy term (reflecting chirality). If the D2d
phase is in a homogeneous state, the linear gradient free energy term always
leads to the tendency of forming localized helical domains.

####  3.1.3 Frustration by an External Electric Field

External electric fields have an orienting effect on LC, in particular the
dielectric anisotropy orients the director of the nematic phase PGdG , while
the tetrahedral structure is aligned by a cubic generalization of the
dielectric energy, Eq. (37), in the Td phase. In the D2d phase both effects
are present simultaneously and the (Legendre transformed) field-induced free
energy reads

$\displaystyle\tilde{\varepsilon}^{E}=$
$\displaystyle-\frac{\epsilon_{a}}{2}n_{i}n_{j}E_{i}E_{j}-\epsilon_{10}T_{ijk}E_{i}E_{j}E_{k}$
(80)
$\displaystyle-\frac{\epsilon_{11}}{3}T_{ijk}(n_{i}E_{j}E_{k}+E_{i}E_{j}n_{k}+E_{i}n_{j}E_{k})(\mbox{\boldmath$n\cdot
E$})$

where, however, the two tetrahedral terms act the same way and can be combined
via $\epsilon_{1}\equiv\epsilon_{10}+\epsilon_{11}$. We note that other
possible terms one might think of such as $\sim T_{ijk}n_{j}n_{k}$ vanish,
since $n_{i}$ has only a $z-$ component in contrast to $E_{i}$. There is no
need to incorporate fourth order terms in $\tilde{\varepsilon}^{E}$ to
guarantee convexity, since we study external fields here. As in the Td phase
the cubic term gives rise to second harmonic generation in optical
applications.

The nematic term is minimized for $n$ parallel or perpendicular to the field
(for ${\epsilon}_{a}\gtrless 0$), while the tetrahedral term forces one of the
tetrahedral unit vectors to be parallel or antiparallel to $E$. However, in
the D2d phase these two cases are incompatible, since the director always
makes an oblique angle (half of the tetrahedral angle, $\phi_{T}/2$, or
$\pi-\phi_{T}/2$), with any of the tetrahedral vectors, disproving the
possibility for zero or 90 degrees. Therefore, the system is frustrated and
the actual equilibrium orientation minimizes the sum of both terms, but not
each term separately. As a result, the orientation depends on the relative
strength of nematic vs. tetrahedral coupling ($\epsilon_{a}/\epsilon_{1}$).
Since these couplings are of different powers in the field strength, $E_{0}$,
the optimum orientation also depends on $E_{0}$.

At small fields the first term is dominant and the director orientation is the
usual nematic one. Above a threshold field strength, ${E}_{c}$, it is
energetically favorable to rotate the D2d structure rigidly such that the
director is tilted away from the dielectrically optimal orientation and, at
the same time, one of the tetrahedral vectors is tilted towards the field by
the same angle. Indeed, minimization of $\tilde{\varepsilon}^{E}$ with respect
to the tilt angle of the director, $\theta_{E}$, leads to

$\theta_{E}=0\quad\quad\mathrm{for}\quad\quad E\leq
E_{c}=\frac{\sqrt{3}}{8}\frac{\epsilon_{a}}{\epsilon_{1}}$ (81)

and

$\sqrt{3}\cos\theta_{E}=\gamma+\sqrt{\gamma^{2}+1}\quad\quad\mathrm{for}\quad\quad
E\geq E_{c}\quad$ (82)

with $\gamma=(1/8)(\epsilon_{a}/\epsilon_{1}E_{0})$ and
$\gamma_{c}=1/\sqrt{3}$. Here, we have assumed positive dielectric coupling,
$\epsilon_{a}>0$, and have chosen, without lack of generality,
$\epsilon_{1}>0$.

There is no jump of the tilt angle at $E_{c}$ and for very large fields
($\gamma\to 0$) $\theta_{E}\to\phi_{T}/2$ meaning one of the tetrahedral
vectors approaches asymptotically the field direction.

If $\epsilon_{1}/\epsilon_{a}$ is large enough for the threshold field to be
within experimental reach, there is a unique way of identifying the D2d phase:
Below the threshold, the director is oriented parallel to $E$. Increasing the
field beyond the threshold, the director turns away to a direction oblique to
the field - something that cannot happen in a conventional uniaxial nematic
phase. The presence of a helix further complicates the behavior. Any
homogeneous external field is incompatible with the combined helical structure
of director and tetrahedral vectors and tends to distort that structure.

For the dynamics discussed below we assume $\epsilon_{a}$ to be positive and
the nematic dielectric anisotropy to be the dominant effect such that the
system is below the threshold for reasonable applied fields. In that case the
symmetry of the D2d phase is preserved and the hydrodynamic description is
valid. In this case rotations of the structure away from the electric field
direction cost energy

$\tilde{\varepsilon}^{E}_{2}=\frac{1}{2}\left({\epsilon}_{a}+\frac{32}{9}\epsilon_{1}\,E_{0}\right)E_{0}^{2}\,(\delta\mbox{\boldmath$n$})^{2}$
(83)

with an effective, field dependent susceptibility (in the big parentheses).
This provides the restoring force for the relaxation of the director.

####  3.1.4 Dynamic Properties

While the dynamic laws for mass, momentum, and entropy density are the same as
in the Td phase, Eqs. (11), (12), and (22), the dynamic equation for the
symmetry variable, $(\partial/\partial t)\Gamma_{i}+Y_{i}=0$, is now written
in terms of $\delta\mbox{\boldmath$n$}$ and $\delta\Omega$

$\displaystyle\frac{\partial}{\partial t}n_{i}+X_{i}$ $\displaystyle=$
$\displaystyle 0$ (84) $\displaystyle\frac{\partial}{\partial
t}\Omega+Y^{\Omega}$ $\displaystyle=$ $\displaystyle 0$ (85)

with $X_{i}=\epsilon_{ijk}n_{j}Y_{k}$ and $Y^{\Omega}=n_{i}Y_{i}$.

In the D2d phase the structure of the reversible director dynamics is the same
as in uniaxial nematics. There are advection and convection and a
phenomenological coupling to symmetric shear flow forsternem ; MPP ;
forsterann ; forster

$X_{i}^{R}=v_{j}\nabla_{j}n_{i}+(\mbox{\boldmath$\omega$}\times\mbox{\boldmath$n$})_{i}-\lambda_{ijk}A_{jk}$
(86)

with
$2\lambda_{ijk}=\lambda(\delta_{ij}^{\perp}n_{k}+\delta_{ik}^{\perp}n_{j})$.
This allows for a steady alignment in simple shear at a tilt angle governed by
$\lambda$, the sole reversible transport parameter. The appropriate back flow
term in the stress tensor, $\sigma_{ij}^{R}=-\lambda_{kji}h_{k}$ guarantees
zero entropy production.

For rotations about the $n$ there is no phenomenological reversible coupling,
only transport,

$Y^{\Omega R}=v_{i}\nabla_{i}\Omega-n_{i}\omega_{i}\ \text{\,.}$ (87)

and, therefore, no flow alignment in the plane perpendicular to $n$.
Obviously, $\delta\Omega$ is not constant under rotations (as a true scalar
is), but behaves like the component of a rotation angle, e.g., as
$\delta\mbox{\boldmath$m\cdot$}(\mbox{\boldmath$n$}\times\mbox{\boldmath$m$})$
in biaxial nematics.

In the Td phase there are reversible phenomenological couplings between flow
and the currents of conserved hydrodynamic variables, Eqs. (29) and (30), that
also exist in the D2d phase. Here they are of the uniaxial form and read for
the heat current and the stress tensor

$\displaystyle j_{i}^{\sigma,ph}$ $\displaystyle=$
$\displaystyle(\Gamma_{21}\delta_{li}^{\perp}+\Gamma_{22}\,n_{l}n_{i})T_{ljk}A_{jk}$
(88) $\displaystyle\sigma_{ij}^{ph}$ $\displaystyle=$
$\displaystyle-(\Gamma_{21}\delta_{lk}^{\perp}+\Gamma_{22}\,n_{l}n_{k})T_{ijk}\nabla_{l}T$
(89)

Appropriate anisotropic generalizations exists for the couplings involving the
concentration current (cf. end of Sec. 2.1) or the electric current, Eq. (44)
and (45), and the stress tensor, with parameters
$\Gamma_{11},\Gamma_{12},\Gamma_{31},\Gamma_{32}$ accordingly. The physical
implications of those couplings has already been discussed in Sec. 2.1.

In the dissipative dynamics there are couplings between scalar hydrodynamic
variables and director rotations of the $\xi$-type, Eq. (76), e.g. described
by the contribution to the entropy production

$R_{T,n}=\Psi^{T}T_{ijk}n_{j}h_{i}\nabla_{k}T.$ (90)

Appropriate couplings exist for the electric field $E_{i}$ or the
concentration current $\nabla_{i}c$ replacing the temperature gradient. There
is no such term for the mass current, since the mass density does not have a
dissipative current. Similar terms would arise, when (gradients of the
thermodynamic conjugates of) the order parameters, $S$ and $N$, are
considered. These terms are the dissipative analog to the static couplings in
$\varepsilon_{c}$, Eq. (77). They are not present in ordinary nematic LC, but
they are similar in structure to the dynamic Lehmann effects in cholesteric LC
leh . There, inversion symmetry is broken by the helical wave vector $q_{0}$,
while here it is the tetrahedral tensor $T_{ijk}$. As a difference, in the D2d
phase transverse bend deformations of the director are involved, while in the
cholesteric case director rotations occur.

Dissipation of the director is isotropic, $2R_{n,n}=(1/\gamma)h^{2}$ as in the
nematic phase. The same is true for the dissipation of the transverse
rotations

$R_{\Omega,\Omega}=\frac{1}{2\gamma^{\Omega}}(h^{\Omega})^{2}$ (91)

with $h^{\Omega}=\partial\varepsilon/\partial\Omega$ that leads to the
dissipative part of the quasi-current $Y^{\Omega
D}=(1/\gamma^{\Omega})h^{\Omega}$. While the Td phase has one rotational
viscosity, $\gamma^{\Gamma}$, there are two in the D2d phase, the nematic one,
$\gamma$, and a second one, $\gamma^{\Omega}$, which are generally different
from from each other due to the anisotropy of the different rotations.

Other dissipative effects are anisotropic as in ordinary uniaxial nematic LC,
e.g., thermal conductivity $2R_{T,T}=\kappa_{ij}(\nabla_{i}T)(\nabla_{j}T)$ or
electric conductivity $2R_{E,E}=\sigma_{ij}^{E}E_{i}E_{j}$ etc.. Only the
viscosity is slightly more complicated in the D2d phase, since for
$2R_{v,v}=\nu_{ikjl}(\nabla_{i}v_{k})(\nabla_{j}v_{l})$ the viscosity tensor

$\displaystyle\nu_{ikjl}$
$\displaystyle=(\nu_{1}-2\nu_{2})\delta_{ik}^{\perp}\delta_{jl}^{\perp}+\nu_{2}(\delta_{ij}^{\perp}\delta_{kl}^{\perp}+\delta_{il}^{\perp}\delta_{jk}^{\perp})$
$\displaystyle+\nu_{3}(\delta_{ij}^{\perp}n_{k}n_{l}+\delta_{il}^{\perp}n_{j}n_{k}+\delta_{kl}^{\perp}n_{i}n_{j}+\delta_{jk}^{\perp}n_{i}n_{l})$
$\displaystyle+\nu_{4}n_{i}n_{j}n_{k}n_{l}+\nu_{5}(\delta_{ik}^{\perp}n_{j}n_{l}+\delta_{jl}^{\perp}n_{i}n_{k})$
$\displaystyle+\nu_{6}n_{p}n_{q}T_{pij}T_{qkl}$ (92)

contains six viscosities as in the case of orthorhombic biaxial nematic LC
mason . The last term in Eq. (3.1.4) vanishes in the uniaxial nematic case.

####  3.1.5 Relative rotations

In the D2d phase the orientation of the director relative to the tetrahedral
structure is fixed. In particular, the director rotates always together with
the tetrahedral structure,
$\delta\mbox{\boldmath$n$}=\mbox{\boldmath$n$}\times\delta\mbox{\boldmath$\Gamma$}$.
This means that any difference between $\delta\mbox{\boldmath$n$}$ and
$\mbox{\boldmath$n$}\times\delta\mbox{\boldmath$\Gamma$}$ vanishes on time and
length scales much shorter than the hydrodynamic ones. Under certain
conditions, however, relative rotations

$\mbox{\boldmath$J$}=\delta\mbox{\boldmath$n$}-\mbox{\boldmath$n$}\times\delta\mbox{\boldmath$\Gamma$}$
(93)

persist and are included as non-hydrodynamic variables in the macroscopic
dynamics.

This is similar to the smectic A phase, where the layer normal and the
director are locked to be parallel. Under certain conditions, like the
vicinity to the nematic phase transition liu  or strong external shear auernh
, this coupling can weaken allowing the two preferred directions to differ
from each other for some time before they have been relaxed back. Another
example of relative rotations arises for mixtures of a rod-like and a disk-
like uniaxial nematic phase mixture . Such relative rotations play a prominent
role in nematic elastomers garmisch ; physica , where they are responsible for
elastic anomalies softmat ; menzel .

The relative rotations $J$ are even under spatial inversion, odd under the
replacement of $n$ by $-\mbox{\boldmath$n$}$, and are invariant under rigid
rotations. They do not involve rotations of the tetrahedral structure about
the director, $\mbox{\boldmath$n$}\cdot\delta\mbox{\boldmath$\Gamma$}$, since
$\mbox{\boldmath$n$}\cdot\mbox{\boldmath$J$}=0$.

The free energy of relative rotations

$\varepsilon_{J}=\frac{1}{2}D_{1}\mbox{\boldmath$J$}^{2}$ (94)

diverges in the rigidly locked case, where the stiffness coefficient
$D_{1}\to\infty$. In Eq. (94) we have neglected some bilinear couplings
between relative rotations and gradients of the director or gradients of
$\Omega$, which are of the linearized form
$J_{x}\nabla_{z}n_{x}-J_{y}\nabla_{z}n_{y}$ and
$J_{x}\nabla_{y}\Omega+J_{y}\nabla_{x}\Omega$. Gradients of electric fields
couple similarly.

The dynamics is given by the balance equation

$\frac{\partial}{\partial t}J_{i}+Y_{i}^{JR}+Y_{i}^{JD}=0$ (95)

The reversible part

$Y_{i}^{J\,R}=v_{k}\nabla_{k}J_{i}+\lambda_{ijk}^{J}A_{jk}$ (96)

contains the transport derivative and a phenomenological coupling to
deformational flow where
$\lambda_{ijk}^{J}=\lambda_{J}(\epsilon_{ikp}n_{p}n_{j}+\epsilon_{ijp}n_{p}n_{k})$
carries one phenomenological parameter. There is no coupling to rotational
flow, since $\delta\mbox{\boldmath$n$}$ transforms the same way as
$\mbox{\boldmath$n$}\times\delta\mbox{\boldmath$\Gamma$}$ under rigid
rotations (cf. Sec.2.1). Lacking the coupling to rotational flow, shear flow
does not lead to a (shear flow) alignment of $J_{i}$.

The dissipative dynamics can be derived from the appropriate part of the
dissipation function, $Y_{i}^{J\,D}=(\partial/\partial L_{i})R^{J}$ where

$R^{J}=\frac{1}{2}\zeta^{\perp}\delta_{ij}^{\perp}L_{i}L_{j}+\zeta_{ij}^{n}L_{i}h_{j}+\zeta_{ijk}^{E}L_{i}\nabla_{j}E_{k}$
(97)

is expressed by $L_{i}=(\partial/\partial J_{i})\varepsilon_{J}$, the
thermodynamic conjugate of the relative rotations. The transport parameter
$\zeta^{\perp}$ governs the relaxation of relative rotations with the
relaxation time $1/(\zeta^{\perp}D_{1})$, which is zero in the rigidly locked
case. The material tensors $\zeta_{ij}^{n}=\zeta_{n}n_{k}\epsilon_{ijk}$ and
$\zeta_{ijk}^{E}=\zeta_{E}(\epsilon_{ikp}n_{p}n_{j}+\epsilon_{ijp}n_{p}n_{k})$
provide dissipative couplings between relative rotations and director textures
or inhomogeneous electric fields.

###  3.2 The non-polar, low symmetry tetrahedral biaxial nematic phases

####  3.2.1 The non-polar tetragonal S4 tetrahedral phase

If one adds two transverse biaxial directors $m$ and $l$ along the $\bar{4}$
tetrahedral directions, the structure is equivalent to a D2d phase. In case
the biaxial directors are rotated in the transverse plane by a finite angle
other than $\pi/4$ and $\pi/2$, as in Fig. 17, an $S_{4}$ symmetric S4 is
obtained. It is obvious to see that due to this rotation the mirror planes are
removed as well as both 2-fold rotation axes ($x,y$ axes). Only the (improper)
4-fold symmetry axis ($n$ or $z$ axis) is left. Due to the existence of an
improper rotation axis, there is no chirality.

Figure 17: The projection of the 4 tetrahedral vectors
$\mbox{\boldmath$n$}^{\alpha}$ ($\alpha=1,2,3,4$) into the $x,y$ plane
perpendicular to the director $n$ in the S4 phase. A circle (asterisk) denotes
tetrahedral vectors that also have a component sticking out of (pointing into)
the projection plane. The tetragonal nematic directors $m$ and $l$ are
equivalent and rotated by an angle $\Theta\neq 0,\pi/4$, thereby removing any
mirror planes as well as the 2-fold axes. The $z$ axis remains to be an
improper $\bar{4}$ axis. For $\Theta=0$, the structure is equivalent to a D2d
phase - from PB14 .

The hydrodynamics of this phase is rather similar to the D2d phase, in
particular the hydrodynamic variables are the same: Rotations of the preferred
direction $n$ (the tetragonal axis) and a rotation about this axis. The latter
can again be described by appropriate rotations of the tetrahedral structure,
$\delta\Omega\equiv(27/128)n_{i}\epsilon_{ipq}T_{pjk}\delta T_{qjk}$, where we
have used $N=1$ in the normalization relation $27\alpha=32N^{2}$. The only
difference is the reduced symmetry of S4 compared to D2d, which is manifest in
more complicated structures of material tensors of fourth order (and higher)
and, in addition, by a few more non-vanishing elements of the equilibrium
tensor $T_{ijk}$ giving rise to some additional cross-couplings. The form of
the hydrodynamic equations is the same as in the D2d phase and will not be
repeated here.

As an example for more complicated material tensors, the gradient free energy
related to rotations of $n$,

$\displaystyle K_{ijkl}$
$\displaystyle=K_{3}n_{i}n_{j}\delta_{kl}^{\perp}+(K_{1}-2K_{2})\delta_{ik}^{\perp}\delta_{jl}^{\perp}$
$\displaystyle+K_{2}(\delta_{il}^{\perp}\delta_{jk}^{\perp}+\delta_{ij}^{\perp}\delta_{kl}^{\perp})+K_{4}n_{p}n_{q}T_{ijp}T_{klq}$
$\displaystyle+K_{8}\delta^{\perp}_{rk}\delta^{\perp}_{tl}T_{ijp}T_{rtp}.$
(98)

contains five Frank bulk orientational elastic coefficients in the S4 phase
(compared to four in D2d, Eq. (75)). The $K_{8}$ term, which does not exist in
the D2d phase, gives rise to new combinations of director variations, if
linearized, of the form $\cos 2\Theta\sin
2\Theta(\nabla_{x}n_{x}-\nabla_{y}n_{y})(\nabla_{y}n_{x}+\nabla_{x}n_{y})$
where $\Theta$ is the in-plane rotation angle as described in Fig. 17.

On the other hand, the gradient energy involving $\delta\Omega$, Eq. (74), and
the linear gradient term giving rise to ambidextrous helicity, Eq. (76), are
as in the D2d phase. The same applies to the static Lehmann-type energy
contributions, Eq. (77). There is still no linear gradient term w.r.t.
$\nabla_{i}\Omega$, because of the invariance under
$\mbox{\boldmath$m$}\Leftrightarrow\mbox{\boldmath$l$}$.

An example of a more complicated material tensor in the dynamics is the
viscosity tensor, relating the stress tensor with the symmetric flow tensor,
$\sigma_{ij}=-\nu_{ijkl}\nabla_{l}v_{k}$. It contains an additional 7th term
(compared to Eq. (3.1.4)),
$\nu_{7}T_{rtp}(\delta^{\perp}_{rk}\delta^{\perp}_{tl}T_{ijp}+\delta^{\perp}_{ri}\delta^{\perp}_{tj}T_{klp})$
in the S4 phase, which has almost the same structure as the $K_{8}$ Frank term
discussed above. All other material tensors occurring in the static or dynamic
part of the S4 hydrodynamics are of a rank less than four and have the same
structure as in the D2d phase.

Among the cases, where additional non-vanishing elements of the equilibrium
tensor $T_{ijk}$ are relevant, is the linear gradient term, Eq. (76),
$(\nabla_{x}n_{y}+\nabla_{y}n_{x})\cos
2\Theta+(\nabla_{x}n_{x}-\nabla_{y}n_{y})\sin 2\Theta$. It involves director
structures not present in the D2d phase.

In the reversible hydrodynamics there are couplings between shear flow and
currents of temperature, concentration and charge, and vice versa, between
shear stresses and gradients of temperature and concentration, and electric
fields, Eq. (88) and (89), which in the S4 phase have the form

$\displaystyle j_{x}^{\sigma,R}$ $\displaystyle=$
$\displaystyle\Gamma_{21}(\cos 2\Theta A_{yz}+\sin 2\Theta A_{xz})$
$\displaystyle j_{y}^{\sigma,R}$ $\displaystyle=$
$\displaystyle\Gamma_{21}(\cos 2\Theta A_{xz}-\sin 2\Theta A_{yz})$
$\displaystyle j_{z}^{\sigma,R}$ $\displaystyle=$
$\displaystyle\Gamma_{22}(\cos 2\Theta A_{xy}+\sin 2\Theta\,[A_{xx}-A_{yy}])$
(99)

and

$\displaystyle\sigma_{xz}$ $\displaystyle=$ $\displaystyle-\Gamma_{21}(\cos
2\Theta\,\nabla_{y}T+\sin 2\Theta\,\nabla_{x}T)$ $\displaystyle\sigma_{yz}$
$\displaystyle=$ $\displaystyle-\Gamma_{21}(\cos 2\Theta\,\nabla_{x}T-\sin
2\Theta\,\nabla_{y}T)$ $\displaystyle\sigma_{xy}$ $\displaystyle=$
$\displaystyle-\Gamma_{22}\cos 2\Theta\,\nabla_{z}T$
$\displaystyle\sigma_{xx}$ $\displaystyle=$
$\displaystyle-\sigma_{yy}=-\Gamma_{22}\sin 2\Theta\,\nabla_{z}T$ (100)

with appropriate terms for the concentration and electric degrees of freedom
involving $\Gamma_{31,32}$ and $\Gamma_{11,12}$. All terms $\sim\sin\Theta$
are new in the S4 phase and not present in D2d. They comprise hyperbolical
flows and stresses and oblique currents.

The dissipative Lehmann-type couplings of the D2d phase, Eq. (90), acquire
more coupling elements in the S4 phase

$\displaystyle j_{x}^{\sigma,D}$ $\displaystyle=$ $\displaystyle-\psi^{T}(\cos
2\Theta\,h_{y}+\sin 2\Theta\,h_{x})$ $\displaystyle j_{y}^{\sigma,D}$
$\displaystyle=$ $\displaystyle-\psi^{T}(\cos 2\Theta\,h_{x}-\sin
2\Theta\,h_{y})$ (101)

and

$\displaystyle\dot{n}_{x}$ $\displaystyle=$ $\displaystyle-\psi^{T}(\cos
2\Theta\,\nabla_{y}T+\sin 2\Theta\,\nabla_{x}T)$ $\displaystyle\dot{n}_{y}$
$\displaystyle=$ $\displaystyle-\psi^{T}(\cos 2\Theta\,\nabla_{x}T-\sin
2\Theta\,\nabla_{y}T)$ (102)

with similar sets of equations involving the concentration and the electric
degrees of freedom.

The orientation of the director (and the tetrahedral structure) in an external
electric field is basically the same as in the D2d phase, Eq. (80), including
dielectric anisotropy and the cubic tetrahedral orientation leading to
frustration. Assuming that the dielectric anisotropy effect is the dominant
one orienting the director (and thus the $\bar{4}$ axis) along the $z$
direction, a small oscillating electric transverse field will lead to a
reorienting force on $n$ of the transversely isotropic form
$E_{x}^{2}+E_{y}^{2}$ due to the dielectric anisotropy, while for the
tetrahedral orientation the reorientation force is, in the S4 phase, of the
form $\cos 2\Theta E_{x}E_{y}+\sin 2\Theta(E_{x}^{2}-E_{y}^{2})$. Thus, this
response to an external field can experimentally reveal the transverse
anisotropy in the S4 phase different from the D2d case.

####  3.2.2 The non-polar orthorhombic D2 tetrahedral phase

Figure 18: Two projections of the 4 tetrahedral vectors
$\mbox{\boldmath$n$}^{\alpha}$ ($\alpha=1,2,3,4$) and the orthorhombic
directors $\mbox{\boldmath$l$},\mbox{\boldmath$m$},\mbox{\boldmath$n$}$ in the
D2 phase. Stars and circles as in Fig. 17. Only three 2-fold symmetry axes
(the $x,y,z$ axes) are left, but no mirror planes - from PB14 .

A D2 phase can be viewed as an orthorhombic biaxial nematic of $D_{2h}$
symmetry (with mutually orthogonal directors
$\mbox{\boldmath$l$},\mbox{\boldmath$m$},\mbox{\boldmath$n$}$) that are
rigidly attached to the three (improper) $\bar{4}$ axes of the tetrahedral
structure, cf. Fig. 18. As a result, only proper (2-fold) symmetry axes are
left, but no mirror planes rendering the phase chiral. The latter is expressed
by the pseudoscalar quantity
$q_{0}\equiv\mbox{\boldmath$n\cdot$}(\mbox{\boldmath$m$}\times\mbox{\boldmath$l$})n_{i}m_{j}l_{k}T_{ijk}$
or equivalently by
$q_{0}^{\prime}\equiv\mbox{\boldmath$n\cdot$}(\mbox{\boldmath$l$}\times\mbox{\boldmath$m$})n_{i}m_{j}l_{k}T_{ijk}=-q_{0}$
indicating that both kinds of handedness are present. This ”ambidextrous
chirality” G1  is of structural origin, in contrast to the molecule-based
chirality of the chiral T phase, Section 2.4.

The hydrodynamics of the D2 phase is that of orthorhombic biaxial nematics
Nbiax ; biaxnemliu , amended by effects of the broken inversion symmetry and
chirality due to $T_{ijk}$. Like in the D2d and in the biaxial nematic phases,
the hydrodynamic degrees of freedom (Goldstone modes) are the three
independent rotations of the rigid structure. They can be realized by the
three rotations $\delta\mbox{\boldmath$n$}$, $\delta\mbox{\boldmath$m$}$, and
$\delta\mbox{\boldmath$l$}$ that preserve their mutual orientation due to the
conditions $m_{i}\delta n_{i}+n_{i}\delta m_{i}=0$, $l_{i}\delta
n_{i}+n_{i}\delta l_{i}=0$, and $m_{i}\delta l_{i}+l_{i}\delta m_{i}=0$. The
tetrahedral structure follows those rotations rigidly. Alternatively, one
could use rotations of the preferred axis $\delta\mbox{\boldmath$n$}$ and the
rotation about this axis
$\delta\Omega=(\mbox{\boldmath$n$}\times\mbox{\boldmath$m$})_{i}\delta m_{i}$
as variables. There are 12 bulk Frank-like quadratic rotation elastic
coefficients, cf. BP26 .

In the following we will concentrate on the $T_{ijk}$\- and $q_{0}$-induced
effects. There are six linear gradient terms in the free energy, not present
in ordinary biaxial nematics

$\displaystyle\varepsilon_{l}$ $\displaystyle=$ $\displaystyle
T_{ijk}(\xi_{1}n_{i}\nabla_{j}n_{k}+\xi_{2}m_{i}\nabla_{j}m_{k}+\xi_{3}l_{i}\nabla_{j}l_{k})$
(103) $\displaystyle+$ $\displaystyle
q_{0}\epsilon_{ijk}(k_{1}n_{i}\nabla_{j}n_{k}+k_{2}\,m_{i}\nabla_{j}m_{k}+k_{3}\,l_{i}\nabla_{j}l_{k})\quad$

The first line is the generalization of Eq. (76) of the D2d phase describing
ambidextrous helicity (cf. Sec.3.1.2), since the inverted structure is
different from the non-inverted one, but leads to the same energy reduction.
The second line contains the linear twist terms of biaxial cholesterics PB90 .
In the D2 phase, they describe ambidextrous chirality, since $q_{0}$ and
$-q_{0}$ are equally likely to occur. It is not possible to individually
minimize each of the six terms for steric reasons. In that sense, the D2 phase
is frustrated.

If there are linear gradient terms, there are also static Lehmann-type energy
contributions leh ; Lehmann , bilinear in linear director gradients and
variations of the scalar variables, $\gamma\in\\{\sigma,\rho,c\\}$

$\varepsilon_{c}=\sum_{\gamma}(\delta\gamma)\bigl{(}T_{ijk}g_{ijk}^{\gamma}+q_{0}\epsilon_{ijk}G_{ijk}^{\gamma}\bigr{)}$
(104)

with
$g_{ijk}^{\gamma}=\xi_{1}^{\gamma}n_{i}\nabla_{j}n_{k}+\xi_{2}^{\gamma}m_{i}\nabla_{j}m_{k}+\xi_{3}^{\gamma}l_{i}\nabla_{j}l_{k}$
and $G_{ijk}^{\gamma}$ as $g_{ijk}^{\gamma}$, but with the coefficients
$\xi_{1,2,3}^{\gamma}$ replaced by $k_{1,2,3}^{\gamma}$. Again, they are
either a generalization of the D2d case, Eq. (77) or of the chiral nematic
case.

The dissipative Lehmann-type terms also come in two classes, either due to the
lack of inversion symmetry (as in the D2d phase, Eq. (90)) or due to chirality
(as in cholesterics), for
$\nabla_{k}Q\in\\{\nabla_{k}T,\nabla_{k}\mu_{c},E_{k}\\}$,

$2R_{L}=\sum_{Q}(\nabla_{k}Q)\bigl{(}T_{ijk}h_{ij}^{Q}+q_{0}\epsilon_{ijk}H_{ij}^{Q}\bigr{)}$
(105)

where
$h_{ij}^{Q}=(\psi_{1}^{Q}m_{j}m_{p}+\psi_{2}^{Q}l_{j}l_{p})n_{i}h_{p}^{n}+\psi_{3}^{Q}l_{j}l_{p}m_{i}h_{p}^{m}$
and $H_{ij}^{Q}$ as $h_{ij}^{Q}$, but with coefficients $\psi_{1,2,3}^{Q}$
replaced by different ones $\Psi_{1,2,3}^{Q}$. The molecular fields,
$h_{i}^{n}$, $h_{i}^{m}$, follow from the Frank gradient energy in the
standard way. The chirality-based static and dynamic Lehmann-type
contributions lead to rotations of the directors due to applied thermodynamic
forces leh ; Lehmann  and the inverse effects inverse , while the
$T_{ijk}$-based ones do not have such a simple geometric interpretation.

The phenomenological reversible couplings between e.g. the heat current and
deformational flow, which is characteristic for tetrahedral systems

$\displaystyle j_{i}^{\sigma,ph}=\Gamma_{ip}^{\sigma}\,T_{pjk}A_{jk}$ (106)
$\displaystyle\sigma_{ij}^{ph}=-T_{pij}\Gamma_{kp}^{\sigma}\nabla_{k}T$ (107)

contain one and two parameters for the Td and the D2d phase, respectively,
while there are 3 for the D2 phase, since

$\Gamma_{ij}^{\sigma}=\Gamma_{1}^{\sigma}n_{i}n_{j}+\Gamma_{2}^{\sigma}m_{i}m_{j}+\Gamma_{3}^{\sigma}l_{i}l_{j}$
(108)

is of the standard orthorhombic form. The same holds for the appropriate
couplings to the concentration and the electric current.

Due to the chirality there is a phenomenological, reversible contribution to
the tetrahedral rotation, Eq. (85), that couples to deformational flow

$\displaystyle Y^{\Omega R}=$ $\displaystyle
q_{0}\lambda^{\Omega}n_{i}T_{ijk}A_{jk}$ (109)
$\displaystyle\sigma_{ij}^{ph}=$ $\displaystyle
q_{0}\lambda^{\Omega}n_{k}T_{ijk}\,h^{\Omega}$ (110)

where $h^{\Omega}$ is the conjugate to $\Omega$, Eq. (73). The stress tensor
carries the appropriate counter terms that guarantee zero entropy production.
There is one material parameter, $\lambda^{\Omega}$, involved, which is
generally different from $\lambda$, the parameter that governs the flow
alignment of the preferred direction $n$, Eq. (86). Together with the
(linearized) response to rotational flow, $Y^{\Omega R}=-\omega_{i}n_{i}$,Eq.
(87)), there is now also a stationary alignment of the tetrahedral rotation in
planar simple shear flow, in contrast to the D2d phase. This is similar to the
case of the chiral T phase, Eq. (69).

There are 4 orientational viscosities

$\displaystyle 2R_{\gamma}=$
$\displaystyle\bigl{(}\frac{1}{\gamma_{1}}m_{i}m_{j}+\frac{1}{\gamma_{2}}l_{i}l_{j}\bigr{)}h_{i}^{n}h_{j}^{n}++\frac{1}{\gamma_{3}}l_{i}l_{j}h_{i}^{m}h_{j}^{m}$
$\displaystyle+\frac{1}{\gamma^{\Omega}}(h^{\Omega})^{2}$ (111)

The orientation by external fields of the combined nematic and tetrahedral
structure in the D2 phase has several different origins

$\displaystyle\tilde{\varepsilon}^{E}=$
$\displaystyle-\frac{1}{2}\epsilon^{a}_{ij}E_{i}E_{j}-\frac{1}{2}\chi^{a}_{ij}H_{i}H_{j}-\epsilon_{10}T_{ijk}E_{i}E_{j}E_{k}$
$\displaystyle-$
$\displaystyle\frac{\epsilon_{11}}{3}T_{ijk}(n_{i}E_{j}E_{k}+E_{i}E_{j}n_{k}+E_{i}n_{j}E_{k})(\mbox{\boldmath$n\cdot
E$})$ $\displaystyle-$
$\displaystyle\frac{\epsilon_{12}}{3}T_{ijk}(m_{i}E_{j}E_{k}+E_{i}E_{j}m_{k}+E_{i}m_{j}E_{k})(\mbox{\boldmath$m\cdot
E$})$ $\displaystyle-$
$\displaystyle\frac{\epsilon_{13}}{3}T_{ijk}(l_{i}E_{j}E_{k}+E_{i}E_{j}l_{k}+E_{i}l_{j}E_{k})(\mbox{\boldmath$l\cdot
E$})$ $\displaystyle+$ $\displaystyle
q_{0}\,\epsilon_{ipq}T_{ijk}\left(\chi_{jp}^{E}E_{k}E_{q}+\chi_{jp}^{H}H_{j}H_{k}\right)$
$\displaystyle+$ $\displaystyle
q_{0}\epsilon_{pik}\zeta_{jp}^{EH}E_{i}H_{j}H_{k}$ (112)

where all second rank tensors in Eq. (3.2.2) are of the form
$\epsilon^{a}_{ij}=\epsilon^{a}_{1}n_{i}n_{j}+\epsilon^{a}_{2}m_{i}m_{j}$.
There is the electric and magnetic anisotropy, and the tetrahedral
orientation, described by quadratic and cubic field-dependent energies,
respectively. Compared to the energy expressions for the Td and D2d phase,
Eqs. (37) and (80), respectively, there are more coefficients involved due to
the biaxiality. In addition, there are terms quadratic in the field due to the
chirality, anisotropy, and tetrahedral order in the D2 phase. Obviously, there
is no orientation that minimizes those contributions individually, and
orientational frustration has to be expected.

###  3.3 The polar, low symmetry tetrahedral biaxial nematic phases

####  3.3.1  The polar trigonal C3v tetrahedral phase

Figure 19: The projection of the three tetrahedral vectors,
$\mbox{\boldmath$n$}^{2,3,4}$, into the x/y plane. They all have a component
pointing into that plane (asterisk). The tetrahedral vector
$\mbox{\boldmath$n$}^{1}$ is sticking out of the plane along the z- axis,
which is the polar, 3-fold symmetry axis of the C3v phase. The nematic
director $n$ is along the z-axis (red dot). This representation of the
tetrahedral vectors corresponds to Eq. (7).

This phase is obtained, when the nematic director is along one of the
tetrahedral vectors, $\mbox{\boldmath$n$}\parallel\mbox{\boldmath$n$}^{1}$
($\parallel\mbox{\boldmath$e$}_{z}$ in Fig. 19). The polarization
$P_{i}=T_{ijk}Q_{jk}=P_{0}p_{i}$ is along that direction. We take the absolute
value of the polarization $P_{0}=|T_{ijk}Q_{jk}|=(4/9)SN$ as a constant. The
hydrodynamic variables are the rotations of the polarization $\delta p_{i}$
with $\mbox{\boldmath$p\cdot$}\delta\mbox{\boldmath$p$}=0$ as in polar nematic
LC polnema  and, in addition, rotations of the tetrahedral structure about the
polar direction
$\delta\Omega\equiv\mbox{\boldmath$p\cdot$}\delta\mbox{\boldmath$\Gamma$}=(1/4\alpha)p_{i}\epsilon_{ipq}T_{pjk}\delta
T_{qjk}$, similar to the D2d phase. However, here $\delta\Omega$ is odd under
spatial inversion, in contrast to the D2d case, Eq. (72), and the rotation
axis is a tetrahedral direction and not a $\bar{4}$ axis as in the D2d case.

The structure of the hydrodynamics is therefore rather similar to that in the
D2d phase, although there are subtle differences due to the polarity and the
3-fold rotational symmetry (about the polar axis) in the C3v phase. In
particular, the linear gradient term,

$\varepsilon_{l}=\xi_{p}^{\prime}T_{ijk}p_{i}\nabla_{j}p_{k}=\xi_{p}\nabla_{i}p_{i}$
(113)

describes splay, as in polar nematic LC polnema , allowing for spontaneous
splay phases. In the C3v phase there is no ambidextrous helicity related to
this linear gradient term. Similarly, the static Lehmann-type energy

$\varepsilon_{c}=(\xi^{\sigma}\delta\sigma+\xi^{\rho}\delta\rho+\xi^{c}\delta
c)\nabla_{i}p_{i}$ (114)

is as in the polar nematic case. The Frank orientational elasticity tensor
carries 8 coefficients as in the S4 phase, Eqs. (3.2.1).

The dissipative Lehmann-type terms have the form

$2R_{L}=\delta_{jl}^{\perp}p_{i}T_{ijk}(\Psi^{T}\nabla_{k}T+\Psi^{c}\nabla_{k}\mu_{c}+\Psi^{E}E_{k})h_{l}^{p}$
(115)

where $h_{l}^{p}$ is the conjugate to $\delta p_{i}$.

Third rank material tensors, e.g. reversibly relating flow with gradients of
temperature, which contain three material parameters in the polar nematic case
polarchol  and two in the D2d phase (cf. e.g. Eq. (88)), have 5 coefficients
in the C3v phase

$\displaystyle j_{i}^{\sigma,ph}$ $\displaystyle=$
$\displaystyle[(\Gamma_{21}\delta_{li}^{\perp}+\Gamma_{22}\,p_{l}p_{i})T_{ljk}$
$\displaystyle+\phi_{1}p_{i}p_{j}p_{k}+\phi_{2}p_{i}\delta_{jk}^{\perp}+\phi_{3}(p_{j}\delta_{ik}^{\perp}+p_{k}\delta_{ij}^{\perp})]A_{jk}$
$\displaystyle\sigma_{ij}^{ph}$ $\displaystyle=$
$\displaystyle-[(\Gamma_{21}\delta_{lk}^{\perp}+\Gamma_{22}\,p_{l}p_{k})T_{ijk}$
$\displaystyle+\phi_{1}p_{i}p_{j}p_{l}+\phi_{2}p_{l}\delta_{ij}^{\perp}+\phi_{3}(p_{j}\delta_{il}^{\perp}+p_{i}\delta_{jl}^{\perp})]\nabla_{l}T$

with $\delta_{ij}^{\perp}=\delta_{ij}-p_{i}p_{j}$. Appropriate equations are
obtained for concentration and electrical current.

The fourth rank viscosity tensor contains, as in the D2d phase, six
coefficients, cf. (3.1.4) with $n_{i}$ replaced by $p_{i}$. The 7th term found
in the S4 phase, $\sim\nu_{7}$, is zero due to the trigonal symmetry of
$T_{ijk}$, Eq. (7). On the other hand, second rank material tensors are of the
standard uniaxial form, since the third rank tensor cannot influence second
rank material properties.

There is flow alignment of the preferred direction, as in polar nematic LC and
in the D2d phase, cf. Eq. (86). The dynamics of rotations, $\Omega$, about the
preferred direction (not present in polar nematic LC) is as in the D2d phase,
with no alignment due to shear flow, and no alignment in an electric field.
There is one rotational viscosity with respect to $\Omega$, which is different
from that for $\delta\mbox{\boldmath$p$}$.

The orientation of the preferred axis in an external electric fields is
dominated by the polarization, which is along the field in equilibrium. There
is no orientational frustration by an external field, since with the
polarization also one of the tetrahedral vectors is along the field.
Deviations of the polar direction from the field cost energy,
$\tilde{\varepsilon}^{E}_{2}=\tfrac{1}{2}P_{0}E_{0}(\delta p_{i})^{2}$, and
lead to a relaxation, which is linear in the field amplitude $E_{0}$, in
contrast to the D2d phase, Eq. (83).

####  3.3.2 The polar orthorhombic C2v tetrahedral phase

Figure 20: Structure of the orthorhombic C2v phase: The tetrahedral vectors
are as in Fig. 18 (upper graph), corresponding to Eq. (6). The biaxial nematic
directors $m$ and $l$ are in the
$\mbox{\boldmath$n$}^{1}/\mbox{\boldmath$n$}^{4}$ and
$\mbox{\boldmath$n$}^{2}/\mbox{\boldmath$n$}^{3}$ planes, which are mirror
planes. The perpendicular z-axis is polar, since $m$ and $l$ are not
equivalent, and is the only (2-fold) symmetry axis left - from PB14 .

This phase can be viewed as an orthorhombic biaxial nematic with the three
mutually orthogonal, non-equivalent directors $n,m,l$, where in addition
inversion symmetry is broken due to the tetrahedral structure $T_{ijk}$. A
possible spatial representation of the structure is shown in Fig. 20. It is
polar with the polar direction $p_{i}\sim T_{ijk}(m_{j}m_{k}-l_{j}l_{k})$
along the director $n$. The absolute value of the polarization is taken as
constant.

The hydrodynamic variables are the rotations of the polar direction, $\delta
p_{i}$, as in a polar nematic LC, and the rotation,
$\delta\Omega=p_{i}\delta\Gamma_{i}$, of the tetrahedral structure about the
polar direction. Since the latter is a $\bar{4}$ axis of the tetrahedral
structure, this variable is somewhat similar to the appropriate one in the D2d
phase and, in particular, there is no flow alignment of such rotations.
Another way of setting up the hydrodynamics is the use of the rotations of the
directors, $\delta\mbox{\boldmath$m$}$ and $\delta\mbox{\boldmath$l$}$ that
preserve their mutual orientation, as in the D2 phase described in Sec. 3.2.2.
The tetrahedral structure follows those rotations rigidly.

The system shows four linear gradient terms

$\varepsilon_{l}=T_{ijk}(\xi_{1}p_{i}\nabla_{j}p_{k}+\xi_{2}m_{i}\nabla_{j}m_{k}+\xi_{3}l_{i}\nabla_{j}l_{k})+\xi_{p}\nabla_{i}p_{i}$
(118)

The first three terms are linear gradient terms similar to that of the D2d
phase favoring spontaneous ambidextrous helical rotations of the tetrahedral
structure about the different $\bar{4}$ axes. The last one is the standard
linear splay term of polar systems. All the spontaneous structures favored by
the individual linear gradient terms are mutually incompatible, and a rather
complicated, inhomogeneous ground state may occur.

The static Lehmann-type energy has 4 coefficients for the thermal, solutal,
and electric degree of freedom, each ($\gamma\in\\{\sigma,\rho,c\\}$)

$\varepsilon_{c}=\sum_{\gamma}(\delta\gamma)\bigl{(}T_{ijk}g_{ijk}^{\gamma}+\xi_{p}^{\gamma}\nabla_{i}p_{i}\bigr{)}$
(119)

with
$g_{ijk}^{\gamma}=\xi_{1}^{\gamma}p_{i}\nabla_{j}p_{k}+\xi_{2}^{\gamma}m_{i}\nabla_{j}m_{k}+\xi_{3}^{\gamma}l_{i}\nabla_{j}l_{k}$
combining the tetrahedral and the polar contributions.

A similar combination of polar and tetrahedral effects are found for the
reversible flow/temperature gradient

$\displaystyle j_{i}^{\sigma,ph}$ $\displaystyle=$
$\displaystyle(\phi_{1}^{\sigma}p_{i}p_{j}p_{k}+\phi_{2}^{\sigma}p_{i}\delta_{jk}^{\perp}+\phi_{3}^{\sigma}[p_{j}\delta_{ik}^{\perp}+p_{k}\delta_{ij}^{\perp}]$
(120) $\displaystyle+\Gamma_{iq}^{\sigma}\,T_{qjk})A_{jk}$
$\displaystyle\sigma_{ij}^{ph}$ $\displaystyle=$
$\displaystyle-(\phi_{1}^{\sigma}p_{i}p_{j}p_{k}+\phi_{2}^{\sigma}p_{k}\delta_{ji}^{\perp}+\phi_{3}^{\sigma}[p_{j}\delta_{ik}^{\perp}+p_{i}\delta_{kj}^{\perp}]$
(121) $\displaystyle\,\,\,\,+\Gamma_{kq}^{\sigma}T_{qji})\nabla_{k}T$

with $\Gamma_{ij}$ given by Eq. (108). There are analogous couplings of flow
to concentration gradients and electric fields.

The dissipation of the rotation $\delta p_{i}$ and of $\delta\Omega$ is as in
the D2 phase, Eq. (3.2.2). The dissipative Lehmann-type terms are the same as