Liquids with Chiral Bond Order
Randall Kamien
To cite this version:
Randall Kamien. Liquids with Chiral Bond Order. Journal de Physique II, EDP Sciences,
1996, 6 (4), pp.461-475. <10.1051/jp2:1996192>. <jpa-00248309>
HAL Id: jpa-00248309
https://hal.archives-ouvertes.fr/jpa-00248309
Submitted on 1 Jan 1996
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J.
Phys.
France
II
Liquids
(1996)
6
Chiral
with
Randall
School
(Received
July 1995,
12
find
elastic
the
elucidate
and
chiral
and
grain
K2
of
its
hexatics
boundary
Introduction
1.
phase,
constants
possibility
topological
the
describe
I
conical
a
and
K3.
and
to
of
that
and
USA
crystals
the
the
chiral
a
liquid crystal
chiral
a
of
discuss
I
propose
of
08540,
helium-3
in
inequality
constraint
relating defects
physical meaning. Finally
phase
NJ
accepted 9 January 1995)
January1996,
8
liquid
of
vortices
phases
new
satisfying
form
Princeton,
devices
similar
>
Study,
Advanced
final
in
models
and
Ferroelectric
Abstract.
I
for
Institute
Textures
PACS.85.50.L
Order
received
Theory
PACS.61.30.Cz
PACS.67.57.Fg
461
PAGE
(*)
Sciences,
Natural
1996,
APRIL
Bond
Kamien
D.
of
461-475
cholesteric
a
role
for
of
in
an
fluctuations
magnetic
the
in
the
bond
order
I
make
the
analogy
ground
state,
field
to
to
In
of
order.
for
Frank
phase
addition
textures
the
hexatic
field
of this
context
between
akin
and
nematic
sufficiently long polymers.
in
defect-riddled
with
applied
the
I
and
derive
nematic
liquid crystals
Renn-Lubensky twist
smectic
smectics.
Summary
prediction of the hexatic liquid-crystalline phase ill, there has been a
order [2j.
experimental
evidence
for
two-dimensional
bond-orientational
Aside
from the possibility of a new type of liquid crystal,
hexatic-like
be
order
intermediate
can
an
freezing of a liquid. Recently there has been a plethora of new results
contin~o~s
stage in the
for chiral phases in both two [3-5j, and three
dimensions
both a
[6, 7j. In this paper I propose
uniform
and defect laden ground state of a liquid crystal with both hexatic
order and chirality.
Toner [8j has proposed that
liquid crystals, upon cooling, could form a liquid crysnematic
talline
phase with
order
and
hexatic
order in the plane
perpendicular to it.
This
nematic
"N + 6" phase may have been recently
observed in a mesophase composed of highly aligned
B-DNA [9j. In Section 2 1 will consider the melting of a chiral
columnar phase [7j into a chiral
liquid with hexatic order. I discuss the melting of a columnar crystal through dislocation loop
unbinding and derive the Landau theory of the chiral liquid. When a proliferation of edge and
dislocations
develops, the crystal melts, leaving a normal liquid with hexatic order and
screw
residual
with the crystal rigidity. Typically, long thin
constraints
associated
molecules
do
no
columnar
phases, but freeze via smectic phases. I expect then that chiral, disc shaped
not have
molecules
could form the mesophases
discussed
here.
It is well known [10j that
cholesteric
director, under an applied magnetic field can unwind as
well as tip up and display a conical
However, the conical phases exist only when the
texture.
Since
the
good
deal
(*
PA
Current
19104,
e-mail:
g
theoretical
Les
of
address:
Department
of
Physics
USA
[email protected]
l~ditions
de
Physique
1996
and
Astronomy,
University
of
Pennsylvania,
Philadelphia,
JOURNAL
462
Frank
satisfy fi2
constants
there is
discontinuous
a
aligned along
effect
K3> which
>
field in the I
the
from
transition
PHYSIQUE
DE
seldom
is
the
cholesteric
a
direction.
magnetic field, although it does not
of the
appropriate inequality, I adapt the
the
as
realization
polymer nematics ii ii
anomalously large elastic
constants
N°4
experimentally.
When K2 < K3
in the xv-plane to a nematic
case
with
Section
In
II
director
find
3
that
the
unwind
known
hexatic
order
pitch.
helical
consequences
the chiral
of
the
has
As
same
possible
a
fluctuations
and
polymer hexatic. I
find
which depend on the polymer length. For sufficiently
long polymers, K2 will eventually grow larger than K3. If liquid crystal phases can still exist
for these long polymers, the conical phase would
This
would be
evidence of hexatic
appear.
other properties of the chiral
When
order. Turning back to the general case, I consider
hexatic.
of
nonlinearities
the
director
nematic
has
conical
a
chiral, an electric polarization
crystalline behavior ii 3j.
the
Since
must
be
there
is
that
of the
the
A
relation
Mermin-Ho
ground
states
of the
Section
in
chiral
of free
absence
the
nematic
texture
the
molecules
i1, its value
director
of the
are
liquid
ferroelectric
director.
Because
field 96> the
theory is similar to
non-chiral
relating the
discuss the geometric
constraints
will
41
field in
the
to
the
account
bond-angle
a
and
texture,
exhibit
phase could
this
into
to
smectic-C*
a
Thus
takes
bond-order
the
to
in
as
[12]
plane perpendicular
in the
i1and
phase of ~He. In
director
texture,
which
way
a
director
nematic
a
nematic
in
cholesterics
develop.
can
defined
order is
bond
determined
and
~He [14j. It plays an important
N + 6 phase and
elucidates
the
disclinations.
role
in
structure
This
of the
known
is
understanding
the
allowed
as
allowed
defects.
large variety of phases similar in structure
a
molecules
participate in N + 6 order, two kinds of
order, leading to cholesteric
nematic
states, and
twisting of the bond order, leading to braided moirA states. The twisting bond order, with an
increasing angle, is analogous to the layered order of smectics with the pitch of the bond
ever
This allows me to
defect phases
order equivalent to the layer spacing of the smectic.
consider
TGB
this
similar to the Renn-Lubensky
chiral
I
discuss
Section
in
smectics.
in
5.
state
With chirality added the N + 6 order can
When
chiral
known phases of smectics.
to
twisting of the
twisting are allowed [7j
Theory
Landau
2.
first
derive
Fluctuations
the
free
the
in
where K~
define
triad
are
the
Frank
(61, 62, fi),
(1fi6(e~~~6 is
defined
where
62
I
=
~~
~~ ~~~
~~
and 2x /qo
everywhere in
constants
must,
A
x
AI
In
this
jj
1fi61~)
the
as
Frank
~ ~°~~ ~
sum
free
Phase
of its
energy
constituent
parts.
density:
~~ ~~~ ~~~~~~
equilibrium
is the
~~~
pitch. In
cholesteric
order
define a right-handed,
orthonormal
space,
the usual
hexatic
order
parameter, ~fi6
case,
by
Columnar
phase
by the
N + 6
described
are
~~~ ~
elastic
Hexagonal
the
chiral
director
~~ ~~
bond-angle order,
of
of the
energy
nematic
~~
to
Melting
and
exhibit
12)
~~~~'~~~
"
=
IEB(r)
where
9~(r)
is the
angle
between
the
particle
I and
the
basis
vector
AI
as
measured
around
r
sign of the angle determined by A, and B(r) is the set of particles that are the nearest
[lsj) of the particle at r.
neighbors (determined by the usual Voronoi polyhedra
construction
While
choose the reference
throughout
the
physics
A
imposes
-fi
symmetry,
vector AI
space.
(modulo
hold
for
the
One
possible
choice
of
need
rotations
such
Ai
constructs
symmetry
A~.
no
neighbors.
precisely,
More
by 2x/6) is for it to point to one of the ground state
nearest
with
the
I
-
fi
~~~j~ ~ j~~j
~
141
tin
x
in
rll
r)I
j~~
LIQUIDS
N°4
and
the
(2)
sum
bond-angle
the
account
parameter
nematic
I will
now
naive
Because
definition
the
(3) implies
that 9~
that
energy
free
-
nesses.
to
there
final
is
the
j~ll
j~i
~
~
preferred
free
that
the
total
free
free
of
At
0~u~
not
from
away
density of
e~~0~w~~
=
-i1
must
with
powers
even
well
a
as
of i1 and
chiral
new
term:
the
(4)
possibility of anisotropic stiffbond-angle order parameter
Finally
there
additional
are
x
Ail
©'V96
+
V
(5)
xl
=
~ia~~tj
=
bn~
so
dislocations.
running
With
-a~~.
becomes
bn~
instead
crystal
=
e~jo~12
locked
are
e~0~bnj
commute
dislocations
96 and bn~
included
i1 - -A,
constructing
be preserved.
When
K)#oA V96
pitch 2x/#o i7,16].
two-dimensional
and
=
do
(V96)~
tendency for the
the
with
terms
the
long wavelengths 96
0~~~. Thus 20~96
derivatives
fi
-
as
will
chiral
>c~ys~~i
bn~
any
into
transformation
lfii).
-
term
The
employed.
justify the
be
and
0.
take
=
coming from
energy
j~i
+
one
must
4
the
spin-stiffnesses
© IA V96) IA IV
=
of
sum
theory
Considering only
include
must
f d~x (FR + Fo~ + FAO~).
F
polymer crystals has been formulated [7j. One would imagine
from the melting of such a crystal. Indeed, this is the case.
resulted
energy
fluctuations
quadratic
around a ground state
with ill
additional
I, the
is the
energy
the
Recently
of i1, under
r) #
(r~
x
fi and 96 18,17]:
between
FRe~
The
indicates
director
Section
i1
A, I have
direction
and
chiral
a
1fi6
can
Air)
result,
derivative
it in
to
nematic
(i1 V96 )~
=
As
symmetry
overall
energy
for 96 includes
theory
wave
nematic
couplings
non-chiral
-
a
return
463
that
of i1.
-96 (or equivalently,
of 96> the free
is
term
around
rotate
a
the
96>
such
r~
covariifit
direction
the
on
hence 96
of 961
will
ORDER
BOND
particles at position
depends on the texture
depends
of 96
includes
Fo~
The
now
-9~ and
Owing to the definition
96 together. The spin
where, since
over
CHIRAL
taking derivatives
complication and
when
ignore this
analysis a posteriori.
For
a
only
taken
is
order
WITH
=
As
in
in
the
w~~
into
When
0.
0~0~ #
lam.
In
)ui~
direction
and 96
=
+
~1/ia~~i
crystal
this
case
a
with
)e~j wq
field
are
I
is
161
by 96
introduced
introduce
dislocation
u
bni12
deformations
dislocations
1introduce
[18]
~1
+
displacement
w~~
density
Burgers vector in
the long wavelength
~e~fuj
=
into
the
which
is
tensor
the I
equal
a~~,
direction
constraint
and
crystal,
to
the
and
between
[7]
20~96
eqfbnj
=
-Tr[a]
(7)
anti-symmetric
In particular (7)
constrains
the two
tensor.
only by the density of screw
dislocations.
Since a proliferation of edge
dislocations
running along the I direction would be sufficient
two-dimensional
columnar
crystal, one might think that the
would not
constraint
to melt a
be altered
and that the free energy of the liquid state
would be augmented by it.
However,
this is not the
of
two-dimensional
melting iii dislocation
In the
usual
pairs
scenario
case.
crystal, parallel
dislocation
unbind, leading to a hexatic liquid. In the case of a columnar
unbind, leading to a hexatic.
tines
columnar
crystal, defects running parallel to
In a
can
(I) are edge dislocations.
columnar
Those
perpendicular
the
direction
defects lying in the
plane (xy) are either edge or screw dislocations, depending on whether the Burgers vector is
normal or parallel to the
dislocation
line, respectively. Thus
dislocation
contribute
loops which
where
eq
chiral
terms
is
the
to
two-dimensional
differ
JOURNAL
464
PHYSIQUE
DE
N°4
II
through edge dislocations along I will contain, as well, screwSince the
in the plane.
unit length of a pair of
energy
per
edge dislocations parallel to I is finite, infinite dislocations, ending at the boundaries, will
proliferate. Instead,
dislocation
loops will unbind [18] leading to a proliferation of screw
not
of screw
dislocations.
To incorporate the
dislocations I add to (6) a term
representing
presence
free energy density [19]:
the
dislocation
melting
the
to
like
of
crystal
the
edge-like
and
dislocations
I>
where
the
represents
E~~~j
from
the
pure
crystal the
core
theory
elastic
continuum
energy
unit
per
m
E~~~ a~iqj a~~ jqj
18j
The elastic
energy of the defects.
energy of the
for ~q along with the
constraint
e~~~0~w~~ =
length of an unpaired edge dislocation parallel
defect
comes
-a~~. In the
I-axis
to the
diverges
logarithmically with the system size.
Dislocation
loops, however will have a finite
where
energy, scaling like R In R where R is the size of the loop. Below the melting transition,
the defect loops are bound, the free energy will favor a~~
0 and hence Tr[a]
0 everywhere.
Thus 20~96
V x in. However, when
dislocation loops unbind, the effective
quadratic energy
for a~~ will be finite.
The diagonal part of (8) becomes
=
=
=
~~diag
Thus
the
eqfbnj [18, 20]
precisely
rameter.
magnetic
order
For
parallel
conical
q
and #
shifts
into
a
in
and
so
vector
V~96
=
0
coordinates
as
v
=
vo
as
into
K2 and ©.
Thus
the
with 20~96
crystal
melted
=
is
energy.
function
a
K)
K(
=
KA and
=
take ©
likewise
0. I
=
consider
as
a
of
class
ground
proposed by Meyer [10]:
free
and
(9)
bn)~
preference for configurations
K),
=
[cos #
cos
qz,
parameters.
constant
cos
# sin
divergence-free.
i1 is
so
e
v
(4)
and
V96 is a
1have
(5)
Thus
constant
the
qz, sin
extremal
The
-1(AV~96 -1(A40V
the
X
of fi>
the amplitude of the hexatic
order paclosely follow those found by Meyer for a cholesteric in a
The analysis here, however,
the bond
involves
to the pitch axis [10j.
configuration as well as that of the nematic
its equilibrium
director.
states
are
merely
to
i7
section
i1
where
Esc~ew (20~$6
"
Phase
states
96 and
I take
demoted
liquid crystal free
our
of this
field
(Tr(a))~
absorbed
Conical
results
the
be
ground
for
parameter
states
by
The
simplicity
will be
can
Uniform
look
now
and
described
The
3.
(7)
constraint
Escrew
"
i1=
ii
ii 0)
equation for 96
is
ill)
0
only solutions for 96 are linear
Inserting the ansatz (10) and
vector.
of
functions
a
constant
,
where
Q is the
l/Iinimizing
with
volume
of
respect
space
to
vo
and
the
oscillating
terms
drop
out
upon
integration
oirer
space.
I find
vo
+1796
=
ijo
sin
#
j13)
LIQUIDS
N°4
thus, with
and
~~
FeR
This
precisely
is
(o/$.
H
=
along
with
effective
K2
q
In
<
=
there
to
an
the
energy
the
/G40
of the
phase
for
as
the
nematic
at
=
For
well.
cholesteric
a
in
qofi.
grows
addition
in
~~
jo/qo
If
6#o sin #
in
to
a
is
unlikely that #o
coupling qo. However,
cholesteric
generated (o>
depends
depend
on
on
phase
can
which
form
the
>
K3
which
however,
enhancement
of
Vi
x
case
still
bn the
F
non-chiral
/
~
=
2
be
part of the
met.
is
while
for
this
ground
such
phase.
order
The
conical
state
well, with pitch
should
to
polymer
since
there is
found
"
at
(o
of
it
0 will
=
tedious.
conical
the
Long
acts
now
yet
that
light
to
cholesteric
though
case.
In
akin
0, the ©
straightforward
phase.
conical
a
from 11?]
energy
be
#o # 0, we have
experimental
the
in
Indeed
would
out
be
the
phase diagram
The
a
hexatic
as
state
as
the
as
symmetry that allows
same
expect © # 0. In the case (o
have a magnetic field type term,
constants
free
0,
=
continuously
appear
Hence
I
form
may
elastic
hope that this inequality could always
with
around
Turning back
is usually not
PHASE.
phases,
nematic
fluctuation
the
on
"N + 6"
for K2
should
fixed #o.
eventually,
nematic.
rotating bond order
a
#, which should produce a conical
all of K2, K3, 1(A and ©, and mapping it
persist
there
(16)
[19, 21j.
by the
allowed
I will
sense.
some
texture
fluctuations
is
K3
-1~3
phase
and,
conical
director
if this
11
K3
of KA for
and
q
POLYMERIC
3. I.
in
"
even
state
(15)
conical
the
function
become
it is
as
>
for
that
(off
K2
the
I find
persists,
state
the
constant,
between
In
=
a
smectic
0
considered
was
=
Thus
as
rational,
not
It is
a
x
smectic-C*-like
incommensurate
an
e
qoK2llG.
<
of fo, or, alternatively,
the
cholesteric
will
state
has,
those
fi40
<
q
K2
For
=
~~fi~(qo/qo)
qo/G
magnetic field of strength
a
unwinding by changing
fi40
qofi the
persists.
state
#
COS~
as
2
cholesteric
and
function
x)1(14)
~~(((l
x)
2
x/2)
fi40 <
ii
state
q
for
465
this
Here
state.
qol(2/fi
>
~~q~x(I
+
by Meyer
studied
possibility
nematic
pure
a
ORDER
happen:
because
must be
never
vo
can
always be parallel to the pitch axis. Translating Meyer's results
fi40 from the pure cholesteric
with increasing
transition
a
is
intermediate
qo)~
2
ii 0j
field will
K3
qo)
(qx
Q
=
conical
the
BOND
cos~ #
e
x
CHIRAL
WITH
polymers
work
recent
ill,12j, one might
constraint locking 20z96
nematics
no
l~~ould be
~
d~x
lE
Vi
u
(0zu)~
2
+
uJ
+
Gp((0~uJ)~
+
11?)
Fi [0~uj
bn,
compression modulus, po the average areal polymer density, 0~u
typical polymer length and FA is expanded to quadratic order in bn
l~~here i1 re I + do.
found in [12j
It was
that for polymers longer than to, E, 1(2 and 1(3
took on
anomalous, t-dependent values.
For polymers with only steric
entropic
interactions
L( la~ where Lp is the polymer persistence length and a is the average areal interpolymer
to
it /to)°'~~ to two loops in a d 4 e
it /to)°'~° and K3 Iii
spacing. It was found that 1(2 Iii
expansion at d
3.
where
G
=
E is the
kTt/po,
bulk
t is
=
the
=
+~
=
+~
=
JOURNAL
466
the
To
free
and
and
add
I
the
K)
K(
=
KA.
=
argument
of [12j
consider
corrections
to
easy
that,
see
only
bn
from
lead
I
and,
in
fact,
finite
non-linearities
stiffness
Fo~. Expanding
hexatic
stiffness
KA
find
that
their
values.
in
of qo, I
only
if I
This
itself;
term
K(
consider
no
K)
is
because
other
term
around
the
i1
*
limit.
0
ii?)
to
KA does
acquire
not
qo
with
As in
the
I do
not
Thus
coupling KA400zu
the
can
renormalize
KA and (o.
to
the (o
I I
=
couplings
chiral
anomalously
adding Fe~
not
changes
in (o>
#
as
consider
consider
linear
corrections
as
I
arising from
propagators
chiral
illustrate,
To
well
KA and #o do
fluctuation-induced
renormalizations.
the
in
hexatic
,
even
N°4
the
non-renormalization
consider
I
II
the
[12j
in
quadratic
to
since
to
to
interested
am
for the
renormalization,
in
ii?)
in
energy
possible corrections
#o. Following the analysis
suffer only finite shifts in
consider
PHYSIQUE
DE
V196. It
the
corrections
from
fluctuations
the
corrections
to #o
must
can
generate
the
is
infinite
any
appropriate
come
chiral
coupling. The tadpole graph arising from this interaction is proportional to (bn2(r)) and thus
only contributes to a finite shift in KA40 in three dimensions, where nematic order persists.
Alternatively, by a judicious choice of rescalings [22j, one can show that the longitudinal part of
of the Frank
irrelevant
renormalization
and thus the new coupling, which
to the
constants,
u is
only involves the longitudinal part of u, will not change the results in [12j Thus the mean field
conical
holds when I replace the
Frank
with
analysis of the uniform
state
constant
constants
t-dependent Frank
Hence, for sufficiently long polymers, it may be possible to have
constants.
(t/to)°'°~> a twofold increase in the
of K2 > K3, though since K2/K3
the
unusual
situation
10~to.
implies
the
polymer
ratio of Frank
be
For instance, in DNA Lp
600 I
constants
must
351 lead to to * 18 ~lm. With typical chain lengths on the order of centimeters,
and a
leads one to hope that
this can lead to a 37Yo increase in the ratio of K2 to K3. This
increase
Thus, it is possible, due to fluctuation
the conical phase could persist and could be observed.
enhancements, for K2 > K3 in polymer systems.
~
=
=
light scattering behavior of this phase due to its
this phase through
crossed
polarizers. As
see
will be that of a
order grows past the
cholesteric-to-conical
point, the nematic
hexatic
texture
smectic-C".
In particular, as I will show in Section 5, the analogy to the
smectic
is precise
of the ubiquitous focal conic
fan-like
and I would
textures
present in
expect the
appearance
modulation
forming, in
without
density
contradistinction
smectics [23j. This will happen
to
a
the smectic-C*
phase which would have a similar behavior. The absence of a density modulation
could be detected
microscopy and X-ray scattering, thus distinguishing this new
~ia electron
phase from the smectic. As in the moir4 phase of chiral columnar crystals [7j, in each constant
function
will be, in
that of a hexatic
the
Fourier
six broad
section
structure
cross
space,
z
of the bond
of the bulk sample, however, the
When looking at the
rotation
structure
spots.
will
order will merge these spots into a ring in the q~-qy plane. The periodicity of the
rotation
where
additional
+1, +2,.
lead to
6n(o sin #
rings at q~
n
exhibit
ferroelectric
behavior.
conical
In addition, as in a smectic-C*, this uniform
texture
can
This
electric dipole
P to form.
The
moment
symmetry of the phase allows for a spontaneous
electric field through a term in the
be seen
through the allowed coupling to an external
can
3.2.
N
Aside
PROPERTIES.
BULK
[8j,
character
+ 6
should
it
from
the
possible
be
to
=
free
energy
F(~
This
E
a
=
-
term
that
I
P
=
interaction
extra
-E.
E
=
have
is
E
in
these
iv
x
Al
=
~1(o sin
#cos # [Ey
cos
qz
E~ sin qzj
(18)
symmetry and is chiral, since under parity,
normal"
constructing (18). There is, however,
to be v in
Since the
chiral phases that has no analog in the
smectic-C".
preserves
the "layer
taken
allowed
~1iv fi)
the
nematic
LIQUIDS
N°4
of
direction
by the
determined
is
v
CHIRAL
BOND
coupling (o
and
WITH
chiral
Fflc
iv
liE
=
x
ORDER
fi,
467
non-chiral
a
allowed
is
term
(19)
fi)
Under
IA and v both change sign) and is not chiral.
changes sign, as does E. This term is allol~~ed because
configuration with v and -v.
They differ by
between
there is a difference, for fixed fi,
a
of the
helix.
smectic-C~
the layer normal
does not have an
the
handedness
hexatic
In a
because of the up-down symmetry of the layered
unambiguously defined direction
structure.
The
that
ferroelectric
liquid crystal does not have a density wave.
Unlike a smectic, this
wave
will make
hexatic
order along v. In Section 5
is one of twisting
supports the conical
texture
and chiral
the analogy between
smectics
hexatics
precise. Note that Fflc also implies that
more
applied electric field will favor configurations in which the hexatic order twists in planes
an
perpendic~tar to i1.
the
which
preserves
spatial
reflection
v
Topological
4.
Up
this
to
symmetry
nematic
fi
x
unambiguously
Constraints:
Mermin-Ho
The
point I have ignored the topological
of 96 arising
orientational
from
(3),
order
it is
that
clear
parameters
of ~He-A
related
relation
in fi
textures
are
Relation
each
to
96 and i1.
between
influence
can
other
in
Saddle-Splay
and
much
the
the
From
local
the
value
The
two
that
the
two
way
same
definition
of 96.
related, namely, through the
Mermin-Ho
relation [14j. I will
Let (Ai,A2> A) be a right-handed,
orthonormal
If I consider a reference
consider a nearby triad, it will be rotated
from the
triad.
triad and
reference
triad by an
azimuthal
angle ii around A, followed by a rotation of b9 about the
intermediate
of bill about the new 6i
order in these
To lowest
rotation
62> and finally by a
small angles,
order
parameters
sketch
here
a
brief
are
derivation,
following [24].
A[
=
Al
=
fi
and
b9Ai
@2
ii
m
Ai
bA2>
b61
~S
bA2
*
+ b~fi62
Taking
the
curl
of
both
sides
+ b9A
b#Ai
Mermin-Ho
V~#
e
using ki
and
iv
celebrated
-b#62
bififi
(21)
e[V~e(
(22)
or
Q~
x
ni~
relation.
A
=
x
=
A2
=
A,
have
we
which
vector
defined
is
=
the
relation
between
96 and Q holds.
phase,
disclinations
simple topological
of
consequence
state
a
so
of the
N + 6
123)
je~~penp~n~a~nPapn~,
plane defined by 61 and 62 will change not only because its true
fixed triad) but also because the basis
change. A vector
vectors
Q. In the ground
fact be
in space if and only if V96
constant
and
(20)
bififi
b9Ai
A
=
b9fi
+
so
Thus
the
b#A2
ki
relative
changes (relative
li
96ki
=
state
cos
of
of this
relation
by the
non-cylindrically-symmetric
nematic
in 96
The
are
content
forced
in
spatially varying
the
to
direction
a
9662
hexatic
li
is
that
in
the
is
molecules.
+ sin
texture.
For
It
to
a
will in
constant
ground
quantifies
concreteness,
JOURNAL
468
PHYSIQUE
DE
_~/
Fig.
Escape
i.
into
escape
thicker
the
the
line is
nematic
the
nematic
axis
and
the
of
small
axis
while
the
shorter,
director
object.
biaxial
a
the
small
N°4
Q-
dimension
dimension,
normal
is
third
the
into
third
II
director
have
director.
plane
the
to
At
of the
tries
to
long
the boundary
boundary.
disclination.
+i
a
small
normal
and
disclination
nematic
+I
a
to
line is the
thinner
uniform
is
As
forced
is
The
boundary
One might
conditions.
well
known
However,
is
it
imagine a +I
center
run
up
along the disclination.
However, if
dimension"
that such a geometry will "escape into the third
the molecules
have the same (biaxial)
orientation
at the cylinder wall, when they
escape they
take
biaxial
a
molecule
in
line
compelled to force
in Figure I.
will be
cylindrical
a
disclination
is
section, I took V
(10)
When
states
ground
from
energy
not
unlike
Gaussian
and
for
this
is
0, ~N.hich,
=
differ
look
I
excited
This
v
in
the
order
up
parameter
(23).
in
~feyer ansatz,
of the
case
the
configurations I
ground state by the
with
phases
hexatic
system size,
the
flexible
the
in
that
Note
center.
in
This
the pre~~ious
justified by simply plugging
is
on
fi
F~
~
=
of
terms
general,
and
A3
e
Q~
the
V96, for
fi
term
always
should
~
mct~de
i
instance,
Q
with
were
be
hexatic
enforced.
forced
in
by
96Ai + sin 96A2
cos
by (AI, k2, A), I find
=
determined
Hi sin 96
cos
nj~
(24)
96
free
elastic
~ jvo~
=
li
director
basis
Hi
appropriate
The
~ jD~Njj~
+
local
[V96
A~
"
~jj
i
j lD~Njj~
li,
one
e[0~e[.
=
hexatic
-6~ [V96
=
D~N2
as
should
relation
The
account.
In the
disclinations
There
the
into
disclination.
energy
a
Mermin-Ho
the
the
in
constraint
free
of
membranes.
li
of
derivatives
covariant
the
take
must
so
[25]. In particular, if I take
D~NI
In
summed
is
state
the
diverges
curvature
compute
where
biaxial
the
into
frustration
This
cylinder.
(23).
into
phase,
x
normal
of the
the
disclination
a
illustrated
with
geometry
will
~jj
+
~
i
IA
~
n~eqN~D~Nj
becomes
of 96
derivatives
would
energy
as
e
be
n)j~
jvo~
Q). Thus, in
wobbling
rotation
true angle of
bond-angle field in the
fi
(V96
of the
consequence
a
j25)
of the previous section~ 96 is, in fact, the
the
calculation
bond-angle in fixed space. It is useful to distinguish the
frame of the (Ai 62> fi) triad 96 and the bond-angle field in space ii-e- the lab frame) #6. The
about
of fi.
In
hexatic
of the
covariant
of the
derivative
bond-angle field 96
triad-fixed
D96
absorbed
where I have
is
the
constant
phase
Section
is
correct,
3 is
bonds
as
triad
the
rotation
not
rotate.
are
its
observable
is
the
V96
Q
Q into #6
=
consequences.
rotation
space-fixed
(26)
V96
(which
Indeed, then, the
do
precisely #6 and
=
is
I
naive
can
do
since
calculation
V
x
Q
for the
=
0).
uniform
When
#6
conical
The
bond-angle field 96 appearing in
angle of the hexatic order. In addition,
LIQUIDS
N°4
CHIRAL
WITH
BOND
ORDER
change of basis vectors 61(r) and 62(r) should
of 96 through
correlations
only "gauge-invariant"
because
jr; r)
Ge~o~
/
exp(I
e~~~6~~)
=
change
not
a
consider
469
physics
the
of 96>
1will
(27)
dt)e~~~~6~°)
Q
r
where
f is
path connecting
a
separations
r,
transformations
of 96.
interpretation
the
texture,
contribution
destroy, long-range
unbound
Since
equilibrium.
as
free
a
in
the
fi
that
the
case,
twist
texture
If such
normal.
z
around
traversed
is
=
6
with
field of
surface
V
at
to
r
normal
-V
x
V96)z
I
=
fi need
director
of
a
0~bn~0ybny
=
if the
appear
amount
around
be the
blue
molecule
of
a
neighbors since the saddle-splay
being straight. It is not
center
one
the
director
be
tangent
to
the
lines.
This
in
in
(28)
e~jk0jbn~0kbny
m
directors
relation
there
of
is
chiral
N + 6
double
twist
is
texture
us
is
a
to
2x
a
which
to
that
has
these
double
would
19
be
Indeed,
as
be
polymers, just
in
the
nematic
are
circuit
a
fewer
braided
in
the
in
directors
that
there
twist
must
the
shows
Since
be such
/6. However,
surface.
system, (29) implies
cylinder,
Figure 2
director.
saddles
multiple of
surface
saddle-surface.
to
the
in
the
that
normals
no
positive.
necessary
normals
deformations
tells
collection
the
are
boundary
their
(29)
0~bny0ybn~
saddle-like
phases
present in the
central
of
Mermin-Ho
not
liquid crystal
were
order
by
constrained
V96)z
to
normal.
is
jeqke~p~n°0jnP0kn~
are
x
surfaces, Air) iv x Q(r)j is the
Rewriting the curvature, I find
saddle-splay term of nematic
well known
saddle-splay elastic
constant.
vectors
Air)
which
nearest
nematic
and
a
texture
a
a
greatly reduce,
can
and
wobbling
possibly
to
of the
the
rich
corresponds
pointing along
N + 6
double
curvature
definition
usual
with
=
words,
disclinations
must
saddle-splay term
the
measures
single-valued on a surface, the
the integral of their saddle-splay
96 is
the
to
disordered,
V
other
The
and
0
=
to
=
these
iv
in
Q
highly
a
where
screen
are
iv
Disclinations
nematic
to
is
[(i1. VIA
fi iv fi)j, the
I can identify ©' with K24 the
understood
geometrically: to lowest
be
can
the
cost
regions
to
curvature
A
This
from
l&iriting 96
iv x Q) e
liquid crystals. Thus
that
nematic
invariant
locked
chosen
that
case
the
large
for
be
logarithmically-divergent energy they will not appear in
divergent energy
0 have precisely the
x Q #
same
9(~°°th + 9(~~ l~~e can expand around a background of
Q, leaving only smooth
in
regions with V9[~~
variations
Q
according
Mermin-Ho
relation
(23).
that
the
Notice
to
x
dislocations
Gaussian
=
leading
space,
will
order.
disclinations
disclination.
The
96
hexatic
Q
x
function
arbitrary function of space.
These
an
change V x Q and hence do not change
there
obstructions
0 everywhere,
are
no
not
when
(27)
to
However
disclinations
that
Note
V
non-vanishing
is
correlation
is
uJ
do
function
Greens
This
where
When
in
constant
this
order.
which
those
of 96.
be
to
When
hexatic
96 by w,
and
structure
choosing A~(r)
to
VUJ
precisely
are
disclinations
the
and 0.
r
long-range
is
by
of Q
shifts
under
there
than
polymers
that
field
the
as
phases of cholesteric liquid crystals [26, 27j. The integrated trajectories are also
4-dimensional
known to be the projections of great circles of 53> the surface of a
sphere, onto
flat
3-dimensional
The
of
implies
that
the
coordination
number
53
is less than
curvature
space.
(29).
which
with
This
the
molecular
6
illustrated
Figure
Here
is
consistent
is
in
3.
28j
[27,
z
(represented by dots) lie on a local triangular lattice, as in the hexatic phase. When
centers
the polymers are tipped over, the
distance
This
between
their
increased.
is necessarily
centers
reduces the allowed packing density leading to
coordination
numbers
less than z
6.
found
in
blue
=
=
JOURNAL
470
Fig.
z
2.
A
6
=
nearest
equal
local
the
to
braided
with
state
neighbors.
The
nematic
double
a
lines
director
twist
The
texture.
II
N°4
line is
center
long polymers
represent
may
PHYSIQUE
DE
or
merely
straight
and
has
fewer
tangent
with
curves
than
vectors
(the integral curves).
0.'
~.
--
L-i
a~
Fig. 3.
tipped
of the
Illustration
the
over,
and
leads
5.
The
to
distance
coordination
Equivalence
DERIVATION OF
uniform
with
state
A
different
lattice
A [6j
field
in
scenario
less
than
Smectics
and
HEXATIC
texture
z
=
molecules.
This
FREE
can
ENERGY.
the
when
allowed
molecules
are
packing density
lD~b61~
I
have
shown
that
for K2 > K3
sufficiently strong hexatic ordering.
Abrikosov flux
defect state, analogous to the
smectictwist grain boundary phase of chiral
isotropic hexatic in a background
nematic
persist
possible, leading to a
isomorphic to the
Landau
theory for an
the
=
that
the
Hexatics
Chiral
and
Fhex
Note
reduces
6.
is also
first
tilted
increases.
of
centers
smectic-C*
a
superconductors
Consider
numbers
CHIRAL
5.I.
a
their
of
packing
frustrated
between
for
+ ri~b61~ + ~l~b61~
130)
LIQUIDS
N°4
where1fi6
is
hexatic
the
order
(/G/6)e~~°°, and,
usual,
as
the gradient of 96.
Q from
Adding
the
this
to
~~ex
Recall
under
that
symmetry.
energy
(T-Tc).
cc
the
to
-fi,
1fi6
ill
++
and
first
zero,
/
((D
d~x
"
6i(off)
18(()
~fi6(~ + IT
2
shifting Tc upwards by 18((, the free energy
of a chiral
between
smectic [29] where 2x/6(o> the spacing
orientation, is the equilibrium smectic layer spacing. There
the
after
the
in
latter.
While
fi
in
invariance
identical
for
the
96
ordered
and
be
to
+ [4l
state
i1]
r,
(e.g.,
fi
x
corresponds
the
is
~~~~
(31)
respects
up
that
of
that,
so
ii,
=
different
to
so
I
the
may
nematic
write
the
energetics
phases.
of
but
the
that
between
of Q
shifts
is not
the
surfaces
is
a
space,
rotation
precisely
that
bond-order
same
difference
ferroelectric
between
term
(19)
-40 and 96 - -96 (1fi6
ill ).
-96 must be accompanied by
-#o in the hexatic that does
-
++
the
the
on
between
Q and
transformations.
The
distinction
true
gauge
symmetry, but
in
Under
is
the
non-chiral
note
are
hexatic
with
important
an
96 #o and
to
by 0uJ
sign of
rotations
chiral
a
Ffi1(32)
~(~fi6(~ +
rather
hand,
other
by angle 4l,
fi
redundancy
responsible
a
are
4l
i1+
-
x
fi
and
spatially-uniform,
rotation
is a
symmetry of the theory. A rotation is physical
equivalent physical systems [30j. In the highly aligned limit,
the
x
Q
chiral
=
order
the
fluctuations
in
tJ(bn2)
from
that
than
recall
important
also
is
lowest
to
I, V
m
(32)
of
(o
under
invariant
(~fi6 (~ +
the
allowed
difference
a
transformations
bn is small and n~
derivative is much smaller
smectic
there
what
is
is
symmetry
a
analogous smectic. It
accompanied by
the
when
that
physics
the
words,
which
physical descriptions. Rigid
gauge-like coupling of 96 to fi.
of
-
other
hexatic
smectic,
of 96 by uJ
under these
Shifts
96
a
chiral
appears
In
exist
not
fi.
In
that
-fi.
-
the
and
smectic
split
~)
as
Fhex
Thus,
(il~
in
term
be
will
m
subtracts
which
have
I
~
the
so
term, though identically
Fo~,
in
approximately1fi6
is
derivative,
covariant
4~(1fi6(~
~~#0il'ilfi~~1fi6 ~1fi6i~lfiij
471
phase,
broken
the
in
refers
D
ORDER
BOND
(covariantly modified)
term
"
-
second
The
free
total
fi
which,
parameter,
r
chiral
CHIRAL
WITH
chiral
phases
hexatic
hence
and
term
should
of i1
around
a
the
contribution
in the
derivative
similar
be
to
jobn.
that
of
the
covariant
Thus
I expect
the
analogous
phase. In the
there is pure chiral
hexatic order, with 96
nematic
(oz. This is equivalent
state
smectic-A
shown in Figure 4. Upon reduction of the nematic order, the
nematic
to a
as
phase. The chiral
director tips out of the I direction
and the phase is equivalent to a smectic-C*
order acquires a longer pitch, which is equivalent to a larger smectic layer spacing. Finally, as
further, the director lies down into a cholesteric
and there
the hexatic
order is reduced
texture
The
hexatic
order.
This is just a pure
cholesteric.
layer spacing has gone
is no chiral
smectic
there is no
order.
smectic
to infinity and
It is
now
clear
5.2.
DEFECT
Mermin-Ho
the
that
(sin #
phase
=
conical
state
is
just
one
ii,
PHASE:
relation
uniform
possible
from
=
THE
and
smectic
the
RENN-LUBENSKY
covariant
derivative,
Ignoring
STATE.
this
model
has
the
been
complications of the
extensively [6]
ground
with
state
a
studied
and, in the type-II regime (where u is sufficiently large) one expects
proliferation of defects. I propose then a new twist grain boundary state of hexatics.
It
a
will
of regions of twisted
of
hexatic
N + 6 separated by grain
boundaries
made
consist
up
disclinations.
Within in each region the bond order will twist with pitch 2x /(o and the director
will be well aligned in a nematic
As one
along a pitch axis perpendicular to the
state.
moves
JOURNAL
472
DE
PHYSIQUE
N°4
II
2ir
6[
~f~
z
Fig.
Model
4.
Between
the
planes
The
of
chiral
a
planes the
analogous
are
fit mod 2x/6.
In each plane the
bond-order
parameter is 96
along the average
uniformly
director, i1
nematic
fi.
precesses
planes, though there is no density wave in this liquid crystalline
hexatic.
bond
#
order
=
smectic
to
phase.
direction
will jump by some
finite angle
grain boundary, the nematic
which,
long
distances,
will
be
cholesteric,
but
at
to
appear
a.
a
pure
will in fact have regions with rotating
hexatic
order.
Itenn-Lubensky state
discussed
The defect free regions of the
above
consists of the
nematic
x/2 in (10) of the uniform conical phase. It is also possible, however, that an
limit ii
conical phase ii < x/2) fills the regions
the defects.
In this
intermediate
between
#n~
case
would be the equilibrium conical angle and the
director
would relax to this
This
exotic
cone.
phase would be similar to the TGBC phase [31] in which the defect-free regions are not smecticA, but rather smectic-C, though in this case it may be more
appropriate to think of the clean
instead.
regions as being smectic-C*
director,
nematic
This
across
lead
will
a
state
to
=
might speculate
One
behavior
same
related
to
think
to
smectic
i-elation
a
TGB
energy
in the
state
are
associated
hexatic
l~~here 9 is
the
requires
bond
field 96 has
order
much
the
N + 6 phases should be in
words an N + 6 phase is closely
in other
with
proposition
smectics it is a savory
configurations
still
possible.
with
the
order, 96 will
azimuthal
DEFECTS.
of chiral
without
in 96
are
not
to
of the
energy
of the
"curvature"
wind
a
required
The
coordinate.
In
around
The
in
be
core
nematic
each
equations
light
of the
Configurations
logarithmically
clarification.
some
disclinations
hai~e
can
TOPOLOGICAL
OF
defects
However,
the
structures
nematic.
STRUCTURE
of the
structure
length.
of
space.
THE
5.3.
blue
limit
in
92> a
the
Since
parameter,
order
myriad of blue phase textures:
Owing to the
connection
phases made of smectic layers, where now the phase 92 present in the highlamellar layers of the
blue phase [26] could be interpreted as ticking off the
a
This
prospect is under investigation [32].
with
biaxial
a
of
chirality
Ho
as
analogy
close
possible phases.
other
on
biaxial
innocuous.
so
of
satisfy the
divergent energy
The
will be
modified, of
field.
Considering
xv-plane,
motion
relation,
Mermin-Ho
which
defects
independent of
imply that at
by
screw
z,
per
unit
for
necessary
course,
a
the
Mermin-
and
radii
the
extra
dislocation
so
96
larger
=
9,
than
LIQUIDS
N°4
@@/40
the
will
field
nematic
CHIRAL
WITH
lock
473
"superfluid velocity"
the
to
ORDER
BOND
v
Vi 96/40, and
=
thus
~
bn
(33)
m
qor
where
vii IT is the unit azimuthal
(xfi
vector in cylindrical
only
Mermin-Ho
depend
the
component of the
not
on
z
1find,
leading
order
bn, that
Using
in
(29)
to
component.
does
the
z
iv
at
large
for
Thus
the
r
vorticity
infinity. Thus,
the
equal
and
the
to
rate
ignoring the effect of the
defect itself, and certainly
of the
energy
(r~l
since
r
which is
which
at
nematic
does
is
non-zero
(34)
qo
r~~,
field Q falls off like
gauge
off
falls
that
is
=
~
the
that
relation
-(
Hi
x
Note
coordinates.
e
bn
than
adds
defect
defects
in
the
relation.
I
to
rate
at
which
40bn) falls
energy
become
an
the
cause
the
(Vi 96
difference
curvature
not
faster
much
the
off
comparable to
energetically
prohibited.
Turning
which
back
are
relation
Ho
Meyer
the
to
find
to
a
ansatz,
and
free
disclination
family of
might
one
satisfy
hence
defects
in
[cos #(r)
cos
consider
Mermin-Ho
the
conical
a
Using
state.
uniform
can
radially
a
solve
conical
the
state
Mermin-
dependent
ansatz, I
take
i1
In
the
of the
center
#(cc)
#~.
=
this
In
qoz,
#(r)
cos
#~0
VI
X
Iv
iv
vj
x
I
a
to
region which
conical
a
does
state,
I.e.
or
136j
or
r
CDs
j oj
r
or
~
v
region
relaxes
j oj
CDs
=
~
(35)
# 0~
r
xqo
vi
x
CDS
=
~
for
#(r)]
defect, #(0)
cholesteric
0 and I have a pure
Moreover, at infinity, the director
order.
to
coinpute
case, it is straightforward
[~
solving
sin qoz, sin
=
hexatic
support
not
=
have
~~~
v
~~~~
+ qo
=
(sin #~
sin
#(r))
in
cylindrical coordinates, and V96
equilibrium value #o sin #~i in a
I +
(37)
V96
r
where
conical
the
sin
more
origin.
Hence
precise,
Since
sin
from
Far
#
0
m
and,
order
in
#(r
cos
#(r)/r
the
is
of the
variation
state.
core,
yl)/r
(xfi
e
non-singular
-
azimuthal
unit
bond-angle
vector
direction,
with
of the
-
=
=
is well
behaved
at
r
=
0.
In
the
6
/
dt
=
dill
/
v
=
M
dxdy iv
x
vj~
defect
of this
core
nematic
~~~
a
40sin#~i and is not singular. Inside
core, as r
cc, v - V96
To be
minimizing the free energy for V96> the bond order is constant.
configurations,
have
field
V
diverge
nonsingular
to
must
not
at the
x v
0)
1, (36) implies that 0~# must go to 0 no slower than linearly in r.
the
field, while outside the core there is a
In considering configurations (37), it is
essential
note
to
closed loop of radius R in an arbitrary
slice, 96
constant-z
by 2x/6 and hence
order
is
pure
=
2x
pure
that,
must
there
uniform
is
[sin #(R)
state.
integrated
single-valued up
when
be
cholesteric
pure
conical
sin
#(0)]
around
to
a
shifts
(38)
JOURNAL
474
Thus,
nematic
were
to
This
N°4
are
quantization
have
II
configurations without any free disclinations, then sin #(r) must take on
integer multiples of1/6.
This
would
prevent fi from relaxing back to a
configuration on a longer length scale than the disappearance of (1fi6(. In other words,
which
there
PHYSIQUE
consider
if I
values
the
DE
have
1fi6
tilt.
constant
the
usual
enforces
(of
quantization
a
radius
+~
This
is
as
defects
type II
buckling
disclination
of 96
vanished
circulation
which
in
a
prevents
of the
of the
core
a
transition
if the
London
of the
TGB
in
hexatic
~),
that
This
membranes
tilt
sort
of
smaller
were
if
would
nematic
restriction
This
[33].
Therefore,
of fi.
the
core
depth I
penetration
state.
the
in
outside
than
(.
reminiscent
is
type of defect
has
a
perhaps an annular region in which the tilt of the
or
changes gradually by 2xm/6, for integer m. Outside the core there is hexatic order as
nematic
well as
order.
Inside there is a gain in cholesteric
nematic
nematic tips over and
energy as the
exploits its chirality, while there is a loss of hexatic energy as its order is destroyed. A detailed
theory must be performed in order
energetic calculation, using the entire rotationally
invariant
determine
whether
this non-topological defect is stable.
to
in
core
which
there
is
hexatic
not
order,
Acknowledgments
It
is
T.C.
pleasure
Lubensky,
National
acknowledge
to
a
stimulating
discussions
with
Nelson, P. Nelson, and J. Toner.
RDK
Foundation, through Grant No.
PHY92-45317.
D-R-
Science
P.
Aspinwall,
l~ms
supported
M.D.
part
in
Goulian,
by the
References
ill Halperin
[2j
[3j
[4]
[5j
B.I. and Nelson D.R., Phys. Re~. Lett. 41 (1978) 121; Nelson
D.R. and Halperin
B-I-, Phgs. Re~. B19 (1979) 2457.
For a review, see Standburg K.J., Rev. Mod. Phys. 60 (1988) 161.
Langer S. and Sethna J., Phys. Rev. A 34 (1986) 5035; Hinshaw G.A. Jr., Petschek R.G.
and
Pelcovits R-A-, Phys. Rev. Lett. 60 (1988) 1864.
Selinger J.V. and Schnur J.M., Prigs. Rev. Lett. 71 (1993) 4091; Selinger J-V-, Wang Z.-G.,
R.F. and Knobler C.M., Prigs. Rev. Lett. 70 (1993) l139.
Bruinsma
Nelson P. and Powers T., Prigs. Rev. Lett. 69 (1992) 3409; J. Prigs. II France 3 (1993)
1535.
[6j
Renn
S.R.
Lubensky T.C., Prigs.
41 (1990) 4392.
and
Rev.
A
38
(1988) 2132; Lubensky
T.C.
and
Renn
S.R., Phgs. Rev. A
[7j
Kamien
R-D-
and
D-R-, Phgs. Rev.
Nelson
Lett.
74
(1995) 2499; Phgs.
Rev.
E 53
(1996)
650.
J., Phys. Rev. A
[8j
Toner
[9]
Podgomik R., Strey H.H.,
Proc.
Nat.
Acad.
Sci.
in
27
(1983)
press
1157.
Gawsrich
K., Rau D-C-,
Rupprecht
A.
and
Parsegian V.A.,
(1996).
Meyer R-B-, Appt. Prigs. Lett. 12, 281 (1968); Appl. Phys. Lett. 14 (1969) 208.
Toner J., Prigs. Rev. Lett. 68 (1992) 1331.
It.D. and Toner J., PAYS. Rev. Lett. 74 (1995) 3181.
[12] Kamien
R.B.,
Liebert L., Strzelecki L. and Keller P., J. Prigs. France Lett 36 (1975) L-69.
Meyer
[13]
[10]
[iii
[14] Mermin N.D. and Ho T.L., Phgs. Rev. Lett. 36 (1976) 594.
Transitions
and
Critical
Phenomena,
[15] See, e.g., Collins It., in Phase
Green
Eds.
(Academic,
New
York, 1972), Vol. 2.)
C.
Domb
and
M.S.
LIQUIDS
N°4
WITH
CHIRAL
BOND
ORDER
(16] Terentjev E.M., Europhgs. Lett. 23 (1993) 27.
[17] Gianessi C., PhYs. Rev. A 28 (1983) 350; PAYS. Rev. A 34 (1986) 705.
M-C- and Nelson D-It-, Phgs. Rev. B 41 (1990) 1910.
Marchetti
[18j
discussions
this point.
thank J. Toner for
[19j
on
[20] Nelson D-R- and Toner J., PAYS. Rev. B 24 (1981) 363.
[21j Ramaswamy S. and Toner J., J. PAYS. Suppt. A (1990) 275.
thank Tom Lubensky for many
discussions
this point.
[22j
on
[23j de Gennes P.G. and Prost J., The Physics of Liquid Crystals, Second
University Press, New York, 1993).
[24j
[25j
[26j
[27j
[28]
[29j
[30j
[31]
[32j
[33j
475
Edition
(Oxford
III (World Scientific, Singapore, 1992).
(1987) 1085; see also David F., Statistical
Mechanics of Membranes
and Surfaces :
Jerusalem
Theoretical Physics,
Winter
School for
D.R. Nelson, et. at. Eds. (World Scientific, Singapore, 1989).
Meiboom S., Sethna J.P.,
Anderson
P-W- and
Brinkman
W.F., PhYs. Rev. Lett. 46 (1981)
1216; see also Wright D-C- and Mermin N-D-, Rev. Mod. PAYS. 61 (1989) 385.
KlAman M., J. PAYS. France 46 (1985) 1193.
KlAman M., Rep. Frog. PAYS. 52 (1989) 555.
de Gennes P-G-, Solid State
Commun.
14 (1973) 997.
Halsey T-C- and Nelson D-R-, PhYs. Rev. A 26 (1982) 2840.
Renn S.R. and Lubensky T.C., Mot. CrYst. Liq. CrYst. 209 (1991) 349.
Kamien R-D- and Lubensky T.C., unpublished.
Seung S. and Nelson D-R-, Phys. Rev. A 38 (1989) 1005; Seung S., Ph.D. Thesis, Harvard
University (1990); see also [25j.
VoJovik
Nelson
G.E.,
D.R.
Exotic
and
Peliti
Properties
of
L., J. PAYS.
3He, Chap.
France
48