Molecular theory of order electricity
M. Osipov, T. Sluckin
To cite this version:
M. Osipov, T. Sluckin. Molecular theory of order electricity. Journal de Physique II, EDP
Sciences, 1993, 3 (6), pp.793-812. <10.1051/jp2:1993168>. <jpa-00247872>
HAL Id: jpa-00247872
https://hal.archives-ouvertes.fr/jpa-00247872
Submitted on 1 Jan 1993
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J.
Phj's.
France
II
(1993)
3
793-812
1993,
JUNE
PAGE
793
Classification
Physics
Abstiacts
61.30
68.10C
theory
Molecular
Osipov ('. *)
M.
A.
ii)
Institut
Faculty
electricity
order
and
T.
Theoretische
for
J.
(2)
Sluckin
Physik,
Technische
Studies,
University
Hardenbergstr.
Berlin,
Universitht
36,
Germany
Berlin,
D-1000
(2)
of
of
Mathematical
Southampton,
of
Southampton
SNH,
S09
Great-
Britain
(Received
12
1992,
November
accepted
final form
in
Mai<.h
1993)
Barbero
and
been
employed by Durand,
liquid
crystal
equilibrium
conical
anchoring
at
to
from a
molecular
point of view, using the density
examine
this concept
interfaces.
In this paper
we
force
of the
electrostatic
functional
theory of liquid crystals. We show that the long range
nature
formal
problems with rather profound
molecules
with
quadrupoles
between
creates
permanent
formulations
of liquid crystal
microscopic and macroscopic
the link
between
consequences
on
employed
with
Gennes
gradient expansion
be
result
is that
the
Landau-de
theory. One
must
analogues in the
nematic.
These
formal
problems
have
caution
in an
inhomogeneous
extreme
explored by Ewald long ago. In addition we derive from a statistical
theory of dielectrics
and
were
describe
order
electricity,
and
phenomenological
relations
used
mechanical
viewpoint the
to
interface
and at a
of order
electricity at an isotropic-nematic
explore in detail the
consequences
The
Abstract,
colleagues
order
particular,
in
the
has
of
existence
interface.
nematic-substrate
1,
of
concept
explain,
electricity
5
Introduction.
liquid
Nematic
ordering
order
vector
is
crystals
characterized
parameter
n(r) is the
is
uniaxial
director
and
ordering
orientational
possess
the
by
tensor
can
be
and S is the
order
parameter
written
scalar
as
S~p
nematic
=
quadrupolar
with
S~p. Far
S(n~ np
order
from
(1/3) &~p ),
parameter.
induced
parameter
S(r ). Note
(*)
Alexander
von
Academy of Sciences,
that
the
Humboldt
Leninsky
local
Fellow.
orientation
of the
Permanent
Prospekt 59,
Moscow
director
address
:
117333,
n
jr
Institute
Russia.
is
very
of
the
tensor
the
where
unit
liquid crystals
polarization can be
Nematic
nonpolar in the ground state.
Nevertheless,
without
external
field a
even
by inhomogeneities in the tensor
order
S(r), leading to
parameter
~°~~ ~" ~ ~P~&,
P'
In the
inhomogeneous
nematic
liquid crystals the spatial
variation
of
determined
by the
variation
of the
director
n(r) and of
parameter S~ p (r) is
are
This
symmetry.
boundaries
relation
a
the
tensor
the
sensitive
of
the
order
scalar
order
the
action
to
Crystallography,
Russian
794
JOURNAL
PHYSIQUE
DE
II
6
N°
conditions
boundary
scalar
order
by contrast,
is
parameter,
; the
more
easily
induce
macroscopic
polarization
proportional
gradients
of
the
to
can
one
a
director.
This
corresponds to the
well-known
flexoelectric
effect, first
the
described
by Meyer
in 1969
which
extensively
Ill,
has
been
studied
According to II ], an
recent
over
years.
inhomogeneous
alignment in a nematic liquid crystal gives rise to an average
polarization
of
fields
external
stable.
and
Thus
Pi
where
and
ej,
On the
ei~
inhomogeneities in the
additional
expect, in these regions, an
been
di~cussed
recently by Durand,
efe<.fi.I<.itj,
aider
term
contribution
the
to
polarization
Barbero
polarization
total
S
where
is
nematic
the
order
g,
=
proportional
and
Vs
[?, 3],
one
can
Thi~
has
propo~ed
the
of S.
have
who
According to [?], the
the following
general
only
but
electric
order
form
gjn(nV)s,
+
and
parameter
gradients
the
to
co-workers
phenomenon.
expressible in
is
P~~
II
n
S may also be positionally
dependent,
defects.
such
boundaries
Thus
a~
or
medium
this
describe
to
(nV
en
+
parameter
strong
near
(Vn
coefficients.
order
scalar
the
n
flexoelectnc
two
are
hand,
other
e
=
gj,
g~
II)
independent
two
are
order-electric
coefficient~.
Flexoelectricity
appears
be
to
experiments
important
very
which
deal
with
nematics
of
property
a
which
be
must
taken
inhomogeneous
director
orientation.
The
an
coefficients
flexoelectric
the
important
of
the
liquid
crystalline
most
are
among
parameters
materials
u~ed in
various
electro-optic cells with inhomogeneous
alignment. On the other
hand, the
phenomenon of order electricity can also be very important for some liquid crystal
properties, including, in particular,
interface
properties [2].
However,
the existing theory of
electricity [2-4] is based mainly on simple
order
phenomenological
and
arguments,
(a,
we
shall see
below) has to be supplemented by a more general microscopic
statistical
mechanical
description.
important
difference
between
flexoelectricity lies in the scale of the
The
and
order
most
effects.
In real liquid crystals
the
director
varies
macroscopic
scale
(determined
njr)
on
a
u~ually by the typical
dimension
of the cell, droplet,
hand
scalar
etc.). On the other
the
order
in the bulk, but varies rapidly
surface or close to a
constant
parameter S is approximately
near
a
correlation
defect.
In the general
S varies
the
length ;cale f this is normally at least
case
on
typical length ~cale
order
of magnitude
smaller
than
the
associated
with
director
two
into
in
account
all
variations.
Thus,
from
usually
i~
contribution
that
noted
the
point
associated
to
the
as~ociated
with
the
of
with
~urface
surface
Up
thermodynamics,
defect.
or
polarization
flexoelectricity.
Indeed,
order
same
(2)
and
of
that
the
the
corresponding
The
of the
nematic
energy
»sociated
with
order
g,, g~ and ej j, e~~ are of the
from
equations jl)
follow~
director
macroscopic
of
view
a
the
typically
order-electric
and
magnitude (see [2]
(If w I,
P~~/P,
and
energy
It should
defect.
of the
energy
electricity is
to
or
polarization
electric
order
electrostatic
larger
much
flexo-electric
Sect.
where
also
than
that
It
paper).
typical
the
a
be
coefficients
2 of this
f is
make<
length
then
of
variations.
to
this
and
natural.
with
order
Firstly,
parameter~.
consequent
point
development
the
However,
difficult
crucial
A
localization
thermodynamic
then
variable
construct
to
property
in
:
a
of
a
order
a
variable
of
concept
to
stage
consistent
boundary
thin
such
of the
neceswry
At this
polarization.
electric
it i~
it i~
define
electricity
order
the
free
energy
formal
considerable
definition
of the
polarization is
region. It is therefore
electric
is
associated,
in
seems
of
to
nematic
a
problems
corresponding
its rapid ~patial
difficult
classical
to
be
very general
liquid crystal
appear,
thermodynamic
variation
consider
thermodynamic~,
it
and
a~
a
with
its
true
the
MOLECULAR
6
N°
OF
THEORY
ORDER
795
ELECTRICITY
macroscopic work produced by a corresponding extemal field. At the same time it is obviously
mechanical
framework,
statistical
within
possible to define this short-range polarization
a
scale.
length
related
averaging
which is not
to any
using ensemble
contribution
of the
calculation
problem is related to the
fundamental
and
second
A
more
qualitative
physical
Using
of
crystal.
liquid
from the order electricity to the total free
a
energy
[2, 3],
arguments
write
can
one
Fo
F
Fo
where
is the
Landau-de
usual
free
polarization
electric
of the
energy
expansion)
Gennes
defined
(which
nematic
V~ is the
equation (2) :
(8
V~
£
'
)~
gr
=
example,
for
can,
electrostatic
and
in
(3)
V~,
+
=
represented by
be
associated
energy
the
with
the
order
(4)
iE~ E_
~
k
Here
E is
taken
the
over
electric
average
normal
modes
without
media
In
4 grP,
in
medium,
the
electric
extemal
the
and
is
fields,
the
are
4
Substitution
V~
2
£
gr
=
(k
of
tensor
and
the
E
0.
curl
induction
E+
D=
equations yield
These
(5)
~
expression
an
is
sum
E~ and P~.
transforms
k~
the
medium.
the
=
=
for the
V~
energy
in
of
tennis
(kP~)(kP_~).
k)~
I
E,
0 and
equation (5) into equation (4) yields
polarization equation (6)
of
order-electric
dielectric
field
(kP~
«k
=
polarization,
electric
order
the
by static Maxwell equations div D
Fourier
simple relation
the
between
E~
the
with
related
following
the
associated
field
(6)
k
Note
equations
that
electricity
polarization
but
of
from
Landau-de
rather
simply
make
a
separate
to
answer
interaction
equation (6)
of
higher
order
range
of the
character
liquid
nematic
condensed
crystals
with
the
with
medium
order
spontaneous
derivatives
of the
of the
is
the
latter
energy
a
also
the
(b)
derive
to
field
E is
contributions
other
one
explain why
might pose
the
the
coefficients
of
to
as
a
question
is
function
of
nonanalytic
determine
which
these
to
the
related
the
at k
with
energy
properties
interactions
question
the
why the
of
the
nonanalytic
be
to
electrostatic
expansion,
Gennes
Landau-de
usual
fact
in
need
the
free
gradient expansion by the
inadequate in this
context.
vector
~
interaction
energy
same
0.
of the
token
in
k
At
respect
depend
electrostatic
total
to
wave
contribution
interaction
diverge. These
corresponding
interaction
usual
problem
a
electric
contribution.
electrostatic
the
from
renormalize
contains
electrostatic
expansion
of
case
any
the
when
energy,
In particular,
all.
not
general
form
the
but it is
correct
simpler
and
do
electrostatic
the
free
at
is
interaction
this
separately
derivative
to
for
equation (3) in a consistent
field
produced by an extemal
electric
the
field
is
produced
by
gradients
of
the
order
parameter,
case
The
corresponding
electrostatic
interaction
is now just a part of the total
therefore
systematically
It is
the
system.
separate
necessary : (al to
the
l3ennes
interactions
in
intuitively
our
contribution
V~
general
dielectric.
the
The
limited
in
situation
Hamiltonian
but
not
valid
are
fact, the
way.
Eo. However, in
In
treated
are
be
P.
Equations (3-6)
inside
(3-6)
should
to
a
[5].
zero
k is
potential.
the
wave
Thus
cannot
traditional
be
the
second
the
vector
way
the
of
potential
the
determined
not
fundamental
nematic
properties
Indeed,
and
on
the
the
long
contribution
represented
Landau-de
in
Gennes
796
JOURNAL
PHYSIQUE
DE
II
6
N°
systematic
derivation
of equation (3) depends crucially
the
on
long-range
(electrostatic)
interactions
liquid
with
crystal~
in
In fact the necessity of performing
realized in the early
such a separation
was
of
the
by
Ewald
[6]
of
in
the
the
of
dielectric
theory
century
context
years
properties of crystals.
The
detail~ of the
mathematical
techniques
which
accomplish this ~eparation in the theory of
dielectric
~u~ceptibility in cry~talline
solid~
described,
for
example, in the
cla~sical
are
monograph [5 (the application of these ideas to the theory of
ferroelectric
ordering is pre,ented
It is
therefore
clear
separation of the
electricity.
order
in
the
and
[7] ).
important
It is
dipolar
electrostatic
paper
equatiofi~ (2),
13 ) and
separation
the
interactions
In
contribution
the
(61 using the
aho
u~ed
(electrostatic
in
~hort-
and
theory
the
charged,
of
these
long-range
of
role
been
the
to
be
cannot
long-range
between
has
fluids, the
electrostatic
interaction
I
gi~es rise
~
liquid-vapor
surface
tension.
In thi;
the
too,
ca~e
represented in the form of the u~ual gradient
e~pansion.
[8, 9].
fluid~
discuss
we
that
stericl
and
contribution
this
In
also
note
Waals
quadrupolar
important
and
separate
a
to
der
(van
range
to
that
short-range
micro~copic
electro~tatic
~tati~tical
force~
in
and
nematics
derive
theory of li~uid cry~tal~.
mechanical
These
equations
of the
the
which
lie at the
heart
phenomenological theory of order
are
electricity [2, 3]. In this derivation
shall
follow the main
ideas of the microscopic theory of
we
dielectric
susceptibility and of
ferroelectric
ordering [5, 7]. These ideas will then be combined
with the general density
functional
approach to the theory of inhomogeneous
fluidq
with longThis
9].
[8,
forces
range
paper
arranged
is
electro~tatic
range
in
with
nematics
for
energy
functional
follows.
as
forces
liquid
order
In
theory of nematic~ with
electricity. In sections
the properties of the
nematic-isotropic
underlying
In
substrate.
section 6 the
interaction
~mectic
in the theory oi
describe
section
2.
order
7
Theory
brief
we
present
of
inhomogeneous
some
section
2
discus~
we
and
4
5
interface
of
role
di~cuss
we
and
of
we
the
on
A-smectic
the
interaction
free
general density
basic
equation~ which
of order
electricity on
influence
electricity and
C phase
transition~
long-
the
present
order
of
Gennes
role
Landau-de
the
derive
the
the
detail
more
in
crystals and the
derivation
electricity. In section 3
long-range
interaction
and
of the
with
nematic
an
quadrupole-qua(lrupole
discussed.
Finally, in
the
i~
conclusions,
nematics
in
the
of
presence
long-range
electrostatic
interactions.
of long-range
interaction~
electrostatic
require~
of
theory
condensed
media,
Without
this
formal
care
care
~usceptibility
dielectric
in
the
those
described
in
the
theory
of
[5]
difficulties,
similar
to
or
For
free
of
example,
the
total
polarised
theory of charged fluids [9j. can
energy
appear.
a
in a system
with only
dielectric
depends on the shape of the sample ; by contrast,
~hort-range
proportional
limit the free
is
always
olume.
interactions, in the thermodynamic
the
to
energy
i
potential
from
shortthe
long-range
of
interaction
the
Thus it is important to
the
part
separate
the
methods
statistical
thermodynamic~.
before
using
formal
of
one
range
It is
well
that
known
special
in the
separation
This
fluctuating
between
parts
average
is
affected
by
be
performed
can
contains
the
treatment
mechanical
of the
molecular
field
theoretical
the
statistical
local
electric
field
multipoles and the
low-q components.
of
state
if
which
average
Such
medium
the
distinguishes
one
far
acts
on
a
electric
molecule.
field
the
long
be
to
appear~
average
Then
obeys the Maxwell
given point including
electric
t'rom
the
between
given
range
field
the
and
the
interaction
if
the
and
equation~
boundary
the
conditions.
In
the
which
is
finds
one
compared
of
case
polarization)
to
the
of
nematic~
the
order
non-zero
molecular
with
of
the
electricity
order
inverse
gradients of
length L or
S.
the
dimension
This
direct
the
of
lowe~t
the
dimen~ion
correlation
fi
region
can,
of
the
close
however,
length fo,
to
be
In
field
(or
average
the boundary in
relatively
~uch
a
c»e
large
it
i~
indeed
also
carried
in
out,
field
medium)
in the
ORDER
797
ELECTRICITY
long
corresponding
the
separate
to
necessary
electric
average
the
OF
THEORY
MOLECULAR
6
N°
from
the
range part (which is equal to the energy of
interaction
potential. This separation is
total
susceptibility
exactly analogous fashion, in the theory of dielectric
solids [5, 8] or in the theory of charged fluids [9]. In
of crystalline
interactions
difficulties
in
discuss
how
the long-range
create
can
we
gradient expansion of the nematic free energy.
ordering
ferroelectric
following
section
Gennes
Landau-de
and
the
the
LANDAU-DE
The
specific
GENNES
GRADIENT
EXPANSION
ITS
LIMITATIONS.
AND
long-range
electrostatic
forces in
nematic
liquid crystals can be represented in a simple
traditional
derivation
of the
Landau-de
Gennes
free
expansion.
way by following the
energy
This expansion can be derived in the most general way using the density
functional
approach to
the theory of liquid crystals [11, 12]. In this approach the liquid crystal free
is a
energy
functional
of the
distribution
function
one-particle
fj(r, WI, which depends both on the
position r and on the orientation w of the molecule. The general
functional
of this
is
structure
2,I
THE
role
of
not
known,
related
of
to
fluid, it will be
g(1, 2)
g(rj~,
It is
then
its
around
wj,
expansion
and
Fi
+
and
of the
free
the
The
direct
integral
corresponding
the
to
of
medium.
convolution
a
energy
are
correlation
the
to
pair
and
function
correlation
observed
factor
structure
known
are
of
a
function
scattering
in
po
terms
of
the
temperature
in
T
can
k~ T
Taylor expansion of the free energy of the nematic
expansion is, in fact, a generalization of the usual
order
The
free
density of a nematic at
parameter,
energy
be
written
approximately
now
as :
functional
a
isotropic phase.
in the
Landau
=
w~)
by
related
-13].
possible to perform
density
F~
is
derivatives
functions
I I
value
po
functional
the
correlation
recalled,
=
experiments
but
course,
direct
the
ldw
This
fi (w )(In fj (w
pjk~
(1/2)
U~(w )/k~ T)
A +
dw, dw~ C~(w,, w~) Afj(wj) Afi(w~)~+
T
(7)
,
where Fj is the free
density of the isotropic phase, Afj
fj
I/4 gr, C~(1, 2) is the
energy
two-particle direct
correlation
function
of the isotropic phase, U~(w
is the
extemal
potential,
Boltzmann's
and k~ is
and A is a
The higher
order
in equation (7)
constant
constant.
terms
depend on higher-order
direct
correlation
functions.
The
Landau-de
Gennes
expansion of the free energy can now be derived from equation (7),
by taking into
distribution
the fact that the one-particle
depends on the nematic
order
account
consider
for simplicity a nematic
composed of uniaxial
molecules
and in
parameter S~ p. Let us
the
absence
of an
extemal
field. Then the
distribution
function fj (r, w
fj (an (r iii depends
only on the unit vector a in the direction of the long molecular
axis. It is now
possible to expand
=
=
the
distribution
fj
function
of
components
a
[14]
with
fi(w)
where
a[j
is
the
respect
symmetric
=
(I/4
grill
traceless
a~
=
expansion (8) is convenient
expressed in terms of the tensor
distribution.
Cartesian
for
order
our
(15/2)a)~) S~p
+
tensor
a[j
The
irreducible
to
constructed
tensors
from
the
:
),
+
(8)
:
ap
(1/3) &~p
since
purposes
parameter
and
is
(9)
the
not
distribution
limited
to
the
function
case
is
of the
already
uniaxial
798
JOURNAL
Substituting
expansion
the
free
nematic
PHYSIQUE
DE
equation (7).
j8) into
derive
we
N°
II
following
the
expression
ior
6
the
energy
~~
F
F
+
=
(4
8
PI
~
gr
)~
~
u~ ~~
L~p~~(r
dr dr~
kT
S~p (r) S~~ (r~
r')
l10)
+
with
Lap y& lr
r~ )
d~aj d~a~
=
where
have
we
the
represents
free
is
r')
r
4
+
7r6 (r
d~aj
r~
a,
~,
p
aj
I1)
~~
the
quadratic
in S~p.
terms
equivalent to the
Landau-de
Gennes
it
expansion
functional
of S~p (r). One
sample as a nofilo<.uf
arrives
after performing a gradient
expansion of equation (10).
only
yet
whole
not
of the
a~,
theory only
substituting
involves
s(r~)
s(r)
=
expansion
Using
the
F
F (r) dr
=
C~ jai,
account
(10)
energy
Gennes
Landau-de
This
into
taken
equation
that
the
a,
~
Note
at
=
(12),
il/2
F (r)
the
density
energy
This
(1/2
+
r~
free
total
F (r).
A~p~~ S~,p S~~
=
(1/2)((r
+
transform
we
free
the
over
r~)V)sir)
(jr
+
)V)~ s(r)
to
energy
written
be
can
(12)
+
integral
volume
a
as
(1~)
V~S~ p v~,S~~
L~~p~~~
with
this
From
schematic
procedure is
compared to the
this
expansion
be
that
seen
system
make
remains
order
order
case
the
are
expected
is
The
of the
point,
L~p~~ (r).
Gennes
free
S(r)
to
direct
correlation
one
can
slowly
this
this is
radius
This
is the
with
the
full
the
direct
for
large
fo in
case
readily
varying
does
case
expected
length.
contrasts
(15)
energy
is a
C~(r). Only in
higher order terms
be short-ranged
molecular
and
r,
parameter
function
in this
critical
d~r i~
Landau-de
the
interactions.
is of the
or
of the
when
Ci(r,~)
function
and
" PI
16
114)
indeed
even
small.
the
such
correlation
gradient
the
be
to
that
see
function
case
It
can
for
a
always
vicinity of
systems
in
the
radius
f.
which
[15].
defined,
is
only
d~r L~p~~lr)
p~)
4
correlation
direct
transition
However,
C~
=
short-ranged
finite
phase
diverges
L~p~~~~.
only
sense
the
with
a
=
derivation
valid
~~~
A~p~a
can
C~(1,
interaction
in
a
long-range forces,
long-range
tail ».
Indeed,
a
Vtjl, 2)/k~T [8],
where
i't(1,
system
possess
2j=
potential.
question
with
«
defining
2)
is
formally
asymptotic expre~sion for
long-range part of the pair
correlation
I
the
function.
the
interaction
kind of
potentials should be
considered
equivalently,
which
problem,
potential~
produce
respect to
present
or
difficulties
derivation
this
question
in the
of the free
expansion.
The
simple
is
to
energy
answer
that the elastic
in
(13)
coefficients
gradient
L
equation
(the
of
the
expansion)
should
constants
exist,
which in turn requires that the integrals in equation (15) should
[9j. It is now
converge
quadrupole-quadrupole
interaction
easily seen
that
the
is long
enough
produce a
to
range
A
crucial
long-range
with
involves
the
what
N° 6
MOLECULAR
divergence of
integral
the
r~ V (r) dr
like
ORDER
OF
V(1, 2 )/k~ T.
m
799
ELECTRICITY
the
integral
over
This
integral
diverges
equation (15). Indeed,
in
C~(1, 2)
when
THEORY
equation (15)
rj~ in
as
logarithm
a
looks
of the
r~~
interaction
potential
quadrupole-quadrupole
interaction
potential
long-range » dipole-dipole
It is interesting to note also that the
more
interaction
produce any divergence in equation (15). The polar asymmetry of this
does
not
orientations.
On the other hand, it
molecular
after averaging
the integral to vanish
over
causes
contribution
interaction
makes a shape-dependent
is well
known
that the dipole-dipole
to the
ground state free energy of a polar dielectric.
intermolecuThus,
the regular gradient expansion fails for systems with long-range
because
longer
Landau-de
Gennes
free
traditional
derivation
of the
interactions,
the
lar
energy
no
sample
size
2) is the
V(1,
when
«
holds.
This
does
not
of
mean,
However,
systems.
that
course,
has
one
first
theory cannot
short-range
expansion holds for
Gennes
Landau-de
the
long-range
the
separate
to
applied
be
of
pans
and
to
such
the
total
with
Gennes
the
Landau-de
potential. The usual
system
remaining long-range part of the potential must be
interactions.
The
effective
short-range
of the
contribution
total free
additional
treated
separately. It gives rise to an
to the
energy
medium.
field in the
In the
sample. This
contribution
is nothing but the
average
energy of the
makes
in a thin boundary region and
of order
electricity this field is
concentrated
a
case
interaction
contribution
the
to
surface
of
energy
nematic.
the
In polar
NEMATICS.
INHOMOGENEOUS
OF
between
potential corresponds to the interaction
determined
dipoles. The average polarization is then
by the average
induced
(or
nematic
liquid crystals with an inhomogeneous order
dipole
On the other hand, for
moment.
determined
polarization can be
by the gradient of the
distribution,
the
parameter
average
quadrupole density. In particular, as shown in equation (2), this includes the gradient
average
interaction
between
long-range potential is the
of S. In this case the corresponding
permanent
is a
medium.
This
difference
of the
quadrupoles in the inhomogeneous
symmetry
consequence
quadiupole density in the ground state. By
of the
nematic
phase, which
possesses
a
non-zero
dipole density. It is worth noting, however,
that
polar crystals possess a
contrast,
non-zero
quadrupole density and therefore one could find
crystalline solids can also possess the
nonzero
polarization near the surface (as in nematics) which is
in crystals the
spontaneous
type of
same
of
the
quadrupole
density. As far as we know, these problems have
proportional to the gradient
LONG-RANGE
2.2
crystals,
the
been
not
should
interaction
dielectric
us
consider
in
studied
traditionally
It
CONTRIBUTIONS
long-range
permanent)
on
be
theory
the
the
TO
ENERGY
crystalline
of
of
calculation
THE
interaction
part of the
solids
since
the
interest
the long-range parts of the dipole-dipole and
contributions
similar
give rise to qualitatively
of
the
since they both
the
represent
average
energy
which
dipole-dipole
interaction
energy
can
average
noted
the
(V~~)
=
(1/2)
focused
been
quadrupole-quadrupole
that
potentials
media,
has
properties.
bulk
PI ld~rj
the
to
electric
be
energy
of
Indeed,
let
free
field.
written
as
:
d~r~ dw, dw~ f~(1, 2) V~~(1, 2),
(16)
with
~'dd(1, 2)
where
f~(1, 2)
is
the
"
~12~((dl 'd2)
two-particle
f2(1, 2j
distribution
=
3
(dl
'~12)(d2'~12) ~12~),
function
fi(i j fi(2j(g~(1,
which
2jj,
can
be
represented
(17)
as
:
(18j
800
JOURNAL
g~(I. ?)
where
;ubsection,
The
h~jl,
~
=
which
tend~
to
?)
is
dipole-dipole
average
in
long
j
can
distance
limit.
be
written
now
j',(2)il~~~(1,
II
N°
di~cussed
function
correlation
pair
the
interaction
(f,(I
(i'~j~,)
the
unity
PHYSIQUE
DE
2))
2)Jij(1,
as
(f~(I
+
=
of
~um
a
in
j,(2)
6
pre»iou~
the
terni~
two
i'ddll. 2))
i19)
with
(.. )
The
correlation
equation
like
second
d~rj d~r~ dw, dwjl..
?
is expected to decay
regular
contribution
since
a
potential.
short-range
j19) can be
rewritten
in equation
term
then
effective
an
The
h~j
function
(19)
PI
jl/2j
in the
I.
the
makes
«fj
f, j2 j V~~(1,
2
))
long-range limit, The fir~t
product i'~wil, 2 h~( I, ?
term
in
beha,~e~
as
d~r d~r'P~~ jr) (~p
(1/2
120)
).
jr
P
r
p
(r'),
(21)
with
Tap
Plr)
where
ln
~dw
po
=
Fourier
fj (w,
equation
space
(r)
r)d(w
(?I)
i~
=
is
the
be
can
~16
2
where
the
Fourier
Thus
energy
the
the
of
the
average
field in
electric
(j'j(i
l'j(~j1/~~(i~ ~))
of the
the
total
#
field
(23
E
~
E~
is
given by
15)
for
from
the
interaction
media.
long
This
is equal to
energy
con~istent
with
is
the
range
part
d~rd~r Q,,p(rji~,, ~~Tp~(r
(i,")
2
PI (k)
only
polarizations
d~kE~
gr
equation
of
the
quJdrupole-
the
r')Q~z(r'i
=
=
The
medium.
the
form
dipole-dipole
nonpolarizable
separation in equation (3).
phenomenological
similar
expression can also be obtained
A
potential
interaction
quadrupole
~'l~h
of
d~k(kP~)jkP ~)L~~
electric
average
density
media.
long range part
macroscopic
the
of
of
component
the
7r
(1/8
l~~l
polarization
in
written
=
nonpolarizable
la lpi
average
Vddll, 2))
(fi(I)fi(2)
3
off
w
d~k(kP[)(kP[~i
Ill)
k
kg Qafl (k).
difference
P and
P'.
equations (23)
between
The~e
are
quadrupole density,
polarization P is a ;um
by
respectively.
determined
average
of
variou~
and
the
e~pressions for
dipole den~ity~ and by
(24) iie~ in the
average
Equation (21)
contributions
oi
also
diver~e
ha~
general
origin.
the
average
gradient
validity when
the
Density
3.
THEORY
MOLECULAR
NO 6
3. I
theory
functional
ELECTROSTATICS
liquid
rupoles.
of
DIPOLAR
oF
The
crystals
molecules
of
with
forces.
electrostatic
We
NEMATICS.
with
start
of
model
a
dipoles and quadpermanent
possess
nematics
considered
have been
by Prost and
which
model
such
801
ELECTRICITY
QUADRUPOLAR
AND
composed of rigid
polarization properties
crystal
a
liquid
nematic
ORDER
OF
of
flexoelectricity.
discuss
the
Here
more
we
which
includes
spatial variations of the scalar order parameter S.
general case
liquid crystals are nonpolar. However, as a result of flexo- or order
Homogeneous
nematic
electricity, or in the presence of an externally imposed electric field, an average dipole density
On the other hand, the quadrupole density Q~p (r) is always
P~(r)
nonappear.
po (d~) can
in the
nematic
phase and can be written as :
zero
[16],
Marcerou
the
in
theory
the
of
context
=
Q«p (r)
where
q~ p
order
=
In
is
q~p
the
Qo(a~ ap
(q«p)
quadrupole
molecular
Qo s~p (r),
Po
=
(1/3 ) &~ ), Qo is the
p
phase with an
density gives rise
In
tensor.
(25)
the
of
case
quadrupole
molecular
inhomogeneous
nematic
a
to
an
~
general
the
polarization P~(r)
the
case
flexoelectric
uniaxial
S~p
and
moment
molecules
is
the
nematic
can
p
(r)
polarization
(r)
P
«
p
=
last
the
o
gradient
of
the
average
:
(26)
Qo V~S
be
represented
of
=
P
the
o
Qo(n(nV))
po
=
S
po
nematic
then
a
as
sum
of the
order
electric
and
vp (q~p (r ))
(d~
Qo((nV)
ii
+
Pk
4
Pk
~
wk
=
(?
(27)
(Vn)) S
n
E in
in
(kp~)(k
i )(k
(e~p
gr
effect
term
The
E~
+
reads
in equation (28)
the
represents
polarization
the
field
P
and
electric
average
Maxwell
equations, equation (3). Taking into account
the
electric
field and
polarization
average
average
where
the
parameter
contributions
P~, (r)
total
=
~
order
polarization
effective
V pQ~
P
The
Po
parameter.
quadrupole
In
=
k
(28)
,
polarizability
the
medium
the
Ep (r
of
related
are
)~
k)~
k
media.
the
by the static
expressions for
equation (28), we derive
the
inhomogeneous
nematic
P
I
of
~p
:
(29)
'
(30)
(kpk
,
with
Po(d)~
Pk
equation (28)
that
Note
symmetry
between
arguments.
the
is in
contradiction
with
equation (2) implies
density and the gradient
Indeed,
polarization
equation (28) (which
(#)~.
poook
=
(31)
equation (2), derived
assumption that
an
of
the
order
polarizable
with
the
there
is
parameter
S.
help
a
By
local
of
simple
relation
contrast,
the
medium) is nonlocal
since the
field E is
electric
related
the
polarization density by the
nonlocal
to
propagator
average
relation
nonlocal
between
the polarization P and the gradient of S can be readily
T~p (r). The
from
equation (30). In the nematic,
also
composed of nonpolar
molecules,
only the
seen
gradient of the average quadrupole density p~ (see Eq. (31)) can always be written in the form
third
term
in
is
nonzero
in the
JOURNAL
802
equation (2)
of
should
in
while
noted,
be
simple
one
gradient
equation (2j
director
the
at
the
In
this
appears
nematic
the
case
incorrect,
surface.
(31)
These
remarks,
phenomenological theory
of
order
electricity
the
polarization
for
the
electrostatic
but
in
the
when
do
to
the
of
case
they
since
complicated. It
equation (2)
polarization is parallel to
the gradient of S. Thus
more
form
based
of
main
the
the
on
of
orientation
tilted
influence
not
are
is
the
to
parallel
the
however,
the
density
reduced
be
can
example,
for
N° 6
polarization P~
full
the
(30),
II
p~ is parallel to k, I-e-,
should
also be
director
vector
be
to
p~ and
between
equations
that
when
case
of S.
the
relation
the
however,
PHYSIQUE
DE
the
re~ults
e~pre~sions
of
for
not
energy.
that the longsubsection
in the previous
~een
corresponds
the
interaction
of
the
molecular
to
range
multipoles with the average
electric
field E, given by equation (?9).
What
remain~
of the
including the fluctuating part of the local field,
interaction,
be
short-ranged.
It is
to
appears
reasonable
long-range
from
interaction
potential
I';
subtract
the
the
total
thi;
to
part
now
can
be
represented then as a sum of two
terms
3.2
THE
of
electrostatic
total
the
have
We
FUNCTIONAL.
ENERGY
FREE
part
interaction
V,
V
=
V~
+
(32)
V~ is the remaining short-range part of the potential, and the potential V~ is equal to the
of the
field given by equation (3).
Sub;titution
of equation128)
electric
into
energy
average
where
equation
V~
yields
(3)
2
~j (kP~ )(kP_
w
=
~
k
~
=
L
=2wp~(~ ((k(d)~)(k(d)_~)~Q((k. (()~
(j)
k)(k.
k)
~
L
Qji[(k(d)~)(k
k)
(k(d)_~)(k
(()~
the
long-range parts
energies.
depend
the
not
on
of
(()
k)j).
1331
~
three
The
equation
in
terms
quadrupole-quadrupole
important
It is
the
free
total
observe
to
of
energy
the
~ystem
k~
F
=
F,
where
It
F,.
I;
i~
now
should
It
density
functional
written
usual
however,
that
be
written
V/~T)
F,
dF
sfioit-iaiige
the
the
fatal
free
variables
l'~,
+
;
(34)
potential i'~.
expansion
Gennes
Landau-de
fluctuating
a~
=
with
the
theory ~tates
den~ity
iunctional
of the
in
does
~
exp(-
T In
perform
stressed,
be
can
dipole-dipole,
the
interaction
therefore
calculated
energy
to
to
energy
of
the
contain;
F
the
free
energy
additi~nal
il~.
contribution
The
free
the
possible
correspond
(33)
dipole-quadrupole
that the potential ~
and
the
one-particle
the
that
p lr,
free
).
w
energy
Thus
the
of the
free
~ystem
energy
can
F in
be repre,ented
equation (341 can
a~
a
be
form
F
F
=
[p (r,
w
)]
p
(r,
w
U~ (r,
w
135 )
dr,
dw
~
with
F,[Pl
F[Pl
=
where
U~(r,
co
is the
external
potential.
The
+
VE[Pl,
electrostatic
l~61
energy
i'c
is
given by
equati~n
133j
N°
MOLECULAR
6
depends
and
density
one-particle
the
on
=
via
(RI
"
should
be
functions
of
noted
with
the
average
jr,
w
4(W )P (r,
w
803
ELECTRICITY
dipole
and
quadrupole
moments
:
dw ;
(37)
dw
j38)
[p
~
short-range
effective
the
of F
derivatives
functional
the
that
media
the
ORDER
id(w) p
(d)
It
OF
THEORY
related
are
potential V~.
correlation
direct
the
to
example,
For
~2yz
6~
6P (1)
subscript
where
the
potential V~.
At
the
of
energy
In the
specific
4.
We
the
full
following
free
two
sections
the
surface
of
nematic-isotropic
consider
now
interface.
molecules
;
shall
we
is
calculated
p(r,
w)=po fj(r, w)
given by equation (35),
consider
properties
is
with
the
by
determined
which
includes
the
of
influence
the
of
order
electricity
in
the
liquid crystals.
nematic
interface.
influence
of
order
we
shall
shall
we
density
function
V[p].
simplicity
the
For
~~~~
corresponding
the
functional
energy
field
electric
average
that
one-panicle
the
the
contexts
The
IN-I)
of
indicates
»
time
same
minimization
s
«
~~~~~'~~'
(2)
be
thus
electricity
solely
concerned
properties
the
on
that
assume
nematic
the
with
effect
the
of
nematic-isotropic
of nonpolar
quadrupolar order
electric
of the
composed
is
polarization.
order
In
consider
to
gradient expansion,
equation (13). This
nematic-isotropic
the
derivation
the
of
approximation
which
interface,
was
shall
we
outlined
in
section
use
Landau-de
the
2. I,
Gennes
and, in particular,
in
physical quantities
such as density and local
order
length scale. In fact.
parameter change slowly on a molecular
for our
model
shall
that density is not a
relevant
although the
parameter,
purposes,
we
suppose
inclusion
of density
difficulty in principle. It is worth mentioning that in other
presents
no
liquid interface problems [17], even when the relevant physical quantities do change rather
rapidly than strictly permitted by our formal understanding of the gradient approximation,
more
surprisingly good results for quantities such as the surface
tension
be
obtained.
can
We
equation (36) for the free energy of the nematic ; this includes the long-range
use
now
electrostatic
contribution.
The « short-range »
contribution
F~~p
be
expanded in the
can
gradients of the order parameter S~
yielding
the following expression for the total free
energy
p,
density in the interfacial region
F
(r)
=
Fo(S)
+
that
supposes
(1/2) Lj V~S~p V~S~p
in
interface
the
(1/2) L~ V~S~~ Vpsp~
+
+
region,
(1/8
w
e~p E~ (r)
Ep(r ).
(40)
The
elasticity
coefficients
Lj
and
L~
be
can
Lj
=
expressed
L~o
as
11 8]
(41)
L~/3
,
with
L2
L20
~~~
"
~
"
P((4
w
7
4
P((4
w
)~
)-
~
~
ir~
r~
i~~
a(/
aj'j a[f
C~ ~(aj,
a(~/ C~ (aj,
~
a~,
a~,
r) d~r d~aj d~a~
r) d~r d~aj d~a~
;
(42)
(43)
804
JOURNAL
free
The
S,~p and
with
sy~tems
of
energy
direct
the
(~9j
state
Ci,(1,
function
short-range
effective
an
equations15).
homogeneous
the
correlation
(31j
and
Finally
augmented
Let
order
consider
now
u~
S,,p,
parameter
N-I
a
one
interiace
cos
director
the
6
we
are
the
free
den~ity
be
a~
E(r)
field
I,
gi,
by
en
the
expre~sion
energy
Then,
neglecting
aide
[19.
Genne~
term.
plane.
v
i
(44)
well-known
iield
S(ii~ fig
=
assumed
I/I j
depend
to
will
the
iii
biaxial
of
F~~(S
=
the
and
by
+
the
[20].
angle
tilt
that
interface
We
obtain
now
S(=)
~ El
~
local
the
Marcus
parameter
L~jj
where
phy~ics of
ba~ic
~tudied
order
14~j
=,
the
~'~
ll'2
+
on
affect
not
the
&,,p ),
only
specifically
of
functional
a
F (z
in power~
of
calculated
for a
(43j is
and
electric
average
be
written
a~
can
the
=
energy
expansion
an
as
6
write
can
can
n
presented
The
cour~e,
ii= (= j. Ignoring the biaxiality
studying : the biaxiality has been
(z ) j
N°
equations (42)
electro~tatic
in
S,, p
where
in
transform
equation (40j i~, of
by the long-range
that
note
we
but
here
2
II
rkp~joo(k.I.k)(k./.kj'
E~=4
20],
is
potential.
Fourier
its
PHYSIQUE
DE
1461
(=1
with
~.
4
~
Qo
"Po
~
(
co~2(o
F~-
~~~
and
~--
have
also
~pecific
order
where
we
The
contribution
using
thi~
of
The
the
free
+
F
Fi
dispersion
the
electric
semi-empirical
hi~
results
alter
to
neglected
=
contribution
(4~)
COh~ ( 6 ),
F
~
of
the
~usceptibility
dielectric
from
comes
third
the
interface
has previously
been
energy of the N-I
method.
The
microscopic
derivation
oi this term
orientation
become~
146j.
is
of the
one
Thi~
(3 ],
Durand
central
paper.
important
of equation (46) is that the
influence
consequence
equilibrium
orientation
of the
director
the
N-I
interface.
at
i~
everywhere
uniform~
minimization
of the
free
the
particularly simple.
Without
the
final
the
obtain
term,
we
of
most
the
I(kj.
equation
obtained
by
in
term
If
electricity
order
we
energy
usual
can
that the
suppose
in
equation146j
de
Genne;
result
l191:
~
L~ sin (2 8j
=
Thus
We
the
equilibrium
tilt
recall
further
may
the
coefficients
Q«p~
Kjj~ K~~, K~~ (in
this
angle
In
fact~
WI?
=
if L~
~
0,
only take~
one
Li
be
expressed
in
L~,
term~
can
appro~imation Kjj
K~~) [19]
L~
equilibrium
&~~
if
that
empirically~ it
tilt angle
&~~
is
=
usually
WI?.
=
Ki~j/S~
(K~~
the
case
that
(~01
o.
=
whereas
into
Lj
K~~
of
=
&~~
account
the
ela;tic
if L~
0
=
the
~
0.
quadratic
constant~
of th~
terms
in
nemztic
Kj~/,12
~Kjj
[20]
and
thus
Lj
~0
and
the
MOLECULAR
N° 6
THEORY
805
ELECTRICITY
ORDER
OF
established
by de Gennes [19]. It is sufficiently important to restate
first
result
was
nematic
completely
defined by the Saupe ordering matrix Q«p (I.e.
precisely.
For
tensor
more
a
quantities such as R~ p ~~ associated with the nematic are neglected) whose gradient terms are of
director
conclusion
of equation (50) is
inevitable.
In other words, a tilted
the form Lj, L~, the
at
restrictions
holds
theoretically
impossible. Although clearly none of these
interface
is
the
qualitative
restrictions
affect
intuitively one feels that these
should
exactly,
nevertheless
not
theoretical
(conical boundary
conditions) is observed,
conclusions.
If a tilted
director
a
likely weak link in the set of the assumptions would
explanation is necessary.
The
most
seem
description of the nematic in terms of the Saupe dyadic ; in other words it would be
to be the
This
necessary
explicitly
to
polynomial,
and
However,
the
merely
not
this
changed by
consider
is
result
influence
the
the
F
Legendre
against the
quadrupoles.
molecular
Fo(S)
(z)
the
of
order
~~
S.
=
effects
electric
equation (46)
Indeed,
Legendre
fourth
average
polynomial P~
introduction
(1/2) i~i~
+
=
P~,
parameter
second
average
robust
not
of
order
is
and
rewritten
be
can
as
~,
:
(51)
with
L~~i
AP((cos
L$~~ +
=
&)(I
aP~(cos
+
(52)
&)),
and
~l
~
~~ff
"
~2
~
+
~21~2(C°S ~)
+
~~~~
'
A
wp( Q(/P,
4
(54)
=
(2/3
a
where
is
F~
Fjj
(2
P=
F~
the
ejj)/3
anisotropy
polynomial.
It is important
minimized
when
the
is
+
of
P~(cos
8)
=
second
and
«
=
L~, A,
,~2«(1+
with
=
order
the
limit
In
~(cos &),
P
x
the
electric
large
general
of
the
aL~/A.
«
The
limit of small
; in the
they can be ignored.
K
case
equation (56)
of
he
solutions
these
according
(I.e.
following
tilt
angle
to
the
«
of
according
chosen
AF).
a
One
+
K)
~
(l
»
(56)
=o,
«
electric
+
of the
measure
effects
dominate,
K)-~'~
the
to
readily
can
Legendre
size
whereas
of
in
:
(i
=
AF
=
second
the
non-dimensional
a
order
lal-'
~
(I
is
is
nematic,
the
(57)
sign
see
of
that
the
anisotropy
(56) yields the
dielectric
equation
:
cos
if
«
the
solutions
two
«~~±
be
must
sign
has
a-i
~)x+
parameter
=
The
a.
«)+2(1+
effects
xi.~
One
and
of
equation (51) is always positive. It is
of the tilt angle
magic
value
the
side of equation (51)
the right hand
on
equilibrium tilt angle is determined by
in
term
0 ; this
corresponds to
of the total expression
cos~ 8~~
1/3. The
minimum
depends on the ratios of coefficients
following equation :
the
susceptibility
P~(cos 8)
dielectric
susceptibility,
the
that
observe
to
,
average
the
(55)
A El P
=
a/2)~,
and
&~~
8
w/2
=
'(i
~z-
=
if
(I
+
K
(i
~
(l
+
K
)- ~'~),
a/2)~.
(58)
JOURNAL
806
Note
that
#
K
angle
tilt
the
anisotropy
dielectric
cos~
Thus
and/or
~mall
the
on
when
II
parameter
which
K
quadrupolar
the
is
coefficient
A i~
8~~
L~/A
=
+
small
when
large.
At
(59)
coefficient
quadrupole
magic
A w L~ the tilt angle i~ close to the
influence
anisotropy does not
the tilt to any great degree.
the
dielectric
anisotropy is small. I-e- a w I, but the quadrupole
coefficient
Lj, the equilibrium tilt angle is still given by equation (59) if L~
A. For
When
large, A
A
L~,
value
»
«
A is
not
;maller
~
&
~
the
~mall
(L~/A /
a
very large
dielectric
for
the
6
N°
finds
One
and
depends
strongly
&~~
is
A~/P
PHYSIQUE
DE
WI?.
=
The
most
anisotropy
complicated
ca~e
small
and
i~
not
corresponds to
angle
tilt
relatively
the
6
large
appears
to
~alues
of
l.
K
In
this
be
temperature
the
parameters
the
case
dielectric
dependent
~mce
S.
~F
We
make
now
expressions
estimates
for the
tilt in
of
the
numerical
equation~ (58)
(59)
and
the
important
most
which
the
enter
quadrupole
is the
these
expressed in terms of the
10-~ dyne. The
K~~j/S~
~10-~
2A~/3
also
for
nematics
known;
composed of nonpolar
P is
parameter
a
a
~0.5
for
strongly polar
molecule~
[19]. Let us
molecules
and
estimate
-1
the
now
a
coefficient
of mesogenic
A given by equation (54). The quadrupole
molecule,
moments
not
are
known
estimate
obtained
by Prost and
Marcerou
[16j in their
exactly but one can use the
analysis of the
flexoelectric
effect.
For
typical
nematic
such
MBBA
Qo~4x
a
a~
10 ~~ e.s.u.
value
plausible ~mce, for example, the quadrupole
In fact this
of
moment
seems
cyclohexane
(which is a much
smaller
molecule) is equal to 16 x lo- ~~ e-s-u.
[22]. Taking
arrive
2 x 10~~ cmand putting P
3, we
the
estimate
2 x 10- ~ dyne.
A
at
po
This
indicates
that the
quadrupolar
coefficient
A is relatively large (A
Lzl and
estimate
thus
the
equilibrium tilt angle can be clo~e to the
magic » value.
Indeed,
for a
0?,
10- ~ dyne and A
? x
10- ~ dyne the
10therefore
and
the tilt angle i~
parameter
L~
«
given by equation j59) in which the last term can be neglected.
Durand
[3] assigned the universally
observed
tilt of the
director
nematic-isotropic
at a free
quantitative
credence
order-electric
thi;
effects.
In this
~ection
have given
interface
to
to
we
coefficient
elastic
A.
have
We
that
above
seen
Kjj, Kj~, Ki~.
constants
the
coefficients
typical
For
Lj, L~
of
nematics
L~
i
be
can
(Ki~
=
=
«
=
point
will
=
=
view.
of
be
very
caution
of
note
interfacial
~trong
Indeed
not
far
in
is
region,
effect
on
we
from
may
the <~
order
which
the
this
at
orientation
»
of
in
example,
for
The
not
considered
the
director.
materials
of
number
a
value
stage.
have
we
that
e~pect
magic
for
between
explicitly in
Consequently
only be valid for nematic liquid crystals composed of
Finally in this section we observe that in some real liquid
this is
because
electricity may not after all be so strong
the
.w.reened by charged impurities.
However,
complete screening
length of the
order
parameter
variation
8~~
permanent
65°.
=
interaction
should
characteristic
equilibrium
the
MBBA
is
much
with
a
the
present
dipoles
paper,
angle
tilt
However,
in
a
the
have
can
a
magic angle tilt beha,<iour
weakly polar
molecules.
crystals the
influence
of order
the
electric
is
field
impossible
smaller
than
the
may
in
be
partially
nematic~
the
Debye screening
length.
5.
Nematic
liquid
crystal
substrate
interface.
Recently the properties of
nematics
contact
m
investigated in detail theoretically using Landau-de
theories
27]. In this section we shall relate these
paper.
Gennes
to
the
flat
structureless
[23, 24] and
order
electric
wall
molecular
ideas
been
ha~e
theory [25-
discu,;ed
in
thi;
MOLECULAR
N° 6
We
with
start
primary
The
substrate
brief
a
of
aim
a
extemally imposed
usually described
are
discussion
of the
ORDER
of
status
the
field
which
in
alters
bulk
the
theories
continuum
807
ELECTRICITY
theories
theory is usually to
equilibrium
surface
tilt,
of the
understand
molecular
is,
that
OF
THEORY
and
the
the
of
response
properties of
surface
configuration. In fact,
of an anchoring
energy
relevant for applications
terms
tilt
the
to
an
properties
these
director
in
interface.
nematic-substrate
anchoring
the
which
is
confined
in the
continuum
microscopic origin is not
intermediate
understand
this
the
regime. So a molecular
theory usually aims to
concept
definition
justify.
A
probable
whose
microscopic
is
rather
harder
anchoring
to
energy
anchoring energy is that, in some
all
precondition of the existence of a
well-defined
sense,
with,
changes.
forces
of
short
when
compared
order
relevant
parameter
are
say,
range
Presumably if there are surface forces in the problem with widely varying ranges, it might be
difficult
Dubois-Violette
and de
Gennes
define
the anchoring
As long ago as 1975
to
energy.
[28] pointed out already that there might be competition between long and short range surface
interface,
the
at
and
whose
forces.
Inclusion
both
of
electricity
and
Landau-de
LANDAU-DE
GENNES
first
discuss
5. I
was
used
theory, which
particularly
Nether
of
to
occurring
anchoring
to
of
electric
the
anchoring
Note
that
within
into
the
Landau-de
Gennes
theory
version
of this
question
is
due
In
this
been
both
recent
phase,
is
first
tum.
which
et
subsection
discussed
within
in
and
Teixeira
to
which
nematic
the
has
A
nematic
account.
of
energy
this
The
phenomena,
effects
order
described
discuss
by Sheng [30].
nematic
a
energy
them
free
shall
we
SUBSTRATE.
A
in
and
discuss
we
determined
by
is
[24].
al.
Barbero
by
and
[29].
Durand
In
take
electricity.
order
To
processes
origins
the
contribution
quadrupolar
cLosE
surface
the
to
approach,
ordering
on
with
difference
a
Gennes
THEORY
theories
these
makes
surface
concentrate
concemed
additional
an
the
order
molecular
Landau-de
the
wall
can
theory
Gennes
written
be
as
free
the
of the
energy
nematic
in
boundary region
the
near
:
F
~(s(o))
F
=
+
~
j~
(60)
(z ),
dz F
o
the
interactions
between
by contact
region,
which
is
interfacial
density
in
the
molecules
and the wall, and F (z)
energy
section.
the
previous
derived
in
(46-48),
already
equations
described
by the same
formally
from
additional
contribution
Substitution
of equations (46-48) into (60) yields the following
solid
interface
of
nematic
polarization
the
anchoring
order
electric
at a
to
energy
a
F~(S(0))
where
A«~
where
Qo is
where,
The
order
the
2
surface
the
energy
is the free
II
dz
e
quadrupole
molecular
0 is
the
angle
this
in the
favours
it
°
(z))(1
tYP2(C°S
+
parameter
nematic
the
(61)
(z)))
°
of the
given by equation
is
a
director
given by equation (61)
magic angle tilt, just as
contribution
case
the
and
between
contribution
wall
F~ (dS/dz ) (as
and
be
F~(S(0)).
energy
never
interactions
only short-range
taken
are
orientation
The
equilibrium
of the
director
determined
equations (60) and (61). These
from
minimal
into
at
the
at
As
the
not
director
only by
is
tilted
by
tilt
at
However,
interface.
which
shown
(55),
and
z-axis.
the
encourages
at the N-I
counterbalanced,
may
nematic-isotropic
interface)
orientation,
also by the
but
surface
free
Sullivan
[25], this surface free energy is
provided
~ Pi(C°S
,
electric
of the
determined
free
~
«Pi Qi
Broadly speaking
vicinity
energy
the
=
before,
as
interface.
is
the
elastic
minimal
of
in
free
planar
for
Tijpto-Margo
orientation
the
and
director
the
account.
wall
expressions
cannot,
however,
depend
the
on
local
be
directly
angle
tilt
808
JOURNAL
PHYSIQUE
DE
II
N°
6
region. At the ~ame time equations (60) and (61)
~urface
W(8~j which depend~
macroscopic
to
energy
can,
Thi~
however,
angle
measured
in
the
experiment.
procedure,
the
tilt
requires
on
calculation
vicinity
of
the
in
wall.
Such
determination
profiles
the
of the order
the
parameter
a
electricity.
interface
without
order
Teixeira
[24] for the nematic
carried
has been
et al.
out by
calculation.
simple
but is by no
means
a
contribution
anchoring energy, oi the
significance of the quadrupolar
Neverthele~s,
the
to the
the
Gennes
framework,
be
determined
by e,timating
Landau-de
nematic,
within
the
can
performed in [?9], the value of the
integral in equation j62). According to the ~imple e~timate,
6 (=),
which
A«~
is
T~).
This
found
doe~
5?
MOLECULAR
shall
relevant
8.
the liquid crystalline
potential between the
because
at
effect
of
The
~urface
a
=
p
32
~
oi
conclu~ion~
the
Ml
/
S
result
~~
+
free
w>~
of the
at
sub~ection
thi~
we
theorie~
thme
><hich
ask
is
within
in
are
is
a
of the
function
behind
that
~
+
and
b) the
the
latter
potential
(I.e.
the
potential felt
interparticle
important
are
compete~
the
ith
~
wall.
the
7
external
wall,
external
i'~(202
approximation
a) the
reader
the
how
of
the
tilt
(6?j
of the
influence
2
which
Fowler
the
terms
molecule~
Sil~ (222
~
P~ (cos Ml
remind
We
J
energy
write
may
wall)
the
near
themselves.
are
~
u,~
In
INTERFACE.
calculated
been
question to
liquid crystalline
formulae
relevant
oi
(cos
~
crucial
of
absefiie
the
P
u.~
=
molecule~
the
near
term
order
density
particles as a
by
to
(i.e.
electric
)25] and by
Tjipto-Margo and
Sullivan
theory of anchoring phenomena. We
molecular
of the
features
have
u-j and u>~
of the
constant
condition~
order
by
reached
studies
favourite
contribution
nematic~.
all
electricity.
described
by a surface
independent of 8, we
quantities
The
under
Gennes
~EMATIC-SUBSTRATE
study of
Js
appro~imation
for
those
to
erg/cm~
lo-
x
Landau-de
A
[26] in their
present
2
conclu~ions
the
attention
the
of
terms
anchoring may be
ignoring the term
Surface
angle
of
draw
to
other
the
the
that
THEORY.~i
Sluckin
and
order
the
indicates
some
then
calculate
dominate
not
review
Teixeira
of
interfacial
the
in
used
be
to
estimate
least
~hall
rapidly
varies
principle~ be
macroscopic
in
psi'~,~
163j
~~~
and
~
~>~
/~~7
64
=
(pS)~ Vi(224)
of the effect of
is an integrated
i'~,~
mea~ure
of the
quantitie~ I',,(i~, i~, ii are the
moment~
potential
repre~entation of the pair
intermolecular
w,h~re
coefficients
the
,4r
For
the
Vj(202
part of the
may
of
purposes
essentially
i~
this
paper
it is
i~~ (222 ), and the Frank
potential with quadrupolar
actually
indistinguishable
due
be
in
the
to
long
explict
distance
the
wall
spherical
p~tential
harmonic,
useful
to
the
ela~tic
the
onset
of the
are
corre~pond~
quadrupoles
due
remind
constants
symmetry
limit
1651
o
governed by V~(2?0 j,
and
part oi the
of
Sj'~dii>'v(iji~i;11,
i'~(iji~ii=
tran~ition
ani~otropic
the
~~,~
and
164)
to
in
the
induced
reader
that
the
nematic-i,otropic
A
phase
governed by
fourth
~mectic
to
I'(224
the
molecules
off-axi~
or
dipoles,
r)
it
governed by
is
Thi~
term.
be
may
which
The
moments.
we
a
di;cu~~
term
term
in
N°
6
MOLECULAR
slightly
and
the
detail
more
such
in the
do
terms
calculation
exist
it
be
also
be
may
molecules.
for
verified
be
even
if w~ is
that
does
term
equation (63)
anchoring
the
that
energy.
which
zero,
However,
out
naturally if the
so
discussion
of the
depends
assigned
model
be
can
it is
It has
been
behaviour
possible
on
a
be
more
balance
fine
actual
[25, 26] that
using this picture. It
down
whereas
in
w~
order
and
to
if
w~
write
as
of
the
and w,~,
alone.
of w~
effects
some
is
the
long
so
more
or
either
at
does
determines
as
important
the
Frank
the
magnitude
less
corresponds
&
0
=
at
the
of
sense
remark,
interaction
Frank
less
&
or
elastic
zero
surface
conical
which
comes
it
in
or
angle
will
the
in
this
be
if it
experimental
anchoring
only
short-ranged
enough.
however,
is
the
of
depending
interface,
N-I
in
elastic
to
w,
=
equilibrium anchoring
which
w~ is positive,
qualitative
interesting to
the
to
effects,
steric
to
are
contribute
not
it
molecular
down
terms
have an equilibrium
magic angle condition
to
the
slowly,
value
elsewhere
made
derive
to
changes
The
quadrupolar
to
difficult
parameter
subsection.
last
write
to
However,
order
shown
can
much
due
term
range
these
crucially
it
quadrupole limit, then the anchoring energy is minimized
is it possible
the sign of w~. Only if w~ is
non-zero
on
texture.
short
a
practice
In
809
ELECTRICITY
ORDER
this
event
any
in
seen
expression
easily
can
In
OF
It
cylindrical
moments.
and it can
constants,
the w~ term
in the
Now
section.
next
in
of the
THEORY
it
constants
that it is
necessary
was
to
successfully write down
in the
terms
anchoring energy expansion, the quadrupolar
interaction
is quite happily short-ranged enough
So the order
A dipolar
would
to
term, by
contrast,
present problems
ensure
convergence.
electricity is being accounted
for by making a short-range
expansion, albeit a rather
different
from the usual
gradient expansion. If, however, we were to ask how the surface ordering
one
would be
affected by changes of the order
close to the
interface,
thus giving rise to a
parameter
surface
of order
functional
short-range
parameter gradients, then we would find that the natural
expansion
would
diverge. We would
then
forced
calculations
of the
be
to
resort
to
type
subsection.
employed in the previous
Finally we note that there are in the literature
examples of conical anchoring [31numerous
33]. There
be
of
explanations
which
with the
shortage
render plausible effects
to
no
can
seems
observed
of
magnitude.
Nevertheless,
far
from
explanation
in
orders
is
particular
a precise
one
surfaces
exotic
which
often
involve
rough
surfaces
by
prepared in rather
cases,
or
ways
rubbing or evaporation of a substrate
oblique
angle.
at
an
that
suppose
6.
The
role
interactions
the
of
order
off
fell
electricity
in
in
r~ ~,
the
smectic
order
A-C
to
transition.
phase the average
quadrupole density is homogeneous in
polarization thus only appears
boundaries
defects.
By
near
or
phases the average quadrupole density Q (r ) oscillates rapidly in phase with
In
nematic
the
electric
wave
the
the
and
in
smectic
order
smectic
density
:
Q (r)
Then,
proposed
as
proportional
to
the
by Barbero
gradient of
P~(r)
In the
theory
=
and
Qi
+
[4],
Durand
VQ (r)
=
(ko
cos
quadrupole
this
=
Qb
ko
one
density
*
Q
+
r
sin
w ).
(66)
formally
can
define
a
polarization
:
(ko
r
+
(67)
[4], the
electrostatic
associated
with this
energy density
form
general
given
by
equation
(6).
It
thus
the
can
possesses
same
the tilt angle of the
director
the
corresponding order electric
in the free
on
term
as
of the N-I
interface.
This is given by the final
in equations (46) and (51). Just as
terms
interface
problem, the corresponding energy is minimal for a director
orientation
tilted to
polarization
dependence
energy
in the
bulk
contrast,
of
be
Barbero
written
and
in the
Durand
810
JOURNAL
plane
smectic
the
interpretation
On
the
normal,
of
hand,
other
between
permanent
sufficient
to
that
see
two
papers
using
Indeed,
quadrupoles
Poniewierski
the
is
consider
(24),
Assuming for simplicity
the perfect
smectic,
that
the
we
only
N°
and
Barbero
[34]
[4]
Durand
propme
to
Using
qualitatively
the
2
one
Fourier
of this
result~
transition
same
between
energy
d~k(kQ,
w
=
an
the
interaction
average
recently shown that
aniwtropic
molecular
have
by
modulated
transition.
A-C
(V~~(i, j ))
in
Sluckin
and
quadrupoles,
smectic
equation
given by
by
used
was
ii
a
simple
k
component
)(kQ-
L
of the
k) k-
paper
one
mechanism.
interaction
shape, is
readily
can
molecular
permanent
168)
~
quadrupole density
is
non-zero
obtain
(V~~(I,
j
))/B
=
(2 WI-
Q(
d-
P((cos
8),
(69)
period and B is the surface
A very
similar
be
area,
expres~ion
can
using the general
e~pression
for
the
electrostatic
given in
energy
equation (6), and then substituting the oscillating polarization P (=) given in equation (67).
smectic
A-C phase
transition
is in order,
The
only be really
caution
however.
A note of
can
electric
interpreted in terms of order electricity in a very qualitative way. Indeed, the order
polarization and
electric
field
in
only
for
smectics
large
average
are
non-zero
very
wave
molecular
length, For such values of k, it is impossible to make
k
L- ', where L is the
vector~
consistent
separation between the contribution
from this polarization, and
contributions
from
a
other
intermolecular
interactions
which
always couple to the
quadrupole-quadrupole
interaction.
In fact, in this
there is no long
electrostatic
interaction
which
part of the
case,
range
effect
determines
the
and the separation is rather
arbitrary,
However,
the general
idea of the
quadrupole
of the
smectic
A-C
fruitful
transition
thi~
transition
nature
to be
appears
very
indeed
interactions
if
the
be
determined
by
which,
quadrupole-quadrupole
itself.
to
not
seems
the
For
example,
in
the
successful
model
of
der
Meer
and
at least
symmetry,
possess
same
van
Vertogen [35] the tilt of the director in the smectic C phase is
determined
by the
induction
interaction
the
between
off-center
molecular
dipole and the polanzability
of the
transverse
neighbour
molecule.
This
interaction,
however,
is equivalent
interaction
between
to
an
an
effective
quadrupole in the center of one
molecule
and the
polarizability of the other. Thu~,
according to thi~ model, the A-C
is also governed by the quadrupolar
transition
in
interaction
this
interaction
induced
by an
between
and an
quadrupole.
permanent
case,
a
where
d i~
obtained
7.
In
smectic
the
directly,
Discussion.
thi~
paper
we
ha~e
di~cus~ed
of
liquid crystal~,
variou~
a~pect~
of
the
impact
of
electrostatic
force~
on
the
phenomenon wa~ named
order electricity
by Durand. The
important property is that a rapid gradient of electric
quadrupole density gives ri~e to electric
effect~
which
be
for using
conventional
idea~ oi the theory of liquid~, Rather,
accounted
cannot
ideas
analogou~
tho~e
used
Ewald
in
his
of
dielectric
by
to
treatment
must
~j,stem~
one
use
early in the century.
These
intermolecular
force~
ideas
involve
a.ieparatioii of the
longinto
forces,
contributions.
In
nematics,
quadrupolar
though
fall
off
and
short-range
they
range
even
i-~
in,lalidate
~ufficiently
long-range
the
usual
gradient
expansion
of
the
Landau
to
are
known in the liquid crystal world as the
Landau-de
Genne~
expansion, Extra term~ must
type,
introduced
for these long range
force~,
be
in order to
Similar
idea~
have
used
been
account
recently to discuss charged and dipolar fluids,
more
behaviour
6
transition.
A-C
molecular
produce
these
idea
This
smectic
the
PHYSIQUE
DE
Thi~
«
»
N° 6
MOLECULAR
Durand
these
able
was
liquid theory.
A
crucial
change.
to
idea
We
however,
have
we
tried
us
and
to
test
to
phenomenology
do is to clothe
rather
better the
these
extent
811
ELECTRICITY
ORDER
OF
to
ideas
to
information
about
respectability of
the previous order
modern
much
extract
in the
which
electric
applies.
has
been
to
orient
itself
in
sections
seen
have
applying
enables
effect
this
how
see
have
indeed
prefers
intuition
use
we
This
phenomenology
parameter
to
What
systems.
THEORY
seen
that
in the
competes
that
the
at
4 and 5 that
quantitatively
if the
order
electric
quadrupolar forces,
magic angle » with respect
of
presence
so-called «
there
is
much
to
this
idea,
short-range
parameter gradient is too large,
seriously.
too
with
other
and
a
to
we
effects.
one
changing
the
have
In
should
order
direction
been
of
able
section
5.2,
show
great
magic angle ideas
that the origin of the
director
We have
also
tilt with
respect to the order
parameter
seen
harrnonic
intermolecular
potential
gradient lies in the V(224) spherical
contribution
the
to
electrostatic
interaction,
In practice, this is the « long-range »
although there are less
energy.
important short-range
contributions
resulting from steric effects. However, the formal
structure
which
which
involves
separation into long and
have set up
apart from in section 5.2
we
short range
contributions,
does not really explicitly recognise this fact. It is almost certainly the
interactions,
which
have a long-ranged
tail, also give rise to
that
screened
do
not
case
effects.
analogous, if weaker,
quasi-order-electric
In this
have
been
able to
not
paper
we
effects
address
the question of how such
treated, and how, in particular,
may be consistently
the limit of the Debye screening length going to infinity can be
understood.
These
and other
related
problems are currently under study.
care
in
the
Acknowledgments.
acknowledges the
financial
of a
fellowship from the
support
his
during
in
of
this
work was
carried
Germany.
Part
stay
von
mutual
visits by etch of the
coauthors
other's
institutions.
the
The
Humboldt
out during
to
Foundation
financially supported a visit by M. A. Osipov to Southampton. T. J. Sluckin thanks
the Royal Society for
financial
which
enabled
him to visit
Moscow.
support
The
authors
grateful
Prof.
G.
Durand
for
stimulating
conversations.
M. A. Osipov
to
are
thanks
S. Hess
for
Prof.
interesting
discussions
thanks
and
le
Ministkre
l'Education
de
Nationale, de la Recherche
France for
financial
enabled
him to
et de l'Espace de
support which
visit Orsay.
M.
A.
Alexander
Osipov
gratefully
Humboldt
Foundation
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6