25. September 1992, accepted 28. September 1992). Abstract,. The influence of the flexoelectric effect on the phase diagram relevant to the order transition.
Phys.
J.
France
II
(1993)
3
147-164
1993,
JANUARY
PAGE
147
Classification
Physics
Abstracts
61.30
Flexoelectricity liquid crystals
alignment
and
(1, ~), G.
R.
Barbed
(t)
Dipartimento
di
Barbero
Fisica,
3),
(1>
Universith
(2) Facolth di Ingegneria, Calabria, Italy (3) Dipartimento di Fisica,
phase
di
Politecnico
di
and
Zvezdin
A.
Calabria, 87036 di Rende Arcavata Calabria, Via Cuzzocrea 48,
delta
Universith
(1. 4)
Gabbasova
Z.
nematic
in
transitions
Reggio
Torino,
Corso
Duca
degli
24,
Abruzzi
(3, 5) (CS), 899128
Italy Reggio Torino,
10129
Italy (4)
Physics
Theoretical
Department,
Bashkir
University,
State
Uliza
Frunze
32,
Ufa,
450074
Russia of (5) Institute Russia Moscow,
(Received13
Physics,
General
March
1992,
Academy
Russian
revised
25
September
of
1992,
Sciences,
Uliza
accepted 28
38,
Valivola
September
117942
1992)
the phase diagram order relevant to the on considered. liquid crystal is The analysis shows on a influence that this be important. It is found that the order phase transition between the initially can transition undistorted and distorted configurations always takes place. In contrast, the phase the distorted saturated configurations is possible only if the dielectric anisotropy is between and the of a tricritical large enough. The stability of the phases is analysed. The existence point is predicted. The dependence of the tricritical point on the flexoelectric coefficient discussed is too. critically analysed. The limits of our calculations and the performed simplifying hypotheses are influence
The
Abstract,
by
induced
transition
an
of the
electric
flexoelectric
field
effect
nematic
Introducdion.
1.
materials
Nematic
parallel
constant
ei.
follows
that
nematic
is
this
nematic
uniform,
parallel (e~ This
well
0)
particular their dielectric perpendicular to it one negative. From this anisotropy, it
from
different
can ej be positive can ei oriented by means
ejj
=
properties Ill-
be
axis
is
position
0)
to
it,
an
Fr6edericksz
depends on the elastic properties and sample and on the surface properties. attention has been particular special
or
of
an
In
the
extemal
electric
field.
If
the
independent, and the electric field is transition order phase is expected [2]. transition. critical field giving rise to The the anisotropy of the medium, the on on A lot
of
devoted
papers the to
are
devoted undistorted
this
to -
order
distorted
JOURNAL
148
[3] (usual
transition
field
transitions
these
on
expression In
for
this
field
will
[8],
analysis
the
the
In
approximation equations relevant to frame.
usual
discussed. conclusions
2.
of
Dielectric
Let
the
to
the
In
transition the
paper
our
surfaces
in the
flexoelectric
phase
diagram
the
and
results
effect
is
reported
is
coefficient
have
follows.
are
and
few
a
analysed.
and
section
In
2 the
rapidly peculiar
section
In
will
be
considered
be
to
nematic
considered
energy
[9].
effect
flexoelectric
obtained
free
5
as
main
discussed
aspects main
the
field.
liquid crystal
d and
the
~~ =
of
director
I.e.
the
small
e~
[10]
dz
fj
n
total
the
Let
the
be
parallel
free
energy
form
) e~E~ cos~
)~
dz
limit
In the
the
~~
K( 2
F
in
written
anisotropy.
dielectric
orientation
field.
extemal
be
by positive planar,
characterized nematic
initial
of
absence can
flexoelectric
the
versus
surface
the
stressed.
nematic-electric
sample
nematic
of the
3 the
for
paper is organized as field with the extemal
Our
ones.
are
nematic
a
thickness
an
configuration
saturated
-
expression
the
strength Consequently the
interaction
the
use
account
energy
true
section 4
be of
the
of
of the
interaction
consider
us
sample
The
distorted
-
We
into
take
we
anchoring independent.
section
In
undistorted
considered.
be
and
thickness
first
in the
n
one.
saturated
-
director) is parallel
(the
by different authors proposed a long time
energy
phase diagram
the
proposed
considered
free
critical
a
[4]. In distorting to influence of the surface properties [5] by using a phenomenological ago by Rapini and Papoular [6], or
distorted
the
to
axis
symmetry than
discussed
been
surface
the
paper electric
applied recently
has
and
N°
II
[7].
of it
variants
a
transition),
Frdedericksz
everywhere the nematic case, when the applied field is larger
last
the
PHYSIQUE
DE
2
+
f)
+
(1) ,
_~~
where
K
:
nematic
elastic
electric
field,
z
=
tilt
nematic d/2
z
constant, e~ coordinate
=
extemal
E
=
nematic
=
normal
limiting
the
to
=
anisotropy
dielectric
angle (see Fig, I), fj, f) surface free energy respectively. By minimizing (I) and z + d/2
densities
the
to
for
(n
z)
surface
at
cos~
=
relative
obtains
one
=
(supposed positive),
surfaces,
differential
the
equation
~~
f~~
with
boundary
the
K
(~ z
(2)
In =
In
stable.
Vz
d/2),
(- d/2,
e
0 =
at
d/2,
z
and
(~
K
=
do
+
(2)
f=/~(I/E)
is
the
coherence
/~
(ar/d)
length threshold
is the
which field
~~~
0
at
=
z
+
=
(3)
d/2
do
z
(dlar )(E/E ). E~ in the strong anchoring f
0, =
conditions
~~~ +
(2
sin
2
dz
it
useful
is
rewrite
to
for the
Frdedericksz
(± d/2 )
are
transition
=
the
following we Consequently fj
situation. suppose and
In that
f)
equations (3), H* in
have
the a
minimum
=
of
extemal
for
H~
field
H+ =
the
surface
the
planar
ar/2
hence
tilt
angles.
orientation
is
=
df)
ldfj do
absence
as
[I I
"
wQ
0,
j
do
"
wQ
(4)
N°
FLEXOELECTRICITY
I
TRANSITION
PHASE
AND
IN
149
NLC
z
d/2
a)
z
dh
b)
'
'
x
'
'
d/2
C)
(sin H, 0, nematic thickness d is considered. slab of n angle. a) In the absence of the extemal distorting field (parallel field E, larger than a critical x-axis (planar orientation). b) If an extemal to the z-axis) n is parallel to the field larger than the sample (distorted phase). c) If an extemal larger uniform across one is applied, n is no the sample and everywhere parallel to the distorting field saturation field is applied n is uniform the across (homeotropic orientation).
Fig. cos
1.-
Geometry
is the
nematic
of
problem.
the
director,
His
the
A
=
tilt
and
~~f+
ld2fd0
y/-
S
Furthermore
we
assume
~
that
~
d0
~
w12
fj
and
f]
have
y~+
S
~
a
~
~
~
~~~
~
maximum
w12
for
H~
0.
+ =
=
This
hypothesis
150
JOURNAL
PHYSIQUE
DE
II
N°
implies
ldfl d°+
ldfi ~ o
~
~~~
'
o
and
0
l~2fd~-2
~
~
-
=
(ar/2 )
considering
and
7~,
fl/+
S
~~+~
'
Finally the surface energies are supposed monotonic df)/dH* Consequently the surface vanish torques alignment. The threshold field for the undistorted by putting
0
~f+
fl/-
s
the
case
~
~
~~~
of (0, ar/2). functions decreasing e and planar only for the homeotropic distorted transition is obtained order phase 0. In this limit equation (2) reduces 7~ -
to
~ih)
j+
d~
where
E/E~
h
is
applied
reduced
the
~
7~=0,
d
dz
field.
Solution
(8)
of
(8)
is
=
7~
(z)
A
By substituting (9) into (3), taking of kind (9) exists only if the h'given by
I hz
cos
=
into
(4) and (5),
account
field
reduced
the
"
which
is
a
in
generalization the simple
The
saturation
equation
(2) in
field
well
in
case
for
limit
the
of the
W+
W~ =
distorted
the
of H
0.
-
In
this
I
solution
electric
field
is
d
that the
larger
is
a
called
+
+
=
is
a
(lo)
by Rapini
long
called
phase reduces
time ago Frdedeflcksz's
transition
and
field.
is
by
obtained
to
0,
(11)
#~
h"
d
e
field.
of
the
saturation
Equations
(lo)
I hz
(12)
(3), and taking into (- d/2, d/2 ), is stable if the
field and
equations
account
reduced
applied
given by
1) ~
arh"
K
sh
conditions
0, Vz
value
h
,,
generalization saturation
(z)
critical
a
#~
which
boundary
solution
than
I hz
ch
=
(12) into
solution
~~ ~~"~
h" is
=
field
"
order
~
solution
threshold
a
is
(z )
By substituting (6, 7) one obtains
h)
d
nontrivial
a
than
~,)~j,
$~ K
equation (2)
case
j dz
h'
saturated
-
d~~
whose
~~~~.
that
larger
is
equation proposed
known
which
(9)
deduce
we
h
(~ )~~ ~,ji(1-~~'
tg(arh')=
Papoular
I hz
B sin
+
ii'~
ii'~ K~
d
~~~~
arh"
given by NehfIng, Kmetz general, and they are
(13)
~ '
and
hold
Sheffer in the
[13]. case
in
N°
FLEXOELECTRICITY
I
f)
which
functions
monotonic
are
TRANSITION
PHASE
AND
of H,
having
to
consider
minimum
a
IN
for
NLC
lsl
ar/2
and
in
which
for
maximum
a
=
0. =
following
the
In
ourselves
limit
we
Ki~ and K~~
where
surface
the
free
observed
=
=
of
energy
surface
order
(lo)
equations
(14),
(14)
[15]
transition
An
expression
for
interpret recently By using for appendixA).
proposed [14]
been
(13)
and
dependent.
temperature
parameters
(14) has
kind
the
cos~
K~~
+
case
,
phenomenological
are
spontaneous
f~ expression
Ki~ cos~
f)
f/
symmetric
the
in
(see
order
to
written
are
2(81/h') ~~~
~~
,
~~~~
(&i/h')~'
l
and
~~~~~~~~~~~~~
tgh(arh")=
[(81
+
2
+
(16)
8~)/h"]~
,
where ~ ~~~
81
d
~~d,
~
8~
and
=
=
(17)
surface anchoring taking into the parameters Ki~ and parameters account properties of the medium K and the sample thickness d. fields The standard analysis performed in this section to deduce the threshold and saturation obtained boundaries. However result the limited only to the vicinity of the phase is are of H, having a monotonic functions surface energies are supposed meaningful, because the adimensional
are
the
K~~,
elastic
minimum
ar/2,
for
and
the
diagram
Flexoelectric
by
obtained
interaction
the previous section proportional to the density proportional
0.
The
flexoelectric
the
kind
same
effect
equations
of
means
interaction
nematic-electric
only
In
is
(15)
is
of
analysis
will
be
considered.
(16)
and
has
been
recently
the
As is
well
known
interaction
dielectric
of the
square the to
field.
electric
dielectric
anisotropy.
considered.
been
has
and it
field
it
gives a bulk effect, resulting in a torque 0) this compensated nematic In (e~ =
vanishes.
In this
flexoelectric
the
section
distorted
a
for =
which
in
[16].
considered
3.
section,
next
maximum
a
=
performed in The phase
derivative
of the
interaction
usually
nematic
nematic
presents
considered.
be
polarization
shown a long time proportional to the flexoelectric polarization, As
polarization
electric
an
This
director.
will
is
called
ago
[17]
spatial and it is
given by P= where
The
and
ejj
extemal
electric
electric
elastic
kind
constants
[18],
(18)
coefficients. The coupling of this polarization with an subject of many theoretical investigations. and experimental density connected the coupling of with P the extemal to energy
the
been the
to
P
ndivn-e~~nxrotn,
flexoelectric
the
has
contribution
field E is of the the
e~~ are field
en
E.
The
which
flexoelectric can
be
polarization
neglected
if
we
are
introduces
looking
also for
a
renormalization
instabilities
of
separating
152
JOURNAL
undistorted
an
from
state
flexoelectric
the
effect
the
case
Simple
of
can
[20]
reduces
In
one.
that,
show
~l
E dz
P
-
e
en + em. that the analysis
=
follows
It
Fj
reported
~
fj
=
+
particular by assuming
In
f)
for
2(Ki~
(13) the
equation #7~
2 =
(21)
Equations surface
the
which
in one.
>
0,
flexoelectric
the
+
2 K~~
the
and
for
~~
2
stabilizes
(22),
+
equations (lo) expression
the
eE
introduces
surface
lower
to
put
(21)
2 K~~
analysing.
are
the
into
field
we
(Kis
effect
flexoelectric
problem
have
we
given by
now
=
by (20)
)
eE),
(Ki~
are
ii7+
the
saturation
the
ii7+
and
and
in the and
2 =
ii7~
effect
(21)
equations
W+
+
equation (lo)
in
substituted
are
(2
cos
(14),
form
and
of
presence
~
f] =
eE)
+
unchanged if f)
]
F
~19)
)i
~2 ~
C°S
remains
2
and
eE)
parameters
flexoelectric
the
field
orientation,
undistorted
functional
+
consequently
and
By substituting
Fr£edericksz
(Ki~
that
show
energy, E
a
iC°S ~2 ~+
)
0
the
=
in
as
detail
in
discussed
is
section
in
(2
cos
W~
whereas
unidimensional problem is by Durand [19]. In this energies. surface renormalization of the the
which
in
event
the
near
N°
II
to
~l~~
where
the
integrated, polarization
be
flexoelectric
the
calculations
contribution
distorted
a
contribution
PHYSIQUE
DE
In
asymmetry
an
particular
in the
destabilizes
the
and
(13)
and
(22)
).
we
obtain
in event
upper for the
2(81/h')
~~~'~
~
(81/h')~
l
(4
+
~~~~
e~/Ke~l'
and ~ ~~~~ ~ ~ ~~~~~~~
tg h(arh")
=
[(&1
+
respectively.
By
equations
(-
&i
(23) and (arh')
cotg
=
(24) +
(4 e~/Ke~)
&~)/h"]~
2
+
simple
give
calculations
~/cotg~(arh')
[1
+
+
(24) ,
(4
ehKe~)])
h',
(25)
and &1
Equation (25)
2 &~ +
=
limit
in the
(cotg e~
-
h
( ark
gives
0
~/cotg h~(arh" )
"
critical
for the
field
(4 e~/Ke~)j) h
[1 E'
of the
flexoelectric
"
instability
(26) the
value ~
E' =
which
equation
generalizes (27)
we
a can
formula estimate
2 ed
obtained the
order
Ki~
(2 in
~
d K
a
of
+
I
lj
in
(27) ,
different
magnitude
way some years of the threshold
ago [21]. By using for the flexoelectric
FLEXOELECTRICITY
N°
instability
e
voltage
threshold
Frdedericksz
considered
in
as
our
known,
can
present
in
references,
bidimensional
and
this
detail
of the
instability
refers
which
153
K~ for
the
of
the
voltage
threshold
en
field
the
at
appears
This
nematic
bidimensional
the
than
director
planar
the
# e~~,
lower
references
one-dimensional
to
instability.
in
connected
one
[23, 24]. As it is
cell
pattem, the
to
in the
shown
voltage V~ given by
threshold
2 arK
e*(I
pl'
+
e~K/4are*~
p
section in
event
in
~
e*=en-e~~
same
threshold
a
~
where
order
bidimensional
a
bidimensional
the
the
in
the
effect, may have instability, as discussed
one-dimensional
quoted
presented
well
analysis
NLC
IN
nematics.
usual
analysis
it is
TRANSITION
(e~~0) nematic liquids crystals. By assuming 10~~ erg/m~ [13] and d lo ~m obtains Ki~ one
flexoelectric
the
to
in the
that
But
PHASE
(l-2) V, I.e, of
E'd =
out
deformation. due
V'
transition
point
We
jfnsated dyn [22],
nearly c 10~~
in
10~~ dyn [2],
AND
the
At
the
threshold
of
wave-vector
the
=
q~ is
structure
arji-J1 ~~~
qc"j it
As p
for
and
4ekKe~=0.9
Et
0, is given by
(cotg In
in
contrast,
figure increasing
4
In
the are
=
h
the
phenomenon
81(h")
and
is
8~ < 0, to h" case reported the lines 81 electric
field
Planar
0
=
the
phase
for
case
&)~~'the order.
The as
Of
&i
transitions >
distorted course
coordinates shown
&)~~'the
in
of
are
phase is analysis our of the
figure
6.
second
situation never
is
tricritical
planar,
field
0.25.
Note
that
nor
is
by considering
3
in
not
h
cases
the
case
8~
2
"
a
the
=
=
this
case
considered.
In
=
Distorted
In
by
Homeotropic
-
h"
order.
is
In
figure
sbnilar
to
and
the
stable,
P
5 the
only
for
8~
event
previous
the -
H
order
81 > 2 8~ ), as point TCP in figure 5 depend
valid
not
2) 8~). corresponds 81 81(h') and 81 81(h") for 8~ > 0. observes the order phase transitions
h'
All
figure
in
is
extemal
=
=
one
-
sample the
to
ekKe~)])
(4
[1
8~.
the
connected
reported furthermore 8~
=
2
field
extemal
~/cotg h~(arh" )
(arh")
applied
the
of
0
~
one.
phase by
stated on
the
0 is In
contrast,
transition means
is of
flexoelectric
for of
the
this
&j~ first
equation (7). coefficient
JOURNAL
156
PHYSIQUE
DE
N°
II
61
/
h
a)
61
/ / 1
/ /
/ /
/
/ / / / / / /
I'
/
/
h
b)
Fig. 2.
Phase
flexoelectric
the
diagram 31 parameter
4
=
3
j
(h') for the
e~/Ke~.
configuration
4
e~/Ke~
Planar 0.9 =
Distorted
-
(continuous
phase transition for different values of curve), 4 e~/Ke~ 0.1 (dashed curve). =
(Planar) one. For h ~ h' the stable configuration is distorted Note that for large 31 (I.e. for strong the a) OS &j w10. anchoring) the h' is nearly one. sub-case independent of the flexoelectric refers to very weak anchoring. 1. This parameter. b) 0 w i w equation (25). For h'~ 0, &1= (w/2)[1 + (4 e~/Ke~)] h'~, as follows from For h
~
h'
the
stable
undistorted
is the
E. They are experimental meaningful only K~~, e~ e are investigations [25] show that the anchoring parameters Ki~ and K~~ seem to depend strongly on reference [26] this is connected to long shown in thickness of the sample. As it has been the selective adsorption. Furthermore, as forces, ions electrostatic due like surface forces, to range flexoelectric also depend the thickness, coefficient [27], the discussed in reference can on as a scalar of the nematic order variation of the spatial parameter. consequence first considered phase-diagram presented in our paper have to be It follows that the as a if Debye underline that the is important However it of the approximation to true ones. We
underline
that
the
if Ki~,
phase-diagram 31 and
E
are,
actually,
thickness
phase
diagram
independent.
d
Recent
PHASE
AND
FLEXOELECTRICITY
N°
TRANSITION
IN
61
NLC
157
/ ~
/ /
/
/ / /
/ / / / ~ /
o
Fig.
&i
values
4e~/Ke~
curve),
whereas
h'
=
distorted
0.1
(dashed
(the
critical
h~h"
for
one
meaningful
in
the
for
the
-Homeotropic
Distorted
=
=
0,
&1
for
and
&i(h")
diagram different
Phase
3.-
0.25
h
the
of
curve). field)
the
region &1~
zero
h"
(the
at
configuration
stable 0.5
to
when
that
Note
goes
4e~/Ke~.
coefficient
flexoelectric
is
0.
field)
For h
~
h"
homeotropic
the
different
is
stable
the
The
one.
32
for
"
(continuous
=
saturation -
phase transition e~/Ke~ 0.9
4
from
for
zero
configuration is phase diagram
the is
(see the text).
61
P D
H
0
h
Fig. 4. Phase diagram &i &i(h) positive &2 (&~ 0.25 ) and 4 e~/Ke~ The phase diagram is meaningful in
for
screening anchoring of our
length
0.1.
=
Distorted Homeotropic phase phases transition P D and D H are of region &1~ 0 (see the text). the
=
=
the
Planar
-
transition
-
The
-
-
second
for
order.
enough the dependence on the thickness of the sample of the disappear. Furthermore, the dependence on the thickness of the sample flexoelectric coefficients the is important only for very small thicknesses [28]. Consequently results expected to work well for thicknesses of the sample larger than a few microns. are
5. The
small
Conclusions. influence
extemal weak
is
parameters
of the
flexoelectricity
field-anchoring anchoring, the
parameter flexoelectric
on
has
the been
effect
phase diagram considered. can
change
of the
transition
order
analysis drastically
The
shows the
that
phase
on
plane
the
in the
limit
transition.
of In
JOURNAL
158
PHYSIQUE
DE
~
II
N°
h i
D ~~
P
H
~
Fig.
5.
first
h~~~
at
As
order.
~
figure 4,
in For
&j
with
~
the
O-S
(see
0 (&~
~
the
0.25 ).
=
order
transitions
tricritical
point.
the
of
coordinates
are
&~
&)~~ all
"~
°~
"~
Note
that
P-D
for &1 ~ &)~~ the P - H second of D-H are
transition
is of
order.
&)~~,
and
phase
The
'
h
diagram
meaningful
is
in
the
region
text).
the
fit
6z"~0.25
~r=O r=O&9
h
Fig.
6.
of the
Tricritical
region &j
6.
in the
dielectric shown
been
for
the
coefficient
4
(see
0.5
~
particular the
line
flexoelectric
and
and its
0.25.
&~
It
=
for 0
w
4
represents
e~/Ke~
w
0.99.
points
the
tricritical
The
phase diagram
different
for is
meaningful
values in the
text).
the
limit
one,
case
e~/Ke~,
of
very hence
weak the
connection
flexoelectric anchoring, the is strongly polar. The
effect with
the
flexoelectric
effect
interaction existence
of
is a
more
important point
tricritical
of has
discussed.
Acknowledgments.
Many relevant
thanks to
are
the
due
to
the
phase-diagrams,
referees, and
of on
Joumal
the
de
Physique II,
bidimensional
for
flexoelectric
important
instability.
suggestions
N°
AND
FLEXOELECTRICITY
I
Appendix
in
useful
very It
has
set
that
terms
the
substrate
can
be
supposed
On
the
other
normal
K.
parameter
Q;~
~
S(T)(n,
n~
found
by
surface
the
and
of
by liquid
obtained
is
nematic
the
not
appear
can
and
the
pm (K; Q 0, by equations (82) we obtain that
F]
from
for
for
E~EI
=
Kile.
On
the
contrary
H+
0 =
160
JOURNAL
F),
minimizes
for E
E~
>
(Ki~
=
K~~)le.
2
+
PHYSIQUE
DE
Ei
For
II
E
=
F](0) (Ki~
are
(K~~
)le,
for
shown
=
that
7~+
(e/2K~~)(E-Ei),
=
(e/2K~~)(E~
0+
figure
in
-E).
The
ar/2 is
stable
for E
In
range
E~
direction
easy
Simple
the
calculations
0-E>
2ed
I,I ~
(ar&i-I)
~
H(ar/2,ar/2)>0-E
E'
the
that
the
~
H-phase
~/(ar&i-1)~-l. is
) given by (C3) for 7~ for E < E', G(7~ +, 7~ minimum of G (7~ +, 7~ )
never
stable
different
for
values
) has a unique corresponds to
JOURNAL
162
PHYSIQUE
DE
II
N°
fl'
E/ELo5 a)
n'
n.
[ ©
a5
n~
E/E'=
E /E'= lo
2
c)
b)
Fig. E
~
~+ ~
+
8.-
compensated surface tilt G for a nematic the energy vs. minimum for ~ + 0 b) E'. G of G is reached (P-phase). E ~ ~ different from (D-phase). c) The minimum E»E'. ~~ zero Total
The
E'.
=
and ar/2
and
~~
0.
The
alignment
of
the
angles reaches
~+ its
and
~~.
al
minimum
for
corresponds
to
=
nematic
is
nearly hybrid.
of
G
FLEXOELECTRICITY
N°
and
+
7~
different
7~
minimum
its to
large
a
whereas
7~
field
is
analysis
This
tends
reported
in
that
for
a
field
a
with
configuration.
anisotropy flexoelectricity.
be
can
is
+
7~
larger
little
E', G (7~ +,
to
compensated
a
that
shows
163
NLC
IN
respect
that
hybrid
the
assume
dielectric
the
the
of
effect
the
overcomes
to
This
very large fact shows
figure 9. Figure 9a
maximum
a
shows
0.
7~
~
presents
7~
and
field
field is
If the
zero.
w/2
+
electric
the
vs,
7~
for
from
TRANSITION
PHASE
AND
E',
than
trend
The
monotonic
a
after
responsible
) reachs
7~
nematic
it
that
submitted
of 7~+ and function of E, tends
to
zero.
stability only
of the
if it
n[n
a
b
O
Fig.
9.
~
E
vs.
angle
tilt
Surface
phase
order
a)
transition.
presents
~
*
~ +
maximum
a
the
vs,
E is
vs.
2
for
an a
applied field. increasing
field
to
near
E/E'
4
3
the
E'
For E
E'.
large E,
For
of ~
trend
*
tending
function
monotonic
tends
~
vs.
E is
to
ar/2
to
typical for
of
second
large
field.
Eds.
(Plenum
b)
zero.
References
[ii
Introduction
to
Liquid
Crystals,
E.
Priestley,
B.
Wojtowicz
P. J.
Ping Sheng
and
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[2] [3] [4]
The
N.
field is
saturation
distorting [5]
field.
T. J.,
SLUCKiN
defined Its
as
value
PONIEWIERSKI
the
field
is
finite
A.,
Fluid
above
only
if
which the
Interfacial
the
nematic
anchoring
is
everywhere is
energy A. Croxton
Phenomena,
1974). oriented
along
the
finite. Ed.
(Wiley,
Chichester,
1986). [6]
RAPiNI
A.,
[7]
A
few
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Phys.
C., J.
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Colloq. France (1984) 1087. Rapini-Papoular expression for M.,
PAPOULAR
ROSENBLATr
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BARBERO
G.,
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the
Liq. Cryst,
on
165
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MADHUSUDANA
surface
(1988)
properties
265.
30
(1969)
C4-54
;
45
N.
of
V.,
surface
the
DURAND
Liquid Crystal
is
energy
of
nematic
are
Nail~rforsch 39 (1984) given by YOKOYAMA H.,
G., Z.
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1066 Mol.
Cryst.
164
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[8]
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DE
II
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M., J. Phys. II France 1 (1991) 691; Microscopic theory of anchoring transitions surfaces of pure at the Liquid Crystals and their mixtures (to be published). [9] MEYER R., Phys. Rev. Lett. 29 (1969) 918. [10] Equation (I) is valid in the limit of small e~, or near the undistorted configuration. Since in the following we are interested iri the evaluation of the threshold field, separating saturation or undistorted phase from a distorted equation (I) is a good approximation. an one, [I Ii See f.I. CHANDRASEKHAR S., Liquid Crystals (Cambridge, 1977).
G.,
BARBERO
[12]
Z.,
GABBASOVA
P. I. C.,
TEIXEIRA
OsiPov
T. J.,
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G., MEUTI M., ii Nltovo Cimento llD (1989) 367 H., KOBAYASHI S., KAMEI H., J. Appl. Phys. 61 (1987) 4051. [13] NEHRING J., KMETz A. R., SCHEFFER T. J., J. Appl. Phys. 47 (1976) 850 YANG K. H., J. Phys. France 44 (1983) 479 BARBERO
YOKOYAMA
See
also
and
449 ; the paper
the
by
YANG
determination
II 4]
M.,
KAGANov the
by
review
(1987) [15]
CHIARELLI
Di
(1984) Lisi G.,
[16] [17] [18] [19] [20]
BARBERO
j25] j26] j27] j28] j29]
K.
(1980)
P.,
SONIN A.
C., Appl. Phys.
Lett.
A., Usp. Fiz.
Nauk
(1982) 62,
43
for
152
(1987)
experimental
surface
free
given by equation (14), and proposed working on magnetic materials. See f-I-
energy
people
among
773
ZVEzDIN
K.,
A.
KADOMTSEVA
M.,
A.
Soviet
Sci.
Rev.
1061
and
Phys.
Section
A 9
S.,
P., FAETrI 31;
C.,
ROSENBLATr
K.,
KomiTov
(1991)
198
L.,
FRoNzoNi
Phys.
France
LAGERWALL
T.,
S.
(1983)
44
UMA HART, Liq.
A.,
GRIFFIN
L.,
J.
Cryst. B.,
STEBLER
(1990)
359
STRIGAzzI
A.~
7
Mol.
Lett.
101A
Cryst. Liq.
l19.
G., ZVEzDIN A., to be published in JETP Lett. France 39 (1978) 273. A., PETROV A. G., MlTov M. D., J. Phys. (Academic Press, N-Y-, Liebert Ed. H. J., Solid State Phys., Sltppl. 14 L. G., Mol. Cryst. Liq. Cryst, l13 (1984) 237.
DERzHANKI DEULING
DURAND In
the
event
can
[21]
expression for the popular very
BELOV
E. I.,
parameters.
is
54
KATz
ROSENBLATr
surface
JETP
in
which
screening length,
[22] [23] j24]
M.,
L.
II?-
FLATISCHLER
Cryst.
BLiNov
H.,
K.
the
[8],
reference
in
or
of
phenomenological
The
by
review
recent
play
an
the
the
important
deformation
is
role.
In
large
and
of the
renormalization our
case
localized
elastic in
over
constant
which
only
renormalization be neglected. considered, this can 1234. G., DURAND G., Phys. Rev. 35A (1987) BARBERO N. V., DURAND G., J. Phys. Lett. 46 (1985) MADHUSUDANA
a
distance
smaller
due to the threshold
and
the
Debye
polarization
saturation
fields
(1979)
C3-331.
L-195.
C3-247. 40 (1979) M., J. Phys. Colloq. France France BOBGLEV Y. P., CHIGRINOV V. G., PIKIN S. A., J. Phys. Colloq. A., SONIN A. A., Liq. Cryst. 5 (1989) 645. KABAENKOV L. M., BLiNov France 51 (1990) 281. BARBERO G., DURAND G., J. Phys. Crystallogr. 33 (1989) 641. TERENTIEV E. M., PIKIN S. A., Sov. Phys. TERENTIEV E. M., PIKIN S. A., Liq. Cryst. 8 (1990) 587. BLiNov
than
flexoelectric
1978).
L.
YOKOYAMA
H.,
VAN
SPRANG
H.
A.,
J.
Appl. Phys.
57
(1985)
4520.
40
are
