Phys.
J.
II
France
(1996)
6
753-765
Switching
Nonlinear
MAY
in
Ferroelectric
Stabilized
Surface
1996,
PAGE
753
Liquid
Crystals E-I-
Demikhov
(~)
Institute
of
(~)
Institute
of
(~)
Institute
of
(Received
4
(~>*), S.A.
(~) and E-S-
Pikina
(~)
Physical Chemistry, University of Paderborn, 33095 Paderborn, Germany Crystallography, Russian Academy of Sciences, 117333 Moscow, Russia Oil and Gas Problems, Leninsky pr. 63, 117917 Moscow, Russia
May 1995,
PACS.64.70.Md
revised
Transitions
Liquid
PACS.68.15.+e
Pikin
in
thin
September1995,
22
accepted
1
February 1996)
liquid crystals
films
Dynamics of director field switching in the vicinity of the SmA-SmC* phase tranby means of light scattering. The dependence of scattered intensity resonance the electric field frequency is observed. The scattering disappears at lower temon resonance The observed anomaly is consistently explained in the framework of the nonlinear peratures. The results of this work of kink switching model. demonstrate that in the materials with case high spontaneous polarization this mechanism is predominant. Abstract.
sition
studied
is
Introduction
1.
The
dynamics of
collective
The
main
[1-12].
liquid crystals
ferroelectric
excitations
which
exist
eigenmodes
collective
in
the
in the
in
system
an
electric
in
equilibrium
phase damping
SmC*
are
the
field is or
diffusive
by the produced in
determined can
be
Goldstone
and
soft
inherent a
field. modes
process of small pulses of the order paramphase. The soft mode corresponds to a similar process for the order parameter amplitude. Generally, the rotation angle of the director in the a-cfield is not small and other electric should be considered. Because stabilized excitations the switching angle in the surface geomefor the conditions director the substrate and in the bulk anchoring motions try is large and on equivalent, the switching process may involve the production of inhomogeneities in not are the director which orientation propagate from one sample boundary to the other. One of the simplest forms of such inhomogeneities is so-called kink of the director field orientation across the film [13-16]. Mathematically, kinks are of the nonlinear solutions equation of motion of director in the static or alternating electric field. Such kinks can form on one of the boundthe where the coupling energy of the director is locally aries decreased with respect to the average by their width and propagation velocity, which depend on value. The kinks characterized are fields and the cell geometry. Obviously, if the kink propagation time the material parameters, coincides with the period of the a.c. electric field, light scattering of the type can resonance be expected.
The
Goldstone
mode
describes
the
eter
(*) Author
©
Les
for
(ditions
correspondence
de
Physique
(e-mail:
[email protected])
1996
JOURNAL
754
In
the
field
present
in
a
spontaneous
a
polarization
is
electric
and
theory with anisotropy
be
can
spontaneous
needed
the
dynamics
II
N°.5
switching process in Qualitatively, a high
of the
polarization.
the
electric
value
of the
the conic spiral in the lower electric fields. We suppress in the low frequency region which not be explained can
to
profile and its dependence on temperature resonance by the kink switching model. The good correlation of of important estimations material parameters such as polar
mode.
The
well-described
experiment enabled viscosity.
and
Experimental
2.
studied
Goldstone
the
field
have
high
light scattering
resonant
scattering by
as
we
with
observed
have
paper,
substance
PHYSIQUE
DE
Results
4~(25,35)-2- [chloro-3-methylpentanoyloxy]-4-'heptyloxybiphenyl (C7) has been studied. the following liquid~crystalline phases: isotropic (62 °C) smectic A possesses (54.6 °C) smectic C* (43 C) smectic G [17]. The spontaneous polarization in the smectic C~ phase of pure chiral C7 varies between 130 and 290 nC/cm~ with decreasing temperature ii?]. Cells 25 ~tm in thickness electrodes The polyimide with polyimide covered ITO used. were coating was uniaxially rubbed to produce planar boundary conditions. This coating played two Chiral
substance
This
°
produced the book-shelf geometry of the director field in the SmA and SmC~ phases, the injection of free charge carriers. The cells were placed in the inverted Olympi~s-PMG-3 polarized microscope and thermostated SmA-SmC* phase within +2mK. The registered visually transition in each temperature to was electric field from a function the cell. experiment. A sine wave generator was applied across field was The amplitude of the applied a-ceither kept with a magnitude about constant varied in the illuminated interval by a halo1.3 V/~tm or was up to 4 V/~tm. The films were lamp slightly decrossed polarizers and the light beam collimated The in 600 to was gen ~tm. connected intensity of reflected light beam was registered by a photomultiplier to a lock-in at the frequency of the electric field. The generator frequency change and data acquisition in the kHz has been carried The experimental out by a computer set-up range 10 Hz-100 program. enabled simultaneous optical observations and reflectivity in visible region of measurements roles:
and
it
prevented
wavelengths. Figure field
shows
with
possess
width
a
Figure
the
constant
2
shows
dependence of the amplitude. The
about
IA
the
curves
run
intensity
through
on
the
frequency
maximum
a
at
of the about
dependence of the
electric
field.
in
When
kHz
and
amplitude and frequency. Res-
resonance
just above TAG in the SmA and SmC* phases. In the disappears 1.5 degrees above the The observation transition.
observed
the
electric
a-c-
2.7
kHz.
temperature
scattering is onance phase the resonance ing in the SmA phase
scattering
the the
bookshelf
temperature
geometry is
is
due
decreased
to
the
the
induction
scattering
of the
amplitude
tilt runs
smectic of
A
switch-
angle by through
Approximately 4 °C below the disappears. The frequency shows a gradual decrease with transition the resonance resonance decreasing temperature. The temperature of the amplitude maximum approximately coincides of the frequency in the vicinity of frequency dependence. But the with a variation minimum of frequency of this point is of the order of the determination. the accuracy resonance frequency Figure 3 shows the dependence of the and amplitude the a.c. electric resonance on field strength. The magnitude of the frequency sharply by increasing the electric field increases value and reaches a saturation region at approximately 0.8 V/~tm. The scattering amplitude steeply displays and saturation region at about 3 V/~tm. Corresponding increases a more scattering curves are presented in Figure 4.
a
maximum
close
below
the
transition
and
then
gets
down.
N°5
SWITCHING
NONLINEAR
SSFLC
IN
755
(arb. units)
INTENSITY
2.5
2.0
1.5
1,o
o.5
AT=i.i°c
O-O 0
2
3
(kHz)
FREQUENCY
Fig.
Temperature
I.
AT
-0A
=
and
I-I
Theoretical
3. In
the
to
the
dependence
where
field
electric
a-c-
frequency
for
(Fig. 5), when the external under field E is parallel consideration electric planes and induces the polarization vector rotation by azimuth angle ~2(r, t) in planes, the equation of director and polarization motion takes form [13,14j geometry
~2
is the
angle,
angle
~t is the
sin
between
~2
Ub~
sin ~2
P
vectors
E,
and
modulus, K
piezoelectric
cos
P
~ ~
Kb~
~2 +
=
~/b~
~~,
ii
at
polarization, b is the spontaneous constant, ~/ is the viscosity coefficient.
~tb is the
=
is the
elastic
quantity
eaE~
U is the energy of anisotropy, ea I.e., the preferential orientations
ii that the angle ~2 varies right side of equation ii
is
the
the
viscous
torque
anisotropy.
dielectric
which
the
hinders
ii
)
"
b
f2
values ~2 which is
))
~2,
=
of
12
#
=
We
shall
0 and ~2
assume
to
invariant
an
We
~.
=
perpendicular of
existence
rotation
cos
(2)
=
correspond to along axis y-axis corresponds to the
~flfi for
on
Model
FE
The
intensity
scattering
the
°C.
smectic
these
tilt
of
4
the
that suppose
cell
f
b sin
~2
in
positive, equation
surfaces.
The
expression
~fiftiy vector
U is
13) with
components
(4)
JOURNAL
756
PHYSIQUE
DE
II
N°5
MAXIMUM FREQUENCY
INTENSITY
(kHz)
(ah. units) 7
6 ~
SmC"
SmA
+
+ + o
oo
+
o
o
_oo + o
~~+o
o
-
+
o
+±
+
o
o
o+(
o
I ~
~ ;+
+
++
+~
+ ~
~+~
~F
2.4 5
3
4
0
2
2
iT-T*j (oc)
Fig.
Temperature
2.
Here
f
shall
the
is
director
order
an
expression
for
think
corresponds that
It is
such
to
time
to
introduce
d is the cell
solution
of
thickness.
so
is
such
a
case,
we
(5)
=
U
being
model
omitted.
We
considerations.
=
One
"
pfi )< can
~
fi
d "
~
~
"
mentioned
assumptions,
the
exact
equation (1) [16]
~~~'~~ where
describe
also
can
in
h#~,
ea(E~)
averaging of U, the second harmonic in is sufficiently good for the qualitative y/~ and parameters variable s
~
where
=
flcos(uJt), but,
anisotropy U,
of
energy
E
approximation
an
convenient
the
m
field
(+).
amplitude
and
Equation (I)
C [5].
smectic
alternating
U
which
frequency (.)
resonance
ferroelectric of
action
approximate
the
use
the
of the
of
parameter
under
motion
dependence
a
spreading along
constant.
axis y
(7) velocity
Solution with
~~~~~~
sinh(s
describes
u
=
the
~~~
) sin(uJt))'
so
solitary
~acos(uJt).
kink
of
function
~2(s,t)
which
is
(8)
SWITCHING
NONLINEAR
N°5
SSFLC
IN
757
FREQUENCY
MAXIMUM INTENSITY
(kHz)
(arb. units)
~
$ .
.
O.
.
O
O +
. +
+
+
O
~
+
+
°
++
2
+
0
0
Fig.
A-c-
3.
strength
field
dependence
of the
frequency (.)
resonance
amplitude
and
(+)
at
AT
=
°C.
-0.5
If the
electric
describe
the
field E does kink
motion
not change with [13,14]
velocity u ~a. (7, 9) demand,
uJ
~
0, then
equations (7, 8)
so
~~~'~~ with
t, I.e., frequency
time
~~~~~~
sinh(s
~~~
at)
=
Solutions
in
fact, boundary
conditions
=
0
or
~2
=
~
at
very
large
values
~~>
~~ 80
8
~2
of
(10)
"
~l
I-e-,
at
large
very
boundary
displacements
kink
conditions
for
the
film
yo
y
with
Pi
at
into
values
y
account
=
0 and y
the
=
d, where 4
is
compared with kink width ~. In practice, d take, for example, the form
as
finite sin ~2
Kb~ =
effective
an
the
thickness
ill
~~
electric
potential of
the
cell
surface.
Taking
relation
~
"~
A
e
cosh(s ~
A(t)
=
so
sin(uJt)
A)
~
~~'
~~~~
(13)
uJ
or
A(t)
=
at
(14)
JOURNAL
758
PHYSIQUE
DE
N°5
II
(arb. units)
INTENSITY
f
5
~~~~~~
[
#
/
~
i j
~
~
V/#m
0.4
2
i
0
2
3
(kHz)
FREQUENCY
Fig.
Dependence
4.
of the
resonance
curves
on
field
electric
a-c-
4
at
AT
=
-0.5
°C.
y
~Q
ecosw
X :
8
Geometry
Fig.
5.
~ve
conclude
that
of
8sinq
P
model.
the
condition
ill)
is
equivalent
cosh(s at
is
the
sol
cell ~
boundaries.
(Ai
r~
a/uJ
The are
latest
much
less
condition
than
to
so
is
condition
~
A)
approximately
fulfilled
parameter
b
(is)
o
=
d/~
that
is
if the
possible
displacements large frequencies.
kink at
SWITCHING
NONLINEAR
N°5
general,
In
kinks
of cell.
bulk
in
If the
our
time
Such
case.
of
ordinary ferroelectrics, that the places so with the formation of "weak" places may exist at the cell
"weak"
certain
starts
orientational
759
by analogy with
assume, in
we
reversal
tion
SSFLC
IN
(kink)
nucleus
formation
smaller
is
than
of process nuclei which boundaries
the
polarizathe
are or
characteristic
the
in time
of
polarization reversal we can underline consider large angles ~2 that reorientation to we necessary ~, therefore this case differs from ordinary dielectric and in principle. Besides, one measurements orientational should that we consider polarization reversal between homogeneous states, note ferroelectric orientational helix is absent in this case. Usually, in chiral C, the smectics I-e-, an helix is untwisted by the cell boundaries if the cell thickness d is sufficiently small, for instance the helix at a certain threshold value of E untwists I ~tm. The applied electric field at d < 0.I (Euntw), which is a function of temperature and vanishes at the phase transition second-order consider the untwisted temperature TAG (5]. Therefore, even in thick films, we can states at the close to TAG under the action of electric field strong enough. temperatures The total intensity of scattering light is proportional to the integral of the scattering amplia(y, t) is proportional to the magnitude a(y, t) squared variables y and t. The function over tude ilhe(I, t)f, where I and f are the initial and final light polarizations, /hI is the perturbation of dielectric permeability tensor for a given light frequency. In this case, the magnitude ilhif difference of products of the order is proportional to the parameter components displacements, qualitatively. It is
hope that
then
kink
solution
therefore
scattering
the
/
I
According
intensity
dt of
light
denote
can
along
axis
incident x
x,
for
the
on
direction
then
=
=
1-
In
lily> 0)(f(Y,0)j~.
b
~2(y,t), (2(y,t)
cos
axis y and for the direction 2. If the
quantity I~~ from
/
2r/w
dt
#
=
/
b sin ~2(y,
the
and the
electric
light
(18)
directed along polarized initially
field is
b~
/
2r/w
dt
o
the
case
of
crossed
I~z
axis
and
y,
we
finally
equation (17):
d
dy(COS~ ~2(y,t)
COS~
~2(y,
0))~.
(19)
o
final
=
II?)
t),
light polarizations directed along axis z, we obtain projection of director n on the z-axis is approximately b2 cos2 which, when it is squared, results in integral (19). ~2 the case of crossed polarizers iiix and fiiy, we have quantity initial
I~~, since
I~~ In
magnitude
to
dYi(i(Y,t)(f(Y,t)
and y for
b~
(16)
d
o
For
the
(z(y, 0)(f(y, 0),
proportional
is
along
cell I
obtain
we
I~~
Izz
describe
components
(1(Y,t) for the
/
2r/w
"
definition
the
to
will
r~
(1(Y,t)(f (y,t) and
(7)
/
the
same
equal
to
quantity
quantity
d
dy(sin ~2(y, t)
cos
~2(y,
t)
sin ~2(y,
0)
cos
~2(y, 0))~.
(20)
o
polarizers iiix and f
=
b~
/ o
2r/w
dt
/ o
i
z,
we
obtain
the
quantity
d
dy(cos ~2(y,t)
cos~2(y, 0))~.
(21)
JOURNAL
760
Using the
~~~~i ~
the of
N°5
constant
(22)
omitted,
which
integrals
take
is
i~~~
not
factor,
a
coordinate
over
y.
one
We
2~/"
I~~(uJ)
Izz (uJ)
=
Iz~ (uJ)
tanh~ (b
A) 3
~~
~~~( tan
~~~~
~
tanh~
dt(
tanh~
~
A +
tanh b
-b~~
/
)
tanh A
(23)
~
~°~)~ [tanh b A
~~~
~~~~
tanh(b
A)
~~~~ ~~~ ~'
~~~~
2r/w
dt[tanh
=
+
A)] ),
A)
2 ~
A
b
sinh
tanh b tanh A
l
[tanh
btanhA)
tanh
in
3
tanh b tanh
tanh~ (b
ln(I
sinhA
tanh~ A)
3
tanh~
tanh A
(19-21)
A
ln(I
tanh A
b +
3
~
I~z(uJ)
b +
+
~
+
~~~j tanh
3
(I
tanh~
sinh
~~
o
+
obtain
~
A)
expressions
rewrite
can
then
3
b~~
=
dt(tanh
b~~
tanh(b
+
+
=
~~
~~~~~~
sinjuJt),
~
is
so
and
c°S~2iYt~
~~
_
terms
II
relations
~~~~~~,~~
where
PHYSIQUE
DE
+
tanh A
+
o
ln(I
~
(25)
A)].
tanh b tanh
Integrals (23-25) cannot be expressed as elementary functions, and, in addition, at very low frequencies uJ, the limits of integration should be changed for the following reasons. It is clear physically that when the period of electric field oscillations is much more than the time of kink through the cell, I-e-, when 2~/uJ » d/u, the perturbations of dielectric permeability motion during Therefore, in the low frequency limit, when A(t) m at, we must time d/u only. occur between take the integrals the limits 0 and bla. In fact, time magnitudes I(uJ) ~ 0 t over tend to finite limits effectively. At very high frequencies uJ, when the amplitudes of kink ~a/uJ, of the of they less cell oscillations order much than thickness d, intensities I I-e-, are are calculated diminish. The decay I(uJ) ~ co can be exactly by the expansions in terms of must magnitude A. For example, from (23-25), we obtain: I~
Ix~(uJ
=
~
Izz(uJ
co)
I~z(uJ
~
m
~
co)
m
b~~(~ tanh~
b~~(6 tanh
co)
m
~
b
3
b
tanh~
5
7tanh~
b~~(tanh b
5
tanh~ 3
(26)
),
uJ
~~ +
~
b)(~~
~
tanh~ b)(~~ ).
(27)
uJ
~
b)(~~ ). uJ
(28)
SWITCHING
NONLINEAR
N°5
(26-28),
equations
From
I(uJ
0) by
~
small
at
of
values
SSFLC
IN
parameter
b,
761
we
also
estimate
can
the
limits
substitution
the
~~~~
~(
A(t)~dt
a~
~
=
~~~
(29)
=
uJ
o
~~
t~dt
a
o
,
I-e-, I~~
Izz(uJ
=
0, b
~
~
~ ~ ~~ ~
0)
(30)
r~
,
I~~(uJ
~
0,
~
0)
I~z(uJ
~
0, b
~
0)
~4~~4
(31)
~,
r~
~2~~4
(32)
r~
equations (30-32) that the limit values of magnitude I(uJ) are small at (in practice, a > 10~ s~~ ). At intermediate values of uJ, we should a of a that the values existence maximum in I(uJ). It is clear physically maximum expect the of the when the kink travels a distance scattering intensity must correspond to a situation d during a time of the order of half-cycle ~/uJ, because, in such a case, the perturbations of dielectric permeability are large anywhere in the bulk of cell. Thus, we can make the estimate: Thus,
large
one
from
see
can
values
of
parameter
~~
~
~ +W
frequency
the
The
value
uJ~xt
uJ~xt
a/b In(I
relation
tanh~(b
~
A) and
inequalities
I,a
»
~eXi
being corresponding be obtained by can
tanhbtanhA) s~~ and
I
»
~~~~
I >
(const
~~ ~~°~~~ ~
where uJ~xt
const
a/b,
is of
and
the
order
the
maximum
~
of I. of
I~~,max
~
~
~/~~
b
~t ~
~fbd
Izz,max
~
+
can
b(I
The
has
maximum
a
is of the
order
const[2i~
I~y,max(Ld
+
u~
a/uJ) la. Thus,
(b
=
2/3, I-e-, have:
we
ea'
~fbd
~~~~ ~
Ldext)
of integral (25) shows that, at frequencies by equation (35), quantity I~z can be written
(-3~
at
of db~
/~ ~
estimate
~~~~
m
~~~~
~
11
determined
i~)]du)
cos
~~~'
~
~t~l~ ~
one
=
of the
uJ
Expression (34) integrals (23, 24)
a
~°~~~
~
/~
_~
'~
)buJ~t~)i,
tanh[(b
+
Imax I(uJ~xt). integrals oftanh(b-A), of Assuming the existence integrals (23, 24) as estimate
extremum
(23-25).
expressions +
(~~)
+W
the intensity rough estimates
to
]
~~~
~dext
,
the
in
((b
a
~
l.~.,
+W
~eXi
+ 4
a
in
'~~~~. ~E
13s)
vicinity of magnitude
uJ
r~
uJ~xt
as
In(exp(-2b)
+
exp(-2
~
)])
m
uJ
~~ ~ ~
m a
[const
+ 8b
3(1
+
i~)~j,
(36)
JOURNAL
762
where
u
-a/uJ. Thus, quantity
=
uJ~xt
PHYSIQUE
DE
a16 again
"
~~~~
One
can
-the
increase
see
-values
uJ~xt
integral
that
these
of
thickness
almost
are
identical
another
/~'
~~~~~
the
have
~~~~
and in the
identical
the
at
ones
qualitative
certain
a
of them
increase
in the
to
quantity I~z,max has
'~
characteristics
results
d
N°5
I, but
at b »
estimate:
~~~'~~~'~
II
values
similarity:
decrease
of ratio
of uJ~xt,
a16.
difference between the field dependences of the quantities I~z,max. The first three decrease with increasing field, but the last if the tilt angle does not change. At small values of amplitude E, one remains constant one from equations (30-32) that intensity I decreases, at least as E~ at decreasing fl. see can dependences of magnitudes uJ~xt and Imax are determined The by the tempertemperature dependence of tilt angle b. For example, uJ~xt decreases but Imax increases at decreasing ature T, I-e-, at increasing tilt angle b. In thick films, a helix in the chiral phase temperature occurs there
But
also
is
essential
an
I~,max, Izz,max, I~~,max
C if T is
close
not
the
to
transition
untwisting field Euntw.
critical
chirality
b
when
T)~/~
A, when
C and
E >
phase A, when
E
