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Volume 156, number 1,2
PHYSICSLETTERSA
3 June 1991
Low-frequency ac electric field and temperature dependence
on the helical pitch in a ferroelectric liquid crystal
Z.M. Sun, Z.H. Wang, X. Zhang and D. Feng
Laboratory of Solid State Microstructures, Nanfing University, Nanfing 210008, China
Received 12 September 1990;accepted for publication 5 April 1991
Communicatedby J. Flouquet
The helical pitch as a functionof low-frequencyac electric field at different temperatures has been investigatedin a ferroelectric
liquid crystal both experimentally and theoretically. A simplified model has been presented to calculate the variation of helical
pitch under suchfields, and the results obtained are in goodagreementwith the experiment. The frequencydispersion oftbe pitch
has also been measured.
In the ferroelectric SmC* phase the tilted chiral molecules precess around the normal going from one smectic
layer to another, which results in a helical structure. In a thick planar sample such structure displays a parallel
stripe pattern in the field of the polarizing microscope. These stripes can be explained as line defects (disclination lines) situated near the sample surfaces by means of the model put forward by Glogarova et al. [ 1 ].
The distance between two neighbour stripes near one of the two boundary surfaces gives the helical pitch [2].
Under the action of an external electric field the helical pitch varies owing to the coupling between the field
and the permanent dipoles as well as the dielectric anisotropy. For the case of dc field, the electric field dependence of the pitch has been obtained experimentally [ 3-5 ], and calculated theoretically by Kai et al. [ 6]
from the free energy approximation. On the other hand, the variation of the pitch under ac electric field has
also been measured by Parmar et al. [ 3 ] and Rout and Choudhary [ 4 ]. However, so far no theory is available
for ac electric unwinding because of the complexity of the problem. Recently we studied ac electric unwinding
in a planar sample of the SmC* phase. We found several instabilities appearing in the sample at certain frequencies. But the situation is relatively simple for the case of low-frequency weak electric field. In this Letter
we present a simplified model to calculate the variation of the pitch under the action of low-frequency ac electric field and compare the calculation with the experiment. Besides, we have also observed the dispersion of
the pitch.
The material used is a chiral ferroelectric mixture FCS l 01, kindly provided by the F. Hoffmann-La Roche
Company, Switzerland. This material presents a SmC* phase (between 15.5 and 5 5.4 °C) and a SmA phase
(between 55.4 and 76.1 °C). The experimental setup and the preparation of the sample with planar and homogeneous alignment have been described in detail in a previous work [7 ]. The cell thickness is 25 gm, determined by a mylar spacer. The monodomain chosen for measurement has a dimension of 2 X 1 mm 2 (fig.
1 ). The pitch is measured from the whole length of 30 stripes with a micrometer eye-piece attached to the
microscope.
When an electric field is applied to the SmC* liquid crystal parallel to the smectic layers, the helical pitch
increases with field and finally disappears at the critical field. Fig. 2 shows the variation of pitch under the
action of a low-frequency ( f = 20 Hz) ac electric field at three different temperatures. As shown in fig. 2, we
see that the pitch does not vary if the field E is lower. When E reaches a certain value, the pitch increases
rapidly. The lower the temperature, the higher the field at which the pitch starts to increase. The field depen1 14
0375-9601/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
Volume 156, number 1,2
PHYSICS LETTERS A
3 June 1991
T-Te=-$.8
T-Te_--5.0 oC
T-Te=-&O °C
o,oo,
.....
.....
6.00
*
/
I
/
..
/
/
/
/ /.
/. /
4,00
-q.O0
~.0~
'
o.oo
1.6o
~.6o
2.60
EYJ~,CTRIC FIELD ( 1 0 5 V / m )
Fig. 1. Monodomain of SmC* helical texture.
[/
0.041
.oooo fml0Bs
:::::
5.50
Fig. 2. Helical pitch as a function of ac electric field at various
temperatures.f= 20 Hz.
7.00
0+*** T-Tem-$.0 *C
. . . . . T-Tcffi-4.0 oC
6 • 8 ,. a T-Tem-6.0°C
8,0~
•
A
~
0
5.00
5.00
4.50
0
A
4.00
4.00
0
--
- - O - -
--
--
0
3.00
3.50
3.00
2.00
0.00
~
0.50
1.00
1.50
2.00
E.50
3.00
ELECTRIC FIELD ( 1 0 ~ V / m )
Fig. 3. Electric field dependence of helical pitch at different frequencies. T - To= - 4 ° C .
.... ,
10
........
,
..........................
,
100
1000
100000
10000
FREQUENCY (Hz)
Fig. 4. Frequency dependence of helical pitch under a fixed field
E = 2 . 0 × 105 V/m at different temperatures. The dashed lines
represents the calculation values derived from ref. [ 10 ].
d e n c e o f the p i t c h at v a r i o u s f r e q u e n c i e s is s h o w n in fig. 3 at T - Tc = - 4 ° C. W e can see t h a t w i t h i n c r e a s e o f
the f r e q u e n c y it b e c o m e s m o r e a n d m o r e difficult for t h e field to u n w i n d the helix. S u c h e x p e r i m e n t a l results
h a v e also b e e n r e p o r t e d by R o u t a n d C h o u d h a r y [ 4 ]. Besides, we m e a s u r e the f r e q u e n c y d e p e n d e n c e o f the
pitch u n d e r a fixed field E = 2.0 X 105 V / m at t h r e e d i f f e r e n t t e m p e r a t u r e s , as s h o w n in fig. 4. W e n o t i c e t h a t
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Volume 156, number 1,2
PHYSICS LETTERSA
3 June 1991
at low frequencies ( f < 100 Hz) the pitch decreases sharply with increase of the frequency, but it tends to be
saturated at high frequencies (f> 1000 Hz). We call such frequency dependence of the pitch the dispersion of
the pitch.
Up to date the dynamics of a SmC* helix under ac electric field has not yet been understood, although Ostrovskii et al. [ 8 ] tried to explain it by considering the dynamic viscosity effect on the helical pitch. Recent
experiments show that in the limit of weak field at low frequencies the variation of the pitch is comparatively
simple, and no complex instability patterns occur in the sample. Here we shall make a theoretical analysis on
this problem.
In a thick planar sample, suppose the helix axis along Z, and a uniform electric field g applied normal to
the glass plates and in the plane of the smectic layers (i.e. along X), 0 being the azimuth angle between the
permanent polarization Ps and the electric field E, 0 being the tilt angle of the molecule with respect to the
layer normal. Then we have a free energy density at equilibrium as follows,
f = ½KzOZ(dO/dz-qo)2 + ( ~a/8n)E202 cosEO-PsEcos (b ,
( 1)
where qo=2n/po, Po is the normal helical pitch at zero field, ~a the dielectric anisotropy, Kz the twist elastic
constant. Here we have neglected the effect of flexoelectricity. The equation describing the distribution of the
angle 0 in an external field is found by minimization of function ( 1 ) and is of the form
d20
~'1 dO
dz z - K2 dt
ea E 2
P~E
41tK2 cosOsinO+ K---~ s i n 0 .
(2)
In eq. (2) we add a viscous term by considering the effect of the dynamic viscosity on the helical pitch, where
7~ is the shear viscosity.
Now let us study the case of low-frequency ac weak field E=Eoexp(icot), o) is the circle frequency of the
field. Different from the case of a high-frequency electric field, the molecules can follow the low-frequency field.
Under a certain field E, the pitch starts to vary from Po and reaches an equilibrium value p(E). Here we suppose that the fundamental structure of the helix does not change obviously, although p ( E ) is larger than Po.
Under the action of the field all molecules vibrate in the vicinity of the equilibrium position with the field
frequency because of the dynamic viscosity effects. Each molecule stays on the conic surface when it vibrates.
In this way we suppose the solution of angle 0 can be expressed as
O=q(E) Z + C e x p [ i ( m T + a ) ] sin[q(E)Z] ,
(3)
where q(E) = 2n/p(E) is the wave number under the field, different from qo, a is the phase difference between
the vibration of molecules and the vibration of the field, C is a constant. In eq. (2), the first term on the righthand side is a position term, and the second a vibration one. Note that C must be small, otherwise it will destroy
the helical structure.
This proposal is consistent with the experimental facts. Experimentally in the case of a low-frequency weak
field we have observed that the intensity of light transmission varies with the same frequency as that of the
field, meanwhile the pitch stripes are stable. These facts can be explained by the suggestion mentioned above.
In the location of pitch stripes (disclination lines), i.e. qZ=n~, the vibration term in eq. (3) is zero. This
satisfies the condition that at the pitch stripes 0 keeps unchanged (~ = n ~). That is to say, the pitch stripes are
stable. On the other hand, at another place, qZv~ n~, the vibration term in eq. (3) is not zero. The molecules
vibrate with the same frequency as that of the field. Such vibration will give rise to a variation of the refraction
index, and then a variation of the transmission intensity. Therefore the oscillation frequency of the transmission intensity is the same as the vibration frequency of the molecules, and then as the frequency of the field.
Substituting eq. (3) into eq. (2) and considering that both C and Eo are small, we obtain
K202Cq 2 exp [i(tot+ a ) ] +Y102ia~Cexp[i(o)t+a) ] +P~Eo exp (itnt) = 0 ,
then we have
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(4)
Volume 156, number 1,2
PHYSICS LETTERSA
C exp ( i a ) = - P~Eo/(K202q 2+ ito?, 02 ).
3 June 1991
(5)
Hence both C and a can be derived as
C=-
PsE°/02
(6)
(K22q4+to272),/2,
or= -arctg(yl/K2q 2) .
(7)
Substituting eqs. (6) and (7) into eq. (3), then substituting the expression of ~ into eq. ( 1 ), we can derive
the average free energy density,
p
2~:
1
f= 2~p(E--------Sf dz f d(tot){½K202[q-qo +Cqcos(tot+ot)cos(qz) ]2
0
0
-PsEo cos(tot) cos[qz+Ccos(tot+a) sin(qz) ]}
= ½K202 [ ( q - qo)2+ ~C2q 2 ] + ~P~EoCcos a .
(8)
From the condition Of/Oq= 0, the wave number q can be derived as
q=qo -
( PsEo/ O2) 2( K2 qS-to2y2 q )
4(K2q4+to2y~)2
(9)
Eq. (9) is an eighth-order algebraic equation and we can only obtain the numerical solutions by microcomputer.
As shown in eq. (9), the pitch is a function of Eo, to if Ps, 0, K2, 7, are known. In fact, the latter four parameters are all related to the temperature. For this reason, the pitch is surely dependent on temperature. In
this Letter, since K2 and Yl of FCS 101 is unknown, we take K2 = 5.5 X 10-7 dyne and Yl = 0.3 poise and neglect
their variation with temperature. The values of both Ps and 0 are taken from the measurements reported in
ref. [9].
Using numerical calculation we can derive the field dependence of pitch at f = 20 Hz at three temperatures
(fig. 2), the field dependence of the pitch at T - T o = - 4°C at three frequencies (fig. 3) and the frequency
dependence of the pitch under a fixed field E = 2 . 0 × l0 s V / m at three temperatures (fig. 4). In these three
figures solid lines correspond to the theoretical calculation. Notice that in fig. 4 the calculation is made merely
at f < 100 Hz. We find that the calculation results are in good agreement with the experimental data. This indicates that our scenario and approaches are effective for the case of a low-frequency weak field. In addition,
it should be pointed out that eq. (9) does not hold for the high-frequency region in fig. 4. We have shown in
a separate paper that at high frequency the field at which the pitch starts to increase rapidly is very strong
( ~ l06 V / m ) [ 10]. But the present field ( E = 2 . 0 × I0 s V / m ) is weak enough, so the pitch should remain unchanged and be equal to the normal pitch at zero field.
In conclusion, the influence of low-frequency ac electric field and temperature on the helical pitch has been
investigated both theoretically and experimentally in this Letter. Using a simplified model we have calculated
the variation of the pitch under the action of the field at different temperatures. It is worth noticing that this
model can only be used at f < 100 Hz for FCS 101. In this frequency region the molecules can follow the field,
and the vibration frequency of molecules is the same as that of the field. We have also considered the case of
high frequencies (f> l 0 kHz) in ref. [ 10 ]. However, at mediate frequencies ( f = 100-10000 Hz ), the situation
is much more complicated. The electrohydrodynamical effects must be carefully studied. A satisfying theory
is expected to be established to explain the experimental results.
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Volume 156, number 1,2
PHYSICS LETTERS A
3 June 1991
T h e a u t h o r s are grateful to Dr. S c h a d t for k i n d l y s u p p l y i n g g o o d q u a l i t y samples. T h i s w o r k was s u p p o r t e d
by t h e N a t i o n a l N a t u r a l Science F o u n d a t i o n o f C h i n a u n d e r g r a n t N o . 1870714.
References
[ 1 ] M. Glogarova, J. Fousek, L. Lejcek and J. Pavel, Ferroelectrics 58 (1984) 161.
[2] M. Brunet and N. Isaert, Ferroelectrics 84 (1988) 25.
[3 ] D.S. Parmar, K.K. Raina and J. Shankar, Mol. Cryst. Liquid Cryst. 103 (1983) 77.
[4] D.K. Rout and R.N.P. Choudhary, Ferroelectrics 82 (1988) 157.
[5] Ph. Martinot-Lagarde, R. Duke and G. Durand, Mol. Cryst. Liquid Cryst. 75 ( 1981 ) 249.
[6] S. Kai, M. Takata and K. Hirakawa, Japan. J. Appl. Phys. 22 (1983) 938.
[7] Z.M. Sun, X. Zhang and D. Feng, Europhys. Lett. 11 (1990) 415.
[8] B.I. Ostrovskii, A.Z. Rabinovich, A.S. Sonin and B.A. Strukov, Sov. Phys. JETP 47 (1978) 912.
[9] D.K. Rout and R.N.P. Choudhary, Ferroelectrics 82 (1988) 149.
[ 10 ] Z.H. Wang, Z.M. Sun and D. Feng, to be published.
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