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Torsion pendulum oscillations caused by dissipative structures (DSS) in nematic liquid crystals (NLC) with homeotropic boundaries
0038-1098/82/410627-03503.0010
Solid State Communications, Vol. 44, No. 5, pp. 627-629. ]Printed in Great Britain.
Pergamon Press Ltd.
TORSION PENDULUM OSCILLATIONS CAUSED BY DISSIPATIVE STRUCTURES (DSS) IN NEMATIC LIQUID CRYSTALS (NLC) WITH HOMEOTROPIC BOUNDARIES* J. Grupp Physikalisches Institut, Westfalische Wilhelms-Universit~it, Domagkstr. 75, D-4400 Mtinster, West Germany
(Received 19May 19S2 by B. Miihlschlegel) A new effect occurs with a thin film of a NLC, sandwiched between two horizontal glass plates: DSS, produced by an electric field, give rise to torsion pendulum oscillations of the upper circular glass disc. The oscillations end up in limit cycles, which depend strongly on the applied voltage. 1N THE LAST fifteen years a lot of papers have appeared which deal with DSS in physical systems and many of them are concerned with DSS in NLC. A review on the state of knowledge on the problems of DSS in NLC is given recently by Manneville [ 1]. In liquid crystal physics most of the work has been done in the field of DSS due to electric fields, commonly called EHD-instabilities [2]. These instabilities occur in a thin NLC-film with certain properties, when an appropriate a.c.-voltage is applied [3]. Mostly the experimental investigations have been carried out through optical methods. In the present investigations we have used additionally a new mechanical method: measurement of the tangential surface forces (effecting a torque in the used geometry) exerted on a glass plate. A detailed report on a more general application of this method to NLC will be given in a lbrthcoming paper [4]. In the present paper we deal with a new oscillatory effect, which is caused by the DSS. In our experiments we have used N-(p-metho xybenzylidene)-p-b utylaniline (MBBA) having a negative dielectric anisotropy. Since the investigations on a sample can take a long time (sometimes several months) MBBA with a lower clearing temperature (T e = 40.3°C) has been used, because the properties of such a sample change more slowly with the time. The cut-off frequency [5] fc of the material was about 1 kHz. All the measurements were carried out at a constant frequency ( f = 60 Hz) in the conducting regime [5] (f~< re) and at a constant temperature [TM = (25.6 + 0.1)°C]. An approximately 150/am thick MBBA-film was sandwiched between two horizontal circular glass plates with transparent electric coatings. The surfaces of the plates had been treated with lecithin to promote homeotropic alignment. The upper glass plate (1 mm thick disc, 3 cm in diameter) was suspended with rotational symmetry from a 30 pm thick quartz-thread, which had been * Part of thesis, MOnster (1981).
coated with an electric conducting thin film of gold. A detailed description of the apparatus will be given elsewhere [6]. The sample cell was placed between crossed polarizers in a parallel monochromatic light beam. During the measurements the sample could also be observed optically. By applying a voltage greater than a critical value Uc to the sample one observes DSS. Simultaneously to the appearance of the DSS the glass disc begins to rotate around the vertical axis. The angular motion is obviously caused by the EHD-instabilities. The quartz-thread gets twisted by the rotation and a restoring torque is built up. With increasing twist angle q5 the angular velocity q~(= dq~/dt) decreases. At a certain twist angle q~max the rotation reverses its direction. When the disc is rotating backwards the thread gets untwisted and, after it has crossed the initial position, it gets twisted in the opposite direction. At another twist angle ~lmin < 0 (IqTlmi~l < q~lma~) the rotation reverses again. After a few initial cycles a stable and reproducible oscillatory motion (limit cycle [7]) is established. The limit cycle depends strongly on the applied voltage. In Fig. 1 limit cycles for several voltages are shown. The period and the amplitude of limit cycles are presented as functions of the applied voltage in Fig. 2. The observed non-harmonic angular motion of the glass disc can be expressed by the following equation of torques, neglecting inertial effects: k~+D~
= M,
(1)
where k is the friction coefficient, D the torsion coefficient of the thread and M the torque due to the DSS. The quantities k and M are functions o f ~ [k = k(4~) and M = M(~)], which depend on the applied voltage. In Fig. 3, the limit cycles are shown in another presentation corresponding to equation (1): twist angle of the thread as a function of the angular velocity ~. The limit cycles are presented for the same constant voltages as in Fig. 1. Obviously the angle q~is an 627
628
TORSION PENDULUM OSCILLATIONS CAUSED BY DSS IN NLC I
I
i
i
Vol. 44, No. 5
- -
f =60Hz - u:g.53V
3
"U=24,38V
v"
?.°\
oU=1501V
•
"
~ -
.........
-
-
• b:953V
~. 'sc
~
o U:1501V
o q
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ro~
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'
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l
i 5000
+
I 10000
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. . . .
. . . . . . . . . . . .
:-.-
2
15000
f
t/~ 3
Fig. 1. Angle ~b as function of time over one period for several limit cycles. The origin of the time scale has been shifted to the positive amplitude of the angle. If the curves are shifted by half of their period and then reflected at the #)-axis the transformed curves are identical with the original ones.
-15
-10
-5
0
5
10
15
Fig. 3. Limit cycles in the ¢-~-plane.
16
/ 103s
/'
/
if
~10
\,
× •
10-4rod
/
I
!2
24
u/v
k
Fig. 4. "Remanent" angular velocity q~o (at ¢ = 0) as function of the applied voltage U.
\.
I
£
I
10
16
22
~
--
28 --1
i
rod
/ s
•
a,
~
g
~/
L 10
_
_
L
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22
28
,Jr Fig. 2. Period r and amplitude ~max of the limit cycles as functions of the applied voltage. antisymmetric relation of ¢. At the amplitude Cmax there occurs a jump of q~. These jumps are indicated by arrows in Fig. 3. The torsion coefficient of the quartz thread certainly influences the amplitudes and periods of the limit cycles. The effect of the thread on the angular motion disappears at zero ang!e (~b = 0). Here the "remanent" angular velocity ¢o [= q~(¢ = .0)] is caused purely by the DSS in the NLC. In Fig. 4, 4)0 is presented
.s
~
;
;
s
~o
,5
+h° 4~ Fig. 5. Limit cycle (U = 29.95 V) fitted by a power series including terms up to fifth order. The d.otted parts denote the unstable branches. The jumps of 4~ are indicated by arrows. as a function of the applied voltage• The voltage of the onset of ¢0 is identical with that of the onset of the EHD-instabilities as could be observed optically. Now we describe the development of the oscillation to the limit cycle beginning at the origin of the presentation (Fig. 3). On the application of the voltage (U > Uc) the rotation of the disc begins and then grows rapidly, crossing the limit cycle. At the subsequent
Vol. 44, No. 5
TORSION PENDULUM OSCILLATIONS CAUSED BY DSS IN NLC
cycles the path is surrounding the limit cycle and approaching it from turn to turn. The limit cycle is reached with a sufficient accuracy after several turns whose number depends on the applied voltage. The functions M(~) and k(~) in equation (2) can be expressed as a power series. Taking into account that M ( - - ~ ) = --M(~) and k(-- ~) = k(~) we obtain for the torques: M(~) = AI~ + A3~3 + As~ s + . . . =
+
3 +
and + ....
= C~6 + C3~ 3 + G ~ s + . . . .
occurrence of DSS in NLC-films. Further experiments on DSS in NLC-films have been carried out too. Their results suggest a qualitative explanation of our torsion pendulum oscillation. These experiments and conclusions will be published in a forthcoming paper [8]. Acknowledgements I want to express many thanks to Prof. Dr F. Fischer for suggesting these experiments and for numerous discussions. This research was supported in part by the Deutsche Forschungsgemeinschaft.
(2)
Inserting equation (2) into equation (1) one gets 4~ as a power series of~: (3)
where C i = (A i --Bi)/D , i = 1, 3,5 . . . . . Figure 5 shows a limit cycle fitted by such a power series including the terms up to fifth order. Although such a treatment allows to fit the limit cycles, yet the driving and the friction terms (A i and Bi) cannot be separated. In this paper we have given a phenomenological description of torsion pendulum oscillations due to the
629
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
P. Manneville, Mol. C~st. Liq. Cryst. 70,223 (1981). Dubois-Violette et al., Liquid Crystals (Edited by L. Liebert), Sol. State Phys. Suppl. 14. Academic Press, New York (1978). P.G. de Gennes, The Phys&s o f Liquid Crystals. Clarendon Press, Oxford (1974). J. Grupp (to be published). Orsay Liquid Crystal Group, Mol. Cryst. Liq. Cryst. 16,229 (1972). J. Grupp (to be published). H. Haken, Synergetics. Springer-Verlag, BerlinHeidelberg (1978). J. Grupp (to be published).
Solid State Communications, Vol. 44, No. 5, pp. 627-629. ]Printed in Great Britain.
Pergamon Press Ltd.
TORSION PENDULUM OSCILLATIONS CAUSED BY DISSIPATIVE STRUCTURES (DSS) IN NEMATIC LIQUID CRYSTALS (NLC) WITH HOMEOTROPIC BOUNDARIES* J. Grupp Physikalisches Institut, Westfalische Wilhelms-Universit~it, Domagkstr. 75, D-4400 Mtinster, West Germany
(Received 19May 19S2 by B. Miihlschlegel) A new effect occurs with a thin film of a NLC, sandwiched between two horizontal glass plates: DSS, produced by an electric field, give rise to torsion pendulum oscillations of the upper circular glass disc. The oscillations end up in limit cycles, which depend strongly on the applied voltage. 1N THE LAST fifteen years a lot of papers have appeared which deal with DSS in physical systems and many of them are concerned with DSS in NLC. A review on the state of knowledge on the problems of DSS in NLC is given recently by Manneville [ 1]. In liquid crystal physics most of the work has been done in the field of DSS due to electric fields, commonly called EHD-instabilities [2]. These instabilities occur in a thin NLC-film with certain properties, when an appropriate a.c.-voltage is applied [3]. Mostly the experimental investigations have been carried out through optical methods. In the present investigations we have used additionally a new mechanical method: measurement of the tangential surface forces (effecting a torque in the used geometry) exerted on a glass plate. A detailed report on a more general application of this method to NLC will be given in a lbrthcoming paper [4]. In the present paper we deal with a new oscillatory effect, which is caused by the DSS. In our experiments we have used N-(p-metho xybenzylidene)-p-b utylaniline (MBBA) having a negative dielectric anisotropy. Since the investigations on a sample can take a long time (sometimes several months) MBBA with a lower clearing temperature (T e = 40.3°C) has been used, because the properties of such a sample change more slowly with the time. The cut-off frequency [5] fc of the material was about 1 kHz. All the measurements were carried out at a constant frequency ( f = 60 Hz) in the conducting regime [5] (f~< re) and at a constant temperature [TM = (25.6 + 0.1)°C]. An approximately 150/am thick MBBA-film was sandwiched between two horizontal circular glass plates with transparent electric coatings. The surfaces of the plates had been treated with lecithin to promote homeotropic alignment. The upper glass plate (1 mm thick disc, 3 cm in diameter) was suspended with rotational symmetry from a 30 pm thick quartz-thread, which had been * Part of thesis, MOnster (1981).
coated with an electric conducting thin film of gold. A detailed description of the apparatus will be given elsewhere [6]. The sample cell was placed between crossed polarizers in a parallel monochromatic light beam. During the measurements the sample could also be observed optically. By applying a voltage greater than a critical value Uc to the sample one observes DSS. Simultaneously to the appearance of the DSS the glass disc begins to rotate around the vertical axis. The angular motion is obviously caused by the EHD-instabilities. The quartz-thread gets twisted by the rotation and a restoring torque is built up. With increasing twist angle q5 the angular velocity q~(= dq~/dt) decreases. At a certain twist angle q~max the rotation reverses its direction. When the disc is rotating backwards the thread gets untwisted and, after it has crossed the initial position, it gets twisted in the opposite direction. At another twist angle ~lmin < 0 (IqTlmi~l < q~lma~) the rotation reverses again. After a few initial cycles a stable and reproducible oscillatory motion (limit cycle [7]) is established. The limit cycle depends strongly on the applied voltage. In Fig. 1 limit cycles for several voltages are shown. The period and the amplitude of limit cycles are presented as functions of the applied voltage in Fig. 2. The observed non-harmonic angular motion of the glass disc can be expressed by the following equation of torques, neglecting inertial effects: k~+D~
= M,
(1)
where k is the friction coefficient, D the torsion coefficient of the thread and M the torque due to the DSS. The quantities k and M are functions o f ~ [k = k(4~) and M = M(~)], which depend on the applied voltage. In Fig. 3, the limit cycles are shown in another presentation corresponding to equation (1): twist angle of the thread as a function of the angular velocity ~. The limit cycles are presented for the same constant voltages as in Fig. 1. Obviously the angle q~is an 627
628
TORSION PENDULUM OSCILLATIONS CAUSED BY DSS IN NLC I
I
i
i
Vol. 44, No. 5
- -
f =60Hz - u:g.53V
3
"U=24,38V
v"
?.°\
oU=1501V
•
"
~ -
.........
-
-
• b:953V
~. 'sc
~
o U:1501V
o q
• U=24 38V
2r
• L, : 2 9 95'¢ +
ro~
0
21
1
I
'
[~ 15000
I
I
-10000
5000
0
l
i 5000
+
I 10000
•- - 4
. . . .
. . . . . . . . . . . .
:-.-
2
15000
f
t/~ 3
Fig. 1. Angle ~b as function of time over one period for several limit cycles. The origin of the time scale has been shifted to the positive amplitude of the angle. If the curves are shifted by half of their period and then reflected at the #)-axis the transformed curves are identical with the original ones.
-15
-10
-5
0
5
10
15
Fig. 3. Limit cycles in the ¢-~-plane.
16
/ 103s
/'
/
if
~10
\,
× •
10-4rod
/
I
!2
24
u/v
k
Fig. 4. "Remanent" angular velocity q~o (at ¢ = 0) as function of the applied voltage U.
\.
I
£
I
10
16
22
~
--
28 --1
i
rod
/ s
•
a,
~
g
~/
L 10
_
_
L
I
l
16
22
28
,Jr Fig. 2. Period r and amplitude ~max of the limit cycles as functions of the applied voltage. antisymmetric relation of ¢. At the amplitude Cmax there occurs a jump of q~. These jumps are indicated by arrows in Fig. 3. The torsion coefficient of the quartz thread certainly influences the amplitudes and periods of the limit cycles. The effect of the thread on the angular motion disappears at zero ang!e (~b = 0). Here the "remanent" angular velocity ¢o [= q~(¢ = .0)] is caused purely by the DSS in the NLC. In Fig. 4, 4)0 is presented
.s
~
;
;
s
~o
,5
+h° 4~ Fig. 5. Limit cycle (U = 29.95 V) fitted by a power series including terms up to fifth order. The d.otted parts denote the unstable branches. The jumps of 4~ are indicated by arrows. as a function of the applied voltage• The voltage of the onset of ¢0 is identical with that of the onset of the EHD-instabilities as could be observed optically. Now we describe the development of the oscillation to the limit cycle beginning at the origin of the presentation (Fig. 3). On the application of the voltage (U > Uc) the rotation of the disc begins and then grows rapidly, crossing the limit cycle. At the subsequent
Vol. 44, No. 5
TORSION PENDULUM OSCILLATIONS CAUSED BY DSS IN NLC
cycles the path is surrounding the limit cycle and approaching it from turn to turn. The limit cycle is reached with a sufficient accuracy after several turns whose number depends on the applied voltage. The functions M(~) and k(~) in equation (2) can be expressed as a power series. Taking into account that M ( - - ~ ) = --M(~) and k(-- ~) = k(~) we obtain for the torques: M(~) = AI~ + A3~3 + As~ s + . . . =
+
3 +
and + ....
= C~6 + C3~ 3 + G ~ s + . . . .
occurrence of DSS in NLC-films. Further experiments on DSS in NLC-films have been carried out too. Their results suggest a qualitative explanation of our torsion pendulum oscillation. These experiments and conclusions will be published in a forthcoming paper [8]. Acknowledgements I want to express many thanks to Prof. Dr F. Fischer for suggesting these experiments and for numerous discussions. This research was supported in part by the Deutsche Forschungsgemeinschaft.
(2)
Inserting equation (2) into equation (1) one gets 4~ as a power series of~: (3)
where C i = (A i --Bi)/D , i = 1, 3,5 . . . . . Figure 5 shows a limit cycle fitted by such a power series including the terms up to fifth order. Although such a treatment allows to fit the limit cycles, yet the driving and the friction terms (A i and Bi) cannot be separated. In this paper we have given a phenomenological description of torsion pendulum oscillations due to the
629
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
P. Manneville, Mol. C~st. Liq. Cryst. 70,223 (1981). Dubois-Violette et al., Liquid Crystals (Edited by L. Liebert), Sol. State Phys. Suppl. 14. Academic Press, New York (1978). P.G. de Gennes, The Phys&s o f Liquid Crystals. Clarendon Press, Oxford (1974). J. Grupp (to be published). Orsay Liquid Crystal Group, Mol. Cryst. Liq. Cryst. 16,229 (1972). J. Grupp (to be published). H. Haken, Synergetics. Springer-Verlag, BerlinHeidelberg (1978). J. Grupp (to be published).