RAPID COMMUNICATIONSPHYSICAL REVIEW E 78, 050701共R兲 共2008兲

Kerr constant and third-order nonlinear optic susceptibility measurements
in a liquid crystal composed of bent-shaped molecules
Surajit Dhara,1,* Fumito Araoka,1 Mongryong Lee,1,† Khoa Van Le,1 Lingfen Guo,1 B. K. Sadashiva,2 Kigook Song,3Ken Ishikawa,1 and Hideo Takezoe1,‡
1Department of Organic Polymeric Materials, Tokyo Institute of Technology, 2-12-1-S8-42 O-okayama, Meguro-ku,Tokyo 152-8552, Japan
2Raman Research Institute, C. V. Raman Avenue, Sadashivanagar, Bangalore-560080, India
3
Department of Display Materials Engineering, Kyung Hee University, Gyeonggi-do 446-701, Korea
共Received 21 August 2008; published 19 November 2008兲
We report the determination of the Kerr constant 共B兲 and the real part of the third-order nonlinear optic
susceptibility 共␹共3兲兲 above the nematic-isotropic phase transition temperature 共TNI兲 of a liquid crystal composedof bent-shaped molecules. The values of B and ␹共3兲 just above 共⬃0.3 ° C兲 TNI are ⬃8 ⫻ 10−12 m / V2 and 5⫻ 10−20 m2 / V2, respectively. The estimated critical temperature TC* is about 1.5 ° C below TNI indicating thatthe nematic-isotropic 共NI兲 transition is weakly first order as in the case of calamitic liquid crystals. The
temperature-dependent Kerr constant is found to be in good agreement with the predictions of the Landau–deGennes theory. The experimental results are compared with those in a calamitic liquid crystal material withnegative dielectric anisotropy in the nematic phase.
DOI: 10.1103/PhysRevE.78.050701

PACS number共s兲: 77.84.Nh, 78.20.Jq, 61.30.Gd, 42.65.Hw

The electro-optical studies on liquid crystals are important
both from the technological as well as fundamental points ofview. The main objective of such studies is to manipulate theoptical properties with the help of an applied electric field or
to investigate the correlation between the molecular structureand electric, optical properties. Most of the investigations
have been made in the liquid crystalline phase. The effect ofshort-range order and the fluctuation of orientational orderparameter just above the nematic 共N兲 to isotropic 共I兲 transition temperature 共TNI兲 have been theoretically discussed byde Gennes 关1兴. Since then the interest on the Kerr effect in
the isotropic phase of liquid crystalline materials has increased many folds to search for new materials with large
Kerr constants and nonlinear optic susceptibility. The number of experimental reports on the electro-optical Kerr effectof calamitic liquid crystals are too large to be cited here infull. In all of these studies the effect of molecular structures,
permanent dipole moments, etc., on the short-range order in
the isotropic phase has been discussed.
The discovery of electro-optical switching in bent-coreliquid crystals has created immense interest in the scientificcommunity 关2兴. Recently some interesting experimental results are reported on the nematic as well as above thenematic-isotropic phase transition temperature of bent-coreliquid crystals. Domenici et al. 关3兴 have reported about thealignment of the bent-core molecules under magnetic fields
关3兴. The critical behavior of the NI transition has been investigated by Wiant et al. 关4兴 from magnetic birefringence andlight scattering experiments. A nonstandard electroconvection in the nematic phase has been reported by Tanaka et al.

*On leave from School of Physics, University of Hyderabad,Hyderabad-500046 India.
†On leave from Department of Display Materials Engineering,Kyung Hee University, Gyeonggi-do 446-701, Korea.
‡Corresponding author; takezoe.h.aa@m.titech.ac.jp1539-3755/2008/78共5兲/050701共4兲关5兴 and Wiant et al. 关6兴. Very recently Wiant et al. has also
observed a transition between two optically isotropic phases
above the clearing point, which is explained by a model for
tetrahedratic orientational order in bent-core liquid crystal
关7兴. All of these reports show that there are some unconventional aspects of NI transition of bent-core liquid crystals
than calamitic liquid crystals though the transitions are similar from the symmetry point of view. These studies inspired
us to carry out the Kerr measurements on bent-core liquid
crystal exhibiting NI transition. There are also various typesof interesting electro-optical studies in the mesophases exhibited by such compounds owing to the shape, polarity, andchirality 关8,9兴. However, there is no report on the measurement of Kerr constant and nonlinear static susceptibility
above TNI in liquid crystals composed of bent-core or bentshaped molecules exhibiting the direct nematic to isotropictransition. The Kerr effect measurements are important tofind the effect of bent-shaped structure on the short-rangeorder and fluctuation of orientational order parameter in theisotropic phase of bent-core liquid crystals that show the direct nematic to isotropic phase transition. In this Rapid Communication we present the measurement of the electrooptical Kerr constant 共B兲 and the third-order nonlinear
susceptibility 共␹共3兲兲 of a bent-core liquid crystal in theisotropic phase.
The chemical structure of the compound is shown at thetop of Fig. 1. The details of the synthetic scheme will bereported elsewhere 关10兴. The compound exhibits the following phase sequence: Cry-97.0 ° C-共Colr-81.5 ° C兲-N107.2 ° C-I. We prepared sandwiched cells with comb-type
interdigited electrodes on one of the substrates. The cell
thickness 共l兲 was ⬃2 ␮m. The width of each stripe in thecomb was 10 ␮m and the gap between the two consecutivestripes was 15 ␮m. For Kerr effect measurements, a He-Nelaser 共␭ = 632.8 nm兲 was used to illuminate the cell. The Kerrcell was kept between crossed polarizers making the direction of an applied field 45° with respect to the polarizer. An050701-1©2008 The American Physical Society

RAPID COMMUNICATIONSPHYSICAL REVIEW E 78, 050701共R兲 共2008兲

DHARA et al.
O
H3COOOH共3兲
共3兲
B = 共24␲/n␭兲关␹1212
共␻1,− ␻1,0,0兲 + ␹1221
共␻1,− ␻1,0,0兲兴.

OOOClClO

共4兲

CH3H

C10H21O

OC10H21

By applying the Kleinman symmetry 关13兴 due to the experimental far resonance condition in the isotropic medium wecan write
共3兲
共3兲
␹1212= ␹1221⬅ ␹共3兲 .

共5兲

Intensity (arb. units)Thus the nonlinear susceptibility can be estimated by
B = 共48␲/n␭兲␹共3兲 .

1.2

0.8

0.4

00

0.6

1.2

1.8x10-13

Square of the electric field (V2/m2)
FIG. 1. Transmitted intensity of the modulated signal 共Iac兲 as afunction of square of the applied field measured at a frequency
2 kHz. The solid line is the best fit obtained from a least-meansquares method. The sample used is also shown at the top of thefigure.

ac high field of a frequency f = 1 kHz was applied to the cell.
The introduction of a ␭ / 4 plate between the analyzer and thecell enables us a linear detection of the modulated ac component of the transmittance 共Iac兲 due to the electric-fieldinduced birefringence 共⌬n兲 in the sample. The secondharmonic component 共2f = 2 kHz兲 of the transmitted
intensity was amplified by a lock-in amplifier. The dc component was measured with a digital multimeter. The induced
optical phase difference ␦ was estimated by using the relation 关11兴

␦=2␲l⌬n Iac= .
␭Idc

共1兲

According to the Kerr law, ⌬n is given by⌬n = B␭E2 ,共3兲
共␻1,− ␻1,0,0兲
⌬n = 共24␲/n兲关␹1212
共3兲
共␻1,− ␻1,0,0兲兴E2共0兲,+ ␹1221

The refractive index of the compound in the isotropic phase
was measured as a function of temperature by using interference fringes of the reflected light from empty and filled cells.
Using the experimental values of B and n we calculated ␹共3兲
at various temperatures above TNI.
In order to confirm the appropriateness of using
2-␮m-thick cells, first we measured the modulated ac transmitted intensity from different cell thicknesses of a standard
compound 5CB. It was found that at a fixed temperature theintensity scarcely depended on the cell thickness despite thatthe linear dependence is expected by Eq. 共1兲, indicating thatthe in-plane field in comb-type cell is confined within a thinlayer. The Kerr constant was also measured using a
2-␮m-thick cell as a function of temperature and a good
agreement was found with the previously reported measurements at all temperatures 关14,15兴. Therefore, we used thickness of ⬃2 ␮m for all of the measurements.
Next the modulated ac signal intensity was measured as afunction of square of the field up to ⬃1.8⫻ 1013 V2 / m2. In
further experiments, however, we restricted our measurements upto ⬃1.8⫻ 1012 V2 / m2 which is 10 times smaller toavoid any thermal and chemical degradation under high andprolonged applied field in the isotropic phase. The linear
relationship shown in Fig. 1 certifies that the electro-optic
response is due to the Kerr effect. The variation of inducedbirefringence 共⌬n兲 as a function of the square of the appliedfield 共E2兲 at various temperatures is shown in Fig. 2. It is
obvious that ⌬n increases linearly with E2 at every temperature. The experimental data at various temperatures were fitted with Eq. 共2兲 by a least-square-fitting procedure to obtain
Kerr constant. The variation of the Kerr constant as a function of temperature is shown in Fig. 3. It is seen that the Kerr
constant is strongly temperature dependent and follows theLandau–de Gennes theory given by 关1兴

共2兲

where B is the Kerr constant. Since nonvanishing elementsof a static third-order nonlinear susceptibility tensor ␹共3兲 in共3兲
共3兲
共3兲
共3兲
, ␹1122, ␹1212, and ␹1221, wherethe present system are ␹1111
共3兲
共3兲
共3兲
共3兲
␹1111 = ␹1122 + ␹1212 + ␹1221, ⌬n at a static field E共0兲 is givenby 关12兴

共3兲

where n is the refractive index of the sample. Using Eqs. 共2兲
and 共3兲 the Kerr constant can be written as

共6兲

B = KB/共T − TC* 兲,

共7兲

where TC* is the critical temperature at which the nematiccoherence length 共␰兲 diverges to infinity and KB is a constant.
To obtain TC* , we plot the inverse of the Kerr constant as afunction of temperature. TC* was found to be 1.5 ° C below
TNI. Such typical values are also reported in several othercalamitic nematogens 关16,17兴.
To calculate ␹共3兲, we measured refractive index of the medium in the isotropic phase as a function of temperature. The
temperature variation of ␹共3兲 is shown in Fig. 4; ␹共3兲 is about
6 ⫻ 10−20 m2 / V2 at 0.3 ° C above TNI and is reduced withincreasing temperature. To compare with the compound ex-050701-2

RAPID COMMUNICATIONSKERR CONSTANT AND THIRD-ORDER NONLINEAR OPTIC …

PHYSICAL REVIEW E 78, 050701共R兲 共2008兲

6.0x1020

115.0oC
113.0oC
111.0oC
110.0oC
109.0oC
108.0oC
107.5oC

9.0

6.0

χ(3)(ω1,-ω1,0,0) (m2/V2)

Birifringence Δn12.0x106

3.0

B1N

5.0

MBBA

4.0
3.0
2.0
1.000

0.60

1.8x1012

1.2

2

468

TNI-T (oC)Square of the electric field (V2/m2)
FIG. 2. Induced birefringence ⌬n as a function of square of theapplied field at various temperatures. Solid line is the best fit obtained from a least-mean-squares method.

6.0x1011

Kerr Constant (m/V2)

8.0x10-12

6.0

4.0

4.0
TNI

2.0

T*c

2.0

00
104

106

108

110

112

114

116

Inverse of Kerr Constant (V2/m)

hibiting negative dielectric anisotropy, reported ␹共3兲 values inMBBA 关18兴 are also shown in Fig. 4. It is noticed that thevalues for the present compound are almost double at alltemperatures. The marked difference is that the temperaturedependence of the bent-core compound is much slower as
the TNI is approached compared to MBBA. Actually, estimated KB values are 2.3⫻ 10−11 mK/ V2 and 7.4⫻ 10−12 mK/ V2 for the present compound and MBBA,respectively.
It has been reported that usually the Kerr constant issmaller in materials with negative dielectric anisotropy thanthose with positive dielectric anisotropy; 共B = 1.1⫻ 10−10 m / V2兲 in 5CB 关14,15兴 and 共B = 4 ⫻ 10−12 m / V2兲 inMBBA 关18兴. The reason is very simple: The director of nematic cybotactic clusters in the isotropic sea gets aligned along
the field direction in positive-type materials, since the average dipole is parallel to the director. In contrast, negativetype materials make their director perpendicular to the field,

118

Temperature (oC)FIG. 3. Solid squares represent the variation of Kerr constant
共B兲 as a function of temperatures. The inverse Kerr constant 1 / B is
also plotted by solid circles to show the Landau–de Gennes-typetemperature dependence. Actually 1 / B ⬀ 共T − TC* 兲 is shown by thebest fit to the data points from a least-mean-squares method. TC* isfound to be almost 1.5 ° C below TNI.

FIG. 4. Variation of nonlinear static susceptibility关␹共3兲共␻1 , −␻1 , 0 , 0兲兴 as a function of temperature. Data corresponding to the solid squares are obtained from the present bent-corecompound and those of solid circles are from Ref. 关18兴. Solid lines
are drawn as guides to the eye.

so that we observe the reduced anisotropy by rotational average of the cluster 共director兲 orientation about the field direction. Now let us discuss the different Kerr constants inliquid crystals composed of bent-shaped molecules and calamitic molecules; i.e., B or ␹共3兲 of the former is about
double those of the latter. ⌬n is given by 关19兴⌬n = ⌬n0␳⌬⑀E2/2␲a0共T − TC* 兲,

共8兲

where ⌬n0 is ⌬n of a perfectly aligned liquid crystal, ␳density, ⌬⑀ dielectric anisotropy, and a0 coefficient of Landau expansion. Since the refractive index anisotropies of thebent-core mesogen and MBBA are both about 0.1 just below
TNI, only the difference arises from ⌬⑀’s. They are −0.8 and−0.4 for bent-core mesogen and MBBA just below TNI, respectively. Thus, the different B or ␹共3兲 originates from thedielectric anisotropy. This conclusion is consistent with thefact that the present bent-core material seems to show negligible biaxiality, because the formation of cybotactic clusters
with a biaxial nematic order and their field-induced orientation would give smaller B or ␹共3兲 by rotational average about
the axis of second largest refractive index parallel to an electric field.
In conclusion, we performed the Kerr effect measurement
and estimated the Kerr constant and the static nonlinear opticsusceptibility as a function of temperature above TNI of aliquid crystal composed of bent-shaped molecules. The
temperature-dependent Kerr constant follows the behavior
predicted by the Landau–de Gennes theory, where TC* isabout 1.5 ° C below TNI indicating that the NI transition is
weakly first order as in the case of calamitic liquid crystals.
The static susceptibility is larger almost by a factor of 2
compared to that of the negative-type calamitic material
MBBA, and is explained by the different dielectric anisotropy. These results are useful for investigating other physical phenomena of bent-core liquid crystals in the vicinity of050701-3

RAPID COMMUNICATIONSPHYSICAL REVIEW E 78, 050701共R兲 共2008兲

DHARA et al.

NI transition. For example, it is also possible to form a polar
cluster owing to the polar packing of the bent-shaped molecules under high electric field and hence a linear couplingof electric field with the order parameter can give rise to anoptical second harmonic generation in the isotropic phase. At
present we are pursuing some investigation to explore suchpossibilities. We expect our present results will inspire more
electro-optic Kerr measurements on the bent-core liquid
crystals to investigate the effect of molecular shape andstructures on the short-range cybotactic clusters.

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This work is partly supported by Grant-in-Aid for Scientific Research 共S兲 共Grant No. 16105003兲 from the Ministry ofEducation, Culture, Science, Sports and Technology of Japan. One of the authors 共S. D.兲 gratefully acknowledges thesupport from the DST, Government of India for the Contract
No. SR/FTP/PS-48/2006 under the SERC Fast Track
Scheme.

050701-4