RAPID COMMUNICATIONSPHYSICAL REVIEW E 81, 010702共R兲 共2010兲

Splay bend elasticity of a bent-core nematic liquid crystal
P. Sathyanarayana, M. Mathew,* Q. Li, V. S. S. Sastry, B. Kundu, K. V. Le, H. Takezoe, and Surajit Dhara†
School of Physics, University of Hyderabad, Hyderabad 500046, India
and Department of Organic and Polymeric Materials, Tokyo Institute of Technology, 2-12-1-S8-42 O-okayama, Meguro-ku,
Tokyo 152-8552, Japan
共Received 6 October 2009; published 28 January 2010兲
We measured the splay 共K11兲 and bend 共K33兲 elastic constants in the nematic phase of a bent-core liquid
crystal. In the vicinity of the nematic-isotropic transition temperature K33 is proportional to the square of theorder parameter. In the nematic range K11 increases monotonically with decreasing temperature, whereas K33
is practically independent of temperature and is smaller than K11. K33 exhibits a pretransitional slow divergence
toward the transition temperature to the smectic phase and becomes slightly larger than K11. The small K33 is
explained on the basis of strong coupling of the bent shape of the molecules with the bend distortion.
DOI: 10.1103/PhysRevE.81.010702

PACS number共s兲: 61.30.Eb, 61.30.Gd

Thermotropic liquid crystals exhibit a variety of phases asfunctions of temperature. Among the known phases, nematic
共N兲 is the simplest one. It exhibits a long-range orientational
order about an axis called director and it is apolar 共n̂ = −n̂兲.
The electro-optic properties of rodlike N liquid crystals are
exploited particularly for flat panel display devices. Whenthe translational symmetry of rodlike molecules is broken it
gives rise to smectic liquid crystals. In general N and smectic
A 共Sm-A兲 liquid crystals are uniaxial since the molecules,
although they are not cylindrical symmetric, freely rotate
about the long molecular axis. In the past few years, a new
type of molecular shape, namely, bent-core 共BC兲 liquid crystals 共with a bent angle of ⬃120°兲, has created immense interest in the scientific community because it gives rise to
several new liquid-crystalline phases along with some known
phases observable in calamitic liquid crystals 关1,2兴. Many
physical properties of bent-core liquid crystals related to thebent structure are not yet understood.
The N phase has been chosen for various physical measurements in order to investigate the structure-property relation in bent-core liquid crystals. Recently some interesting
experimental results are reported in the N phase of BC liquid
crystals, i.e., large flexoelectric coefficient 关3,4兴, unusualelectroconvection 关5–7兴, and Kerr effect 关8兴. There are also
some reports on the measurement of splay 共K11兲 and bent
共K33兲 elastic constants in the mixtures of rod and BC liquid
crystals. Dodge et al. showed that dissolving a small concentration of BC molecules into a compound with rodlike molecules can reduce K33 by a factor of 2 or more 关9,10兴. Kundu
et al. reported anomalous temperature dependence of K11 andK33 in the mixtures of rod and BC molecules and explained iton the basis of mutual orientation of rod and BC molecules关11兴. On the other hand Olivares et al. 关12兴 conducted a
dynamic light scattering measurement in a pure BC compound with oxazole heterocyclic ring in the central core andsuggested that the temperature dependence of elastic constants is weak with no signature of pretransitional behavior

toward the transition temperature to the smectic 共Sm-C兲
phase. Recently Dozov theoretically showed that a spontaneous symmetry breaking phase transition inside the N phase ispossible because of mutual tendency of banana-shaped molecules to induce a local bend director with a negative bendelastic constant 关13兴. Therefore the knowledge about theelasticity in pure BC liquid crystals is important. However,until now no direct measurement of the temperaturedependent elastic constants for pure BC liquid crystals has
been reported. In this Rapid Communication we carry out thetemperature-dependent measurements of K11 and K33 of apure bent-core liquid crystal and report that the temperaturedependence of K11 and K33 is significantly different from thatin rodlike molecules; i.e., K33 is much lower compared to K11
except very near the Sm-C phase transition.
The chemical structure of the compound is shown in Fig.
1共a兲 共inset兲. It has the following phase transitions: Sm-C,
118.5 ° C N, 176.5 ° C I. The compound exhibits a large temperature range of the N phase. It is also stable as there wasno noticeable change in the transition temperatures after several cycles of heating and cooling. Planarly aligned cells
were prepared by using two indium tin oxide coated glass
plates. These plates were treated with polyimide and cured at
180 ° C for 1 h and rubbed antiparallel to each other. For
homeotropic cells, JALS-204 was used as an alignment layer
and cured at 200 ° C for 1 h. The cell thickness was measured by a spectrometer. The typical cell thickness used forthe experiments was ⬃5 ␮m. The cell was filled with thesample in the isotropic phase and a well-aligned N sample
was obtained. The parallel 共⑀储兲 and perpendicular 共⑀⬜兲 components of dielectric constants were measured in homeotropic and planar cells, respectively, by using a LCR meter
共Agilent 4980A兲. The optical retardation was measured byusing a phase modulation technique 关14兴 with the help of ahelium-neon laser, a photoelastic modulator 共PEM-100兲, anda lock-in amplifier 共SR-830兲. A sinusoidal voltage with afrequency of 4111 Hz and an amplitude of up to 20 V at steps
in the ranges 0.05–0.5 V were used for the measurements byusing the same LCR meter. The retardation and also thesample capacitance were measured simultaneously as functions of temperature and voltage. The capacitance value obtained in planar cells at various temperatures far below theFreedericksz threshold voltage also provides the perpendicu-*Present address: Liquid Crystal Institute, Kent State University,
Kent, Ohio 44242, USA.
†
Corresponding author; sdsp@uohyd.ernet.in
1539-3755/2010/81共1兲/010702共4兲

010702-1©2010 The American Physical Society

RAPID COMMUNICATIONSPHYSICAL REVIEW E 81, 010702共R兲 共2010兲

SATHYANARAYANA et al.

lar component 共⑀⬜兲 and the extrapolated value at high voltage provides the parallel component 共⑀储兲 of dielectric constants. The measured value agrees well with themeasurements in the independent cells. The splay elastic
constant K11 was obtained directly from the Freedericksz

冕 冑

␲/2
2
V
= 冑1 + ␥ sin2共␾m兲
Vth ␲
0

冢

nd
␭

␦ = 2␲ e

冕 冑
冕 冑
␲/2

0

␲/2

0

1 + ␬ sin2共␾m兲sin2共␺兲d␺ ,关1 + ␥ sin 共␾m兲sin2共␺兲兴关1 − sin2共␾m兲sin2共␺兲兴2

关1 + ␥ sin2共␾m兲sin2共␺兲兴关1 + ␬ sin2共␾m兲sin2共␺兲兴d␺
关1 − sin2共␾m兲sin2共␺兲兴关1 + ␯ sin2共␾m兲sin2共␺兲兴关1 + ␥ sin 共␾m兲sin 共␺兲兴关1 + ␬ sin 共␾m兲sin 共␺兲兴d␺
关1 − sin2共␾m兲sin2共␺兲兴
22

where d is the cell thickness, ␾m is the tilt angle at the
middle of the cell, and the other terms of the reduced quantities are ␥ = 共⑀储 / ⑀⬜兲 − 1, ␬ = 共K33 / K11兲 − 1, ␯ = 共n2e / n2o兲 − 1, andsin共␾兲 = sin共␾m兲sin共␺兲. The sample retardation at higher voltage is fitted to the above equations by an iterative procedure
to get the bend elastic constant K33 关11,16兴.
We first show the variation of birefringence 共⌬n兲 in the Nphase as a function of temperature in Fig. 1共a兲. It is noticed
that birefringence appears abruptly at the N-isotropic 共NI兲phase transition and gradually increases as the temperature

Sm-X

Birefringence (∆n)

0.15
0.10

Birefringence
Order Parameter

0.8
0.6

Sm-C

Nematic
I
O

0.05

CH3 O

O

N

N

R1=C8H17

R1

0.4

O

N

0.2
R2

Ordere Parameter (S)

1.0

(a)

0.20

R2=C12H25

0.0
10

0.00
-80 -70 -60 -50 -40 -30 -20 -10 0o

T-TNI ( C)

(a)

(b)

Dielectric Constant

5.2

4.4
4.0

Nematic
Sm-C

-60

I

ε⊥

3.6
-80

(b)

ε||

4.8

-40

-20

threshold voltage of the voltage-dependent retardation databy the equation K11 = ⑀o⌬⑀共Vth / ␲兲2. The voltages above thethreshold voltage Vth and the retardation ␦ are given by theparametric equations 关15兴0o

T-TNI ( C)

FIG. 1. 共Color online兲 共a兲 Variation of birefringence and orderparameter as a function of reduced temperature. Chemical structureof the compound used in the experiment is also shown in the inset.
共b兲 Variation of parallel ⑀储 and perpendicular ⑀⬜ components ofdielectric constant as a function of temperature.

2

2

−

冣

no
,
ne

共1兲

共2兲

decreases in the N phase. A small jump in the birefringencedata is also observed near the N-Sm-C phase transition at
T − TNI = −64°. The temperature-dependent birefringence in
the N liquid crystals can be approximated by the formula⌬n = ⌬no共1 − T / Tⴱ兲␤, where Tⴱ and ␤ are the adjustable fit
parameters and ⌬no is the birefringence of the perfectlyaligned sample. The equation fits well in the entire nematic
range with fit parameters ⌬no = 0.28 and ␤ = 0.21. The orderparameter S was estimated using the relation S = ⌬n / ⌬no.
The temperature variation of the calculated order parameteris also shown in Fig. 1共a兲. The order parameter in the Nphase just below the NI phase transition was approximately
0.2 and reached 0.7 with a small jump at the N-Sm-C phasetransition. The temperature-dependent S was also measured
by Weissflog et al. 关17兴 and Dong et al. 关18兴 in the N phaseof banana liquid crystals using nuclear magnetic resonancespectroscopy, and similar temperature dependence withslightly smaller values of S was reported.
The variation of parallel 共⑀储兲 and perpendicular 共⑀⬜兲 components of dielectric constants as a function of temperature isshown is Fig. 1共b兲. In the N phase, the dielectric anisotropy
is small positive and changes sign in the Sm-C phase. The
temperature-dependent dielectric anisotropy can be understood as follows: the molecules have both longitudinal andtransverse dipole components mainly due to the ester linkages. In the N phase, molecules freely rotate about their long
axis, so that the effect of transverse dipoles is suppressed.
But the longitudinal dipoles respond to an electric field dielectrically. Hence, positive dielectric anisotropy results in
the N phase. The change of sign of dielectric anisotropy fromthe N phase to the Sm-A phase of rodlike molecules wasreported by de Jeu et al. 关19兴 and was explained by an increased dipolar correlation between the molecules in thesame layer. In addition to such an effect, we should consider
the bent shape of the molecules, i.e., local packing of thebent-shaped molecules in the Sm-C layers hinders the freerotation about their molecular long axis, leading to quasimacroscopic transverse dipoles, which respond to the electric

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RAPID COMMUNICATIONS

8
6
4
2
0

5
10
15
Applied Voltage (V)

20

16

K11

12
8

16
14
12
10
8
6
4
2
0

K11
K33

0.08

K33

0.12
0.16
Birefringence(∆n)

4
0

Sm-C

Elastic Constants (pN)

Retardation (∆φ)

10

Elastic Constant (pN)

PHYSICAL REVIEW E 81, 010702共R兲 共2010兲

SPLAY BEND ELASTICITY OF A BENT-CORE NEMATIC…

Nematic

I-70 -60 -50 -40 -30 -20 -10

0oFIG. 2. 共Color online兲 Retardation as a function of applied electric field at 157 ° C. The best fit to the theoretical retardation obtained by using Eq. 共2兲 is given by a solid line. K33 is determined bybest fit.

field, contributing the increase in ⑀⬜. The increasing ⑀⬜ withdecreasing temperature is caused by larger rotational hindrance. As discussed by de Jeu et al. 关19兴, the decrease in ⑀储
is explained by the increased antiparallel correlation between
the dipoles along the molecular long axis. This could bepossible even in the local packing condition. In addition, thelayer formation suppresses the dielectric response about themolecular short axis, contributing the decrease in ⑀储 in the
Sm-C phase. The sharp variation of dielectric constants
across the N to Sm-C transition 关Fig. 1共b兲兴 in the present
bent-core liquid crystal compared to the slow variation in
rodlike molecules reported 关19兴 is due to the contribution ofrotational hindrance.
Let us describe the K11 and K33 measurements. First, theobserved retardation is shown in Fig. 2 as a function of theapplied voltage at 157 ° C. From the critical voltage 共2.85 V兲
of the Freedericksz transition we estimated K11 as 6.5 pN. A
typical best fit of the overall data to Eq. 共2兲 with K33 as afitting parameter is shown in Fig. 2, giving K33 = 3.7 pN. The
temperature dependence of K33 and K11 is shown in Fig. 3.
K11 and K33 were also plotted as functions of thebirefringence. As shown in the inset, K33 is proportional to⌬n2 共⬀S2兲 in the vicinity of the NI transition as predicted by
the simple mean-field theory. The value of K11 however
monotonically increases as the temperature is lowered andno pretransitional divergence is observed. K11 is also fittedwith K11 ⬃ ⌬nx and the value of x ⬃ 2.9. Slightly lower valueof x 共⬃2.5兲 was reported in the rod and bent-core mixtures关11兴. However, the value of K33 at a given temperature isalways lower than K11 except when very to close the
N-Sm-C transition. For example, at T − TNI ⬇ −1 ° C. i.e., justbelow the NI transition K33 / K11 ⬃ 0.7 and reduces to ⬃0.4 at
about T − TNI ⬇ −45 ° C. Then because of pretransitional increase in K33 below T − TNI ⬇ −50 ° C, K33 / K11 increases to⬃1 with approaching to the N-Sm-C transition. In the case of
rodlike molecules 关e.g., octyl cyano biphenyl 共8OCB兲兴 it is
known that K33 is always higher and the ratio is ⬃2, very
close to the NI transition and increases to ⬃6 as the Sm-A
phase is approached 关9,11兴. Interestingly the values of K11 inboth 8OCB and the present compound are comparable, although the temperature dependence of K11 is stronger in thepure bent core than in 8OCB as seen with the larger value ofx.

T-TNI ( C)

FIG. 3. 共Color online兲 Variation of splay 共K11兲 and bend 共K33兲
elastic constants as a function of reduced temperature. Temperaturedependent K11 and K33 are shown against ⌬n in the inset. Solidlines are best fit to K11 and K33 vs ⌬nx. x = 2.9 and 2 in cases of K11
and K33, respectively.

The sign inversion of elastic anisotropy 共K33 − K11兲 andthe temperature dependence of these two elastic constants are
expected to originate from the bent structure of the molecules. A schematic representation of splay and bend distortion of bent-core molecules is shown in Figs. 4共a兲 and 4共b兲.
The almost linear increase in K11 can be attributed to theincrease in the order parameter as seen in the birefringencemeasurement 关Fig. 1共a兲兴. Here, the bent shape does not facilitate the splay distortion in the medium 关Fig. 4共a兲兴. However, as the temperature is lowered the molecular bent shapecoupled to the bend distortion easily facilitating the bent distortion. As a result K33 becomes lower in the N phase. In thiscase the strain in the bend distortion partly relieved bychanging the equilibrium distribution of the molecules with
one side “up” or “down” as shown in Fig. 4共b兲. As the temperature is lowered toward the Sm-C phase, because of
smectic short-range fluctuations, K33 starts to change thetemperature dependence at T − TNI ⬇ −30 ° C. The divergenceof elastic constants near the Sm-C phase has been theoretically studied by de Gennes 关20兴 and Chen and Lubensky
关21兴. According to both theories all three elastic constants
show pretransitional divergence. In the present compound wefind no pretransitional divergence in K11 and the exponent ofdivergence of K33 is ⬃0.43, which is much lower than thepredicted value 共⬃0.7兲 indicating that the existing theoreticalmodels are insufficient to explain the pretransitional behavior
near Sm-C of bent-core liquid crystals. It may be pointed outthat there are also some experimental reports on the elastic
constant measurements of some binary mixtures by using a

(a)

(b)

FIG. 4. Schematic representation of 共a兲 splay and 共b兲 bend distortion in bent-core nematic liquid crystals.

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SATHYANARAYANA et al.

light scattering technique above the N to Sm-C transition
关22,23兴. They reported that K33 obtains a maximum value
just above the N to Sm-C transition point and such a behavior is not anticipated by any of the existing models.
Previously the compound was reported to show theuniaxial to biaxial N transition 关24兴 at T − TNI ⬇ −30°, although recently Van Le et al. 关25兴 showed that the medium isuniaxial. Interestingly the temperature at which K33 starts
changing the temperature dependence 共T − TNI ⬇ −30 ° C兲 coincides with the Nu-Nb transition temperature reported 关24兴,suggesting the onset smectic short-range fluctuation at thattemperature. Thus, the biaxial-like order seen in the x ray asreported in Ref. 关24兴 could be due to the presence of smectic
short-range fluctuations, as suggested 关26,27兴.
In conclusion we measured the K11 and K33 elastic con-stants as functions of temperature in a pure bent-core liquid
crystal. Our measurement shows that K33 is much lower thanK11 except for the temperature regions close to the N to
Sm-C transition. The temperature dependence clearly showsa strong coupling of bent shape with the bend distortions inpure compound. Thus, various physical properties and theirtemperature dependence related to K33 are strongly influenced and further the bent shape has to be taken into accountin understanding the physical properties in bent-core molecules.

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One of the authors 共S.D.兲 gratefully acknowledges the
support from the DST, Government of India for Project No.
SR/FTP/PS-48/2006 under the SERC Fast Track Scheme.

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