RAPID COMMUNICATIONSPHYSICAL REVIEW E 82, 050701共R兲 共2010兲

Splay-bend elasticity of a nematic liquid crystal with T-shaped molecules
P. Sathyanarayana,1 M. C. Varia,2 A. K. Prajapati,2 B. Kundu,1,* V. S. S. Sastry,1 and S. Dhara1,†
1

School of Physics, University of Hyderabad, Hyderabad 500046, India
Department of Applied Chemistry, Faculty of Technology and Engineering, M. S. University of Baroda,
Vadodara 390001, India
共Received 14 April 2010; revised manuscript received 14 September 2010; published 10 November 2010兲
2

We measured the splay 共K11兲 and bend 共K33兲 elastic constants in the nematic phase of a liquid crystal withT-shaped molecules. We find that the ratio, K33 / K11 ⯝ 1 in the entire nematic range except very close to thenematic to Sm-A 共SN兲 transition. Both K33 and K11 show pretransitional divergence as the SN transition isapproached from higher temperature. The ratio, K33 / K11 suggests that the length 共L兲 to effective width 共D兲
ratio 共i.e., L / D兲 is significantly smaller due to the presence of long and flexible lateral group, compared to thatof rigid rodlike molecules. It is argued that apart from the extra contribution to the elasticity the long andflexible lateral group also has a significant contribution to the suppression of the splay fluctuations in the onsetof smectic short-range fluctuation. The structure of the Sm-A phase is investigated by using small angle x-ray
diffraction, and a possible arrangement of the molecules in the Sm-A layer is proposed.
DOI: 10.1103/PhysRevE.82.050701

PACS number共s兲: 61.30.Eb, 61.30.Gd

Interest in the synthesis and physical characterization ofliquid crystals with chemical structure similar to the shapesof many English letters such as V, T, H, W has increased
after the discovery of electro-optical switching in bent-coreliquid crystals 关1–3兴. The structure-property relationship ofsuch unconventional liquid crystals has always been interesting from both technological and fundamental points of view.
It is observed that the physical properties and, in particularelectro-optical responses, in these unconventional liquid
crystals are very different compared with the calamitic liquid
crystals, and the molecular shape has significant effect onphase structure and macroscopic physical properties 关4兴. For
investigating the structure-property relation, nematic phase
共N兲 has been chosen due to its simplicity. There are many
interesting physical properties reported in the nematic phaseof bent-core liquid crystals among the unconventional liquid
crystal molecules mentioned above. It has been found that
bent-core nematic has a number of extraordinary properties
such as the sign inversion of elastic anisotropy 共K33 − K11兲
关5兴, unusual electroconvection 关6兴, giant flexoelectricity 关7兴,and large viscosity 关8兴. In this paper we report measurementof order parameter, splay, and bend elastic constants of aliquid crystal comprising of T-shaped molecules. These arefound to be significantly different from the conventional calamitic and bent-core liquid crystals. Further, from the x-ray
diffraction measurements we propose a possible orientationof these T-shaped molecules in the Sm-A layers.
The compound used in the present study was synthesizedin our laboratory and the details will be reported elsewhere.
The chemical structure of the compound is shown in Fig.
2共a兲. It has the following phase sequence: Cr 90 ° C Sm-A
108 ° C N 128 ° C I. Planar aligned cells were prepared byusing two glass plates coated with indium tin-oxide. The
plates were treated with polyimide and cured at 180 ° C for
one hour and rubbed antiparallel to prepare cells with planaralignment. For homeotropic cells, JALS-204 was used andcured at 200 ° C for 1 h. Typical cell thickness used for theexperiments was d ⬃ 5 ␮m. The parallel 共⑀储兲 and perpendicular 共⑀⬜兲 components of the dielectric constant were measured as a function of temperature in homeotropic and planarcells, respectively, with a LCR meter 共Agilent 4980A兲 at
4111 Hz, applying a voltage of 0.3 V. The optical phaseretardation was measured by using the phase modulation
technique 关9兴. The setup comprises a Helium-Neon laser 共␭= 632.8 nm兲, photoelastic modulator 共PEM-100兲 and alock-in amplifier 共SR-830兲. The retardation and sample capacitance were measured simultaneously as functions of temperature and voltage up to 20 V in steps ranging from 0.05 V
to 0.5 V. The x-ray diffraction measurements were made using Cu K␣ radiation from a rotating anode. The splay elastic
constant K11 was measured directly from the Freedericksz
threshold voltage 共Vth兲 of the voltage dependent retardation
data using the equation K11 = ⑀o⌬⑀共Vth / ␲兲2, where ⌬⑀共=⑀储− ⑀⬜兲 is the dielectric anisotropy. The voltage dependent optical phase difference 共␦兲, and voltages above the thresholdvoltage Vth, are given by following parametric equations
关10,11兴:*Present address: Liquid Crystal Institute, Tokyo University of
Science, Yamaguchi, Japan.
†
Corresponding author; sdsp@uohyd.ernet.in
1539-3755/2010/82共5兲/050701共4兲

FIG. 1. 共Color online兲 共a兲 Schlieren texture in nematic phase 共at
123 ° C兲 and 共b兲 Focal conic texture in Sm-A phase 共at 103 ° C兲.

050701-1©2010 The American Physical Society

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SATHYANARAYANA et al.

冕 冑

␲/2
2
V
= 冑1 + ␥ sin2共␾m兲
Vth ␲
0

nd
␭

␦ = 2␲ e ⫻

冦

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

共1兲

冧

共2兲

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 quan⑀Kn2tities are: ␥ = ⑀⬜ − 1, ␬ = K3311 − 1, ␯ = ne2 − 1, sin共␾兲o= sin共␾m兲sin共␺兲. The sample retardation at higher voltages
are fitted to the above equations by an iterative procedure to
get the bend elastic constant K33 关10,11兴.
A typical schlieren texture in the nematic phase at 123 ° C
and a focal conic texture in the Sm-A phase at 103 ° C areshown in Fig. 1. A uniform planar alignment of the director
共n̂兲 is obtained in rubbed cells. The variation of birefringence共⌬n兲 in the N and the Sm-A phase as a function of temperature is shown in Fig. 2共a兲. ⌬n develops sharply at the NI
phase transition as expected and gradually increases as thetemperature decreases in the N phase. A significant increasein ⌬n is also observed in the Sm-A phase 共below 108 ° C兲.
The temperature dependent ⌬n in N liquid crystals can be储FIG. 2. 共Color online兲 共a兲 Variation of birefringence 共⌬n兲 andorder parameter 共S兲 as a function of temperature. Chemical structure and the energy minimized structure obtained from quantum
chemical calculations using GAUSSIAN-03 共inset兲. 共b兲 Variation ofparallel 共⑀储兲 and perpendicular 共⑀⬜兲 components of the dielectric
constant and the dielectric anisotropy 共⌬⑀兲 as a function oftemperature.

2

2

−

no
,
ne

approximated by the formula ⌬n = ⌬no共1 − T / Tⴱ兲␤, where Tⴱand ␤ are the adjustable fit parameters and ⌬no is the birefringence of the perfectly aligned sample. The equation fits
well in the entire nematic range with fit parameters ⌬no= 0.31 and ␤ = 0.21. The order parameter S was estimatedusing the relation S = ⌬n / ⌬no. The temperature variation of
the calculated order parameter is also shown in Fig. 2共a兲. The
jump in the order parameter at the NI transition was 0.25 andit increases to 0.65 共at 95 ° C兲 with a significant increase⌬SSN ⬃ 0.06 共at 107 ° C兲 in the Sm-A phase. There have been
many experimental and theoretical studies on the N to Sm-A
共SN兲 transition in compounds with rodlike molecules. The
coupling between the smectic order parameter ␺ and thefluctuations in the nematic order parameter S can cause acrossover toward a tricritical point leading to a first ordertransition as the McMillan parameter ␣共=TSN / TNI兲 approaches 1 关12兴. The theoretical value of ␣ for the crossoverto the tricritical point is ␣ ⬃ 0.83. However, the experimental
value of ␣ is not universal and is slightly higher 关13兴, and it
also depends on the type of the Sm-A phase 关14,15兴. The
value of ␣ for the present compound is 0.95 and is well
above the tricritical point value. The large value of ⌬SSN共⬃0.06兲 in addition to large ␣ suggests that the present SNtransition is first order in nature. A small increase in S in the
SN transition is also known in many other calamitic liquid
crystals such as 8OCB 共4-octyloxy 4⬘-cyanobiphenyl兲 关11兴,
8CB 共4-octyl-4⬘-cyanobiphenyl兲 关10兴 and in some banana
shaped compounds 关5兴. In the case of 8CB the jump in theorder parameter ⌬SSN is ⬃0.01 and the SN transition is considered to be a weakly first order 关16兴. In the present compound ⌬SSN 共⬃0.06兲 is much larger than that of 8CB. We
also measured the enthalpies of NI 共⌬HNI兲 and NS transitions 共⌬HSN兲 using a differential scanning calorimeter. Theyare ⌬HNI = 2.0 kJ/ mol and ⌬HSN = 1.6 kJ/ mol, respectively.
Thus enthalpies of both these transitions are comparable forpresent compound whereas ⌬HSN is ⬃11 and ⬃22 times
smaller than ⌬HNI in cases of 8CB and 8OCB, respectively
关17兴. The large values of ⌬SSN and enthalpy 共⌬HSN兲 further
support first order nature of the SN transition.
The temperature variation of parallel 共⑀储兲 and perpendicular 共⑀⬜兲 components of the dielectric constant is shown in
Fig. 2共b兲. The dielectric anisotropy is small and positive共⌬⑀ = 0.9 at T = 115 ° C兲 in the N phase, and reduces in the
Sm-A phase. The temperature dependent dielectric anisotropy is interpreted on the basis of the role of permanent
dipoles and the molecular shape. The molecules have both

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SPLAY-BEND ELASTICITY OF A NEMATIC LIQUID…

FIG. 3. 共Color online兲 共a兲 Variation of K11 and K33 as a functionof temperature. Dashed line is a theoretical fit to K33 = K033
+ A关共T / TNS兲 − 1兴−x. 共Inset兲 Retardation as a function of applied voltage at 126 ° C. Continuous line is a theoretical fit to Eqs. 共1兲 and共2兲. 共b兲 共i兲 Schematic of the splay fluctuations at the onset of shortrange smectic order. Note the tilting of the molecule 共splay deformation兲 is associated with a change in the layer spacing in 共b兲共II兲.

longitudinal and transverse dipole components mainly due tothe azo and ester linkages. The partial suppression of thetransverse dipole moment in the nematic phase is made possible by largely unhindered rotations about the long molecular axis. This leads to a positive ⌬⑀ in the N phase. In the
Sm-A phase the dielectric anisotropy is reduced because therotations about their long axes are more hindered due to thepresence of a long side chain. This results in developing aquasimacroscopic transverse dipoles as suggested by Sathyanarayana et al. 关5兴. Further, the increased in-layer dipolar
correlation as reported by de Jeu et al. 关18兴 has significant
contribution in reducing the dielectric anisotropy in the
Sm-A phase.
To measure K11 and K33, the optical phase difference共⌬␾ = 2␲⌬nd / ␭兲 of the sample was measured as a functionof voltage. A typical variation of experimental data ⌬␾ at atemperature 126 ° C and the best fit to Eq. 共2兲 are shown in
Fig. 3共a兲. K11 was measured directly from the Freedericksz
transition as mentioned earlier. The experimental data at
higher voltage was fitted to the parametric equations 关Eqs.
共1兲 and 共2兲兴 to get K33 as a fit parameter. The temperaturedependent variation of both K11 and K33 is shown in Fig.
3共a兲. The variation of K33 near TNI is proportional to S2 andK33 increases as the temperature is reduced similar to its
variation as in the case of many calamitic liquid crystals,
diverging rapidly as the SN transition is approached. Interestingly we find that the temperature dependence of K11 is
the same as that of K33. The ratio K33 / K11 ⬃ 1 at all temperatures except very close to the SN transition. This ratio varies
from 2 to 6 in case of rodlike molecules 共e.g., 8OCB兲 as thetemperature is reduced from the NI to the SN transition. K33
shows strong pretransitional divergence as predicted and is00+ A关共T / TNS兲 − 1兴−x 关19兴, where K33
is thegiven by K33 = K33

background nematic contribution and x is the critical exponent. In the present compound x = 1 ⫾ 0.09 which is similar tothe value reported for 8CB 关10兴. It is known that for rigid
rodlike molecules K33 / K11 ⬃ L / D where L / D is the length towidth ratio of the molecule 关20兴, and it is found to decrease
when flexible groups are present in the molecules. Using asemiempirical relation for flexible molecules, the ratio
K33 / K11 can be related to the molecular properties and isexpressed as 关20兴: K33 / K11 = 共关1 + ⌬ − 3⌬⬘ P4 / P2兴兲 / 共关1 + ⌬+ 4⌬⬘ P4 / P2兴兲; where ⌬ = 共2R2 − 2兲 / 共7R2 + 20兲, ⌬⬘ = 9共3R2− 8兲 / 16共7R2 + 20兲. R = 共L − D兲 / D, where L and D are thelength and width of the molecule. Since the molecules havelong flexible lateral group, we calculated the radius of gyration along the long axis to estimate their width. It turns outthat this width 共⯝9 Å兲 is about a third of the experimentallymeasured length of the long axis 共28.8 Å, discussed below兲,i.e., L ⯝ 3D. Assuming P4 / P2 is small 共⬃0.1兲 and with L⯝ 3D 共i.e., R ⯝ 2兲 and the ratio, K33 / K11 ⯝ 1 which is close to
the experimental value obtained 关Fig. 3共a兲兴.
The divergence of K11 is a consequence of suppression of
splay fluctuations as Sm-A phase is approached. This looks
unusual since the splay fluctuations are taken to be compatible with the constant layer spacing requirement of smectics
in conventional liquid crystals. However in the present case,the molecule has a long flexible lateral group and is undergoing largely unhindered rotations along the long axis. This
seems to modify the aspect ratio of the molecules, effectively
making it ⬃3 共i.e., L ⯝ 3D兲, depicted as a cylinder in Fig.
3共b兲-I. Splay fluctuations involving such objects lead to significant changes in the layer spacing as shown schematicallyin Fig. 3共b兲-II, necessitating relatively large elastic energy.
Hence the splay fluctuations are considerably suppressedleading to the observed divergence in K11. Further, comparedto more rigid molecules the flexible terminal alkyl chains andlong flexible lateral groups have the potential to partiallyhinder the mutual sliding of the molecules along each other.
The consequent close proximity of aromatic cores of neighboring molecules may also be responsible for larger effective
width, thus supporting the experimental observations.
Finally we discuss the arrangement of T-shaped moleculesin the smectic layers as suggested from the x-ray diffraction
measurements. Typical intensity profile as a function of angle
␽ is shown in Fig. 4. Two sharp reflections in the small-angle
region with spacing 28.8 Å, 15.1 Å, as well as an additional
broad diffuse wide-angle reflection at about 4.6 Å can be
seen. Ratio of the two spacings 1: 0.5 clearly indicates alayered structure. Length of different segments of a single
molecule was also estimated from energy minimized structure by considering different dihedral angles. The average
length of the particular segment, containing three phenyl
rings, azo and ester linkages with two terminal alkyl chains,is L ⯝ 31 Å 关Fig. 2共a兲兴. This suggests that L is parallel to thesmectic layer normal and the flexible chains are randomly
oriented in the plane of the smectic layer as schematicallyshown in Fig. 4. Slightly lower value of the experimentallymeasured length 共i.e., 28.8 Å兲, compared to the computed L
value 共=31 Å兲 from the energy minimized structure, suggests that the molecules is not fully stretched in the Sm-A
phase.
In conclusion, we measured elastic constants K11 and K33050701-3

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SATHYANARAYANA et al.

flexible lateral group. Small angle x-ray diffraction and energy minimized structure suggest that the three phenyl rings
and two linkages 共azo and ester兲 and two terminal alkyl
chains form the long axis. This axis is parallel to the Sm-A
layer normal and the flexible lateral groups are in the plane.
This system may prove to be a convenient medium for testing theoretical models based on one elastic constant approximation. Detailed theoretical investigations can through lightto understand the observed divergence of the spay elastic
constant of such compounds from the perspective of elastic
free energy considerations near the SN transition.
FIG. 4. 共Color online兲 X-ray intensity as a function of angle ␽.
Arrow indicates a small second order reflection. 共Inset兲 Schematicarrangement of molecules in the Sm-A layer.

ACKNOWLEDGMENTS

as a function of temperature in a liquid crystal with T-shaped
molecules exhibiting both first order NI and SN transitions.
Our measurements show that K33 / K11 ⬃ 1 and both diverge
as the Sm-A is approached. These results suggest that thelength to effective width ratio of the molecule is smaller thanthat of rigid rodlike molecules due to the presence of long

One of the authors 共S.D.兲 gratefully acknowledges the
support from the DST, Govt. of India for the Project No.
SR/FTP/PS-48/2006 and CAS School of Physics. We thankProfessor V. A. Raghunatha for helping in x-ray measurements. We also gratefully acknowledge Professor A. Jakli,
Professor H. Takezoe, and Professor N. V. Madhusudana for
useful discussions. P. S. acknowledges CSIR-UGC for support.

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