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Splay-bend elasticity and rotational viscosity of liquid crystal mixtures ofrod-like and bent-core molecules

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P. Sathyanarayana,a B. K. Sadashivab and Surajit Dhara*a
Received 25th April 2011, Accepted 23rd June 2011
DOI: 10.1039/c1sm05751k
We report temperature dependent measurements of birefringence (Dn), splay (K11), bend (K33) elastic
constants and rotational viscosity (g1) of mixtures of rod-like and bent-core molecules. K11 and Dnremain almost constant for all the mixtures whereas K33 is always lower than K11 and further decreases
with increasing concentration of bent-core molecules in the mixtures. Apart from the large jump in Dn
at the nematic-isotropic transition (NI) a significant jump in Dn at the nematic-smectic transition (NS)is also observed that increases with increasing mol% of the bent-core compound in the mixtures.
Rotational viscosity increases by a factor of four when the concentration of the bent-core compoundincreased from 0 to 10 mol%. The results are discussed on the basis of their structural similarity andmutual alignment in the nematic phase.

I.

Introduction

The interests in investigating the structure-property relation in
unconventional liquid crystals has increased significantly in thepast decade after the discovery of the electrooptical switching in
bent-core liquid crystals.1 Several new and complex structures
known as ‘banana’ or B-phases (e.g., B1, B2, etc) owing to theirshape in such unconventional liquid crystals are already reported.2–4 Apart from the B-phases these bent-core compounds
occasionally exhibit some phases of calamitic liquid crystals likenematic and smectic. The physical properties of the nematic phasecomposed of bent-core molecules have been reported to show
several interesting and new results. For example, it has been foundthat bent-core nematic liquid crystals exhibit sign inversion ofelastic anisotropy,5,6 unusual electroconvection,7–9 giant flexoelectricity,10,11 large viscosity,12,13 Kerr coefficient,14 induced15
and spontaneous biaxiality.16,17 Nematic phases composed ofa mixture of rod-like and bent-core compounds are also interesting as they often exhibit unusual physical properties.18–22 For
example, there are some reports on the unusual elastic properties
in the mixture of rod and bent-core liquid crystals. In particular,
Dodge et al. showed that addition of a small concentration ofbent-core liquid crystals in a calamitic compound can reduce K33
by a factor of two or more.18,19 Kundu et al. reported anomalous
trends in the temperature dependence of splay (K11) and bend(K33) elastic constants in the nematic phase of mixtures of 4-noctlyoxy 4-cyanobiphenyl (8OCB) and the present bent-corecompound (Fig. 1(a)) and explained the results on the basis ofaSchool of Physics, University of Hyderabad, Hyderabad, 500046, India.
E-mail: sdsp@uohyd.ernet.inb
Raman Research Institute, C. V. Raman Avenue, Bangalore, 560080,
India

8556 | Soft Matter, 2011, 7, 8556–8560

their mutual orientation20 in the presence of short-range smectic
fluctuations. In this paper, we present the measurements of Dn,
K11, K33 and g1 as a function of temperature and show that these
physical properties of the mixture are significantly different thanthose previously reported in other binary systems due to theFig. 1 (a) Chemical structures of the compounds and their phase transition temperatures. (b) Phase diagram of the mixtures of the above twocompounds. Vertically down red arrows indicate the mol% chosen for theexperiments.

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structural similarity, i.e., the aromatic and aliphatic parts of therod are matched with one half of the bow in the present system.

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II. Experimental
Liquid crystal cells were prepared by using two indium tin oxide(ITO) coated glass plates. These plates were treated with polyimide (AL-1254) and cured at 180  C for 1h and rubbed antiparallel for planar or homogeneous alignment of the sample. The
cell thickness was measured by using a fibre optic spectrometer(Ocean Optic, HR4000). Typical cell thickness used in theexperiment was 9 mm. The cell was filled with the sample in theisotropic phase and a well aligned nematic phase was obtained oncooling. The perpendicular component of the dielectric constant(3t) was measured in a planar cell below the Freedericksz
threshold voltage by using a LCR meter (Agilent 4980A). To
measure the parallel component of dielectric constant (3k) the
effective dielectric constant was measured from 0.02 to 20V. The
experimental variation of the voltage dependent dielectric
constant tends to saturate at high voltages. The linear partof dielectric constant is plotted against 1/V and extrapolated to1/V ¼ 0 to obtain 3k at various temperatures. The measured value
is in good agreement with the measurements made in an independent cell.
The optical retardation was measured by using a phasemodulation technique23 with the help of a Helium-Neon laser,a photoelastic modulator (PEM-100) and a lock-in amplifier(SR-830). A sinusoidal voltage of frequency 4111 Hz andamplitude up to 20 V at steps 0.02 V were used for themeasurements by using the LCR meter. The retardation and also
the sample capacitance were measured simultaneously as functions of temperature and voltage. The splay elastic constant (K11)
is obtained from the Freedericksz threshold voltage (Vth) and isgiven by K11 ¼ 3oD3(Vth/p)2, where D3 ¼ 3k  3t is the dielectricanisotropy. At strong surface anchoring conditions, the voltages
above the threshold voltage Vth and the retardation (Df) are
given by the parametric equations:24
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
V2¼
1 þ g sin2 ðfm ÞVth pð p2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1 þ k sin2 ðfm Þsin2 ðjÞ dj (1)2
1 þ g sin ðfm Þsin2 ðjÞ 1  sin2 ðfm Þsin2 ðjÞ00I(t) ¼ Iosin2{[Dtot  d(t)]/2},

1



ð p2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2
2
2
2
1 þ g sin ðfm Þsin ðjÞ 1 þ k sin ðfm Þsin ðjÞ
B
C

 dj

B
C
2
2
2
2
B 0
ðf
Þsin
ðjÞ
1
þ
n
sin
ðf
Þsin
ðjÞ
1

sin
n
oC
m
m
B p sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
C

Bð
ne C
B 2 1 þ g sin2 ðfm Þsin2 ðjÞÞð1 þ k sin2 ðfm Þsin2 ðjÞÞ
C
@
A
dj
2
2
ðð1  sin ðfm Þsin ðjÞÞ0(2)
where d is the cell thickness, fm is the tilt angle at the middle ofthe cell and the other terms of the reduced quantities are: g ¼ (3k/3t)  1, k ¼ (K33/K11)  1, n ¼ (n2e/n2o)  1, sin(f) ¼ sin(fm)sin(j).
The sample retardation at higher voltage is fitted to the above(3)

where Io is the maximum intensity change and Dtot is the total
phase difference. The optical phase difference d(t), for small
director distortion can be approximated as d(t) ¼ doexp(2t/so),
where do is total phase difference of liquid crystal under bias
voltage (Vb) that is not far from the Freedericksz thresholdvoltage (Vth). In the case where do is close to np, d(t) becomesd(t) ¼ doexp(4t/so). The slope of the plot ln[do/d(t)] with time (t)yields the relaxation time (so). The rotational viscosity (g1) of theliquid crystal is given by:
g1 ¼

so K11 p 2
d2(4)

where d is the thickness of liquid crystal sample. This procedure
was repeated in the entire nematic range for all the mixtures.

III.

nd
Df ¼ 2p elThis journal is ª The Royal Society of Chemistry 2011

equations by an iterative procedure to get the bend elastic
constant K33.20,25
To verify the anchoring condition we estimated the pretilt
angle (a) of the mixtures by putting the lower limit of the integration in eqn (1) and (2) as a fit parameter a. In all the mixtures
it was found to be less than 1 indicating the strong anchoring at
the surface hence a was assumed to be zero in further analysis. It
may be pointed out that a pure bent-core nematic also exhibitsa very low pretilt angle on the same alignment layer.13,26 Independent measurements were also carried out to measure thetemperature dependent anchoring energy of the mixtures which
also indicated strong anchoring and the details will be publishedelsewhere.
The phase-decay-time measurement technique was used tomeasure the rotational viscosity (g1). A signal generator (Tektronix, AFG3102) was used to apply voltage at a frequency of
4.111 kHz and a photomultiplier tube (Hamamatsu, H6780-01)
was used as a detector to measure the time dependent transmitted
intensity. A small voltage (Vb) corresponding to the first maxima
or minima was applied depending on transmission intensity suchthat the total phase retardation of sample was np, where n is an
integer. At time t ¼ 0, the bias voltage (Vb) was removed and therelaxation transmission intensity change of the liquid crystal cell
was measured with an oscilloscope (Tektronix, TDS 2012B).
The time dependent intensity at a particular temperature isgiven by:27

Results and discussion

We studied various mixtures of compounds 4-biphenylyl-400 -nundecycloxybenzoate
and1,3-phenylene-bis[4-(3-methylbenzoyloxy)]-40 -n-dodecylbiphenyl-40 -carboxylate. The chemical
structures of the compounds and their individual phase transition
temperatures are shown in Fig. 1(a). The mixture exhibits somenew phase transitions and the detailed phase diagram is reportedby Pratibha et al.28,29 The part of phase diagram showing NI phasetransition is reproduced (Fig. 1(b)) and is the interest of thepresent study. The NI transition temperature decreases slightly(Fig. 1(b)) and the N-SmA2 transition temperature increases withincreasing mol% of bent-core compound and SmA2b appears
beyond 4 mol%. Very recently we have made detailed calorimetric measurements and showed that the critical heat anomaly
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associated with the SmA2 to SmA2b transition is described witha Fisher-renormalized form of the usual scaling expression.30 The
temperature dependent variation of the birefringence (Dn) of thepure compound and various mixtures are shown in Fig. 2(a). Dn
jumps to 0.06 from 0 at NI transition in the pure as well as in all
the mixtures and increases with decreasing temperature in thenematic phase. Dn is almost constant in all the mixtures at a fixed
temperature which suggests that the orientation order, S (f Dn) is
not affected by the inclusion of the bent-core molecules. A
significant jump in Dn is also observed at the NS transition in all
the mixtures including the pure compound and indicate that theNS transition is first order and consistent with the calorimetricmeasurements.30 Further, the relative jump in Dn across the NS
transition appears to increase with increasing mol% of the bentcore compound and indicates the increase in the coupling of thesmectic order parameter j and fluctuation in the nematic orderparameter S as the McMillan parameter (a) increases from 0.96 to
0.97 in the present mixtures. Measurement of Dn, K11 and K33 of
mixture of the present bent-core compound with 8OCB werereported by Kundu et al.20 They reported inverse effect i.e., Dn
decreases significantly with increasing concentration of bent-corecompound.
The temperature dependent variation of dielectric anisotropy
for the various mixtures (D3 ¼ 3k  3t) is shown in Fig. 2(b). The
dielectric anisotropy is positive and relatively small and decreases
with increasing mol% of bent-core compound at a fixed
temperature. The resultant dipole moment of the rod-like

Fig. 2 Variation of (a) birefringence (Dn) and (b) dielectric anisotropy(D3) of the mixtures as a function of temperature. Continuous lines are
drawn as a guide to the eye.

8558 | Soft Matter, 2011, 7, 8556–8560

molecules is along the long axis whereas the bent-core molecules
have transverse dipole moments. The reduction in dielectricanisotropy with increasing concentration of the bent-corecompound in the nematic phase, keeping the orientational orderalmost the same, suggests that the long axes of the bent-coremolecules on an average are parallel to the long axes of the rodlike molecules as shown schematically in Fig. 3(a).
The optical phase difference (Df) of the sample was measured
as a function of applied voltage to measure splay (K11) and bend(K33) elastic constants at various temperatures. A typical variationof experimental data together with the best fit to eqn (1) and (2) isshown in Fig. 4(a). The temperature-dependent variation of splay(K11) and bend (K33) elastic constants are shown in Fig. 4(a) and
Fig. 4(b) respectively. K11 increases with decreasing temperature in
the nematic phase and show a little change of curvature as thesmectic phase is approached. Interestingly at a fixed temperature
there is no significant change in K11 with increasing mol% of thebent-core compound similar to Dn (Fig. 2a). Since Dn(f S) do notchange with the mol% of the bent-core compound, K11(f S2 f
Dn2) is also not affected. On the other hand K33 decreases significantly (Fig. 4(b)) with increasing concentration of the bent-corecompound. Variation of K11 and K33 at two temperatures namely
T  TNI ¼ 3.8 and 7.8 is shown in Fig. 4(b) (inset). K11 remains
constant and K33 is reduced by a factor of 1.7 and tends tosaturate as the concentration is increased from 0 to 10 mol%.
Measurement of K11 and K33 in the mixture of the present bentcore compound with 8OCB was also reported by Kundu et al. andthey found that both decreases with increasing concentration ofbent-core compound in the mixtures.20 The strikingly different
results i.e., both Dn and K11 remains constant in the present
mixtures is attributed to the structural similarity, i.e., the aromaticand aliphatic parts of the rod are matched with one half of thebow. Further as mentioned earlier the bow axes of the bent-coremolecules are parallel to the long axes of the rod-like molecules(Fig. 3(a)), and the bent molecules do not facilitate splay fluctuations hence K11 remains almost the same for all mixtures. The
decrease of bend elastic constant (K33) is due to the coupling of thebent shape of the bent-core molecules with bent distortion as

Fig. 3 (a) Mutual alignment of rod-like and bent-core molecules innematic phase. Dotted region show a temporary cluster in the nematic
phase. Schematic representation of (b) splay and (c) bent distortion of thedirector in the mixtures.

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eqn (4) by measuring the relaxation time so and K11 of several
mixtures. The temperature-dependent variation of g1 is shown in
Fig. 5(b). g1 of the pure compound is relatively low andcomparable to many known calamitic compounds.32 It increases
with decreasing temperature in the nematic phase as expected
due to the increase in the orientational order parameter (S).33 In
the mixture g1 increases significantly compared to the purecompound with increasing mol% of bent-core compound. For
example, at two temperatures namely T  TNI ¼ 1.8 and 6.8 , itincreases by a factor of four when the concentration of the bentcore compound is increased from 0 to 10 mol% (Fig. 5(b), inset).
It may be mentioned that in the mixture the increase in g1 can besolely attributed to the increase in so since K11 almost remains
constant. The temperature dependent variation of g1 wasreported to be higher in the pure bent-core compounds thanconventional calamitic compounds.13,34,35 In some cases it wasreported to be 10 times higher and was explained due to theformation of temporary smectic clusters that originate from thebent-shape of the molecules.6,34 Existence of such clusters in
the nematic phase is revealed in dynamic light scattering36 andrheological studies.37 In the present study g1 is almost 4 times
larger even at a relatively small concentration (10 mol%) of thebent-core compound and expected to be contributed by such

Fig. 4 (a) Variation of splay elastic constant (K11) of various mixtures asa function of temperature. (Inset) Retardation as a function of applied
voltage at T  TNI ¼ 7 . Continuous green line is a theoretical fit to eqn(1) and (2). (b) Variation of bend elastic constant (K33) of various
mixtures as a function of temperature. (Inset) Variation of both K11 andK33 with the concentration of bent-core compound. Continuous lines are
drawn as a guide to the eye.

described by Sathyanarayana et al.5 They suggested that the strain
in the bent distortion is partly relieved due to the coupling and as
a result K33 can be significantly lower than K11. A schematic
representation of the splay and bend distortions in the mixture isshown in Fig. 3(b) and 3(c) respectively. The pretransitional
divergence of K33 in pure compound as the N-SmA2 transition isapproached is due to smectic short-range fluctuations in the sensethat the bent fluctuations are suppressed in the onset of smectic
short-range order. Such pretransitional divergence of K33 in the
mixture is also observed in other calamitic and unconventional
mesogens.5,19,25,31 Furthermore, it is noticed that the relative
divergence in K33 in the mixtures compared to the pure compoundas the N-SmA transition is approached is reduced with increasing
mol% of the bent-core compound. This is a consequence of the
reduced K33 in the sense that bent fluctuations are relatively larger
in the mixtures compared to the pure compound.
Finally we discuss the rotational viscosity (g1) of the mixturesin the nematic phase. The normalised intensity of the transient
response of various mixtures is shown in Fig. 5(a). Timedependence of the retardation was estimated from the normalised intensity by using eqn (3). A representative variation of ln[do/d(t)] with time (t) of experimental data (10 mol%) together withthe best fit is shown in Fig. 5(a)(inset). g1 was estimated from
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Fig. 5 (a) Time dependent normalised transmitted intensity after theremoval of the bias voltage Vb at T  TNI ¼ 2 for various mixtures.
(Inset) Linear variation of ln[do/d(t)] with time (t) at the same temperature
for the mixture with 10 mol % bent-core compound. (b) Variation of therotational viscosity (g1) of mixtures as a function of reduced temperature.
(Inset) Variation of g1 with the concentration of bent-core molecules at
two different reduced temperatures. Continuous lines are drawn asa guide to the eye.

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temporary clusters of bent-core molecules. A schematic representation of such temporary clusters in the nematic phase isdepicted in Fig. 3(a). The size and the number of clusters canincrease with increasing concentration of bent-core moleculesand g1 of the mixture can increase.

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IV. Conclusion
In conclusion, we have measured the birefringence, splay andbend elasticity and rotational viscosity of several mixtures of rodlike and bent-core molecules in which the aromatic and aliphatic
parts of the rod are matched with one half of the bent-coremolecule. We observed that the birefringence and hence theorientational order (S) remain the same in all mixtures. The
dielectric anisotropy decreases with an increase in concentrationof the bent-core molecules indicating that bow axes of bent-coremolecules align parallel to the long axes of rod-like molecules.
The splay elastic constant remains constant, whereas the bendelastic constant decreases significantly with increasing concentration of bent-core molecules suggesting the coupling of thebent-shape with the bend distortion. The increase in rotational
viscosity with concentration suggests the presence of temporary
smectic clusters of bent-core molecules in the mixtures.
It may be mentioned that the flexoelectric effect in such
mixtures could be significant and can affect K33 measurements. It
was shown by Brown et al.38 from voltage dependent dielectricmeasurements that it can slightly underestimate K33 when the
flexoelectric effect is neglected. Further it was shown by Kundu
et al.21 that the mixture of bent-core and rod-like nematic samples
exhibits homeotropic alignment because of a large positive selfenergy due to a high flexoelectric coefficient at higher concentrations of the bent-core molecules. In our present study the highest
concentration of bent-core molecules is higher than their mixtures
but no such effect was observed suggesting that the flexoelectriccoefficient may not be significantly large. In our experiment weused retardation data to measure K33 and found it is reduced bya factor of 1.8 times at 10 mol%. Such a large reduction in thepresent data can not be accounted for by merely the flexoelectriceffect. A detail theoretical investigation is needed to find the effectof flexoelectric as well as order electric polarization on the voltage
dependent retardation measurements.

Acknowledgements
One of the authors, S. D., gratefully acknowledges the support
from UGC-CAS School of Physics. P. Sathyanarayana
acknowledges CSIR-UGC for the fellowship. B. K. S. thanks the
Indian National Science Academy, New Delhi for financial
support.

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