RAPID COMMUNICATIONSPHYSICAL REVIEW E 87, 030501(R) (2013)

Active and passive viscosities of a bent-core nematic liquid crystal
S. Dhara,1,* Y. Balaji,1 J. Ananthaiah,1 P. Sathyanarayana,1 V. Ashoka,1 A. Spadlo,2 and R. Dabrowski2
2

1
School of Physics, University of Hyderabad, Hyderabad 500046, India
Institute of Chemistry, Military University of Technology, 00-908 Warsaw, Poland(Received 4 February 2013; published 29 March 2013)

We report the measurements of active and passive viscosities of a bent-core nematic liquid crystal. The active
viscosity is measured using a rheometer and the passive viscosities are measured by measuring the self-diffusion
coefficient of a microsphere in the aligned sample. The effective active viscosity is larger and the effect ofpresmectic fluctuations are observed much higher than the N –Sm-C transition temperature than commonly seen
in calamitic liquid crystals. The temperature dependence of passive viscosities is stronger than that of active
viscosities. The self-diffusion of the microsphere is reduced and appears to be non-Gaussian as the N –Sm-C
transition is approached. The study of Brownian fluctuations is useful to understand the smectic fluctuations inbent-core nematic liquid crystals.
DOI: 10.1103/PhysRevE.87.030501

PACS number(s): 64.70.mj, 83.80.Xz, 66.20.Ej

The viscosity of liquid crystals strongly depends on the
microscopic structure of the constituent molecules and themesophases. There are several theoretical and experimental
studies of the low molecular weight liquid crystals where themolecules have mostly cylindrical symmetry [1–3]. It is known
that the nematic N phase has three principal viscosities called
Miesowicz viscosities, namely, η1 , η2 , and η3 , depending on
the direction of the shear and the orientation of the liquid
crystal director (the average alignment direction of the longaxes of the molecules). A schematic representation of thedirector orientation in a calamitic (rodlike) nematic liquid
crystal with respect to the shear and corresponding viscosities
is shown in Fig. 1. These viscosities can be expressed in termsof Leslie coefficients α1 –α6 , which are used to define thenematic stress tensor. In nematic liquid crystals, both α2 andα3 are negative close to the nematic-isotropic (N -I ) transition
and the director √is aligned along the flow direction with anangle θ = tan−1 ( α2 /α3 ) [4,5]. However, if the compounds
also have a smectic-A (Sm-A) phase below the N phase, α3
is renormalized (i.e., α3R > 0) a few degrees below the N-I
transition temperature. As a result, the ratio α2 /α3 becomes
negative and the director changes its orientation to the neutral
direction (x axis). As the temperature is further reduced it has
been shown theoretically that γ̇ τ → 1, where γ̇ and τ are
the shear rate and relaxation time of the critical fluctuations
and consequently α3R tend to diverge a few degrees above the
N –Sm-A transition [6]. As a result, the director undergoes
precessional motion with an angular frequency ω0 along theneutral direction that can be described by the equation of anellipse n2y (t)/n2y0 + n2z (t)/n2z0 = 1, where ny (t) = ny0 cos(ω0 t)
and nz (t) = nz0 sin(ω0 t) are components of the director n(t) =[nx (t),ny (t),nz (t)]. Several steady state structures are observed
experimentally [6], depending on the relative strength of thedirector components, and are denoted by am (ny0 > nz0 ), as(ny0 = nz0 ), a(b) (ny0 < nz0 ), and ac (ny0 $ nz0 ). The above
structural features are also reflected in the static dielectricmeasurements on a few calamitic compounds exhibiting a
N–Sm-A phase transition [7,8]. Liquid crystals of bent-core*

Corresponding author: sdsp@uohyd.ernet.in

1539-3755/2013/87(3)/030501(5)molecules have created immense interests [9,10]. Usuallythese molecules have a strong tendency to form smectic phases.
However, they occasionally exhibit a nematic phase in which
the physical properties are reported to be significantly different
from conventional calamitic nematic liquid crystals [11–19].
The presence of temporarily fluctuating smectic clusters is also
evident in many experiments. In this context the measurementof flow or shear viscosity in the bent-core liquid crystals isvery rare. The major problem in such measurements is therequirement of a large sample, which in general is not easy
to get. In the present paper we report on the measurement ofactive and passive viscosities of a bent-core liquid crystal. We
show that the temperature-dependent behaviors of the active
and passive viscosities are noticeably different and the studyof self-diffusion of a microsphere is a sensitive technique toinvestigate the presmectic fluctuations in bent-core nematic
liquid crystals.
The chemical structure and the phase transition temperatures of the bent-core nematic (BCN) are shown in Fig. 2(a),which exhibits the following phase transitions (cooling): I at
176.4 ◦ C to N at 118.3 ◦ C to Sm-C at 101.3 ◦ C to Sm-X at
90.4 ◦ C to Sm-Y at 60.2 ◦ C in Cr. The purity of the compound
was confirmed by high-performance chromatography andother techniques. The active viscosity measurements weremade using a rheometer (Anton Paar MCR 501) in plate-conegeometry with a plate diameter of 50 mm and a cone angle of1◦ . The plates were not treated for any specific orientation of
the molecules. The temperature of the sample was controlledwith a Peltier temperature controller within an accuracy of
0.1 ◦ C. We used a videomicroscopy technique to measure theself-diffusion coefficients of a tiny silica microsphere (witha diameter of 0.98 µm) and hence the passive viscositiesof the sample parallel and perpendicular to the director.
To promote a specific molecular alignment on the surface
the silica microspheres were coated with octadecyldimethyl(3-trimethoxysilylpropyl) ammonium chloride (DMOAP) before mixing with the liquid crystal [20]. The Brownian
fluctuations of an isolated microsphere in a planar cell (d =23 µm) was recorded and the position was determined with thehelp of an appropriate computer program. The histogram ofthe microsphere displacements were fitted to Gaussian with aprobability P that the particle would diffuse a certain distance030501-1©2013 American Physical Society

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S. DHARA et al.

FIG. 1. (Color online) Schematic representation of the threefundamental director orientations in the nematic phase with respect
to the direction of shear. Miesowicz viscosities corresponding to eachorientation are designated by η3 , η2 , and η1 , respectively.

δ in the time interval τ is P (δ|τ ) = P0 (τ )exp(−δ 2 /(2 (τ ),where ((τ ) is the width of the distribution and P0 (τ ) is
the normalization constant. The self-diffusion coefficients ofthe microsphere parallel (D|| ) and perpendicular (D⊥ ) to thedirector are obtained by using the relation D||,⊥ = (2||,⊥ /4τ .
The corresponding viscosities are estimated by using the
Stokes-Einstein relation η||,⊥ = kB T /6π rD||,⊥ , where r is theradius of the microsphere [20].
We show the variation of shear stress with shear rate
in Fig. 2(b) at various temperatures in the nematic phase.
The sample was mounted on the rheometer in the nematic
phase at a temperature of 140 ◦ C. It is observed that theshear stress is proportional to the shear rate except in a low
shear rate range (γ̇ ! 2 s−1 ) where the stress is constant,

Sm-C

(2nd)
(1st)

FIG. 2. (Color online) (a) Chemical structure and the firsttwo phase transition temperatures of the bent-core compound.
(b) Variation of shear stress with shear rate at various temperatures.
The sample was mounted at 140 ◦ C. Here 1st and 2nd indicate thetwo consecutive measurements at the same temperature.

FIG. 3. (Color online) Temperature variation of ηeff in both 8OCB
and BCN compounds at a shear rate γ̇ = 50 s−1 . Upper and lower x
axes correspond to the temperature scale of the BCN (open squares)
and 8OCB (solid circles) samples, respectively. Common featuresindicating various steady state structures as described in the text are
indicated in the shaded regions. The notation used in the shaded
regions follows the notation that was used in Ref. [6] to describe thedifferent dynamical response regimes of 8CB. In the inset, the arrow
indicates a small slope change in the viscosity data of the BCN liquid
crystal.

suggesting a shear thinning behavior. This could be due tothe shear-induced realignment of the nematic director within
the domains that are separated by disclination lines [21]. Thiseffect is more prominent in the first measurement [140 ◦ C(1st)], i.e., immediately after mounting the sample. Beyondγ̇ ' 2 s−1 , the shear stress is proportional to the shear rate,
hence the BCN shows the behavior of a Newtonian fluid within
the experimental limit (γ̇ = 1000 s−1 ). The shear-dependent
viscosity of a bent-core nematic was also studied using ananoliter rheometer by Bailey et al. [22] in the shear rate
range of 500–10 000 s−1 . They reported Newtonian behavior
in the lower shear rate range (!2000 s−1 ) and non-Newtonian
behavior at higher shear rates. Thus our results are consistent
with their data in the lower range of the shear rates. In
Fig. 3(a) we show the variation of the effective viscosityηeff with shifted temperature (i.e., T − TN-I ) at a shear rateγ̇ = 50 s−1 . The value of TN-I was determined from thetemperature-dependent measurement of the viscosity. Herethe effective viscosity means the bulk viscosity measured in
the cone-plate arrangement. We notice that ηeff is reduced
below the N -I phase transition and increases with decreasing
temperature. At T − TN-I = −58 ◦ C, it sharply changes slope
and increases very rapidly, indicating a N to Sm-C phasetransition. For the purpose of comparison we also measuredηeff for the 4( -octyloxy-4-cyanobiphenyl (8OCB) sample andpresent the results in Fig. 3. The behavior of ηeff has somecommon features in both compounds. It is reduced below theN -I transition, increases with decreasing temperature, anddiverges at the N to Sm-A–Sm-C transition. In addition there
is a small slope change in the viscosity data near the N to
Sm-A–Sm-C transition in both samples. The data in the case

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ACTIVE AND PASSIVE VISCOSITIES OF A BENT-CORE . . .

of the BCN are shown in the inset of Fig. 3 for clarity. In
contrast, the ηeff behavior has at least two uncommon anddistinct features. First, ηeff in the BCN is significantly larger
8OCB
than in 8OCB. For example, at T − TN-I = −2 ◦ C, ηeff=
BCN
10.2 mPa s and ηeff = 18.7 mPa s for 8OCB and BCN liquid
crystals, respectively. Assuming we are essentially measuring
η2 (since this temperature is close to N -I ), we assert that thisviscosity is almost twice as high in the BCN than in the 8OCB
liquid crystal. Second, the small slope change in ηeff of the
BCN (Fig. 3) appears at a much higher temperature than thatin the 8OCB liquid crystal. For example, it appears at 24.5 ◦ C
and 2 ◦ C above the N to Sm-A–Sm-C phase transition for the
BCN and 8OCB samples, respectively. A similar slope change
in the viscosity data of 4( -octyl-4- cyanobiphenyl (8CB) in
conjugation with x-ray [6] and rheodielectric data [23] was
identified as the onset of precessional motion of the director
at which γ̇ τ → 1 and several steady state structures appear as
discussed in the previous section. Considering the similarity
to the previously measured viscosity data of 8CB and 8OCB,
we conjecture that the precessional motion of the director in
the present BCN sample begins far away, i.e., 24.5 ◦ C above
the N –Sm-C transition. Previously, some studies involving
x-ray measurements indicated that this compound exhibits
a uniaxial to biaxial nematic transition around T − TN-I '−30 ◦ C [24,25]. However, concurrently various experiments
indicated that this is due to the onset of smectic fluctuations
and no phase biaxiality was observed [26,27]. Moreover,recently, in the same compound around a similar temperature
we observed that the bend elastic constant as well as the rotational viscosity starts to increase more rapidly with the temperature [18,28]. We believe that this small slope change(Fig. 3, inset) indicates the temperature at which γ̇ τ → 1.
The study of Brownian fluctuations helps us understand theeffect of smectic fluctuations on the passive viscosity of BCNs.
The histograms of the particle displacement δ|r(t + τ ) − r(t)|in both x (||n) and y (⊥ n) are shown in Fig. 4. It is evident
that (|| > (⊥ , implying D|| > D⊥ , as expected. The ratio
D|| /D⊥ varies with temperature and changes from 1.4 to 1.7,
which is close to the number (1.6) reported in calamitic liquid

FIG. 4. (Color online) Histogram of particle displacements parallel (circles) and perpendicular (stars) to the director for τ = 1 s. The
solid lines are Gaussian fits.

FIG. 5. (Color online) (a) Variation of ηeff (stars), η|| (squares),and η⊥ (circles) as a function of shifted temperature. The inset shows
the two-dimensional projection of trajectories of a microsphere closer
to the N -I transition (T − TN-I = −7.6 ◦ C) and closer to the N –SmC transition (T − TN-I = −47.6 ◦ C).

crystal, e.g, 4( -pentyl-4- cyanobiphenyl with dipole defects[29]. In Fig. 5 we show the temperature variation of η|| andη⊥ of the BCN liquid crystal as determined from D|| andD⊥ using the Stokes-Einstein relation. We note that in theisotropic phase η|| = η⊥ and is comparable to ηeff measured
in the rheometer. Just below the N -I transition the director isaligned along the flow direction and η|| is nearly equal to the
Miesowicz viscosity, i.e., η|| ∼= η2 = (α3 + α4 + α6 )/2. In thenematic phase η|| < η⊥ (∼= η3 = α4 /2) and both increase as thetemperature is reduced; ηeff also increases with temperature,
but the relative rate of increase of the former two is larger. Two
considerations can be made to understand their temperaturedependence. First, apart from the temperature effect, theviscosity can also increase as the smectic phase is approached
due to the occurrence of temporary smectic clusters in the bulk.
Second, the colloids are coated with DMOAP, which promotes
homeotropic alignment of the bent-core molecules. On thesurface of the microsphere the free rotation of the moleculesalong the long axis is restricted due to steric hindrance;
as a result they are pinned in a facilitating formation of
permanent smectic clusters or layers around the microspheres.
As a result, the hydrodynamic radius of the microsphere
could be significantly larger than the actual radius due to thelarge cloud of the distorted nematic around the microsphere.
Assuming η|| = ηeff = 21.6 mPa s near the N -I transition(T − TN-I = −18 ◦ C), the estimated hydrodynamic radius of
the microsphere is 0.68 µm, which is about 0.2 µm larger
than the actual radius. A schematic molecular orientation ofbent-core molecules and some clusters around the microspherewith a dipolar defect is shown in Fig. 6. In fact, it was
shown that both diffusion coefficients (D|| and D⊥ ∝ 1/η)
decreases when the size of the microsphere is greater than
0.5 µm, which is attributed to the effective size of the
microsphere, which is much bigger than the geometric size[20]. In addition, the smectic layering effect can increase
with decreasing temperature due to a kind of condensationof the clusters around the microsphere effectively increasing

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S. DHARA et al.

FIG. 6. (Color online) Schematic molecular orientation aroundthe microsphere with a dipolar defect configuration. It may be notedthat the clusters attached to the microsphere are permanent, whereas
they are temporal in nature in the bulk. The small dot near the top
indicates a hyperbolic hedgehog.

the diameter of the microsphere with decreasing temperature.
As the temperature is further reduced and we approach the
Sm-C phase, interestingly we notice that η|| tends to diverge
around 24 ◦ C above the N –Sm-C transition. To understandthis we look at the Brownian trajectory of a microsphere. The
two-dimensional projection of the trajectories of a microspherenear N-I (T − TN-I = −7.6 ◦ C) and N –Sm-C (T − TN-I =−47.6 ◦ C) transitions are shown in the inset of Fig. 5. We noticethat the fluctuations are highly restricted near the N –Sm-C
transition compared to the N -I transition. Since the cell isrubbed, it is expected that the smectic layer planes in thetemporarily fluctuating clusters are oriented perpendicular to
the rubbing direction. In this case the fluctuating clusters arelarge enough and the self-diffusion of the microparticle inthe direction perpendicular to the layer plane is expected to
be lower than that in parallel direction and as a result η|| candiverge.
Finally, we would like to comment on the Brownian
fluctuations near and far from the N–Sm-C transitions forlonger time intervals (3 s). In Fig. 7 we show the histogramsof the Brownian fluctuations of the microsphere parallel andperpendicular to the director at a time interval τ = 3 s. The
perpendicular component appears to fit reasonably well with

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FIG. 7. (Color online) (a) Histograms of particle displacement
parallel (||) and perpendicular (⊥) to the director for τ = 3 s. The
solid lines are Gaussian fits to the data. The tail parts that significantly
deviate from the fittings are highlighted by shaded regions.

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We gratefully acknowledge the support from the DST(Grant No. SR/NM/NS-134/2010) and CSIR (Grant No.
03(1207)/12/EMR-II and PURSE, UOH). Y.B. acknowledges
ACRHEM for financial support.

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