Soft Matter
PAPER

Cite this: Soft Matter, 2020,
16, 7556

Microrheology to probe smectic clusters inbent-core nematic liquid crystals†
Sathyanarayana Paladugu, *ab Supreet Kaur, c Golam Mohiuddin,
Ravi Kumar Pujala, b Santanu Kumar Pal c and Surajit Dhara *acMany bent-core nematic liquid crystals exhibit unusual physical properties due to the presence ofsmectic clusters, known as ‘‘cybotactic’’ clusters, in the nematic phase. Here, we investigate the eﬀect ofthese clusters on the complex shear modulus (G*(o)) of two asymmetric bent-core liquid crystals using
a microrheological technique. The compound with a shorter hydrocarbon chain (8OCH3) exhibits only anematic (N) phase whereas the compound with a longer chain (16OCH3) exhibits both nematic (N) andReceived 2nd May 2020,
Accepted 1st July 2020

smectic-A (SmA) phases. The rheological results are correlated with the measurements of curvature

DOI: 10.1039/d0sm00796j

N phase transition temperature, is strikingly diﬀerent than that of 8OCH3, owing to the smectic clusters.

rsc.li/soft-matter-journal

An approximate size of the clusters is estimated using a simple model. Therefore, microrheological studies on
bent-core nematic liquid crystals are very useful in extracting information about underlying smectic clusters.

elastic constants. Our results show that the directional shear modulus of 16OCH3, just above the SmA to

1 Introduction
Bent-core (BC) liquid crystals have captivated the interest of
many scientists as they continue to exhibit new and interesting
physical properties compared to conventional calamitic liquid
crystals made of rod-like molecules.1,2 For example, bent-corenematic liquid crystals exhibit negative bend-splay elastic
anisotropy3–10 i.e., dK31 = K33  K11 o 0, giant flexoelectricity,11–13
large rotational viscosity,4,14–16 and biaxiality.17–24 Since the freerotation of the molecules around the bow axis is hindered, BC
molecules have a strong tendency to form nano-sized smectic
clusters, known as cybotactic clusters.3,25–31 The unusual properties, some of which are reported above, have been attributed to theeffect of these clusters. The size of the stable smectic clusters has
been measured experimentally using cryo-transmission electron
microscopy (cryo-TEM).32 These smectic clusters are expected tooffer both bending modulus and layer compressional modulus
which would impact the complex shear modulus of the nematic
phase. However, measurements using conventional rheological
techniques involving bulk samples would smear out their effect.
Therefore, microrheological techniques, involving a very small
amount of sample (typically a few microlitres), are important.33

In this paper, we report on the measurements of flow viscosities
and the complex directional shear modulus and correlate
the results with the splay and bend elastic constants of two
asymmetric bent-core liquid crystals. Our study shows thatmicrorheological approach using self-diffusing microparticles
is a sensitive technique to investigate the cybotactic clusters inbent-core nematic (BCN) liquid crystals.

2 Materials and methods
2.1

Materials

Two homologous bent-core compounds, namely 8OCH3 and
16OCH3, with non-polar (–OCH3) moieties were synthesized in
our laboratory, where 8 and 16 represents the length of hydrocarbon chain R (see Fig. 1). The details of the synthesis werereported previously by us.34 The chemical structures of themolecules and the phase transition temperatures of the compounds are shown in Fig. 1. They exhibit the following phaseaSchool of Physics, University of Hyderabad, Hyderabad, India.
E-mail: sathyapaladugu@gmail.com, sdsp@uohyd.ernet.inbSoft and Active Matter Group, Department of Physics, Indian Institute of Science
Education and Research (IISER), Tirupati, Andhra Pradesh 517507, Indiac
Department of Chemical Sciences, Indian Institute of Science Education andResearch (IISER) Mohali, Sector-81, Knowledge City, Manauli, 140306, India† Electronic supplementary information (ESI) available. See DOI: 10.1039/
d0sm00796j

7556 | Soft Matter, 2020, 16, 7556--7561

Fig. 1 Chemical structures of the asymmetric bent-core molecules andphase transition temperatures.

This journal is © The Royal Society of Chemistry 2020

Papertransitions in cooling: 8OCH3: I 150.1 1C N 66.7 1C Cr and
16OCH3: I 126.7 1C N 97.7 1C SmA 73.4 1C Cr.

2.2

Experimental

The experimental cells for aligning the liquid crystal director
were prepared with two patterned indium-tin-oxide (ITO)
coated glass plates. The ITO plates were spin coated withpolyimide AL1254 and baked at 180 1C for 1 h. The AL1254
coated glass plates were then rubbed with a rubbing machine
and assembled in an antiparallel way to ensure uniform planaralignment of the director. The glass plates were glued together
with UV-curable adhesive (NOA65-Norland Products, Inc.) withsilica beads as spacers. The thickness of the cells (d) was
measured using an interferometric method and the typical cell
thickness used in the study was 8 mm.
The dielectric constant as a function of voltage (0.02–20 V) at
different temperatures in the nematic phase was measured at
4.111 kHz with an LCR meter (Agilent 4980A). The perpendicular component of the dielectric constant (e>) was obtained
from the dielectric measurement below the Freedericksz
threshold voltage (Vth) and the parallel component (e8) was
obtained by extrapolating the voltage dependent dielectric
constant to V - N, i.e., 1/V - 0. Simultaneously, the voltage
dependent optical retardation was measured with the help of ahome built electrooptic setup, consisting of a photo-elastic
modulator (Hinds Instruments) and a lock-in amplifier (SRS-830).
The experimental diagram of the setup is shown in Fig. 2(a). From
the Freedericksz threshold voltage (Vth) of the retardation measurement, the splay elastic constant (K11) was obtained, using therelation K11 = e0De(Vth/p)2, where De = e8  e> is the dielectricanisotropy. The bend elastic constant (K33) was obtained by fitting
the voltage dependent optical retardation, following the method

Soft Matterdescribed in ref. 35–37. The accuracy in the measurement of
the elastic constant is within 6%.
The flow viscosities parallel (Z8) and perpendicular (Z>) to
the director were measured by measuring the self-diﬀusion of asilica particle of diameter 3 mm.38,39 Before dispersing in LCs,
the particles were treated with octadecyldimethyl(3-trimethoxysilylpropyl)ammonium chloride (DMOAP), which aligns the
director normal to the surface.40 Normal anchoring induces apoint defect and stabilises a dipolar director field surrounding
the particles, as a result of which the particles are levitated in
the bulk due to elastic repulsion from the surface.41 This isadvantageous as a long duration (about several hours) of datacollection is required. The dynamics of the Brownian motion of
an isolated silica particle in the BCN was recorded through a
60 water immersion objective using a CCD camera (iDs-UI)
connected to a Nikon inverted microscope with a frame rate of
20 fps.42 The diagram of the experimental setup is shown in
Fig. 2(b). The trajectory of a particle is extracted from therecorded sequence of images using a Python routine known as
Trackpy.43 The mean squared displacements (MSDs) parallel andperpendicular to the director were measured with the lag time t.
The diﬀusion coeﬃcients parallel (D8) and perpendicular (D>) to
the director were measured from the respective MSDs as afunction of the lag time t, using the equations h(x(t + t) x(t))2i = 2D8t and h(y(t + t)  y(t))2i = 2D>t. The corresponding
viscosities are calculated using the Stokes–Einstein equation,
Z8,> = kBT/6pRD8,>, where kB is the Boltzmann constant, T is theabsolute temperature and R is the radius of the microparticle.40
The complex viscoelastic shear modulus G*(o) = G 0 (o) +iG00 (o), where G 0 (o) is the storage modulus and G00 (o) is the loss
modulus in the frequency (o) domain, was calculated using thegeneralized Stokes–Einstein equation (GSER).44 The GSER in
the Fourier domain can be expressed as:G ðoÞ ¼ G0 ðoÞ þ iG00 ðoÞ ¼

kB T
pRioFf o Dr2 ðtÞ 4 g(1)

where F denotes the Fourier transform and R is the radius of
the microparticle. The parallel (G8*(o)) and perpendicularcomponents (G>*(o)) of the complex shear modulus G*(o)were obtained from the mean squared displacement (MSD)
parallel and perpendicular to the nematic director by thekB T
and G? ðoÞ ¼
equations Gk ðoÞ ¼
2pRioFf o Dx2 ðtÞ 4 g
kB T, respectively.45
2pRioFf o Dy2 ðtÞ 4 g

3 Results and discussion
3.1

Fig. 2 (a) Schematic diagram of the experimental setup for measuring
birefringence, and splay and bend elastic constants. (b) Setup for microrheological studies. (c) Trajectories and time evolution of the MSD (meansquared displacement) of a microparticle at 99 1C and 110 1C of compound 16OCH3. The diﬀusion coeﬃcients D8, D> at 110 1C and 99 1C (in1015 m2 s1) are 1.90, 1.46 and 0.32, 0.48, respectively.

This journal is © The Royal Society of Chemistry 2020

Splay and bend elastic constants

We begin with the discussion of the splay (K11) and bend (K33)
elastic constants of the compounds. Both the compounds
exhibit positive birefringence (Dn) and dielectric anisotropy(De) (see the ESI†). The temperature dependence of K11 andK33 of both the compounds is shown in Fig. 3(a) and (b). The
bend-splay anisotropy (dK31 = K33  K11) of compound 8OCH3 isnegative in the entire nematic range and its magnitude continues

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Paper
where Kni, n = 2, 3, 4 are fitting parameters. The continuous line in
the inset of Fig. 3(a) is a theoretical fit to eqn (2). Considering that
the K(2)–(4)s are temperature independent, the fitting parametersi(measured in pN) for 8OCH3 are found to be: K(2)1 = 14.3  1.1,
(4)
(2)
(3)
K(3)
1 = 13.7  1.2, K1 = 0.7  0.1, and K3 = 6.4  1.1, K3 = 17.2 (4)
1.2, K3 = 1.8  0.1.
The temperature dependence of K11 and K33 of compound
16OCH3 is shown in Fig. 3(b). In this compound, the bend-splay
anisotropy, dK31 = K33  K11, is also negative, except very close
to the N–SmA phase transition temperature. In particular, as
the N–SmA transition is approached, dK31 tends to change signfrom negative to positive. This feature is explained by the strong
pre-transitional divergence of K33 as the bend deformation isincompatible with the equidistance of smectic layers. The dependence of Kii on S is shown in the inset of Fig. 3(b). Excellent fits are
obtained with the fit-parameters for 16OCH3 (measured in pN):
(3)
(4)
K (2)
1 = 46.5  1.1, K 1 = 67.5  1.2, K 1 = 26.3  0.1 and
(2)
(3)
(4)
K 3 = 30.9  1.1, K 3 = 48.1  1.2, K 3 = 10.8  0.1.
3.2

Fig. 3 (a) Temperature dependent splay (K11) and bend (K33) elastic
constants of 8OCH3. (Inset) Kii as a function of order parameter S.
(b) Temperature dependent K11 and K33 of 16OCH3. (Inset) Kii as a functionof order parameter S. Continuous lines are theoretical fits to eqn (2).

to increase with decreasing temperature. Similar negative anisotropy has been reported in many symmetric and asymmetric bentcore nematic liquid crystals.8,10,15,46 The negative anisotropy is
attributed to the coupling of the bent-shape of the molecules withthe bend distortion.3 For most of the liquid crystals made of rodlike molecules, Kii p S2, where S is the orientational orderparameter. To see how Kii of these compounds depends on theorder parameter, we plot Kii as a function of S as shown in the inset
of Fig. 3(a). The order parameter S was obtained from the temperature dependent birefringence (ESI†). It is noticed that K33
shows antagonistic dependence on the order parameter and simply
can not be fitted to second-order in S. In particular, K33 of 16OCH3
tends to decrease after a pronounced maximum. The dependenceof Kii on S could be better explained using more detailed mean-fieldcalculations given by Berreman and Meiboom.47 They introduce athird-order dependence on S given by:
ð2Þ

ð3Þ

ð4Þ

Kii ¼ Ki S 2 þ Ki S3 þ Ki

7558 | Soft Matter, 2020, 16, 7556--7561

S4
ð1  SÞ2(2)

Flow viscosities

In what follows we measured the flow viscosities of thesecompounds from the self-diﬀusion of microparticles following
the technique described in Section 2.2. The measurements are
restricted only to the nematic phase as the particles tend to
sediment quickly in the isotropic phase. Some representative
trajectories of a microparticle and corresponding MSDs at two
temperatures, namely 99 1C and 110 1C, for compound 16OCH3
are shown in Fig. 2(c). It is found that at 110 1C, D||/D> C 1.3,
whereas at 99 1C, which is very close to the N–SmA transition
temperature, D||/D> C 0.7.
The temperature dependent viscosities parallel and perpendicular to the director, namely Z8 and Z>, of compound 8OCH3 below
the isotropic to nematic transition temperature (TNI) are shown in
the inset of Fig. 4. Both Z8 and Z> increase with decreasing
temperature as expected. In the entire nematic range Z> 4 Z8,thus the viscosity anisotropy (dZ = Z>  Z8) is positive. For example,
at a temperature T = 70 1C, dZ = Z>  Z8 C 100 mPa s.
The temperature dependent viscosities of compound
16OCH3 in the nematic phase are shown in Fig. 4. Both Z8
and Z> increase with decreasing temperature as expected.
However, the rate of increase of the viscosities at lower temperatures of 16OCH3 is much faster than that of 8OCH3. This is
expected due to the increasing amplitude of fluctuations in
smectic order as the N–SmA transition is approached, similar to
those observed in ordinary rod-like LCs showing the sametransition. Interestingly, it is found that for 16OCH3, below
T E 104 1C, dZ changes sign from positive to negative. For
example, at T = 114 1C, dZ C 40 mPa s, whereas at T = 98 1C,
dZ C 110 mPa s. The sign change of the anisotropy can notsimply be explained based on the presmectic fluctuations asthat would affect both the viscosities equally. We attribute
this effect to the cybotactic clusters of compound 16OCH3.
These clusters are stable compared to the unstable clusters inordinary liquid crystals with rod-like molecules as the freerotation of the former around the bow axis is reduced sufficiently due to the steric hindrance. Moreover, the size and theThis journal is © The Royal Society of Chemistry 2020

PaperSoft Matter

Fig. 4 Temperature dependent flow viscosities parallel (Z||) and perpendicular (Z>) to the director of 16OCH3. (Inset) Temperature dependent Z||
and Z> of 8OCH3. Error bars represent the standard deviation of the meanvalue. TNI indicates the isotropic to nematic phase transition temperature.

volume fraction of these clusters near the N to SmA transition
are expected to grow. The resulting smectic planes of theclusters are oriented perpendicular to the director in a planarcell. These oriented clusters effectively reduce the self-diffusionof the microparticle parallel to the director (perpendicular tothe smectic planes) compared to the perpendicular direction(parallel to the smectic planes), resulting in negative viscosity
anisotropy. Further comparative studies on cluster size andstability between bent-core and ordinary rod-like LCs exhibiting
an N–SmA transition could be reassuring.
3.3

Complex shear modulus

The temperature dependent elastic constants and flow viscosities demonstrated the presence of stable cybotactic clusters in
the nematic phase of compound 16OCH3 that contributes
greatly to these properties, especially near the N to SmA phasetransition. As a next step, we investigate the eﬀect of theseclusters on the complex shear modulus G*(o). Fig. 5 and 6 show
the frequency dependence of the storage (G|| 0 (o) and G> 0 (o))
and loss modulus (G800 (o) and G>00 (o)) parallel and perpendicular to the nematic director at two diﬀerent temperatures.
Overall, both the parallel and perpendicular components
exhibit fluid-like behaviour: a storage modulus lower than thecorresponding loss modulus below a critical frequency. In the
low frequency range one observes a behaviour typical of a
Maxwell-fluid: G8,>00 (o) p o and G8,> 0 (o) po2 (see Fig. 5(a)and 6(a)).
In the nematic phase of 8OCH3 the perpendicular components
are greater than the corresponding parallel components, i.e.,
G> 0 (o) 4 G|| 0 (o) and G>00 (o) 4 G800 (o) (see Fig. 5(a) and (b)).
Similar behaviour is also seen in compound 16OCH3 at T =120 1C (Fig. 6(a)). However, at lower temperature (T = 100 1C),
just above the N–SmA transition temperature, the trend isquite opposite when the relative magnitudes of the parallel

This journal is © The Royal Society of Chemistry 2020

Fig. 5 Frequency dependent storage (G 0 (o)) and loss modulus (G00 (o)) of
8OCH3. The subscripts || and > denote the components parallel andperpendicular to the director at (a) 130 1C and (b) 90 1C. A and B represent
the exponents of o corresponding to the loss and storage moduli,
respectively, in the Maxwell-fluid model.

and perpendicular components are compared. In particular,
G|| 0 (o) 4 G> 0 (o) and also G800 (o) 4 G>00 (o) (see Fig. 6(b)),
which is just opposite to the behaviour seen at higher temperature(T = 120 1C). In addition, the magnitudes of the directional moduli at
T = 100 1C are almost one order of magnitude larger than thosemeasured at T = 120 1C. To bring out the contrasting behaviour
we show the ratios G80 (o)/G>0 (o) and G800 (o)/G>00 (o) at a fixed o(2 rad s1) in Tables 1 and 2. In compound 8OCH3, at two different
temperatures G80 (o)/G>0 (o) o 1 and also G800 (o)/G>00 (o) o 1(Table 1). For compound 16OCH3 at higher temperature (T =120 1C), both the ratios are also less than 1 (Table 2). However, atlower temperature (T = 100 1C), G80 (o)/G>0 (o) 4 1 and G800 (o)/G>00 (o) 4 1, which are opposite to the features observed at T =120 1C. Thus, the parallel components of the complex viscoelastic
shear modulus of 16OCH3 exceed the perpendicular components
just above the N–SmA phase transition temperature.
This distinct feature in the nematic phase can be explained
considering the contribution of stable cybotactic clusters.

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consequently the contribution due to the layer compressional
modulus B is negligible. As the temperature is reduced towards
the N–SmA transition, the clusters become stable and their size
and the volume fraction increase,50 as a result of which G 0
is enhanced. Taking G 0 = 2 Pa (Fig. 6(b)), K = 5 pN (Fig. 3(b)),
C E 0.249 and assuming a typical layer compressional modulus
B E 106 Pa,51 the calculated cluster size L E 50 nm, which isvery close to the size measured by cryo-transmission electron
microscopy (cryo-TEM) in similar bent-core liquid crystals.32

4 Conclusions
In conclusion, we studied the elastic properties, flow viscosities
and directional shear modulus of two bent-core liquid crystals.
The bend-splay anisotropy (dK31 = K33  K11) of compound
8OCH3 is negative at all temperatures, whereas, for 16OCH3,
dK31 tends to change sign just above the SmA–N transition
temperature. For compound 8OCH3, the flow viscosity anisotropy dZ = (Z>  Z||) 4 0 at all temperatures. In the case of
16OCH3 it changes sign just above the SmA–N phase transition
temperature. The parallel components of the complex shear
modulus of compound 16OCH3 exceed the perpendicular components, just above the N–SmA phase transition temperature.
This special feature is explained considering the contributionof the layer compressional modulus of stable smectic clusters
in the nematic phase. Our experiment provided an approximate
estimate of the size of the smectic clusters which is comparable
to the size measured by using the cryo-TEM. The microrheology
experiments require a few microlitres of sample and can beused for probing the effects of nanoscale heterogeneity due toshort-range order in liquid crystals and in other soft materials.
Fig. 6 Frequency dependent storage (G 0 (o)) and loss modulus (G00 (o)) of
16OCH3. The subscripts || and > denote the components parallel andperpendicular to the director at (a) 120 1C and (b) 100 1C. A and B
represent the exponents of o corresponding to the loss and storage
moduli, respectively, in the Maxwell-fluid model.

Conflicts of interest

Table 1 Ratio of the real and imaginary components parallel andperpendicular to the director of compound 8OCH3 at o = 2 [rad s1]

Acknowledgements

Temp. [1C]G8 0 (o)/G> 0 (o)G800 (o)/G>00 (o)

130
90

0.5
0.3

0.6
0.6

Table 2 Ratio of the real and imaginary components parallel andperpendicular to the director of compound 16OCH3 at o = 2 [rad s1]

Temp. [1C]G8 0 (o)/G> 0 (o)G800 (o)/G>00 (o)

120
100

0.1
1.5

0.6
1.3

In analogy with smectic droplets, the storage modulus of thepﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
cybotactic clusters can be written as G0 ¼ C ðKB=LÞ where K is
the bending modulus, B is the layer compressional modulus,
L is the cluster size and C is a constant.48,49 At higher temperature (near the NI transition) the clusters are unstable and7560 | Soft Matter, 2020, 16, 7556--7561

There are no conflicts to declare.

SD acknowledges financial support from SERB (Ref. No. CRG/
2019/000425) and DST-FIST-II, School of Physics. SKP thanks
SERB (file no. CRG/2019/000901/OC) for funding. SP thanks
IISER Tirupati for the fellowship and research support. RKP
acknowledges Department of Science and Technology, India,
for INSPIRE Faculty award grant DST/INSPIRE/04/2016/002370.
SK acknowledges IISER Mohali for a PhD fellowship. GM
acknowledges DST-SERB, India, for NPDF File No. (NPDF/
2016/000560).

Notes and references
1 H. Takezoe and Y. Takanishi, Jpn. J. Appl. Phys., 2006, 45,
597–625.
2 A. Jákli, O. D. Lavrentovich and J. V. Selinger, Rev. Mod.
Phys., 2018, 90, 045004.

This journal is © The Royal Society of Chemistry 2020

Paper
3 P. Sathyanarayana, M. Mathew, Q. Li, V. S. S. Sastry,
B. Kundu, K. V. Le, H. Takezoe and S. Dhara, Phys. Rev. E:
Stat., Nonlinear, Soft Matter Phys., 2010, 81, 010702.
4 P. Sathyanarayana, S. Radhika, B. K. Sadashiva andS. Dhara, Soft Matter, 2012, 8, 2322–2327.
5 V. Görtz, C. Southern, N. W. Roberts, H. F. Gleeson andJ. W. Goodby, Soft Matter, 2009, 5, 463–471.
6 P. Tadapatri, U. S. Hiremath, C. V. Yelamaggad and
K. S. Krishnamurthy, J. Phys. Chem. B, 2010, 114, 1745–1750.
7 M. Majumdar, P. Salamon, A. Jákli, J. T. Gleeson and
S. Sprunt, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys.,
2011, 83, 031701.
8 S. Kaur, J. Addis, C. Greco, A. Ferrarini, V. Görtz, J. W. Goodby
and H. F. Gleeson, Phys. Rev. E: Stat., Nonlinear, Soft Matter
Phys., 2012, 86, 041703.
9 S. Kaur, G. Mohiuddin, P. Satapathy, R. Nandi, V. Punjani,
S. K. Prasad and S. K. Pal, Mol. Syst. Des. Eng., 2018, 3, 839–852.
10 N. Avci, V. Borshch, D. D. Sarkar, R. Deb, G. Venkatesh,
T. Turiv, S. V. Shiyanovskii, N. V. S. Rao and O. D. Lavrentovich,Soft Matter, 2013, 9, 1066–1075.
11 J. Harden, B. Mbanga, N. Éber, K. Fodor-Csorba, S. Sprunt,
J. T. Gleeson and A. Jákli, Phys. Rev. Lett., 2006, 97, 157802.
12 P. Sathyanarayana and S. Dhara, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2013, 87, 012506.
13 K. V. Le, F. Araoka, K. Fodor-Csorba, K. Ishikawa and
H. Takezoe, Liq. Cryst., 2009, 36, 1119–1124.
14 P. Sathyanarayana, T. A. Kumar, V. S. S. Sastry, M. Mathews,
Q. Li, H. Takezoe and S. Dhara, Appl. Phys. Express, 2010,
3, 091702.
15 P. Sathyanarayana, V. S. R. Jampani, M. Skarabot, I. Musevic,
K. V. Le, H. Takezoe and S. Dhara, Phys. Rev. E: Stat., Nonlinear,
Soft Matter Phys., 2012, 85, 011702.
16 E. Dorjgotov, K. Fodor-Csorba, J. T. Gleeson, S. Sprunt andA. Jákli, Liq. Cryst., 2008, 35, 149–155.
17 J. A. Olivares, S. Stojadinovic, T. Dingemans, S. Sprunt andA. Jákli, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2003,
68, 041704.
18 B. R. Acharya, A. Primak and S. Kumar, Phys. Rev. Lett., 2004,
92, 145506.
19 L. A. Madsen, T. J. Dingemans, M. Nakata and E. T. Samulski,
Phys. Rev. Lett., 2004, 92, 145505.
20 Y. Jang, V. P. Panov, A. Kocot, J. K. Vij, A. Lehmann and
C. Tschierske, Appl. Phys. Lett., 2009, 95, 183304.
21 Y. Xiang, J. W. Goodby, V. Görtz and H. F. Gleeson, Appl.
Phys. Lett., 2009, 94, 193507.
22 M. Lehmann, Liq. Cryst., 2011, 38, 1389–1405.
23 H. Yoon, S.-W. Kang, M. Lehmann, J. O. Park, M. Srinivasarao
and S. Kumar, Soft Matter, 2011, 7, 8770–8775.
24 T. B. T. To, T. J. Sluckin and G. R. Luckhurst, Phys. Rev. E:
Stat., Nonlinear, Soft Matter Phys., 2013, 88, 062506.
25 S. Stojadinovic, A. Adorjan, S. Sprunt, H. Sawade and A. Jákli,
Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2002, 66, 060701.

This journal is © The Royal Society of Chemistry 2020

Soft Matter
26 P. Sathyanarayana, B. K. Sadashiva and S. Dhara, Soft
Matter, 2011, 7, 8556–8560.
27 C. Tschierske and D. J. Photinos, J. Mater. Chem., 2010, 20,4263–4294.
28 R. Y. Dong, Int. J. Mod. Phys. B, 2010, 24, 4641–4682.
29 C. Keith, A. Lehmann, U. Baumeister, M. Prehm and
C. Tschierske, Soft Matter, 2010, 6, 1704–1721.
30 O. Francescangeli and E. T. Samulski, Soft Matter, 2010, 6,
2413–2420.
31 V. Domenici, Soft Matter, 2011, 7, 1589–1598.
32 C. Zhang, M. Gao, N. Diorio, W. Weissflog, U. Baumeister,S. Sprunt, J. T. Gleeson and A. Jákli, Phys. Rev. Lett., 2012,
109, 107802.
33 C. Bailey, K. Fodor-Csorba, J. T. Gleeson, S. N. Sprunt andA. Jákli, Soft Matter, 2009, 5, 3618–3622.
34 S. Kaur, G. Mohiuddin, V. Punjani, R. K. Khan, S. Ghosh and
S. K. Pal, J. Mol. Liq., 2019, 295, 111687.
35 H. J. Deuling, Mol. Cryst. Liq. Cryst., 1972, 19, 123–131.
36 H. Gruler, T. J. Scheﬀer and G. Meier, Z. Naturforsch., A:
Phys., Phys. Chem., Kosmophys., 1972, 27, 966.
37 G. Barbero and L. R. Evangelista, An Elementary Course on
the Continuum Theory for Nematic Liquid Crystals, World
Scientific, 2000.
38 J. C. Loudet, P. Hanusse and P. Poulin, Science, 2004,
306, 1525.
39 H. Stark, Phys. Rep., 2001, 351, 387–474.
40 M. Škarabot and I. Muševič, Soft Matter, 2010, 6, 5476–5481.
41 O. P. Pishnyak, S. Tang, J. R. Kelly, S. V. Shiyanovskii andO. D. Lavrentovich, Phys. Rev. Lett., 2007, 99, 127802.
42 J. C. Crocker and D. G. Grier, J. Colloid Interface Sci., 1996,
179, 298–310.
43 D. Allan, C. van der Wel, N. Keim, T. A. Caswell, D. Wieker,
R. Verweij, C. Reid, Thierry, L. Grueter, K. Ramos, apiszcz,
zoeith, R. W. Perry, F. Boulogne, P. Sinha, pfigliozzi, N. Bruot,
L. Uieda, J. Katins, H. Mary and A. Ahmadia, soft-matter/trackpy:
Trackpy v0.4.2, 2019, DOI: 10.5281/zenodo.3492186.
44 T. G. Mason, K. Ganesan, J. H. van Zanten, D. Wirtz andS. C. Kuo, Phys. Rev. Lett., 1997, 79, 3282–3285.
45 J. He, M. Mak, Y. Liu and J. X. Tang, Phys. Rev. E: Stat.,
Nonlinear, Soft Matter Phys., 2008, 78, 011908.
46 B. Kundu, R. Pratibha and N. V. Madhusudana, Phys. Rev.
Lett., 2007, 99, 247802.
47 D. W. Berreman and S. Meiboom, Phys. Rev. A: At., Mol., Opt.
Phys., 1984, 30, 1955–1959.
48 P. Panizza, D. Roux, V. Vuillaume, C.-Y. D. Lu and
M. E. Cates, Langmuir, 1996, 12, 248–252.
49 S. Fujii, S. Komura, Y. Ishii and C.-Y. D. Lu, J. Phys.:
Condens. Matter, 2011, 23, 235105.
50 Y. P. Panarin, S. P. Sreenilayam, J. K. Vij, A. Lehmann and
C. Tschierske, Beilstein J. Nanotechnol., 2018, 9, 1288.
51 M. Benzekri, T. Claverie, J. P. Marcerou and J. C. Rouillon,
Phys. Rev. Lett., 1992, 68, 2480–2483.

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