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11, 7674

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Orientation, interaction and laser assisted
self-assembly of organic single-crystal
micro-sheets in a nematic liquid crystal
M. V. Rasna,a K. P. Zuhail,a U. V. Ramudu,b R. Chandrasekar,b J. Dontabhaktunicand Surajit Dhara*a
Colloidal self-assembly has been one of the major driving themes in material science to obtain functional
and advanced optical materials with complex architecture. Most of the nematic colloids reported so far are
based on the optically isotropic spherical microparticles. We study organic single crystal micro-sheets andinvestigate their orientation, interaction and directed assembly in a nematic liquid crystal. The micro-sheets

Received 10th August 2015,
Accepted 13th August 2015

induce planar surface anchoring of the liquid crystal. The elasticity mediated pair interaction of micro-sheets

DOI: 10.1039/c5sm01991eand the angle between two micro-sheets in a homeotropic cell are supported by the Landau-de Gennes

shows quadrupolar characteristics. The average orientation angle of the micro-sheets in a planar cell
Q-tensor modeling. The self-assembly of the micro-sheets is assisted by a laser tweezer to form larger

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two-dimensional structures which have the potential for application of colloids in photonics.

1 Introduction
Nematic colloids prepared by dispersing micrometer sized
colloidal particles in nematic liquid crystals (LCs) received
considerable attention due to their potential application in
the field of soft materials and photonics.1–7 The inclusion offoreign particles disturbs the uniform director field (average
direction of molecular alignment) and induce point or line
defects and the corresponding elastic distortion spreads over a
few micrometer range.8–14 The defect formation is due to thecompetition between the bulk elastic property of the medium
and the anchoring of LC molecules on the particle’s surface. In
the case of normal surface anchoring, each spherical microparticle forms a source of singularity (radial hedgehog defect)
and for the conservation of topological charge, a hyperbolic
hedgehog defect appears in the director field and known
as elastic dipole.14 For weak anchoring the defect appears in
the form of a thin disclination line encircling the colloidal
particles (Saturn ring) and known as quadrupole.14 Similarly
with planar surface anchoring, the microspheres induce two
surface point defects at the poles along the director and are
known as boojums.14–16 The interaction among these colloids
is mediated by sharing the topological defects and there are many
theoretical and experimental reports on the colloidal interactionsaSchool of Physics, University of Hyderabad, Hyderabad-500046, India.
E-mail: sdsp@uohyd.ernet.inbSchool of Chemistry, University of Hyderabad, Hyderabad-500046, Indiac
School of Natural Sciences, Mahindra Ecole Centrale, Hyderabad-500043, India

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in nematic liquid crystals.1,2,8–13 Colloidal self-assembly mediatedby the nematic elasticity is important for photonic applications
such as photonic band gap crystals, metamaterials, and 3D
photonic crystals.17–21
Most of the reports on nematic colloids are concerned on
the inclusion of spherical microparticles. In the recent years
significant interest is observed on the dispersion of nonsphericalmicroparticles such as rod-shaped, star-shaped, triangular, square,bullet and doughnut shaped in nematic liquid crystals.23–33
Lapointe et al. showed that polygonal platelets exhibit either
dipolar or quadrupolar interactions in nematic liquid crystals
depending on whether the platelets have an odd or even numberof sides.25 Very recently it has been shown by computer simulation that quasicrystalline colloidal lattices can be obtained innematic liquid crystals using pentagonal colloidal platelets.22 In
previous experiments, the microparticles used were either silica
or made of amorphous materials, which were optically isotropic.
In this paper we report on the orientation, interactions and laser
assisted self-assembly of organic single crystal micro-sheets in anematic liquid crystal.

2 Experiment
Micro-sheets were prepared as per the reported procedure.34
They grow either in rectangular or hexagonal shape. The typicalsize of the rectangular platelets i.e., lengths and widths is in therange of 9 to 10 mm and 4 to 5 mm respectively. The average
thickness of the platelets is about 200 nm. The micro-sheets

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have a monoclinic crystal structure and are optically birefringent(Dn = 0.25 ! 0.05). They emit fluorescence bands at 719 and653 nm upon excitation at 431 nm. They can be used as organic
waveguides and wavelength filters at different wavelengths.34
The platelets are dispersed in 5CB (pentyl cyanobiphenyl) liquid
crystal by physical mixing. The experimental cells with various
spacing were fabricated using glass plates that are spin coatedwith polyimide AL-1254 and cured at 180 1C for 1 h. They wererubbed in an antiparallel way for homogeneous alignment of
the nematic director. For homeotropic cells, we used polyimide
JALS-204 and cured at 200 1C for 1 h. A laser tweezer was built on
an inverted microscope (Nikon eclipse Ti-U) using a CW diodepumped solid state laser operating at 1064 nm. The trajectoriesof the particles are video recorded using a CCD camera (Pixelink
PLB 741F) and an appropriate computer program was used to
track the centers of the micro-sheets.

3 Numerical modelling
We employed the Landau-de Gennes phenomenological theory tomodel our experimental system.35 The free energy of the systemconstitutes contributions from elasticity of the nematic medium,
temperature dependent bulk order and surface anchoring. It is
written in terms of tensorial order parameter Qij:
"2 1
"2
1 !
1
1 !
f ¼ L @Qij =@xk þ AQij Qji þ BQij Qjk Qki þ C Qij Qij þfS
2
2
3
4
(1)
The contribution from the surface of the platelets inducing#$2~ij $ Q~? ,degenerate planar anchoring is given by fS ¼ W Q
ij~ij $ Q~?
where W is the anchoring strength, while the term Q
ij
imposes the orientational in-plane ordering. The free energy is
minimized using the explicit finite difference relaxation method
on a cubic mesh. Rectangular micro-sheets of length l = 360 nm
and breadth b = 180 nm are considered for simulations in a
micron thick liquid crystal cell maintaining the same aspectratios as that of the experiments. Degenerate planar surface
anchoring of strength W = 10$3 J m$2 is induced by all thesurfaces of the microsheets. Top and bottom substrates of theliquid crystal cell induce strong uniform planar anchoring in
the case of planar cells and periodic boundary conditions are
assumed along XY directions. In the case of homeotropic cells,
the top and bottom substrates induce strong homeotropic anchoringof strength W = 10$3 J m$2. All the material parameters consideredin the simulations are that of a standard nematic liquid crystal.35

Fig. 1 (a) Optical microscopy image of rectangular single crystal microsheets. Polarised optical microscopy images of a micro-sheet with the longaxis (b) making 451 with respect to the polariser and (c) parallel to the polariser.

in a planar cell. It is observed that the long axes of the microsheets are making some angle with respect to the rubbing
direction. Fig. 2(b) shows the fluorescence image of the microsheets when illuminated with blue filter. The wavelength of the red
colour is 653 nm. Fig. 2(c) shows the histogram of the angles oflong axes (y) with respect to the rubbing direction. The maximum
number of particles are orientated making about 331 with respect
to the rubbing direction. Using Landau de-gennes Q-tensor
modelling we calculated the free energy for various in-planeorientations. The variation of free energy with y is shown in the

4 Results and discussion
Fig. 1(a) shows the optical microscopy image of a large numberof micro-sheets on the glass slide. Fig. 1(b) and (c) shows thepolarising optical microscopy images of a rectangular microsheet at two different orientations. It is clearly observed that the
micro-sheets are optically anisotropic and the optical axis iseither parallel or perpendicular to the long axis. Fig. 2(a) shows
the image of a few micro-sheets dispersed in 5CB liquid crystal

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Fig. 2 (a) Micro-sheets dispersed in aligned nematic liquid crystals. The
white arrow at the right corner indicates the rubbing direction (optical axis
of LC). (b) Fluorescence image of the micro-sheets illuminated with a blue
filter. (c) Distribution of angles of the long axes of the platelets with respect
to the rubbing direction. (inset) Variation of free energy of a micro-sheet
for various in-plane rotations.

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inset of Fig. 2(c). It is observed that the energy is minimum at
y C 351 which is very close to the average orientation angleof the rectangular micro-sheets. The wide distribution of y in
Fig. 2(c) could be due to the non-uniformity in size, shape andedge roughness of the micro-sheets.
To understand the molecular orientation of the liquid crystal
around the micro-sheet, images were taken between crossed
polarisers with a l-plate (red-plate) inserted between the sample
and the polariser. Fig. 3(a) shows the red-plate image of a microsheet. The plane of the micro-sheet is parallel to the substratesurface. The bluish and yellowish colors around the micro-sheet
correspond to the clockwise and anti-clockwise rotation of thedirector from the rubbing direction. It indicates the planaranchoring of the director on the surface of the micro-sheet.
The strongest distortion of nematic is at the edges and thecorners. Experimentally, it is difficult to precisely locate thedefects on the micro-sheets. The calculated director field surrounding the micro-particle is shown by head-less vectors in
Fig. 3(b). The director profile apparently has well-defined winding numbers reminiscent to +1/2 and $1/2 defect lines similar tothat reported in the case of square-shaped particles.4 In the caseof rectangular micro-sheets, there is an asymmetry about they-direction (perpendicular to the rubbing direction) which isevident from the simulated director profile and indicated by
red-lines. Fig. 3(c) shows the image of a micro-sheet in a homeotropic cell. The plane of the micro-sheet is vertical with respect tothe substrate surfaces. The brighter edges in Fig. 3(c) indicatethat the long axis of the micro-sheet is tilted with respect tothe vertical direction. The variation of calculated free energy of a
micro-particle in a homeotropic cell at various tilt angles withrespect to the z-axis (normal to the cell substrate) is shown in
Fig. 3(d). It is observed that the minimum energy is obtained at
an angle of 221.
The interaction between two micro-sheets is studied by thevideo microscopy technique. In order to measure the elastic
interaction, the viscous drag coefficient of the micro-sheets is

Fig. 3 (a) Polarising optical microscopy image of a micro-sheet with a
red-plate (l-plate, 530 nm) in a planar cell. Cell thickness: 12 mm. The
direction of l-plate is shown by a black arrow. (b) The calculated director
field surrounding the micro-particles is shown as head-less vectors inwhite. The red lines indicate the asymmetry of the director field. (c) A
micro-sheet in a homeotropic cell. (d) Variation of free energy at various
title angles with respect to the z-axis. Cell thickness: 15 mm.

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Fig. 4 (a) Histograms of displacement parallel and perpendicular to thedirector obtained from the trajectory for time delay t = 0.05 s. The redlines are the Gaussian fits to the data. Note that the histogram is displacedalong the positive x-direction due to the macroscopic flow.

calculated. The drag coefficients are determined directly fromthe study of Brownian motion of the micro-sheets in the liquid
crystal.25,36 In a homogeneous cell, the Brownian motion of apreselected micro-sheet which is far away from the other microsheets is video recorded and the trajectory of the micro-sheet isdetermined from the particle tracking. Fig. 4 shows two typical
histograms of displacement parallel and perpendicular tothe director (n) obtained from the trajectory for a time delayt = 0.05 s. The histograms of the displacement along (x-axis)
and perpendicular to the director (y-axis) show the anisotropy
in the motion. Here the x-displacement is shifted along thepositive x direction, which indicates macroscopic flow. Similar
behavior was also reported by Tkalec et al., in the case of microrod dispersed in nematic liquid crystals.23 The mean squaredisplacement along the perpendicular (s>2) and parallel (sJ2) tothe director varies linearly with t and the slope of the linear fits
gives the respective diffusion coefficients (DJ and D>). At room
temperature, from the linear fit, we get DJ = 6.2 % 10$3 mm2 per s
and D> = 4.5 % 10$3 mm2 per s. The drag coefficients are calculated
from the Einstein relation, zi = kBT/Di, where the subscript i
stands for J or > to the director. At T = 298 K, we get, zJ = 0.66 %10$6 kg per s and z> = 0.92 % 10$6 kg per s.
The elastic force is balanced by the viscous drag force in thenematic media which is given by Fdrag = $zdR(t)/dt, wheredR(t)/dt is the time derivative of inter-particle separation. Whenthe elastic force (Fel) is balanced by viscous drag force, the equationof motion is given by Fel + Fdrag = 0. The micro-sheets havequadrupolar interaction and the corresponding force is given
by Fel = $k/R6, where k is a constant that depends on the meanelastic constant K, geometry and the size of the micro-sheet. The
time dependent particle separation is given by R(t) = (R07 $ 7at)1/7,
where a = k/z and R0 is the initial separation at time t = 0.
The strength of elastic interaction between the micro-sheets
is studied by positioning two micro-sheets in the defined
locations namely along the broadside-on (Fig. 5(a)) and end-on(Fig. 6(a)) positions by using the laser tweezer. After removing
the laser power the trajectories of the micro-sheets are video
recorded and their time dependent position is determined byparticle tracking. The instantaneous force on the particle iscalculated from the numerical differentiation of the trajectoryThis journal is © The Royal Society of Chemistry 2015

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Fig. 5 Elastic interaction of the micro-sheets along the broad-side on
position. (a)–(c) Sequence of CCD images of the micro-sheets taken at different
times. (d) Variation of inter-particle separation (R(t)) with time (logarithmic scale).
Blue line shows the nonlinear least square fit to R(t) = (R07 $ 7at)1/7. (inset) The
elastic interaction potential as a function of separation.

and the interaction potential is calculated by integrating theforce over the distance travelled.
Fig. 5(d) shows the inter-particle separation (R(t)) of the
micro-sheets along the broadside-on position. The least square
fit provides a = 3.7 % 105 mm7 per s. A few CCD images atdifferent times are also shown in Fig. 5. Using this value of aand the drag coefficients the calculated maximum attractive
force for separation R = 6.5 mm is 4 pN. The inset of Fig. 5(d)shows the elastic interaction potential obtained by integrating!
"Ðthe force, U ¼ Fdrag & dR , as a function of separation R. The
lowest potential energy is about $1600kBT. Fig. 6(d) shows thetime variation of inter-particle separation of the micro-sheets
along the end-on position. The maximum attractive force for the
separation R = 13.4 mm is 1.2 pN. The inset of Fig. 6(d) shows thevariation of interaction potential as a function of R. The lowest
potential obtained in this position is about $700kBT. We observe

Fig. 6 Elastic interaction of the micro-sheets along the end-on position.
(a)–(c) Sequence of CCD images of the micro-sheets taken at different
times. (d) Variation of inter-particle separation (R(t)) with time (logarithmic
scale). Blue line shows the nonlinear least square fit to R(t) = (R07 $ 7at)1/7.
(inset) The elastic interaction potential as a function of separation.

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Soft Matter

Fig. 7 Elastic interaction of the micro-sheets in a homeotropic cell.
(a)–(c) Sequence of CCD images of the micro-sheets taken at different
times. The angle between two micro-sheets in (c) is about 601. (d) Variationof inter-particle separation (R(t)) with time (logarithmic scale). Blue line
shows the nonlinear least square fit to R(t) = (R07 $ 7at)1/7. (inset) The elastic
interaction potential as a function of separation.

that the interaction potential along the broadside-on position isstronger than that of the end-on position. Similar behavior wasobserved in the case of micro-rods with homeotropic surface
anchoring dispersed in nematic liquid crystals.23 This is due tothe larger reduction of elastic distortion associated with thebroadside-on position of the micro-sheets.
We also studied the interaction between two micro-sheets ina homeotropic cell. A few CCD images at different times arealso shown in Fig. 7(a)–(c). Fig. 7(d) shows the inter-particle
separation (R(t)) of the micro-sheets with time. The least square
fit provides a = 4.8%105 mm7 per s and the calculated maximum
attractive force for separation R = 4.8 mm is 3.8 pN. The inset of
Fig. 7(d) shows the variation of elastic interaction potential as afunction of separation R. The lowest potential energy is about$1200kBT. We observe that the equilibrium angle between thetwo micro-sheets varies within 55 to 601 (Fig. 7(c)). We numerically calculated the interaction between two micro-sheets. In
simulations the sheets are kept at a fixed distance with theirsmaller edges parallel to the plane of the substrate and theinteraction potential is calculated by varying the mutual angle(see Fig. 8). The angle is zero when both the sheets are parallel toeach other with their large surfaces facing towards one another.
The nematic distortions around the micro-sheets control the angle
between them. As the angle increases, the distortions reduceand the sheets stabilise at 551 to each other.
The micro-sheets have an inherent tendency to self-assemble
due to the topological defect mediated long-range attractive
interactions as discussed. We used the laser tweezer to assist
the self-assembly of the micro-sheets to form a larger crystal.
For the purpose of demonstration of this idea, we show theassembly of a few rectangular and hexagonal micro-sheets in
Fig. 9. It may be mentioned that micro-sheets absorb 431 nm
and exhibit two fluorescence bands in the red region (653 nm).
Hence they appear reddish under white light illumination.

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Fig. 8 Variation of potential energy with angle between the two microsheets in a homeotropic cell. The minima of the free energy are marked bya downward arrow.

Paper

and anisotropic. The equilibrium orientation angle of a microsheet in a planar cell, and the angle between two micro-sheets ina homeotropic cell are qualitatively in agreement with theLandau de-Gennes Q-tensor modelling. We demonstrated laser
assisted 2D self-assembly exploiting the long-range elastic interaction. Our experiment opens up the possibility for making
larger organic 2D crystals using micro-sheets as building
blocks. The laser assisted self-assembly of micro-sheets withboth shape and optical anisotropy could be useful for various
optical applications. These particles also show interesting multiaxis rotations under applied electric field, and the details will bereported elsewhere. In addition, the high birefringence of theparticles may result in large torque acting to spin and align themin the optical field which can be useful as torque sensors.37,38

Acknowledgements
We gratefully acknowledge the support from DST-PURSE.
MVR acknowledges UGC-BSR for fellowship. We thank Prof.
Holger Stark and Prof. Ken Ishikawa and Dr VSR Jampani foruseful discussions.

References

Fig. 9 Polarising optical microscopy images of laser assisted self-assemblyof (a) rectangular and (b) hexagonal micro-sheets in 5CB liquid crystal.
The double headed white arrows at the right corners show the far fieldalignment of the director.

The slight variation in the intensity of color could be due todifferent sheet thickness and slight difference in the molecularorientation on the micro-sheets. Thus, the single crystal microsheets could be used as building blocks for making larger 2D
colloidal crystals. It may be mentioned that the micro-sheets
were crystallized in tetrahydrofuran/water (THF/H2O) solvents.
They have some edge roughness as a result there are some voidsin the assembled structure (Fig. 9(a)). However, it is possible
to obtain micro-sheets with smoother edge under controlled
conditions. Some preliminary experiments show that the hexagonal micro-sheets have better edge smoothness and consequently the packing is better (Fig. 9(b)).

5 Conclusions
In conclusion, we observed that single crystal micro-sheets
induce planar anchoring of the nematic liquid crystal and theinteraction potential between two micro-particles is quadrupolar

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1 P. Poulin, H. Stark, T. C. Lubensky and D. A. Weitz, Science,
1997, 275, 1770.
2 T. C. Lubensky, D. Pettey, N. Currier and H. Stark, Phys. Rev.
E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1998,
57, 610.
3 I. Musevic, M. Skarabot, U. Tkalec, M. Ravnik and S. Zumer,
Science, 2006, 313, 954.
4 J. Dontabhaktuni, M. Ravnik and S. Zumer, Soft Matter,
2012, 8, 1657.
5 I. Musevic and M. Skarabot, Soft Matter, 2008, 4, 195.
6 M. Skarabot, M. Ravnik, U. Tkalec, I. Poberaj, D. Babic,
N. Osterman and I. Musevic, Phys. Rev. E: Stat., Nonlinear,
Soft Matter Phys., 2007, 76, 051406.
7 K. P. Zuhail, P. Sathyanarayana, D. Sec, S. Copar, M. Skarabot,
I. Musevic and S. Dhara, Phys. Rev. E: Stat., Nonlinear, Soft
Matter Phys., 2015, 91, 030501(R).
8 A. Nych, U. Ognysta, M. Skarabot, M. Ravnik, S. Zumer andI. Musevic, Nat. Commun., 2013, 4, 1489.
9 O. Guzman, E. B. Kim, S. N. Grollau, L. Abbott and
J. J. Pablo, Phys. Rev. Lett., 2003, 91, 235507.
10 M. Ravnik, M. Skarabot, S. Zumer, U. Tkalec, I. Poberaj,
D. Babic, N. Osterman and I. Musevic, Phys. Rev. Lett., 2007,
99, 247801.
11 M. Vilfan, N. Osterman, M. Copic, M. Ravnik, S. Zumer,
J. Kotar, D. Babic and I. Poberaj, Phys. Rev. Lett., 2008,
101, 237801.
12 T. A. Wood, J. S. Lintuvuori, A. B. Schofield, D. Marenduzzo
and W. C. K. Poon, Science, 2011, 334, 79.
13 K. Takahashi, M. Ichikawa and Y. Kimura, J. Phys.: Condens.
Matter, 2008, 20, 075106.
14 H. Stark, Phys. Rep., 2001, 351, 387.

This journal is © The Royal Society of Chemistry 2015

View Article Online

Published on 13 August 2015. Downloaded by University of Hyderabad on 26/09/2015 05:39:17.

Paper

15 T. C. Lubensky, D. Petty, N. Currier and H. Stark, Phys. Rev. E:
Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1998, 57, 610.
16 Q. Liu, B. Senyuk, M. Tasinkevych and I. I. Smalyukh, Proc.
Natl. Acad. Sci. U. S. A., 2013, 110, 9231.
17 Y. A. Vlasov, X.-Z. Bo, J. C. Sturm and D. J. Norris, Nature,
2001, 414, 289.
18 J. D. Joannopoulos, P. R. Villeneuve and S. Fan, Nature,
1997, 386, 143.
19 V. G. Veselago, Phys.–Usp., 1968, 10, 509.
20 J. B. Pendry, Phys. Rev. Lett., 2000, 85, 3966.
21 K. J. Stebe, E. Lewandowski and M. Ghosh, Science, 2009,
325, 159.
22 J. Dontabhaktuni, M. Ravnik and S. Zumer, Proc. Natl. Acad.
Sci. U. S. A., 2014, 111, 2464.
23 U. Tkalec, M. Skarabot and I. Musevic, Soft Matter, 2008, 4, 2402.
24 C. P. Lapointe, K. Mayoral and T. G. Mason, Soft Matter,
2013, 9, 7843.
25 C. P. Lapointe, T. G. Mason and I. I. Smalyukh, Science,
2009, 326, 1083.
26 M. A. Gharbi, M. Cavallaro, G. Wu, D. A. Beller, R. D. Kamien,
S. Yang and K. J. Stebe, Liq. Cryst., 2013, 40, 1619.
27 J. S. Evans, C. N. Beier and I. I. Smalyukh, J. Appl. Phys.,
2011, 110, 033535.

This journal is © The Royal Society of Chemistry 2015

Soft Matter

28 D. A. Beller, M. A. Gharbi and I. B. Liu, Soft Matter, 2015,
11, 1078.
29 A. Martinez, T. Lee, T. Asavei, H. Rubinsztein-Dunlop and
I. I. Smalyukh, Soft Matter, 2012, 8, 2432.
30 D. Andrienko, M. P. Allen, G. Skacej and S. Zumer, Phys. Rev.
E: Stat., Nonlinear, Soft Matter Phys., 2002, 65, 041702.
31 B. Senyuk, M. C. M. Varney, J. A. Lopez, S. Wang, N. Wu and
I. I. Smalyukh, Soft Matter, 2014, 10, 6014.
32 B. Senyuk, Q. Liu, E. Bililign, P. D. Nystrom and
I. I. Smalyukh, Phys. Rev. E: Stat., Nonlinear, Soft Matter
Phys., 2015, 91, 040501(R).
33 B. Senyuk, Q. Liu, S. He, R. D. Kamien, R. B. Kusner,
T. C. Lubensky and I. I. Smalyukh, Nature, 2013, 493, 200.
34 N. Chandrasekhar, S. Basak, M. A. Mohiddon and
R. Chandrasekhar, ACS Appl. Mater. Interfaces, 2014, 6, 1488.
35 M. Ravnik and S. Zumar, Liq. Cryst., 2009, 36, 1201.
36 S. Dhara, J. Ananthaiah, P. Sathyanarayana, V. Ashok,
A. Spadlo and R. Dabrowski, Phys. Rev. E: Stat., Nonlinear,
Soft Matter Phys., 2013, 87, 030501(R).
37 T. A. Nieminen, N. R. Heckenberg and H. Rubinsztein-Dunlop,
J. Mod. Opt., 2001, 48, 405.
38 M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg and
H. Rubinsztein-Dunlop, Nature, 1998, 394, 348.

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