Temperature dependence of equilibrium separation and lattice parameters of nematic
boojum-colloids
K. P. Zuhail and Surajit Dhara
Citation: Applied Physics Letters 106, 211901 (2015); doi: 10.1063/1.4921634
View online: http://dx.doi.org/10.1063/1.4921634
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APPLIED PHYSICS LETTERS 106, 211901 (2015)

Temperature dependence of equilibrium separation and lattice parameters ofnematic boojum-colloids
K. P. Zuhail and Surajit Dharaa)
School of Physics, University of Hyderabad, Hyderabad 500046, India(Received 23 January 2015; accepted 13 May 2015; published online 26 May 2015)
We report experimental studies on the temperature dependence of equilibrium separation of a pairof nematic boojum colloids. The interparticle separation is linearly proportional to the elastic
anisotropy and the orientation angle with respect to the rubbing direction. The latter result is inagreement with the recent prediction of computer simulations. We prepared two-dimensional
colloidal crystals assisted by a laser tweezer and measured the temperature dependence of thelattice parameters. We observe a structural transition in boojum colloidal crystal across the N-SmA
transition. The structural transition is mediated by the transformation of boojum defects into focal
conic lines across the phase transition. The hexagonal close pack structure of the colloidal crystal
C 2015 AIP Publishing LLC.
in the SmA phase could be useful for designing photonic crystal. V[http://dx.doi.org/10.1063/1.4921634]

Photonic crystals resemble solid crystals, where theatoms are replaced by larger dielectric particles with high refractive index and the electrons are replaced by photons.1
Like solid crystals photonic crystals also have the band structure and band gap, consequently, the light with a range ofwave vectors corresponding to the band gap cannot propagate. The photonic crystals with capability of varying band
gap are known as tunable photonic crystals. One way of tuning the band gap is by changing the lattice spacing of thecrystals. This can be done by several ways such as chemical
tuning,2,3 temperature tuning,4,5 and field tuning.6,7
Colloidal particles dispersed in isotropic fluids show
two or three dimensional aggregations due to the delicate
balance of attractive dispersion forces (e.g., van der Waals)
and the repulsive forces (e.g., Coulombic and Steric). In
recent years, colloidal particles in liquid crystal (LC) haveattracted a lot of scientific interest because of its possibility
for the bottom up assembly of photonic crystal. Whenmicron sized spherical colloidal particles are dispersed inanisotropic fluids, such as in nematic liquid crystals, stabledefects are accompanied with each colloid.8,9 It can give twostable defect configurations in the nematic phase when they
are treated to promote vertical alignment of the molecules on
its surface.9–11 If the far field director (the average directionof molecular orientation) has the similarity to the field lines
of electric dipole, in which a point-like hyperbolic hedgehog
defect is associated with each colloidal particle, it is called adipolar colloid. If the far field director has the similarity to
the field lines of electric quadrupole, in which a ring ofstrength 1/2 defect is formed around the equatorial planeof the colloids, they are known as quadrupolar colloids.11
When the colloids with planar anchoring of LC molecules
are introduced, the particle creates a pair of surface defects
called boojum, and the particle defect pairs are called as boojum colloids.11–14 The interaction between two boojum colloids are similar to the quadrupolar interaction and thepotential has r5 dependence.15,16 There are many reports ona)Electronic mail: sdsp@uohyd.ernet.in

0003-6951/2015/106(21)/211901/4/$30.00

the theoretical and experimental studies on the associateddefects and the colloidal interactions.17–26 The long range
structural elastic forces (typical binding energy of several
1000 kBT per micron-sized particle) of the medium tends tobind the colloidal particles together. On the other hand, theaccompanied point-like defects or loops give rise to a shortrange repulsive force. The appropriate balance between these
two forces can create diverse structures such as chains, twoand three dimensional crystals.8,9,27,28
Most of the experimental reports are on the sphericalparticles dispersed either in nematic or in cholesteric liquid
crystals. Very recently, we showed that the hyperbolic
hedgehog defects of nematic colloids are strongly influenced
by the elasticity and the onset of smectic-A (SmA) layering
across the nematic to smectic-A (N-SmA) phase transition.
We showed that the hyperbolic hedgehog defect is transformed into a focal conic line in the SmA phase. The phasetransition of liquid crystal has a strong influence on the pairwise colloidal interaction.29 In this letter, we report on thetemperature dependence of equilibrium separation of a pairof isolated nematic boojum colloids. We show that the interparticle separation is linearly proportional to the orientation
angle with respect to the rubbing direction and is the directverification of the recent prediction of computer simulation.
The hexagonal close pack crystal structure in the SmA phase
could be used as a building block to make three-dimensional
colloidal crystals.
We used silica microspheres of diameter 2R0 ¼ 5.2 lm
and 2.34 lm obtained from Bangs Laboratory. Microspheresare coated with MAP (N-Methyl-3 aminopropyl trimethoxysilane, ABCR GmbH and Co-KG) to induce the planaranchoring of LC molecules on the particle’s surface. These
particles are dispersed in 8CB liquid crystal (40 -n-octyl4cyano-biphenyl), which is obtained from Sigma Aldrich.
The sample is filled in a cell made of thin glass plates sealedwith optical adhesive. To induce planar alignment of liquid
crystal molecules, glass plates are coated with a polyimide(AL-1254) and rubbed in antiparallel way after curing at
180  C for 1 h. 8CB liquid crystal exhibits the following

106, 211901-1

C 2015 AIP Publishing LLC
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K. P. Zuhail and S. Dhara

phase transitions: Cr 21.5  C, SmA 33.5  C, and N 40.5  C I.
A laser tweezer was built around an inverted microscope(Nikon eclipse Ti-U) using a CW diode pumped solid statelaser operating at 1064 nm (Aresis: Tweez 250si). A computer interfaced acousto-optic deflector was used for the trap
movement. The temperature of the sample was controlled bya heater and proportional-integral-derivative (PID) controller
with an accuracy of 60.1  C. The trajectories of the particles
were video recorded with a CCD camera at 20 frames per
second. The center-to-center distance between the particles
was measured by analysing the video using a particle tracking software with a resolution of 610 nm.
Figure 1 shows the polarising optical microscope imagesof a boojum colloid and the transformation of boojum defectacross the N-SmA phase transition. A video clip of the defecttransformation is given in the supplementary material(video-1).31 Just above the N-SmA transition, the area of distorted director field around the colloidal particle is extended
along the rubbing direction (Fig. 1(b)). The extended regions
are observed as a high contrast tails with dark straight lines
on both poles of the microsphere. The reshaping of the boojum defect is somewhat similar to that observed in case of
hyperbolic hedgehog defect in dipolar colloids across the
N-SmA transition.29 In the latter case, using Landau-deGennes Q-tensor modeling (in the N phase), we showed thatthe increase of bend elastic constant (K33) replaces theenergetically costly bend distortion by more favorable splay
distortion, concentrated in an elongated tail. However, one
important difference is that, in case of dipolar colloid, thepoint defect moves toward the surface of the particle justbefore the transition, whereas in case of boojum defect similar movement is not observed. In the SmA phase, the brighter

FIG. 1. Polarising optical micrographs of a boojum colloid in a planar cell(a) in the N phase, (b) at the N-SmA phase transition, and (c) in the SmA
phase. (See supplementary material, video-1).31 Photomicrographs of a boojum defect (diameter is 2.34 lm) with a k plate between the analyser and thesample (d) in the N phase, (e) at the transition, and (f) in the SmA phase.
Polarisers are marked with a cross and the k-plate with a red line in
Fig. 1(d). (g) Schematic diagram of the director orientation around the particle with boojum defect. The lines represent the LC director. (h) Schematic
representation of SmA layers in the vicinity of the particle with focal conic
line defects. A representative director orientation within SmA layers is also
shown in a small circular area. Cell thicknesses for particles 5.2 lm and2.34 lm are 8 lm and 4.5 lm, respectively.

Appl. Phys. Lett. 106, 211901 (2015)

regions in the neighborhood of the microsphere indicate thatthe layers are strongly bent. To construct the molecular orientation around the microsphere, we observed images
between crossed polarizers with an additional k-plate(530 nm) inserted between the sample and the polariser. The
red color corresponds to a horizontal orientation of the longaxes of the molecules (Figs. 1(d)–1(f)), whereas the bluish
and yellowish colors correspond to the clock-wise and anticlockwise rotation of the director from the rubbing direction.25 The schematic representations of LC director aroundthe microspheres in the N phase (with boojum) and in the
SmA phase (with focal conic lines) are shown in Figs. 1(g)and 1(h), respectively. The preexisting nematic distortion
around the particle constrains the equal layer spacing
requirement of SmA and hence the tails are transformed into
focal conic lines with the cusp angle in the layer structure.
These focal conic lines are parallel to the surrounding director, being less and less visible far from the microsphere. Our
experimental observations of SmA focal lines around themicrosphere are almost similar to the phenomena observedin case of spherical microdroplets of glycerin embedded inaligned SmA liquid crystal by Blanc and Kleman.30 They
calculated the curvature energy due to the spherical inclusion
in aligned SmA liquid crystal and showed that the angulardiscontinuity of layers decreases continuously far from theinterface.
In the nematic phase, boojum colloids are manipulatedusing a laser tweezer. Figure 2 shows the microscope imagesof a pair of boojum colloids in the N and SmA phases of
8CB liquid crystal. The focal conic line defects on both sidesof the pair are observed in the SmA phase (Figs. 2(b) and2(d)). The temperature of the sample was decreased at therate of 0.3  C per minute and the centre to centre separation(D) of two boojum colloids is measured as a function of temperature using an appropriate particle tracking program.
Figure 3(a) shows the temperature dependence of separation
across the N-SmA transition of two colloidal particles. Theinterparticle separation (D/R0, where R0 is the radius of theparticle) decreases with decreasing temperature in thenematic phase. For example, above the N-SmA transition
temperature (T  TNA ’ 5  C), the equilibrium separationis 6.76 lm and it decreases to 6.2 lm in the SmA phase.
A sharp discontinuity in the interparticle separation isFIG. 2. Photomicrographs of a pair of colloidal particles before and after the
N-SmA transition with and without polarisers. (a),(c) N phase (35.5  C)(b),(d) SmA phase (33  C). Particle diameter and cell thickness are 5.2 lm
and 8 lm, respectively.

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K. P. Zuhail and S. Dhara

FIG. 3. (a) Variation of center to centre separation (D) between a pair of colloidal particles across N-SmA phase transition.31 See supplementary material, video-2. (Inset) Variation of D/R0 above the N-SmA phase transition as
a function of the elastic constant anisotropy (1  K11/K33). The red line isthe best fit with a slope a ¼ 0.09. The scale on the top shows the corresponding temperature range. Particle diameter and cell thickness are 5.2 lm
and 8 lm, respectively. (b) Variation of h with D/R0 obtained from Video-2.
The vertical arrow indicates a slope change at the N-SmA transition. (Inset)
Computer simulated variation of h with D/R0. The simulated points are collected from Ref. 12.

observed, marking the discontinuity of the elastic constants
and the onset of smectic layering. In case of dipolar colloids,
the interparticle separation shows opposite behavior in thesense that it increases with decreasing temperature.29
Recently, James and Fukuda32 theoretically studied theeffect of anchoring strength and elastic anisotropy on theseparation of a pair of dipolar colloids in the N phase. They
showed that in the strong anchoring regime (K33/
WR0  0.04, where K33 being the bend elastic constant andW, the anchoring energy), the equilibrium separation
increases linearly with the elastic anisotropy. Very recently,
it was experimentally verified by us.29 However, theoretically similar aspects of the boojum colloids have not beenstudied so far. In order to bring out the effect of elasticity,
we collected temperature dependent splay and bend elastic
constants (K11 and K33) of 8CB liquid crystal from Ref. 33
and interpolated to obtain elastic constants for all temperatures. The inset of Fig. 3(a) shows the variation of D/R0 as afunction of elastic anisotropy (1  K11/K33). Interestingly,
we observe that the equilibrium separation decreases linearlywith increasing elastic anisotropy with a slope of 0.09.
This is in contrast to the positive slope observed in dipolar

Appl. Phys. Lett. 106, 211901 (2015)

colloids.32 It would be interesting to supplement our experimental results either by computer simulation or by theory.
Above the N-SmA transition temperature (T  TNA’ 5  C), the line joining the centers of two colloidal particles
makes an angle h ¼ 33 with respect to the rubbing direction,
which is consistent with the previous reports.16,26 When thetemperature decreases, consequently the separation and also
the angle decreases. Figure 3(b) shows the variation of angle(h) with D/R0 across the N-SmA transition. It decreases withdecreasing temperature with a slope change at the transition.
For example, in the nematic phase, h ¼ 33 6 1 and it
decreases to 22 in the SmA phase. Recently, Mozaffari et al.,studied the interaction of nematic boojums and dependence ofh on the interparticle separation by computer simulations.12
They showed that the particles attract each other at h ¼ 60
and h ¼ 30 , with a strong attraction in the latter case. For
smaller angles (h  15 ), the interparticle interaction exhibitsa non-monotonous behavior as a function of interparticle separation. They also calculated the free energy of two nematic
colloids at various values of D/R0 and h. From the global freeenergy minimum they found that h decreases with D/R0. In
order to compare, we collected the simulated points from
Ref. 12 that coincides with our experimental range of D/R0
and are shown in the inset of Fig. 3(b). It is observed that theangle decreases linearly with D/R0 in the nematic phase. Thus,
our experimental results qualitatively verified the above prediction. Experimentally, h decreases more rapidly than predicted in the simulation and this could be due to the specificchoice of surface anchoring energy or one elastic constant
approximation in the simulation.
Finally, we show the variation of lattice parametersacross the N-SmA phase transition in 2D boojum colloidal
crystal. We assembled 36 nematic boojum colloids using thelaser tweezer and made a 2D crystal in the N phase(Fig. 4(a)). In the N phase, it forms a parallelogram unit
cell with lattice parameters a ¼ b ¼ 6.7 6 0.05 lm andc ¼ 74 6 1 . When the temperature of the sample is reduced,
there is a structural change in the crystal lattice (see thesupplementary material, video-3).31 Fig. 4(b) shows theimage of 2D crystal in the SmA phase that is obtained bycooling the sample from N phase at the rate of 0.3  C/min.
Images in Figs. 4(c) and 4(d) are taken using a color camera
by placing the sample in between crossed polarisers. In the
SmA phase, it shows almost a hexagonal close pack unit
cell. For example, in the SmA phase the lattice parameters
a ¼ b ¼ 6.1 6 0.05 lm and c ¼ 60 6 1 (Fig. 4(b)). We measured the temperature variation of all the lattice parametersacross the N-SmA phase transition using the particle tracking
program. Figure 4(e) shows the temperature variation of aand b in both the phases. It is observed that the lattice parameters a and b are almost equal and decrease with decreasing
temperature with a sudden jump at the N-SmA phase transition. For example, both the lattice parameters decrease by
about 9% in the SmA phase compared to the nematic phase.
Figure 4(f) shows the temperature variation of the angle (c)across the N-SmA transition. It is observed that in the nematic phase c decreases almost linearly with a distinct slopechange at the transition. For example, in the nematic phasec ¼ 74 and it decreases to 60 in the SmA phase. In addition, the area of the 2D crystal also decreases from 1082 lm2This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 14.139.69.34
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211901-4

K. P. Zuhail and S. Dhara

Appl. Phys. Lett. 106, 211901 (2015)

hexagonal close pack structure in the SmA phase showing a
discontinuous change in the lattice parameters at the transition. The structural transition is driven by the transformationof topological defects. The tuning of lattice parameters bytemperature could be useful for controlling photonic bandgap. In addition, by measuring the interparticle separationand studying the topological defect transformation across thephase transition it could be possible to detect subtle phasetransitions in various other liquid crystals.
S.D. acknowledges support from DST-PURSE, UPE-II(UoH), DST (SR/NM/NS-134/2010), CSIR (03(1207)/12/
EMR-II). K. P. Zuhail acknowledges UGC for fellowshipand thank Dr. V. S. R. Jampani for help.
1

FIG. 4. Optical photomicrographs of a 2D boojum crystal in the (a),(c) Nphase (37  C) and (b),(d) SmA phase (32.7  C) of 8CB liquid crystal. (a) and(b) without polarisers (c) and (d) with crossed polarizers.31 See supplementary material, video-3. (e) Temperature variation of the lattice parameters aand b across N-SmA phase transition. (f) Temperature variation of c acrossthe N-SmA phase transition. Microsphere diameter and cell thickness are
5.2 lm and 8lm, respectively.

(N phase) to 910 lm2 (SmA phase), which is about 16%
reduction compared to the initial crystal area. Thus, the sizeof the crystal can also be tuned by changing the temperature.
In conclusion, we studied interparticle separation of nematic boojum colloids across the N to SmA phase transition.
The dependence of angle of two assembled boojum colloidson the interparticle separation qualitatively agrees with theprediction of computer simulation. The point defects on thepoles of the spherical particle in the N phase are transformed
to focal conic lines in the SmA phase. The 2D boojum crystal with oblique lattice in the nematic (N) phase changes to a

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