PHYSICAL REVIEW E 92, 052501 (2015)

Spherical microparticles with Saturn ring defects and their self-assembly acrossthe nematic to smectic- A phase transition
K. P. Zuhail,1 S. Čopar,2 I. Muševič,2,3 and Surajit Dhara1,*
1

2

School of Physics, University of Hyderabad, Hyderabad-500046, India
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
3
Condensed Matter Physics Department, Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia(Received 4 August 2015; published 2 November 2015)

We report experimental studies on the Saturn ring defect associated with a spherical microparticle across thenematic (N ) to smectic-A (SmA) phase transition. We observe that the director distortion around the microparticle
changes rapidly with temperature. The equilibrium interparticle separation and the angle between two quadrupolar
particles in the N phase are larger than those of the SmA phase. They are almost independent of the temperaturein both phases, except for a discontinuous jump at the transition. We assembled a few particles using a laser
tweezer to form a two-dimensional colloidal crystal in the N phase. The lattice structure of the crystal dissolves
irreversibly across the N -SmA phase transition. The results on the pretransitional behavior of the defect aresupported by the Landau–de Gennes Q-tensor modeling.
DOI: 10.1103/PhysRevE.92.052501

PACS number(s): 61.30.Jf, 64.70.M−

I. INTRODUCTION

the pretransitional structure of a pair of quadrupolar colloids
with Landau–de Gennes Q-tensor modeling. We find that thetemperature dependence of interparticle separation and theangle of their joining line with respect to the rubbing direction
are insensitive to the elastic anisotropy. This is in sharp contrast
with our previous observations on the dipole and boojum
defects.

Dispersion of particles with predefined surface anchoringin an aligned nematic liquid crystal (NLC) creates elastic
distortion of the director around the particle [1–6]. The
distortions culminate in the appearance of a topological defectin the neighborhood of each particle and mediate long-range
forces, which opens up new directions in colloidal assembly [7–12]. These have been the subject of intensive researchduring recent years [13–21]. Such colloidal assemblies are
significantly different from the spatial aggregation of colloidal
particles in isotropic fluids, which are mostly governed by
the electrostatic, sterical, and van der Waals interactions. The
forces in liquid crystals are of elastic origin and extend over
several micrometers [1–4].
When the microparticles are treated to promote the perpendicular alignment of liquid crystal molecules, two different
defect configurations can be stabilized. In analogy with theelectric field, depending on the far-field symmetry, these“elastically charged” objects are called elastic dipoles andquadrupoles [2–4]. In the elastic dipole a hyperbolic hedgehog
defect of strength −1 is associated with each microparticle,
whereas in the case of the quadrupole, a Saturn ring defectof strength −1/2 encircles the microparticle. The interaction
potential between a pair of dipoles and quadrupoles varies asr −3 and r −5 , respectively [10].
Most of the experimental investigations are carried out innematic or cholesteric liquid crystals. We recently studied
colloidal particles with hyperbolic hedgehog and boojum
defects across the N -SmA phase transition. We showed thatboth types of nematic defects are transformed into focal conic
line defects in the SmA phase [22,23]. The phase transition
has a strong influence on the pairwise colloidal interaction
and is responsible for structural transitions in two-dimensional
colloidal crystals. In this paper, we report on studies ofcolloidal particles with Saturn ring defects across the N -SmA
phase transition. We investigate the effect of elasticity on*

Email: sdsp@uohyd.ernet.in

1539-3755/2015/92(5)/052501(5)II. EXPERIMENTAL

We used Silica microspheres of diameter 5.2 and 8 µm
(Bangs Laboratory, USA) coated with octadecyldimethyl3-trimethoxysilylpropyl-ammonium chloride (DMOAP) to
induce strong homeotropic anchoring of liquid crystal (LC)molecules. The particles were dispersed in 8CB liquid crystal(4’-n-octyl-4-cyano-biphenyl) which exhibits the following
phase transitions: Cr (crystal) 21.5 ◦ C SmA (smectic-A)
33.5 ◦ C N (nematic) 40.5◦ C I (isotropic). The colloidal
mixture is introduced into a glass cell coated with polyimide
Al-1254 and rubbed in an antiparallel way to induce planaralignment of LC molecules. A laser tweezer was built aroundan inverted optical polarizing microscope (Nikon Eclipse
Ti-U) using a cw solid-state laser operating at 1064 nm (Aresis,
Tweez 250Si). An acousto-optic deflector interfaced with acomputer was used for trap movement. The temperature of thesample was controlled by a heater and a proportional-integralderivative (PID) controller with an accuracy of ±0.1 ◦ C. A
camera (CCD, iDs-UI) with a frame rate of 20 frames per
second was attached to the microscope for video recording. A
particle tracking program was used to measure the particles’
coordinates in time with a resolution of ±10 nm.
III. RESULTS AND DISCUSSION

Figure 1 shows the transformation of a Saturn ring defectacross the N-SmA phase transition. There are four visible
colored lobes of distortion around the particle in the Nphase, and these lobes elongate as the N -SmA transition
point is approached [Figs. 1(a) and 1(b)]. Just below thetransition, the lobes shrink to narrow stripes emanating from052501-1©2015 American Physical Society

PHYSICAL REVIEW E 92, 052501 (2015)

ZUHAIL, ČOPAR, MUŠEVIČ, AND DHARA

5 μm

(a)

5 μm

5 μm

35 OC (f)(a)

34 oC (b)33.7 oC (c)

(e)

(f)

33.5 oC

5 μm(b)33.5 OC (g)(d)
K33/K11 = 2(c)

33 C (h)(d)

32 C (i)(e)(j)

K33/K11 = 6

K33/K11 = 8

O

(g)

(h)

(i)

O

FIG. 2. (Color online) (a)–(c) Optical photomicrographs just before the N -SmA phase transition of 8-µm homeotropic microspherein a planar cell. Cell thickness: 11 µm. (d)–(f) Photomicrographs with
red wave-plate between the sample and the analyzer in the N phase.
(See Supplemental Material, video 2 [26].) (g)–(f) Simulated images
with the wave plate at various ratios of K33 /K11 .

FIG. 1. (Color online) (a)–(d) Optical photomicrographs (obtained from polarizing optical microscopy) of a quadrupolar colloidal
particle of diameter 8 µm in an 8CB filled cell of thickness 11 µm
under crossed polarizers at various temperatures across the N -SmA
phase transition. See Supplemental Material, video 1 (f)–(i) [26].
Corresponding images with the red wave-plate (green line denotes
the slow axis) between the analyzer and the sample. Note that theblue and yellow colors have exchanged their positions around themicrosphere in the SmA phase [in Fig. 1(h)] due to higher-ordereffects. (e) The streamlines of the director above the transition. (j) In
the smectic phase, the boundary condition is incompatible with thelayer structure at the equator, where the Saturn ring was present in thenematic phase. Note that the layer thickness is greatly exaggerated.

a barely visible ringlike remnant of the nematic defect on thesurface of the microparticle [Fig. 1(c)]. Further down from thetransition temperature the ring almost disappears [Fig. 1(d)].
A few images were taken between crossed polarizers, with anadditional λ plate (red wave-plate) inserted between the sample
and the polarizer to reconstruct the molecular orientation
around the microsphere. The red color corresponds to ahorizontal orientation of the long molecular axes, whereas
bluish and yellowish colors correspond to anticlockwise andclockwise rotation of the director from the rubbing direction,
respectively. By analyzing these images we can conclude thatthe director distortion around the microparticle expands before
the N-SmA phase transition, and then in the SmA phasethe distortion is highly confined into a narrow region at theequator of the particle, as layer stiffness prevents an extensive
deformation [Fig. 1(c)]. Video I in our Supplemental Material

clearly shows this transformation [26]. As the temperatureis reduced further into the SmA phase, the ring fades away
and two horizontal faint bands appear below and above themicroparticle (see video I [26]). These bands are the distorted
regions in which the SmA layer normal is slightly tilted withrespect to the rubbing direction. This appears due to theimposition of strong homeotropic anchoring of the SmA layers
on the surface of the microsphere. Careful observation suggeststhat the SmA layers in the bands have opposite tilt. A schematic
representation of the director in the N phase corresponding to
the Saturn ring shown in Fig. 1(a) and the confined Saturnring in the SmA phase corresponding to the state shown in
Fig. 1(c) are shown in Figs. 1(e) and 1(j), respectively. The
shaded equatorial band in Fig. 1(j) roughly marks the thinnear-surface region where the boundary condition violation isreconciled in a way that cannot be directly deduced from themicrographs.
In the nematic phase, a pair of colloidal particles withquadrupole defects are manipulated using laser tweezers
and observed across the N -SmA phase transition. A few
representative images are shown in Fig. 2. The size of thelobes representing the director distortion increases as the
N -SmA transition point is approached. Since the splay elastic
constant (K11 ) remains constant, the pretransitional structureof the quadrupole is due to the increase in the bend elastic
constant near the N-SmA phase transition. To confirm thatthe increase in the elastic constant ratio is the main reason,
we performed numerical simulation using Landau–de Gennes
Q-tensor modeling in the N phase, as in Ref. [22]. The free

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SPHERICAL MICROPARTICLES WITH SATURN RING . . .

FIG. 4. (Color online) Variation of the free energy with the angle(θ ) at two different ratios of K33 /K11 obtained by Landau–de Gennes
modeling for the nematic phase. The free energy minimum is around79◦ for both ratios.

FIG. 3. (Color online) (a) Variation of center-to-center separation (D) between a pair of colloidal particles across the N -SmA phasetransition. The red and black color corresponds to particle diametersof 8 and 5.2 µm, respectively. (b) Variation of θ with temperatureacross the N -SmA phase transition.

energy density in the form
f =

B
C
A
Qij Qj i + Qij Qj k Qki + (Qij Qj i )2
2
3
4
L1
L3
2
Qij Qkl,i Qkl,j+ (Qij,k ) +
2
2

(1)

was minimized on a finite difference grid. The phase parameters were set at A = −0.172 × 106 N m−2 , B = −2.12 ×
106 N m−2 , C = 1.73 × 106 N m−2 . The tensorial elastic constants Li relate to the Frank constants via the relations L1 =
2
4
(2K1 + K3 ), L3 = 27S
3 (−K1 + K3 ), assuming K2 = K1
27S 2
and S being the equilibrium bulk order parameter [24,25].
We set L1 = 4 × 10−11 N and varied the ratio K33 /K11 . The
simulated cell was 1.4-µm thick and the particle diameter was
0.9 µm. The corresponding images with increasing ratio of
K33 /K11 are shown in Fig. 2, where the effect is qualitatively
reproduced.
We also measured the equilibrium interparticle separation(D) across the N -SmA phase transition. The temperature
variation of D is shown in Fig. 3. We observe that above
T − TNA = 3 ◦ C, D is independent of temperature and it
increases slightly below this temperature. At the N-SmA
phase transition D changes discontinuously, marking thediscontinuity of the elastic constants of the N phase. For
example, in the N phase (5.2-µm particles), 4 ◦ C above the
N-SmA transition point, D $ 6.8 µm and in the SmA phase,D $ 6.2 µm. This can be contrasted with the past experimentswith different defect states accompanying the particles: fordipolar colloids [22], it was found that the interparticle
separation increases linearly with the elastic anisotropy. In

the case of boojums [23], D decreases linearly with the elastic
anisotropy. Thus the separation between a pair of microspheresis governed by the type of the associated defect and the elastic
anisotropy.
We also measured the angle between the rubbing direction
and the joining line of the centers of the two particles withtemperature [Fig. 3(b)]. The angle (θ ) is larger in the N phasecompared to the SmA phase and discontinuously changes
across the phase transition. For example, in the nematicand SmA phases, the angles are θ $ 79 ± 1◦ and 74.5 ± 1◦ ,respectively. We calculated the free energy profiles for two
different ratios of K33 /K11 in the N phase using Landau–deGennes theory (Fig. 4). It is observed that the equilibrium
angle in both cases is about $79 ± 3◦ , which is comparable tothe experimental values.
When a pair of colloidal particles are assembled and cooled
into the SmA phase, two long and dark tails, asymptoticallyparallel to the rubbing direction, appear [Fig. 5(a)]. The
brighter regions on both sides of the tails indicate that the layers
are strongly bent. The tails are focal conic line defects similar
to those reported recently [22,23], which is supported by themolecular orientation inferred from the photomicrograph withan inserted red-plate between the sample and the polarizer[Fig. 5(b)]. A schematic representation of the SmA layer
structure around the colloids is shown in Fig. 5(c).
Finally, we show the effect of the N -SmA phase transition
on a two-dimensional quadrupolar colloidal crystal. We
5 μm

(a)

5 μm

32 oC (b)(c)

FIG. 5. (Color online) Optical photomicrographs of a pair ofcolloidal particles in the SmA phase (at T -TNA = −1 ◦ C), without(a) and with (b) the wave plate. The blue and yellow colors aroundthe microsphere in the SmA phase are second-order colors. Particle
diameter: 8 µm. (c) A sketch of the smectic layer arrangement withmarked focal lines that are visible in the photomicrographs.

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ZUHAIL, ČOPAR, MUŠEVIČ, AND DHARA

5 μm

SmA phase completely breaks apart into chains of various
lengths perpendicular to the rubbing direction. There are somekind of bondlike links observed between the neighboring
microspheres in the SmA phase. Careful observation suggeststhat in the bond region the SmA layers are parallel between thetwo surfaces of the microparticles. Using a particle tracking
program, we measured the lattice parameters, shown as a, b
in Fig. 6(c). The lattice parameter a remains nearly constant,while b increases rapidly and reaches to a maximum value
before decreasing in the SmA phase. In the N phase there is astrong bend deformation between the neighboring chains [2].
The divergence of b is expected due to the expulsion of the bend
distortion as the SmA phase is approached. This is somewhat
similar to what was observed in case of dipolar colloidal
crystal [22,23]. However, there are certain different features ofthis structural transition compared to that observed in the caseof dipole and boojum colloidal crystals. The main difference isthat in the previous systems we observed a reversible structuraltransition in both heating and cooling. The structure changed
from oblique in the N phase to a hexagonal lattice in the SmA
phase. In the case of quadrupolar crystal, it is irreversible, i.e.,the crystal lattice is broken in the SmA phase while cooling andit does not rearrange while heated back to the N phase. This
could be due to the much lower binding energy (∼10 times)of the quadrupolar crystals compared to dipolar crystals.

5 μm

γ
b a

(a)

36 oC (b)5 μm32.5 oC

5 μm(c)

36 oC (d)32.5 oC(e)IV. CONCLUSIONS

FIG. 6. (Color online) Optical photomicrographs of a 2D
quadrupolar crystal in the (a, c) N phase (36 ◦ C) and (b, d) SmA
phase (32.5 ◦ C) of 8CB liquid crystal; (a,b) without polarizers and(c, d) with crossed polarizers. Particle diameter 5.2 µm, and cell
thickness 8 µm. (e) Temperature variation of lattice parameters a andb across the N -SmA transition (see Supplemental Material, video
3 [26]).

assembled 36 quadrupolar particles with the help of a laser
tweezer and prepared a two-dimensional (2D) quadrupolar
crystal following the procedure reported in Ref. [2]. The
sample was cooled across the N -SmA phase transition at
a rate 0.3 ◦ C/min (see Supplemental Material, video 3).
Figure 6 shows the optical micrographs of the 2D crystal in
the N and SmA phases. In N phase [Fig. 6(a)] the crystal
exhibits an oblique unit cell with lattice parameters a = 6 ±0.05 µm, b = 6.3 ± 0.05 µm, and γ = 64◦ ± 1◦ , and this isconsistent with the previous report [2]. When the temperatureis reduced, the crystal expands laterally and then in the[1] P. Poulin, H. Stark, T. C. Lubensky, and D. A. Weitz, Science
275, 1770 (1997).

In conclusion, we studied the transformation of a Saturnring defect across the N -SmA phase transition. The director
distortion around the particle changes continuously due to thepretransitional increase in the ratio of the bend and splay elastic
constants. The Saturn ring in the SmA phase transforms into abarely visible localized near-surface defect region. The sharp
discontinuity of the nematic elastic constants across the phasetransition is reflected in the equilibrium separation and angle
between the two particles. For a pair of particles the experimental observations are supported by numerical simulations in thenematic phase. In the SmA phase long focal conic line-defects
are observed on both sides of the dimer. The long-range
distortion in the direction of the focal lines interferes with
the integrity of the oblique lattice of 2D quadrupolar particles,
which breaks irreversibly at the transition into the smectic
phase. The irreversible structural transition indicates that theinteraction between two neighboring chains of particles in a
2D crystal is weak and short range.
ACKNOWLEDGMENTS

We gratefully acknowledge support from the Departmentof Science and Technology, Govt. of India (SR/NM/NS134/2010) and DST-PURSE. K.P.Z acknowledges the UGC
for a fellowship. The authors also acknowledge support fromSlovenian Research Agency under Contracts No. J1-6723(I.M.), No. P1-0099 (I.M. and S.Č.), and No. Z1-6725 (S.Č.).

[2] I. Muševič, M. Škarabot, U. Tkalec, M. Ravnik, and S. Žumer,
Science 313, 954 (2006).

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[3] H. Stark, Phys. Rep. 351, 387 (2001).
[4] R. P. Trivedi, D. Engstrom, and I. I. Smalyukh, J. Opt. 13, 044001(2011).
[5] O. V. Kuksenok, R. W. Ruhwandl, S. V. Shiyanovskii, and E. M.
Terentjev, Phys. Rev. E 54, 5198 (1996).
[6] P. Poulin, V. Cabuil, and D. A. Weitz, Phys. Rev. Lett. 79, 4862
(1997).
[7] T. C. Lubensky, D. Pettey, N. Currier, and H. Stark, Phys. Rev.
E 57, 610 (1998).
[8] V. G. Nazarenko, A. B. Nych, and B. I. Lev, Phys. Rev. Lett. 87,
075504 (2001).
[9] Y. Gu and N. L. Abbott, Phys. Rev. Lett. 85, 4719
(2000).
[10] J. C. Loudet, P. Poulin, and P. Barois, Europhys. Lett. 54, 175(2001).
[11] P. G. Petrov and E. M. Terentjev, Langmuir 17, 2942
(2001).
[12] M. Yada, J. Yamamoto, and H. Yokoyama, Phys. Rev. Lett. 92,
185501 (2004).
[13] A. B. Nych, U. M. Ognysta, V. M. Pergamenshchik, B. I. Lev, V.
G. Nazarenko, I. Muševič, M. Škarabot, and O. D. Lavrentovich,
Phys. Rev. Lett. 98, 057801 (2007).
[14] I. I. Smalyukh, O. D. Lavrentovich, A. N. Kuzmin, A. V.
Kachynski, and P. N. Prasad, Phys. Rev. Lett. 95, 157801(2005).

[15] R. W. Ruhwandl and E. M. Terentjev, Phys. Rev. E 55, 2958
(1997).
[16] M. Škarabot, M. Ravnik, S. Žumer, U. Tkalec, I. Poberaj,
D. Babič, N. Osterman, and I. Muševič, Phys. Rev. E 77, 031705(2008).
[17] B. I. Lev and P. M. Tomchuk, Phys. Rev. E 59, 591 (1999).
[18] O. Guzman, E. B. Kim, S. Grollau, N. L. Abbott, and J. J. de
Pablo, Phys. Rev. Lett. 91, 235507 (2003).
[19] M. Ravnik, M. Škarabot, S. Žumer, U. Tkalec, I. Poberaj,
D. Babič, N. Osterman, and I. Muševič, Phys. Rev. Lett. 99,
247801 (2007).
[20] I. Muševič, M. Škarabot, D. Babič, N. Osterman, I. Poberaj,
V. Nazarenko, and A. Nych, Phys. Rev. Lett. 93, 187801 (2004).
[21] M. Vilfan, N. Osterman, M. Čopič, M. Ravnik, S. Žumer, J.
Kotar, D. Babič, and I. Poberaj, Phys. Rev. Lett. 101, 237801(2008).
[22] K. P. Zuhail, P. Sathyanarayana, D. Seč, S. Čopar, M. Škarabot,
I. Muševič, and S. Dhara, Phys. Rev. E 91, 030501(R) (2015).
[23] K. P. Zuhail and S. Dhara, Appl. Phys. Lett. 106, 211901 (2015).
[24] M. Ravnik and S. Žumer, Liq. Cryst. 36, 1201 (2009).
[25] C. Blanc, D. Svenšek, S. Žumer, and M. Nobili, Phys. Rev. Lett.
95, 097802 (2005).
[26] See Supplemental Material at http://link.aps.org/supplemental/
10.1103/PhysRevE.92.052501 to access video clips of the phasetransition.

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