RAPID COMMUNICATIONSPHYSICAL REVIEW E 91, 030501(R) (2015)

Topological defect transformation and structural transition of two-dimensional colloidal crystals
across the nematic to smectic- A phase transition
K. P. Zuhail,1 P. Sathyanarayana,1 D. Seč,2,3 S. Čopar,2,3 M. Škarabot,3 I. Muševič,2,3 and S. Dhara1,*
1

School of Physics, University of Hyderabad, Hyderabad-500046, India
2
J. Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
3
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia(Received 22 April 2014; revised manuscript received 21 November 2014; published 10 March 2015)
We observe that topological defects in nematic colloids are strongly influenced by the elasticity and onsetof smectic layering across the nematic (N ) to smectic-A (SmA) phase transition. When approaching the SmA
phase from above, the nematic hyperbolic hedgehog defect that accompanies a spherical colloidal inclusion istransformed into a focal conic line in the SmA phase. This phase transformation has a strong influence on the
pairwise colloidal interaction and is responsible for a structural transition of two-dimensional colloidal crystals.
The pretransitional behavior of the point defect is supported by Landau–de Gennes Q-tensor modeling accounting
for the increasing elastic anisotropy.
DOI: 10.1103/PhysRevE.91.030501

PACS number(s): 61.30.Jf, 82.70.Dd

Topological defects [1,2] are point-, string-, wall-, or
looplike singularities of physical fields that are studied in very
different systems, such as the early Universe or condensed
matter. In condensed matter, topological defects are observed
as vortices in superfluid helium [3], Abrikosov vortices in
type-II superconductors [4], domain walls in soft ferromagnets[5], etc. In liquid crystals, topological defects are generated by
the symmetry breaking phase transition from the isotropic to
the liquid crystalline phase [6]. They can also be produced
and manipulated in a controlled way by introducing foreign
particles into the uniformly aligned liquid crystal [7,8].
For example, microspheres with perpendicular (homeotropic)
surface alignment of nematic liquid crystal (NLC) molecules
are accompanied by topological point defects, thus forming
elastic dipoles in the bulk NLC. Depending on the anchoring
strength and confinement, defects may also appear in the formof Saturn rings and boojums [9]. Due to elastic distortionof the NLC around each particle, nematic colloids experiencelong range forces [10], which opens new directions in colloidal
assembly. There are many theoretical and experimental studies
of topological defects and colloidal interactions in the NLCs
[10–16], mainly motivated by fascinating self-assembled
colloidal superstructures, such as chains [7], two- [8] andthree-dimensional crystals [11], entangled clusters [13,17,18],
knotted and linked particles [19–22], and chiral nematic
colloids [23,24].
However, little is known about the behavior of topologicaldefects and colloidal structures when crossing the phase transition point, such as the N -SmA phase transition. In this Rapid
Communication, we study the transformation of a topologicaldefect across the N -SmA phase transition, where, in addition
to the long range orientational order of NLC molecules, a onedimensional positional order of NLC molecules is establishedand results in smectic layering on the nanoscale. We observea structural transition in a two-dimensional (2D) dipolar
colloidal crystal, where defect transformation, interparticle
separation, and lattice rearrangement are governed by theelasticity of the nematic and SmA phases.

In experiments, we used silica microspheres, coatedwith DMOAP [octadecyldimethyl (3-trimethoxysilylpropyl)
ammonium chloride] to obtain perpendicular (homeotropic)alignment of LC molecules on the surface [8]. The microspheres were dispersed in 8CB (4! -n-octyl-4-cyano-biphenyl)
liquid crystal. 8CB exhibits the following phase transitions:
Cr (crystal) 21.5 ◦ C SmA (smectic-A), 33.5 ◦ C N (nematic),
40.5 ◦ C I (isotropic), and a positive elastic anisotropy, i.e.,!K = K33 − K11 > 0, where K33 and K11 are the bend andsplay elastic constants. The experimental details and additional
measurements are stated in the Supplemental Material (SM)[25].
Figures 1(a), 1(c), and 1(e) show the transformation ofa hyperbolic point defect, associated with an isolated microsphere, across the N -SmA phase transition in a thick planar cell
(15 µm). One can clearly see three stages in the transformationof a point defect in the N phase into a line defect in the SmA
phase. As the temperature is decreased from the nematic into
the SmA phase, a point defect in the nematic 8CB LC isforced to move towards the microsphere’s surface [compare
Figs. 1(a) and 1(c)], and a high-contrast “tail” emanatesfrom the point defect. To confirm that the elastic constantanisotropy is the main contributor to the transformations seenin the experiment, this effect was qualitatively reproduced byLandau–de Gennes Q-tensor modeling [26,27] of the N phase.
Lowering the temperature increases the K33 (bend) elastic
constant relative to K11 (splay). Figures 1(b) and 1(d) show
simulated optical transmission micrographs between crossed
polarizers for two elastic constant ratios in the N phase. The
increase of K33 [Figs. 1(c) and 1(d)] replaces the energeticallycostly bend distortions around the particle by more favorable
splay distortions, concentrated in an elongated tail, forming
a sort of a “splay soliton.” A video (Video-1) of topologicaldefect transformation across the N -SmA transition and thedetails of the simulation are given in the SM [25].
In the SmA phase we observe a long and dark tail-likestructure in place of the splay soliton seen already in the
N phase [Fig. 1(e)]. To find the molecular orientation in
the tail, we used a full wave retardation plate (530 nm)
between the analyzer and the sample as is explained in the
SM. This method helped us to reconstruct an approximate*

sdsp@uohyd.ernet.in

1539-3755/2015/91(3)/030501(4)030501-1©2015 American Physical Society

RAPID COMMUNICATIONSPHYSICAL REVIEW E 91, 030501(R) (2015)

K. P. ZUHAIL et al.

(a)

N

(b)

5 µm
(c)

N-SmA

(d)

(e)

SmA

(f)

FIG. 1. (Color online) (a),(c),(e) Experimental optical micrographs of 7 µm homeotropic microspheres in a thick planar 8CB
cell between crossed polarizers in the nematic phase, (a) just before
the N -SmA transition (c) and after the transition (e). Cell thickness
is 15 µm. The hyperbolic hedgehog defect in the nematic phaseis marked with an arrow (a). (b), (d) Simulated micrographs of
the N phase for (b) equal Frank elastic constants K33 = K11 andfor (d) increased bend elastic constant (K33 = 8K11 ) which pushes
the hedgehog towards the particle’s surface. (f) The sketch of areconstructed cross section of smectic layer structure in the vicinityof the microsphere.

geometry of the layer structure, as shown schematically in
Fig. 1(f). The requirement for equally spaced layers in the
SmA phase geometrically constrains the preexisting nematicorder; because the splay soliton in the form seen in the N phase
does not allow smooth equidistant layers, the tail is forced
into a focal line with the cusp angle in the layer shape that
falls off slowly with the distance from the particle [Figs. 1(e)and 1(f)]. The layer incompressibility further decreases therange of influence of the boundary conditions imposed by theparticle. The homeotropic boundary condition is reconciled
in a zone next to the microsphere surface, too narrow to bediscerned optically. Outside this energetically costly boundary
zone, the layers are only moderately bent compared to the far
field. On the defect-adjacent hemisphere, the layers outside
the boundary zone closely resemble the configuration for theplanar surface anchoring [28], while the opposite hemisphereobeys the homeotropic anchoring and the boundary zone is notneeded. The particle effectively “lenses” the layers into a cusp,
similar to the effect of curvature in 2D smectic structures [29].
Because the elasticity and smectic layering have such astrong influence on topological defects, we expect that theforces between colloidal particles should be strongly influenced by the N -SmA transition as well. We first investigate
a pair of particles, which spontaneously chain together in
the N phase, as shown in the right-hand inset in Fig. 2(a).
The microspheres attract via the elastic deformation of the
NLC but are separated by a topologically required hedgehog
point defect in between, which stabilizes the structure. The

FIG. 2. (Color online) (a) Variation of the center to center separation (D) of two homeotropic microspheres across the N -SmA
transition in 8CB. See SM (Video-2). The microsphere diameter is
5.2 µm, and the cell thickness is 7.7 µm. Insets show experimental
images of a pair of dipolar microspheres before and after N -SmA
transition. (b) Variation of D/R0 above the N -SmA transition as afunction of the elastic constant anisotropy (1 − K11 /K33 ). The redline is the best fit to Eq. (1) with a slope 0.06. The scale on the top
shows the temperature range that corresponds to the anisotropy range
on the principal axis. (c) Variation of the separation with reduced
temperature below the transition (logarithmic scale). The red line
shows the best fit to the power law (D − D0 ) ∼ |χ |α with α = 0.38.

equilibrium separation of the pair is governed by the nematic
elasticity and should therefore be highly sensitive to the onsetof N -SmA phase transition.
Figure 2(a) shows the temperature dependence of microsphere separation (D) across the N -SmA transition. We
see that D increases with decreasing temperature in the Nphase, which is followed by a sharp jump at the N -SmA
phase transition. In 8CB, we find 5 ◦ C above the N -SmA
transition, D = 2.3R0 , which is close to the previous reports[30–32]. Recently, James et al. theoretically studied the effectof anchoring strength and elastic anisotropy on the separationof a pair of dipolar nematic colloids [33]. In 8CB, considering R0 = 2.6 µm, the anchoring strength W % 10−4 J/m2 ,and the bend elastic constant around K33 = 10 pN, we get
K33 /W R0 ! 0.04, which is in the limit of strong surface
anchoring, justifying the approximation for the equilibrium
separation (D) with a linear relation
D/R0 % 2.439 + 0.0878(1 − K11 /K33 ).

(1)

In order to verify that, we collected the temperaturedependencies of K11 and K33 for 8CB from Ref. [34] and030501-2

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TOPOLOGICAL DEFECT TRANSFORMATION AND . . .

interpolated for all temperatures of interest. Since K33 depends
on temperature, we selected an appropriate temperature range(considering the size of the microsphere) in which theassumption of strong anchoring is valid. Figure 2(b) shows
the variation of D/R0 as a function of elastic anisotropy(1 − K11 /K33 ) with the best fit to Eq. (1). The experimentallymeasured slope of D/R0 for 5.2 µm particles is 0.06, which isslightly lower than the predicted value. In the selected regime,
the anisotropy is small and the linearity holds. Closer to thetransition, an increase of K33 makes the surface anchoring
comparatively small, which is another explanation for why thehedgehog can move closer to the surface.
The N-SmA phase transition is reflected in a rather sharp
discontinuity of the interparticle separation D, marking thediscontinuity of the elastic constants and the onset of smectic
layering. Furthermore, in the SmA phase [Fig. 2(a)], D isincreasing with decreasing temperature from the N -SmA
transition point and tends to saturate at lower temperatures.
This indicates that the layer compressibility is important,because it is known that in the SmA phase, the layer
compression elastic modulus (B) varies as B ∼ χ α , whereχ ≡ (1 − T /TNA ) is the reduced temperature, and the criticalexponent α is between 0.39 and 0.42. To identify the role of thelayer compression modulus, we fitted the temperature variationof separation to (D − D0 ) ∼ χ α , where D0 is the minimum
value just below the transition. As shown in Fig. 2(c), thebest fit gives critical exponent α = 0.38. This is in the range(0.36–0.38) of reported values measured by a second-soundresonance technique [35]. To confirm the generality of ourfindings, we studied particles with different size in 8CB andalso in a different liquid crystal, namely, 4! -butyl-4-heptylbicyclohexyl-4-carbonitrile (CCN-47). It may be mentionedthat CCN-47 exhibits a negative elastic anisotropy (!K < 0).
The results are presented in the SM (also see Video-3) [25].
Finally we show the effect of N -SmA phase transition on
the 2D dipolar colloidal crystal, assembled by laser tweezers.
The crystal was prepared following the procedure reported inRef. [8]. Figure 3(a) shows a 2D dipolar crystal made of 64
assembled microspheres with diameter 5.2 µm in the nematic.
Figure 3(b) shows the image of a 2D crystal in the SmA phaseand it was obtained by cooling the sample from the nematic
phase at a rate of 0.3 ◦ C/min. The corresponding images
under crossed polarizers are given in the SM. In the nematic
phase it shows an oblique lattice with lattice constants a =6.18 ± 0.05 µm (% 2.38R0 ), b = 6.51 ± 0.05 µm (% 2.5R0 ),and γ = 61◦ ± 1◦ , which are comparable to those reported byMuševič et al. [8] in 5CB (4-cyano-4! -pentylbiphenyl) NLC.
As the temperature is reduced towards the SmA phase, thecrystal expands in a direction perpendicular to the chains
and the point defects are transformed into focal conic lines
at the transition; consequently, the crystal collapses into ahexagonal structure in the SmA phase (see SM Video-4) [25].
The transition is reversible in the sense that the 2D dipolar
crystal is almost restored and the hyperbolic hedge defects
reappear at the same places when the sample is again heatedback to the nematic phase (see SM Video-5) [25]. Carefulobservation indicates that in the SmA phase, the microspheres
are slightly displaced from the common plane (∼1 µm) inan alternating striped pattern. This suggests that it is morefavorable to buckle the crystal into the third dimension, which

FIG. 3. (Color online) Optical photomicrograph of a 2D dipolar
crystal of particles with diameter 5.2 µm in the N phase (35.8 ◦ C)
and SmA phase (32.7 ◦ C) of 8CB. (a),(b) Between crossed polarizers;(c),(d) without polarizers. See SM video-4. (e) Temperature variationof the lattice parameters a and b across the N -SmA transition.
Inset: Temperature variation of b with the best fit (red line) tob = b0 + C|χ |−β with β = 0.8 (logarithmic scale). Diameter of the
microspheres is 5.2 µm and cell thickness is 9.3 µm.

will play a more important role in thicker cells. The areaof the 2D crystal decreases from 2400 (nematic phase) to1560 µm2 (SmA phase) which is about a 35% reduction of
the initial crystal area. This is due to the absence of hedgehog
point defects and out of plane shift of some particles. Usinga particle tracking method, we measured the temperaturedependence of the lattice constants, shown as a and b in
Fig. 3(c). Interestingly, the lattice parameter a (measuring theseparation along the chain) remains nearly constant, while b
exhibits the presmectic divergence and finally both jump toa smaller value in the SmA phase. We also measured theangle γ [Fig. 3(a)] and no noticeable change is observed
across the phase transition. In the N phase there is a strong
bend deformation of the director between the neighboring
chains [8]. This bend deformation is expected to get expelled
due to the pretransitional divergence of the bend elastic0
constant, which is given by K33 = K33+ A|χ |−1 [36]. In
order to identify the role of bend deformation, the temperaturedependence of b is fitted to b = b0 + C|χ |−β , where b0 and
C are constants. The fit [inset of Fig. 3(c)] gives the exponent
β % 0.8, close to the theoretical exponent of pre-smectic bendelastic constant and suggests that the divergent behavior ofb, consequently the anisotropic crystal expansion is due tothe expulsion of bend deformation. Further, we studied thestructural transition in CCN-47 on both cooling and heating(Video-6 and Video-7 in SM) [25]. The structural transition and030501-3

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K. P. ZUHAIL et al.

the temperature dependence of lattice parameters are almost
similar to that observed in 8CB. The experimental results arepresented in the SM. This suggests that the structural transitionof the 2D dipolar crystal across the N -SmA phase transition isdriven by transformation of topological defects and controlled
by the elasticity of the respective media.
In conclusion, we observed the transformation of thehyperbolic hedgehog point defect into a focal conic line
across the N-SmA phase transition. This transformation isdue to the N-SmA pretransitional divergence of the bendelastic constant and the emergence of smectic layering.
It strongly affects not only pair colloidal interactions, but
has a profound effect on the restructuring of 2D colloidal

crystals across the N-SmA phase transition. We believe thisstudy of transformation of topological defects across thephase transition will initiate new theoretical and experimental
investigations on the topology and phase transitions in soft
matter.

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S.D. acknowledges the support from DST-PURSE and
UPE-II. K.P.Z. acknowledges UGC for support through afellowship. I.M., S.Č., and D.S. acknowledge Slovenian
Research Agency (ARRS) for support under Contracts No. J16723 (D.S.) and No. Z1-6725 (S.Č.). D.S. also acknowledges
funding by EU MC program FREEFLUID. We thank Prof.
Slobodan Žumer for useful discussions.

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