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JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 21 (2009) 505103 (5pp)

doi:10.1088/0953-8984/21/50/505103

Thermotropic uniaxial nematic liquid
crystal with ±1/2 strength disclinations:defect–antidefect production andcorrelation
Surajit Dhara1 , T Arun Kumar1, Ken Ishikawa2 andHideo Takezoe2
1

School of Physics, University of Hyderabad, Hyderabad 500046, India
Department of Organic and Polymeric Materials, Tokyo Institute of Technology,
2-12-1 O-okayama, Meguro-ku, Tokyo 152-8552, Japan
2E-mail: sdsp@uohyd.ernet.in and takezoe.h.aa.@m.titech.ac.jp

Received 29 September 2009, in final form 30 October 2009
Published 23 November 2009
Online at stacks.iop.org/JPhysCM/21/505103
Abstract
We report the observation of mostly wedge disclinations of strength ±1/2 in a uniaxial nematic
liquid crystal. The dense network of the disclination is stabilized in cells of thickness (∼6 μm)
treated with perfluoropolymer. The distribution of net defect strength was found to be Gaussian.
The scaling exponent obtained from the width of the distribution is ν = 0.29 ± 0.08 and is ingood agreement with the prediction based on the cosmological Kibble mechanism. The
exponent and the constant of the scaling relation are compared with the values reported forlyotropic liquid crystal with half-strength disclination by Mukai and co-workers.
(Some figures in this article are in colour only in the electronic version)

crystals with the nematic to isotropic (NI) phase transition
have proven to be simplest, and are also the systems inwhich it is easiest to form and study the defect–antidefectpairs. Here antidefects are the defects with opposite sign of
the winding number and are generally written with a minus
sign. In the nematic case, the tensor order parameter Q i j
breaks the rotational symmetry of the isotropic liquid phaseand the defect–antidefect pairs are formed spontaneously.
There are a few reports on studies of the formation of
defect–antidefect pairs and their correlation in thermotropic
nematic liquid crystals with defect strength 1 [8] and inlyotropic liquid crystal with half-strength defects [9]. Thus
it is important to investigate the correlation in various othersystems with different defect strengths. In particular, there
are no reports on the observation of mostly half-strength
defects in the thermotropic nematic liquid crystals. In thispaper we report on a unique system in which mostly halfstrength defects in a uniaxial nematic liquid crystal arestabilized in perfluoropolymer treated cells. We also studied
the defect–antidefect correlation from the cosmological pointof view and a good agreement was found with the theoreticalprediction [10].

1. Introduction
Thermotropic liquid crystals exhibit a rich variety of textures
under the polarizing microscope [1, 2]. The different phases
exhibit their own characteristic defect structures under thepolarized optical microscope. For example, nematic liquid
crystals are characterized by observing schlieren texture.
Smectic and cholesteric liquid crystals are characterized by
focal conic and fingerprint textures. Usually the schlieren
texture in the nematic case contains point or line defects
with the appearance of four and two brashes under crossed
polarizers, respectively [2]. These defects belong to aclass, called the topological defects, in a system with broken
symmetry. The vortices in superfluid helium and in the XY
model, the dislocation in periodic crystals, and the cosmic
strings in the early Universe also belong to the topologicaldefects and have been a common interest of scientists invarious branches in physics. There are several systems incondensed matter reported in which topological defects areformed in the laboratory from the symmetry breaking phasetransition, such as superfluid helium [3], thermotropic liquid
crystals [4–6] and superconductors [7]. Among these, liquid0953-8984/09/505103+05$30.00

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J. Phys.: Condens. Matter 21 (2009) 505103

S Dhara et alhysteresis, a bistable device has also been proposed byus [13]. Very recently we also reported on the shock freehomeotropic alignment of SmA and SmC∗ phases in thisalignment layer [14]. In the present experiment the typical cell
thickness used was ∼6 μm. All the observations were carriedout while cooling the sample. The temperature of the sample
was controlled with a Mettler Hotstage and a controller to an
accuracy of 0.1 ◦ C. Texture observation was carried out withthe help of an optical polarizing microscope.
Figure 1. (a) Structure of the perfluoropolymer alignment layer.
(b) Schematic of the molecular structure CCN-47 used in theexperiment.

3. Results and discussion
In figure 2, we show the textures obtained for somecommon liquid crystals with positive and negative dielectricanisotropies in CYTOP coated cells. 8CB (octyl-cyanobiphenyl), a liquid crystal with a positive dielectric anisotropy( = ∼8 at 30 ◦ C), exhibits a homeotropic alignment(figure 2(a)). MBBA (methoxybenzylidene butylaniline) liquid
crystal with a negative dielectric anisotropy ( = −0.6 at30 ◦ C) shows homeotropic alignment just below the nematic–
isotropic transition temperature (figure 2(b)) and the director
gradually tilts from the vertical direction as the temperature
is lowered [12]. In the case of MLC-6608, a large negativedielectric anisotropy ( = −10 at 25 ◦ C) liquid crystal
mixture, we noticed a planar texture with mostly ±1 defects(figure 2(c)). However, the single-component liquid crystal
CCN-47, with a similar large negative dielectric anisotropy,exhibits mostly ±1/2 defects and the defects are stable(figure 2(d)).
The temporal variation of the typical textures of CCN47 on CYTOP just after the transition is shown in figure 3.
The sample is cooled at the rate of 0.2 ◦ C s−1 in a 6 μm
thick cell. We noticed that mostly ±1/2 and a few ±1
disclinations appeared during the phase transition. After 5 s,some of the ±1 defects disappeared from the field of view
(figure 3(b)). After 15 s, no textural change occurs in the2. Experimental details
We used perfluoropolymer, poly[perfluoro(4-vinyloxy-1-butene)], known as CYTOP (Asahi Glass Co., Ltd), as an alignment
layer. The chemical structure is shown in figure 1(a). Thepolymer was dissolved in CT-Solv-180 at a weight ratio of 1:2
and the solution was spin coated on ITO (indium tin oxide)
coated glass substrates by using a conventional spin coating
method. Then it was baked at 100 ◦ C for 30 min. The
typical polymer film thickness was ∼100 nm. The chemicalstructure of the compound 4 -butyl-4-heptyl-bicyclohexyl-4carbonitrile (CCN-47) is shown in figure 1(b). The compoundexhibits the following phase transition temperatures undercooling: Cr 25.6 ◦ C SmA 28.2 ◦ C N 57.3 ◦ C I. It shows alarge negative dielectric anisotropy ( = −5.7 at 30 ◦ C).
The chemical structure and some physical properties of thepresent compound have already been reported [11]. CCN47 in CYTOP coated cells shows a planar texture in thenematic phase. It exhibits a discontinuous anchoring transition
from planar to homeotropic at ∼45 ◦ C with a large hysteresis(∼5◦ ) [12] in heating and cooling. Exploiting this large

Figure 2. (a) Homeotropic texture of 8CB at 37 ◦ C. (b) Homeotropic texture of MBBA at 38 ◦ C. (c) Schlieren texture with 1 defects inMLC-6608 at 85 ◦ C. (d) Schlieren texture with 1/2 defects in CCN-47 at 50 ◦ C. Crossed arrows denote the crossed polarizers.

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S Dhara et al

implication is that the director is in the plane, as any tilt of thedirector in such a case is topologically forbidden [19].
It is also clear from figure 3 that the interdefect separation
is sufficiently large and we can identify the strength of thedefect by the rotation of the polarizer. If the brashes rotate
in the same direction as the rotation of the polarizer then it isassigned a positive defect; otherwise it is assigned a negative
defect. In figure 4 we show a small part of the texture before
and after rotating the polarizer by 20◦ . The open squares andopen circles denote the defect with positive sign and negative
sign respectively. We have taken about 25 images (textures)from the different parts of the sample at the same temperature(TNI − T = 0.5◦ ). Each texture was divided into several small
square regions of equal size. In each square, suppose there aren + of the +1/2 defects and n − of the −1/2 defects. Then thetotal number of defects is N = n + + n − and the net defectstrength is n = n + − n − . The net defect strength in eachsquare region was determined for the statistical analysis of all
the 25 textures. For the present size of the square region wecan get three different average values for the total of defect–antidefect pairs ( N ), namely N = 8, 12, and 16. The numbersof regions obtained in each case were about 96, 52 and 32
respectively. For the present size of the square, not enoughdata were available for statistical analysis with N < 8 or
N > 16. In order to find the correlation we followed theprocedure mentioned in [9]. For each square region the net
defect strength (n ), i.e. the number of defects minus thenumber of antidefects, is determined. Then the occurrence
distribution, f (n), for each n was determined. In figure 5,we show the variation of f (n) versus n for N = 8, 12,and 16. A nonlinear least squares program is used to fit thefollowing equation [8]:f (n) = a exp[−(n − n)2 /2σ 2 ](1)where σ is the variance of the distribution. Good Gaussian
fittings are obtained for various values of N . The fit parameterswith standard error are summarized in table 1. The maximumof the Gaussian is centered at zero and consistent with the fit
parameter n . The widths of the Gaussian distribution σ for
N = 8, 12, and 16 are significantly different to those for thesystem with ±1 defects as measured by Digal et al [8]. The
σ values for our system are comparable to the values reported
for N = 10, 14, and 18 in the case of lyotropic liquid crystals
with ±1/2 by Mukai et al [9]. Interestingly our measured σvalues are almost half the values predicted for N = 10, 20,and 30 for a square shaped elementary region. In order tofind the correlation, the σ values were fitted with the followingequation:
σ = C Nν
(2)

Figure 3. Textures observed under crossed polarizers in a 6 μm thickcell at (a) 2 s, (b) 5 s and (c) 15 s after the TNI = 57.3 ◦ C transition(cooling rate 0.2 ◦ C).

cell (figure 3(c)). No noticeable change was observed evenafter several hours of waiting. This shows that the disclination
network is very stable. The abundance of ±1/2 strength
disclinations is sometimes regarded as evidence of a biaxial
system [15], but we confirmed that the present medium isdefinitely uniaxial, as is clear from the completely dark viewin the homeotropically aligned state. In the nematic phase of alyotropic liquid crystal, similar texture was also observed [9].
The energy per unit length of an isolated disclination, from thesimple one-elastic-constant approximation, can be written as
W = Wc + π K S 2 ln(R/rc ) [16], where Wc is the energy perunit length of the core, R is the sample size, rc is the radiusof the core and S is the strength of the defect. Thus for agiven sample size R , the disclination with strength ±1 will cost
four times the energy of those with strength ±1/2. However,experimentally, in the nematic phase ±1 defects are more often
seen in the literature [8, 16–19] and to our knowledge thisis the first example of a case of thermotropic nematic liquid
crystal exhibiting mostly ±1/2 strength defects. The physicalwhere C is a constant and the exponent ν determines thecorrelation. The theoretical value of exponent ν is 1/2 and
1/4 in cases where the defects and antidefects are uncorrelated
and correlated respectively. The width of the distribution (σ )for different N values and the fitting with equation (2) is shownin figure (6). For the present system we find ν = 0.29 ± 0.08
and C = 0.35. This is similar to the value reported by Mukaiet al [9] and almost half of the value reported by Digal et al[8]. This difference with respect to the case of ±1 defects may
be attributed to the strength of the disclination.
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S Dhara et al

Figure 4. (a) Part of the texture obtained from figure 2 between the crossed polarizers at 53 ◦ C. (b) Texture after rotating the crossed
polarizers by an angle of 20◦ . Squares and circles represent the +1/2 and −1/2 defects, respectively.

Figure 5. Plot of the frequency f (n) versus n .

Figure 6. A log–log plot for determining the critical exponent ν .
Table 1. Fit parameters obtained by fitting equation (1).

4. Conclusions
In conclusion, in the present work we reported that mostly±1/2 defects can be stabilized over a large area inuniaxial nematic liquid crystal on surfaces treated withperfluoropolymer. We described the evolution of defects
from the nematic–isotropic phase transition. We found thatthe defect–antidefect correlation exponent ν = 0.29 andthis is consistent with the Kibble mechanism [10] for thedefect–antidefect correlation in the early Universe. Thisvalue is also consistent with the previous reports on uniaxial
thermotropic liquid crystal with ±1 and lyotropic liquid crystal
with ±1/2 strength defects. The present study together withthe results of [8] and [9] clearly suggests that irrespectiveof the strength of the defects and the experimental systems,
the critical exponent of the defect–antidefect correlation isuniversal in nature. Commonly ±1 defects are observed inNI quench studies. In the present case it is not yet clearly
understood why mostly ±1/2 defects are observed in thepresent compound in cells treated with perfluoropolymers.
However, the molecular dynamics studied by Billert et al [20]and recent computer simulations by Bradac et al [21] on thekinetics of the NI quench suggest that the point defects areNa8121651.7 ± 3.7
24.6 ± 2.2
13.5 ± 1.6

n−0.02 ± 0.07
−0.06 ± 0.08
−0.04 ± 0.011

σ0.65 ± 0.050.73 ± 0.07
0.80 ± 0.10

much more mobile compared to the line defects, and easier
to annihilate than the lines. It may be mentioned that thesurface energy of perfluoropolymer is much smaller comparedto the other surfaces [14], hence, the point defects are highly
mobile on a perfluoropolymer surface and as a result they areannihilated easily. Therefore, more ±1/2 strength defects areexpected in NI quench systems. A systematic measurementof the anchoring energy and the knowledge of some physicalproperties are needed for further understanding of this specificsystem. Finally we expect that the present study will inspire
various other experiments on defect–antidefect formation andcorrelation.

Acknowledgments
One of the authors, S Dhara, gratefully acknowledges
the support from DST, Government of India, for the4

J. Phys.: Condens. Matter 21 (2009) 505103

S Dhara et al

project Number SR/FTP/PS-48/2006 under the First Track
Scheme. The authors also gratefully acknowledge the valuablesuggestions made by the referees.

[8] Digal S, Ray R and Srivastava A 1999 Phys. Rev. Lett. 83 5030
[9] Mukai H, Fernandes P R G, de Oliveira B F and Dias G S 2007
Phys. Rev. E 75 061704[10] Kibble T W B 1976 J. Phys. A: Math. Gen. 9 1387
[11] Dhara S and Madhusudana N V 2008 Phase Transit. 81 561[12] Dhara S, Kim J K, Jeong S M, Kogo R, Araoka F,Ishikawa K and Takezoe H 2009 Phys. Rev. E 79 060701R[13] Kim J K, Araoka F, Jeong S M, Dhara S, Ishikawa K andTakezoe H 2009 Appl. Phys. Lett. 95 063505[14] Jeong S M, Kim J K, Shimbo Y, Araoka F, Dhara S, Ha N Y,Ishikawa K and Takezoe H 2009 Adv. Mater. 21 1
[15] Chandrasekhar S et al 1998 Liq. Cryst. 24 67[16] Chaikin P M and Lubensky T C 1998 Principles of Condensed
Matter Physics (Cambridge: Cambridge University Press)[17] Chandrasekhar S 1977 Liquid Crystals (Cambridge: CambridgeUniversity Press)[18] Brown G H (ed) 1975 Advances in Liquid Crystals vol 1(New York: Academic) p 267[19] Madhusudana N V and Pratibha R 1982 Curr. Sci. 57 877[20] Billert J L, Smondyrev A M, Loriot G B and Pelcovits R A
1999 Phys. Rev. E 60 6831
[21] Bradac Z, Kralj S and Zumer S 2002 Phys. Rev. E 65 021705

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