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Effect of an electric field on defects in a nematic
liquid crystal with variable surface anchoring
a

a

b

b

c

T. Arun Kumar , V.S.S. Sastry , Ken Ishikawa , Hideo Takezoe , N.V. Madhusudana &Surajit DharaaaSchool of Physics, University of Hyderabad, Hyderabad, 500046, Indiab

Department of Organic and Polymeric Materials,, Tokyo Institute of Technology, 2-12-1S8-42 O-okayama, Meguro-ku, Tokyo, 152-8552, Japanc

Raman Research Institute, C.V. Raman Avenue, Bangalore, 560080, India

Available online: 12 Aug 2011

To cite this article: T. Arun Kumar, V.S.S. Sastry, Ken Ishikawa, Hideo Takezoe, N.V. Madhusudana & Surajit Dhara (2011):
Effect of an electric field on defects in a nematic liquid crystal with variable surface anchoring, Liquid Crystals, 38:8,
971-979
To link to this article: http://dx.doi.org/10.1080/02678292.2011.587962

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Liquid Crystals,
Vol. 38, No. 8, August 2011, 971–979

Effect of an electric field on defects in a nematic liquid crystal with variable surface anchoring
T. Arun Kumara , V.S.S. Sastrya , Ken Ishikawab , Hideo Takezoeb , N.V. Madhusudanac and Surajit Dharaa *aSchool of Physics, University of Hyderabad, Hyderabad-500046, India; b Department of Organic and Polymeric Materials, Tokyo
Institute of Technology, 2-12-1-S8-42 O-okayama, Meguro-ku, Tokyo-152-8552, Japan; c Raman Research Institute, C.V. Raman
Avenue, Bangalore-560080, India

Downloaded by [University of Hyderabad] at 03:42 12 August 2011

(Received 22 January 2011; final version received 10 May 2011)
We report experimental studies on defects in a nematic liquid crystal with negative dielectric anisotropy mounted
in a cell with perfluoropolymer-coated surfaces. The sample exhibits a discontinuous anchoring transition fromplanar to homeotropic on cooling at zero or a small electric field, and above a cross-over voltage a continuous‘inverse Freedericksz transition’, at which the director starts tilting in opposite directions at the two surfaces.
Defects of strength ±1/2 are either annihilated or expelled when the director tilts. On the other hand, disclination lines of ±1 which end in partial point defects (boojums) at the surfaces in the planar alignment regime acquirepoint defects of strength ±1 at the midplane of the cell when the director tilts. At a low enough temperature, thehomeotropic anchoring becomes strong, and an electric field above the Freedericksz threshold generates the usual
umbilic defects, which follow the dynamic scaling laws found in earlier studies.
Keywords: defects; surface anchoring; annihilation; boojums

1.

Introduction

Liquid crystals exhibit a variety of colourful textures under a polarising optical microscope [1]. These
textures are useful for the preliminary characterisation of the liquid crystalline phases. For example,
nematic liquid crystals generally exhibit schlieren textures. Cholesteric and smectic liquid crystals exhibit
fingerprint and focal conic textures. These textures
are composed of different types of defects [2, 3].
In nematics the characteristic defects are disclinations which are basically line defects. In smectics,
in addition to the focal conic defects, dislocations
exist as line defects, although they cannot be seen
optically. The line defects are topological defects inorder parameter space. They are also seen in various other systems: for example, vortices in superfluid
helium, dislocations in crystals [4]. The disclinations
seen in nematics have been studied extensively, boththeoretically and experimentally [5–15]. Two disclinations attract each other when they are ‘unlike’ andrepel when they are ‘like’, mimicking the interactions
between electrical charges. In a classic paper Nehringand Saupe [6] presented detailed observations on
schlieren textures of nematics and calculated theforce of interaction between two disclinations. They
showed that annihilation of unlike half-strength
defects and creation of integer-strength defects frommerger of like half-strength defects sometimes occur∗ Corresponding author. Email: sdsp@uohyd.ernet.in
ISSN 0267-8292 print/ISSN 1366-5855 online
c 2011 Taylor & Francis

DOI: 10.1080/02678292.2011.587962
http://www.informaworld.com

very near the nematic–isotropic phase transition temperature. Lavrentovich et al. [16] have studied boththeoretically and experimentally different types ofdefects and their structures in various liquid crystals.
There are also several experimental and theoretical studies on the defect–antidefect annihilation
dynamics in nematic liquid crystals [17–20]. The
annihilation dynamics have been studied as functionsof time, when the director field evolves after rapid
quenching from isotropic to nematic phase. Bogi
et al. have studied the influence of surface anchoring on the annihilation dynamics of two parallel ±1/2 defects. They found that when the defect lines
are far from each other, the elastic interaction iscompletely screened by anchoring energy [17]. The
dynamics of ±1/2 defects under an external electricfield was investigated by Blanc et al. [18]. They foundthat the relaxation is governed by the π wall connecting two opposite charges and that the +1/2 defects
move faster than −1/2 defects.
Recently we reported the observation of a strong
discontinuous anchoring transition at ∼47.5◦ C fromplanar to homeotropic on cooling, and the reverse
transition at ∼50.6◦ C on heating in a nematic liquid crystal (CCN-47) on perfluoropolymer-coatedsurfaces [21, 22]. In the cooling mode, the sample exhibits a texture with a large number of ±1/2
defects above ∼47.5◦ C, and we have reported earlierDownloaded by [University of Hyderabad] at 03:42 12 August 2011

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about the defect–antidefect correlations in this system [23]. There are several experimental investigations reported on CCN-47 [24–30]. Very recently,
using the same compound, we have fabricated abistable device by exploiting the large hysteresis
region of the discontinuous anchoring transition[24, 25] in a rubbed cell. In this paper we reportobservations on the merging of two like half-strength
defects to form integer strength (±1) defects andannihilation of two unlike half-strength defects when
the director tilts away from planar alignment. The
tilting occurs below the anchoring transition temperature under applied electric fields which are large
enough to change the transition to a continuous
one. We argue that the defects of strength 1 which
are seen just below this temperature have a new
structure, with either a + or a − point defectlocated at the midplane of the sample, and associated with two boojums lying at the two surfaces.
Their number density decreases as the temperatureis reduced. At a temperature ∼6◦ below the anchoring transition temperature, the anchoring strengthfor homeotropic alignment grows to a large value,and sample leads to the formation of umbilics.
We find that these field-induced defects follow the
dynamic scaling laws found earlier for both umbilics and schlieren defects near the NI transition
temperature.

2.

Experimental details

We used a perfluoropolymer, namely, poly[perfluoro(4-vinyloxy-1-butene)] known as CYTOP (Asahi
Glass Co., Ltd) deposited on indium tin oxide (ITO)coated glass plates as the alignment layer. CYTOP
is commonly used as an antireflection coating fororganic light-emitting devices. We have reported
recently that it can be used as a perfect homeotropicalignment layer for the smectic liquid crystals [26].
In the present experiment we spin coated CYTOP on
ITO-coated glass plates which were used to constructexperimental cells. Typical thickness of the cells usedin the experiments was ∼5.2 µm. The nematic liquid
crystal, 4 -butyl-4-heptyl-bicyclohexyl-4-carbonitrile(CCN-47) was filled in the cell in the isotropic phaseby capillary action. The chemical structures of CCN47 and CYTOP are shown in Figure 1(a). The liquid
crystal compound exhibits the following phase transitions: Cr 25.6◦ C Sm–A 28.2◦ C N 57.3◦ C I and alarge negative dielectric anisotropy ( = −5.7 at
30◦ C). Texture observations were made by using apolarising optical microscope (Olympus BX-51). The

temperature of the sample was controlled by mounting it in a Mettler hotstage. A signal generator wasused to apply voltages at a frequency of 3.11 kHz.

3.

Results and discussion

3.1 Textures without an electric field
We first show the textures observed under a polarising microscope in the absence of an electricfield. The compound exhibits a planar texture justbelow the nematic–isotropic phase transition temperature (57.3◦ C). A typical planar texture at 52.2◦ C
with mostly half-strength disclinations is shown in
Figure 1(b). The discontinuous anchoring transition,
the details of which have been described earlier [21],starts at ∼47.5◦ C (Figure 1(c)) in a few locations, andthe sample becomes completely homeotropic as it iscooled down (Figure 1(d)). The system also exhibitsa large hysteresis of a few degrees Celsius in theanchoring transition temperature.
We observed textures as the sample was heated at
a rate of 1◦ per minute from the homeotropic state(Figure 2(a)). During heating the sample starts toundergo the discontinuous anchoring transition at
50.6◦ C from homeotropic to planar (Figure 2(b)).
Only integer strength defects with many walls associated with homeotropic to planar transitions areseen (Figure 2(c)). As the director reorients from
homeotropic to planar via tilted orientation, only
integer strength defects can be expected to form inthe sample. The defect density is much lower compared with that of the field-induced umbilic defects
which are discussed later. On further heating the dark
brushes become comparatively thicker (Figure 2(d)).
In several heating cycles defects are found to appear
at the same locations showing that the director distortion starts at the glass plates.
On cooling the sample under the action of an
AC electric field, the anchoring transition temperature is found to decrease with the field. The detailsof the theoretical and experimental results have beenreported elsewhere [22]. The transition continues tobe a discontinuous one from planar to homeotropic
up to a cross-over voltage of ∼0.5 V. Above this voltage, the anchoring transition becomes continuous,
the tilt angles at the two surfaces just start to deviate from zero at the transition point, and increasein magnitude as the temperature is lowered further.
Our experimental results could only be explainedby assuming that the tilt angles at the two surfaces
have opposite signs, so that the director retains anorientation parallel to the glass plates at the midplaneof the sample. Thus, the above transition could beDownloaded by [University of Hyderabad] at 03:42 12 August 2011

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973

Figure 1. (a) Chemical structures of the perfluoropolymer alignment layer (CYTOP) and the liquid crystal (CCN-47); (b)photomicrographs of the texture at 52.2◦ C; (c) during discontinuous anchoring transition at 47.5◦ C; (d) homeotropic texture at
45.4◦ C. The observation was made in the absence of an electric field using a 5.2 µm cell (colour version online).

Figure 2. Photomicrographs showing discontinuous anchoring transition on heating the sample from homeotropic state: (a)homeotropic texture at 50◦ C; (b) during the discontinuous anchoring transition at 50.6◦ C; (c) planar texture at 52.4◦ C; and (d)at 54.2◦ C. The observation was made in the absence of an electric field using a 5.2 µm cell (colour version online).

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described as an ‘inverse Freedericksz transition’ [22].
As we shall see, the opposite tilts induced by the fieldat the two surfaces give rise to a new type of defectstructure in the sample.
3.2 Textures with electric field
In Figure 3 we show the textures under the application of a sine wave voltage of 0.9 V at a frequency
3.11 kHz. No observable change in the texture isobserved at temperatures above that corresponding
to the anchoring transition (Figure 3(a)). This is
expected as the dielectric anisotropy of the sample
is negative and the field is perpendicular to the glass
plates. When the temperature is reduced to the transition point, some domains with lower birefringencestart growing in the sample. This is a signature of
the continuous anchoring transition, at which thedirector starts tilting away from the two surfaces(Figure 3(b)). These domains again start from thesame locations at which the discontinuous transition occurs at low voltages, showing that the CYTOP
coating has some non-uniformity. Like half-strength
defects merge together to form defects with integer strength (±1) or are expelled from the tilted
domains. Unlike half-disclinations annihilate eachother (Figure 3(b) and (c)).

On cooling from the isotropic phase, the nematic
director has planar alignment, as evidenced by thepresence of half-strength disclination lines. We concentrate on disclinations of strength ±1, which survive the continuous anchoring transition under theapplied external electric field. The stability of the(non-topological) ±1 disclinations arises because of
the escape of the director in the third direction[7], removing the core of the line defect. Hereafter
we call the ±1 disclination line the line defect,
although it contains no orientational singularity orcore. However, they are lines connecting two point
defects at the top and bottom surfaces (as describedin the following). The director configurations of +1
and −1 line defects confined to a cell of finite thickness are shown in Figures 4 and 5, respectively. The
top and side views of +1 defect are sketched in
Figure 4. The director tilts away from the plane of theglass plates, the nail heads being at lower positions
than the tips in Figure 4(a). The director configuration has rotational symmetry about the defect line
(Figure 4(b)), and incorporates a partial ‘+1 point’
disclination at the lower glass plate and a partial ‘−1point’ disclination at the upper surface. Such surface point defects are called boojums [31]. The topview of the −1 line defect is shown in Figure 5. Notethat there is no rotational symmetry about the line

Figure 3. Photomicrographs showing anchoring transition under an applied AC voltage of 0.9 V and frequency of 3.11 kHz
during cooling: (a) at 52.6◦ C; (b) showing a continuous anchoring transition at 46.6◦ C; (c) at 46.4◦ C; (d) dark field of view witha few four brush defects at 31.9◦ C. Cell thickness 5.2 µm (colour version online).

Liquid Crystals

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Figure 4. Schematic of director orientations of a +1 defect line: (a) top view; (b) side view.

Figure 5. Schematic of director orientations in the topview of a −1 disclination line.

defect, along which there is a two-fold axis. As wemove along the 45◦ line from lower right to upper
left quadrant, the director twists in an anti-clockwise
fashion. Looking along the lower left to upper rightquadrant, the twist is clockwise. On the other hand,along the vertical and horizontal planes, there is only
splay–bend distortion. Again near the upper andlower glass plates, there will be partial point defects
or boojums of opposite signs.
We now focus our attention on a small region of
the sample and carefully observe the merging andannihilation processes. We noticed that at 44.1◦ C
some of the half-strength defects had already merged
to form defects of strength one (Figure 6). There arestill some half-strength defects and in a few cases
the inversion walls connecting them can be clearlyseen (see the [green] arrows [in colour online]). As thetemperature is lowered the positions of one-strength
defects are almost fixed and the half-strength defects
move closer (Figure 6(b) and (c)). The half strength
defects of the same sign also move closer to reduce
the energy of the inversion wall. Finally two like halfstrength defects merge to create a defect of strength
one (Figure 6(d)). In the present case it is a +1. Twounlike half-strength defects are annihilated as shownin Figure 6(d). We also note that in a given pair ofannihilating defects, the +1/2 defects move fastertowards the −1/2 defect (Figure 6(a) and (b)) as
reported by Blanc et al. [18]. Finally no half-strength
defects remain in the texture (Figure 6(d)). Whenthe electric field is switched off at any temperature
below that of the anchoring transition, immediately
homeotropic domains are formed and the field ofview becomes completely dark recovering the discontinuous anchoring transition.
If the homeotropic anchoring at the surfaces isstrong, an electric field of the appropriate magnitude
applied along the director of a nematic with negative
dielectric anisotropy induces the usual Freedericksz
transition. In this case, the director starts tilting
at the midplane of the sample and in view of thedegeneracy in azimuthal angles, line defects of integral strength which are somewhat similar to thosedescribed above form in the sample. The strong
homeotropic anchoring at the surfaces ensures thatno boojums form at the ends of the line defects,
which are called umbilic defects [2]. We describe
observations on umbilic defects at low temperaturesin Section 3.3.
In our sample, the boundary condition changes
from planar to homeotropic below the anchoringtransition temperature, presumably because of thebuild up of smectic-A-like short-range order on theDownloaded by [University of Hyderabad] at 03:42 12 August 2011

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T.A. Kumar et al.

Figure 6. Merging of ±1/2 defects under the applied AC voltage of 1.5 V; (a) pairs of half-strength defects are shown withenclosed dotted white lines at 44.1◦ C; some of the half-strength defects have already merged to form integer-strength defects; (b)one pair of unlike defects (enclosed with dotted lines at the left) annihilated at 44◦ C; (c) both the unlike and like half-strength
defects approach each other; (d) like half-strength defects merge to form an integer defect and unlike half-strength defects areannihilated. White wispy lines pointed by green arrows are inversion walls (colour version online).

flat surfaces of the cell walls as the temperature of
the sample is lowered. As we have discussed earlier[22], this implies that the anchoring strength is weakclose to the anchoring transition temperature. What
will be the structures of the defects of strength 1 when
the director tilts oppositely at the surfaces below thecontinuous anchoring transition temperature underthe AC electric field?
In the case of a disclination line of strength+1, there are two possibilities consistent with the
opposite tilts at the two surfaces: (i) the boojum

with negative sign as in Figure 4(b) can detach fromthe surface and become a point disclination withstrength −1 which settles down at the center ofthe sample, and a new boojum with positive sign
is formed at the surface. Thus the disclination line
now will have a collapse of the director occurringin opposite directions in the two halves of the cell
(Figure 7(a)). (ii) Another possibility is for the positive boojum (of Figure 4(b)) to shift to the midplane
as a +1 point defect, and a boojum of negative
strength to form at the surface (Figure 7(b)). BothFigure 7. Schematic representation of two possible director configurations with opposite tilts at the two surfaces in the case of+1 defect.

Downloaded by [University of Hyderabad] at 03:42 12 August 2011

Liquid Crystalsof these field-induced structures have rotational symmetry about the line defects and exhibit similar dark
brushes between crossed polarisers. As the temperature is lowered, the tilt angle increases near thesurfaces and the overall brightness decreases.
The change in the director field in the presence ofthe external field can be expected to be more complicated in the case of a line defect of strength −1.
The expected director patterns in the XZ and YZ
planes which had only splay–bend distortions above
the anchoring transition temperature are shown in
Figure 8. Again there are only splay–bend distortions in these two planes, but it is clear that in any
given quadrant, the twist distortion of the director
changes sense between the upper and lower halves ofthe sample. The director structure here is more complicated than that of the point defect at the center ofthe sample shown in Figure 7(a) and (b). The projection of the director field in the XY -plane in one of
the quadrants is shown to emphasise that the optical pattern between crossed polarisers will be similar
to that of the line defect of −1 strength in the planar aligned sample. The overall pattern again has
two-fold symmetry about the line defect as before.
When the +1 and −1 line defects with these modified
structures approach each other, it will be energetically less expensive if the +1 defect with structure
as in Figure 7(a) approaches the −1 defect along the
Y -axis, while that with the structure as in Figure 7(b)approaches along the X -axis. The opposite tilts ofthe director at the two plates below the continuous
anchoring transition temperature in the presence ofsufficiently high external electric field gives rise to
the new type of structures illustrated above. Bothline defects of +1 and −1 strength will have point

Figure 8. Some cross sections of the director configuration of −1 disclination, with opposite tilts at the two
surfaces.

977

defects at the midplane of the cell, apart from the two
boojums at the two surfaces. The structures are new,which appear because of the opposite tilts at the two
surfaces realised by the anchoring transition under
an electric field: the field free line defects are notexpected to have the point defects at the center. The
umbilic defects seen above the Freedercksz threshold in materials with negative dielectric anisotropy
in cells with strong homeotropic boundary conditions do not have the boojums. The interactions
between the new types of defects in our sample canbe expected to be dominated by the point defects at
the midplane of the cells.
Although the structures of both +1 and −1line defects change substantially below the inverse
Freedericksz transition point, they continue to produce four dark brushes between crossed polarisers.
On further cooling of the sample, the number of these
defects is also reduced due to the mutual annihilationof defects of opposite strength, which can approacheach other as described above. A typical texture on
cooling the sample to 31.9◦ C which is much lower
than the anchoring transition temperature is shownin Figure 3(d). The texture appears dark with a few
four-brush defects remaining in the background.
3.3 Dynamics of umbilic defects
We further studied the dynamics of the field-induced
umbilics that emerge from the homeotropic state
when the anchoring energy is relatively strong. The
details of annihilation dynamics of umbilic defects
have been reported by Dierking et al. [20] in a commercially available mixture (ZLI-2806) on substrates
treated with JALS-204-R40. In the present study theumbilic defects were induced by sine wave of amplitude 4 V and frequency of 3.11 kHz at the temperature of 42◦ C. As discussed in our earlier paper [22],the anchoring strength A is about 10−5 J m−2 at thistemperature. Further, the bend elastic constant K 33
is 0.8 pN (see [30]). Thus, the extrapolation length
K 32 /A  0.08 µm, which is much smaller than thesample thickness d = 5.2 µm. The anchoring energy
is effectively strong, and the applied field induces
umbilic defects above the Freedericksz threshold as
discussed earlier. The recording and analysis of textures were carried out with the help of a suitablesoftware.
Typical photographs showing the evolution ofumbilic defects with time are shown in Figure 9. It
is seen that initially there are a large number of ±1
umbilic defects (Figure 9(a)), which mutually annihilate with the progress of time to reduce the number
density (Figure 9(b)–(d)). Finally a small number
remains static due to the surface irregularities orDownloaded by [University of Hyderabad] at 03:42 12 August 2011

978T.A. Kumar et al.

Figure 9. Textures (at 42◦ C) showing umbilic defects under an applied voltage of amplitude 4 V and frequency 3.11 kHz;elapsed time after the application of the field: (a) 1 s; (b) 7 s; (c) 13 s; and (d) 25 s. Cell thickness d = 5.2 µm (colour version
online).

impurities (Figure 9(d)). The director field around a+1 strength umbilic defect is shown schematically in
Figure 10. Diagrams showing the variations of defectdensity ρ(t) and the inter-defect separation D(t) as
functions of time are shown in Figure 11(a) and (b),respectively. According to the theoretical predictions[12] the defect density and the inter-defect separation
obey the following scaling laws:
ρ(t) ∝ t−ν(1)

D(t) ∝ (t0 − t)α(2)with ν = 1, andFigure 10. Schematic of director field of an umbilic defectof +1 strength generated by an electric field. Note theabsence of boojums.

where t0 – t is the time to annihilate and α = 0.5.
A nonlinear least squares fitting program was used
to fit the experimental data. The best fits to ρ(t) andD(t) are shown in Figure 11(a) and (b), respectively.
The fit parameters are obtained as ν = 1.1 ± 0.05,
and α = 0.55 ± 0.01, and are very close to the theoretical predictions. Similar values of the above two
exponents have also been reported by Dierking et al.
[20]. This study verifies previously reported results
and suggests that irrespective of the material andalignment layers these critical exponents for umbilic
defects are universal.

4.

Conclusions

In conclusion, we have reported the evolution of line
defects of strength 1 in a planar aligned sample to
new types of defects, consisting of point defects ofintegral strength at the midplane, and boojums at
the surfaces, as the sample is cooled below a continuous anchoring transition temperature under anelectric field. We have recorded the annihilation ofhalf-strength defects of opposite sign and creation of
integer-strength defects from those of the same sign.
We have also studied the dynamics of the umbilic
defects formed under the action of an electric fieldon the homeotropically aligned state at a sufficiently
low temperature, and found that the exponents forLiquid Crystals(a)ρ (mm–2)80

ρ (t ) ∝ t –v

υ = 1.1 ± 0.05
40

0

0

10

20

30

(b) 10080

D (μm)

Downloaded by [University of Hyderabad] at 03:42 12 August 2011t (s)D(t) ∝ (t – t)–α°α = 0.5 ± 0.01

60

40

20

–10

0

10

20
t –t (s)
°

30

40

50

Figure 11. (a) Time evolution of defect density ρ(t). The
continuous line is the best fit to the theoretical Equation(1) with the exponent ν = 1.1 ± 0.05. (b) Time evolution of
defect pair separation D(t). The continuous line is the best
fit to the theoretical Equation (2) with the exponent α =0.55 ± 0.01.

the time dependences of the defect density and theinter-defect separation are consistent with the resultsof previous studies.
Acknowledgements
We thank Professor K.P.N. Murthy, School of Physics,
University of Hyderabad for various useful discussions.
SD and AK gratefully acknowledge the support from the
UGC-CAS, School of Physics. We acknowledge Asahi
Glass for supplying CYTOP.

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