PHYSICAL REVIEW E 82, 011701 共2010兲

Temperature- and electric-field-induced inverse Freedericksz transition in a nematogen withweak surface anchoring
T. Arun Kumar,1 P. Sathyanarayana,1 V. S. S. Sastry,1 Hideo Takezoe,2,*,† N. V. Madhusudana,3,*,‡ and Surajit Dhara1,*,§
1

School of Physics, University of Hyderabad, Hyderabad 500046, India
Department of Organic and Polymeric Materials, Tokyo Institute of Technology, 2-12-1-S8-42 O-okayama, Meguro-ku,
Tokyo 152-8552, Japan
3
Raman Research Institute, C. V. Raman Avenue, Bangalore 560080, India
共Received 2 March 2010; revised manuscript received 12 May 2010; published 2 July 2010兲

2

We report electric field dependence of the anchoring transition in a mesogen on cooling in a cell withperfluoropolymer treated surfaces. Below a crossover voltage Vco the transition is discontinuous between
planar and homeotropic alignments, and as the temperature is lowered, the transition temperature decreases
quadratically with the field. Above Vco the transition is continuous between planar and tilted alignments, thetransition temperature decreasing essentially linearly with the rms field. We develop a simple model to accountfor these results and argue that the higher field regime corresponds to a temperature driven inverse Freedericksz transition in which the director orientation starts tilting at the weakly anchored surfaces while the tilt
angle remains zero at the midplane of the cell.
DOI: 10.1103/PhysRevE.82.011701

PACS number共s兲: 61.30.Jf, 61.30.Hn

I. INTRODUCTION

threshold voltage, around which the bulk free-energy density
is given by

Nematic liquid crystals have an orientational order which
is apolar in nature 关1兴 and described by a dimensionless unit
vector n共r兲 called the director. Spatial gradients in n共r兲 costa very small curvature elastic energy, making the nematic aprime example of a soft material. The apolar director couples
to an external electric field through the dielectric anisotropy⌬⑀ = ⑀储 − ⑀⬜ of the material, the subscripts indicating directions with reference to n. A classic experiment on liquid
crystals concerns the Freedericksz transition, in which a material with positive ⌬⑀ is taken between two indium tin oxide
共ITO兲 coated conducting glass plates, whose surfaces havebeen treated to get a strong planar anchoring of the director,
and separated by a gap d along the z direction. The RapiniPapoular 关2兴 form of the anchoring energy per unit surface
area is given by1
Fs = A sin2 ␺s ,2

共1兲where ␺s is the tilt angle of n with respect to the surface andthe positive coefficient A is the anchoring strength. The anchoring energy is “strong” if the extrapolation length␭ = K11 / A is of molecular dimensions, K11 being the splay
elastic constant 关1兴. If an electric field E is applied between
the ITO plates, the dielectric energy is lowered by a tilting of
the director. As ␺s = 0, the tilt angle can be expected toacquire a maximum value ␺ M at the center of the cell withz = d / 2, and the resulting curvature distortion of n共z兲 costs anelastic energy. Indeed the distortion occurs only beyond a*Corresponding authors.
†

takezoe.h.aa@m.titech.ac.jp
nvmadhu@rri.res.in§
sdsp@uohyd.ernet.in‡

1539-3755/2010/82共1兲/011701共6兲

冉 冊

1d␺共z兲 2 ⑀0
g2共z兲 = K11− ⌬⑀E2 sin2 ␺共z兲.
2
2
dz

共2兲

The corresponding Euler-Lagrange 共EL兲 equation can besolved in the usual manner 关3兴 by noting that d␺共z兲 / dz = 0 at
the center of the sample 共z = d / 2兲. The threshold voltage is
K11
given by 关3兴 Uth = ␲ ⑀0⌬⑀ , beyond which the cell taken between appropriately placed crossed polarizers exhibits anelectro-optic effect. Analogous electro-optic effects in various geometries are exploited in the ubiquitous liquid-crystal
displays 关4兴.
If the anchoring is not strong in the sense mentioned
above, the threshold voltage is reduced 关5兴 and the nonuniform ␺共z兲 profile at any finite value of the electric field canbe calculated by solving the EL equation and explicitly taking into account the surface torque balance conditions at bothz = 0 and d. Usually the anchoring strength A decreases withtemperature T in the nematic range, and A共T兲 depends on
both the surface treatment and the chemical composition of
the nematic 关6,7兴. Some surface treatments give rise to anonzero pretilt angle ␺t, in which case there is no true threshold. ␺t can also vary with temperature, and in general theright-hand side of Eq. 共1兲 should be replaced by
A共T兲sin2关␺s − ␺t共T兲兴 / 2.
We recently reported that the mesogen 4-butyl-4-heptylbicyclohexyl-4-carbonitrile 共CCN-47兲, taken on glass plates
treated with poly关perfluoro共4-vinyloxy-1-butene兲兴 also
known as CYTOP, undergoes an abrupt anchoring transition
共ATr兲 from planar 共␺t = 0兲 to homeotropic 共␺t = ␲ / 2 rad兲alignment as the temperature is lowered to 48 ° C 关8兴 in thenematic range. In the present paper we study this transition
under varying electric fields to find that the character of ATr
remains abrupt below a crossover voltage of ⬃0.5V, while
above that voltage ATr changes to a temperature driven continuous transition. We develop a simple model to account for

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FIG. 1. 共Color online兲 Photomicrographs of the textures aroundthe anchoring transition temperature in CCN-47 between crossed
polarizers at 共a兲 52.2, 共b兲 48, and 共c兲 46 ° C, all under zero field.
Textures in the same region under an electric field 0.29 V / ␮m, as
photographed at 共d兲 47.2, 共e兲 46.8, and 共f兲 29 ° C. White dotted lines
in Fig. 1共d兲 enclose a domain with tilted director. Cell thickness
5.2 ␮m.

these results and argue that the higher voltage regime corresponds to an inverse Freedericksz transition.
II. EXPERIMENTCYTOP is spin coated on ITO-coated glass plates which
are used to construct the liquid-crystal cells. Typical thickness of the cells used in the experiment is ⬃5.2 ␮m.
CCN-47 is filled in the cell in the isotropic phase by capillaryaction. CCN-47 exhibits the following sequence of phasetransitions: Cr 25.6 ° C Sm- A 28.2 ° C N 57.3 ° C I, andhas a large negative dielectric anisotropy 共⌬⑀ = −5.7 at
30 ° C兲. The textures are observed using a polarizing optical
microscope 共Olympus BX-51兲. The temperature of thesample is controlled by a Mettler hotstage to an accuracy of
0.1 ° C. A signal generator is used to apply sinusoidal voltages at a frequency of 3.11 KHz.
III. RESULTS AND DISCUSSION

We first show the textures observed under a polarizing
microscope in the absence of an electric field. The compound
exhibits a planar texture as it is cooled below the isotropic tonematic transition temperature. At 52.2 ° C mostly halfstrength disclinations are observed 关Fig. 1共a兲兴. The physicalinferences from the occurrence of the two-brush defects are
twofold. First the director is apolar, i.e., n and −n are physi-

cally equivalent, and second the director is in the plane parallel to the plates, forming line defects 关9兴. As the temperature is lowered dark domains are nucleated at about 48 ° C ina few locations of the sample and spread with time关Fig. 1共b兲兴 even if the temperature is held fixed. In the dark
domains the director is normal to the substrate, i.e., the alignment is homeotropic. Thus at the ATr the director rotates by
90°, from planar to homeotropic, discontinuously. On further
cooling to about 46 ° C, the entire texture turns completely
dark 关Fig. 1共c兲兴. A careful observation of the sample showsthat the medium is uniaxial. The details of the discontinuous
anchoring transition in the absence of a field have alreadybeen reported 关8兴. Further studies on the sample are carriedout by applying different electric fields and observing theevolution of the textures. Initially the sample is heated to theisotropic phase and slowly cooled 共at 0.2 ° C / min兲 in thepresence of the desired ac field 共at 3.11 KHz兲. Interestingly,
after the transition from isotropic to nematic phase, the appearance of the texture is essentially the same as in the zerofield case. The half-strength and one-strength defects appear
at the same locations in the sample as in the field free run.
For low fields 共ⱕ0.1 V / ␮m兲 homeotropic domains also
nucleated at the same locations as in the zero-field case but
at field-dependent lower temperatures 关Fig. 1共d兲兴. The
growth of any given domain is also similar to that in the field
free sample. The ATr occurs with the formation of dark domains 共i.e., discontinuously兲 up to the field of ⬃0.1 V / ␮mwhen the ATr temperature is 47.2 ° C. Beyond this field the
ATr occurs with the appearance of regions with reduced birefringence, instead of dark domains, but located at the sameplaces. Textures exhibited by the sample which undergoes acontinuous ATr under the field are also shown in Figs.
1共d兲–1共f兲. As shown in Fig. 1共d兲, a domain with lower birefringence appears as the temperature is reduced under anelectric field of 0.29 V / ␮m. This domain grows in size on
further cooling 关Fig. 1共e兲兴. Curiously, most of the defects are
expelled from the growing domains such that their nuclei
collect at the interface between the domains and the surrounding sample which continues to have a planar alignment.
On very rare occasions, a half-strength defect is trapped inside the domain, and a relatively thin wall connects such a
defect to the edge of the domain. These observations indicatethat the director inside the domain is titled with respect to theplane of the sample 关Fig. 1共e兲兴. The tilted director field insidethe domain can merge with the planar director field outside
the domains only if the 1/2 strength defects lie on the boundaries of the domains. If the defect is inside the domain, a wall
across which the tilt angle changes sign has to lie between
the defect and the boundary of the domain 关Figs. 1共e兲 and2共c兲兴. On further cooling the texture finally becomes very
dark but for a few one-strength defects 关Fig. 1共f兲兴 suggestingthat the director is now almost vertical with respect to thesubstrates.
Further careful observations across the periphery of thetilted domain in Fig. 1共e兲 show that the contrast of the texture changes gradually, indicating that the tilt angle of thedirector changes continuously from planar to tilted across theperiphery. The side and the top views of the director orientation across the periphery are shown schematically in Figs.
2共a兲 and 2共b兲, respectively. The reason for choosing the spe-

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TEMPERATURE- AND ELECTRIC-FIELD-INDUCED…

FIG. 3. 共Color online兲 Variation in anchoring transition
temperature with applied electric field. Continuous lines are best
fit to the theoretical Eqs. 共5兲 and 共13兲 with fit parameteru = 1.6⫻ 10−6 J / m2 K. The type 2 and type 1 transitions, shown by
the dashed lines are superceded by transitions of the other typeoccurring at higher temperatures.

FIG. 2. 共Color online兲 Schematic drawings of 共a兲 the side view
and 共b兲 the top view 共say in the upper half of the cell兲 of the director
orientation across boundary between the untilted and tilted regions.
In 共b兲 the length of the line is a measure of the projection of thedirector in the plane of the plate. Regions in 共I兲 and 共III兲 represent
tilted and 共II兲 and 共IV兲 represent planar director orientations. 共c兲
Part of the texture taken from Fig. 1共e兲 including a half-defect connected by a inversion wall and the superimposed director orientations across the boundary between the regions with tilted and planaralignments of the director.

cific director profile shown in Fig. 2共a兲 will be clear later.
The detailed director orientation is superimposed on the texture in Fig. 2共c兲 which includes a boundary between tilted
and non tilted domains and a half-defect connected with aninversion line inside the tilted domain as well.
The variation in anchoring transition temperature, i.e., thetemperature at which the nucleation of the domain starts atsome particular location, is measured as a function of theapplied field from the microscope observation and is shownin Fig. 3. The anchoring transition temperature decreases
with increasing electric field with a change in slope at⬃0.1 V / ␮m, corresponding to the crossover from discontinuous to continuous ATr. It may be mentioned that this is
not a thermodynamic phase transition as the phase remains
nematic before and after the transition; instead it is an anchoring transition bringing about a change in the director
orientation. The lowering of the anchoring transition temperature on the application of the field is a result of thecompetition between the surface anchoring and dielectric response to the electric field. At a temperature lower than
48 ° C, the surface interaction favors a homeotropic alignment, whereas the electric field favors a planar alignment as
the dielectric anisotropy of the material is negative, whose
magnitude increases with decreasing temperature. We develop a theoretical model to account for the above observations by assuming that, in the absence of an electric field, theanchoring energy per unit area can be expressed in the formuFs = 共T − TA兲sin2 ␺s ,2

共3兲where TA = 48 ° C and u is positive. This means that the anchoring strength favoring planar alignment above TA weakens rapidly as that temperature is approached from above,and changes sign at TA to favor homeotropic anchoring
whose strength rapidly increases as the temperature is lowered. A possible origin of the abrupt anchoring transition isthe build up of smectic-A type short-range order near the
CYTOP coated surfaces at temperatures lower than TA,
which favors the homeotropic alignment. An external electricfield favors the tilt angle ␺ to be zero both in the bulk and thesurfaces, as the dielectric anisotropy ⌬⑀ is negative. Whenthe sample is cooled under the action of an electric field, four
possible director profiles are possible below TA, which arise
from the mutually antagonistic preferred orientations arisingfrom the surface anchoring and dielectric energy density.
This can in turn give rise to three different types of anchoring transitions. In type 1 transition, the director can have auniform tilt angle ␺共z兲 = ␺s independent of the position z evenbelow the transition temperature. The curvature elastic energy is zero, and the total energy per unit area of the cell of
thickness d is given by
F1共T兲 = −⑀0共⑀储 sin2 ␺ + ⑀⬜ cos2 ␺兲E2d + u共T − TA兲sin2 ␺ ,2冋

F1共T兲 = u共T − TA兲 −

册

⑀0
⑀0
⌬⑀E2d sin2 ␺ − ⑀⬜E2d. 共4兲
22

As ⌬⑀ is negative, the term in the square brackets of Eq. 共4兲
changes sign at the anchoring transition temperature,
T1at = TA −

⑀0
E 2d
兩⌬⑀兩
.
2
u

共5兲

At temperatures above T1at the term in the square brackets of
Eq. 共4兲 will have a positive sign, and the energy F1共T兲 is011701-3

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冉 冊

dFs
d␺共z兲
= K11
.
d␺共0兲
dz z=0(a)(b)

FIG. 4. Schematic diagram illustrating the director orientations
near the two surfaces and the midplane of the cell below the anchoring transition of type 2 corresponding to 共a兲 the even and 共b兲
the odd solutions of ␺共z兲.

minimized by ␺ = 0, i.e., by a uniform planar alignment of
the director throughout the cell. Below T1at the term becomes
negative, and the total energy is minimized by ␺ = ␲2 , i.e., thedirector adopts a uniform homeotropic alignment throughout
the cell. The anchoring transition is very similar to that in thefield-free sample, and ignoring the temperature dependenceof ⌬⑀, the transition point T1at decreases quadratically withthe applied field E, from TA when E = 0. The fitted line asshown in Fig. 3 takes into account the temperature dependence of ⌬⑀.
As the applied field E is increased, the dielectric energy
density, which depends quadratically on the field, favors the
director tilt angle ␺共z兲 to be zero in the bulk. As we willshow below, this changes the nature of the anchoring transition when the sample is cooled under relatively large valuesof the applied electric field E. The tilt angle ␺共 2d 兲 in thecentral plane of the cell with z = 2d can be expected to besmaller than that at the surfaces, ␺s at z = 0 or d. This meansthat the director profile has a curvature deformation, and theelastic energy of the medium has to be taken into account. In
the present paper our main interest is in the analysis of theanchoring transition, just beyond which the director field has
only a splay deformation. The bulk free-energy density of themedium is again given by Eq. 共2兲 but with a negative ⌬⑀. As
we are interested only in locating the transition at which thetilt angle ␺共z兲 just starts deviating from 0, we linearize theEuler-Lagrange Eq 关5兴, which reads as
d2␺ ⑀0兩⌬⑀兩E2−
␺ = 0.
dz2
k11

共6兲

As the sample is cooled under an electric field E in a temperature range favoring homeotropic anchoring at the surfaces located at z = 0 and d, we can expect that ␺共0兲 and ␺共d兲
start deviating from 0 at the transition. There are two possible solutions: 共a兲 both ␺共0兲 and ␺共d兲 are assumed to havethe same sign, and we look for a solution characterized by␺共z兲 = ␺共d − z兲 关Fig. 4共a兲兴␺共z兲 = ␣ cosh关␤共d/2 − z兲兴,

共8兲A similar equation holds at the surface with z = d. Again linearizing the above equation and using the solution for ␺共z兲,
we getu共T − TA兲cosh共␤d/2兲 + K11␤ sinh共␤d/2兲 = 0.

共9兲

The continuous anchoring transition temperature in this regime 2a is given by
T2eat = TA −冉冑1冑⑀0兩⌬⑀兩K11E2 tanh du2冊⑀0兩⌬⑀兩E2, 共10兲
K11

where the subscript e signifies that ␺共z兲 is an even function.
As tanh共x兲 ⬇ 1 for all x ⱖ 2.5, in this regime the anchoringtransition temperature essentially decreases linearly with increase in rms value of E, reflecting the experimentally observed trend. However, as tanh共x兲 ⬇ x for very small valuesof x 共say ⱕ0.25兲, the above equation actually reduces to Eq.
共5兲 derived for the discontinuous anchoring transition between the planar and homeotropic states, while the derivationof the last equation above is valid only for a continuoustransition, in which ␺共z兲 just starts acquiring a nonzero
value. This is an unsatisfactory feature of the even solution
assumed for ␺共z兲. Thus, even though at first sight Eq. 共10兲seems to mathematically describe both the experimentallyfound distinct regimes as far as the dependences on the electric field are concerned, it is really valid only in the regime 2
in which the transition is continuous. Indeed by equating theanchoring transition temperatures derived for the two regimes we get the condition for the crossover as

冉冑

tanh

d
2

冊 冉冑

⑀0兩⌬⑀兩E2
d
=
K11
2

冊

⑀0兩⌬⑀兩E2
,
K11

共11兲

which is valid only for E = 0. Thus the above solution does
not describe the experimentally found nonzero value of thecrossover field satisfactorily. Hence we look for another possible solution to the linearized Euler-Langrange equation
关Eq. 共6兲兴.
共b兲 We now assume that ␺共0兲 and ␺共d兲 have opposite
signs, and more generally, ␺共z兲 = −␺共d − z兲. This implies thatthe tilt angle of the director at the mid plane of the cell
␺共d / 2兲 = 0 even in the distorted state, which is favored by thedielectric interaction of the medium with the external field关Fig. 4共b兲兴. The solution is␺共z兲 = ␣ sinh关␤共d/2 − z兲兴.

共12兲Proceeding as before, the continuous anchoring transition
temperature corresponding to the above odd solution is given
by

冉冑

1
d
T2oat = TA − 共冑⑀0兩⌬⑀兩K11E2兲coth
u
2

共7兲

冊

⑀0兩⌬⑀兩E2
.
K11
共13兲

where ␣ is a measure of the amplitude of ␺共z兲 and␤2 = 共⑀0兩⌬⑀兩E2 / K11兲. The surface torque balance equation atz = 0 reads as

As coth共x兲 also tends to the constant value 1 for large x, theessentially linear dependence of the anchoring transition
temperature T2oat on the rms electric field E is recovered in

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TEMPERATURE- AND ELECTRIC-FIELD-INDUCED…

the regime 2b. The condition for the crossover between thetwo regimes is again found by equating T1at to T2oat andreads as

冉冑

coth

d
2

冊 冉冑

⑀0兩⌬⑀兩E2
d
=
K11
2

冊

⑀0兩⌬⑀兩E2
.
K11

共14兲

This is satisfied for 共d / 2兲冑⑀0兩⌬⑀兩E2 / K11 ⬇ 1.199 68. The
crossover voltage is thus given by冑

Vco = Ecod ⬇ 2.4

K11.
⑀0兩⌬⑀兩

共15兲

The above equation shows that 共i兲 the crossover occurs at awell-defined applied voltage Vco, which is independent of thesample thickness, and 共ii兲 more remarkably, the crossovervoltage does not depend on the anchoring coefficient u. It
only depends on the material parameters K11 and ⌬⑀. The
temperature dependence of the dielectric anisotropy of
CCN-47 has been measured 关10兴. Only the bend elastic constant K33 of CCN-47 has been measured, and we will discuss
he method of estimating K11 later. Using the data near the
relevant temperature, the calculated value of Vco is ⯝0.5 V,
which agrees extremely well with the measured value. Thus
the odd solution for ␺共z兲 gives the correct description of thedirector distortion below the anchoring transition in the regime 2. We fit the experimental data in the two regimes using
Eqs. 共5兲 and 共13兲, respectively. As K11 has not been measured
for CCN-47, we estimate it using the mean field result, viz.,
K11共T兲 = K0S2, where S is the orientational order parameter
whose temperature dependence has been measured 关10兴. The
estimated value of K0 ⯝ 7.2⫻ 10−12 N. The agreement between the theoretical expressions Eqs. 共5兲 and 共13兲 and theexperimental data is highly satisfactory 共Fig. 3兲. The overall
agreement of the experimental and calculated electric field
variations in the anchoring transition temperatures in bothregimes shows that the assumption made in Eq. 共3兲 about thetemperature variation of the anchoring energy A is appropriate for the ATr in the system studied. The calculations corresponding to the type 2b regime can be extrapolated all theway down to E = 0 共Fig. 3兲. Indeed as coth共x兲 = 1 / x for verysmall values of x, for low values of E, Eq. 共13兲 leads to a
field-independent anchoring transition temperature
T2oat共E → 0兲 = TA −

2K11
.
ud

共16兲

This hypothetical continuous anchoring transition would
have taken place when the temperature is just low enough forthe gain in anchoring energy by a tilting of the director at thetwo surfaces to exceed the positive energy cost of the resulting elastic distortion of the director field. Of course this hypothetical transition is cut off at a higher temperature by thediscontinuous transition of type 1. There is some similarity
between this phenomenon and the nematic to isotropic 共NI兲transition. In the latter case, as was pointed out by de Gennes
关1兴, the thermodynamically first-order NI transition occurs at
a temperature slightly higher than that of a hypotheticalsecond-order transition. The analogy is not perfect, however,as the two types of anchoring transitions discussed by us leadto two very different director distributions in the cell. It may
be noted that the anchoring transition of type 2b occurs as
the sample is cooled under an electric field 兩E兩, with the tilt
angle at the center of the sample ␺m remaining at a value= 0 and the tilt angles at the two surfaces ␺s exhibiting atransition to a nonzero value. The transition occurs as thesurfaces favor homeotropic alignment 共␺s = ␲2 兲 with a very
weak anchoring energy. In contrast, as described earlier, thefamiliar Freedericksz transition in the above geometry occurs
if a sample with positive dielectric anisotropy and a strong
planar anchoring is subjected to an appropriate electric field
across the cell. This transition is characterized by ␺s = 0 and␺共d / 2兲 acquiring a nonzero value. We can thus describe thetype 2 anchoring transition in our system as an inverse
Freedericksz transition. Further, in the usual Freedericksz
transition, the tilt angle at the mid plane ␺共d / 2兲 takes amaximum value, decreasing to 0 at both surfaces if the anchoring is strong, or to some finite nonzero value smaller
than ␺共d / 2兲 if the anchoring is weak. This necessarily requires the solution ␺共z兲 be an even function 关5兴. On the other
hand, in the inverse Freedericksz transition investigated byus, the odd solution is relevant.
IV. CONCLUSIONSIn conclusion, we have shown that the discontinuous anchoring transition of CCN-47 observed on CYTOP becomescontinuous beyond a crossover voltage. The quadratic andlinear variations of the anchoring transition temperature in
the low voltage discontinuous and high voltage continuous
regimes with an applied field have been accounted for by asimple physical model. The latter can be described as aninverse Freedericksz transition. The observations also showthat disclinations tend to get expelled from regions in which
this transition takes place. A half-strength defect trapped inside the tilted region is connected with a tilt-inversion wall.
The effect of an electric field on the homeotropic state 共i.e.,
below TA兲 and the detailed analysis of the evolution of thedefects and the director profile will be published elsewhere.
ACKNOWLEDGMENTS

One of the authors 共S.D.兲 gratefully acknowledges the
support from the DST, Government of India for the Project
No. SR/FTP/PS-48/2006 and Professor A. Pathak, CAS,
School of Physics. S.D. also acknowledges Professor K. P.
N. Murthy for various useful discussions. We acknowledge
Ashai Glass for supplying CYTOP. N.V.M. acknowledges
University of Hyderabad for support.

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关1兴 P. G. de Gennes, The Physics of Liquid Crystals, 2nd ed. 共Clarendon, Oxford, 1993兲.
关2兴 A. Rapini and M. Papoular, J. Phys. Colloq. 30, C4-54 共1969兲.
关3兴 L. M. Blinov and V. G. Chigrinov, Electro-optic Effects in
Liquid Crystals 共Springer-Verlag, New York, 1996兲.
关4兴 B. Bahadur, Liquid Crystals: Applications and Uses 共World
Scientific, Singapore, 1990兲, Vol. I.
关5兴 E. G. Virga, Variational Theories for Liquid Crystals 共Chapman and Hall, London, 1994兲.

关6兴 S. Faetti, M. Gatti, V. Palleschi, and T. J. Sluckin, Phys. Rev.
Lett. 55, 1681 共1985兲.
关7兴 M. Vilfan and M. Copic, Phys. Rev. E 68, 031704 共2003兲.
关8兴 S. Dhara, J. K. Kim, S. M. Jeong, R. Kogo, F. Araoka, K.
Ishikawa, and H. Takezoe, Phys. Rev. E 79, 060701 共2009兲.
关9兴 J. Nehring and A. Saupe, J. Chem. Soc., Faraday Trans. 2 68,
1 共1972兲.
关10兴 S. Dhara and N. V. Madhusudana, Phase Transitions 81, 561
共2008兲.

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