Eur. Phys. J. E 22, 139–149 (2007)
DOI: 10.1140/epje/e2007-00020-3

THE EUROPEAN
PHYSICAL JOURNAL E

Eﬀect of high electric ﬁelds on the nematic to isotropic transition
in a material exhibiting large negative dielectric anisotropy
S. Dhara1,a and N.V. Madhusudana2,b1
2School of Physics, University of Hyderabad, Hyderabad 500046, India
Raman Research Institute, C.V. Raman Avenue, Bangalore 560080, India
Received 30 November 2006 / Received in ﬁnal form 14 February 2007
c EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2007
Published online: 14 March 2007 – 
Abstract. We report the experimental high electric ﬁeld phase diagram of a nematic liquid crystal which
exhibits a large negative dielectric anisotropy. We measure simultaneously the birefringence (∆n) and thedielectric constant (⊥ ) at various applied ﬁelds as functions of the local temperature of an aligned sample.
We also measure the higher harmonics of the electrical response of the medium. The following experimental
results are noted: (i) enhancement of orientational order parameter S in the nematic phase due to boththe Kerr eﬀect and quenching of director ﬂuctuations; (ii) enhancement in the paranematic to nematictransition temperature (TP N ) with ﬁeld; (iii) divergence of the order parameter susceptibility beyond thetricritical point as measured by third harmonic electrical signal; (iv) a small second harmonic electricalsignal which also diverges near TP N , indicating the presence of polarised domains. Our measurements showthat ∆TP N (= TP N (E) − TNI (0)) varies linearly with |E| whereas the Landau de Gennes theory predicts adependence on E 2 . It is argued that the quenching of director ﬂuctuations by the ﬁeld makes the dominant
contribution to all the observations, including the thermodynamics of the transition.
PACS. 64.70.Md Transitions in liquid crystals – 61.30.Gd Orientational order of liquid crystals; electricand magnetic ﬁeld eﬀects on order – 61.30.Dk Continuum models and theories of liquid crystal structure

1 Introduction
Liquid crystals are anisotropic ﬂuids, made of molecules
which lack spherical symmetry. They exhibit a variety ofphase transitions. These transitions can be thermodynamically ﬁrst or second order in nature [1]. The simplest liquid crystal phase known so far is the uniaxial nematic.
When the nematic is made of rod like molecules their longaxes exhibit long-range orientational order along a speciﬁc
direction called the director denoted by n̂, a dimensionless unit vector. n̂ and −n̂ are indistinguishable and thesystem does not show any macroscopic polarisation. In
the uniaxial nematic phase, the magnitude of the orientational order parameter S = 3 cos2 θ − 1/2 [1], where θis the angle between the long axis of a molecule and the
director and angular brackets represent a statistical average. Thermodynamically the isotropic to nematic phasetransition is weakly ﬁrst order with a heat of transitionof ∼1 kJ/mol [1]. Studies on the eﬀects of electric andmagnetic ﬁelds on liquid crystals are interesting becauseof both fundamental and technological importance. The
weak curvature elasticity of the nematic produces large
electrooptic eﬀects exploited by the display industry andalso very strong ﬂuctuations of the director which resultabe-mail: sdsp@uohyd.ernet.in
e-mail: nvmadhu@rri.res.inin a turbid appearance of a bulk sample [1]. There havebeen several reports on the eﬀect of strong electric andmagnetic ﬁelds on the isotropic to nematic phase transition [2–9]. The electric ﬁeld experiments were performed
mainly on systems exhibiting positive dielectric anisotropy∆ = || − ⊥ , where the subscripts refer to directions inrelation to n̂. The dielectric anisotropy is a measure of theorder parameter S  ∆/∆a , where ∆a is the dielectricanisotropy in the fully aligned state. ∆ couples with E 2
to lower the free energy density: FE = −(o ∆a SE 2 )/2
where E is the applied electric ﬁeld and o the vacuum
dielectric constant. The ﬁeld induces a weak orientational
order even in the isotropic phase (paranematic), as a resultof which all the physical properties in the medium become
weakly anisotropic. The ﬁeld-induced birefringence (∆n)
in the paranematic phase due to the electric ﬁeld is called
the Kerr eﬀect, and the analogous eﬀect due to a magnetic ﬁeld is called the Cotton-Moutton eﬀect. Under theaction of the ﬁeld the paranematic to nematic transition
temperature is shifted towards higher values and the orderparameter is increased in the nematic phase also. At low
ﬁelds the transition occurs with a ﬁnite jump in the orderparameter in a ﬁrst order phase transition. With increasing ﬁeld the jump decreases and above the critical ﬁeld
there is a continuous evolution of the order parameter fromparanematic to nematic phase in materials with ∆ > 0.
The symmetries of the two phases are the same and the

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possibility of a second order phase transition can be ruledout. These phenomena resemble the classical liquid to gas
transition under pressure [2]. To shift the transition temperature by a measurable quantity the electric or the magnetic free energy per molecule should be comparable to thethermal energy kB T . The volume diamagnetic susceptibility anisotropy of a typical nematic is 10−6 m3 /mol, andthe required magnetic ﬁeld is 103 tesla, which is beyond experimental realization. On the other hand materials withlarge positive dielectric anisotropy of ∼20 are available
and the required electric ﬁeld is approximately 107 V/m
and attainable in the laboratory. If ∆ > 0, the director
is aligned along the direction of the applied electric ﬁeld
and the enhancement in the order parameter is due to theKerr eﬀect as well as quenching of director ﬂuctuations
but the system remains uniaxial [2]. Studies on the eﬀectof strong electric ﬁelds on liquid crystals with negative dielectric anisotropy are also interesting because the phase
diagram is very diﬀerent in nature compared to that in asystem with ∆ > 0. A strong dipolar group which makesa large angle with the long axis of the molecule is needed
to get a material with ∆ < 0. To study the electric ﬁeld
eﬀect, the nematic liquid crystal is sandwiched between
two ITO (indium-tin oxide) coated glass plates, which aretreated for planar alignment of n̂ along the X-axis. The
ﬁeld is applied along the Z-axis as shown in Figure 1a.
The dipoles of the molecules tend to align along the ﬁeld
direction. As a result the long molecular axes tend to beperpendicular to the ﬁeld direction. In the paranematic
phase the distribution of azimuthal angles of the moleculesin the (XY ) plane perpendicular to the ﬁeld is random.
The projections of the long axes of the molecules in the
XY plane are also shown in Figure 1a. The distributionfunction f (θ) of the long axes around the ﬁeld direction
shows a small peak at θ = π/2. With increasing ﬁeld thepeak height increases as schematically shown in Figure 1b.
The paranematic phase (NU − ) is uniaxial with a negative
order parameter. When the system is cooled under theﬁeld below the paranematic to nematic transition temperature the long axes of the molecules tend to align ina preferred direction (X-axis say)which can be controlled
by the surface interaction with the treated plates. In thenematic phase there is a partial quenching of the director
ﬂuctuations due to the electric ﬁeld in the ZX plane. Thus
the ﬂuctuations in the ZX plane are smaller than those in
the plane (XY ) perpendicular to the ﬁeld as shown in Figure 1a. The diﬀerential quenching of ﬂuctuations leads to
induced biaxiality under ﬁeld [3,10]. Therefore, under theﬁeld the paranematic to nematic transition corresponds
to uniaxial nematic (NU − ) to biaxial nematic (NB ) transition. With increasing ﬁeld the NU − to NB transition
temperature is shifted to higher values and the order parameter (see Eq. (1) in Sect. 2) is increased in the NB
phase. The jump in the order parameter reduces and ﬁnally the transition between the two phases with diﬀerent
symmetries becomes second order in nature above the tricritical ﬁeld.
The main problem in conducting high electric ﬁeld experiments on liquid crystals is the heating due to ionicFig. 1. (a) Schematic representation of the distribution ofmolecules in the ﬁeld induced paranematic (NU − ) and biaxial
nematic (NB ) liquid crystals with ∆ < 0. The relative sizes ofthe molecules are exaggerated for clarity. (b) Schematic representation of the distribution function in the presence of electricﬁeld above TP N . Dotted line represents the distribution function without ﬁeld in the isotropic phase, and the continuousline that in the paranematic phase with applied ﬁeld. The peakaround π/2 is exaggerated for clarity.

conductivity of the sample as well as dielectric relaxation.
There can also be a hydrodynamic instability [1] for systems with ∆ < 0. The ﬁrst qualitative observation of thecritical ﬁeld for a positive ∆ material was due to Nicastro
and Keyes [4] who used a DC external ﬁeld. They alsomeasured the DC Kerr eﬀect in the isotropic phase of anegative ∆ material. To minimise the heating eﬀect Durand et al. [5,6] applied an electric ﬁeld of short pulse
width (∼10 to 100 µs ) with long oﬀ period i.e. with asmall duty cycle. In this technique it takes a long time(∼40 min) to collect each data point. We have adapted atechnique developed by Basappa et al. [7]. In this, the local temperature of the sample is measured under ﬁeld andthe data points are collected continuously. This techniqueenables us to perform optical as well as electrical measurements simultaneously on materials with ∆ < 0.

S. Dhara and N.V. Madhusudana: Eﬀect of high electric ﬁelds on the nematic to isotropic transition

In this paper we report the following results on a material with ∆ < 0: (a) electric ﬁeld induced enhancementof birefringence in the nematic phase, (b) dielectric measurement of the ﬁeld induced order in the paranematic
phase, (c) an essentially linear dependence of the uniaxial
paranematic to biaxial nematic transition temperature on
the modulus of the ﬁeld, (d) divergence of the order parameter susceptibility above the tricritical point as measured by 3rd harmonic electrical signal and (e) detection ofa small 2nd harmonic electrical signal at a high ﬁeld. The
result (c) has already been reported in a brief letter [11].

141Fig. 2. Schematic representation of the h(∝E 2 ), T phase diagram of a nematic with ∆ < 0. Thick line indicates ﬁrst ordertransitions, and dashed line second order transitions: TCP denotes the trictrical point. (adapted from Ref. [2])2 Theoretical background
The application of an external electric ﬁeld to a nematic
liquid crystal with a negative dielectric anisotropy induces
biaxial ordering. Considering up to the fourth order termin the expansion, the Landau de Gennes free energy inthe presence of an external electrical ﬁeld can be written
as [2,3]F =−o∆a Eα Eβ Qαβ
2
1 
1 
1 
+ A T rQ2 + B T rQ3 + C (T rQ2 )2
2
3
4


(1)



where Q is the tensor order parameter and A (=a (T −T ∗ ), B , C are Landau coeﬃcients and T* the lowest
supercooling limit of the isotropic phase. The tensor orderparameter can be deﬁned as⎞⎛ 1− 2 (x + y)00(2)Qαβ = ⎝
0
− 12 (x − y) 0 ⎠00x
where x is related to S deﬁned earlier (x = 2/3S) andy is the biaxial order parameter which distinguishes between two directions orthogonal to the director. The equilibrium order parameters can be found by minimising thefree energy density F . The calculated phase diagram using this model for a nematic liquid crystal with negative
dielectric anisotropy is schematically shown in Figure 2 inwhich h = o ∆a E 2 /3. The dependence of TP N on ﬁeld
can be expressed analytically only when it becomes second ordered, above the tricritical point. At such high ﬁelds,∆T = TP N − TN I where TN I is the ﬁeld free isotropic tonematic transition point, is found to depend both on | E |as well as E 2 [2,3]. However, as discussed in reference [3],at lower ﬁelds, the first order transition point can only becalculated numerically. ∆T has a practically linear dependence on E 2 [1,3] as shown schematically in Figure 2. For
small E values, ∆T and ∆x = x(E) − x(0), the change inthe order parameter in the nematic phase at the transition
point are also small. Expanding the free energy to linear
order in ∆x and ∆T , it can be shown that ∆T ∝ E 2 .

3 Experimental
In order to measure the local temperature of the sample
we use an evaporated nickel (Ni) ﬁlm to design a tem-

perature sensor. Ni shows high temperature coeﬃcient ofresistance (∼6180 ppm/K). A thin ﬁlm of nickel is vacuum coated on a plane glass plate. A zigzag pattern ofthe nickel ﬁlm, which has a strip width ∼200 µm and total
length ∼0.05 m is etched using a photolithographic technique. The resistance of the pattern is ∼100 to 200 Ω. The
Ni thermometer is calibrated in each individual cell. The
thermometer is covered with a thin layer of insulating SiO
on which a circular aluminum (Al) electrode (0.5 cm diameter) is vacuum evaporated. In order to reduce the ﬁeld
gradient at the edge, a guard ring is provided just outside
the Al electrode with separation of 100 µm as shown in
Figure 3a. The top electrode is an ITO coated glass plate.
Finally a thin layer of SiO is coated on both the plates at
a grazing angle of 30◦ using a vacuum evaporation technique. The director is aligned orthogonal to incident direction of the SiO beam [1]. Mylar spacers are used toobtain the required cell thickness. The sample thickness ismeasured outside the electrode area by an interferometric technique. The typical cell thickness is ∼16 µm. The
electrical connections to the two plates are made throughcopper wires which are soldered using an ultrasonic soldering gun. The cross section of a typical cell is schematicallyshown in Figure 3b. The cell is mounted in an Instec hotstage (HS1) which itself is placed on the rotating stage ofa microscope (Leitz, Orthoplan). The temperature is controlled to an accuracy of 0.008C. The cells are ﬁlled withthe sample in the isotropic phase and on slow cooling to
the nematic range, well-aligned samples are obtained.
The block diagram of the experimental set up is shownin Figure 4. The output voltage of a lock in ampliﬁer (LIA,
model SRS 830) is connected to a high gain voltage ampliﬁer (TREK, 601-2). The output of the ampliﬁer is connected to one of the two branches as shown in Figure 4.
In one branch there is a potential divider circuit, which
is made of two resistors (100 Ω and 1 MΩ) connected inseries. The potential divider circuit is used to measure thephase and amplitude of the ampliﬁed voltage. In anotherbranch the sample cell in connected in series with a capacitor Cm (∼1 µF). A manual DPDT switch is used to switch
between the two branches. The signal across the capacitor
Cm is used to measure the phase and amplitude of thecurrent ﬂowing in the cell. Using an impedance analysis
technique [7] we measure the capacitance (CS ) and resistance (RS ) of the sample. If V0 and φ0 are the amplitude

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surements in the transmission mode. As we have used an
ITO coated glass plate as the top electrode, the opticalmeasurements can be made in the reﬂection mode of the
microscope. A laser beam (He-Ne, λ = 632.8 nm) is passed
through a polariser (POL) and made to be incident on thesample. The bottom Al electrode reﬂects the laser beam.
The reﬂected beam is passed through an analyser (ANL)
which is crossed with respect to the polariser. A photodiode (PD1, model Centronics OSD-5) is used to measure
the reﬂected intensity. The stability of the laser intensity
is monitored by another photodiode (PD2). A multimeter(MUL, Keithley 2000) is used to measure the output voltages of both the photodiodes as well as the resistance of
the Ni thermometer. The temperature variation of the optical intensity is measured. The intensity in this geometryis given by the relation
I=

Fig. 3. (a) Schematic diagram of the structure of the lower
electrode. There is an insulating SiO coating between Ni thermometer and the Al electrode. Note the zigzag pattern of theNi thermometer. (b) Schematic of the side view of the cell (notto scale).

Fig. 4. Schematic diagram of the experimental setup. PD1,PD2 (Photodiodes). POL (Polariser), ANL (Analyser), MUL(Multimeter), AMP (Voltage Ampliﬁer), LIA (Lock in ampliﬁer), R1 , R2 (Resistances), COM (Computer).

and phase at the output of the ampliﬁer and Vm and φm
are those measured across the capacitor Cm , it can beshown that the capacitance and resistance of the sample
are given by [7]: RS = Y /(ω sin α), CS = X/Y where
X = cos α − Q, Y = (sin2 α + X 2 )/(Cm Q), Q = Vm /Vo ,α = φ0 − ψm and ω = 2πf , f is the frequency of the applied signal. The dielectric constant of the sample is given
by CS /C0 , where C0 is the capacitance of the empty cell.
The lower electrode is opaque due to the Al and Ni
coatings, and it is not possible to perform optical mea-sin2 2ψ
(1 − cos ∆φ)2(3)

where ψ is the angle made by the polariser with the optic
axis and the phase diﬀerence
∆φ =

2π
∆n2d
λ

(4)

where ∆n = ne − no , ne and no are the extraordinary and ordinary refractive indices of the liquid crystal medium. d is the sample thickness and the factor 2
arises because the light travels twice across the sample
thickness in the reﬂection mode. The angle ψ is set at45◦ to optimise the measurements. The birefringence iscalculated from the measured intensity. The frequencies
and the voltage ranges used in the experiments are 4111,
15111 Hz, and 5 to 270 V respectively. The sample used is
4 -butyl-4 heptyl-bicyclohexyl-4-carbonitrile (CCN47) obtained from Merck. It has the following phase transitions:
Cr 28 ◦ C SmA 30.6 ◦ C N 59.7 ◦ C I. This compound exhibits room temperature liquid crystalline phase with alarge negative dielectric anisotropy (−8.0 at 20 ◦ C). The
measurements are completely controlled by a computer,using a suitable program. All the experiments are performed on cooling the sample from the paranematic phase.

4 Results and discussion
Variations of birefringence (∆n) as well as dielectric
constant (⊥ ) are shown as functions of local temperature(measured by the Ni resistance thermometer) at various ﬁelds in Figures 5a and 5b respectively. The following
important features are noted from Figure 5a: (i) with increasing ﬁeld (a) ∆n is enhanced in the nematic phaseand (b) the paranematic to nematic (NU − − NB ) transition temperature (TP N ) is shifted toward higher values;(ii)∆n rises from zero relatively sharply as the temperature is lowered in the phase transition region; (iii) separations among the curves are quite wide close to the transition point and are reduced as the temperature is loweredin the nematic phase. The experiments were conducted on

S. Dhara and N.V. Madhusudana: Eﬀect of high electric ﬁelds on the nematic to isotropic transition

143

Fig. 6. Variation of paranematic-nematic transition temperature (TP N ) as a function of applied ﬁeld. Data points are
obtained from the results shown in Figure 5a.

Fig. 5. (a) Variations of ∆n as well as S across theparanematic-nematic transition region as functions of local
temperature at 1.5 × 105 V/m (squares), 3.3 × 106 V/m (circles), 6.5 × 106 V/m (triangles), 9.8 × 106 V/m (diamonds),
1.3 × 107 V/m (hexagons), 1.6 × 107 V/m (stars). Frequencyof the applied ﬁeld: 4111 Hz. Cell thickness: 16.4 µm. (b) Variation of ⊥ across the paranematic-nematic transition region
as a function of local temperature at the same ﬁelds measured
simultaneously with ∆n shown in (a). Dotted line connects thetemperatures (designated as TP N ) at which the ∆n starts torise from zero value.

three independent samples and the results are similar in
all the cases.
The temperature at which ∆n starts rising from zero
value is a measure of the paranematic to nematic transition temperature (TP N ). At low ﬁelds, the paranematic tonematic transition is ﬁrst ordered in nature, and one would
expect a sharp jump in the birefringence at TP N . In our
experiments, the temperature resolution in 8 mK, and as
can be seen from Figure 5a, the width of the transition region is ≤0.06 K, for ﬁelds upto 9.75 × 106 V/m. The widthis similar for the four curves, which implies that it does notarise from ﬁeld gradients within the area of illuminationby the laser beam. The width may partly arise from dissolved impurities. Adiabatic calorimetric measurements
have shown that even a highly stable low temperature nematogen like hexyl cyanobiphenyl exhibits a two phase coexistence of 30 mK around TN I [12]. An additional contribution to the width can arise from the nature of our experiment. The transmitted intensity acquires a nonzero value

as soon as there is some alignment along the x-axis. The
alignment naturally is initiated at the two treated surfacesof the 16 µm thick cell. The alignment can be expected
to pervade the entire cell at a slightly lower temperature,
which should be considered as the true bulk paranematicto nematic transition point. In any case, as all the experimental runs have been made under similar conditions
on cooling the sample, the relative shifts should not besensitive to the method of identifying the transition temperature. The well separated temperatures at which thebirefrigence becomes nonzero can be taken as a measureof TP N .
The variation of TP N obtained from Figure 5a is shown
as a function of ﬁeld in Figure 6. A large shift in TP N(∼0.3◦ ) is measured between 1.5 × 105 to 1.6 × 107 V/m.
The important result is that TP N varies linearly with therms ﬁeld. It may be recalled that a quadratic variation of
TP N is predicted by the Landau de Gennes free energy
as shown in Figure 2. This has to be contrasted with theclearly quadratic dependence of the shift in the smectic
A-nematic (AN) transition point with ﬁeld in the case ofoctyloxy cyanobiphenyl [7]. The linear variation of TP N as
a function of ﬁeld indicates that the quenching of director
ﬂuctuations not only increases the order parameter but
also has a strong inﬂuence on the phase transition. This
point was not recognised in earlier high electric ﬁeld experiments on systems with ∆ > 0 [5]. In the case of AN transition, both bend and twist elastic constants diverge as thetransition point is approached from the nematic side, andcorrespondingly, the director ﬂuctuations tend to zero andthus do not inﬂuence the thermodynamics of the transition. It should be mentioned that on prolonged application of high ﬁelds the sample degraded slightly reducing
the nematic to isotropic transition temperature. We havecarried out experiments only for a few ﬁelds to complete
the experiment quickly (in ∼8 h) and hence avoided suchdegradation.
In the nematic phase, for example at 59.6 ◦ C (seeFig. 5a) the enhancement in ∆n is measurable up to∼9.75 × 106 V/m. Beyond that the curves in the nematic
phase are crowded though they are well separated in thetransition region. Visual observations of the sample were144

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made between crossed polarisers in the nematic phase withthe principal director oriented parallel to the polariser (seeFig. 4). Up to a ﬁeld of ∼1.1 × 107 V/m, the ﬁeld of viewis uniformly dark. Beyond this ﬁeld, some birefringent regions are seen along with the motion of some dust particles. In spite of using a guard ring there may be a ﬁeld
gradient at the edges of the sample. The ﬁeld may also
be nonuniform inside the cell due to slight variations of
the local thickness or the presence of the dust particles
in the sample. Such ﬁeld gradients which may not be in
the region in which the laser beam is incident can cause
physical motion of the medium at high ﬁelds. As a result
the sample can get misaligned and lead to light transmission. The merging of the curves beyond 107 V/m probably
occurs due to such a misalignment.
From the dielectric data (shown in Fig. 5b) several
important points can be noted: the width of the transition is somewhat larger than that found in the opticalexperiments. The increase of the paranematic to nematictransition temperature with ﬁeld is however seen clearly.
The temperatures which are connected by a dotted line
in Figure 5b are obtained from Figure 5a and correspondto those at which ∆n starts to rise from zero. The larger
width of the transition may arise due to the fact that ⊥is measured over a large area (2 × 10−5 m2 ) over which
the temperature may have some nonuniformity. With increasing ﬁeld, ⊥ is increased in the NU − phase (see rightside of the dotted line in Fig. 5b). This shows that the induced order parameter is also increased. We can calculate
the uniaxial negative order parameter in the paranematic
phase from the dielectric data, which will be discussed
later. At temperatures well below the transition region ⊥
decreases slowly with decreasing temperature and the enhancement of ⊥ with increasing ﬁeld is also very small.
On the other hand, with increasing ﬁeld a substantial enhancement of ∆n which is directly proportional to S isobserved well below the transition temperatures also.
The orientational order parameter for uniaxial nematic
liquid crystals can be well approximated by S ≈ ∆n/∆n0 .
The birefringence of the compound used is very small, andhence the internal ﬁeld corrections are negligible. Thus it isa legitimate approximation to calculate the order parameter directly from ∆n data. In the nematic phase under anelectric ﬁeld (as ∆ < 0 ) the director ﬂuctuations in theplane containing the ﬁeld are reduced compared to thosein the orthogonal plane. As a result the system becomes
biaxial, which is described by the order parameter S andthe biaxial order parameter P , which is related to y (seeEq. (2)). In our experiment we measure the birefringencein the plane orthogonal to the electric ﬁeld. This is a measure of (S − P ) rather than S. However, the ﬁeld induced
biaxial order parameter P ∼ 10−3 (at ∼3 × 106 V/m) in asimilar sample [10] whereas S ∼ 0.2. Hence we use ∆n as
a measure of the order parameter S. We have measured∆n over the entire nematic range of the ﬁeld free sample
to estimate ∆n0 , the value for perfect orientational order(S = 1) by using the Haller extrapolation procedure [13].
The scale corresponding to S = ∆n/∆n0 is also marked in
Figure 5a. The Landau de Gennes theory [1] gives rise to

Fig. 7. Variations of order parameter at diﬀerent ﬁelds inthe nematic phase within 0.15 ◦ C of the transition point. Continuous lines are the theoretical ﬁts to equation (5). Verticaldotted lines indicate the temperatures corresponding to theﬁtted values of T ∗∗ .

the following form of the temperature dependence of theorder parameter in the ﬁeld free uniaxial nematic liquid
crystal:
βT(5)
S − S0 = α 1 − ∗∗T
where T ∗∗ is the absolute limit of superheating of the nematic phase, S0 is the order parameter at T ∗∗ , β = 0.5,
and α is a proportionality constant.
According to the Landau de Gennes theory [1] (T ∗∗ −TN I ) is 8 times smaller than (TN I −T ∗ ). Experiments showthat T ∗ ∼ TN I − 1. Thus T ∗∗ is ∼0.1◦ above TN I , according to the theory. As is well known [1]experimental resultson most nematogens yield β ≈ 0.2, which reﬂects the inadequacy of the meanﬁeld model. In our samples there is asmall width of the paranematic to nematic transition. We
can ﬁt the temperature variation of the order parameterby treating equation (5) as an empirical relation which
is valid even in the presence of the ﬁeld. The ﬁeld dependence of T ∗∗ can be expected to reﬂect the variationof the bulk transition temperature. We use the data in
the nematic phase within only 0.15 ◦ C from the transition point in this analysis. As we have mentioned, there issome misalignment of the sample beyond ∼107 V/m andhence we will consider the order parameter data only up
to 9.75 × 106 V/m. In Figure 7 experimental values of theorder parameter in the nematic phase within 0.15 ◦ C fromthe transition point as well as the calculated variations using equation (5) are shown. The comparison between thetwo is reasonably good up to 6.5 × 106 V/m, and not sogood for 9.75 × 106 V/m. The ﬁtted values of S0 , α, βand T ∗∗ are shown in Table 1. T ∗∗ at diﬀerent ﬁelds arealso indicated by vertical dotted lines in Figure 7. The
values of TP N which are deﬁned as the temperature below which the birefringence becomes nonzero (see Fig. 5a)
are also shown in Table 1 to compare with T ∗∗ . From Table 1 it is seen that T ∗∗ is ∼0.07 ◦ C lower than TP N at
1.5×105 V/m and the diﬀerence is reduced to 0.04 ◦ C at aﬁeld of 9.75 × 106 V/m. This small diﬀerence between T ∗∗

S. Dhara and N.V. Madhusudana: Eﬀect of high electric ﬁelds on the nematic to isotropic transition
Table 1. Parameters which ﬁt equation (5) at diﬀerent ﬁelds.
Field in V/m
5

1.5 × 10
3.27 × 106
8.51 × 106
9.75 × 106

S0

α

β

0.12
0.11
0.09
0.08

0.37
0.40
0.47
0.52

0.17
0.17
0.17
0.16

T

∗∗

◦

( C)

◦

TP N ( C)

59.68
59.73
59.80
59.88

59.75
59.78
59.86
59.92

Fig. 8. Dependence on electric ﬁeld of ∆TP N = (TP N (E) −TNI (0)) (triangles), where TP N (E) is the temperature at which
the birefringence starts to rise from zero (see Fig. 5a), andof ∆T ∗∗ = T ∗∗ (E) − T ∗∗ (0) (circles, see Fig. 7). Variation of∆TP N (squares) of 5CB obtained from Figure 3 of reference [5]is also shown.

and TP N is a measure of the width of the transition, discussed earlier. When the ﬁeld is increased from ∼1.5 × 105
to ∼9.75 × 106 V/m, S0 decreases from 0.12 to 0.08, indicating that the strength of the transition is decreased
as the ﬁeld is increased as expected. The value of β isalmost constant (0.17) and decreases slightly to 0.16 at
9.75 × 106 V/m. In order to measure ⊥ , the lowest ﬁeld
applied was 1.5 × 105 V/m. As the ﬁeld is too small tocause any measurable change in the properties near TN I ,we assume that T ∗∗ for 1.5 × 105 V/m is T ∗∗ (0). The ﬁeld
variation of ∆T ∗∗ (=T ∗∗ (E)−T ∗∗ (0)) is shown in Figure 8.
The amount of shift is ∼0.2 ◦ C which is similar to the shiftof ∆TP N when the ﬁeld is increased from 1.5 × 105 V/m
to 9.75 × 106 V/m. The important observation is that thevariation of ∆T ∗∗ is essentially linear with |E|. The electric ﬁeld phase diagram of a nematogen with ∆ > 0 has
been published [5]. However, ∆TP N (= TP N (E) − TN I (0))
has not been plotted as a function of |E| in this study.
We use the data given in Figure 3 of reference [5] andplot the shift ∆TP N of pentyl cyanobiphenyl (5CB) alsoin Figure 8. It is clear that ∆TP N shifts linearly with |E|in this case as well.
4.1 Order parameter enhancement by the ﬁeld
Nematic Phase. The increase in order parameter due
to the application of electric ﬁeld arises because of two
physical mechanisms. One mechanism is the macroscopic
quenching of thermal ﬂuctuations of the director [1]. As
in the case of materials with ∆ > 0 [6] the enhancementof order parameter for a system with ∆ < 0 due to this145

eﬀect alone can be written as
3KB T(o ∆)1/2 |E|
8π(K)3/2(6)= Cl |E|

δSl = S(E, T ) − S(0, T ) =where K is the average curvature elastic constant, S(E, T )is the order parameter in the presence of ﬁeld and S(0, T )is the same in the absence of ﬁeld at the temperature
T . The subscript l indicates a linear variation with |E|.
The electric ﬁeld quenches director ﬂuctuations only in theplane containing the ﬁeld. The ﬂuctuation components inthe orthogonal plane are not quenched, and as such thenumerical value of Cl is smaller than in materials withpositive ∆ [1]. In the mean ﬁeld approximation K ∝ S 2 ,and as ∆ ∝ S, Cl ∝ S −5/2 . As we shall see below, close
to the NI transition point, the variation of Cl is muchsteeper, and we can write Cl ∝ S −x , in general. As discussed by Lelidis et al. [6], the enhancement of the orderparameter due to the quenching of thermal ﬂuctuations inturn generates a saturating term in second order, which is
given by:C2
(7)
δSlq = Clq E 2 = −x l E 2S(0)The other mechanism for the enhancement of order parameter under the application of ﬁeld is the Kerr eﬀect,
which is microscopic in origin. The increase in order parameter due to this eﬀect alone can be written as
δSkq = Ckq E 2(8)where Ckq is an appropriate susceptibility. This eﬀect isquadratic in E, indicated by the subscript q. The enhancement in the order parameter due to both the eﬀects canbe written as
δS(E) = δSl + δSq = Cl |E| − x

Cl2 E 2
+ Ckq E 2
S(0)

(9)

The variations of measured δS(E)(=S(E)−S(0)) at diﬀerent temperatures very close to TN I are shown in Figure 9a.
Using a least squares ﬁtting procedure, the enhancementof the order parameter is ﬁtted with equation (9). The
temperature variations of the ﬁtted parameters Cl andCq (=Clq + Ckq ) are shown in Figure 9b. Cl decreases and
Cq increases as the temperature is lowered in the nematic
phase [6,8]. The measurements which have been madewithin ∼0.2 ◦ C of TN I show that x  5, which is muchlarger than the mean ﬁeld value of 5/2. The rapid increaseof Cl as the temperature is increased leads to an evenmore rapid increase in the negative contribution to thequadratic term as given in equation (9). Thus, though Ckq
also increases with temperature, the contribution from thequenching of ﬂuctuations dominates, and leads to a lowering of the net Cq as TN I is approached. The overall ﬁeld
induced enhancement of the order parameter of course increases as TN I is approached and is dominated by δSl .
Paranematic Phase. In the ﬁeld induced paranematic
phase the symmetry axis is parallel to the laser beam and146

The European Physical Journal E

Fig. 10. Variation of uniaxial negative order parameter (SU − )in the paranematic phase as a function of applied ﬁeld at atemperature 60.1 ◦ C. Circles are experimental data. Continuous line is the ﬁt to the functional form S = Cq E 2 , where
Cq = 2.8 × 10−16 (m/V)2 .

cell. The ﬁt parameter Ckq = 2.8 × 10−16 (m/V)2 and iscomparable to the value of Ckq in the nematic phase.
4.2 Variation of paranematic to nematic transition
point with ﬁeld: inﬂuence of director ﬂuctuations
Fig. 9. (a) Variation of enhancement of the order parameter (δS) as a function of ﬁeld at diﬀerent local temperatures. Continuous lines are ﬁt to the functional form of δS =Cl |E| + Cq E 2 . (b) Variations of the ﬁtted parameters, Cl andCq as functions of local temperature. Dotted lines are drawn
as guides to the eye.

therefore we can not measure the order parameter using the optical technique. However we can calculate theuniaxial negative order parameter (SU − ) from the dielectric data taken from Figure 5b. Thus the measurement
2
yields P|| =  + 3 ∆a SU − where the superscript signiﬁes paranematic phase and the subscript refers to the direction which is parallel to the symmetry axis. On theother hand, in the nematic phase the principal director
is perpendicular to the ﬁeld and we measure N⊥ which is
=

−
∆
S/3,
ignoring
biaxiality
which isgiven by Na⊥very small. (Above the tricritical point the paranematicN
nematic transition is second order and, P|| = ⊥ and hence
it is expected that SU − = −S/2 at TP N .) We have used⊥ at 1.5×105 V/m and the value of S obtained by opticalmeasurement at (TP N − 2) ◦ C to evaluate ∆a = −9.97.
The variation of uniaxial order parameter SU − is shown as
a function of the applied ﬁeld at T = 60.1 ◦ C in Figure 10.
The enhancement of order parameter in the paranematic
phase arises mainly due to the Kerr eﬀect and we use aleast squares ﬁtting procedure to ﬁt the data with theequation S = Cq E 2 . The agreement with the measured
data is not very good. As we discussed earlier the dielectric constant is measured over a large area (2 × 10−5 m2 )and the ﬁeld and temperature may not be uniform in the

As shown in Figure 8, TP N as well as T ∗∗ vary linearlywith |E|. This is true for systems with both ∆ > 0 as
well as ∆ < 0. This clearly shows the importance of thequenching of director ﬂuctuations even in the thermodynamics of the paranematic to nematic (PN) transition under electric ﬁelds.
We may point out an analogous problem of the undulation interaction, which arises in lamellar systems. In
this case, the layer ﬂuctuations are restricted due to thepresence of the neighbouring layers and the corresponding
increase in the free energy was calculated by Helfrich [15].
In the case of nematic liquid crystals the amplitudes ofthe director ﬂuctuations are reduced in the presence of
the ﬁeld. We can use a dimensional analysis to estimate
the corresponding increase in the free energy density. Our
experiments were conducted on samples of ∼16 µm thickness. Theoretical calculations [10,16] show that at sucha thickness, the director ﬂuctuations are not signiﬁcantly
suppressed by the boundary conditions. However, Dunmur
et al. [10] found that the measured quenching of ﬂuctuations by an external electric ﬁeld is signiﬁcantly smaller
than the theoretical estimate for a sample of thickness of∼12 µm. We have not carried out measurements on the
thickness dependence of the electric ﬁeld eﬀects. Our earlier studies have shown that in the absence of externalﬁelds, there is a signiﬁcant increase in the measured order parameter when the sample thickness is reduced to amuch smaller value, ∼1 µm [16], implying a strong suppression of the ﬂuctuations in the latter case. We do notconsider the inﬂuence of the sample boundaries in thefollowing analysis of the electric ﬁeld eﬀects. There are
two important length scales in this problem. One is the 121electric coherence length ξ(E) = oK∆|E| , which isS. Dhara and N.V. Madhusudana: Eﬀect of high electric ﬁelds on the nematic to isotropic transition147

a length such that the director ﬂuctuations with wavelengths longer than ξ are essentially suppressed by theﬁeld [1]. The other length gives rise to the cut oﬀ wave
vector qc = 2π/l, for the applicability of the elasticity
theory, where l is a typical molecular dimension. As the
additional contribution arises from an entropic origin, we
write the additional free energy density with the following combinations of the thermal energy and the above
two lengths: ∆F = kB T qc2 /ξ + kB T qc /ξ 2 . In the meanﬁeld model K = Ko S 2 and ∆ = ∆a S, and we can write
 12

√
o
. Thusξ(E) = γ S|E|, where the constant γ = oK∆
a
νT 2
µT
E
∆F = √ |E| +
S
S

(10)

where µ = kB qc2 /γ, and ν = kB qc /γ 2 , are constants andνµ. We have shown elsewhere [11] that the above formof ∆F can be obtained from a simple physical model treating the director ﬂuctuation as a random variable. Calculations based on the total free energy including the director ﬂuctuations show that both δS(E) and ∆TP N (E)predominantly vary as |E| [11].
4.3 Order parameter susceptibility: generationof third harmonic component of the electrical signal
We measure the low frequency dielectric response of themedium. As we will show below, the nonlinear (ﬁelddependent) dielectric constant generates higher harmonics
of the low frequency response. Considering only the capacitive response, the current through the cell is given by
I = d/dt[CS V (t)] where CS = C0 ⊥ , C0 being the capacitance of the empty cell and V (t) = V0 sin ωt the applied
voltage, ⊥ =  − 13 ∆a S(E), andS(E) = S(0) + Cl |E| + Cq E 2(11)|E| = |Eo sin ωt| can be expanded in a Fourier series:2
2
2
Eo 1 − cos 2ωt −cos 4ωt + ...
π
3
15
(12)
The capacitive current ﬂowing through the cell can now
be calculated. The voltage Vm measured across the capacitance Cm (see Fig. 4) is given by Vm = I/ωCm , and thecomponent at the frequency of the applied voltage is givenby:|E| = |Eo sin ωt| =

8
V0
C0 V0
∆a V02∆a
S(0) −∆a Cl −
Cq
−
¯
Cm
3
9π
d
4 d2
(13)
The quantity in brackets is ⊥ , which is enhanced with theapplied voltage as shown in Figure 5b. The third harmonic
signal, which has contributions from both the 2Ω and 4Ω
components in equation (12), is given by
V1ω =

V3ω =

8
∆a C0
Cq 3
Cl Vo2 +V
d Cm 15π4d oFig. 11. (a) Variations of the third harmonic (3f ) electrical signals as functions of local temperature. Applied ﬁeld:
1.6×107 V/m (circles) and 1.3×107 V/m (squares), frequency:
1111 Hz. (b)Variations of transmitted intensity as well as thethird harmonic electrical signal as functions of local temperature at 1.6 × 107 V/m: dotted vertical line denotes the PN
transition temperature corresponding to the peak in the 3f
signal.

(14)

V3ω depends only on the ﬁeld enhancement in the orderparameter, and hence on the susceptibilities Cl and Cq .
We have simultaneously measured the ﬁrst (f ), second(2f ) and third (3f ) harmonic electrical signals as well as
the transmitted intensity as functions of local temperature near the paranematic-nematic transition point. The
3f signal shows a clear peak only at relatively high ﬁelds,
indicating that the transition has become second orderedin nature. Variations of 3f signals at 1.3 × 107 V/m and1.6 × 107 V/m at the frequency of 1111 Hz are shown in
Figure 11a. This frequency was chosen to get reasonablehigher harmonic signals. As we mentioned earlier, there issome misalignment of the director above 1.1 × 107 V/m.
However, this does not aﬀect the electrical measurements
which essentially sense only ⊥ at high ﬁelds. The 3f signal at 1.3 × 107 V/m shows a nearly symmetric narrow
peak at 59.89 ◦ C. Its value at the peak (∼2.8 × 10−4 V)is ∼6 times larger than that of the background signal.
At 1.6 × 107 V/m the peak occurs at a higher temperature (60.08 ◦ C). The signal at the peak (∼4 × 10−4 V)is ∼ 8 times larger than the background signal. We have

148

The European Physical Journal E

Fig. 12. Variations of third (3f ) and second (2f ) harmonic electrical signals across the paranematic-nematic transition region as functions of local temperature. Applied ﬁeld:
1.3 × 107 V/m, frequency: 1111 Hz.

transition. Due to the slow dynamics near the transition
point the domains may not be able to reorient at the frequency of the applied ﬁeld. As a result a second harmonic
electrical signal (2f ) can be generated [8]. The variationof measured 2f component of the electrical signal acrossthe paranematic-nematic transition at 1.3 × 107 V/m isshown in Figure 12. There is some background signal dueto the nonlinearity of the voltage ampliﬁer. However near
the PN transition point, a small peak in 2f signal whichis ∼20% higher than the background is seen and it occurs at the same temperature as that of the peak in the
3f component (Fig. 12). It probably indicates the presence of polarized domains that do not reorient with theﬁeld [8].

5 Conclusions
estimated the tricritical ﬁeld to be 1.7 × 107 V/m for∆a = −10, from the electric ﬁeld phase diagram given inreference [3]. Optical measurements could not be used tomeasure the order parameter S beyond 1 × 107 V/m, andhence we could not precisely locate the tricritical point.
As we mentioned earlier the steep variation of ⊥ upto
9.8 × 106 V/m indicates that the transition is ﬁrst ordered whereas the variation of ⊥ beyond 1.3 × 107 V/m
is smooth and gradual, indicating that the transition has
become second ordered (see Fig. 5b). Further the electrical measurement shows a clear peak in the third harmonic
signal and hence we believe that the ﬁeld induced NU − to
NB transition becomes tricritical near 1.3 × 107 V/m. The
lower experimental value of the tricritical ﬁeld comparedto the calculated one probably arises from the contribution of the ﬁeld quenching of the director ﬂuctuations to
the thermodynamics of the PN transtion.
From the data shown in Figure 11 we ﬁnd that in thenematic phase V3ω is ∝Vo2 rather than Vo3 . For example at
59.4 ◦ C, V3ω /Vo2 is 8.5 ×10−10 V−1 at 1.3 × 107 V/m and8.4 × 1010 V−1 at 1.61 × 107 V/m. At 59.6 ◦ C this ratio increases to 1.1×10−9 V−1 and 1.0×10−9 V−1 . On the other
hand, in the paranematic phase, V3ω is ∝V03 . For example
at 60.3 ◦ C, V3ω /V03 is 1.4 × 10−11 and 1.3 × 10−11 V−2respectively for the same applied ﬁelds. Thus it is clearthat even at the high ﬁelds which are above the tricriticalvalue, the main contribution to the susceptibility arises
from the quenching of director ﬂuctuations in the nematic
phase. On the other hand, in the paranematic phase, theKerr eﬀect makes the dominant contribution to the susceptibility.
We show the variations of the 3f signal as well as thetransmitted intensity in Figure 11b as functions of thelocal temperature at 1.6 × 107 V/m. The temperature atwhich the 3f signal shows a peak (60.08 ◦ C) is indicatedby a vertical dotted line (see Fig. 11b). It occurs at the
middle of the steep variation of the transmitted intensity
which is hence the actual bulk PN transition temperature.
Under a strong electric ﬁeld, the PN transition becomes second ordered and the dipole moments of themolecules may get correlated over large domains near thisWe have presented the high electric ﬁeld phase diagram ofa nematic liquid crystal with ∆ < 0. In the nematic phasethe ﬁeld induced enhancement in the order parameter isanalysed using the Kerr eﬀect as well as the quenching ofmacroscopic director ﬂuctuations by the ﬁeld. The temperature variations of the relevant susceptibilities (Cq andCl )indicate that the latter contribution dominates. The
uniaxial negative order parameter (SU − ) in the paranematic phase is estimated from the dielectric constant data.
It varies as E 2 indicating that the contribution comesfrom the Kerr eﬀect only. Our measurements as well as
the data available in the literature show that under anelectric ﬁeld the paranematic-nematic transition temperature (TP N ) varies lineraly with |E| in both materials with∆ < 0 and ∆ > 0. It is argued that the quenching of
the director ﬂuctuations under strong electric ﬁelds signiﬁcantly contributes to the thermodynamics of the nematic to paranematic transition. At relatively high ﬁelds,
the third harmonic electrical signal which is a measure of
the susceptibility of the medium exhibits a peak at thetransition temperature indicating that the transition has
become second ordered. Analysis of the third harmonic
signal also shows that near TP N the main contribution tothe order parameter susceptibility of the nematic phase
comes from the quenching of the director ﬂuctuations evenat high ﬁelds. The detection of a very small, sharp peakin the second harmonic signal which arises at the same
temperature where the third harmonic signal diverges
indicates the presence of polarised domains near the transition point.
Our results illustrate the dominant contribution madeby the quenching of (transverse) director ﬂuctuations by
the external ﬁeld to various measured properties. A rigorous theoretical model of the electric ﬁeld phase diagram
should take into account this eﬀect, even when the transition is ﬁrst ordered. Near the critical point in materials
with ∆ > 0 and above the tricritical point in materials
with ∆ < 0, the ﬂuctuations in the magnitude of orderparameter also should contribute to the problem.

S. Dhara and N.V. Madhusudana: Eﬀect of high electric ﬁelds on the nematic to isotropic transition

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