EUROPHYSICS LETTERS

1 August 2004

Europhys. Lett., 67 (3), pp. 411–417 (2004)
DOI: 10.1209/epl/i2004-10078-8

Influence of director fluctuations on the electric-field
phase diagrams of nematic liquid crystals
Surajit Dhara 1 (∗ ) and N. V. Madhusudana 1
1

Raman Research Institute - C.V. Raman Avenue, Bangalore 560 080, India(received 5 January 2004; accepted in ﬁnal form 24 May 2004)
PACS. 64.70.Md – Transitions in liquid crystals.
PACS. 61.30.Gd – Orientational order of liquid crystals; electric and magnetic ﬁeld eﬀects onorder.
PACS. 61.30.Dk – Continuum models and theories of liquid crystal structure.

Abstract. – We report the ﬁrst experimental high–electric-ﬁeld phase diagram of a nematic
liquid crystal with negative dielectric anisotropy. The variation of the paranematic-nematictransition temperature (TPN ) is essentially linear with |E|, whereas the Landau-de Gennes
theory predicts a quadratic variation. It is argued that the quenching of director ﬂuctuations
by the ﬁeld contributes signiﬁcantly to the thermodynamics of the transition.

Introduction. – Nematic liquid crystals have a long-range orientational order of moleculeswith shape anisotropy. Conﬁning our attention to rod-like molecules, the local average orientation direction of the long axes is denoted by a dimensionless unit vector n̂ called the director.
It is apolar in nature which requires that the orientational order parameter of a nematic isa second-rank tensor given by Qαβ = S(3nα nβ − δαβ )/2, where S = 3 cos2 θ − 1/2 is themagnitude of the order parameter, θ being the angle made by the long axis of the moleculewith n̂. The Landau-de Gennes free-energy expression describing the nematic-isotropic (NI)phase transition will then have a nonzero cubic term (see eq. (1) below), implying that thetransition is thermodynamically first ordered in nature with S jumping from a ﬁnite value( 0.3) to 0 [1]. The dielectric anisotropy ∆ (=  − ⊥ , the subscripts referring to directions
in relation to n̂) is proportional to S. As the orientational part of the dielectric energy density
20
under an external electric ﬁeld E is fe = − ∆
2 (n̂ · E) , where 0 is the vacuum dielectric2
constant, E is conjugate to S. If E is large, S is increased and further a weak orientational
order is induced even above the NI transition temperature (TNI ), giving rise to a paranematic(P) phase. The PN transition temperature increases with ﬁeld. If ∆ > 0, N and P phases
have the same symmetry and beyond a critical ﬁeld (Ec ) there is a continuous evolution of S
between N and P phases in analogy with the liquid-gas transition under varying pressures [2].
Typically Ec  2 × 107 V/m and the electric-ﬁeld phase diagrams of a couple of nematogens
with positive ∆ have been studied experimentally [3–6].
(∗ ) Present address: Department of Physics, Birla Institute of Technology and Science - Pilani 333 031, India.
c EDP Sciences
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The nematic order also gives rise to a curvature elastic response to director-distortionsin the medium which can be decomposed into three basic deformations viz splay, twist and2
bend [1]. The relevant energy density in the one-constant approximation is fd = K
2 (∇n) ,−12
newtons. Consequently, there are large
where K, the elastic constant, is rather small, ∼ 10
director ﬂuctuations in the medium, giving rise to a strong scattering of light. The external
electric ﬁeld E quenches the director ﬂuctuations as well, thus increasing the measured valueof the order parameter S, as pointed out by de Gennes [1]. The relevant increase in the orderparameter, viz (S(E) − S(0)), is ∝ |E|, varying linearly with the modulus of the ﬁeld. At
low ﬁelds this director ﬂuctuation-quenching eﬀect dominates, as was found experimentallyusing both electric [4, 6] and magnetic [7] ﬁelds. Many compounds exhibit a transition fromthe N phase to the layered smectic-A (SmA) phase as the temperature is lowered. The SmAN transition can be second-ordered in nature and, as de Gennes pointed out [8], there isa formal analogy between the SmA-N and superconductor-normal metal transitions. There
are two types of smectics viz type-I and type-II just as in superconductors. Interestingly
Halperin, Lubensky and Ma (HLM) [9] pointed out that in type-I materials of both systems,
gauge ﬂuctuations (which are the director ﬂuctuations in the nematic) couple to the (smectic)
order parameter, making the relevant transition weakly ﬁrst-ordered in nature. The HLM
eﬀect has been clearly demonstrated in the liquid-crystal case [10–13]. In particular, thequenching of the director ﬂuctuations due to a strong electric ﬁeld has been shown to restore
the second-order character of the SmA-N transition [13].
Returning to the PN transition, for materials with negative ∆, the phase diagrams can be
expected to have a diﬀerent character from those for positive ∆ materials [2]. The long axesof the molecules in the ﬁeld-induced paranematic (P) phase tend to orient orthogonal to E but
without any preference in azimuthal angles. Thus the P phase is uniaxial with a negative sign
(NU− ) with S < 0. In the N phase the medium becomes weakly biaxial under E, which has acontribution from the diﬀerential suppression of the ﬂuctuations of n̂ in the plane containing E
and n̂ compared to that in the orthogonal plane [14]. The ﬁrst-ordered biaxial nematic NB to
NU− transition is expected to go over to a second-ordered one above a tricritical ﬁeld Etc [2,15].
In the following we describe the ﬁrst experimental electric-ﬁeld phase diagram of a nematicwith negative ∆. Our main result is to show that the quenching of thermal ﬂuctuations of thedirector signiﬁcantly contributes to the ﬁeld-dependent shift of the PN transition temperature.
Experimental results. – Application of a large electric ﬁeld to a nematic heats the sample
both due to ionic currents and dielectric relaxation. Durand et al. [3, 4] used short highfrequency electric pulses with long time intervals between the pulses to overcome this problem.
Basappa et al. [5] developed a technique of measuring the local temperature using a patternednickel thermometer. We adopted the latter technique which allows us to measure both ⊥ andthe birefringence ∆µ(= µ − µ⊥ ) using a light beam travelling along the ﬁeld direction of aplanar aligned sample taken between an indium tin oxide-coated top plate and an aluminium(Al)-coated bottom plate. The Al coating is separated from the nickel thermometer by athin insulating layer. The experiment was controlled by a computer and the details will
be published elsewhere [16]. The optical measurement is made in the reﬂection mode of apolarising microscope (Leitz; Orthoplan). The laser beam which is polarised at an angle of45◦ to the director is reﬂected from the bottom aluminium electrode after passing throughthe sample, and the intensity transmitted through a crossed polariser is measured using aphotodiode. The sample is 4 butyl-4 heptyl-bicyclohexyl-4-carbonitrile (CCN-47) obtained
from Merck. It has the highly polar cyano group making a large angle with the long axisof the molecule and two fully saturated cyclohexane rings. Thus the medium has very low
birefringence (∆µ). The phase sequence of the compound is Cr 28 ◦ C SmA 30.6 ◦ C N 59.7 ◦ C I.

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413

Fig. 1 – (a) Variation of order parameter (calculated using S = ∆µ/∆µ0 ) at 15 × 104 V/m (opensquares), 3.3 × 106 V/m (open circles), 6.5 × 106 V/m (upward triangles), 9.8 × 106 V/m (diamonds),
1.3 × 107 V/m (downward triangles). (b) Variation of order parameter in the nematic phase within
0.15 ◦ C of the transition point. Data points are those below the temperatures limited by the dashedline shown in (a). Continuous lines are the theoretical ﬁts to S = S0 + α(1 − T /T ∗∗ )β . Vertical dottedlines indicate the temperatures corresponding to the calculated values of T ∗∗ .

The measured optical phase diﬀerence is 2∆φ = 4π
λ (∆µ)d, where d is the sample thickness( 16.0 µm, measured using an interferometric technique), and the factor 2 comes from the
double passage of the light beam through the sample in the reﬂection mode. The medium
exhibits a wide nematic range and the measured temperature variation of ∆µ [16] is used toestimate ∆µ0 , the value for perfect orientational order (S = 1) by using the Haller extrapolation procedure [17]. As ∆µ is very small, the order parameter is given to a very good
approximation by S = ∆µ/∆µ0 . Under an electric ﬁeld, the medium becomes biaxial and ∆µis a measure of S − P , where P is the biaxial order parameter which is two orders of magnitude lower than S [14] and is ignored in further discussion. The data collected at diﬀerent
sinusoidally varying ﬁelds at a frequency of 4111 Hz over narrow temperature ranges aroundthe PN transition temperatures are shown in ﬁg. 1(a) as functions of the local temperature
measured by the nickel thermometer to an accuracy of ∼ 8 mK. In the paranematic phase(NU− ), with the orientational order induced by the electric ﬁeld, the optic axis is deﬁned by
the ﬁeld, and no birefringence is measured as the light beam travels along the optic axis. A
simultaneous measurement of the dielectric constant shows an enhancement (∝ E 2 ) reﬂecting
the induced order in the paranematic phase [16]. The obliquely coated SiO on the electrodes
induces a surface order above TNI even in the absence of the ﬁeld. This gives rise to a measurable birefringence below some temperature. As seen in the data at the lowest ﬁeld (ﬁg. 1(a)),
the thickness-averaged value of S tends to diverge as the temperature approaches the transition point to the bulk nematic phase, as predicted by theory [18]. Below the bulk transition
temperature (TPN ), the order parameter variation has the opposite curvature, tending tosaturate as the temperature is lowered (ﬁg. 1(a)). The dotted line in ﬁg. 1(a) connects thetemperatures up to which the smooth bulk-like variation of S is seen in diﬀerent voltage runs.
In order to get a quantitative measure of TPN , we ﬁtted the data within a range of 0.15 ◦ C
below the temperature lying on the dotted line to the functional form S = S0 + α(1 − T /T ∗∗ )βin which S0 is the value at T ∗∗ , the superheating temperature of the nematic, and α and β
are constants. This form is predicted by the Landau-de Gennes theory with β = 0.5, but,
as is well known [1], experimentally β is found to be much smaller. Our data can be ﬁtted
reasonably well up to a ﬁeld of 107 V/m to the above form with β  0.17 (ﬁg. 1(b)). T ∗∗ isvery close to the bulk transition temperature TPN and can be taken as a measure of the latter.
We plot both ∆T ∗∗ (= T ∗∗ (E) − T ∗∗ (0)) and ∆T corresponding to the temperature at which

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Fig. 2 – (a) Dependence on electric ﬁeld of ∆T = T (E) − T (0) (ﬁlled triangles), the temperaturesat which the birefringence starts to rise from zero (see ﬁg. 1(a)), and of ∆T ∗∗ = T ∗∗ (E) − T ∗∗ (0)(ﬁlled circles, see ﬁg. 1(b)). Variation of ∆TPN (ﬁlled squares) of 5CB obtained from ﬁg. 3 of ref. [3]is also shown. (b) Variation of the measured enhancement of order parameter ∆S = S(E) − S(0) at
a temperature of 59.6 ◦ C in CCN-47. The lines are guides to the eye.

∆µ becomes nonzero, as functions of the RMS values of the applied ﬁeld in ﬁg. 2(a). The
most important observation, which has also been conﬁrmed on an independent sample [16], isthat the variations have ﬁnite slopes at zero ﬁeld, and increase essentially linearly with |E|.
The electric-ﬁeld phase diagrams of nematogens with positive dielectric anisotropy have been
analysed [3] by using the standard Landau-de Gennes model (see eq. (1) below). However,
TPN has not been plotted as a function of |E| in these studies. We use the data given in ﬁg. 3
of ref. [3] and plot the shift ∆TPN (= TPN (E) − TNI ) of pentyl cyanobiphenyl (5CB) also in
ﬁg. 2(a). It is clear that TPN shifts linearly with |E| in this case as well. The Landau-deGennes free energy density of a nematic with ∆ > 0 subjected to an electric ﬁeld is given by
FLdG = F0 − hSE 2 +a(T − T ∗ ) 2 B 3 C 4
S − S + S ,
2
3
4

(1)

where a, B, C are the Landau coeﬃcients, T ∗ is the maximum supercooling temperature of theisotropic phase, and h = 0 ∆0 /3, where ∆0 is the value of ∆ for S = 1. As eq. (1) contains
a term quadratic in the ﬁeld, it can only predict that ∆TPN ∝ E 2 [19]. The linear dependence
on |E| clearly shows the importance of the director ﬂuctuations even in the thermodynamics
of the PN transition under electric ﬁelds.

Fig. 3 – (a) Calculated variation of ∆TPN as a function of electric ﬁeld. (b) Calculated variation of∆S in the nematic phase at TNI − 0.1◦ as a function of ﬁeld.

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Surajit Dhara et al.: Influence of director fluctuations

Indeed, as we mentioned earlier, the quenching of director ﬂuctuations in the nematic
phase gives rise to an enhancement in the measured value of S, viz S(E) − S(0), which is∝ |E|. We show in ﬁg. 2(b) the order parameter of CCN-47 at the ﬁxed local temperature of
59.6 ◦ C as a function of ﬁeld. The variation is linear with |E| at low ﬁelds and the quadratic
contribution becomes more visible at high ﬁelds.
Discussion. – For modest ﬁelds, such that the PN transition has a ﬁrst-order character,the ﬂuctuations in the magnitude of the order parameter S can be expected to be relatively
small. The long-wavelength director ﬂuctuations are partially quenched by the electric ﬁeld,
thus decreasing the entropy of the medium. The resulting increase in the free energy has notbeen taken into account in eq. (1). We may point out an analogous problem of the undulationinteraction, which arises in lamellar systems. In this case, the layer ﬂuctuations are restricted
due to the presence of the neighboring layers and the corresponding increase in the free energy
was calculated by Helfrich [20]. In the case of nematic liquid crystals, the amplitudes of thedirector 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. There are two
1 1, whichlength scales in this problem. One is the electric coherence length ξ(E) = ( 0K∆ ) 2 |E|
is a length such that the director ﬂuctuations with wavelengths longer than ξ are essentiallysuppressed by the ﬁeld [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:
q2∆F = kB T ξc + kB T ξq2c . In the mean-ﬁeld model K = K0 S 2 and ∆ = ∆0 S, and we can√1write ξ(E) = C S/|E|, where the constant C = (K0 /0 ∆0 ) 2 . Thus,
βT 2
αT
E ,
∆F = √ |E| +
S
S

(2)

α. The contribution of thewhere α = kB qc2 /C, and β = kB qc /C 2 are constants and β
above unconventional expression to ∆F is zero when E = 0. In order to justify eq. (2), we
ﬁrst consider a material with positive ∆ for the sake of simplicity. Treating the director
ﬂuctuation as a random variable [1], the ﬂuctuation amplitude n⊥ in a plane perpendicular
to the director can be assumed to have a Gaussian distribution:


n2⊥
1
exp
−
W (n⊥ ) =  ,(3)2n2⊥ C 2πn2⊥ 1where C  is a normalising constant which can be found by setting 0 W (n⊥ )dn⊥ = 1. It should
be pointed out that the limit of the integration is taken from 0 to 1 instead of 0 to ∞ as for ausual Gaussian distribution. In the absence of the ﬁeld, the ﬂuctuation amplitude is given byn2⊥ 0 = (kB T /2π 2 K)qc [1]. We estimate n2⊥ 0  0.14 in the nematic phase at 50 ◦ C, assumingl  10 Å, and the elastic constant K  5 × 10−12 newtons. For this value of n2⊥ 0 , 95% of thearea of the variation of W (n⊥ ) is covered within n⊥  0.8 and hence the upper limit of integra
tion can be taken to be inﬁnity and the normalising constant is then C  1. With increasingﬁeld, n2⊥ E decreases and hence the approximation is more justiﬁable. The entropy due to1this distribution is given by ζ = −kB 0 W (n⊥ ) ln(W (n⊥ ))dn⊥ . This can be simpliﬁed to get
 
 2
n⊥ E
∆ζ = ζE − ζ0 = kB ln

− ln



 2
n⊥ 0


,

(4)

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where ζE and ζ0 are the entropies and n2⊥ E and n2⊥ 0 are the mean-square ﬂuctuation
amplitudes in the presence and in the absence of electric ﬁeld, respectively. The sum of the
distortion and electric free energy densities fd and fe for each Fourier mode can be equatedwith the thermal energy 1/2kB T (equipartition theorem). The mean-square ﬂuctuation amplitude is then given by [1]
 2

1
kB T
,
n⊥ (q) E =2
V K(q + ξ −2 )(5)where V is the volume of the sample and ξ the electric coherence length deﬁned earlier. In
real space the ﬂuctuation amplitude is given by


 2
1
kB T
1 π
n⊥ E =
−
qc −
.
(6)
2
2π K
ξ 2
qc ξ
The excess free energy density is given by ∆F = −T ∆ζ. Using eqs. (4) and (6), the simpliﬁed form of the free energy can be recast in the form of eq. (2). Using eqs. (1) and (2), the total
free energy density is given by F = FLdG +∆F . Close to the critical point n2⊥  can be large andthe above approximation is no longer valid, and further the ﬂuctuations of order parameter S
should also be taken into account. In the paranematic phase there is no director in the absenceof ﬁeld and hence the free energy density in that phase is given by eq. (1). The equilibrium
value of S minimises the appropriate free energies. The paranematic-to-nematic transition
temperature is found numerically by comparing the two minimised energies (F and FLdG ). For
illustration, we use the Landau coeﬃcients which are known for 5CB [3]: a = 0.13×106 J/Km3 ,B = 1.6 × 106 J/m3 , C = 3.9 × 106 J/m3 , and h = 6 × 10−11 . The parameters α and β are
estimated to be 3×10−8 and 10−15 , respectively. We ﬁnd that ∆TPN shows practically a linear
variation up to 5× 106 V/m, and beyond that the inﬂuence of the quadratic component is seen(ﬁg. 3(a)). This reﬂects the trend seen in ﬁg. 2(a) qualitatively. The calculated variation of order parameter in the nematic phase (ﬁg. 3(b)) also has a trend similar to that of the measured
variation shown in ﬁg. 2(b). The experimetnal data reported in this letter pertain to a material
with negative ∆. In this case, the ﬁeld is applied orthogonal to n̂. The director ﬂuctuations
are suppressed only in the n̂ − E (i.e., ẑ, x̂)-plane, but not in the orthogonal (ẑ, ŷ)-plane. Unlike in a material with positive ∆, the distribution function is not cylindrically symmetric forE = 0. It can be written as W(nx , ny ) = W1 (nx )W2 (ny ). The width n2x E is given by eq. (6),while n2y E = n2y 0 . The net result is that the change in entropy due to the suppression of
director ﬂuctuations is reduced by a factor of 2 compared to that in a material with positive∆. This may partly account for the higher slope seen in ﬁg. 2(a) in the latter case.
Conclusion. – To conclude, our measurements as well as an analysis of the data available
in the literature show that under an electric ﬁeld the paranematic-nematic transition temperature (TPN ) varies lineraly with |E| in both materials with ∆ < 0 and ∆ > 0. It is argued thatthe quenching of the director ﬂuctuations under strong electric ﬁelds signiﬁcantly contributes
to the thermodynamics of the nematic-to-paranematic transition. Using a dimensional argument as well as a simple physical model, we have calculated the entropic contribution to
the free energy density. The calculated trends in ∆TPN as well as S(E) − S(0) reﬂect theexperimental ones. The quenching of the director ﬂuctuation thus plays a signiﬁcant role in
the thermodynamics of both SmA-N and PN transitions. As mentioned earlier, for materials
with negative ∆ the ﬁeld-induced NB to NU− transition becomes second-ordered above a
tricritical ﬁeld. This has been explored using the 3rd harmonic of the electric current which isa measure of the susceptibility of the sample [6]. This result will be discussed elsewhere [16].

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