Eur. Phys. J. E 13, 401–408 (2004)
DOI 10.1140/epje/i2003-10084-8

THE EUROPEAN
PHYSICAL JOURNAL E

Enhancement of the orientational order parameter of nematic
liquid crystals in thin cells
Surajit Dharaa and N.V. Madhusudanab
Raman Research Institute, C.V. Raman Avenue, Bangalore 560 080, India
Received 25 September 2003 and Received in ﬁnal form 22 February 2004 /c EDP Sciences / Società Italiana di Fisica / Springer-Verlag 2004
Published online: 24 May 2004 – 
Abstract. We report measurements of birefringence (∆n) of several nematic liquid crystals having transverse as well as longitudinal dipole moments in thin (1.4 to 2.3 µm) and thick (7 to 16 µm) cells. Rubbed
polyimide-coated glass plates are used to get planar alignment of the nematic director in these cells. We
ﬁnd signiﬁcant enhancement (6 to 18%) of ∆n (∝ S, where S is the orientational order parameter) in thincells in all compounds with aromatic cores even at temperatures far (∼ 20 ◦ C) below the nematic-isotropictransition point. The enhancement is larger in compounds having several phenyl rings and lower if thenumber of phenyl rings is reduced. In a compound that does not have an aromatic core no signiﬁcantenhancement is observed, implying that the strength of the surface potential depends on the aromaticityof the cores. Assuming a perfect orientational order at the surface, calculations based on the Landau-deGennes theory show that the thickness averaged enhancement of S is sharply reduced as the temperature
is lowered in the nematic phase. The measured order parameter S is further enhanced in thin cells becauseof the stiﬀening of the elastic constant which reduces the thermal ﬂuctuations of the nematic director. The
combined eﬀect is however too small at low temperatures to account for the experimental data.
PACS. 61.30.-v Liquid crystals – 61.30.Pq Microconﬁned liquid crystals: droplets, cylinders, randomly
conﬁned liquid crystals, polymer dispersed liquid crystals, and porous systems – 61.30.Hn Surface phenomena: alignment, anchoring, anchoring transitions, surface-induced layering, surface-induced ordering,wetting, prewetting transitions, and wetting transitions

1 Introduction
Organic molecules with anisotropic shape exhibit a variety of liquid-crystalline phases. The least ordered of liquid
crystals is the nematic (N), in which the long axes of therod-like molecules are spontaneously aligned on averagealong a speciﬁc direction called the director, n̂, a dimensionless unit vector [1]. It is apolar in nature which requiresthat the orientational order parameter of a nematic is asecond rank tensor given by Qαβ = S(3nα nβ − δαβ )/2,
where S = 3 cos2 θ − 1/2 is the magnitude of the orderparameter, θ being the angle made by the long axis of themolecule with n̂. In many experimental studies nematic
liquid crystals are sandwiched between two parallel glass
plates, which are pretreated for planar alignment [1]. Suchcells are also used in display (LCD) applications. The typical thickness used for this purpose is ∼ 5 µm. In the reﬂective mode displays, it is about 2 µm. Because of the ﬂuid
nature of the medium the response time is relatively slowaPresent address: Department of Physics, Birla Institute ofTechnology and Science, Pilani 333 031, India.
be-mail: nvmadhu@rri.res.in

and is given by τ = ηd2 /π 2 K [1] where η is an eﬀective
viscosity, K an eﬀective curvature elastic constant and dthe sample thickness. The response time can be reduced
considerably by reducing the sample thickness. In bistabledisplays, which are being investigated in recent years, thethickness has to be quite small, about 2 µm [2]. When thesample thickness is reduced the surface-to-volume ratio isincreased and the bulk properties are signiﬁcantly inﬂuenced by the conﬁning surfaces.
Studies of liquid crystals in conﬁned geometries arevery important both from fundamental and technologicalpoints of view. Several studies have been undertaken to
make physical measurements and to investigate the inﬂuence of diﬀerent types of conﬁnement [2]. There are several
theoretical and a few experimental studies on the conﬁnement eﬀects on the orientational order parameter of nematic liquid crystals [3–7]. Ping Sheng [3] calculated theorder parameter proﬁle in thin cells using the Landaude Gennes theory assuming a surface potential, which
enhances the order parameter at the surface. He foundthat within a limited range of surface potentials a surface
transition can occur at a temperature higher than that of
the bulk transition. Close to the nematic-isotropic (N-I)

402

The European Physical Journal Etransition point, the surface-induced order decays to thebulk value over a length scale, which is an order of magnitude larger than the order parameter correlation length[3,4]. In the nematic phase the order parameter at the surface of the substrate as well as in the interior of the sample increases with decreasing cell thickness. Experimentally, Miyano [5] measured the wall-induced birefringencein pentyl cyanobiphenyl (5CB) above the N-I transition
temperature both in homeotropically and planar aligned
samples. In the former case, the sample was viewed at anangle with respect to the optic axis. It was found that the
pretransition birefringence diverges in the homeotropic
cell while there is no such dependence in the planar cell.
Mada et al. [6] reported the measurement of surface andbulk order parameters in the nematic as well as in theisotropic phase as functions of temperature using an interferometric method [8] in heptyl cyanobiphenyl (7CB).
They used plates which were pretreated by oblique evaporation of SiO. The surface order parameter was foundto be ∼ 0.15 just above TNI and had ﬁnite values evenat temperatures further above TNI . In the nematic phasethe surface order parameter was found to be always 25%
higher than the bulk order parameter.
Sobha et al. [7] reported that ∆n is enhanced by a measurable amount in thin cells (1.5 µm) compared to thatin thick cells (25 µm) in p-cyanophenyl p-n–heptylbezoate(CP7B). This compound is highly polar and the authors
found a nematic-nematic (N-N) transition in thin cells(3 µm) though such a transition was not found in the bulk.
Interestingly, the N-N transition in CP7B is found to shift
to higher temperatures when the thickness of the cell is
reduced to 1.9 µm. They argued that the shift in the N-Ntransition temperature is a clear consequence of the signiﬁcant enhancement of the order parameter as the cell
thickness is decreased. In the bulk, the N-N transition
which is a consequence of a jump in the short-range order from a monolayer to a partial bilayer structure [1,9]is pre-empted by the crystallisation of the sample. Using
the Landau-de Gennes theory they described a uniaxial
nematic made of biaxial particles, which gives rise to relatively large values for the correlation length of the orderparameter in a suitable parameter range [10]. The large
correlation length can give rise to a signiﬁcant enhancement of the order parameter in thin cells. More recently,
Manjula et al. have conﬁrmed that the N-N transition in
the bulk of CP7B occurs above the ambient temperature
under high pressures [11]. They also found that in a binary
mixture, in which the N-N transition in the bulk occurs
above the ambient temperature, the transition temperature increases as the thickness of the cell is decreased [12].
All the compounds used in the above-mentioned studies have large longitudinal dipole moments. In order tocheck if the enhancement of the orientational order parameter in thin cells is a general feature of nematic liquid
crystals, we have made such measurements on systems
with diﬀerent chemical structures having both longitudinal and transverse dipole moments. In this paper wereport the measurement of ∆n of nematic liquid crystals in thin and thick cells which are pretreated withpolyimide and rubbed unidirectionally. We show that ∆nand hence the orientational order parameter is enhanced
considerably in thin cells with respect to that in thickcells in all the compounds with aromatic cores. The enhancement is higher in compounds having several phenyl
rings like, 2-cyano 4-heptylphenyl-4 -pentyl-4-biphenyl
carboxylate (S1014) and 4-cyanophenyl 3 -methyl 4 -(4 n-decylbenzoyloxy) benzoate (CN) (see Fig. 1). In trans4-pentyl-(4-cyanophenyl)-cyclohexane (PCH5), which has
only one aromatic ring, the enhancement is decreased.
In 4 -butyl-4 heptyl-biclohexyl-4-carbonitrile (CCN47),
which does not have any aromatic core, no signiﬁcant enhancement is observed except close to TAN , the nematicsmectic transition temperature. This implies that the surface potential on the molecules with aromatic cores isstronger than that on molecules without any aromaticcores. We show that the Landau-de Gennes theory canpartially account for this eﬀect if a strong surface potential is assumed which gives perfect order at the surfaces.
This also leads to an increase in the elastic constant which
results in a reduction in the ﬂuctuation amplitude of thenematic director, and hence contributes to the larger measured value of the order parameter in the thinner cells. The
calculated enhancement drastically decreases as the temperature is lowered, in contrast to the experimental trend.

2 Experimental
We have chosen a number of compounds with diﬀerent
chemical structures in which the dipole moments makediﬀerent angles with the long axes of the molecules. The
chemical structures and phase sequences of the compounds are shown in Figure 1. The ﬁrst compound (S1014)
is obtained from Merck. It is highly polar and the directionof the permanent dipole moment makes a large angle withthe long axis and hence the dielectric anisotropy is negative (∆ = || − ⊥ < 0). The nematic phase occurs over awide temperature range and can be supercooled to room
temperature. It also has a smectic-C–like short-range order (cybotactic groups) in the entire nematic range [13].
The second compound (CN) is also highly polar with along alkyl chain. It exhibits the nematic phase at relatively high temperatures. The direction of the permanent
dipole moment in this molecule is along the long axis andhence the dielectric anisotropy is positive. The third compound (BAN) is made of banana-shaped molecules and ishighly biaxial. This compound does not exhibit the nematic phase. The mixture with 50 : 50 wt% of compounds
CN and BAN gives rise to a uniaxial nematic with positive
dielectric anisotropy. This mixture was studied to explore
the eﬀect of molecular biaxiality on the order parametermeasured in thin cells. The compounds CN and BAN weresynthesized in our chemistry laboratory. The fourth compound (PCH5) also obtained from Merck, is highly polar.
It exhibits a nematic phase at room temperature and is
widely used in display applications. The direction of thepermanent dipole moment in this molecule is along thelong axis and hence the dielectric anisotropy is positive.
The last compound (CCN47) was obtained from MerckSurajit Dhara and N.V. Madhusudana: Enhancement of the orientational order parameter...

403

Fig. 2. Variations of ∆n of S1014 in 1.5 µm (open circles) and6.7 µm (open squares) thick cells as functions of temperature.
The dotted line is drawn as guide to the eye. Data corresponding to the ﬁlled squares are obtained from reference [13].

Fig. 1. Chemical structures, phase sequences and transition
temperatures of the compounds used in thin-cell experiments.
The transition temperatures are given in degree Celsius.

and is highly polar with a negative dielectric anisotropy,
but does not have an aromatic core.
The cell is constructed using two ITO (indium tin oxide) coated glass plates which are treated with polyimide
and cured at a temperature of 280 ◦ C for 90 minutes. After curing, the plates are rubbed for homogeneous alignment of the nematic director. Glass beads are mixed with
an epoxy glue, which is spread outside the electrode region to get the required cell thickness > 2 µm. However,
to make cells of thicknesses < 2 µm we do not use any
spacers. The glue used does not aﬀect the liquid crystals
and is usually used for making liquid-crystal displays. The
thickness of the cell is measured carefully at several positions using an interferometric technique, with the helpof the Ocean Optics spectrometer (model, S2000). Several
cells are made at a time and those with uniform thickness(within ∼ 1%) are selected for the experiment. To test
if the cell thickness changes with temperature, we heatedthe empty cell up to 150 ◦ C and measured the thickness
at several temperatures and found no noticeable change.
The cell is ﬁlled with the sample in the isotropic phase bycapillary action. A heating stage (INSTEC HS1) is used tocontrol the temperature to an accuracy of ∼ 10 mK. The
heater is kept on the rotating stage of a polarizing microscope (Leitz, ORTHOPLAN). A helium-neon laser beam
(λ = 632.8 nm) is used to illuminate the sample. A tilting
compensator (Leitz, 5λ) is used to measure ∆n. We could
also measure ∆n from the transmitted optical intensity.
In thick cells the variation of optical intensity shows maxima and minima as a function of temperature between two
crossed polarizers. However in thin cells (< 2 µm) the variation of optical intensity shows only one broad maximum

and it is not possible to calibrate the intensity and hencemeasure ∆n accurately. Therefore, ∆n values in all thecells are measured by using the tilting compensator. The
measurements are made while cooling the samples fromthe isotropic phase.

3 Results and discussion
The variations of birefringence (∆n) of S1014 measured
in two cells of thickness 1.5 µm and 6.7 µm, respectively,
are shown as functions of temperature in Figure 2. The∆n values measured using the tilting compensator in the
thick cell compares well with the data of reference [13](ﬁlled squares, Fig. 2) in which ne (extraordinary refractive index) and no (ordinary refractive index) have beenmeasured separately using a goniometer. ∆n measured in
the thin cell (1.5 µm) is larger at all temperatures thanthat in the thick cell (6.7 µm) and at lower temperatures
the two curves are roughly parallel (Fig. 2). It should bepointed out that S1014 has also a smectic-C–like shortrange order throughout the nematic range [13].
The variations of ∆n of CN as functions of temperature are shown for cells of three diﬀerent thicknesses,
namely 1.5 µm, 2.2 µm and 14 µm in Figure 3. This compound exhibits the nematic phase at high temperatures(105.5 to 154.2 ◦ C). Again, at a ﬁxed temperature, ∆nincreases with decreasing cell thickness. For example, at
T = 142 ◦ C, ∆n is higher by 6% and 11%, respectively,in the 2.2 µm and 1.5 µm cells compared to that in thecell of thickness 14 µm. For this compound we analyse
the temperature dependence of ∆n, which can be approximated well for nematic liquid crystals by the formula ∆n = ∆no (1 − T /T1 )β here T1 and β are adjustableparameters and ∆no is the birefringence of the perfectlyaligned sample [14]. Least-squares ﬁts to the experimental data on CN with the above equation are shown for thethree diﬀerent cell thicknesses in Figure 3.

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The European Physical Journal E

Fig. 3. Variations of ∆n of CN as functions of temperaturein cells of three diﬀerent thicknesses: 1.5 µm (open circles)
2.2 µm (open triangles) and 14 µm (open squares). The continuous lines are the theoretical ﬁts to the functional form of∆n = ∆no (1 − T /T1 )β . The relevant ﬁt parameters are shownin the insets.

Fig. 4. Variations of ∆n as functions of temperature of the
mixture of CN and BAN in three diﬀerent cell thicknesses,
namely 1.4 µm (open circles) 2.3 µm (open triangles), 16 µm(open squares). Dotted lines are drawn as guides to the eye.

It is noticed that the ﬁtted birefringence of the completely ordered sample in 1.5 µm is slightly higher (by 5%)than that in 2.2 and 14 µm cells. The ﬁt parameter T1 ishigher by 0.1◦ in the 1.5 µm cell than that in the 14 µm
cell. This indicates that TNI increases as the thickness isdecreased as predicted by the Landau theory [3] and hence
T1 , which is slightly above TNI , is also increased. However
our main interest is to study the eﬀect of conﬁnement on
the orientational order parameter and we have not carefully measured the enhancement of TNI at reduced thicknesses of the samples. The value of β is 0.13 in the 1.5 µm
thin cell and increases to 0.15 in the 14 µm thick cell. Similar values of β are obtained in several other nematics [1].

Fig. 5. Variations of ∆n of PCH5 measured in 1.4 µm (opencircles) and 7 µm (open squares) thick cells as functions of temperature. Dotted lines are drawn as guides to the eye.

The variations of ∆n in the 50 : 50 wt% mixture of
CN and BAN are shown as functions of temperature in
Figure 4 for three thicknesses. Close to the transition
temperature the data points are noisy because thismixture exhibits a nematic-isotropic coexistence range
 1 ◦ C. Therefore in this mixture ∆n is measured whileheating the sample. Again, at a ﬁxed temperature, ∆nincreases with decreasing cell thickness. For example, at
TNI − T = 12◦ , ∆n is higher by 6% and 13%, respectively,in the 2.3 µm and 1.4 µm cells compared to that in thecell of thickness of 16 µm.
The variations of ∆n of PCH5 measured in two cellsof thickness 1.4 µm and 7 µm, respectively, are shown asfunctions of temperature in Figure 5. The values of ∆nof the 7 µm thick cell agree well with those calculated
from earlier measurements of ne and no [15]. Close to theisotropic-to-nematic transition point (TNI − T < 2◦ ) ∆nmeasured in the two cells are comparable. At lower temperatures, there is a clear enhancement of ∆n in the thinner cell. For example, it is 6% higher than in the thicker
cell at 5◦ below TNI .
The variations of ∆n as functions of temperature areshown for CCN47 in two diﬀerent cell thicknesses, namely
1.5 µm and 6.7 µm in Figure 6. Interestingly, no signiﬁcant enhancement is noticed between the two curves corresponding to the two thicknesses in the nematic phasedown to TNI − T  20◦ . There is an enhancement of ∆nin both the cells as the temperature approaches TAN , due
to the coupling of the orientational and translation orderparameters [1]. The relative enhancement in the thinnercell increases at lower temperatures. For example, at TAN
in the thinner cell ∆n is ∼ 10% higher than that in the
thicker cell.
We summarise the experimental results of ∆n on all
the ﬁve systems studied in Table 1 (at TNI − T = 12◦ ) in
which the relative enhancement of ∆n in thin cells withrespect to thick cells is shown. We notice that in the ﬁrst
four systems ∆n is enhanced considerably in thin cells
compared to that in thick cells. It is also noticed from theSurajit Dhara and N.V. Madhusudana: Enhancement of the orientational order parameter...

405

Table 1. Comparison of ∆n in thin and thick cells.
CompoundThickness of the thinand thick cells (µm)(TNI − T ) in ◦ C∆nthin −∆nthick× 100∆nthick

S1014
CN
Mixture (50 : 50) wt%
PCH5
CCN47

1.5 and 6.7
1.5 and 14
1.4 and 16
1.4 and 7
1.4 and 7

12
12
12
12
12

15%
11%
13%
6%
no signiﬁcant enhancement

Fig. 7. Schematic representation of a few allowed ﬂuctuation
modes between the two plane surfaces.

thermal ﬂuctuations of the director on the magnitude of
the scalar order parameter in thin cells.
Fig. 6. Variations of ∆n of CCN47 in 1.5 µm (open circles) and 6.7 µm (open squares) cells as functions of temperature. Note that the separation between the two curves increases as TAN (smectic-to-nematic transition temperature) is
approached. Dotted lines are drawn as guides to the eye.

table that due to the addition of biaxial molecules in the
mixture, the enhancement of ∆n in thin cells does notchange signiﬁcantly from that of CN. Thus, the eﬀect ofmolecular biaxiality on the orientational order parameterin thin cells may not be signiﬁcant.
The orientational order parameter in thin cells in theﬁrst three systems studied is considerably larger compared to that in the thick cells irrespective of the molecular structure, temperature range and orientation of thepermanent dipole moments in the molecules. In PCH5,
which has only one phenyl ring, the relative enhancement
is smaller than in the other compounds with more than
one phenyl ring. CCN47 has no aromatic rings and also
does not exhibit any signiﬁcant dependence of ∆n on the
thickness except close to TAN . The data on ∆n in the
thicker cells of both S1014 and PCH5 agree well withthose calculated from the ne and no measurements of earlier workers. This shows that in the cells used by us thenematic has a planar alignment, without any signiﬁcantpretilt. To understand the enhancement of the orientational order parameter in thin cells two possibilities canbe invoked: A) The suppression of the thermal ﬂuctuationsof the director in thin cells can enhance the order parameter compared to that in the thicker cells; B) The eﬀectof a strong surface potential which increases the order parameter at the surfaces. First, we will discuss the eﬀect of

3.1 Quenching of director ﬂuctuations in thin cells
In the nematic phase thermal ﬂuctuations of the director
are quite strong. These ﬂuctuations cause strong scattering of light and the nematic is turbid in appearance. In
the one elastic constant approximation the mean-square
ﬂuctuation at any wave vector q is given by [1]|n⊥ (q)|2  =kB T,V Kq 2(1)

where kB is the Boltzmann constant, T the temperature,
V the volume, and K the elastic constant. It is well known
that under the application of an external strong ﬁeld the
director ﬂuctuations are partially quenched and the order parameter is enhanced as a result [16,17]. In a recent
study we have shown that the quenching of director ﬂuctuations also signiﬁcantly inﬂuences the thermodynamics
of the nematic-paranematic phase transition under an external electric ﬁeld [18]. In that case the ﬂuctuation amplitude is reduced due to the external ﬁeld. In thin cells
the number of modes are reduced due to the ﬁnite cell
thickness. The wave vector q can be decomposed into qz ,along the z-axis and q⊥ , which is in the xy plane (Fig. 7).
When the thickness is decreased the number of allowed
modes along the z-axis is reduced. Any mode which has
wavelength longer than twice the cell thickness d is notallowed, due to the conﬁnement along the z-direction.
In principle, to calculate the r.m.s. ﬂuctuation in real
space, we have to sum over the allowed modes along thez-direction [19]. However, as the smallest gap d used inour experiment is > 1 µm which is much larger than the406

The European Physical Journal Eintermolecular distance a(≈ 2 × 10−7 cm), we will replace
the summation by an integration over the appropriate limits. In real space the mean-square ﬂuctuation amplitudeis given by qmax qmax2πq⊥ dq⊥
kB T
dq
(2)
|n⊥ (r)|2  =z
2 + q2 ,3
8π K π/dq⊥0zwhere qmax ≈ 2π/a, is the cut-oﬀ wave vector. As d ≈1 µm, qmax d ≈ 103 . We use this to simplify the above
integral to get


π
kB T
2
ln
2
+
|n⊥ (r)|  ≈qmax
8π 2 K
2


 
π
2π
+
−1 .
(3)
ln
d
dqmax
It is noticed from equation (3) that the ﬂuctuation amplitude is reduced with the reduction of the sample thickness. Using equation n2z  = 1 − 2n2⊥ , the measured
value of the order parameter can be written as Smeas =S0 (3n2z  − 1)/2, where S0 is the order parameter in theabsence of director ﬂuctuations as calculated, for example, in a mean-ﬁeld theory. We get the order parameter ina cell of ﬁnite thickness d as Smeas = S0 (1 − 3|n2⊥ (r)|).
Using equation (3) it can be written as



π
3kB T
Smeas = S0 1 − 2qmax ln 2 +
8π K
2


 
π
2π
+
−1
.
(4)
ln
d
dqmax
Using K  5 × 10−7 dynes, a  20 Å, and T = 330 K, we
ﬁnd that the order parameter increases by ∼ 0.1% in a
1 µm cell compared to that in a 10 µm cell. On the other
hand, experimentally, we ﬁnd in the ﬁrst four compounds(see Tab. 1) that the enhancement of the order parameter in thin cells is nearly 6 to 13% compared to that in
the thicker cells. As such, the enhancement of the orderparameter due to the partial quenching of the director
ﬂuctuations in thin cells cannot explain our experimental
result. Now we consider the eﬀect of conﬁnement on theorder parameter of nematic liquid crystals due to the surface potential. As mentioned earlier using the Landau-deGennes theory Ping Sheng calculated the order parameter
proﬁle in thin cells [3]. He restricted the calculation between TNI −0.12◦ to TNI +0.36◦ . We extend the calculation
to a few degrees below TNI .
3.2 Eﬀect of a large surface orientation potential
We consider a nematic liquid crystal conﬁned between twoplane parallel surfaces separated by the distance d. The
solid-liquid crystal interfaces are deﬁned by z = 0 andz = d, respectively. The sample is assumed to be uniform in the x and y directions (Fig. 7). The substrates
are treated such that the molecules experience a uniaxial aligning potential in a particular direction. It can beassumed that the surface potential averaged over many
molecules over a small volume is given by [3]V = ν(θ, z) = −Gδ(z)P2 (cos θ) = −Gδ(z)S,(5)

where the angular brackets represent a local average. As
we have experimentally found that the enhancement of
the order parameter in thin cells is very substantial, we
assume the surface potential is so large that there is perfect order at the surfaces, i.e. S(0) = S(d) = 1. Selinger
et al. [20] calculated the variation of the order parameter from the surface to the bulk assuming a strong surface potential (VS = 10Vo , where Vo is the Maier-Saupeorientation potential between the molecules). They foundthat at the surface the order parameter is saturated i.e.
S(0) = 1. However, they did not extend the calculation
to the nematic phase. Following Sheng [3], we write theLandau-de Gennes free energy density

2
dS
,
(6)
φ = f (S) + L
dz
wheref (S) =aB
C(T − T ∗ )S 2 − S 3 + S 4 ;
2
3
4

(7)

f (S) is the bulk free energy density and a, T ∗ , B, C are
material parameters. L is the bare elastic constant. The
total free energy density Φ which is the volume integral of
φ can be minimised with the help of the Euler-Langrange
equation, ∂φ/∂S = ∇ · [∂φ/∂∇S]. The boundary conditions are: S = 1 at z = 0, d and (dS/dz)z=d/2 = 0 andS(d/2) = Sm , where Sm is the minimum order parameter
which is obtained at the mid-plane of the sample. Using
the above boundary conditions and following the procedure given by Sheng [3], it can be shown that
z
=
ξ0

 1
S(z)

1
dS,
F (S) − F (Sm )(8)

where ξ0 = L/aTNI is the bare correlation length and
F (S) = f (S)/aTNI and F (Sm ) = f (Sm )/aTNI . The material parameters a, B, C for PCH5 are assumed to
be a = 0.10 J/cm3 K, B = 1.5 J/cm3 , C = 3.8 J/cm3 ,T ∗ = 327.2 K, TNI = 328.5 K. The ﬁrst three coeﬃcientsare of the same order as in other nematics [3]. The elastic constant L, is found by using the formula [21] L =(K1 + K2 + K3 )/6S 2 where K1 , K2 and K3 are the splay
twist and bend elastic constants and taken from reference [22] for PCH5. L is estimated to be  3×10−13 J/cm.
The estimated bare correlation length ξ0 ∼ 10 Å. Using
these parameters we have numerically calculated the S(z)proﬁle for half the cell thickness at a few temperatures in
the nematic phase. The variation of the calculated S(z)at TNI − 1◦ is shown as a function of z/ξ0 in Figure 8.
It is noticed that the order parameter decays to a value
hardly diﬀerent from Sm when z/ξ0 ∼ 50. The thickness
averaged order parameter in the cell is calculated using
1 d
S=
S(z)dz .
(9)d 0Surajit Dhara and N.V. Madhusudana: Enhancement of the orientational order parameter...

407

Table 2. Comparison of calculated values of the order parameters in 1 µm cell using diﬀerent approximations and the bulk
values at a few temperatures.
Temperature

Sb

S

S1

TNI − 0.2◦
TNI − 1◦
TNI − 5◦
TNI − 10◦

S−Sb
× 100
Sb

S 1 −Sb
× 100
Sb

0.297
0.373
0.566
0.714

0.310
0.383
0.571
0.716

0.313
0.385
0.572
0.717

4.4%
2.7%
0.9%
0.3%

5.2%
3.2%
1.1%
0.4%

Smeas (10 µm) Smeas (1 µm)
0.091
0.209
0.460
0.631

0.119
0.228
0.467
0.635

Smeas (1 µm)−Smeas (10 µm)
× 100
Smeas (10 µm)

30%
9%
1.5%
0.6%

2

We can introduce an additional elastic term L S ∂S
in∂zequation (6), where L is also an elastic constant. This
term is also allowed by the symmetry of the medium. Following a procedure similar to that given above and assuming L = L , we ﬁnd
z
=
ξ0

 1
S(z)

1 + S2dS .
F (S) − F (Sm )(10)Using the same material parameters a, B, C, T ∗ we calculate the variation of the order parameter S1 (z) at a few
temperatures where the suﬃx 1 indicates that an additional elastic term is used. The S1 (z) proﬁle, which is
2term incalculated at TNI − 1◦ including the L S ∂S∂zequation (6), is shown by a dotted line in Figure 8. We
notice that the overall variation of S1 (z) is similar to thatof S(z). The order parameter S1 (z) is somewhat higher
upto z/ξ0  50 than that of S(z). We summarise thecalculated values of the order parameter in the bulk Sb ,obtained from equation (7), and in the 1 µm cell at a few
temperatures in Table 2. Close to TNI the enhancement
is large and decreases rapidly as the temperature is lowered. This trend is expected as the perfect order at thesurface has a larger eﬀect when the bulk order parameter is lower. The bulk order parameter is increased withdecreasing temperature and the eﬀect of surface order isreduced. On the other hand, experimentally, we ﬁnd thatthe enhancement is not signiﬁcantly reduced at lower temperatures (see Tab. 1).
The enhancement of the order parameter due tothe reduction of the cell thickness has two aspects as
discussed above. Firstly, the number of ﬂuctuation modes
are reduced (see Sect. 3.1). This increases the orderparameter by ∼ 0.1% when the thickness is decreased
from 10 to 1 µm. Secondly, with a strong surface ordering
potential, the thickness averaged value of the orderparameter S in thin cells is enhanced. This increased
value of S, in turn, leads to an increase in the eﬀective
curvature elastic constant (K ∝ S02 ) which results in a
reduction in the ﬂuctuation amplitude (see Eq. (3)). To
estimate this eﬀect, we simply use the average value S1
in place of S0 in equation (4) and use K = K0 (S 1 )2 to
calculate Smeas . In this calculation we use K0 = 2 × 10−6
dynes and a = 20 Å. The calculated values of Smeas incells of thicknesses 1 µm and 10 µm at a few temperatures
are also shown in Table 2. Smeas is less than the meanﬁeld order parameter S0 due to the director ﬂuctuations
in both the 10 and 1 µm cells. However, the relative
enhancement of Smeas in a cell of thickness 1 µm is larger

Fig. 8. Variations of the calculated order parameters (continuous line: S(z) and dotted line: S1 (z)) as functions of z/ξ0
near one wall of the thin cell. Variations are shown only up toz/ξ0 = 60 for clarity. (1 µm corresponds to z/ξ0 = 1000.)

than that of S1 . Thus, the combined eﬀect of the surface
potential and the stiﬀening of the elastic constant has
ampliﬁed the enhancement of Smeas in thin cells. Closeto TNI the enhancement in Smeas is very large and again
decreases sharply as the temperature is lowered. On theother hand, the experimentally measured enhancement
remains high as the temperature is lowered in the nematic
phase (Figs. 2–5). Thus, the large enhancement well below
TNI is not accounted for by the model.
Our experiments show that the orientational order parameter of nematic liquid crystals in thin cells is enhanced
considerably with respect to that in thick cells in all thecompounds with aromatic cores. In PCH5, which has only
one phenyl ring, the relative enhancement is smaller thanthat in compounds with more than one phenyl ring. In
CCN47, which has no phenyl rings in the molecular structure, no enhancement is observed except close to TAN .
This implies that the surface potential on the moleculeswith aromatic cores on the polyimide coating is muchlarger than that on molecules without any aromatic cores.
The Landau-de Gennes theory for the uniaxial medium
partially accounts for the enhancement of the order parameter if the surface potential produces perfect order.
Both the number of ﬂuctuation modes as well as the ﬂuctuation amplitude are reduced in thin cells. The lattereﬀect is due to the stiﬀening of the elastic constant due tothe enhanced order and has a larger inﬂuence on the “measured” order parameter. However, the calculated values of408

The European Physical Journal E

the enhancement of the order parameter decrease sharply
as the temperature is lowered from TNI . On the other
hand, the experimental values do not decrease so drastically. Curiously, in CCN47 there is a signiﬁcant enhancement of the experimentally measured order parameter in athin cell near TAN . The reason for this enhancement is notclear. Further theoretical work is required to understandthese experimental results.
We would like to thank Prof. B.K. Sadashiva for giving us twocompounds (CN and BAN).

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