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Critical behavior of optical birefringence at the nematicsmectic
A phase
transition in a binary liquid crystal system
Sudipta Kumar Sarkar, Malay Kumar Das
Department of Physics, North Bengal University, Siliguri 734
013, West Bengal, India
a b s t r a c ta r t i c l e i n f o
Article history:
Received 1 July 2014
Received in revised form 6 September 2014Accepted 23 September
2014
Available online 26 September 2014
Keywords:
Phase transition
Binary liquid crystal mixture
Optical birefringence
Critical behavior
Tricritical point (TCP)
We report the measurement of optical birefringence (n) of a
binary liquid crystal system consisting of
decyloxycyanobiphenyl (10OCB) and heptylcyanobiphenyl (7CB) by
means of a high resolution temperature
scanning technique. The birefringence data are found to be
rather successful in studying the nature of the tran-
sition and the critical behavior at the nematicsmectic A (NSmA)
phase transition in these mixtures. In the
vicinity of NSmA phase transition the optical birefringence (n)
data exhibit strong pretransitional
behavior which gets enhanced as the nematicregion diminishes.The
critical exponent (), when plottedagainst
the McMillan ratio (TNA/TNI), a uniform crossover from second
order to rst order behavior have been observed
with a tricriticalpoint (TCP) atx10OCB = 0.587. The3D-XYlimit
for the NSmA phase transition would hypothet-
ically reach at the McMillan ratio equal to 0.937 for this
binary system.
2014 Published by Elsevier B.V.
1. Introduction
The liquid crystalor mesophase is a distinct state of matter
observedbetween the isotropic liquid and the crystalline solid.
These compounds
exhibit different types of phase transitions and have been found
good
model systems for testing the general concept of phase
transitions and
critical phenomena. Two of the more common mesophases are the
ori-
entationally ordered nematic (N) and the layered smectic A
(SmA)
phases [1]. The nature of the nematicsmectic A (NSmA) phase
transi-
tion in liquid crystals has been a subject of extensive
theoretical and ex-
perimental studies due to its several interesting features.
Duringthe last
four decades, numerous efforts have been made to determine the
uni-
versalityclass of the NSmA phase transition,yet it remains a
majorun-
solved problem in the eld of soft condensed matter systems. From
the
mean eld approach Kobayashi [2] and McMillan [3] suggested that
the
NSmA phase transition can either be ofrst order or second order
de-
pending on the nematic range. According to McMillan [3], the
nature of
the NSmA phase transition is governed by the parameter
TNA/TNI,
whereTNAandTNIare the nematicsmectic A (NSmA) and nematic
isotropic (NI) phase transition temperatures respectively. When
the
McMillan ratio (TNA/TNI) exceeds the limiting value 0.87, the
nematic
smectic A (NSmA) phase transition becomes rst order otherwise
it
will be of second order. The crossover from second order to rst
order
nature takes place at a tricritical point (TCP). On the basis of
coupling
between the nematicand smectic order parameters and Landaufree
en-
ergy expansion, de Gennes[1,4]predicted the NSmA phase
transition
to be in the three dimensionalXY(3D-XY) universality class in
analogywith the normal-superconducting phase transition. However,
Halperin,
Lubensky and Ma (HLM) [5] envisaged that the NSmA phase
transition
is always rst order in nature owing to the coupling between the
ne-
matic director uctuations and the smectic A order parameter. So
the
idea of tricritical point (TCP) does not arise in this case.
Alben[6]sug-
gested the existence of tricritical point (TCP) in liquid
crystal binary
mixtures. In recent years, several works have been performed on
ne-
maticsmectic A (NSmA) phase transition and its critical
behavior
but different experimental results reveal that the crossover
behavior
from second order to rst order nature of the NSmA phase
transition
is not universal. Even if, the theoretical limiting value of the
McMillan
ratio is 0.87 for the tricritical point (TCP), experimentally it
is found to
lie between 0.942 and 0.994[710].
In this work, we reportthe phase diagram of a binarysystem
consisting
of two terminal polar liquid crystal compounds
decyloxycyanobiphenyl
(10OCB) and heptylcyanobiphenyl (7CB). From the high
resolution
temperature scanning technique, the optical birefringence (n)
for
seven different concentrations of this binary system has been
mea-
sured. In order to shed some light on the critical nature of the
NSmA
phase transition, the high resolution birefringence data have
been
used. The primary interest is to extract the critical exponent
()
which determines the order character of the nematicsmectic A
(N
SmA) phase transition and also allow us to nd out the 3D-XYas
well
as the tricritical limit of the NSmA phase transition in this
binary
system.
Journal of Molecular Liquids 199 (2014) 415418
Corresponding author.
E-mail address:[email protected](M.K. Das).
http://dx.doi.org/10.1016/j.molliq.2014.09.040
0167-7322/ 2014 Published by Elsevier B.V.
Contents lists available atScienceDirect
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a temperature range of 2 K is shown. As shown in Fig. 3, on
both
sides ofTNA, there is a signicant pretransitional change innon
ap-
proaching the phase transition. This type of pretransitional
behavior
near TNAhas been clearly observed for the other mixtures also.
For
all the mixtures studied, the change in birefringence(n),
between
the nematic and smectic A phases near the transition
temperature
(TNA) varies from 0.0008 to 0.0184. A plot of(n) against the
mole
fraction of 10OCB is shown in the inset ofFig. 3which evidently
indi-
cates that the effect of NSmA coupling increases as the nematic
range
is reduced.
3.3. Critical behavior at the nematic
smectic A (N
SmA) phase transition
The high resolution birefringence (n) data have beenusedto
inves-
tigate the critical behavior at theNSmA phase transition. At the
transi-
tion temperatureTNA,nvalue do not exhibit a sharp discontinuity,
so
to identify the exact transition temperature we have used
theminimum
value of the temperature derivative of birefringence (n). In
order to
extract the critical exponent () from the birefringence data, a
new pa-
rameter have been dened with the following form[13,14]:
Q T n T n TNA
TTNA1
where n(TNA) is the birefringence value at the transition
temperature
TNAas obtained by differentiating n. An overview of the
temperature
dependence ofQ(T) as determined from Eq. (1), in the immediate
vicin-
ity ofTNAfor different mixtures are shown inFig. 4. It should be
noted
that the peak height ofQ(T) near the transition temperature
gradually
increases with an enhancement in mole fraction of 10OCB. This is
obvi-
ously due to an increase in(n) near the NSmA phase
transitiontem-
perature, which again is induced by an enhancement in strength
of the
coupling between the nematic and smectic order parameters. To
deter-
mine the critical exponent (), a systematic analysis of theQ(T)
data
have been performed by using a simple power law expression
with
the following form[13]:
Q T A tj j
B 2
whereA+and Aare the critical amplitudes and B+and Bare the
background terms above and below the nematicsmectic A phase
tran-sition temperature (TNA), is the critical exponent similar to
the specic
heat critical exponent[13,14]and t= |(TTNA)/(TNA)| is the
reduced
temperature. For all the mixtures under study, the Q(T) values
are
well portrayed by Eq.(2)as indicated by the solid lines in Fig.
4with
thet parameters listed inTable 1. It should be mentioned that
some
data points very close to the transition temperature (TNA) have
been ex-
cluded from thetting in order to obtain thebestt value of the
critical
exponent ().
The qualities of thets have been tested by calculating the 2
value
on either sides of the transition temperature. The2 is
determined by
323 324
0.155
0.160
0.165
T/ K
0.2 0.4 0.60.00
0.01
0.02
n)
(
x10OCB
x10OCB= 0.387
n
(n)
Fig. 3. Birefringence (n) as a functionof temperaturein
thevicinity ofnematicsmecticA
phase transition (TNA) temperature forx10OCB= 0.387. The solid
lines are lineart to the
birefringence (n) data. Vertical arrow indicates the change in
the birefringence value at
the transition temperatureTNA. In the inset the change in
birefringence (n) at TNAis
plotted against the mole fraction of 10OCB.
300 310 320 330 3400.00
0.02
0.04
0.06
0.08
T / K
0.701
0.587
0.502
0.3870.300
0.255
Q(T)
0.205
Fig. 4.Temperature variation of the parameter Q(T) near the NSmA
phase transition at
different mole fractions of 10OCB. Solid lines are t to
Eq.(2)for different mixtures near
the NSmA phase transition. The tting parameters are given
inTable 1.
Table 1
The besttted parameter values forQ(T) near NSmA phase transition
obtained from t to Eq.(2)and the corresponding 2 associated with
the t.
x10OCB or+ or+ 2 No. of data points
0.205 Tb TNA 0.00025 0.00002 0.263 0.018 0.00040 0.00001 1.53
107
T NTNA 0.00034 0.00001 0.266 0.012 0.00030 0.00002 1.47 101
0.255 T bTNA 0.00043 0.00004 0.330 0.022 0.00095 0.00006 1.17
102
T NTNA 0.0005 0.00003 0.331 0.006 0.00065 0.00003 1.26 105
0.300 T bTNA 0.00023 0.00001 0.371 0.028 0.00157 0.00004 1.21
137
T NTNA 0.00026 0.00003 0.372 0.013 0.00162 0.00008 1.15 127
0.387 T bTNA 0.00043 0.00001 0.420 0.004 0.00012 0.00001 1.16
201
T NTNA 0.00023 0.00002 0.422 0.008 0.00269 0.00007 1.22 120
0.502 T bTNA 0.00049 0.00004 0.467 0.045 0.00031 0.00001 1.20
160
T NTNA 0.00036 0.00003 0.477 0.008 0.00442 0.00013 1.12 80
0.587 T bTNA 0.00062 0.00004 0.498 0.014 0.00012 0.00001 1.17
207
T NTNA 0.00039 0.00004 0.501 0.016 0.0080 0.0007 1.16 61
0.701 T bTNA 0.00144 0.00002 0.511 0.023 0.0029 0.00008 1.19
214
T NTNA 0.00136 0.00007 0.511a
0.00146 0.00008 1.46 40
a
was keptxed at 0.511.
417S.K. Sarkar, M.K. Das / Journal of Molecular Liquids 199
(2014) 415418
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the ratio of variance of the t (s2) and the variance of the
experimental
data (2), and can be written as follows[15]:
2
s2
2
1
Np
2
X
i
yifi 2
3
where Nis thetotal numberof data points,p is thenumber of
adjustable
parameter andfiis theitht value corresponding to the
measurement
yi. The2 value equal to unity yields an ideal t but values lying
be-
tween 1 and 1.5 correspond to good ts. For our present ts,
the2
value lies between 1.12 and 1.53 which indicates a good t to
theQ(T)
data. The values of the ratio of critical amplitudes (A
/A+) is in the
range 0.73
1.86, while the critical exponentvaries from 0.263 to0.511
within the error limit. The critical exponent was adjusted
freely
in thetting procedure on either sides of the NSmA phase
transition
temperature except forx10OCB= 0.701, for which the value was
kept xed to 0.511 for TNTNAin order to obtain best t to the
data
points. It is seen that the ts yield nearly equal value ofon
both
sides of the transition temperatureTNA. An average value of the
critical
exponent when plotted against the McMillan ratio (TNA/TNI) shows
a
denite pattern as shown inFig. 5. For the mixturesx10OCB= 0.205
to
0.502, values are less than that for the tricritical limit (0.5)
indicating
a second order nature of the nematicsmectic A (NSmA) phase
transi-
tion. On the other hand, for the mixture x10OCB= 0.587 a very
careful
analysis yields 0.5 withinthe error limit,whichindeed
thetricritical
limit of NSmA phase transition. An inspection ofFig. 5reveals
that as
the McMillan ratio increases or the nematic range of the
mixtures de-creases the critical exponent increases due to the
strong coupling be-
tween the nematic and smectic A order parameters and the
narrowing
of the nematic range results in a crossover behavior at a
tricritical
point (TCP) from second order to rst order transition. In this
work,
the nematicsmectic A (NSmA) phase transition approaches the
tricritical point (TCP) almost linearly. Therefore, the
tricritical point
(TCP) for this binary system is found to observe atx10OCB= 0.587
for
the McMillan ratio equal to 0.993. An extrapolation of the
polynomial
t to thevalues (shown by the dashed line in Fig. 5) towards
the
lower end of McMillan ratio yields a TNA/TNI of about 0.937
correspond-
ing to the 3D-XYlimit (= 0.007) of the NSmA phase
transition.
4. Summary and conclusions
In this paper a detailed investigation of optical birefringence
in the
nematic and smectic A phases of a binary system were performed
by
means of a high resolution temperature scanning technique.
Particular
emphasis was given to study the pretransitional behavior of
birefrin-
gence in the vicinity of NSmA phase transition. Near the
NSmA
phase transition all the mixtures under study shows a strong
pretransitional effect which gets enhanced as the nematic region
is de-
creased. Moreover, the high resolution birefringence data have
been
used to assess the order of the NSmA phase transition. The
nature of
nematicsmectic A (NSmA) phase transition remains continuous
up
tox10OCB= 0.502, but it becomesrst order atx10OCB= 0.587. The
en-
hanced coupling between the nematic and smectic order
parameters
change the nature of the N
SmA phase transition from second orderto rst order with a
tricritical point (TCP) atx10OCB= 0.587 for the Mc-
Millan ratio 0.993. The 3D-XYlimit of the nematicsmectic A
(NSmA)
phase transition for thisbinary system would hypothetically be
reached
at the McMillan ratio equal to 0.937. Therefore, it can be
concluded that
like adiabatic scanning calorimetry and volume thermal expansion
coef-
cient our high resolution optical birefringence data can also be
suc-
cessfully applied to study the order character of the NSmA
phase
transition in liquid crystals.
Acknowledgement
We gratefully acknowledge nancial support from Department of
Science and Technology, New Delhi (Project No:
SB/EMEQ-290/2013).
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0.90 0.92 0.94 0.96 0.98 1.00-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
3D-XYUniversality
1st
order
Critica
lexponent()
McMillan ratio (TNA/TNI)
TCP
Fig. 5.The critical exponent is plotted against the McMillan
ratio (TNA/TNI). () repre-
sents the values averaged over the two values obtained from
tting above and below
the transition temperatureTNA. The vertical bars indicate the
error associated with the
values. Dashed line represents a polynomialt to the values.
Upper and lower head ar-
rows denote the tricritical point (TCP) and the 3D-XYlimit of
NSmA phase transition
respectively.
418 S.K. Sarkar, M.K. Das / Journal of Molecular Liquids 199
(2014) 415418
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