Min Sang Park,1Beom-Jin Yoon,1Jung Ok Park,1,2Veena Prasad,4Satyendra Kumar,5and Mohan Srinivasarao1,2,3
1School of Polymer, Textile and Fiber Engineering, Georgia Institute of Technology, Atlanta, Georgia, USA
2Center for Advanced Research on Optical Microscopy (CAROM), Georgia Institute of Technology, Atlanta, Georgia, USA
3School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia, USA
4Center for Liquid Crystal Research, Jalahalli, Bangalore 560013, India
5Department of Physics, Kent State University, Kent, Ohio, USA
(Received 14 January 2009; published 8 July 2010)
Polarized Raman spectroscopy was used to investigate the development of orientational order and the
degree of phase biaxiality in a bent-core mesogenic system. The values of the uniaxial order parameters
hP200i and hP400i, and biaxial order parameters hP220i, hP420i, and hP440i, and their evolution with
temperature were determined. The temperature dependence of almost all order parameters reveals a
second order transition from the uniaxial to biaxial nematic phase with hP220i increasing to ?0:22 before a
first order transition to the smectic-C phase, upon cooling.
DOI: 10.1103/PhysRevLett.105.027801 PACS numbers: 61.30.?v
The biaxial nematic, Nb, phase has been one of the most
elusive liquid crystalline phases since its prediction by
Freiser [1,2] in 1970 based on his generalization of the
Maier-Saupe model  of the uniaxial nematic, Nu, phase.
This prediction was followed by a number of important
microscopic [4–6] and phenomenological [7–9] models,
and by simulations  which yield interesting phase dia-
grams with one biaxial nematic, Nb, phase and two (pro-
late and oblate) uniaxial nematic, Nu, phases. In the micro-
scopic models, the biaxiality was either due to molecular
shape biaxiality [4,5] or a consequence of mixing rodlike
and platelike objects . Quite remarkably, they all lead to
phase diagrams with similar topology having two Nu
phases, one (Nuþ) with positive S ½¼ 0 ! þ1? and the
other (Nu?) having negative S ½¼ 0 ! ?1=2?. The tran-
sitions between the isotropic (I) and the Nu?=Nuþ(or,
Nu?) phases are predicted to be first order while the
transitions from the Nu?to the Nbphase form lines of
second order transitions.
Encouraged by the success of the work of Saupe and co-
workers in discovering the Nbphase in a lyotropic liquid
crystalline system , several attempts were made to
verify its existence in thermotropic systems [12–14] which
were proved unsuccessful . However, recent discov-
eries of the Nbphase using x-ray diffraction  and NMR
 have rekindled the scientific interest in nematic biax-
iality. These discoveries have inspired theorists [18,19] as
well as experimentalists [20–23] to inquire into the physics
of this hitherto elusive but scientifically interesting phase.
At the transition to the Nbphase, the growth of (biaxial)
orientational order (OO) in a direction perpendicular to n
is described by a two-component order parameter. Here n,
known as the nematic director, describes the average direc-
tion of alignment of the molecular axes. This transition
thus falls in the two-dimensional–XY universality class
. An essential step toward understanding their rich
phenomenology is to quantify the degrees of OO and its
evolution as a function of temperature, through the various
In this Letter, we report the results of our Raman scat-
tering study of the evolution of orientational order in a
bent-core compound A131 , which possesses both the
Nuand Nbphases, thereby permitting measurements of
thermal evolution of distinct orientation modes associated
with the uniaxial and biaxial OO parameters. The valuable
information presented here will help explain phase transi-
tions and dynamics of thermotropic biaxial nematic liquid
Unlike the orientation of the conventional Nuphasewith
perfect symmetry cancellation, in the Nbphase, there is
breaking of the continuous rotational symmetry about the
nematic director, n. Thus, a more elaborate form of the
orientation distribution function (ODF) must be used to
express the degree of orientation in the Nbphase. A set of
orthogonal functions of Euler angle (?, ?, ?) can be used
to construct a simple ODF expansion. This will enable us
to transform the orientation dependence of physical quan-
tities in the macroscopic (or laboratory) frame XYZ to their
orientation dependence in the molecular frame xyz. The
ODF is generally expressed as 
Depending on the values of LðL 2 NÞ, the ODF is made
up of a set of ð2L þ 1Þ2Wigner matrices . However, if
there are certain symmetries in both sample and molecular
systems, a number of components are left out of consid-
eration, so that ODF can be simplified . Figure 1
depicts the experimental geometry where the orientation
of a particular unit with respect to a coordinate system may
be described by the Euler angles, ?, ?, ?. For simplicity, it
is usually assumed that the Raman probe is axially sym-
2L þ 1
mnð?;?;?Þ are the Wigner rotation matrices and
mni indicates the statistical average of DL
PRL 105, 027801 (2010)
9 JULY 2010
? 2010 The American Physical Society
metric. Thus, one may assume a random rotation about the
Euler angle ? and, as a result, the order parameters hDL
only have values other than zero when n ¼ 0. In addition,
mirror symmetry of a sample fixes m to be even. Hence, we
can expand this matrix in terms of the series of Legendre
polynomials, hPLm0i. Since the study of polarized Raman
spectroscopy allows us to obtain quantitative information
about molecular orientation up to fourth orientation order
parameters, ODF in this study can be expanded in terms of
5 coefficients, hP200i, hP400i, hP220i, hP420i, and hP440i.
Note that the two parameters, hP200i and hP400i, commonly
recognized as the uniaxial OO parameters, inform about
the average orientation of molecules distributed along the z
direction with dependence only on the angle ?. In contrast,
hP220i, hP420i, and hP440i characterize symmetry with a
of these biaxial OO parameters are indicative of the exis-
tence of biaxial symmetry. To simplify, the ODF can be
expressed  by fð?;?Þ as
2hP200ið3cos2? ? 1Þ þ 15hP220ið1 ? cos2?Þcosð2?Þ
8hP400ið3 ? 30cos2? þ 30cos4?Þ
2hP420ið?1 þ 8cos2? ? 7cos4?þÞcosð2?Þ
4hP440ið1 ? 2cos2? þ cos4?þÞcosð4?Þ
Based on the geometry in Fig. 1, the intensity of polar-
ized Raman scattering is a function of the electric field and
the ODF integrated over all possible orientations, and can
be expressed as
where k is a proportionality constant that depends on
factors such as incident light intensity, instrumental trans-
mission, light collection efficiency, etc., ? is the angle
between the incident polarization direction and the sym-
metry axis of the Raman probe, and Eijð?;?;?Þ is the
electric field vector with rotational degrees of freedom
about ?, ?, ?. This enables us to obtain coefficients of
ODF from the measurements of Iij. The notation Iijde-
notes the scattering intensity analyzed in the i direction (in
our geometry, i ? Z, Y) with incident polarization in the j
direction (in our geometry, j ? Z, Y). Expanding Eq. (3)
with our suitable geometry gives the reduced expression of
Raman intensities, Ik¼ IZZð?Þ and I?¼ IYZð?Þ, accom-
plished by resolving polarized light intensity into the par-
allel andperpendicular direction
Furthermore, a series of spectra were recorded over the
entire range of 0 to 2? at 10?interval by rotating the
samples relative to the fixed direction of polarization of
the incident light. It is because a range of orientations of
the polarizer and analyzer with respect to the symmetry
axes of the LC provides more detailed information than
only parallel and perpendicular polarizations of laser ex-
citation [28,29]. Consequently, we can derive expressions
for the depolarization ratio, Rð?Þ ¼ I?=Ill, which is of
importance in determining the Raman tensor, and write
where r ¼ ?0yy=?0zzis the Raman tensor ratio.
In this study, 15 ?m thick LC cells with homogeneous
director alignment were used. Homogeneous (uniaxial ne-
matic) director alignment was ascertained under crossed
polarizers using a polarizing optical microscope. The po-
larized Raman spectra were obtained using a 10? dry
objective and a 785 nm laser light source (Kaiser Optic
Systems). We used a compound A131, for which the most
intense peak is the one at 1141 cm?1attributed to the
asymmetric stretching of C-O-C . It is noteworthy
that strong Raman effects correspond to internal vibrations
of the mesogenic units, whose dipole moments are likely to
FIG. 1 (color online).
molecular axes xyz in a macroscopic (laboratory) system of axes
XYZ. The polarization of the Raman scattered light is resolved
by the different direction of the analyzer.
Euler angles ?, ?, ? are defined with
PRL 105, 027801 (2010)
9 JULY 2010
be oriented along their long molecular axes (in the z
direction in Fig. 1). It allows us to assume collinearity of
the principal axes of the Raman tensor with the molecular
axes.Figure2(a) is an example of the experimental data for
IZZð?Þ, IYZð?Þ, and Rð?Þ of A131. IZZð?Þ and IYZð?Þ are
normalized to unity at the peak of IZZ. The pattern of
twofold and fourfold rotational symmetry of IZZð?Þ and
IYZð?Þ is conformable to trigonometric periodicities of ex-
pansion in Eq. (2) for IZZð?Þ and IYZð?Þ, respectively .
These polarized Raman spectra over a wide range of
incident angles were fit to Eq. (4) using the polynomial fit
function in MATHEMATICA. This resulted in six fitting pa-
rameters, namely, uniaxial order parameters, hP200i and
hP400i, biaxial order parameters, hP220i, hP420i, and
hP440i, and r. It is necessary to note that there may be as
many as 6 coefficients from Eq. (4) for a fit, if the evalu-
ation is done without some constraints. Using Eq. (4), the
calculated depolarization ratio, Rð?Þ, is shown in Fig. 2(b),
with the theoretical fit.
We determined temperature dependence of hP200i,
hP400i, hP220i, hP420i, and hP440i by making measurements
of IZZð?Þ and IYZð?Þ and calculating Rð?Þ through the
various phases which A131 exhibits, i.e., I-Nu, Nu-Nb,
and Nb-smectic-C transitions upon cooling. We assumed
that the biaxial terms are negligible in the estimation of the
uniaxial order parameters and Raman tensor ratio, r ¼
?0yy=?0zz. Based on these approximations, biaxial order
terms play a significant role for the fits. Figure 3 is a plot
of various order parameters as a function of temperature,
T ? TNIð?CÞ. Among the biaxial OO parameters, the high
value of hP220i provides quantitative evidence that one of
the molecular short axes is preferentially aligned perpen-
dicular to the ZY plane. The fourth rank order parameters,
hP420i and hP440i, have a more complicated physical inter-
pretation. The observed negative values, in particular, for
hP440i, indicate that the direction of the chain axis is
projected onto the XY plane at approximately ?45?to
the X or Y axis, rather than collinear to them, since the
value of (1 ? 2cos2? þ cos4?) is positive for all ?. The
absolute limits for the 3 biaxial order parameters, hP220i,
hP420i, and hP440i, can be easily calculated, as was done by
Jarvis et al.,  and can be written as
increase until the discontinuous transition to the smectic-C
phase occurs. Judging from the changes in the slope of
these parameters, we conclude that the Nu! Nbphase
transition is of second order. Based on the discontinuous
nature of the variation of hP200i and hP400i, we conclude
Nb! smectic C to be of first order.
An interesting observation is that the value of hP200i
(?0:275) at T ? TNI¼ ?0:1?C is smaller than the ex-
pected value of 0.35–0.4 for conventional uniaxial nem-
atics. Theoretically, it is predicted that the biaxial order
parameter fluctuations may drive the I-Nutransition to be
second order . It is likely that the biaxial order parame-
ter fluctuations in the uniaxial nematic phase are respon-
sible for this behavior. Hence, we compared the thermal
fluctuations of A131 with those of E7 close to TNIand
A131. A detailed analysis of the fluctuation at or close to
It is worth noting that a previous report quantifying biaxial
order parameters was not in agreement with the limit of
analytical values ofEq. (5) .Bycontrast, ourresults for
the biaxial terms, displayed in Fig. 3, lie well within the
absolute limits of the values throughout the temperature
To determine the changes in the order parameters at the
phase transition from the Nu! Nbphase, we performed
analysis at small temperature intervals. From the inset of
Fig. 3, it is clear that an abrupt change in order parameter
does not occur at the (Nu! Nb) transition temperature,
ruling out a first order transition. However, there is a
qualitative change in the temperature dependence of these
parameters at the transition. It is clearly seen in the case of
uniaxial order parameters in Fig. 4. hP200i and hP400i
increase in the uniaxial phase with decreasing temperature
and leveloff, while inthe biaxial phaseboth theparameters
intensity profiles for A131 at 166:5?C
and (b) the corresponding depolarization
ratio profile; measured (r) and theoreti-
cal fit (solid line), using hP200i ¼ 0:381,
hP400i ¼ 0:207, and r ¼ ?0:1983.
(a) The Raman scattered light
PRL 105, 027801 (2010)
PHYSICAL REVIEW LETTERS
9 JULY 2010
the phase transition will be published elsewhere . This Download full-text
is consistent with the results of the study of the order
parameter and director fluctuations by Sprunt et al. .
In summary, the quantitative and qualitative analyses of
OO in the nematic phase strongly support the existence of
the biaxial nematic phase in A131 from ?30 to 58?C
below TNI. We have demonstrated the power of the polar-
ized Raman scattering technique in reliably measuring
nematic order parameters and revealing their thermal evo-
lution across the second order uniaxial to biaxial nematic
phase transition, and at the first order biaxial nematic to
smectic-C phase transition. The technique and the results
presented here have a significant impact on our ability to
quantitatively understand OO in nematic fluids and test
theoretical prediction of shear flow aligning behavior and
calculations of viscosities and elastic constants [36,37], the
thermodynamic properties, and the rheological behavior of
This work was supported, in part, by the Office of Basic
Energy Sciences, Department of Energy, Grant No. DE-
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of Organic Molecules
hP220i (d), hP420i (?), and hP440i (m), obtained by fitting the
depolarization ratio profile, Rð?Þ.
Temperature dependence of biaxial order parameters:
hP200i (?) and hP400i (m), obtained by fitting the depolarization
ratio profile, Rð?Þ.
Temperature dependence of uniaxial order parameters:
PRL 105, 027801 (2010)
PHYSICAL REVIEW LETTERS
9 JULY 2010