VOLUME 75, NUMBER 9 PHYSICAL REVIEW LETTERS 28 AUGUST 1995
Phase Diagram and Orientational Order in a Biaxial Lattice Model: A Monte Carlo Study
F. Biscarini,
1,
* C. Chiccoli,
2
P. Pasini,
2
F. Semeria,
2
and C. Zannoni
1
1
Dipartimento di Chimica Fisica ed Inorganica, Università di Bologna, Viale Risorgimento 4, 40136 Bologna, Italy
2
Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy
(Received 27 March 1995)
We have determined the phase diagram for a lattice system of biaxial particles interacting with a
second rank anisotropic potential using Monte Carlo simulations for a number of values of the molecular
biaxiality. We find increasing differences from mean field theory as the biaxiality increases. We have
also calculated for the first time the full set of second rank biaxial and uniaxial order parameters and
their temperature dependence, and on this basis we comment on the difficulties of measuring phase
biaxiality by NMR.
PACS numbers: 61.30.Gd, 61.30.Cz, 64.70.Md
Liquid crystal phases formed by biaxial particles have
been studied using a number of theoretical methods; e.g.,
without trying to be exhaustive, mean field theory (MFT)
[1–4], counting methods [5], Landau–de Gennes theory
[6,7], bifurcation analysis [8], density functional theory
[9], etc. All the theories mentioned above predict that the
system will exhibit four phases as the molecular biaxial-
ity varies: a positive (N
1
) and a negative (N
2
) uniaxial
phase, respectively, formed by prolate or oblate particles,
a biaxial (N
B
), and an isotropic (I ) phase. The nematic-
isotropic phase transition is expected to be first order and
to weaken as the biaxiality increases until it becomes con-
tinuous at the point (Landau bicritical point) of maximum
molecular biaxiality. At this point the system should go
directly from a biaxial to an isotropic phase. The biaxial-
uniaxial transition is expected to be second order. The
possibility of a biaxial-nematic mesophase has been con-
firmed by some computer simulations of a lattice system
of biaxial particles [10,11], and of a fluid system of biax-
ial spherocylinders [12]. On the experimental side, there
is an increasing number of biaxial lyotropic [13] and poly-
meric [14] phases while the observation of thermotropic,
claimed by a number of authors [15], typically on the ba-
sis of optical observations, is still questioned [16].
Given this extensive activity, it is surprising that a de-
tailed computer experimental study of the phase diagram
and, even more, of the full set of order parameters and
their detailed temperature dependence within the biaxial
and uniaxial phases are not available as yet. This informa-
tion is crucial for the study of static but also, indirectly, of
dynamic properties [17] and in general to validate molecu-
lar theories. In this Letter we make an attack on this prob-
lem and we propose a general prescription for calculating
order parameters from simulations in systems with sym-
metry lower than uniaxial. We base our calculations on
the simplest second rank attractive pair potential [3,10],
Uv
ij
2e
ij
P
2
cos b
ij
1 2lR
2
02
v
ij
1 R
2
20
v
ij
1 4l
2
R
2
22
v
ij
, (1)
with the biaxial molecules, or “spins,” fixed at the sites of
a three-dimensional cubic lattice. The coupling parameter
e
ij
is taken to be a positive constant e when particles i
and j are nearest neighbors and zero otherwise. v
ij
is the
relative orientation of the molecular pair, given by three
Euler angles a, b, and g [18]. P
2
is a second Legendre
polynomial and R
L
mn
are combinations of Wigner functions
D
L
mn
[18] symmetry-adapted for the D
2h
group of the two
particles. Their explicit expressions, for the even terms,
are
R
L
mn
1
2
ReD
L
mn
1 D
L
m2n
. (2)
The biaxiality parameter l accounts for the deviation from
cylindrical molecular symmetry: when l is zero, the po-
tential Eq. (1) reduces to the well-known Lebwohl-Lasher
P
2
potential [19], while for l different from zero the par-
ticles tend to align not only their major axis, but also their
faces. The value l 1
p
6 marks the boundary between
a system of prolate l, 1
p
6 and oblate molecules
l. 1
p
6 and is analogous to the self-dual case de-
scribed by Straley [2]. Thermodynamic results for a pro-
late particle at biaxiality l and T
kT e can be mapped
onto l
0
, T
0
3 2l
p
6
p
6 1 6l, 24T
6l1
p
6
2
for the dual oblate particle.
We have performed Monte Carlo (MC) simulations
[19] for 16 values of the biaxiality parameter l, both on
an 8 3 8 3 8 and a 10 3 10 3 10 lattice. For each l
value, about 40 temperatures have been investigated. For
l 0.3 additional larger size simulations (40 3 40 3
40) have been performed at selected temperatures near the
biaxial-nematic and nematic-isotropic transitions to rule
out system size artifacts. The Metropolis MC method
with periodic boundary conditions and controlled angu-
lar displacement [19,20] has been employed for lattice
updates. We have typically used 30 000 lattice sweeps
(cycles) for equilibration and 20 000 for production. The
40 3 40 3 40 system has required 40 and 30 kcycles.
In Fig. 1 we show the heat capacity C
V
obtained by dif-
ferentiating the energy against temperature as described
in [20] and plotted against temperature for l 0.2, 0.3,
and 0.408 25. Starting with the lower values of l, we
see that C
V
exhibits a small peak at low temperature
0031-9007 95 75(9) 1803(4)$06.00 © 1995 The American Physical Society 1803