Dipartimento di Chimica Fisica ed Inorganica, Università
di
Bologna,
Viale Risorgimento 4, 40136 Bologna, Italy
Istituto Nazionale di Fisica Nucleare, Sezione di Bologna,
Via Irnerio 46, 40126 Bologna, Italy
F. Biscarini
, C. Chiccoli, P. Pasini,
F. Semeria, C. Zannoni
Tue Oct 31 10:19:45 MET 1995
PACS numbers: 61.30.Cz, 61.30.Gd, 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) [][][][],
counting methods [],
Landau - deGennes theory [][],
bifurcation analysis [], density functional theory [] etc.
All the theories mentioned above predict that the system will exhibit
four phases as the molecular biaxiality varies: a positive (N
) and
a negative (N
) uniaxial phase, respectively formed by prolate or oblate
particles, a
biaxial (N
) 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
continuous 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 confirmed by some computer simulations of a
lattice system of biaxial particles [][],
and of a fluid system of biaxial
spherocylinders [].
On the experimental side, there is an increasing number of biaxial
lyotropic [] and
polymeric [] phases while the observation of thermotropic, claimed
by a number of authors [], typically on the basis of optical
observations, is still questioned
[].
Given this extensive activity, it is surprising that a detailed 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 information is crucial for the study of static but also, indirectly, of dynamic properties [] and in general to validate molecular theories. In this Letter we make an attack on this problem and we propose a general prescription for calculating order parameters from simulations in systems with symmetry lower than uniaxial. We base our calculations on the simplest second rank attractive pair potential [][]:
with the biaxial molecules fixed at the sites of a three dimensional
cubic lattice.
The coupling parameter
, is taken to be a positive constant, 
when
particles
and 
are nearest neighbors and zero otherwise.
is the relative orientation
of the molecular pair, given by three Euler angles 
[]. 
is a second Legendre polynomial and
are combinations of Wigner functions 
[]
symmetry - adapted for the
D
group of the two particles. Their explicit expressions, for the even
terms, are:
The biaxiality parameter 
accounts for the deviation from
cylindrical molecular symmetry: when is zero, the potential
eq. 1 reduces to the well-known Lebwohl - Lasher
potential [], while
for different from zero the particles tend to
align not only their major axis, but also their faces.
The value 
marks the boundary between a system of prolate
(
) and oblate
molecules
(
) and is analogous to the self - dual case
described by Straley
[].
Thermodynamic results at biaxiality and 
can be mapped onto
We have performed MC
simulations [] for 16 values of the biaxiality parameter
,
both on a 
and a 
lattice.
For each value, about 40 temperatures have been
investigated. For 
additional larger size simulations
(
) 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 angular displacement [][] has
been
employed for lattice updates. We have typically used 30,000 lattice
sweeps (cycles) for equilibration and 20,000 for production.
The system has required 40 and 30 kcycles.
In fig. 1 we show the heat capacity 
obtained by differentiating
the energy against temperature as described in [] and plotted
against temperature for 
.
Starting with the lower values of , we see that
exhibits a small peak at low temperature and a sharper one at
higher temperature corresponding to
biaxial - uniaxial and uniaxial - isotropic phase transitions
respectively.
The first transition shifts at higher temperature as increases,
until
the two peaks coalesce into a single broad hump near the self-dual case
(fig.1-c). The MC phase diagram shown in fig. 2 has been obtained from
the peaks in
the heat capacity.
We see good agreement with the single result at 
of
ref. [] that was obtained with 512 particles and a
fcc lattice, after scaling to 6 nearest neighbors.
The comparison between MC and MFT results obtained here following the procedure
in Ref. [21] shows a rough qualitative
agreement between the two methods. However, MFT predicts a rather
pronounced increase with , while
the simulations suggest an essentially constant transition
temperature 
.
Above the tricritical point, the transition temperature between 
and
the isotropic phase increases monotonically.
We have used these points, where the variation in transition temperature
is much greater than the error to generate the dual points below
, marked as (+) in fig.2. We see that they confirm
the small variation in .
The MFT predictions worsen from a discrepancy of 17% to about 30% as
the biaxiality increases and the transition becomes more second order.
We now turn to the determination of
the
order parameters and their
temperature variation.
In the principal axis system of a uniformly aligned biaxial phase
the complete set of rank 2 order
parameters consists of the averages 
,
,
non vanishing also in the uniaxial phase, and
,
different from zero only
in the biaxial phase.
In the isotropic phase, all the order parameters are zero. In the absence
of external fields, the location of the principal system is
unknown and
can fluctuate during the simulation and thus has to be determined
using rotational invariants for each
configuration.
For uniaxial systems this determination is usually realized
by diagonalization of a suitable ordering matrix whose largest eigenvalue
is the instantaneous order parameter and then averaging
results for the various configurations.
The calculation of biaxial order parameters is more complicated because
of the need of a suitable prescription for assigning the two
lower eigenvalues
in a way that does not unduly enhance or cancel the phase biaxiality.
We have
attacked the problem using an approach similar to that of
an actual experiment.
We start by
considering the general expression for the eigenvalues of a tensor
property 
of a biaxial molecule, in the (principal) frame of a
biaxial phase in terms of order
parameters []:
where we use lower case to indicate the eigenvalues and 
are irreducible spherical components
of 
of rank 
. The order parameters are
,
,
,
.
Experimentally one would try to
select a sufficient number of
molecular properties 
and
measure their
average values 
.
Then, through a diagonalization of these average tensors
one could determine the order
parameters. The
requirement that these order parameters are the same for different
observables helps in assigning
the correct principal laboratory frame.
As an illustration the explicit expressions for the eigenvalues of a
tensor 
are
The non-trivial problem is finding a consistent way of assigning the
three
eigenvalues 
of the matrix 
to the X,Y,Z axis.
In the uniaxial phase or anyway when
,
and taking 
, we have
and
letting 
is sufficient to assign the axes except for an irrelevant exchange of
and 
.
However, in the biaxial phase it may well happen that
, e.g. when
, and, even
if we assume that and
are
always positive, there is not a unique choice
of axes other than assigning using 
.
In particular the basic assumption used to calculate
in the uniaxial case,
i.e. 
, breaks down.
Fortunately in simulations we can perform more virtual experiments,
determining the average of other probe properties sensitive
to the alignment of the two other molecular axes.
In practice, equations containing
and
as well as
,
are
constructed from the average of two other matrices,
say 
and 
by means of equations 3-5.
The resulting expressions of the order parameters in terms of the average
tensors read:
The normalized eigenvectors of the matrices give the axes of the
reference
system, except for the sign, so there are 
different
systems
corresponding to the eigenvalue permutations.
We choose the eigenvalue
permutation
which satisfies the following conditions:
a) ;
b) the same order parameters must have the same values in all the
ways they
are computed (here, e.g., and
are computed in two
different ways);
c) for each configuration at one temperature the order parameters
must be
as close as possible to the mean value of the order parameters of the
previous temperature (the sum of the differences is minimized).
The above procedure effectively assigns the X and Y axes when the phase
is
biaxial. In the uniaxial phase X and Y are undistinguishable and the
method, even though not needed, is not applicable because it forces a
difference that is completely spurious.
In a similar
way,
application of the usual algorithm for finding
to an
isotropic phase will give a spurious non-zero order parameter (decreasing
with size).
The curves for (Fig. 3-a) exhibit the usual
regular decrease with temperature and show no indication of
the biaxial-uniaxial transition.
While at low biaxiality the nematic - isotropic transition is
sharp (although it appears continuous for these system sizes),
the curve at 
shows a smooth decay consistent
with the expected second order character of the transition.
On the other hand, the molecular biaxiality parameter 
(Fig. 3-b) has a local maximum at the biaxial - uniaxial phase
transition, is non zero also in the uniaxial
phase [], and it increases with temperature
until it reaches a maximum just below the nematic - isotropic transition.
This behavior is qualitatively consistent with the MFT predictions.
In Fig. 4 the phase biaxiality parameters 
,
are shown .
The order parameter (Fig. 4-a) decays
monotonically from
to zero at the uniaxial phase transition. Because of this large
variation
(an order of magnitude larger than ) this order
parameter provides an effective monitor of the biaxial transition.
The other parameter
is rather small, increases to a maximum value (
),
and then decays smoothly to zero at the uniaxial phase transition.
It is worth noting that the quadrupolar asymmetry parameter that
determines the observation of biaxial effects in Deuterium NMR
experiments [22] is estimated to be 
when but for the other values 
is
at the border of the threshold needed for NMR detection [16] (
). It seems that an important probe of phase biaxiality would
instead be the measurement of
, possibly by the use of suitable
biaxial, rather than uniaxial probes.
0.5truecm
We are grateful to MURST, CNR. We thank the EU, HCM project, for partial support and Prof. G.R. Luckhurst (Southampton) for useful discussions.
Present address: Istituto di Spettroscopia Molecolare
and LAMEL - CNR, Bologna
Figure:
Heat capacity 
vs. reduced temperature 
for biaxiality
. MC
results are shown as symbols, MFT as continous, dashed and dotted lines
respectively.
Figure:
Phase diagram showing the reduced transition temperature
vs . MC results are shown as full squares ( ) and circles (). Empty
squares are ( ) points mapped from (
) onto
(
)
(see text). MFT results are shown as
continuous lines. The triangles at are from ref. [10].
Figure: Second - rank order parameters vs. :
(a),
= (b) obtained from
MC for
0.2 (circles), 0.3 (squares),
0.40825 (triangles) together with MFT results (lines as in Fig. 1).
Figure: Biaxial second-rank order parameters vs.
:
(a), 
(b) .
MC results for 0.2 (circles),
0.3 (squares), 0.40825 (triangles). MFT results are shown as
lines (cf. Fig. 1.