Liquid Crystal Phase Transition Lattice Model

I S OT R O P I C - N E M AT I C - L I Q U I D
C RYSTA L P H A S E T R A N S I T I O N : A
L AT T I C E M O D E L
ricardo faúndez-carrasco*
contents
1 Abstract 2
2 Introduction 3
2.1 History of LC’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Generalities about LC’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Chirality in LC’s and helical particles . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Theoretical Model 8
3.1 Basics of the LC model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Generalisation to helicoidal particles . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Computational Methods 12
4.1 General features of the code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2 Monte Carlo method and Metropolis algorithm . . . . . . . . . . . . . . . . . . . 15
4.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.4 Heat capacity and order parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5 Results and Discussion 19
6 Conclusions and Outlook 22
list of figures
Figure 1 Common liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Figure 2 LC’s most common phases . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Figure 3 Cholesterics and chiral molecules . . . . . . . . . . . . . . . . . . . . . . . 6
Figure 4 Features and properties of cholesterics . . . . . . . . . . . . . . . . . . . . 7
Figure 5 Modelisation of helicoidal molecules . . . . . . . . . . . . . . . . . . . . . 12
Figure 6 Periodical Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 17
Figure 7 Rod-like systems reaching estationary phase . . . . . . . . . . . . . . . . 20
Figure 8 Nematic - Isotropic phase transition . . . . . . . . . . . . . . . . . . . . . 21
Figure 9 Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Figure 10 Order Parameter for Nematic - Isotropic phase transitions . . . . . . . . 23
* Tutor: Giorgio Cinacchi. Departamento de Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid, Madrid,
España.
1

abstract 2
1 abstract
A study of the nematic-isotropic phase transition in liquid crystals is made here. Since
this phase transition mainly involves the rotational degrees of freedom of the particles, the
Lebwohl-Lasher lattice model is used. Using an original computer code, it is reported on the
phase transition that takes place in systems of rod-shaped particles arranged on a simple cu-
bic lattice. The computer code written in this thesis implements the Monte Carlo method and
the Metropolis algorithm. The energy, heat capacity and order parameter are calculated as
properties that allow the characterisation of the phase transition and are found in very close
agreement with literature data. This proves the validity of the Monte Carlo computer code
written. The latter thus forms the basis on which to build further developments, namely the
extension of the lattice model to handle helicoidal particles and the study of the chiral nematic
liquid-crystal phases exhibited by them. In this report, a few comments are made on these
further developments.

introduction 3
2 introduction
It is usual to distinguish among three possible states of matter (four if we include plasma
state): gas, liquid and solid. However, there has been a special emphasis in recent years on
several materials whose properties cannot be classified entirely as belonging to just one of the
classic states of matter, such as liquid crystals, the topic of this thesis.
2.1 History of LC’s
Liquid Crystals (LC’s from now on) are materials that may flow like a liquid but at the same
time their molecules or components have orientational and/or (partial) positional order. In
fact at high temperatures they become isotropic conventional liquids. Their history begins in
1888, when an Austrian botanical physiologist named Friedich Reinitzer looked into various
physico-chemical features that exhibited some derivatives of cholesterol. Before that, some
research had already been done on how cholesterol changed its color depending on the tem-
perature, but none of those scientists recognised it as a new phenomenon. Reinitzer found
out that cholesteryl benzoate does not melt as other materials do but instead has two different
melting points at 145.5 oC and at 178.5 oC with clearly differentiable phases. He contacted Otto
Lehmann, a physicist from Aachen (Germany), who confirmed that within the range of tem-
peratures between the two melting points, the material showed crystallites, what suggested
some kind of global ordering or crystal structure rather than just an ordinary liquid. Reinitzer
presented these results at a meeting in Vienna in 1888 [1]. Reinitzer gave up the study of
these materials (which Lehmann would call cholesteric liquid crystals in 1904) after discover-
ing three major features of them: the existence of two melting points and the capacity of these
materials to either reflect circularly polarized light and change the polarization direction of
incident light. Lehmann did comprehend it as a new phenomenon, so his research started
with cholesteric benzoate, a material with capacity to flow and a partial crystalline order [2].
Next years were spent on synthetizing new materials that showed this behavoir [3, 4].
It was not until 1962 that the development of flat panels for electronic devices became im-
portant and the research on liquid crystals had an unexpected boost. In that year Richard
Williams, from RCA Laboratories, applied an electric field to a thin layer of a nematic liq-
uid crytal (para-azoxyanisole of figure 1a, fairly similar to cholesterol derivatives) at a high
temperature to form a regular pattern. Meanwhile George H. Heilmeier used this very same
molecule to create a flat crystal-based panel to replace the cathode ray vacuum tube used in
televisions. Unfortunately, this material and most of those known by then required very high
temperatures to become liquid crystals, what made them unconvenient to use. Nevertheless
Joel E. Goldmacher and Joseph A. Castellano found out that mixtures of what were called
nematic materials (fully described in the subsequent section) with subtle differences in the
number of carbon atoms yielded nematic liquid crystals at room temperatures ( 30 oC) [5], so
the entire field was enhanced. Moreover in 1969 Hans Kelker synthesized MBBA (figure 1b),
one of the most popular samples in liquid crystal investigations [6], which is liquid-crystalline
at ambient conditions.
The growth of the commercialization of these materials led to the investigation of more
stable substances, mostly rod-like particles. Finally, the Nobel Prize was awarded in 1991 to
Pierre-Gilles de Gennes for "discovering that methods developed for studying order phenomena in
simple systems can be generalized to more complex forms of matter, in particular to liquid crystals and
polymers"[7].
2.2 Generalities about LC’s
The various LC’s phases, as soon as they cannot be classified neither as ordinary liquids nor
crystal solids, are called mesophases due to their intermediate behaviour, for a LC is a material
in which some degree of anisotropy is present in at least one spatial direction, together with
properties such as fluidity and ordered structure. Furthermore, LC’s can be distinguished by
the physical quantity that has the biggest effect over the system and its composition. This way,

introduction 4
(a) p-azoxyanisole (PAA) (b) N-(p-methoxybenzylidene)-p-
butylaniline (MBBA)
Figure 1: Common liquid crystal molecules.
we can divide LC’s into thermotropic, lyotropic and metallotropic phases, each one with its own
subclasses [8]. However we will focus on thermotropic LC’s as far as this thesis is based on
them rather than the other categories. We shall explain briefly some characteristics of each of
the above-mentioned categories with special interest in thermotropic LC’s, and within them,
nematics and cholesteric phases will occupy the most relevant part of our treatment.
1. Metallotropic LC’s: Composed by low-melting inorganic molecules like ZnCl2 in ad-
dittion to other soap-like molecules, their properties depend on the organic-inorganic
molecules ratio and the temperature.
2. Lyotropic LC’s: Consist of organic macromolecules that act as LC’s mixed with solvent
molecules filling the space around and giving fluidity to the system, so the properties
vary with the concentration of particles and the temperature.
3. Thermotropic LC’s: Formed by organic molecules, temperature is the physical quantity
that drives the liquid-crystal phase behaviour.
For each of these groups, phases can be nematics, smectics or columnar. The different types of
mesophases can be distinguished by three characteristics: positional order, orientational order
and lenght-scales over which these types of order exist. These characteristics tell us whether
the molecules have a partially positionally ordered arrangement, whether the particles are
pointing toward the same direction and if this order is extended only to short distances (short-
ranged) or to macroscopic lenghts (long-ranged). This gives rise to the next three categories:
(a) Nematic phase (b) Smectic A (c) Columnar phase
(top view)
(d) Columnar phase (side
view)
Figure 2: Common liquid-crystal phases. Images were obtained using the free software QMGA
(qmga.sourceforge.net).
2.2.1 Nematics
The word for nematics comes from the Greek νηµα which means thread because of the thread-
like shape of some defects observed in these materials. Nematics show anisotropic positional
short-range order but orientational long-range order. This means that a nematic material does
not organize its molecules on the sites of a lattice but they are disordered in space, whereas
almost all of them share a preferential direction for their orientation, as seen in figure 2a.

introduction 5
molecules Tipically rod-shaped particles have played a main role in the development of
nematics, although there are also some cases in which disk-like or rectangular particles form
the nematic phase so instead of a uniaxial particles there may be a biaxial nematic. However
we will limit ourselves to the uniaxial cases. We have already seen some of the most known
rod-shaped molecules that form a nematic phase in figure 1. For example, p-azoxyanisole
(PAA) can be considered as a rigid rod of around 20 ˙A long and width 5 ˙A. Recalling the
history of LC’s explained above, we saw PAA turns into a LC only at high temperatures
(between 116 oC and 135 oC) so the discovering of MBBA, which reaches that phase at room
temperature (20 - 47 oC) was regarded as a great advance in the experimentation on LC’s. But
MBBA is not so stable as it would be desired, so nowadays it is more common to use other
molecules such as cyanobiphenyls and cyanobicyclohexyl derivatives for these compounds are
chemically more stable.
properties There are some features that are remarkably important in nematics so they
define whether or not a material is nematic:
1. Molecules have no long-range positional order. This characteristic is also shared with
conventional liquids and allows for the fluidity of nematics.
2. On the other hand, molecules have directional long-range order along a director vector n,
common for all the particles and semi-arbitrary in space (its direction is given by minor
forces like the effect of the walls). All the molecules are completely symmetrical about a
rotation around n. Nematics tend to flow in the direction given by this director.
3. There is no difference in the system between a director vector n or -n.
4. A very important feature as we shall see in the next section is that nematic phases are
only present in systems where each constituent is identical to its mirror image, that
is achiral. There are some cases where chiral particles present nematic behaviour, but
only if there is the same proportion of left-handed and right-handed molecules within
the system. Actually, to a good degree of accuracy, cholesterics and nematics differ
almost only in the chirality of their components but this has profound consequencies on
the structure of the phase. However, particle chirality is a necessary but not sufficient
condition for phase chirality.
2.2.2 Smectics
Its name comes from the Greek word σµηγµα which means soap. Smectics are usually present
at lower temperatures than nematics, and at the same time represent more ordered systems,
defined by one-dimensional quasilong-range order. Smectics are formed by layers of material
with orientational order but often no positional order within the layers and with a interlayer
spacing ranging on the order of the size of the constituent molecules. We shall concentrate
only on the most important classes of smectics known currently: smectics A (figure 2b) and
smectics C, which differ only on how close are the orientations of the particles to a vector
perpendicular to the layers. Smectics A align normal to the surface of the layers, but smectics
C are slightly tilted by a small angle. In particular, if the particles that form a smectic C are not
equal to their mirror image, and helicoidal smectic C* appears. In fact, as we will see below,
chirality has a huge impact on the behaviour of liquid crystals, even giving rise to new phases
like cholesterics.
2.2.3 Columnar phases
These are two-dimensionally ordered systems in three dimensions so they form ordered tubes
as long as they are composed of stacked piles of discoidal organic molecules like hexasub-
stituted phenylesters. A visual representation is showed on figures 2c and 2d. They have
three-dimensional long-range order just as a solid crystal does but molecules can move along
the columns, the latter making a motion resembling that of sliding doors.
LC’s are not just an academical topic, for the uses and applications of LC’s are many: ly-
otropics are fundamental phases in living systems as soon as cell membranes and proteins are

introduction 6
made of phospholipides and similar molecules, which form lyotropic LC’s. On the other hand,
nematics are basic components of modern electronic devices, and most of the screens and pan-
els seen daily around us are made of LC’s due to their extraordinary optical properties. Other
thermotropic mesophases are cholesterics, which can be a good representation of the DNA at
some stages of its development, and where the chiral nature of the constituent molecules play
a major role on their properties.
2.3 Chirality in LC’s and helical particles
As we have seen previously, twisted molecules confer special properties to a LC material and
lead to new phases, in particular state C* in smectics C and cholesterics, if we think of them
as a special case of nematics (in fact, we shall see that except for the case when the pitch p is
too small, cholesterics can be viewed as twisted nematics). Smectics C* and cholesterics differ
in the orientational and positional order of the molecules. As the rest of smectics, smectics C*
show the molecules arranged in layers but with every particle aligned to the same direction,
whereas cholesterics show no long-range positional order within each layer, each of them
oriented by a director vector that changes among layers, as can be seen in figure 3a.
(a) (Left) Cholesteric phase of rod-shaped particles. (Right) Cholesteric
phase of helix-shaped particles. Images obtained using the free
software QMGA
(b) General formula of a cholesterol ester
molecule.
Figure 3: Rod-like and helical particles in a cholesteric and a chiral molecule that forms this phase.
From now on we shall concentrate on cholesterics, named this way because this phase
was first observed for cholesterol derivatives. Cholesteric mesophases need the constituent
molecules to have chirality i.e. the particle is different that its mirror image, although it is
a necessary condition but not sufficient. Most usual molecules in cholesterics are cholesterol
esters, whose general formula is showed in figure 3b. The molecular rings are not aromatic and
the structure is not planar but the rings are rigid and the edges of the particle act somehow as
flexible tails.
Cholesterics and nematics can be viewed (in a very general way) as two subclasses of the
same mesophase, but nematics need achiral molecules or a racemic combination (1:1) of left-
handed and right-handed species, and cholesterics can only take place if the molecules are
chiral. Due to this fact, whereas nematogens (molecules of the nematic) align with respect to

introduction 7
a fixed director vector n in space (usually Z direction is taken as reference), cholesterics align
following a director vector that we will call n that is not fixed in space but rotates between
layers of the material (see figures 3a and 4a) following the general expression
nx = cos (q0z + φ) (1)
ny = sin (q0z + φ) (2)
nz = 0 (3)
Where q0 has units of m−1. We shall call pitch p the distance over which the LC molecules
undergo a full 2π twist. Because the directions given by n and -n are equivalent, the structure
repeats itself every π radians. This pitch changes with temperature, something that makes
these materials quite interesting from the point of view of their optical properties. Typical
values of the pitch are in the range of around 3000 ˙A, much larger than molecular dimensions
but of similar lenghts of optical wavelengths, resulting in Bragg scattering of light beams.
(a) Structure of a cholesteric. (b) Free energy as a function of twisting q. (top)
Nematic phase. (down) Cholesteric phase.
Figure 4: Properties of the cholesteric mesophase. Notice that a cholesteric is defined by the twisting of
the particles vector and the length of the pitch, which depends on the temperature.
Because of the periodicity of these structures, the pitch can also be written in the following
way:
p =
2π
|q0|
(4)
The magnitud |q0| allows to look into various features; for example, the sign of q0 distin-
guishes between right-handed and left-handed helical molecules, and at a given temperature
the sign is always the same for a sample. Actually it may happen that at a certain temperature
T◦, the sign changes. In that very temperature, the system behaves like a conventional nematic
mesophase, and once that point is crossed, the physical properties remain almost unchanged.
Previously we said that in cholesterics it is mandatory not to be racemic, i.e. to have same
proportion of left-handed and right-handed particles impedes the formation of this mesophase.
Figure 4b presents the free energy per unit volume F as a function of q, the wavevector defined
by the general formulas 1, 2 and 3, i.e:
q =
∂θ(z)
∂z
, (5)
with θ(z) = qz + φ. If the constituent molecules of the material are achiral so the image
of a given particle is the same as that of its mirror image, the free energy function F(q) is
symmetrical and then F(q) = F(−q). In nematics, the minimum is located at q = 0, as seen
in the upper curve in figure 4b. On the opposite side, if the molecules are chiral F(q) is not
symmetrical and the minimum value of F(q) does not fall at q = 0. This case corresponds to
cholesterics, shown in the lower curve of figure 4b.

theoretical model 8
3 theoretical model
We would need a consistent mathematical model if we would like to describe with a good
degree of accuracy any physical system. In order to do that, we shall explain two concepts
that will reveal themselves as fundamental in our treatment of LC’s: the existence of a director
vector and an order tensor [8]. Both will allow us to define any LC nematic system.
order tensor Assuming that the molecules within the LC are rigid rod-shaped, we can
introduce a unit vector u(i) along the axis of the ith molecule which describes its orientation.
It is not possible to introduce a vector order parameter analogous to the magnetization vector
in ferromagnetic materials. Rather, a second-rank tensor appears to describe the ordering in
LC’s, and its expression is given in equation 6 .
Qαβ(r) =
1
N
N
i=1
u
(i)
α u
(i)
β −
1
3
δαβ (6)
where the sum is over all the N molecules in the volume located at the point r: indexes
α, β = (x, y, z). There are some properties that make this order tensor interesting to describe a
physical system:
1. As soon as the product u
(i)
α u
(i)
β = u
(i)
β u
(i)
α and δαβ = δβα, it is trivial to show that Qαβ
is a symmetric tensor so:
Qαβ = Qβα (7)
2. It is traceless, i.e. its trace Tr(Qαβ) is equal to zero since u is a unit vector:
Tr(Qαβ) =
α=(x,y,z)
Qαα =
1
N
i
u
(i)
x
2
+ u
(i)
y
2
+ u
(i)
z
2
−
1
3
3 = 1 − 1 = 0 (8)
This way we pass from a 3 x 3 matrix with 9 independent elements to only 5 elements.
3. In the isotropic phase Qαβ = 0. It is easy to understand since every molecule of the
system points to a different orientation, making the average global order vanish.
4. In a perfectly aligned nematic phase in which the molecules tend to align along the z
axis,
Q =


−1/3 0 0
0 −1/3 0
0 0 2/3

 (9)
These properties tell us Qαβ is a very good candidate to describe the order parameter of our
LC systems. It is zero in the isotropic phase when the LC behaves in a conventional liquid-like
way. This tensor is also very sensitive to the average orientation of the constituent molecules.
The next question is: how do we know which is the preferred orientation direction of the
molecules? The answer is given by the director vector.
director vector It is a feature of special interest in cases where global degree of ordering
is not changing very much and therefore only the average orientation of the molecules has
a clear defined meaning. The order tensor explained above can be diagonalised for specific
Cartesian coordenate systems. Once it is diagonalised, we will mainly pay attention to the
biggest eigenvalue and its associated eigenvector, since the first one yields a way of measuring
the degree of alignment of the molecules with respect to the director vector n(r). Actually if
we denote the biggest eigenvalue by λB, the value 3
2 λB can be taken as the order parameter S,
a scalar quantity that meassures the degree of global ordering. It is worth noticing that it is
the same to work either with the vector n or -n, for there is no physical polarization along this
axis.

theoretical model 9
3.1 Basics of the LC model
At this point we are ready to introduce theories based on the features explained before (order
tensor and director vector): we shall begin with a cursory descriptionof the Onsanger theory
and Flory model, and after that the Maier-Saupe will be explained briefly in order to finish the
section with a full description of the Lebwohl-Lasher model.
It is worth noticing that all of the theories we are considering are based on statistical physics
and because of that each model has to characterise both the molecules and the interactions.
Actually all of them consider the interactions among molecules as mean-field forces where
the correlations between two molecules are not studied separately but a mean value of their
interaction is taken into account to calculate the macroscopic properties of the system.
3.1.1 Onsanger’s theory and Flory’s model
It is possible to experimentally prepare solutions of hard-rod macromolecules whose length L
and diameter D are well defined. Even in natural biological systems we can find these kind of
molecules, for example Tobacco Mosaic Virus (TMV). The following assumptions were made in
1949 by Onsanger [9] as fundaments of a statistical theory for LC’s:
1. Rods can not interprenetate each other. The only forces of importance are those cor-
responding to steric repulsions (and if desired, Coulumb repulsion between charged
rods, even though this can be taken into account effectively by increasing the size of the
effective diameter of the molecules).
2. If we call c to the number density of rods, the volume fraction Φ = 1
4 πcLD2 is much
smaller than unity.
3. The rods are very long so L >> D.
4. Let’s call cfadΩ the number of rods per unit volume pointing in a small solid angle dΩ
around a direction labelled by the unit vector a. The sum of these solid angles overall
will give the total concentration c:
fadΩ = 1 (10)
With these assumptions it is obtained a non-linear integral equation, and depending on
the ratio ΦL/D, it can give rise to anisotropic solutions. We simply recall that the Onsanger
solution leads to an sharp transition between the ordered nematic phase and the isotropic
conventional liquid phase. On the other hand, the hard rods become "athermal" molecules in
this model, which is valid for colloidal (lyotropic) LC’s.
The point of view adopted by Flory [10] was similar to the Onsanger one but with subtle
differences: instead of rigid rod-like molecules, a rod is described as a set of points inscribed
on a lattice, and the number density of points acts just the same as the ratio L/D. The interest
for this model comes from that it is possible to calculate exactly the partition function of the
system, something that can not be achieved in Onsanger’s theory. In fact the Onsanger theory
and the Flory’s model complement each other but fail when trying to account for the system
properties along the entire range of possible c’s.
3.1.2 Maier-Saupe Theory
This theory was developed by Alfred Maier and Wilhelm Saupe and can be regarded as the
analogous in nematics of the approximation made by P. Weiss to describe ferromagnetics [11].
There are slight differences depending on wheter the molecules are uniaxial or biaxial, but we
shall focus on the uniaxial case as far as that is the shape of the molecules of our model.
Instead of considering the interaction due to every molecule of the volume this model works
through a mean-field average of these interactions. The details of the calculations can be found
in [8] and [11]. We will not go further into calculations since the purpose of this thesis is to
use the Lebwohl-Lasher model to simulate a LC phase system, which can be considered as

theoretical model 10
the lattice version of the Maier-Saupe model. The latter theoretical model yields a threshold
temperature Tc defined by equation 11 and an associated value of the order parameter just
before the phase transition.
kBTc
U(Tc)
= 4.55 (11)
where U(Tc) is the energy of the system at temperature Tc. In the original presentation it was
assumed that U is due entirely to van der Waals forces. If the temperature of the system T
is below Tc, the nematic phase is the stable one. The higher the temperature rises, the more
disordered the system becomes until the system is completely isotropic and behaves like a
conventional liquid. The order parameter for T just below Tc is
Sc ≡ S(Tc) = 0.44 (12)
This equation yields a universal value for the order parameter at the phase transition. This
has been in part confirmed by experiments on thermotropic nematics [12, 13].
3.1.3 Lebwohl-Lasher Model
This model, which we shall call LL from now on, is particularly useful in the description of
LC’s. It is a lattice version of the Maier-Saupe theory previously mentioned [14, 15]. It is the
one we have used in this study. In the LL model, molecules are placed on the sites of a simple
cubic lattice and described by an unit vector representing the orientation of each one of these
molecules. The interaction among particles is expressed by the equation 13 which is a pair
potential restricted to first nearest neighbours.
H = −ε0
i,j
P2 cos (θi,j) (13)
where ε0 is an energy parameter that measures the strenght of the interaction, and P2 cos (θi,j)
is the second Legendre polinomial with θi,j the angle between two nerby orientation vectors
(equation 14).
P2(z) =
1
2
3z2
− 1 (14)
It is a matter of convenience to define a set of normalised adimensional units. In order to
do so, we shall set the value of ε0 = 1, so from now on, we shall work in units of normalized
temperature T∗ defined by:
T∗
=
kBT
ε0
(15)
This makes possible to work with adimensional units for the energy too.
Beside these considerations about the description of the molecules and their interactions,
we need to consider which boundary conditions are useful as we will investigate this model
via a numerical simulation method. In the LL model periodical boundary conditions are
used. These conditions lack of physical significance but in some way make the lattice expand
to infinite and avoid surface effects. Actually what we do is to establish our built lattice as a
building block of a bigger system; this way the molecules of one edge surface will interact with
their first neighbours, which will be in the very same state as the molecules on the opposite
edge surface.
This model has become a prototype for computer simulations on systems which undergo
a nematic-isotropic-like orientational phase transition. Its role is fairly similar to the one that
Ising and Heisenberg models play in magnetic phase transitions. Note that in the case of
Ising magnetism, we only deal with spins with two possible states whereas in the LL model a
particle can have any orientation, defined either by two polar angles θ and φ. Thus, computer
calculations are heavier to run, as long as for example we have to restrict ourselves to small
systems with a volume size of around 30 x 30 x 30 molecules instead of the million particle
simulations that are possible for the Ising model [16].

theoretical model 11
3.2 Generalisation to helicoidal particles
In order to handle helicoidal particles and characterise the chiral nematic liquid-crystal phases
that these sort of molecules undergo [17], a series of modification must be done on the previous
model, namely the change in the shape of the molecules, no longer rod-like but helix-like,
and because of it also the pair interaction among molecules must change. Additionally, new
boundary conditions are required for the chiral nematic phases to take place.
new molecular shape Every unit vector we are considering as the orientation vector of
every molecule must now take an helix-like shape, and ought to be able to rotate along the
main axis of the molecule as rod-like molecules do. From a computationally point of view the
required steps are listed below:
1. Every rod-like molecule is divided into N equal segments of length L/N being L the total
length of the rod, in our case we can set L = 1. We can think of this stage as the passage
from a single-unit to a multi-unit rod.
2. To modify the multi-unit rod to a multi-unit helix, the N segments have to rotate along
the main axis of the rod, forming helices with n turns (as can be seen in figure 5a. In
the simplest case n = 1, i.e. the segments make a 2π radians twist. The geometry of
the molecules is fixed by specifing the radius r, which is the maximum distance of one
segment in relation to the main axis. The smaller the radius, the larger is the particle
pitch. In the limit when the radius is very small the molecule behaves like a normal
rod-like one.
pair potential Due to the modification in the shape of the molecules the pair potential be-
tween two particles is also affected, so now the energy is given by the pair potential expressed
in equation 16. In this new expression of the potential, the effect of one molecule on another
one is calculated as the sum of the interactions between the segments that form them. This
potential is limited to first neighbours, with an additional condition: the distance that define
if two molecules are first neighbours is given by the distance between the centres of gravity
of two adjacent molecules. If the distance between two segments of different molecules is big-
ger than the distance between the centres of gravity of two molecules, there is no interaction
between those segments even though those molecules are first neighbours. This condition is
expressed by equations 17 and 18.
Ei,j = −
N
ki=1
N
kj=1
εki,kj
P2 cos θki,kj
(16)
where the sum extends to all the segments of two molecules, ε being a quantity that measures
the strenght of the interaction, and P2 cos θki,kj
a second Legendre polynomial for which
θki,kj
is the angle between two normalised segment vectors. On the other hand, ε is given by
the following conditions:
εki,kj
=
ε0
N
if d(ki, kj) L (17)
εki,kj
= 0 otherwise (18)
where ε0 is the strenght we use for the rod-like molecules case.
degrees of freedom When the molecules are rod-shaped, orientational jumps can take
place in any spatial direction. Differently from a Lebwohl-Lasher rod, which can rotate on the
surface of the unit sphere, a helical rod for simplicity is assumed to rotate only in a plane as
well as around its main symmetry axis defined by the vector uh. This feature can be seen in
figure 5b.
It is clear from the pictures in figure 5 and the explanations above that the mechanical state
of an helical rod is described by giving the vectors uh and wh.

computational methods 12
(a) Helical molecule formed by
twisted segments. The orienta-
tional state of the molecule is
defined by two normal vectors,
uh and wh.
(b) Degrees of freedom of one molecule. It can
rotate only on the plane YZ and around its
main symmetry axis uh.
Figure 5: Modelisation of a helix-like molecule.
boundary conditions When the pitch of a helix is unknown or even temperature depen-
dent, usage of periodic boundary conditions may introduce "frustation" effect, that can cause
the system to choose the wrong phase entirely and discontinuos changes in the pitch. A new,
self-determined boundary-conditions are required which are known as "spiraling boundary con-
ditions" [18].
4 computational methods
The aim of this section is to explain the original C++ code we have built and used to obtain
the results that are presented in the next section. In order to make it clear, here we shall
present how the different parts of the model are implemented, showing some lines of the code
when required. The simulation requires the files "nematics.cpp" and "functions.h". The first one
includes the main body of the code, with the implementation of the model system, the identifi-
cation of all the particles, the equilibration of the system through the Monte Carlo method and
the calculation of some properties of the system that cannot be processed through an external
function. When it was possible, those functions were written in the latter file, which contains
the implementation of Marsaglia’s pseudo-random number generator, the calculation of the
change in energy per particle, the Monte Carlo algorithm itself, and the routines for a couple
of properties such as the calculation of the order tensor Qαβ.
4.1 General features of the code
We have used the programming language C++ to build an original code based on arrays and
pointers that implements a Monte Carlo numerical simulation for a 20 x 20 x 20 sized lattice.
As in all the theories explained in the previous section, our molecules will be defined as
rigid rods, using time-dependant unit vectors to modelize them. So built, the molecules lack
of diameter and then they represent infinitely thin rods. The interaction among particles follow
the rule given by equation 13, with the potential restricted to first neighbours. Beside that, we
work with the normalised and adimensional units defined by the normalised temperature of
equation 15. We have also taken kB = 1 for the sake of simplicity, and from now on the
energies will be normalised by dividing by the factor ε0 present in the pair potential.

The initial configuration of our model can be of our choice. Three possibilities were consid-
ered:
1. To start from a perfectly aligned arrangement of molecules. All of them point towards
the z-axis with no deviations.
2. To start from a series of molecules with their orientation vectors distributed pseudo-
randomly over the unit sphere. To achieve this we make use of Marsaglia’s pseudo-random
number generator, which will be explained in detail below.
3. From a previous configuration. It implies to save the vectors of a previous configuration,
load them and restart the system from that point.
Before going further into the description of these possibilities, we shall present how we
ordered the coordinates of the 8000 molecules to arrange them into fixed positions on the
lattice and to associate with each of them an energy and an orientation vector. It is shown
below the code for this feature:
1 const unsigned int N = 20;
2 int Npart = N*N*N;
3 int coord[Npart][3]; //Coordinates of the 8000 particles
4 int ident[N][N][N]; //Auxiliar number to asign a number 0-7999 to each particle
5 for (int i=0;i<N;i++){
6 for (int j=0;j<N;j++){
7 for (int k=0;k<N;k++){
8 ident[i][j][k] = i+N*j+N*N*k;
9 coord[ident[i][j][k]][0] = i;
10 coord[ident[i][j][k]][1] = j;
11 coord[ident[i][j][k]][2] = k;
12 }
13 }
14 }
15 double u[Npart][3]; //Array of orientations
As can be seen, every molecule has an associated identity number between 0 and 7999. That
number depends on the coordinates of each particle (line 8). The arrays called coord and u
store a three dimensional vector with the spatial coordinates and the unit orientational vectors
of each particle, respectively. This identifier number is also valid in the Monte Carlo method,
allowing us to pick a particle and access to its properties immediately.
aligned vectors In the simulations we follow the general convention, and the molecules
are aligned along the Z-axis. This will be the configuration of minimum energy of the system,
for the scalar product between parallel vectors is equal to the unity, and all the orientation vec-
tors are aligned along the very same direction. It is clear that this is also the configuration with
the higher degree of symmetry, so the order parameter should be maximum. It corresponds
to a perfect nematic phase, and is only stable at vanishing temperatures.
randomly on the unit sphere Here it is required to present a fully explanation of the
Marsaglia method for the generation of pseudo-random numbers on the unit sphere. Within
this algorithm we chose two pseudo-random numbers, let’s call them x1 and x2 which are
in the interval [−1, 1]. To obtain these numbers we make use of the simple rand() function
already avalaible in C++, and stating the seed of the pseudo-random number generator on
a null time pointer. We shall restrict these two numbers to the interior of the unit circle, so
numbers which do not fulfill the constraint x2
1 + x2
2 < 1 are automatically rejected. After
that, we simply make use of the formulas suggested by Marsaglia (lines 7,8 and 9). The three
pseudo-randomly generated numbers are saved into a three dimensional array v. The entire
algorithm is shown below:
1 void Marsaglia(double v[3]){
2 double xmar,ymar,zmar,sq,vmar;

3 do{
4 xmar = rand()/((double) RAND_MAX)*2.0-1.0;
5 ymar = rand()/((double) RAND_MAX)*2.0-1.0;
6 sq = xmar*xmar+ymar*ymar;
7 xmar = 2.0*xmar*sqrt(1.0-sq);
8 ymar = 2.0*ymar*sqrt(1.0-sq);
9 zmar = 1.0-2.0*sq;
10 }while(sq>1);
11 vmar = sqrt(xmar*xmar+ymar*ymar+zmar*zmar);
12 v[0] = xmar/vmar; v[1] = ymar/vmar; v[2] = zmar/vmar;
13 }
We cannot simply pick three numbers from the pseudo-random number generator function
rand() included in the standard library of the language because doing so would imply that
statistically we would find that most of our vectors are poiting towards the poles of the unit
sphere.
loading a previous configuration The Monte Carlo algorithm implies very heavy cal-
culations and many time steps (of the order 105 at least), so it is really useful to run the code
once and save the configuration so it is possible to re-run the code again but starting from a
configuration that is closer to, or already has reached the equilibrium, speeding up the time
of convergence of the method. The way we achieve this is simply by saving the 8000 orienta-
tion vectors on an external file that can be loaded when the program is run. The code itself
detects if the file exists, so if it does not and the program cannot load a previous configura-
tion, instantly the system is reinitialized to either an aligned system or a pseudo-randomly
distributed on the unit sphere.
In all of these cases, after setting the array of orientational vectors u, it is still necessary to
calculate the average normalised energy per particle < U∗ >/N. In order to do so, the fol-
lowing algorithm calculates the energy due to the first neighbours of a given molecule. The
implementation operates over the i,j and k directions separately. Here we present the function
for the i-axis, the others being analogous:
1 double calcE(double u[8000][3],int coord[8000][3],int ident[20][20][20],const
2 unsigned int N,int in){
3 double energy = 0.0;
4 // Coordinates of the given particle part
5 int idin = coord[in][0];int jdin = coord[in][1];int kdin = coord[in][2];
6 // calculus along i axis
7 for (int ibour=-1;ibour<=1;ibour+=2){
8 int is = idin + ibour;
9 if (is == N){
10 is = 0;
11 }
12 if (is == -1){
13 is = N-1;
14 }
15 int js = jdin;int ks = kdin;
16 int etivic = ident[is][js][ks];
17 double pr = 0.0;
18 for (int kpr=0;kpr<3;kpr++){
19 pr = pr + u[in][kpr]*u[etivic][kpr];
20 }
21 double eij = -(1.5*pr*pr-0.5);
22 energy += eij;
23 }
24 return energy;
25 }

Between lines 9 and 14 we can see the implementation of periodical boundary conditions. We
will come back to them later. In the code above it is important to notice the contribution from
the neighbours through the pair potential given in 13 in line 21. The exit value of the function
is the value of the energy difference between the old and the new states of the molecule. This
will be later explained better within the Monte Carlo section. This function can be applied
in any situation in which the orientations of the particles are well defined, so these two steps
together are the previous and required step before applying the Monte Carlo algorithm. Once
we know the orientation vectors and the total energy of the system, we can proceed to the next
stage.
4.2 Monte Carlo method and Metropolis algorithm
The entire thesis is based on the application of the Monte Carlo method to a physical model
system in order to simulate the behaviour of a corresponding real system. In what follows, we
shall constraint ourselves to systems in which number of molecules (N), the total volume of
the system (V) are fixed for a given temperature T.
Suppose we call by A a given observable of the system. The classical thermal average of A
is given by
< A >=
dpNdrNA pN, rN exp −H pN, rN /kBT
dpNdrN exp −H pN, rN /kBT
(19)
where the quantity Q, defined by the expression 20 is the partition function of the system,
with rN being the coordinates of all the particles, pN their corresponding lineal momenta and
H the Hamiltonian of the system:
Q = c dpN
drN
exp −H pN
, rN
/kBT (20)
where c is simple a normalisation constant. When we want to obtain an average of a quantity
that depends on the position of the particles the problem becomes usually impossible to solve
by brute force [19]. This trouble gave rise to more creative ways of solving it, and one of them
has become especially important: the algorithm proposed in 1953 by Metropolis et al [20], or
importance sampling algorithm.
The original paper suggested that it was possible to create a general algorithm (and in
fact, they did develope it) valid to calculate the properties of systems formed by individual
molecules that interact among themselves. The details of the mathematical demonstration are
contained in the literature above mentioned, and we shall only comment the features that
are most important for our system. Esentially the algorithm determines whether a virtual
movement of molecules within the system can be considered as a new stable state. The steps
to follow in the method are the following:
1. A random trial move is made on a particle of the system. In our case,the molecules are
fixed at the sites of a lattice, but it is the orientation vector of one chosen molecule what
changes. We have considered completely random orientational jumps.
2. Calculate the the change in energy due to this new configuration, ∆E.
3. In case the new configuration is more stable to the previous one, ∆E < 0, the system is
brought to a state of lower energy and the move is accepted.
4. If on the opposite side, ∆E > 0, we allow the move with probability exp (−∆E/kBT).
In order to do that, we define a new random number ζ between 0 and 1, and if ζ <
exp (−∆E/kBT) the move is accepted and the system is brought to the new configuration,
even though it was not energetically favourable.
5. If the probability given by ζ is larger than the exponential, the move is rejected and the
system goes back to the old configuration.

A very interesting thing of this method is that, either the move is accepted or rejected in a
time step, it is equally considered for the global properties of the system. Now we present the
complete algorithm we have created and which has allowed to obtain all results shown later:
1 double monteCarlo(double u[8000][3],int
2 coord[8000][3],int ident[20][20][20],const unsigned int N,double T){
3 // Step I: select rod
4 int idin = rand()%N;int jdin = rand()%N;int kdin = rand()%N;
5 int in = ident[idin][jdin][kdin];
6 //Step II: calculus of the energy
7 double energyold = calcE(u,coord,ident,N,in);
8 double uxold = u[in][0];double uyold = u[in][1];double uzold = u[in][2];
9 //Step III: generation of a random point on the unit sphere
10 double v[3];Marsaglia(v);
11 //Step IV: change rod’s orientation
12 u[in][0] += 1.0*v[0];u[in][1] += 1.0*v[1];u[in][2] += 1.0*v[2];
13 double umod = sqrt(u[in][0]*u[in][0]+u[in][1]*u[in][1]+u[in][2]*u[in][2]);
14 u[in][0] = u[in][0]/umod;
15 u[in][1] = u[in][1]/umod;
16 u[in][2] = u[in][2]/umod;
17 double uxnew = u[in][0];double uynew = u[in][1];double uznew = u[in][2];
18 //Step V: calculate he energy
19 double energynew = calcE(u,coord,ident,N,in);
20 double diffenerg = energynew-energyold;
21 //Step VI: applying MC criteria
22 if (diffenerg > 0){
23 double alef = rand()/((double) RAND_MAX);
24 if (alef > exp(-diffenerg/T)){
25 u[in][0] = uxold;
26 u[in][1] = uyold;
27 u[in][2] = uzold;
28 energynew = energyold;
29 }
30 }
31 //Step VII: update total energy
32 diffenerg = energynew - energyold;
33 return diffenerg;
34 }
If we look at line 12 of the code above, we will find that when we change the orientation of
a particle, we do it by adding a unit randomly generated vector to the original one and then
re-normalising. If instead of a unit vector we sum a much smaller one (let’s say u = 0.1*v),
we observe that the smaller the orientational jump, the bigger the probability of a move to be
accepted, so much longer simulations are required to obtain the same results.
4.3 Boundary Conditions
As said before, we set periodical boundary conditions for the system. Known the coordinate
of a given particle in one direction, we look for the coordinates of the two first neighbours in
that direction. If the chosen particle is placed on an edge surface of the 20 x 20 x 20 lattice, one
of those two neighbours will be out of the limits (i.e. a maximum of three neighbours can be
out of the lattice if we consider the three dimensional case). In that case, we consider the first
neighbour is not the one that actually should be but the molecule on the opposite edge surface.
For example, if the system consists of a thread of N nodes, the neighbours of the node N are
N − 1 and N + 1, but instead of this last one we will take the node 1 as first neighbour of node
N. This procedure can be applied to a systems which as many dimensions as desired, and
the idea is exactly the same (an illustration for a two-dimensional case is shown on figure 6).
The net effect is that the system becomes in someway infinite, and we avoid the undesirable

effects due to surface effects. At the end, the cell box we use in our simulation becomes a some
sort of brick, with the virtual infinite system composed of these identical bricks. Other way
to imagine the effect of these conditions is to think of the system as an enclosed circle, where
there is no surface but the "end" of the circle is linked to the "beginning".
Figure 6: Periodical boundary conditions applied to a two-dimensional system. The effect of this condi-
tions is that the volume becomes an infinite repetition of the same block, which acts as the unit
building block of the system.
Nonetheless, these boundary conditions do not lack of problems, since they have not a phys-
ical meaning but computational. We present the few lines that actually contain the periodical
boundary conditions in our code (only the conditions for the i-axis are shown):
1 for (int ibour=-1;ibour<=1;ibour+=2){
2 int is = idin + ibour;
3 if (is == N){
4 is = 0;
5 }
6 if (is == -1){
7 is = N-1;
8 }
9 int js = jdin;int ks = kdin;
10 int etivic = ident[is][js][ks];
11 }
4.4 Heat capacity and order parameter
From the very definition of heat capacity, we already know that
Cv =
∂U
∂T
(21)
From statistical mechanics we also know that:
Cv =
< (∆U∗)2
>
T∗2
(22)
where < (∆U∗)2
> are the fluctuations of the energy. The heat capacity can be calculated in
two ways.
We cannot calculate the average energy per molecule until the system has reached the ther-
modynamical equilibrium, for what a previous equilibration simulation of at least 106 time
steps is required. Once the system reaches that equilibrium state, we can sum the energy of
the system at every Monte Carlo step and then divide by that number of time steps. Once we
have done that, and we have a curve that represents < U∗ > as a function of T∗, there are two
ways to calculate Cv.
• The most direct way is to use a built-in function to differentiate the experimental points
obtained computationally, following this way the expression 21.

• Another option is to calculate at every Monte Carlo step the mean standard deviation
from the average, σMSD =
√
σMQD, deﬁned by expression 23. Taking the averaged
σMSD over all the simulation for a given temperature and sampling over a range of tem-
peratures using 22, we should obtain similar results to those calculated by the previous
method.
σ2
MQD =< U2
> − < U >2
=< (∆U∗
)2
> (23)
Regarding the order tensor and the order parameter, it is also divided into two different
steps:
1. Calculate the order tensor Qαβ for a given conﬁguration of the system. It will not be
diagonal in the general case. The expression for the calculus of the tensor is given in 6,
which after some mathematical treatment can be calculated as in the code below.
1 void calcQ(double u[8000][3],int Npart,double Q[3][3]){
2 /**** Calculates the elements of matrix Q (order parameter) ****/
3 for (int i=0;i<3;i++){
4 for (int j=0;j<3;j++){
5 Q[i][j] = 0.0;
6 }
7 }
8 for (int n=0;n<Npart;n++){
9 for (int i=0;i<3;i++){
10 for (int j=i;j<3;j++){
11 Q[i][j] = Q[i][j] + u[n][i]*u[n][j];
12 }
13 }
14 }
15 for (int i=0;i<3;i++){
16 Q[i][i] = 1.5*Q[i][i]/((double) Npart)-0.5;
17 }
18 for (int i=0;i<2;i++){
19 for (int j=i+1;j<i+3;j++){
20 Q[i][j] = 1.5*Q[i][j]/Npart;
21 Q[j][i] = Q[i][j];
22 }
23 }
24 Q[2][0] = Q[0][2];
25 }
2. Diagonalise the tensor and take the biggest eigenvalue, for it is the order parameter of
the system at a given moment. In our case, we have chosen not to diagonalise completely
the matrix but calculate only the biggest eigenvalue (and if desired, the associated eigen-
vector). This saves computational time and since we do not really care about the other
eigenvalues, is a good idea to make easier the programming task. For this purpose,
we have used a method called power’s method [21]. The main feature of this method,
which is not valid for a complete diagonalisation, is that it only serves to calculate the
biggest eigenvalue and its associated eigenvector by an iterating algorithm based on the
product of the matrix by an adaptative vector which ends to be the eigenvector. The
element of this eigenvector with the biggest value is the associated eigenvalue. Then a
renormalisation is done.
1 double eigValue(double A[3][3]){
2 /**** Calculates the biggest eigenvalue of a given 3x3 matrix ****/
3 /******************* POWER METHOD ALGORITHM *********************/
4 double eps = 1e-6;
5 double v0[3] = {1,1,1};
6 double v[3];
7 double dif = 0;

results and discussion 19
8 double max = 0;
9 double maxp = 0;
10 double max2 = 0;
11 double sum = 0;
12 double eig = 0;
13 do{
14 for (int i=0;i<3;i++){
15 if (fabs(v0[i])>max){max = v0[i];}
16 }
17 for (int i=0;i<3;i++){
18 sum = 0.0;
19 for (int j=0;j<3;j++){
20 sum += A[i][j]*v0[j];
21 }
22 v[i] = sum;
23 }
24 max2 = max;
25 maxp = eig;
26 max = 0;
27 for (int i=0;i<3;i++){
28 if (fabs(v[i])>max){max = v[i];}
29 }
30 for (int i=0;i<3;i++){
31 v[i] = v[i]/max;
32 }
33 eig = max;
34 for (int i=0;i<3;i++){
35 v0[i] = v[i];
36 }
37 dif = fabs(max-maxp);
38 }while(dif>eps);
39 return eig;
40 }
5 results and discussion
Next we present the results obtained by running our codes. All the codes were programmed
using a UNIX machine, so the easiest way to compile is to type: g++ nematics.cpp functions.h -o
nematics on the shell, creating a executable named nematics which will be run by typing ./nemat-
ics. The procedure in Windows-based machines is harder in general and not recommended.
Not all the features and graphics are directly calculated at once in the program, and it is up to
the user comment or uncomment the desired lines of the code.
Our first aim was to develop a Monte Carlo method for the system so it should reach an
equilibrium state at some stage of the simulation. In figure 7 some examples are shown. We
can see how the system, starting from a given configuration of the molecules, tend to have an
stationary average energy per molecule. No accumulation of the averages of the properties of
the system can be made until the system is in the stationary phase, what makes this stage of
the work fundamental, and in fact we spent most of the time improving this part of the code.
There are many things to comment on concerning this key point. First of all, the range of
temperatures. Simulations over a large range of temperatures show that the phase transition
takes place within the range T∗ = 1.0 − 1.3. The temperatures chosen in figure 7 correspond
to the limits of pre and post-transitional phases. Second, it is easy to see that it does not
take the same time to reach the equilibrium to all the possible initial configurations, but once
reached, the equilibrium energy is always the same for a given temperature (figures 7a-7c and
figures 7b-7d). This is particularly clear if we compare figures 7a and 7c. In the first case
the equilibrium is reached from a state of minimum energy, for all the molecules are aligned

(a) Initially aligned at T∗ = 1.0. (b) Initially aligned at T∗ = 1.3.
(c) Intially isotropic at T∗ = 1.0. (d) Initially isotropic at T∗ = 1.3.
(e) Initially at equilibrium at T∗ = 1.0. (f) Initially at equilibrium at T∗ = 1.3.
Figure 7: Rod-like systems studied via a Monte Carlo simulation. One can see how, as time increases, all
the systems tend to reach a temperature-dependant average energy independent of the initial
conﬁguration: compare panels (a), (c), (e) and panels (b), (d), (f).

along the same direction from the very beginning. On the other hand, in the latter one the
system starts from a perfect isotropic configuration, the most similar to a conventional liquid
phase, what makes much harder to align the molecules, something that is reflected on the time
steps that are needed. The zero energy at the beginning is easy to understand if we take into
account that no order is present in the system so all the orientation vectors point to different
directions, making the director vector also undefined. At last, in figures 7e and 7f systems in
the equilibrium are shown. It is clear that once the system has reached it, the deviations are
quite small compared to the mean value of the energy.
These results show that any system made by rod-like molecules, independently of whether
the system is in conventional liquid-like phase or starts from an aligned configuration, if the
pair interacion is described by equation 13 tend to align always in the same way and because
of that, to a same average energy per molecule. With this in mind, it is clear that it does
not really matter much if the temperature-dependant experiments are made on one system or
another as long as the system has reached the equilibrium state.
Once known this, we can proceed further in our study on nematic-isotropic phase transitions
in LC’s formed by rod-like molecules. Figure 8 shows the average energy per molecule as a
function of the temperature. The data point we obtained are shown in black and were taken
once the system reached the equilibrium state. It is clearly seen a sharp change of slope,
suggesting that a phase transition takes place. It is also shown the results presented in the
literature [22]. It is observed a slight difference in the phase transition region between our data
and the ones from that reference. This is due to the size of the lattice, because as the system
size increases, the phase transition shifts to lower temperatures according to the periodic
boundary conditions that tend to favour the ordered phase.
Figure 8: Energy per particle versus temperature for the Lebwohl-Lasher model. The points obtained
by ourselves are shown as black dots with their error bars, whereas red squares are the data
taken from the literature [22]. The inset shows data over a larger temperature interval. Figure
obtained with the open-available software xmgrace.
To characterize this phase transition, other properties of the system were studied such as
the heat capacity Cv and the order parameter S, both already described theoretically and
computationally in previous sections.
In figure 9 the heat capacity C∗
v is shown. Figure ?? shows the results achieved by differen-
tiation of the data points obtained in figure 8 after a previous Akima spline smoothing [23].
Even with this treatment, the curve is still full of sharp slopes and discontinuities. This is

conclusions and outlook 22
Figure 9: Curve of heat capacity C∗
v as function of the temperature T∗ by direct differentiation of the
energy data. Figure obtained with the open-available software xmgrace.
due to the fact that it is harder to reach a good accuracy for a derivative quantity like the
heat capacity. None the less,the tendency is right and the predictions are correct. The peak
in the heat capacity locates the phase transition temperature at T∗ 1.129, according to the
literature, where this temperature is estimated to be T∗ = 1.1232 ± 0.0006 for a larger system.
In ﬁgure 10 the average value of the order parameter is shown as function of the temperature.
For every temperature the system reaches the equilibrium, and then for some large number of
time steps, the order tensor Qαβ is calculated and diagonalised, whereas its biggest eigenvalue
is averaged together with the ones calculated for previous time steps.
As expected, at low temperatures the order parameter is maximum, as long as the particles
are oriented better along one axis. As the temperature increases, the system undergoes a phase
transition and the order parameter decreases with a steep slope, rapidly becoming a totally
disordered system. However, order parameter’s value never reaches the zero. This is due to
periodical boundary conditions we have chosen, which do not allow the system to be fully
disorderd. That is why the reason that makes this parameter to have very small values once
the "almost" conventional isotropic phase is reached. In other words, the non-zero value of S
is a ﬁnite system size effect.
6 conclusions and outlook
We have succesfully characterised the properties of the Lebwohl-Lasher model for LC’s. The
transition temperature obtained, is found to be within the range suggested by the literature, in
good agreement with the results in reference [22]. The only slight difference is understood and
due to a smaller system size used that, combined with the usual periodic boundary conditions,
have the effect of shifting the nematic-isotropic transition temperature to lower temperatures
as the size of the lattice increases.
According to the results obtained we can say that our code works correctly, providing quanti-
tatively accurate data for the heat capacity and order parameter during the entire temperature.
More simulations would be needed in order to obtain better results for the heat capacity, and
larger systems would increase the overall accuracy of the various properties.

conclusions and outlook 23
Figure 10: Nematic-Isotropic phase transition from the point of view of the order parameter. Black
points are the average order parameter we obtained whereas red ones were taken from the
literature [22]. The slight difference observed is due to the different size of the lattice. Image
obtained with free software gnuplot.
All of the work done here is a ﬁrst approach and a required initial step towards the general-
isation of the method for more complex systems such as those formed by helix-like molecules.
The main ideas were explained above and their actual implementation is left for future work.

references 24
references
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from the history of liquid crystals. London: Taylor & Francis
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[21] Classnotes of the subject "Computación Avanzada", by Prof. Alejandro Gutiérrez during
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