# Density-functional theory for clustering of two-dimensional hard particle
fluids

Yuri Martínez-Ratón<EMAIL_ADDRESS>Grupo Interdisciplinar de Sistemas
Complejos (GISC), Departamento de Matemáticas, Escuela Politécnica Superior,
Universidad Carlos III de Madrid, Avenida de la Universidad 30, E-28911,
Leganés, Madrid, Spain  Enrique Velasco<EMAIL_ADDRESS>Departamento
de Física Teórica de la Materia Condensada, Instituto de Física de la Materia
Condensada (IFIMAC) and Instituto de Ciencia de Materiales Nicolás Cabrera,
Universidad Autónoma de Madrid, E-28049, Madrid, Spain

###### Abstract

Fluids made of two-dimensional hard particles with polygonal shapes may
stabilize symmetries which do not result directly from the particle shape.
This is due to the formation of clusters in the fluid. Entropy alone can drive
these effects, which represent a challenge for standard theories. In this
article we present a general density-functional theory for clustering effects
in fluids of hard particles in two dimensions. The theory combines a free-
energy functional of the angular distribution function with an association
energy term which qualitatively reproduces the clustering tendencies of the
particles found in Monte Carlo simulations. Application is made to a fluid of
hard right-angled triangles.

Liquid crystals, hard right-angles triangles, virial coefficients, density
functional theory, association theory.

##  I Introduction

Hard particle models continue to attract great interest in the theory of phase
transitions and colloidal physics review . The hard-sphere model in two and
three dimensions demonstrated for the first time the existence of a crystal
phase due solely to purely repulsive interactions HS_crystal1 ; HS_crystal2 ,
with important implications in colloidal science Poon . Anisotropic models
have also proved useful to explain mesogenic formation involving both
translational and orientational order Onsager ; Parsons ; Lee ; Frenkel1 ;
Frenkel2 ; Frenkel3 ; Frenkel4 ; Jack . The appearance of order in these
systems is obviously due to entropic interactions driving the system to
ordered or partially ordered configurations where the free volume available to
the particles is maximized. The role of entropy in the formation of colloidal
crystals has recently been emphasised and explained in terms of the concept of
entropic bonding Glotzer0 .

In systems of identical particles, the extra volume available to each particle
is usually shared equally among all the particles. However, in some systems
this extra volume is not so trivially split GAN . The configurational space is
reorganized into internal coordinates, associated to the formation of
clusters, which internally loose entropy, while the gain in entropy of
collective coordinates associated to the cluster centers of mass more than
compensates the former entropy loss. The net effect is the formation of
differently shaped superparticles which can favour the stability of high-order
symmetries, not evident from the particle shape alone, in the fluid.

This effect may be especially important in two-dimensional (2D) systems made
of polygonally-shaped particles. Some particle shapes seem to be more prone to
exhibit these entropic clustering effects, which have a huge impact on the
stability of highly-symmetric fluid phases. The compartmentalization of
entropy between low- and high-entropy subsystems has been invoked to design
host-guest chemical systems with important consequences for nanoparticle
storage Glotzer1 ; Glotzer2  in colloidal crystals. A general theory to
describe storage and encapsulation in equilibrium mixtures of tailored hard
particles has been proposed recently Wittmann . However, there is evidence
that the entropy-driven self-assembly of identical hard particles into
clusters is also at work in 2D one-component liquid-crystal, as opposed to
crystal phases or mixtures MIG1 ; Proc .

Many-body effects leading to self-assembly in 2D oriented fluids represent a
challenge for standard theories for liquid crystals MAR2 . These theories, the
most successfully of which is Scaled-Particle Theory (SPT), are based on the
Onsager theory Onsager , whose central ingredient is the volume (area)
excluded to a particle by another particle. This is totally equivalent to
considering a virial expansion truncated at second order, i.e. at two-body
correlations. However, the formation of clusters of particles necessarily
involve higher-order correlations. Some proposals of extended theories that
incorporate these correlations systematically, in the context of 2D oriented
fluids, have been made VEL1 . The effect of third- and fourth-order
correlations on the stability of the tetratic (T) phase in fluids of hard
rectangles and hard right-angled triangles have been analysed, with relatively
disappointing results in the latter case MAR2 . In particular, Monte Carlo
(MC) simulations reveal a T phase with strong octatic (eightfold or 8-atic, O)
correlations for right-angled triangles GAN ; MAR3 , but the extended theories
only predict a uniaxial nematic (N) in the whole density range. Even a theory
containing the exact fourth virial coefficient, which would seem at first
glance as an essential ingredient to capture the formation of square dimers
and tetramers (which are behind the stability of O fluctuations in the nematic
director), fails completely in this respect VEL1  It must be said that the
difference between the isotropic (I)-N and I-O bifurcation packing fractions
tend to converge as more virial coefficients are included. Also, the
corresponding theories involve calculations with a huge numerical burden.

Clearly a density-functional theory constructed from few-particle correlations
does not have the correct structure to describe clustering and the relative
equilibrium distributions of even the most basic particle arrangements. A
first step to explore how the new symmetries can be generated considers these
clusters as fixed entities or superparticles. This view leads to an extreme
model, where clusters may orient in such a way as to give rise to high-order
symmetries in the final monomer orientational distribution function, e.g. the
orientational eightfold symmetry of the O phase in the case of right-angled
triangles MAR3 . However, the fractions of the clusters have to be imposed by
hand and are not obtained self-consistently.

A number of particle models are clear candidates for exhibiting strong
clustering effects in their liquid-crystalline and crystal phases. Both
regular and nonregular polygonal shapes are included in the list. As a general
criterion, particles that can be associated into clusters with shapes that
tessellate the area can be considered to be potential candidates. The
clustering tendencies of some of these particles have already been explored
previously. Fig. 1 shows some candidate hard polygonal-particle models for
which one would expect to find strong clustering: rectangles, equilateral
triangles, isosceles triangles with an opening angle of $120^{\circ}$, and
right-angled triangles. Several single particles can be combined into dimers,
trimers, tetramers and larger clusters of different shapes. The correct
criteria to a priori identify the important cluster shapes are not evident;
however, a reasonable criterion should include an element of roundness and
convexity that permits some rotational and translational freedom of the
clusters.

Some of these superparticles may be particularly stable in the equilibrium
fluid and give a dominant contribution to the structure and free energy of the
fluid, to the extent that the global symmetry of the bulk phase may be
determined by the abundance of these clusters with particular symmetries. The
global symmetry may be very different from the one that would directly follow
from the shape of the particle. Particular clusters will be dominant at high
densities and may even characterise uniquely the crystal phase. For example,
rectangles with an aspect ratio of two will form square dimers, which
crystallise into a high-entropy crystal, presumably formed by dimers oriented
randomly. Likewise, equilateral triangles will tend to form hexagonal crystals
made of hexamers with hexagonal shape GAN . Finally, right-angled triangles
form particularly stable tetramers with square shape, which crystallise into a
square lattice GAN .

Figure 1:  Candidate hard polygonal particles to be described by the present
theory for entropic clustering. On the left a single particle is depicted for
each model. (a) Rectangles form rectangular clusters by joining particles side
by side. For a given aspect ratio the most abundant cluster has a shape close
to a square, which favours the T phase. (b) Equilateral triangles may form a
variety of shapes, the most abundant ones being dimers and hexagons made of
six particles. (c) Isosceles triangles with an opening angle of $120^{\circ}$
may form stable rhomboidal clusters made of two particles and equilateral
triangular and trapezoidal clusters made of three particles. (d) Right-angles
triangles form large variety of dimers, including two chiral rhombuses, and a
tetramer with square symmetry. Clusters within a rectangular box are not
intended to be included in the theory in a first implementation.

Some of these systems have already been explored, either by theory, simulation
or experiment. Hard rectangles of aspect ratio $\kappa$ have been studied
using the excluded-volume-based SPT Schlacken ; Effect  and higher-order
virial theories Third , as well as by simulation Woj ; Donev  and experiments
on colloidal particles Chaikin  and on vibrated granular monolayers MIG1 . All
of these studies have shown a tendency for the formation of clusters of
approximately square shape containing a number of particles close to $\kappa$.
These clusters orient at right angles and enhance the stability of the
fourfold T (or 4-atic) phase over the I or disordered phase MAR1 . Rectangular
clusters of larger size, involving a larger number of parallel rectangles, are
also easily observed in simulations and experiments. The size distribution of
these clusters decays exponentially with their size, and extended SPT theories
that consider these clusters to be different species have shown the square
cluster to have the dominant contribution in the free energy MAR3 . These
clusters are the seeds for the formation of large layered regions with smectic
symmetry Narayan ; MIG1 .

Hard equilateral and right-angled triangles have been investigated by MC
simulation by Gantapara et al. GAN . These particles not only exhibit strong
clustering in the crystal phase, but also chiral features: particles associate
into clusters that may rotate in one of two possible senses to give rise to
chiral configurations. For example, in the crystal phase, four right-angles
triangles form tightly-bound square clusters that behave as single
superparticles and collectively rotate in one direction to reduce their
entropy, which compensates the entropy increase of the cluster. Since two
rotations are possible, this is a clear example of chiral phases that coexist
separated by interfaces GAN  in an otherwise achiral global phase.

The clustering properties of the fluid phases of right-angles triangles have
also been theoretically studied in detail using extended SPT and simulations
MAR2 ; VEL1 . A 4-atic phase with strong 8-atic correlations predicted by MC
simulations seems to be the most likely candidate for oriented fluid in this
system. The strong eightfold correlations can only be explained as a result of
the presence of various types of square dimers, square tetramers, and
rhomboidal and triangular dimers in the fluid, causing predominant fourfold
symmetries along with secondary eightfold symmetries (see Fig. 2 where
fractions of the different types of clusters, obtained from MC simulation, are
represented as a function of packing fraction MAR3 ). These symmetries are not
shared by the particle shape and result from the formation of clusters with
symmetries different from that of the individual particles.

Many other possibilities exist. One involves isosceles triangles with an
aperture angle of 120∘. Three such triangles form an equilateral triangle. At
high density these trimers will stabilise a crystal with sixfold symmetry.
Fig. 3 shows a snapshot of a typical configuration of such a crystal, obtained
from our own MC simulations (not previously published). Note that the
partitioning of configurational space is revealed by local rotations of
clusters, which leave an extra area in the interstitial spaces that increase
the collective free area of clusters. At intermediate fluid densities a
6-atic, or triatic, phase can be stabilised. It is not unlikely that these
clusters involving three particles cannot be excited at lower densities, and
that a N or 2-atic phase be the stable mesophase. Clearly subtle entropic
effects, giving rise to unexpected phase diagrams, may be at work.

Figure 2:  Fractions of clusters as a function of packing fraction in the
system of right-angled triangles as obtained by MC simulation (adapted from
MAR3 ). Clusters are represented by different symbols (see key). Open symbols
are expansion runs, while filled symbols correspond to compression runs.
Arrows in the $\eta$ axis indicate the coexistence densities of I ($0.733$)
and T ($0.782$), and the transition density to the crystal ($0.87$), according
to Ref. GAN .  Figure 3: Particle configurations of the crystalline phase of
isosceles triangles with an aperture angle of 120∘. The equilateral triangular
clusters formed by three triangles and with any neighboring clusters having
opposite orientations ($180^{\circ}$ rotations) conform the crystal phase.
Clusters rotate by a small angle, giving rise to locally chiral regions.

In this article we propose and explore a general theory, the Entropic
Clustering Theory (ECT), that attempts to describe liquid-crystal phases with
strong clustering tendencies in systems with arbitrary polygon-shaped
particles. The theory combines a density-functional theoretical setting with
standard approaches for associating fluids, and is inspired by previous
applications to dipolar, polymer, hard-sphere, and patchy-fluid systems
vanRoij ; Segura ; Cuesta ; Tavares ; delasHeras ; Torres . Orientational
order is described through the usual angular distribution function, and a
free-energy functional is written which includes two-body excluded-area
effects. In addition, entropic tendencies of particles to cluster are mimicked
by an association energy. Such an energy incorporates a penalty barrier
against cluster formation, together with an effective coupling with the order
parameters. As shown below, these ingredients are sufficient to describe the
clustering tendencies found in MC simulations. Sizes and shapes of clusters
are defined a priori using common sense, while their statistical abundances in
equilibrium result from a chemical-equilibrium condition. We combine these
ideas with density-functional theory to describe both clustering and
orientational ordering in 2D fluids. As an example, we apply the theory to a
fluid made of right-angled triangles. To complete the analysis, a cell theory
(CeT) is used to approximate the equation of state (EOS) of the crystal phase,
made of tetramers located at the sites of a simple square lattice. As will be
seen, the resulting theory accurately reproduces not only the EOS of the
crystal phase, but also that of the T phase, when compared with simulation.

The remainder of the article is organised as follows. In Section II we
describe the theoretical framework, while results from the theory are shown in
Section III. Some conclusion and prospects for future applications of the
theory to other particle shapes are drawn in Section IV.

##  II Theory

###  II.1 Entropic clustering theory

Despite the fact that the theory is general, we assume for definiteness that
the fluid is made of hard right-angled triangular particles, Fig. 1(d). Our
model consists of a mixture of six species which, apart from the triangular
monomers, are formed by assembling monomeric units into five different
clusters: square dimers, resulting by joining two HRT along their hypotenuses;
triangular dimers, obtained by joining two HRT along their catets, so that the
right angles of both triangles are in contact; two different rhomboidal
dimers, one obtained from the other by specular reflection with respect to the
common contact edge, giving rise to two different enantiomers (these rhomboids
are formed by sticking two HRT through their catets but this time with the
right angles being in contact with the 45∘-angles); and finally square
tetramers, formed by joining four triangles together by their catets with
their right-angle vertexes converging at the center of a perfect square. Note
that there are two more clusters (also enantiomers) composed by three monomers
(enclosed by a rectangular box in Fig. 1(d)) which, for the sake of
simplicity, are not taken into account in our model. It is hoped that their
effects do not significantly change the phase behavior of the system.

Once again we stress the fact that our model is completely general, and valid
for any multicomponent two-dimensional mixture of hard particles including
monomers and a certain number of clusters obtained by self-assembling of
monomers. The criterion to select the geometry of clusters is mainly dictated
by their convexity. Certainly convex clusters will retain a much higher local
entropy than nonconvex clusters, which may interlock with neighbours and
reduce the entropy. However, this criterion also constitutes a convenient
constraint which facilitates the calculation of excluded areas. Also, cluster
fractions are variables in our model and it is convenient not to use too many
species in order to make the calculations feasible. The central quantities
that characterize the mixture are the density angular profiles
$\rho_{i}(\phi)$ ($i=1,\dots,6$), with $\phi$ the angle between the particle
axis and the polar axis of a fixed reference frame.

Clusters interact through hard-core interactions and the corresponding
interaction free-energy will be approximated by that obtained from scaled
particle theory (SPT) extended to a multicomponent mixture. The interaction or
excess part of the free-energy density in reduced thermal units is therefore:

$\displaystyle\Phi_{\rm int}[\\{\rho_{i}\\}]$
$\displaystyle\equiv\frac{\beta{\cal F}_{\rm in}[\\{\rho_{i}\\}]}{A}$ (1)
$\displaystyle=-\rho\log(1-\eta)+\frac{\sum_{i,j}\langle\langle A_{ij}^{(\rm
spt)}\rangle\rangle}{1-\eta},$

where ${\cal F}_{\rm int}[\\{\rho_{i}\\}]$ is the interaction or excess part
of the Helmholtz free-energy of the mixture and $A$ is the total area of the
system. As usual $\beta$ is the Boltzmann factor. The quantity

$\displaystyle\eta\equiv\sum_{i}\rho_{i}a_{i}=\rho_{0}a_{0}$ (2)

is the total packing fraction of the mixture (packing fractions of monomers),
and it is taken as a fixed number, equal to the total number density of
monomers, $\rho_{0}$, multiplied by its area $a_{0}$. The quantity
$\rho_{i}\equiv\int_{0}^{2\pi}d\phi\rho_{i}(\phi)$ is the number density of
the cluster $i$, while $a_{i}$ is the area, equal to the product of the number
of triangles forming the $i$-cluster, $n_{i}$, and the area of the monomers
$a_{0}$: $a_{i}=n_{i}a_{0}$. The total number density of clusters is just the
sum $\rho=\sum_{i}\rho_{i}$. Eqn. (1) defines the double angular average of
the spt-area $A_{ij}^{(\rm spt)}(\phi_{1}-\phi_{2})$ with respect to the
angular profiles $\rho_{i}(\phi_{1})$ and $\rho_{j}(\phi_{2})$:

$\displaystyle\langle\langle A_{ij}^{(\rm
spt)}\rangle\rangle=\int_{0}^{2\pi}d\phi_{1}\int_{0}^{2\pi}d\phi_{2}\rho_{i}(\phi_{1})\rho_{j}(\phi_{2})$
$\displaystyle\times A_{ij}^{(\rm spt)}(\phi_{1}-\phi_{2}).$ (3)

This area in turn is related with the excluded area as

$\displaystyle A_{ij}^{(\rm spt)}(\phi)=\frac{1}{2}\left(A_{ij}^{(\rm
excl)}(\phi)-a_{i}-a_{j}\right).$ (4)

We should note that if $N$ is the total number of components of the mixture
(the total number of clusters plus one) we need to calculate a total of
$N(N+1)/2$ different excluded areas. In this case the number is 21. However
there are some symmetries dictated by the interior angles of the polygonal
clusters which allow to find some relations between the excluded areas.

The ideal part of the multicomponent mixture is

$\displaystyle\Phi_{\rm
id}[\\{\rho_{i}\\}]\equiv\sum_{i}\int_{0}^{2\pi}d\phi\rho_{i}(\phi)\left[\log(\rho_{i}(\phi))-1\right],$
(5)

where the thermal areas have been dropped. Finally, an important part of the
free-energy functional is a term involved in the balance of entropy when
$n_{i}$ monomers assemble into a cluster of index $i$. We call this part the
association energy and make it to explicitly depend on the T order parameters
$Q_{4}^{(i)}$:

$\displaystyle\Phi_{\rm
as}[\\{\rho_{i}\\}]\equiv\sum_{i}\rho_{i}\left[\epsilon_{i}^{(1)}+\epsilon_{i}^{(2)}\left(Q_{4}^{(i)}\right)^{2}\right].$
(6)

This dependence is motivated in our present case by the favoured
clusterization into tetramers when the T order parameter is high. For general
particle symmetries the term enclosed by square brackets can be replaced by a
general function $v_{\rm as}\left(\eta,\\{Q_{2n}^{(i)}\\}\right)$, which could
also depend on the packing fraction and on the rest of order parameters. The
order parameters of the $i$th species are in turn defined as the angular
average of $\cos(2n\phi)$ with respect to the angular distribution function of
the $i$th species,

$\displaystyle
h_{i}(\phi)\equiv\frac{\rho_{i}(\phi)}{\rho_{i}}\Rightarrow\int_{0}^{2\pi}d\phi
h_{i}(\phi)=1,$ (7)

resulting in

$\displaystyle Q_{2n}^{(i)}=\int_{0}^{2\pi}d\phi h_{i}(\phi)\cos(2n\phi),$ (8)

where $n$ determines the symmetry of the phase: $n=2$ and $4$ for N and T
symmetries, respectively.

The equilibrium number-density angular profiles $\rho_{i}(\phi)$ are obtained
from the ECT presented here by minimizing the constrained total free-energy
density functional,

$\displaystyle\Phi(\\{\rho_{i}\\})=\Phi_{\rm id}(\\{\rho_{i}\\})+\Phi_{\rm
int}(\\{\rho_{i}\\})+\Phi_{\rm as}(\\{\rho_{i}\\})$
$\displaystyle-\lambda\left(\sum_{i}\rho_{i}n_{i}-\rho_{0}\right),$ (9)

(with $\lambda$ the Lagrange multiplier) with respect to $\rho_{i}(\phi)$:
$\displaystyle{\frac{\delta\Phi(\\{\rho_{i}\\})}{\delta\rho_{i}(\phi)}=0}$.
This minimization results in

$\displaystyle\rho_{i}(\phi)=\rho_{i}h_{i}(\phi)=z^{n_{i}}e^{-c_{1}^{(i)}(\phi)},\quad
z\equiv e^{\lambda},$ (10)

where we have defined

$\displaystyle c_{1}^{(i)}(\phi)=-\log(1-\eta)+\frac{{\cal
S}_{i}(\phi)}{1-\eta}+\epsilon_{i}^{(1)}$
$\displaystyle+\epsilon_{i}^{(2)}\left[2Q_{4}^{(i)}\cos(4\phi)-\left(Q_{4}^{(i)}\right)^{2}\right],$
(11) $\displaystyle{\cal
S}_{i}(\phi)=2\sum_{j}\int_{0}^{2\pi}d\phi^{\prime}\rho_{j}(\phi^{\prime})A_{ij}^{(\rm
spt)}(\phi-\phi^{\prime}).$ (12)

Defining also the quantities

$\displaystyle T_{i}\equiv\int_{0}^{2\pi}d\phi e^{-c_{1}^{(i)}(\phi)},$ (13)

we obtain

$\displaystyle\rho_{i}=z^{n_{i}}T_{i},$ (14)

and from the conservation number of monomers distributed among the clusters,
we have

$\displaystyle\rho_{0}=\sum_{i}n_{i}T_{i}z^{n_{i}}.$ (15)

In our case this is a fourth-order polynomial with respect to $z$ which is
solved at each step of the numerical procedure to find the functions
$\rho_{i}(\phi)$.

To perform the numerical minimization the angular distribution functions
$h_{i}(\phi)$ are Fourier expanded,

$\displaystyle h_{i}(\phi)=\frac{1}{2\pi}\left[1+\sum_{k\geq
1}h_{k}^{(i)}\cos(2k\phi)\right],$ (16)

with $\\{h_{k}^{(i)}\\}$ the set of Fourier amplitudes. From Eqn. (3) we
obtain

$\displaystyle\langle\langle A_{ij}^{(\rm
spt)}\rangle\rangle=\rho_{i}\rho_{j}\left[A_{ij,0}^{(\rm
spt)}+\frac{1}{2}\sum_{k\geq 1}A_{ij,k}^{(\rm
spt)}h_{i}^{(k)}h_{j}^{(k)}\right],$ (17)

and from Eqn. (12),

$\displaystyle{\cal S}_{i}(\phi)=2\sum_{j}\rho_{j}\left[A_{ij,0}^{(\rm
spt)}+\sum_{k\geq 1}A_{ij,k}^{(\rm spt)}h_{k}^{(j)}\cos(2k\phi)\right],$ (18)

where we have defined the Fourier components of the spt-areas as

$\displaystyle A_{ij,k}^{(\rm
spt)}\equiv\frac{1}{2\pi}\int_{0}^{2\pi}d\phi\cos(2k\phi)A_{ij}^{(\rm
spt)}(\phi).$ (19)

Inserting Eqn. (16) into Eqn. (10), multiplying by $\cos(2k\phi)$ and
integrating over $\phi$, we obtain

$\displaystyle h_{k}^{(i)}=\frac{2\tilde{T}_{i}^{(k)}}{\tilde{T}_{i}^{(0)}},\
\tilde{T}_{i}^{(k)}=\int_{0}^{2\pi}d\phi\cos(2k\phi)e^{-S_{i}(\phi)},$ (20)
$\displaystyle S_{i}(\phi)=\frac{{\cal
S}_{i}(\phi)}{1-\eta}+2\epsilon_{i}^{(2)}Q_{4}^{(i)}\cos(4\phi).$ (21)

The unknowns are $z$, $\\{\rho_{i}\\}$, and $\\{h_{k}^{(i)}\\}$. Taking into
account that $T_{i}$ and $\tilde{T}_{i}$ depend on the latter magnitudes, we
use Picard iteration to solve the set of Eqns. (14) and (20). The quartic
polynomial (15) is then solved for $z$ using the Newton-Raphson method, with a
fixed value of $\rho_{0}$. In this way the equilibrium values of all variables
can be found. We have used a total of 40 amplitudes in the Fourier expansion
of each $h_{i}(\phi)$. Gaussian quadrature is used to calculate the angular
integrals.

From its definition in terms of the total free-energy density $\Phi$,
$\displaystyle{\beta p=\eta\frac{\partial\Phi}{\partial\eta}-\Phi}$, the
pressure is calculated as

$\displaystyle\beta p=\frac{\rho}{1-\eta}+\frac{\sum_{i,j}\langle\langle
A_{ij}^{(\rm spt)}\rangle\rangle}{(1-\eta)^{2}}.$ (22)

The equation of state (EOS) in reduced dimensionless units, $p^{*}\equiv\beta
pa_{0}$, will be plotted in Sec. III for a particular set of model parameters
to compare the results from our model with MC simulations.

Figure 4: Sketch of right triangles forming a perfect T crystal. The sizes of
triangles and the parameter $\Delta$ characterizing the crystal are indicated,
together with an effective square tetramer of size $\sigma=\sqrt{2}l+\Delta$
(the lattice parameter). The free area of a particular triangle is obtained by
joining the three centers of mass of the triangle when in contact with all
pairs formed by its three neighbors, labeled as 1, 2 and 3.

###  II.2 Cell theory

A cell theory can be used to describe the free energy of the T crystal phase.
Fig. 4 shows the particle configurations of a perfectly orientationally
ordered T crystal of right triangles characterized by the parameter $\Delta$
(distance between vertexes of neighbor triangles along $x$ and
$y$-directions). It is easy to check that the free area for the center of mass
of one triangle constrained to move inside the cavity formed by their
neighbors also has a right triangular geometry, with base and height equal to
$2\Delta$ and $\Delta$, respectively. The free area is then $A_{\rm
free}=\Delta^{2}$. Taking the unit cell as a square with lattice parameter
$\sigma=\sqrt{2}l+\Delta$ ($l$ being the common length of the two identical
sides), inside which four triangles are found, the packing fraction of the
crystal is obtained as $\eta=4a_{0}/\sigma^{2}$, which gives

$\displaystyle\eta=\left(1+\frac{\Delta}{\sqrt{2}l}\right)^{-2}.$ (23)

The free energy per particle of the CeT approximation, in reduced thermal
units, is $\varphi=-\log\left(A_{\rm free}/\Lambda^{2}\right)$ (with $\Lambda$
the thermal length). Expressing $A_{\rm free}$ as a function of $\eta$,

$\varphi=-2\log\left[\frac{\sqrt{2}{l}}{\Lambda}\left(\eta^{-1/2}-1\right)\right],$
(24)

The pressure is obtained by differentiating $\varphi$ with respect to $\eta$.
In reduced units,

$p^{*}=\eta^{2}\frac{\partial\varphi}{\partial\eta}=\frac{f_{s}\eta}{1-\sqrt{\eta}},$
(25)

where $f_{s}=1$. This expression is identical to that of a fluid parallel hard
squares in a simple square lattice. This is not simply a coincidence, since
the 2D-periodic extension (with period $\sigma$ along $x$ and $y$) of square
tetramers (shown in Fig. 4 with solid lines) is equivalent to a simple square
lattice of parallel squares.

The CeT expression (25) explicitly assumes that the orientation of the
particles is frozen. In fact this is not the case, the degree of orientation
being described quantitatively by a set of order parameters. An extension of
the CeT approximation consists of allowing for the central particle to orient
along different directions, still keeping the orientation of its neighbors
fixed. The calculation can be done easily using MC integration. In turn, two
situations can be implemented for the orientations: equal probability, or
orientations weighted by the equilibrium distribution obtained from
simulation. These two limits provide corrections to Eqn. (25) as weakly
density-dependent amplitudes $f_{s}$ that range from $\sim 1.5$ in the equal-
probability case to $\sim 1.2$ when the equilibrium distribution is used to
sample the orientations. In the following we set $f_{s}=1.5$, in the
understanding that this is a phenomenological correction (we must note that
the true structure of the tetratic crystal is chiral and involves local
rotations of the triangles, a feature that is not incorporated in the CeT
approximation).

###  II.3 Distribution functions and order parameters

Once the equilibrium angular distribution functions $\\{h_{i}(\phi)\\}$ for
all clusters are found, the corresponding equilibrium angular functions of
monomers (free ones and those that making up different clusters) can be
calculated. We enumerate clusters as follows: cluster 1 corresponds to free
monomers, and their contribution to the total angular function is
$\displaystyle{h_{1}^{(m)}(\phi)=\frac{\rho_{1}}{\rho_{0}}h_{1}(\phi)}$. We
should remember that $h_{1}(\phi)$ quantifies the probability density for the
triangle axis (pointing from the base to the $90^{\circ}$-vertex along the
height) to be oriented with respect to some fixed reference axis at an angle
$\phi$. Note that $h_{1}(\phi)$ contains a factor $\rho_{1}/\rho_{0}$, the
fraction of monomers. The second cluster is the triangular dimer, made from
two monomers, each pointing at angles $3\pi/4$ and $-3\pi/4$ with respect to
the dimer triangular axis. Then
$\displaystyle{h_{2}^{(m)}(\phi)=\frac{\rho_{2}}{\rho_{0}}\left(h_{2}(\phi+3\pi/4)+h_{2}(\phi-3\pi/4)\right)}$.
The third cluster corresponds to one of the rhomboids, made again of two
monomers oriented at angles $\pi/4$ and $-3\pi/4$ with respect to the
rhomboidal axis (parallel to the contact edge). Then
$\displaystyle{h_{3}^{(m)}(\phi)=\frac{\rho_{3}}{\rho_{0}}\left(h_{3}(\phi+\pi/4)+h_{3}(\phi-3\pi/4)\right)}$.
The other rhomboidal cluster is the mirror reflection of the latter, so we
have
$\displaystyle{h_{4}^{(m)}(\phi)=\frac{\rho_{4}}{\rho_{0}}\left(h_{4}(\phi-\pi/4)+h_{4}(\phi+3\pi/4)\right)}$.
The square dimer (the 5th species) if made of two monomers with axes parallel
and antiparallel to any of the diagonal of the square, so we have
$\displaystyle{h_{5}^{(m)}=\frac{\rho_{5}}{\rho_{0}}\left(h_{5}(\phi)+h_{5}(\phi+\pi)\right)}$.
Finally, we have the square tetramers (the 6th species), made of four monomers
at angles $\pm\pi/4$ and $\pm 3\pi/4$ with respect to the square diagonals,
and consequently
$\displaystyle{h_{6}^{(m)}(\phi)=\frac{\rho_{6}}{\rho_{0}}\sum_{k=\pm
1}\left(h_{6}(\phi+k\pi/4)+h_{6}(\phi+3k\pi/4)\right)}$. The total angular
function of monomers is simply the sum

$\displaystyle h(\phi)=\sum_{i=1}^{6}h_{i}^{(m)}(\phi)$ (26)

From this, the order parameters of all the monomeric units can be calculated:

$\displaystyle Q_{2n}=\int_{0}^{2\pi}d\phi\cos(2n\phi)h(\phi),\quad n=2,4.$
(27)

This angular function and the order parameters represent the degree of
ordering of the HRT fluid and will be shown later. We note that the T
directors can change from the original ones when the relative compositions of
clusters change due to the shift of the main peaks of $h(\phi)$ by $\pm\pi/4$.
This could change the sign of $Q_{4}$. In the following we will always plot
the absolute value $|Q_{4}|$ when necessary.

###  II.4 Choice of association energy

The values of the parameters that define the association energy (see Eqn. (6))
were chosen as follows. Note that for free monomers we always set
$\epsilon_{1}^{(1,2)}=0$. In some cases, we also set $\epsilon_{i}^{(2)}=0$
for any $i$, while we considered a repulsive association energy for
$\epsilon_{i}^{(1)}$, i.e. a penalty barrier that prevents the formation of
clusters:

$\displaystyle\epsilon^{(1)}_{2}=\epsilon_{3}^{(1)}=\epsilon_{4}^{(1)}=\epsilon>0,\hskip
8.5359pt\epsilon_{5}^{(1)}=\sqrt{2}\epsilon,\quad\epsilon_{6}^{(1)}=4\epsilon.$
(28)

Therefore, for all dimers joined by the catets this penalty is the same, while
for the square dimer joined by the triangle hypotenuse the penalty is a factor
of $\sqrt{2}$ larger. Finally the penalty for the square tetramer, with four
catet contacts, is 4 times larger. In summary, the penalty energy is taken to
be proportional to the total contact length of the particles. The coupling
with the T order parameter is taken in some cases by selecting

$\displaystyle\epsilon_{i}^{(2)}=\delta<0$ (29)

for any $i\neq 1$, which means that the T ordering enhances the formation of
all clusters in an equivalent way for any $i\neq 1$. In principle a more
meaningful choice is to make $\epsilon_{i}^{(2)}$ to depend on $i$ because
square dimers and tetramers are certainly more abundant than the rest of
clusters in the O and T phases. However we decided for simplicity to use a
single coupling parameter for all clusters.

As shown in Sec. III, excluded-area interactions alone predict the system to
be mainly composed of tetramers even at moderate packing fractions. This is
not seen in simulations, where tetramers appear to grow at high $\eta$. This
in turn can be understood in terms of excluded-area interactions: As we
consider tetramers to be perfect squares (monomers are in contact along their
edges), the total entropic interaction between all clusters promotes a huge
clusterization to the geometry with the largest loss in excluded area, which
is the square geometry made from four monomers. In simulations clusters can be
seen not to be perfect, and they persist during a finite lifetime. Thus,
excluded-area interactions alone overestimate the clusterization of the fluid,
something that can be prevented by selecting the values of
$\epsilon_{i}^{(1)}>0$ to be positive, i.e. creating a penalty barrier for the
clusterization which is higher in the case of tetramers. Also, simulations
show that clusterization is strongly promoted when the T orientational
ordering is high, and consequently the values of $\epsilon_{i}^{(2)}<0$ should
be negative. In principle the $\epsilon_{i}^{(j)}$ parameters should depend on
packing fraction $\eta$. However, for simplicity, they are considered to be
constants.

Finally, to measure the degree of fractionation of monomers among different
clusters, we use the area fraction of cluster $i$:

$\displaystyle\nu_{i}=\frac{\rho_{i}n_{i}}{\rho_{0}}.$ (30)

Figure 5:  (a) EOS obtained from SPT, ECT with $\delta=0$ and $\epsilon=0$ or
$5$, and MC simulations (data extracted from Ref. GAN ). The I branches are
shown by solid lines, while dashed lines represent the N (from SPT) or T (from
ECT) branches. Filled circles indicate the bifurcation points from the I to N
(SPT) and T (ECT) phases. Filled circles joined with a line indicate the I-T
first order transition from MC. The arrow indicates the location of the
T-crystal transition, as obtained from MC simulations GAN . (b) N (dashed) and
T (solid and dotted) order parameters of the monomeric units as obtained from
the SPT (dashed), ECT with $\epsilon=\delta=0$ (solid) and $\epsilon=5$,
$\delta=0$ (dotted).

##  III Results

Figure 6: Area fraction composition of monomers (m), square dimers (sd),
triangular dimers (td), rhomboidal dimers (rd) and square tetramers (st) of
the I phase (those of the T phase are indistinguishable form the I ones) as a
function of the monomer packing fraction obtained from ECT with $\epsilon=0$.
The vertical dotted line shows the I-T second order transition.

We first refer to Fig. 5(a), which shows the equation of state $p=p(\eta)$ as
obtained from SPT and from the new ECT theory, using two values for the
association energy. MC results GAN  are also included for comparison. As can
be seen from the figure, the SPT greatly overestimates the pressure,
predicting a bifurcation from the I to the N phase at a density which is too
high. Fig. 5(b) shows the behavior of the uniaxial order parameter $Q_{2}$
(dashed line) along the N branch, starting at the I-N bifurcation point. Even
worse, the oriented fluid is predicted to be a uniaxial phase, while the MC
simulations predict a T (4-atic) phase. Note that the small difference between
the EOS from SPT (based on the second virial coefficients) and MC simulations
at $\eta\approx 0.3$ implies that virial coefficients higher than the second
are necessary to better describe the EOS even at these low densities.

The results change significantly by considering a mixture of clusters, still
without the association free-energy contribution. In this case the pressure is
severely underestimated at low density, but it seems to nicely converge to the
pressure of the T (either fluid or solid) phase at high density. The tetratic
order parameter $Q_{4}$ (measuring the ordering of the correct liquid-crystal
symmetry) is shown in Fig. 5(b) as a solid line. The predicted bifurcation is
still too high as compared with simulation. Finally, when an association term
is included, with $\epsilon_{i}^{(2)}=0$ but $\epsilon_{i}^{(1)}>0$ (so that
association into clusters is penalised), the pressure improves substantially
in the whole density interval, and the theory correctly identifies the T phase
as the equilibrium oriented fluid phase. However, the location of the
bifurcation still occurs at a high density, and in fact it gets worse with
respect to the previous theory (see dotted line in Fig. 5(b) which represents
the $Q_{4}$ order parameter).

Fig. 6 shows the fraction of clusters as a function of density for the mixture
theory. The vertical dotted line indicates the location of the bifurcation to
the T phase. The monomer fraction decreases with density, while the fraction
of dimers increases. Note that the two enantiomers of the rhombic dimer
clusters (Fig. 1) appear with equal probabilities, making the I phase achiral.
At high density the fraction of tetramers increases strongly, while the rest
of clusters appear in less and less proportions, so that at bifurcation the
system almost completely consists of tetramers. However, as shown in Fig. 2
MAR3 , the monomer association into clusters begins to be important at
densities much higher than shown in Fig. 6. The increased stabilisation of
clusters may explain the dramatic lowering of pressure with respect to
simulation. We will see below that the association term corrects this behavior
quite substantially.

Figure 7: (a): Area fraction compositions of monomers and different clusters
(as they are labeled) as a function of the monomer packing fraction obtained
from the ECT with $\epsilon=5$. With dashed lines (hardly visible) we show the
area fraction compositions of the T branch. The dotted vertical line indicates
the I-T second order transition. (b): Differences between the area fraction
compositions of the T and I phases of monomers and different clusters.

Fig. 7(a) shows the area fraction of clusters (solid lines) as a function of
packing fraction for the I phase, when the repulsive energy is set to
$\epsilon=5$. In this case the fraction of monomers stays at a large value up
to $\eta=0.4$, from which it begins to decrease. Fractions of the other
clusters then begin to increase and reach a maximum in the density interval
$0.7-0.8$, decreasing for larger densities. It is then that the fraction of
tetramers grows and finally saturates to unity for very large packing
fractions. We note that this behavior is qualitatively similar to that
observed in the simulations MAR3 , see Fig. 2. Also note that the fractions of
the two types of rhomboidal clusters are the same, so that the system remains
achiral.

In Fig. 7(b) the difference in cluster area fractions between the T and I
phases (i.e. T minus I) is plotted at the I-T bifurcation density and beyond.
The fraction of free monomers, triangular dimers and rhomboidal dimers is
lower in the T phase, as opposed to square dimers and tetramers which are much
more abundant (except for $\eta>0.917$). This is understandable since square
clusters tend to stabilize the T order (note that at high density the square
tetramers are the one ones present in the fluid).

Figure 8: EOS from the ECT (with $\epsilon=5$ and $\delta=-0.88$) for I (solid
line) and T (dashed line) phases, and from the CeT for the T phase (taking
$f_{s}=1.5$). The MC results are also shown (data extracted from Ref. GAN ).
The filled circle indicates the I-T bifurcation given by the ECT. Filled
circles joined by a line indicate the coexistence packing fraction values of
the I-T transition obtained from MC simulations. Open squares joined by a line
indicate the coexistence values as obtained from the I branch of the ECT and
the T-phase branch obtained from the CeT (with $f_{s}=1.5$). The arrow
indicates the T-crystal transition, as obtained from MC simulations GAN .

We now consider the coupling between the tetratic order parameter and the
association energy. This coupling promotes cluster formation, see Eqns. (6)
and (29). Keeping the penalty barrier as before ($\epsilon>0$), the I-T
bifurcation now occurs at lower densities, compared with the case $\delta=0$.
Fig. 8 shows the resulting EOS from the ECT with $\epsilon=5$ and
$\delta=-0.88$. The I-T second-order transition now takes place rather close
to the coexisting value of the I phase at the first-order I-T transition, as
obtained from MC simulations GAN . The EOS of the T crystal, obtained from the
CeT (with $f_{s}=1.5$), is also plotted in the figure. The agreement between
this EOS and the T liquid-crystal branch from MC simulations is remarkable. If
we take the I branch obtained from the ECT, and the T branch from the CeT of
the crystal phase, and calculate the I-T coexistence, the resulting
coexistence density values are very close to those from MC simulation. These
results highlight the important role of clustering and spatial correlations in
the quantitative prediction of the EOS of the T phase of HRT fluids.

Figure 9: Area fraction compositions from the ECT (with $\epsilon=5$ and
$\delta=-0.88$). I and T branches are shown by solid and dashed lines,
respectively. The vertical dotted line indicates the I-T second-order
transition. Figure 10: Tetratic ($Q_{4}$, solid line) and octatic ($Q_{8}$,
dashed line) order parameters of monomeric units plotted from the I-T
bifurcation point, as obtained from the ECT with $\epsilon=5$ and
$\delta=-0.88$. The inset shows the monomer distribution function for
$\eta=0.876$. Figure 11: Scaled free-energy density minus a straight line,
$\Phi^{*}a=\Phi a+46.989-65.462\eta$, as a function of the monomer packing
fraction for the I, N and T branches, obtained from the ECT with $\epsilon=15$
and $\delta=0$. The filled circle indicates the I-N bifurcation point, while
the N-N and N-T two-phase coexistences are indicated by the filled squares.

Fig. 9 shows the area fraction of clusters as a function of the packing
fraction for the same values as above, $\epsilon=5$, and $\delta=-0.88$. The
overall trend in similar to that obtained for the values $\epsilon=5$ and
$\delta=0$ (see Fig. 8), but with an important difference: slightly above the
I-T bifurcation, the fluid is now enriched in all dimers with approximately
equal proportions, while it is depleted in tetramers. The fraction of the
latter abruptly increases with $\eta$ from the bifurcation. From Fig. 9 it can
be seen that the T phase has a larger proportion of square dimers compared to
the I (metastable) phase (dashed vs. solid lines respectively). Rhomboidal and
triangular dimers follow the opposite trend (with the I phase being more
enriched in them). This behavior of the area fraction of monomers as a
function of $\eta$ in turn explains why the T order parameter of monomers
$Q_{4}$ has the peculiar behavior shown in Fig. 10. Note that the O order
parameter $Q_{8}$ is higher than $Q_{4}$ for some values of $\eta$, which is a
direct consequence of the presence of satellite peaks located at angles
$\pm\pi/4$ off the main peaks of the orientational distribution function of
monomers $h(\phi)$ (see inset). This feature is an indication of the
importance of O correlations in the system.

Figure 12: Area fraction compositions vs. packing fraction along the I (a) , N
(b) and T (c) branches from the ECT with $\epsilon=15$ and $\delta=0$. The
shaded regions indicate the regions of instability of the I (a), N (b) and T
(c) phases due to the second or first-order transitions between different
phases. The regions of stability of all phases are correspondingly labeled. In
panel (b) we show in the inset the area fraction compositions of the two
different rhomboidal dimers: at the left of the breaking-symmetry point they
are identical while at the right they separate, with one of them eventually
dominating as $\eta$ increases.

At this point the success of ECT to describe the phase behavior of HRT is
apparent: (i) It predicts a transition to the T liquid-crystal phase from the
I phase at a packing fraction similar to that found in MC simulations. (ii) At
sufficiently high densities, the T phase is mainly populated by tetramers, and
the ECT can be combined with the CeT to obtain the coexistence densities of
the I and T phases, respectively. (iii) Finally, the strong O orientational
correlations between particles found in simulations are also predicted from
our theory.

However, it is interesting to explore the model in regions of parameter space
other than the one which provides the best agreement with MC simulations. In
particular we are interested into the phase behavior for large values of the
penalty parameter $\epsilon$ and for $\delta=0$, i.e. when the formation of
tetramers is greatly suppressed at moderate densities. As discussed below,
chiral phases can be found in this regime.

Figure 13:  Monomer distribution function with quasi-O symmetry obtained from
the ECT with the values of $\epsilon=15$ and $\delta=0$ and fixing the monomer
packing fraction to $\eta=0.937$. Note that this quasi-O structure is
metastable with respect to the N phase. Figure 14: Phase diagram in the
$\epsilon$-$\eta$ plane from the ECT with $\delta=0$. The regions of stability
of the I, Ni and T are correspondingly labeled. With open circles we show the
calculated two-phase coexistences for a certain set of values of $\epsilon$.
The filled circle label the azeotropic point. With dashed lines we show the
I-N1 and I-T second order transitions. The two-phase coexistence regions are
shaded in grey.

As an example, Fig. 11 shows the free-energy density in reduced units as a
function of packing fraction for $\epsilon=15$. Different energy branches are
plotted, corresponding to the stable or metastable liquid-crystal phases
found. The I phase is stable up to the I-N1 bifurcation point. The $N_{1}$
phase, mainly populated by triangular monomers, is achiral (the proportion of
the two different rhomboidal enantiomers are exactly the same). The stability
of this uniaxial phase, predicted by the standard SPT for HRT, has already
been discussed in MAR2 ; VEL1 ; MAR . In this case the phase is stable due to
the extremely high proportion of free monomers up to densities $\eta\sim
0.825$ (see Fig. 12).

Interestingly, further increasing $\eta$ along the N branch, a symmetry
breaking point is reached at which the fraction of one of the species of
enantiomers increases at the expense of the other species, giving rise to a
chiral N2 phase (see Fig. 12). Following the same branch, the N2 phase becomes
populated by just one of the enantiomers at high densities, with marginal
fractions for the rest of clusters. Fig. 11 shows the clear first-order nature
of the N1-N2 transition.

Finally, at very large packing fractions, the N2 phase becomes unstable with
respect to the T phase of tetramers, with the N2-T transition being of first
order (see Figs. 11 and 12). An example of a monomer distribution function
with strong O correlations in the T phase is shown in Fig. 13, pertaining to
the fluid with $\eta=0.937$. Exactly at this point the T phase becomes
unstable with respect to the N2 phase (this example is shown because, as shown
in Fig. 12(c), one can design a procedure to obtain a function $h(\phi)$ with
this symmetry by combining square dimers and tetramers in similar proportions,
with the remaining clusters being marginal in the mixture).

Fig. 14 shows the phase diagram of the model in the $\epsilon^{-1}-\eta$
plane. Phase boundaries were obtained by calculating the free energies of all
phases involved. Second-order transitions are indicated by dashed lines, while
two-phase coexistence regions are represented by shaded regions. The phase
diagram shows that the first order I-N1 and N2-T transitions coalesce into an
azeotropic point, above which only a second-order I-T transition remains (this
is the phase behavior described before, which corresponds to the values of
$\epsilon$ for which good agreement with MC simulations is found).

##  IV Discussion and conclusions

In this work we have proposed a new theoretical framework (ECT), based on
density-functional theory, to describe the phase behavior of 2D hard
particles, for which standard versions of density-functional theory (such as
SPT) fail to predict even the symmetries of the stable liquid-crystal phases.

The main ingredients of the theory are: (1) Mapping of the one-component fluid
onto a multicomponent mixture of different species, consisting of monomers
(the original particles) and different clusters, each formed by aggregations
of monomeric units. Clusters are selected to be convex, and one of them should
conform with the symmetry of the crystal phase (for example tetramers in the
case of HRT). Clusters are obtained by perfectly joining the edges of original
particles in different conformations. (ii) Clusters interact through hard-core
interactions, so that all the excluded areas between different species are
required. These interactions are approximated through the excess part of the
free-energy given from SPT. Of course clusters observed in MC simulations are
not perfectly shaped, since they form and break with a certain persistence
time, which should depend on the packing fraction. As a consequence, our model
overestimates the formation of those clusters which tend to minimize the total
excluded areas (tetramers in case of HRT). Thus a new term, the association
energy, is added to the free energy. This term renormalizes particle
interactions to better approximate the real situation. In practice the
association energy penalizes the formation of clusters. Also, MC simulations
indicate that a coupling between orientational ordering and association energy
should exist: for example, T ordering favors the formation of square dimers
and tetramers. (iii) An important ingredient of the theory is the use of the
conservation of the total number of monomers as a constraint. This is taken
through a Lagrange multiplier in the free energy. As usual the ideal free
energy is taken as that of the multicomponent mixture of species. Minimizing
the total free energy, the equilibrium cluster densities and their
orientational distribution functions, $h_{i}(\phi)$, are obtained, and from
them all the thermodynamic quantities of the mixture can be derived. (iv)
Finally, from the functions $h_{i}(\phi)$ the corresponding function for
monomers (the free ones and those forming clusters), $h(\phi)$, can be
obtained by simple geometric relations (these depend on the relative monomer
orientations with respect to the main axis of the cluster). In this way the
orientational ordering of the original fluid can be quantified. An example was
given for the case of HRT.

Following this procedure, we applied our model to the case of HRT fluid,
analyzed the phase behavior, and compared the results with MC simulations. The
overall agreement is rewarding: (i) The EOS obtained from the ECT greatly
improves upon the standard SPT treatment. (ii) The symmetry of the stable
liquid-crystal phase (the T one) is now correctly predicted. (iii) The packing
fractions of the I-T phase transition decrease and can be made to be close to
those of simulations. (iv) Using CeT to describe the T branch and ECT for the
I phase we obtained a first-order I-T transition with values for the
coexistence packing fractions in close agreement to those predicted by
simulations.

Clearly a first-principles development for the association energy is needed to
justify our procedure. Progress along this avenue is begin made. We expect
that the present formalism can be applied to any 2D fluid made of polygonal
particles, a program that we are actually implementing to confirm the success
of the ECT in the description of the oriented phases of these fluids.

###### Acknowledgements.

Financial support from Grant No. PID2021-126307NB-C21 (MCIU/AEI/FEDER,UE) is
acknowledged.

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