Capillary ordering and layering transitions in two-dimensional hard-rod �uids
Yuri Martínez-Ratón
Grupo Interdis
iplinar de Sistemas Complejos, Departamento de Matemáti
as,
Es
uela Polité
ni
a Superior, Universidad Carlos III de Madrid,
Avenida de la Universidad 30, E�28911, Leganés, Madrid, SPAIN
(Dated: November 16, 2018)
In this arti
le we 
al
ulate the surfa
e phase diagram of a two-dimensional hard-rod �uid 
on�ned
between two hard lines. In a �rst stage we study the semi-in�nite system 
onsisting of an isotropi
�uid in 
onta
t with a single hard line. We have found 
omplete wetting by the 
olumnar phase
at the wall-isotropi
 �uid interfa
e. When the �uid is 
on�ned between two hard walls, 
apillary

olumnar ordering o

urs via a �rst-order phase transition. For higher 
hemi
al potentials the
system exhibits layering transitions even for very narrow slits (near the one-dimensional limit).
The theoreti
al model used was a density-fun
tional theory based on the Fundamental-Measure
Fun
tional applied to a �uid of hard re
tangles in the restri
ted-orientation approximation (Zwanzig
model). The results presented here 
an be 
he
ked experimentally in two-dimensional granular media
made of rods, where verti
al motions indu
ed by an external sour
e and ex
luded volume intera
tions
between the grains allow the system to explore those stationary states whi
h entropi
ally maximize
pa
king 
on�gurations. We 
laim that some of the surfa
e phenomena found here 
an be present in
two-dimensional granular-media �uids.
PACS numbers: 64.70.Md,61.30.Hn,61.20.Gy
I. INTRODUCTION
The e�e
t of �uid 
on�nement on phase transitions is
nowadays an a
tive line of s
ienti�
 resear
h due to the
dire
t appli
ation of the theoreti
ally predi
ted surfa
e
phase diagrams in the nanote
hnology industry. Con�n-
ing simple �uids, su
h as hard [1, 2℄ or Lennard-Jones
[3℄ spheres, in a narrow slit geometry, results in a ri
h
phase behavior, whi
h has re
ently been studied in de-
tail. Liquid 
rystals 
on�ned in nanopores is another
typi
al example of systems with important appli
ations
in the industry of ele
troni
 devi
es. For this reason they
have been extensively studied using theoreti
al models
based on density-fun
tional theory. In parti
ular, 
apil-
lary phase transitions exhibited by a nemati
 �uid 
on-
�ned between hard walls[4, 5℄, or walls favoring a par-
ti
ular an
horing [6℄, have been predi
ted. When non-
uniform liquid-
rystal phases, su
h as the sme
ti
 phase,
are in
luded in the study of 
on�ned systems, the result-
ing surfa
e phase diagrams display a ri
h phenomenol-
ogy, whi
h in
ludes wetting transitions, the appearan
e
of sme
ti
 defe
ts [7℄, and layering transitions [8℄.
The e�e
t of 
on�nement on two-dimensional �uids is
also an interesting topi
 of resear
h. Langmuir monolay-
ers of lipids on the surfa
e of water have been extensively
studied in the last hundred years [9℄, and the dis
ov-
ery of stru
tures and phase transitions in these systems
has experien
ed a dramati
 evolution driven by the new
experimental te
hniques. Now it is possible to 
on�ne
these two-dimensional systems by external potentials and
study the in�uen
e of the 
on�nement on the mole
ular
Ele
troni
 address: yuri�math.u
3m.es
pa
king of surfa
e monolayers.
Another paradigm of two-dimensional systems where
the 
on�nement plays an important role is the pa
k-
ing stru
tures formed by parti
les in granular media
[10℄. The 
rystallization of a quasi-two-dimensional one-

omponent granular-disk �uid has re
ently been studied
experimentally [11℄. It was found that the properties
of the 
rystal stru
ture obtained (su
h as pa
king fra
-
tion, latti
e stru
ture, and Lindenman parameter) 
o-
in
ide with their 
ounterparts obtained from MC sim-
ulations of a hard disk �uid. Re
ent experiments have
found non-equilibrium steady states in a vibrated granu-
lar rod monolayer with tetrati
, nemati
 and sme
ti
 
or-
relations [12℄. Some of these textures are similar to the
equilibrium thermodynami
 states of two-dimensional
anisotropi
 �uids resulting from density-fun
tional 
al-

ulations [13℄ and MC simulations [14℄. It was shown by
several authors that the inherent states of some frozen
granular systems 
an be des
ribed by equilibrium sta-
tisti
al me
hani
s [15℄. Also, an experimental test of
the thermodynami
 approa
h to granular media has re-

ently been 
arried out [16℄. Con�ning two-dimensional
granular rods in di�erent geometries (
ir
ular, re
tangu-
lar, et
.) results in spontaneous formation of patterns,
with di�erent orientationally ordered textures and de-
fe
ts next to the 
ontainer [17℄. The authors of Ref. [18℄
have 
arried out MC simulations of a 
on�ned hard disk
�uid. They have found that the 
rystal phase fails to nu-

leate due to formation of sme
ti
 bands when the system
is 
on�ned [18℄. It would be interesting to devi
e an ex-
periment with 
on�ned granular disks with the aim of

omparing the properties of the non-uniform stationary
states with those obtained from the statisti
al me
hani
s
applied to the hard disk �uid.
The main purpose of this arti
le is the study of a 
on-
�ned two-dimensional hard-rod �uid. We are interested
http://arxiv.org/abs/0704.1467v1
mailto:yuri@math.uc3m.es
in the 
al
ulation of the surfa
e phase diagram of a hard
re
tangle (HR) �uid 
on�ned by a single or two hard
lines. We 
an think on a HR �uid as an experimental re-
alization of a system of hard 
ylinders 
on�ned between
two plates at a distan
e less than twi
e the 
ylinder di-
ameter. We suggest that some of the surfa
e phase tran-
sitions obtained here by applying the density fun
tional
formalism to a 
on�ned two-dimensional HR �uid should
be similar to the steady states of 
on�ned granular rods.
Some experiments are required to verify this hypothesis.
The paper is organized as follows. In Se
. II we
present the theoreti
al model: the fundamental-measure
density fun
tional applied to a HR �uid in the restri
ted-
orientation approximation. This se
tion is divided into
two subse
tions. In the �rst the model is parti
ularized
to the study of the bulk phases, while in the se
ond part
the theoreti
al expressions used in the 
al
ulations of the
thermodynami
 and stru
tural properties of the inter-
fa
es are presented. The results are presented in Se
.
III. First we study the bulk phase diagram of a HR �uid
with aspe
t ratio equal to 3, and then the resulting sur-
fa
e phase diagrams of a single wall-HR �uid interfa
e
and of the �uid 
on�ned between two hard lines are pre-
sented. Some 
on
lusions are drawn in Se
. IV.
II. THEORETICAL MODEL
In this se
tion we introdu
e the theoreti
al model used
in the 
al
ulations of the bulk and interfa
e equilibrium
phases. To study highly inhomogeneous phases su
h as
those resulting from the 
on�nement of a �uid in a nar-
row slit geometry or the solid phase with a high pa
king
fra
tion, we have used the Fundamental-Measure Theory
(FMT) applied to an anisotropi
 �uid of hard re
tangles.
It is well known that this formalism presents a great ad-
vantage over other te
hniques when dealing with highly
inhomogeneous phases, and that this is mainly due to the
fa
t that a basi
 requirement to 
onstru
t the FMT den-
sity fun
tional is that it 
onform with the dimensional

ross-over 
riterium [19, 20℄. To implement the 
al
u-
lations we have used the restri
ted-orientation approxi-
mation, where the axes of the re
tangles are restri
ted
to align only along the 
oordinate axes x or y. Thus,
the whole system is des
ribed in terms of density pro�les
ρν(r) (ν = x, y).
While the ideal part of the free energy density in re-
du
ed thermal units has the exa
t form
Φid(r) =
ρν(r) [ln ρν(r)− 1] , (1)
the FMT intera
tion part of the 2D HR �uid is approxi-
mated [20℄ by
Φexc(r) = −n0(r) ln [1− n2(r)] +
n1x(r)n1y(r)
1− n2(r)
, (2)
where the weighted densities nα's are 
al
ulated as
nα(r) =
ν=x,y
ρν ∗ ω
(r), (3)
and where the symbol ∗ stands for 
onvolution, i.e.,
ρν ∗ ω
dr′ρν(r
ν (r − r
′). The weights ω
are the 
hara
teristi
 fun
tions whose volume integrals

onstitute the fundamental measures of a single parti
le
(the edge lengths and surfa
e area). They are de�ned as
ω(0)ν (r) =
− |x|)δ(
− |y|), (4)
ω(1x)ν (r) =
− |x|)δ(
− |y|), (5)
ω(1y)ν (r) =
− |x|)Θ(
− |y|), (6)
ω(2)ν (r) = Θ(
− |x|)Θ(
− |y|), (7)
where σνµ = σ+(L− σ)δµν , with L and σ the length and
width of the re
tangle and δµν the Krone
ker fun
tion,
while δ(x) and Θ(x) are the Dira
 delta and Heaviside
fun
tions, respe
tively.
A. The bulk phases
To 
al
ulate the bulk phase diagram we need to mini-
mize the Helmholtz free energy fun
tional βF [{ρν(r)}] =
dr [Φid(r) + Φexc(r)] with respe
t to the density pro-
�les ρν(r). These density pro�les have the symmetries

orresponding to the equilibrium phases, whi
h 
an be
the isotropi
 or nemati
 �uids, the sme
ti
 phase (with
parti
les arranged in layers with their long axes pointing
perpendi
ular to the layers), the 
olumnar phase (with
long axes parallel to the layers), plasti
 solid (parti
les
lo
ated at the nodes of the square grid with averaged ori-
entational order parameter over the 
ell equal to zero),
and oriented solid (with both translational and orienta-
tional order). To take proper a

ount of all these possible
symmetries, we have used a Fourier-series expansion of
the density pro�les:
ρν(r) = ρ0xν
k=(0,0)
k1,k2
cos(q1x) cos(q2y), (8)
where we de�ned k ≡ (k1, k2) [with N = N(1, 1)℄,
q1 = 2πk1/dx, and q2 = 2πk2/dy are the wave ve
tor

omponents parallel to x and y axes respe
tively, and dx,
dy are the periods of the re
tangular grid along these di-
re
tions. α
k1,k2
are the Fourier amplitudes of the density
pro�le of the spe
ies ν with the 
onstraint α
0,0 = 1. ρ0 is
the average of the lo
al density over the 
ell, while xν is
the 
ell-averaged o

upan
y probability of spe
ies ν. The
Fourier series is trun
ated at that value N whi
h guar-
antees that α
N,N < 10
. With this parametrization the
weighted density 
an be 
al
ulated expli
ity as
nα(r) = ρ0
k1,k2
ω̂(α)ν (k) cos(q1x) cos(q2y), (9)
where ω̂
ν (k) are the Fourier transforms of the 
orre-
sponding weights, whi
h have the form
ω̂(0)ν (k) = χ0(q1σ
ν /2)χ0(q2σ
ν/2), (10)
ω̂(1x)ν (k) = σ
νχ1(q1σ
ν/2)χ0(q2σ
ν/2), (11)
ω̂(1y)ν (k) = σ
νχ0(q1σ
ν/2)χ1(q2σ
ν/2), (12)
ω̂(2)ν (k) = aχ1(q1σ
ν/2)χ1(q2σ
ν/2), (13)
Here a = Lσ is the surfa
e area of the parti
le, and
χ0(x) = cosx, χ1(x) = sin(x)/x. We have sele
ted the
orientational dire
tor parallel to y. Thus, the equilib-
rium sme
ti
 (
olumnar) phase should be found by min-
imizing the free energy with respe
t to the Fourier am-
plitudes α
0,k (α
k,0), the sme
ti
 (
olumnar) period dy
(dx) and the order parameter QN ∈ [−1, 1] [related to
the xν 's through the relations x‖,⊥ = (1 ±QN)/2 where
the symbols ‖,⊥ stand for parti
le alignment along y
and x respe
tively℄. For uniform phases [α
k1,k2
∀(k1, k2) 6= (0, 0)℄ QN 
oin
ides with the nemati
 order
parameter. The solid phase is to be found by minimiz-
ing the free energy with respe
t to all the Fourier ampli-
tudes α
k1,k2
, the 
rystal periods dx and dy, and the order
parameter QN in the 
ase of an orientationally ordered
solid. To measure the pa
king stru
ture and the orienta-
tional order of the bulk phases we use the lo
al density
and the order parameter pro�les, ρ(r) =
ν ρν(r), and
Q(r) = [ρy(r)− ρx(r)] /ρ(r) respe
tively.
B. The interfa
ial phases
As we want to study the hard wall-�uid interfa
e or the
HR �uid 
on�ned in a slit geometry, we have introdu
ed
the following external potential:
Vν(x) =
∞, x < σxν/2
0, x ≥ σxν/2,
for the semi-in�nite system, and
Vν(x) =
∞, x < σxν/2 and x > H − σ
0, σxν/2 ≤ x ≤ H − σ
ν /2,
for the slit geometry, where H is the slit width, and
the normal to the wall was sele
ted in the x dire
tion.
Note that this external potential represents a hard line
whi
h ex
ludes the 
enter of mass of parti
les at distan
es
less than their 
onta
t distan
es with the wall. In this
sense we 
an say that the external potential favors par-
allel alignment at the wall. This is in 
ontrast with the
favored homeotropi
 alignment usually 
onsidered in sev-
eral studies of three-dimensional liquid 
rystals 
on�ned
by a single or two walls (in parti
ular that of Ref. [8℄).
The one-dimensional equilibrium density pro�les ρν(x)
were found by minimizing the ex
ess surfa
e free energy
per unit length
+ P −
ρν(x) [µν − Vν(x)]
, (16)
where β = (kBT )
, Φ(x) = Φid(x) + Φexc(x), and µν
are the 
hemi
al potentials of spe
ies ν �xed at the bulk
�uid-phase value at in�nite distan
e from the wall, while
P is the �uid pressure. The 
hemi
al potential of the
bulk �uid phase is 
al
ulated, as usual, as µ =
xνµν ,
with xν the molar fra
tions of spe
ies ν. If the bulk phase
is an isotropi
 �uid then xν = 1/2, and µν = µ, ∀ν.
To measure the degree of interfa
ial order, we will
use the adsorption of the density pro�le, de�ned as
dx [ρ(x)− ρ(∞)], and the order parameter pro�le
Q(x).
The expression (16) 
oin
ides with the de�nition of the
surfa
e tension of the wall-�uid interfa
e for the semi-
in�nite 
ase, whi
h is approximately equal to half the
ex
ess surfa
e free-energy for the slit geometry when the
wall distan
e H is large enough to a

ommodate both
interfa
es.
To minimize the fun
tional given by (16), we have dis-

retized spa
e in the x dire
tion and minimize γ with
respe
t to ρν(xi) (xi ∈ [x0, xN ]) using the 
onjugate-
gradient algorithm.
III. RESULTS
In this se
tion we present the main results obtained
from the appli
ation of the theoreti
al model just de-
s
ribed to the study of surfa
e properties of a 2D HR
�uid. Parti
les were 
hosen to have aspe
t ratio κ ≡
L/σ = 3. This aspe
t ratio is 
hosen be
ause one of the
aims of the present work is the study of layered phases

on�ned by one or two walls. As we will show bellow for
κ = 3 the stable phase is the 
olumnar layered phase.
In the �rst subse
tion we will summarize the results
obtained in the 
al
ulation of the bulk phase diagram of
this system, while in the se
ond subse
tion we will fo
us
on the study of the surfa
e phase diagram.
A. Bulk phase diagram
We have minimized the free energy density of the HR
�uid, de�ned as Φ ≡ V −1
dr [Φid(r) + Φexc(r)], with
respe
t to the Fourier amplitudes, periods, and mean o
-

upan
y probability, as des
ribed in detail in Se
. II A.
The results are plotted in Fig. 1, where the free-energy
densities of all the stable and metastable phases found
are plotted as a fun
tion of the pa
king fra
tion η = ρ0a.
We have found, apart from the usual isotropi
 (I) and ne-
mati
 (N) phases, two di�erent sme
ti
 phases (Sm1, and
Sm2), a plasti
 solid (PS), perfe
tly oriented solid (OS),
and �nally the 
olumnar phase (C), whi
h is the stable
one in the whole range of pa
king fra
tions explored.
0.55 0.6 0.65 0.7 0.75
Figure 1: The res
aled free-energy density Φ∗ = Φ+2.9875−
5.8501η is plotted against the mean pa
king fra
tion for all
the stable and metastable phases found. These are: isotropi
(dashed line), nemati
 (dotted line), sme
ti
-1 and sme
ti
-
2 (dotted and dashed lines), plasti
 solid (dashed line la-
belled as PS), while the perfe
tly oriented solid and the

olumnar phases (labelled in the �gure as OS and C respe
-
tively) are plotted with solid lines. The open 
ir
le indi-

ates the isotropi
-nemati
 bifur
ation point; the open square,
the isotropi
-plasti
 solid bifur
ation point; and the solid 
ir-

les represent the 
oexisting pa
king fra
tions at isotropi
-

olumnar phase 
oexisten
e.
The 
oupling between the spatial and orientational de-
grees of freedom of the parti
les results in the presen
e of
phases (stable or metastable) with di�erent symmetries.
In Fig. (2) we have sket
hed some of the parti
le 
on-
�gurations 
orresponding to phases with 
olumnar (a),
sme
ti
-1 (b), sme
ti
-2 (
), and plasti
 solid (d) sym-
metries found from the numeri
al minimization of the
density fun
tional. The dire
tions of spatial periodi
ities
of ea
h phase have been depi
ted in the �gure.
In Fig. 3 (a) we have plotted the density and order-
parameter pro�les of the 
oexisting 
olumnar phase. The

olumnar phase is orientationally ordered in the y dire
-
tion with the long re
tangle axis pointing along the y axis,
while the periodi
ity of both density and order parame-
ter pro�les (whi
h are in phase) is along the x dire
tion
[see Fig. 3 (a)℄. The mean 
oexisten
e pa
king fra
tions
of the I and C phases are ηI = 0.57058 and ηC = 0.60310,
respe
tively while the period of the C phase, in units of
the HR width, was found to be dx/σ = 1.20102. In Fig.
3 (b) we have plotted the order parameter QN, and the
period of the 
olumnar phase as a fun
tion of the mean
pa
king fra
tion.
(c) (d)
Figure 2: Sket
h of parti
le 
on�gurations 
orresponding to
di�erent phases: 
olumnar (a), sme
ti
-1 (b), sme
ti
-2 (
),
and plasti
 solid (d) phases. The dire
tion of spatial period-
i
ities are labeled in the �gure.
To 
ompare the di�erent pa
kings of HR parti
les in
the metastable phases (found as the lo
al minima of the
free energy density) for a �xed mean pa
king fra
tion
η = 0.7, we have plotted the density and order-parameter
pro�les of the Sm1,2 [Fig. 4 (a) and (b)℄, and PS and OS
[Fig. 5 (a)�(
)℄ phases. As 
an be seen from Fig. 4
(a), the density pro�le of the Sm1 phase has two max-
ima per period. The less pronoun
ed maxima, lo
ated
at the interstitials, re�e
t the high population of parti-

les with long axes oriented parallel to the sme
ti
 layers
[see the sket
hed parti
le 
on�gurations in Fig. 2 (b)℄.
This alignment is also shown in the order-parameter pro-
�le, whi
h rea
hes high negative values at the intersti-
tial positions. This phase bears a strong resemblan
e
to the �ndings of Refs. [21℄ and [22℄ where the parti-

le equilibrium 
on�gurations in the 3D sme
ti
 phases
show the same pattern. As a 
onsequen
e of this (al-
ternating population of parti
les aligned perpendi
ular
�sharpest peak in the density pro�le� and parallel to the
layers), the sme
ti
 period in units of the parti
le length
is dy/L = 1.53025, higher than the sme
ti
 period of the
Sm2 phase (dy/L = 1.17935). The density and order-
parameter pro�les of the Sm2 are shown in Fig. 4 (b).
As 
an be seen from the �gure, these pro�les re�e
t the
usual pa
king in sme
ti
s, 
hara
terized by a single den-
sity peak with vanishingly small population of parti
les
in the interstitials, while the order parameter rea
hes its
maximum value at the position of the sme
ti
 layers [see
Fig. 2 (
) for the sket
hed parti
le 
on�gurations℄.
The density and order parameter pro�les of the PS
phase with mean pa
king fra
tion equal to 0.7 are plot-
-0.5 -0.25 0 0.25 0.5
0.5 0.55 0.6 0.65 0.7 0.75 0.8
Figure 3: (a): density ρ(x) (solid line) and order parameter
Q(x) (dashed line) pro�les of the 
olumnar phase at 
oex-
isten
e with the isotropi
 phase. (b): order parameter QN
and period of the 
olumnar phase against the mean pa
king
fra
tion.
ted in Fig. 5 (a) and (b). The plasti
 solid has the same
periodi
ity in the x and y dire
tion, i.e. dx = dy = d,
and the order parameter averaged over the unit 
ell is
stri
tly equal to zero. As we 
an see from Fig. 5 (b),
while the order parameter at the nodes of the square
latti
e is equal to zero, it rea
hes positive (negative) val-
ues at the (±0.5, 0) [(0,±0.5)℄ positions along the sides
of the 
ell (the same solution with the x and y dire
-
tions inter
hanged was found in the minimization of the
free energy). Finally, the density pro�le of the perfe
tly
aligned two-dimensional solid is plotted in Fig. 5 (
).
Although the phases des
ribed above are metastable
with respe
t to the 
olumnar phase, they 
an be stabi-
lized for di�erent values of the parti
le aspe
t ratio. A
detailed study of the 
omplete phase diagram, ne
essary
to elu
idate this point, is a work in progress.
We now pro
eed to make a 
omparison between the
results for the 2D Zwanzig model with κ = 3 obtained
-0.5 -0.25 0 0.25 0.5
-0.5 -0.25 0 0.25 0.5
Figure 4: Density (solid line) and order parameter (dashed
line) pro�les of the sme
ti
-1 (a) and sme
ti
-2 (b) metastable
phases for a value of mean pa
king fra
tion �xed at 0.7.
above and those for hard parallelepipeds with restri
ted
orientations and the same value of κ [23℄. This 
ompari-
son will show the di�eren
es in phase behaviour between
three and two dimensions as predi
ted by Fundamental-
Measure Theory (whi
h, as already pointed out, 
on-
forms with the dimensional 
rossover 
riterion). As
shown in Ref. [23℄, hard parallelepipeds exhibit a se
ond-
order phase transition between isotropi
 and plasti
 solid
phases. As density in
reases the system goes to a dis
oti
sme
ti
 phase (
on�rmed by simulations) via a �rst-order
phase transition, whi
h in turn dis
ontinuously 
hanges
to a 
olumnar phase and then to an oriented solid. By

ontrast, the present model shows that, in two dimen-
sions, the isotropi
 phase exhibits a �rst-order transition
to a 
olumnar phase that is stable until very high pa
k-
ing fra
tions (more stable that plasti
, oriented solid and
di�erent sme
ti
 phases). As a 
onsequen
e, one expe
ts
that the 
orresponding surfa
e phase diagrams be also
Figure 5: Density (a) and order-parameter (b) pro�les of the
plasti
 solid phase. (
): Density pro�le of the perfe
tly ori-
ented solid.
di�erent.
B. Surfa
e phase diagram
In this se
tion we deal with surfa
e phenomena. In the
�rst part we will 
on
entrate on the semi-in�nite wall-
isotropi
 interfa
e of a HR �uid, while in the se
ond part
we will fo
us on the slit geometry. We will demostrate
the presen
e of 
omplete wetting, 
apillary ordering and
layering transitions in the 
on�ned two dimensional hard
rod �uid. For a detailed dis
ussion on general grounds of
the phase behavior and 
riti
al phenomena of a 
on�ned
by a single wall �uid see Ref. [24℄.
1. The wall-�uid interfa
e
The intera
tion between the isotropi
 �uid phase and a
hard wall was studied by 
al
ulating the one-dimensional
equilibrium density ρ(x) and order-parameter Q(x) pro-
�les through the ex
ess surfa
e free-energy minimization
[see Eq. (16)℄. The 
hemi
al potential µ of the �uid phase
at in�nite distan
e from the wall was varied within the
range of isotropi
-phase stability, i.e. µ ∈ [−∞, µ0] (µ0
being the value at whi
h the I-C phase transition o

urs).
It is well known that the presen
e of a hard wall in a sys-
tem of elongated parti
les indu
es parallel alignment of
the parti
le axes with respe
t to the wall [25, 26℄. This
preferential alignment is a result of the entropi
 depletion
e�e
t. In the parallel 
on�guration, the 
enters of mass
of the parti
les are mu
h 
loser to the wall, so the gain
in volume per parti
le is larger and, as a 
onsequen
e,
the 
on�gurational entropy of the system is maximized.
This e�e
t is responsible for the o

urren
e of a biaxial
nemati
 phase whi
h breaks the orientational symmetry
in a three-dimensional nemati
 �uid [4℄. The same deple-
tion me
hanism is at work in 2D, as we will show below.
The results from the minimization are shown in Figs. 6
(a) and (b) for an undersaturation of β∆µ = −1.1×10−4.
As we 
an see from the �gure, the density and order-
parameter pro�les indi
ate 
olumnar order near the wall,
whi
h propagates several 
olumnar periods into the �uid
phase. The wall-�uid intera
tion enhan
es the orienta-
tional order near the surfa
e and the adsorption of par-
ti
les, 
reating a stru
tured layer with 
olumnar-phase
symmetry whi
h grows in width with in
reasing 
hem-
i
al potential and diverges at µ = µ0. Thus, 
omplete
wetting by a 
olumnar phase o

urs at the wall-isotropi
interfa
e. This result is shown in Fig. 7 (a) where the ex-

ess surfa
e free-energy γ and the adsorption 
oe�
ient Γ
are plotted against ∆µ = µ−µ0. As we 
an see, Γ grows

ontinuously, ultimately diverging logarithmi
ally with
∆µ (see inset of �gure). The ex
ess surfa
e free energy γ
has a maximum, and at this point the adsorption passes
through zero. This result is dire
tly related to the interfa-

ial Gibbs-Duhem equation, Γ = −dγ/dµ, whi
h relates
the adsorption 
oe�
ient with the �rst derivative of the
ex
ess surfa
e free-energy with respe
t to bulk 
hemi
al
potential. At µ0 the ex
ess surfa
e energy is equal to the
wall-isotropi
 surfa
e tension γWI, whi
h is in turn equal
to the sum of wall-
olumnar and 
olumnar-isotropi
 sur-
fa
e tensions, γ(µ0) = γWI = γWC + γCI (the Young's
equation for 
omplete wetting).
0 10 20 30 40 50 60
0 10 20 30 40 50 60
Figure 6: Density (a) and order parameter (b) pro�les of the
wall-isotropi
 �uid interfa
e. The undersaturation is �xed to
β∆µ = −1.1× 10−4.
We have 
arried out a logarithmi
 �t of the adsorp-
tion 
oe�
ient with respe
t to undersaturation β∆µ =
β (µ− µ0), and we �nd that Γσ = τ1+τ2 ln [β|∆µ|], with
τ1 = 0.02387 and τ2 = −0.03396. Then, integrating the
interfa
ial Gibbs-Duhem relation Γ = −dγ/dµ, we �nd
the expression
βγσ ≈ βγWIσ − [τ1 + τ2 (ln (β|∆µ|) − 1)]β∆µ, (17)
whi
h approximates the ex
ess surfa
e free energy near

omplete wetting. The above expression is plotted
against β∆µ in Fig. 7 (b), where the results from dire
t

al
ulation of βγσ, using the equilibrium density pro�les
obtained, are also plotted. As we 
an see the agreement
is ex
ellent even for relatively high values of undersatu-
ration.
0 2 4 6 8
σ 10-4 10-2 100
0.4(a)
0 0.01 0.02 0.03 0.04 0.05
0.136
0.138
0.142
0.144
Figure 7: (a): ex
ess surfa
e free energy (solid line) and ad-
sorption 
oe�
ient (dashed line), in redu
ed units, against
β∆µ. The inset shows Γσ vs. β∆µ in logarithmi
 s
ale. (b):
ex
ess surfa
e free energy vs. βµ in the neighborhood of zero
undersaturation. The open 
ir
les show the values obtained
from the numeri
al minimization, while the solid line repre-
sents the analyti
 
urve obtained by integrating the interfa
ial
Gibbs-Duhem relation with the �tted logarithmi
 dependen
e
of the adsorption 
oe�
ient (see text). The solid 
ir
le shows
the value of the W-I surfa
e tension βγWIσ = 0.13822
To 
al
ulate the stru
tural and thermodynami
 prop-
erties of the 
olumnar-isotropi
 interfa
e, we have imple-
mented a numeri
al s
heme already used in Ref. [27℄,

onsisting of minimizing the surfa
e ex
ess free energy γ
in a box of width h 
ontaining a stripe of a few 
olum-
nar layers surrounded by isotropi
 material with periodi
boundary 
onditions. h is 
hosen su
h that the density
pro�les 
an easily a

ommodate the two interfa
es and
go to the 
oexisten
e �uid density at the periodi
 bound-
ary. A typi
al result from this 
al
ulation is plotted in
Fig. 8 (a) and (b) for the density and order-parameter
pro�les, respe
tively. Thus, the I-C interfa
ial tension

an be 
al
ulated as half the ex
ess surfa
e free energy
resulting from the minimization. We have found a value
of βγICσ = 0.00672.
-60 -40 -20 0 20 40 60
-60 -40 -20 0 20 40 60
Figure 8: Density (a) and order-parameter (b) pro�les of a
numeri
al box 
ontaining two isotropi
-
olumnar interfa
es.
Finally, to verify that Young's law for 
omplete wet-
ting holds, we need to 
al
ulate the surfa
e tension of
the wall-
olumnar interfa
e. To 
onstru
t density pro-
�les 
ompatible with this semi-in�nite interfa
e, one has
to establish a boundary, at the side of the 
omputational
box opposite to the wall, and pla
e, beyond the bound-
ary and into the bulk, a periodi
ally stru
tured pro�le,

hoosing the phase (i.e. the value of the pro�le at the
boundary) arbitrarily within the bulk period. Although
this re
ipe 
an in prin
iple be implemented, we have 
ho-
sen to �x bulk I-C 
oexisten
e 
onditions in a 
on�ned

olumnar phase and 
al
ulate the density pro�le of the
system 
on�ned between two walls; the separation be-
tween the walls was 
hosen large enough so that the ef-
fe
ts of having a �nite interfa
e penetration length 
aused
by the presen
e of the 
on�ned external potential 
an be
negle
ted. Also, in order to ensure that 
ommensurabil-
ity e�e
ts 
an be ignored, the distan
e between the walls
was set to a (large) integer number of equilibrium periods
of the 
olumnar phase. The results from these 
al
ula-
tions are plotted in Fig. 9 (a) and (b). The W-C surfa
e
tension 
al
ulated as half the value of the ex
ess surfa
e
free energy results in βγWCσ = 0.13150, 
ompatible with
Young's law in 
onditions of 
omplete wetting of the W-I
interfa
e by the 
olumnar phase.
0 20 40 60 80 100 120
0 20 40 60 80 100 120
Figure 9: The density (a) and order parameter (b) pro�les
of two wall-
olumnar interfa
es
2. Capillary ordering
This se
tion is devoted to a study of the e�e
t of 
on-
�nement of a 2D HR �uid on the thermodynami
 and
stru
tural properties of the �uid. In parti
ular, we are
interested in the enhan
ement of the orientational and
layering ordering due to 
on�nement, and the 
ommen-
surability e�e
ts exhibited by a layered phase sandwi
hed
between two hard walls at a distan
e that may or may
not 
ommensurate with the period of the bulk 
olumnar
phase. It is well known that, under 
ertain 
ir
umstan
es
(related to the nature of the �uid-�uid and surfa
e-�uid
intera
tions), a �uid inside a pore 
an exhibit 
apillary
�rst-order phase transitions between two di�erent phases
at a 
hemi
al potential below the bulk 
oexisten
e value.
An example of this phenomenon is the re
ently studied

apillary nematization [4℄ and sme
tization [8℄ of a liquid

rystal �uid inside a pore. The bulk 
ondensed phase
may have uniform (nemati
) or nonuniform density pro-
�les. For the latter 
ase, 
apillary layering transitions be-
tween interfa
ial phases with di�erent number of sme
ti
layers [8℄ 
an also be found. Here we will show that these

apillary and layering phase transitions are not unique to
3D system. They are also present in 2D anisotropi
 �uids
whi
h 
an stabilize layered phases with di�erent spatial
symmetries, su
h as the 
olumnar phase.
With a view to �nding the e�e
ts of 
on�nement on

olumnar ordering in a HR �uid, we have minimized the
ex
ess surfa
e free energy with respe
t to the density pro-
�le for the parti
ular 
ase of HR's with κ = 3. The �uid
is 
on�ned by two hard walls at a distan
e H/σ = 30 (in
units of the parti
le width). As already pointed out, hard
walls favor alignment parallel to the wall, as well as ad-
sorption of parti
les at both surfa
es (density and order
parameters at 
onta
t are mu
h higher than their bulk
values). This 
oupled translational-orientational order-
ing near the surfa
es propagates into the �uid, 
reating

olumnar ordering. We have found that for low values
of the 
hemi
al potential of the bath the density pro�le
is stru
tureless (ex
ept just at the wall 
onta
t), similar
to the bulk isotropi
 phase. In
reasing the 
hemi
al po-
tential several damped 
olumnar peaks appear near the
wall in a 
ontinuous fashion, i.e. with their heights in-

reasing 
ontinuously. At some value of the 
hemi
al po-
tential, the system exhibits a �rst-order phase transition
between a phase with highly damped 
olumnar peaks to
a new phase with mu
h stronger 
olumnar ordering even
at the 
enter of the pore. The typi
al density and order-
parameter pro�les of both interfa
ial phases are shown in
Fig. 10 (a)-(d). Although the less-ordered phase exhibits
strong os
illations in both density and order-parameter
pro�les, the peak amplitudes are damped into the pore
faster than those of the higher ordered phase. We will
take the 
onvention to 
all the �rst `isotropi
', and the
se
ond `
olumnar' surfa
e phases. This 
onvention is jus-
ti�ed by the fa
t that, just before the transition des
ribed
above, 
olumnar ordering in
reases 
ontinuously, start-
ing from an isotropi
-like density pro�le, as the 
hemi-

al potential is in
reased. Thus we 
annot tra
e out a
de�nite boundary (a value for µ below that 
orrespond-
ing to �rst order phase transition) below or above whi
h
the pro�le inside the pore 
an be 
onsidered isotropi
 or

olumnar. Only the �rst order phase transition des
ribed
above 
an really distinguish two di�erent surfa
e phases,
one of them less ordered (following our 
onvention, the
isotropi
 phase) than the other (the 
olumnar phase). As
we 
an see in the �gure, the latter has 25 
olumnar peaks.
The transition point is 
al
ulated from the dis
ontinu-
ity in the �rst derivative of the ex
ess surfa
e free en-
0 5 10 15 20 25 30
0 5 10 15 20 25 30
0 5 10 15 20 25 30
0 5 10 15 20 25 30
Figure 10: Isotropi
 (a)-(b) and 
olumnar (
)-(d) phases that

oexist at the same 
hemi
al potential bellow µ0. (a), (
):
Density pro�les. (b), (d): Order-parameter pro�les.
ergy with respe
t to the bulk pa
king fra
tion η. The

orresponding plot is shown in Fig 11 (a). At this point
the adsorption 
oe�
ient jumps dis
ontinuously from the
less- (the damped 
olumnar) to the higher-ordered phase
[see Fig. 11 (b)℄.
This surfa
e transition point is lo
ated below the bulk
isotropi
-
olumnar phase transition [see Fig. 11 (a)℄,
0.54 0.55 0.56 0.57 0.58
0.54 0.55 0.56 0.57 0.58
Figure 11: Ex
ess surfa
e free energy (a) and adsorption 
oef-
�
ient (b) against pa
king fra
tion of the bulk isotropi
 �uid.
In the �gure at top, the solid 
ir
le represents the transition
point between both interfa
ial phases, while open square in-
di
ates the point 
orresponding to the bulk 
oexisten
e value
for isotropi
 and 
olumnar phases.
showing the presen
e of 
olumnar-order enhan
ement in
the pore. On further in
reasing the 
hemi
al potential
up to a su�
iently high value (above µ0), we �nd a �rst-
order layering transition between two interfa
ial 
olum-
nar phases whi
h di�er by just a single 
olumnar layer.
The behavior of the ex
ess surfa
e free energy and the
adsorption 
oe�
ient is similar to that shown in Fig. 11
(a) and (b). Alternatively we 
an �nd the transition from
n−1 to n 
olumnar layers by �xing the 
hemi
al potential
and in
reasing the pore width H .
The two surfa
e phase transitions des
ribed above,
namely �rst-order 
apillary I-C ordering and (n − 1)�
n layering transition, are 
onne
ted in the µ−H surfa
e
phase diagram through the pe
uliar stru
ture shown in
Fig. 12.
The parabola below the bulk transition line 
orre-
sponds to �rst-order transition lines separating regions
of stability of the isotropi
 and the 
olumnar interfa
ial
phases, while the straight lines indi
ate layering transi-
tions. In
reasing the 
hemi
al potential from low values
to those 
orresponding to the parabola, the density pro-
�les always 
hange 
ontinously from a stru
tureless to
damped 
olumnar density pro�le. Both types of transi-
28 28.5 29 29.5 30
24 C25
Figure 12: µ − H surfa
e phase diagram showing �rst-order

apillary 
olumnar ordering and layering transitions. The
pore width 
overs a range whi
h goes from 23 to 25 
olum-
nar layers, as labeled in the �gure. The open 
ir
les indi
ate
the 
ases 
hosen for 
al
ulations, while the solid line is a 
u-
bi
 spline interpolation. The horizontal dashed line shows the
value of the bulk 
hemi
al potential at the isotropi
-
olumnar
phase 
oexisten
e.
tions (the isotropi
-
olumnar and n−1-n layering transi-
tions) 
oales
e in triple points, two of whi
h are shown in
Fig. 12. At the triple points an isotropi
 and two 
olum-
nar interfa
ial phases with n− 1 and n layers 
oexist in
equilibrium. The set of 
onne
ted of Fig. 12 are similar
to those found in MC simulations of the 
on�ned hard-
sphere �uid [2℄. In this work the authors have shown the
existen
e of 
apillary freezing of the HS �uid, 
on�ned
in the slit geometry, for 
hemi
al potential values below
the bulk freezing transition. The transitions lines in the
µ−H surfa
e phase diagram follow the same topology of

onne
ted set of parabolas as found in our system.
Some of the topologi
al features of this surfa
e phase
diagram 
an be elu
idated from the Clausius-Clapeyron
equation as applied to the interfa
ial 
oexisten
e lines.
The ex
ess surfa
e free energy γ(µ,H) along 
oexisten
e
is a fun
tion of two variables, the 
hemi
al potential µ,
and the pore width H . Thus, in�nitesimal 
hanges in
these variables along the 
oexisten
e 
urve are related
through the equation
dγα − dγβ = ∆
dH = 0, (18)
where the 
oexisting 
ondition γα = γβ (for
α, β =I,Cn−1,Cn) was used, and ∆u = uα − uβ for any
fun
tion u. Using the interfa
ial Gibbs-Duhem equation
∂γ/∂µ = −Γ and the de�nition of the solvation for
e
f = −∂γ/∂H , we arrive at
, (19)
whi
h relates the �rst derivative of the 
hemi
al potential
with respe
t to the pore width with 
hanges in the sol-
vation for
e and in the adsorption 
oe�
ient at the tran-
sition point. The negative slope of the layering 
urves is
a dire
t result of Eq. (19), as the in
rement in the ad-
sorption is always positive for the (n − 1) → n layering
transition, while the 
hange in the solvation for
e is also
positive (the latter 
an be interpreted as an in
rement
with respe
t to the bulk of the ex
ess surfa
e pressure,
whi
h is obviously larger for the phase with n layers).
For values of the pore width that 
ommensurate with an
integer number of 
olumnar periods of the bulk 
olumnar
phase, the solvation for
e be
omes zero and we get a min-
imum in the I-C 
apillary transition 
urve (see Fig. 12).
At ea
h side of the minimum the solvation for
e 
hange
the sign to positive (left side) or negative (right side) as
we 
ompress or expand the �lm, respe
tively, while the

hange in adsorption remains positive.
The Kelvin equation for 
apillary 
ondensation in a slit
geometry relates the undersaturation in 
hemi
al poten-
tial with pore width H as
∆µ = µ(H)− µ0 = −
(ρα − ρβ)H
, (20)
where ρα and ρβ are the bulk 
oexisting densities of
phases α and β (α being the 
ondensed phase), while
γαβ is the surfa
e tension of the 
orresponding interfa
e.
It was assumed that 
omplete wetting by the α phase
o

urs at the W-β interfa
e. For a detailed dis
ussion of
the Kelvin equation in the 
ontext of liquid 
rystal phase
transitions see Ref. [28℄. Applying this equation using
H/σ = 28.88 (the lo
ation of the minimum in the µ−H
phase diagram of Fig. 12), we obtain an undersaturation
β∆µ = −0.0429, while its real value is β∆µ = −0.1255.
In the derivation of the Kelvin equation, deviations from
the bulk stru
ture of the density pro�le arising from
the 
on�nement by the external potential are negle
ted.
Also, the elasti
 energy resulting from the 
ompression or
expansion of a layered phase 
on�ned between two walls
is not taken into a

ount. These e�e
ts might be respon-
sible for the di�eren
es found between our 
al
ulations
and the estimation based on the Kelvin equation. We
have 
he
ked that the sequen
e of minima in the µ−H
phase diagram tends to µ0 as H → ∞, a result predi
ted
by Eq. (20).
Refs. [4℄ and [8℄ showed that the 
apillary nemati-
zation line of the 
on�ned liquid-
rystal �uid ends in a

riti
al point for small values of the pore width. In or-
der to study how the topology of the surfa
e phase dia-
gram 
hanges in the regime of small pore widths, we have

arried out the 
orresponding 
al
ulations of interfa
ial
stru
ture. We have found that the I-C 
apillary ordering
transition 
hanges at some parti
ular value of H (near
its maximum undersaturation represented by the mini-
mum in the I-C interfa
e 
oexisting 
urve) from �rst to
se
ond order. For lower values of H two 
riti
al points
emerge from this single point, the distan
e between them
in
reasing. In Fig. 13 one of these s
enarios is shown.
As we 
an see, there is a range of values of H (near
the triple points) where the �rst-order 
apillary order-
ing transitions are still present but, between the 
riti
al
points belonging to di�erent layering bran
hes, 
olumnar
ordering grows 
ontinously from the isotropi
 (damped

olumnar interfa
ial phase) to a highly-ordered 
olum-
nar phase. Layering transitions are always present even
for very small H , as will be shown below. An interest-
ing feature of this phase diagram is that the lo
ation of
the triple points moves above the bulk 
oexisten
e value
µ0. This indi
ates that the interfa
ial 
olumnar phase
just below the triple points 
an be unstable for values of

hemi
al potentials 
orresponding to those of 
olumnar-
phase stability at bulk (similar to the 
apillary evapora-
tion of the 
on�ned �uid). For wide enough slits (those
for whi
h the parabolas are 
onne
ted) the triple points
are pra
ti
ally lo
ated at µ0, as 
an be observed from
Fig. 12.
18 18.5 19 19.5 20 20.5
Figure 13: µ−H surfa
e phase diagram showing 
riti
al points
(�lled 
ir
les). Number of 
olumnar layers are indi
ated as
subs
ripts.
For even smaller values of H , only layering transitions
remain; these end in 
riti
al points lo
ated above µ0, as
Fig. 14 shows.
When the width H is su
h that the pore 
an only a
-

ommodate one parti
le with its long axis perpendi
u-
lar to the wall (or not more than four or three parti
les
aligned parallel to the wall) the system is near the one-
dimensional limit. It is known that hard-
ore systems in
this limit do not exhibit �rst-order phase transitions, but
even for very narrow slits we 
an still �nd �rst-order tran-
sitions at whi
h the density pro�le experien
es an abrupt

hange inside the pore. In Fig. 15 (a) and (b) we show
two 
oexisting density pro�les 
orresponding to oversat-
urations, β∆µ = 0.51760 and β∆µ = 0.72836, and pore
widths H/σ = 4.32 and H/σ = 3.14 in (a) and (b), re-
spe
tively. The �uid inside the pore undergoes a phase
transition, whi
h dramati
ally 
hanges the stru
ture of
the interfa
ial density pro�les by in
reasing the heigh of
11.4 11.45 11.5 11.55 11.6
7.8 7.85 7.9 7.95 8
Figure 14: µ −H surfa
e phase diagrams for small values of
four [Fig. 15(a)℄ or three [(b)℄ density peaks inside the
pore.
IV. CONCLUSIONS
In this arti
le we have shown that 2D �uids 
omposed
of anisotropi
 parti
les intera
ting via hard-
ore repul-
sion and 
on�ned in a slit geometry exhibit a 
omplex
and ri
h interfa
ial phase behavior. Apart from �rst-
order 
apillary 
olumnar ordering, we have also found
layering transitions in this system. These results are sim-
ilar to those found in 3D liquid-
rystal �uids 
on�ned in
a pore, where 
apillary sme
tization and layering phe-
nomena were also found [8℄. In view of these similarities,
we 
an extra
t the 
on
lusion that, independent of the
system dimensionality and the pe
uliarities of the layered
phases, either sme
ti
 or 
olumnar, if the �uid-wall inter-
a
tion enhan
es layered interfa
e ordering (homeotropi
in 
ase of sme
ti
 phases, and the entropi
ally favored
parallel alignment for the 
olumnar phase), 
ompatible
with the equilibrium bulk phase, then the 
on�ned �uid
exhibits the interfa
ial phase transitions des
ribed above.
In this study we have used as a model a hard-re
tangle
�uid, and the density and the order-parameter pro-
�les were 
al
ulated by minimizing the ex
ess surfa
e
free-energy fun
tional resulting from the Fundamental-
Measure Theory applied to the two-dimensional Zwanzig
model. The orientational degrees of freedom were
dis
retized, in order to take advantage of having a
free-energy fun
tional whi
h redu
es to the exa
t one-
dimensional fun
tional when the density pro�le is 
on-
strained to lie along a line. This property is 
ru
ial
to study strongly 
on�ned �uids (as is the 
ase in this
study), in parti
ular when the pore width has only a few
parti
le diameters in width.
As already pointed out in Se
. I, some experiments
had shown profound similarities between parti
le 
on-
�gurations obtained as stationary states of systems of
anisotropi
 grains and those 
orresponding to the equi-
0 1 2 3 4 5
0 0.8 1.6 2.4 3.2
Figure 15: (a): density pro�les of two 
oexisting phases
(shown with solid and dashed lines) at β∆µ = 0.5176. The
pore width is H/σ = 4.32. (b): same as in (a) but for a pore
with H/σ = 3.14 and for β∆µ = 0.72836.
librium states obtained by density fun
tional minimiza-
tion [12℄. These similarities 
an be explained by applying
a maximum-entropy prin
iple on granular 
olle
tions of
parti
les, i.e. for a �xed pa
king fra
tion, externally-
indu
ed vibrational motion for
es the system to explore
those stationary states whi
h maximize the 
on�gura-
tional entropy (sin
e the grains 
annot overlap). Of

ourse, equilibrium statisti
al me
hani
s is unable to pro-
pose an equation of state for granular matter, but it 
ould
be possible to predi
t that granular matter 
omposed of
anisotropi
 parti
les and 
on�ned between parallel walls
may support a stationary texture 
onsisting of layers of
parti
les oriented parallel to the wall. The manner in
whi
h the grain orientations propagate into the 
ontainer
would depend on the average pa
king fra
tion and on
the frequen
y of the external for
e. Only at this qualita-
tive level 
an we give some insight into possible 
omplete
wetting phenomena and 
apillary ordering in granular
rod �uids 
on�ned between two horizontal plates at a
distan
e slightly larger than the parti
le dimensions in
the verti
al dire
tion (thus simulating a two-dimensional
system), and also 
on�ned by one or two verti
al planes
(these playing the role of hard walls).
Some 
al
ulations (not shown here) on the 2D HR �uid
show that, for di�erent aspe
t ratios, 2D sme
ti
 and

rystal phases 
an be stable over some range of pa
k-
ing fra
tions. It would be interesting to explore whether

on�nement suppresses or enhan
es bulk ordering, and
to study the 
hanges in the surfa
e phase diagram when
phases of di�erent symmetries are in
luded. Work along
this dire
tion is 
urrently in progress.
A
knowledgments
I thank D. de las Heras, E. Velas
o, and L. Mederos for
useful dis
ussions, and E. Velas
o for a 
riti
al reading
of the manus
ript. The author gratefully a
knowledges
�nan
ial support from Ministerio de Edu
a
ión y Cien
ia
under grants No. BFM2003-0180 and from Comunidad
Autónoma de Madrid (S-0505/ESP-0299) and (UC3M-
FI-05-007). The author was supported by a Ramón y
Cajal resear
h 
ontra
t from the Ministerio de Edu
a
ión
y Cien
ia.
[1℄ M. S
hmidt, and H. Löwen, Phys. Rev. Lett. 76, 4552
(1996); Phys. Rev. E 55, 7228 (1997).
[2℄ M. Dijkstra, Phys. Rev. Lett. 93, 108303 (2004); A. For-
tini, and M. Dijkstra, J. Phys. Condes Matter 18, L371
(2006).
[3℄ L. Salama
ha, A. Patrykiejew, S. Sokolowski, and K.
Binder, Eur. Phys. J. E 13, 261 (2004); L. Salama
ha,
A. Patrykiejew, S. Sokolowski, Eur. Phys. J. E 18, 425
(2005).
[4℄ R. van Roij, M. Dijkstra, and R. Evans, Europhys. Lett.
49, 350 (2000); M. Dijkstra, R. van Roij, and R. Evans,
Phys.Rev. E 63, 051703 (2001)
[5℄ L. Harnau, and S. Dietri
h, Phys. Rev. E 66, 051702
(2002).
[6℄ I. Rodriguez-Pon
e, J. M. Romero-Enrique, E. Velas
o,
L. Mederos, and L. F. Rull, J. Phys.: Condens. Mat-
ter 12, A363 (2000); I. Rodriguez-Pon
e, J. M. Romero-
Enrique, and L. F. Rull, Phys. Rev. E 64, 051704 (2001)
[7℄ R. E. Webster, N. J. Mottram, and D. J. Cleaver, Phys.
Rev. E. 68, 021706 (2003); Z. Kutnjak, S. Kralj, G. La-
hajnar, and S. Zumer, Phys. Rev. E 70, 051703 (2004).
[8℄ D. de las Heras, E. Velas
o, and L. Mederos, Phys. Rev.
Lett. 94, 017801 (2005); Phys. Rev. E 74, 011709 (2006).
[9℄ V. M. Kaganer, H. Mohwald, and P. Dutta, Rev. Mod.
Phys. 71, 779 (1999).
[10℄ I. S. Aranson, and L. S. Tsimring, Rev. Mod. Phys. 78,
641 (2006).
[11℄ P. M. Reis, R. A. Ingale, and M. D. Shattu
k, Phys. Rev.
Lett. 96, 258001 (2006).
[12℄ V. Narayan, N. Menon, and S. Ramaswamy, J. Stat.
Me
h. P01005 (2006)
[13℄ Y. Martínez-Ratón, E. Velas
o, and L. Mederos, J. Chem.
Phys. 122, 064903 (2005); J. Chem. Phys. 125, 014501
(2006).
[14℄ A. Donev, J. Burton, F. H. Stillinger, and S. Torquato,
Phys. Rev. B 73, 054109 (2006).
[15℄ A. Fierro, M. Ni
odemi, and A. Coniglio, Eurphys. Lett.
59, 642 (2002); Phys. Rev. E 66, 061301 (2002); A.
Coniglio, A. Fierro, and N. Ni
odemi, Eur. Phys. J. E
9, 219 (2002).
[16℄ D. S. Dean, and A. Lefevre, Phys. Rev. Lett 90, 198301
(2003).
[17℄ J. Galanis, D. Harries, and D. L. Sa
kket, Phys. Rev.
Lett. 96, 028002 (2006).
[18℄ D. Chaudhuri, and S. Sengupta, Phys. Rev. Lett. 93,
115702 (2004).
[19℄ Y. Rosenfeld, M. S
hmidt, H. Löwen, and P. Tarazona,
J. Phys.: Condens. Matt. 8, L577 (1996); Phys. Rev. E
55, 4245 (1997); P. Tarazona, and Y. Rosenfeld, Phys.
Rev. E 55, R4873 (1997).
[20℄ J. A. Cuesta and Y. Martínez�Ratón, Phys. Rev. Lett.
78, 3681 (1997); J. Chem. Phys. 107, 6379 (1997).
[21℄ R. van Roij, P. Bolhuis, B. Mulder, and D. Frenkel, Phys.
Rev. E 52, R1277 (1995).
[22℄ J. S. van Duijneveldt, and M. P. Allen, Mole
. Phys. 90,
243 (1997).
[23℄ Y. Martínez-Ratón, Phys. Rev. E 69, 061712 (2004).
[24℄ R. Lipowsky, J. Appl. Phys. 55, 2485 (1984).
[25℄ A. Poniewierski, and R. Holyst, Phys. Rev. A 38, 3721
(1988).
[26℄ A. Chrzanowska, P. I. C. Teixeira, H. Ehrentraut, and D.
J. Cleaver, J. Phys.: Condens. Matter 13, 4715 (2001).
[27℄ Y. Martínez-Ratón, A. M. Somoza, L. Mederos, and D. E.
Sullivan, Faraday Dis
uss. 104, 111 (1996); Y. Martinez,
A. M. Somoza, L. Mederos, and D. E. Sullivan, Phys.
Rev. E 53 2466 (1996).
[28℄ T. J. Slu
kin and A. Poniewierski, Mole
. Cryst. Liq.
Cryst. 179, 349 (1990).

  In this article we calculate the surface phase diagram of a two-dimensional
hard-rod fluid confined between two hard lines. In a first stage we study the
semi-infinite system consisting of an isotropic fluid in contact with a single
hard line. We have found complete wetting by the columnar phase at the
wall-isotropic fluid interface. When the fluid is confined between two hard
walls, capillary columnar ordering occurs via a first-order phase transition.
For higher chemical potentials the system exhibits layering transitions even
for very narrow slits (near the one-dimensional limit). The theoretical model
used was a density-functional theory based on the Fundamental-Measure
Functional applied to a fluid of hard rectangles in the restricted-orientation
approximation (Zwanzig model). The results presented here can be checked
experimentally in two-dimensional granular media made of rods, where vertical
motions induced by an external source and excluded volume interactions between
the grains allow the system to explore those stationary states which
entropically maximize packing configurations. We claim that some of the surface
phenomena found here can be present in two-dimensional granular-media fluids.

Capillary ordering and layering transitions in two-dimensional hard-rod �uids
Yuri Martínez-Ratón
Grupo Interdis
iplinar de Sistemas Complejos, Departamento de Matemáti
as,
Es
uela Polité
ni
a Superior, Universidad Carlos III de Madrid,
Avenida de la Universidad 30, E�28911, Leganés, Madrid, SPAIN
(Dated: November 16, 2018)
In this arti
le we 
al
ulate the surfa
e phase diagram of a two-dimensional hard-rod �uid 
on�ned
between two hard lines. In a �rst stage we study the semi-in�nite system 
onsisting of an isotropi
�uid in 
onta
t with a single hard line. We have found 
omplete wetting by the 
olumnar phase
at the wall-isotropi
 �uid interfa
e. When the �uid is 
on�ned between two hard walls, 
apillary

olumnar ordering o

urs via a �rst-order phase transition. For higher 
hemi
al potentials the
system exhibits layering transitions even for very narrow slits (near the one-dimensional limit).
The theoreti
al model used was a density-fun
tional theory based on the Fundamental-Measure
Fun
tional applied to a �uid of hard re
tangles in the restri
ted-orientation approximation (Zwanzig
model). The results presented here 
an be 
he
ked experimentally in two-dimensional granular media
made of rods, where verti
al motions indu
ed by an external sour
e and ex
luded volume intera
tions
between the grains allow the system to explore those stationary states whi
h entropi
ally maximize
pa
king 
on�gurations. We 
laim that some of the surfa
e phenomena found here 
an be present in
two-dimensional granular-media �uids.
PACS numbers: 64.70.Md,61.30.Hn,61.20.Gy
I. INTRODUCTION
The e�e
t of �uid 
on�nement on phase transitions is
nowadays an a
tive line of s
ienti�
 resear
h due to the
dire
t appli
ation of the theoreti
ally predi
ted surfa
e
phase diagrams in the nanote
hnology industry. Con�n-
ing simple �uids, su
h as hard [1, 2℄ or Lennard-Jones
[3℄ spheres, in a narrow slit geometry, results in a ri
h
phase behavior, whi
h has re
ently been studied in de-
tail. Liquid 
rystals 
on�ned in nanopores is another
typi
al example of systems with important appli
ations
in the industry of ele
troni
 devi
es. For this reason they
have been extensively studied using theoreti
al models
based on density-fun
tional theory. In parti
ular, 
apil-
lary phase transitions exhibited by a nemati
 �uid 
on-
�ned between hard walls[4, 5℄, or walls favoring a par-
ti
ular an
horing [6℄, have been predi
ted. When non-
uniform liquid-
rystal phases, su
h as the sme
ti
 phase,
are in
luded in the study of 
on�ned systems, the result-
ing surfa
e phase diagrams display a ri
h phenomenol-
ogy, whi
h in
ludes wetting transitions, the appearan
e
of sme
ti
 defe
ts [7℄, and layering transitions [8℄.
The e�e
t of 
on�nement on two-dimensional �uids is
also an interesting topi
 of resear
h. Langmuir monolay-
ers of lipids on the surfa
e of water have been extensively
studied in the last hundred years [9℄, and the dis
ov-
ery of stru
tures and phase transitions in these systems
has experien
ed a dramati
 evolution driven by the new
experimental te
hniques. Now it is possible to 
on�ne
these two-dimensional systems by external potentials and
study the in�uen
e of the 
on�nement on the mole
ular
Ele
troni
 address: yuri�math.u
3m.es
pa
king of surfa
e monolayers.
Another paradigm of two-dimensional systems where
the 
on�nement plays an important role is the pa
k-
ing stru
tures formed by parti
les in granular media
[10℄. The 
rystallization of a quasi-two-dimensional one-

omponent granular-disk �uid has re
ently been studied
experimentally [11℄. It was found that the properties
of the 
rystal stru
ture obtained (su
h as pa
king fra
-
tion, latti
e stru
ture, and Lindenman parameter) 
o-
in
ide with their 
ounterparts obtained from MC sim-
ulations of a hard disk �uid. Re
ent experiments have
found non-equilibrium steady states in a vibrated granu-
lar rod monolayer with tetrati
, nemati
 and sme
ti
 
or-
relations [12℄. Some of these textures are similar to the
equilibrium thermodynami
 states of two-dimensional
anisotropi
 �uids resulting from density-fun
tional 
al-

ulations [13℄ and MC simulations [14℄. It was shown by
several authors that the inherent states of some frozen
granular systems 
an be des
ribed by equilibrium sta-
tisti
al me
hani
s [15℄. Also, an experimental test of
the thermodynami
 approa
h to granular media has re-

ently been 
arried out [16℄. Con�ning two-dimensional
granular rods in di�erent geometries (
ir
ular, re
tangu-
lar, et
.) results in spontaneous formation of patterns,
with di�erent orientationally ordered textures and de-
fe
ts next to the 
ontainer [17℄. The authors of Ref. [18℄
have 
arried out MC simulations of a 
on�ned hard disk
�uid. They have found that the 
rystal phase fails to nu-

leate due to formation of sme
ti
 bands when the system
is 
on�ned [18℄. It would be interesting to devi
e an ex-
periment with 
on�ned granular disks with the aim of

omparing the properties of the non-uniform stationary
states with those obtained from the statisti
al me
hani
s
applied to the hard disk �uid.
The main purpose of this arti
le is the study of a 
on-
�ned two-dimensional hard-rod �uid. We are interested
http://arxiv.org/abs/0704.1467v1
mailto:yuri@math.uc3m.es
in the 
al
ulation of the surfa
e phase diagram of a hard
re
tangle (HR) �uid 
on�ned by a single or two hard
lines. We 
an think on a HR �uid as an experimental re-
alization of a system of hard 
ylinders 
on�ned between
two plates at a distan
e less than twi
e the 
ylinder di-
ameter. We suggest that some of the surfa
e phase tran-
sitions obtained here by applying the density fun
tional
formalism to a 
on�ned two-dimensional HR �uid should
be similar to the steady states of 
on�ned granular rods.
Some experiments are required to verify this hypothesis.
The paper is organized as follows. In Se
. II we
present the theoreti
al model: the fundamental-measure
density fun
tional applied to a HR �uid in the restri
ted-
orientation approximation. This se
tion is divided into
two subse
tions. In the �rst the model is parti
ularized
to the study of the bulk phases, while in the se
ond part
the theoreti
al expressions used in the 
al
ulations of the
thermodynami
 and stru
tural properties of the inter-
fa
es are presented. The results are presented in Se
.
III. First we study the bulk phase diagram of a HR �uid
with aspe
t ratio equal to 3, and then the resulting sur-
fa
e phase diagrams of a single wall-HR �uid interfa
e
and of the �uid 
on�ned between two hard lines are pre-
sented. Some 
on
lusions are drawn in Se
. IV.
II. THEORETICAL MODEL
In this se
tion we introdu
e the theoreti
al model used
in the 
al
ulations of the bulk and interfa
e equilibrium
phases. To study highly inhomogeneous phases su
h as
those resulting from the 
on�nement of a �uid in a nar-
row slit geometry or the solid phase with a high pa
king
fra
tion, we have used the Fundamental-Measure Theory
(FMT) applied to an anisotropi
 �uid of hard re
tangles.
It is well known that this formalism presents a great ad-
vantage over other te
hniques when dealing with highly
inhomogeneous phases, and that this is mainly due to the
fa
t that a basi
 requirement to 
onstru
t the FMT den-
sity fun
tional is that it 
onform with the dimensional

ross-over 
riterium [19, 20℄. To implement the 
al
u-
lations we have used the restri
ted-orientation approxi-
mation, where the axes of the re
tangles are restri
ted
to align only along the 
oordinate axes x or y. Thus,
the whole system is des
ribed in terms of density pro�les
ρν(r) (ν = x, y).
While the ideal part of the free energy density in re-
du
ed thermal units has the exa
t form
Φid(r) =
ρν(r) [ln ρν(r)− 1] , (1)
the FMT intera
tion part of the 2D HR �uid is approxi-
mated [20℄ by
Φexc(r) = −n0(r) ln [1− n2(r)] +
n1x(r)n1y(r)
1− n2(r)
, (2)
where the weighted densities nα's are 
al
ulated as
nα(r) =
ν=x,y
ρν ∗ ω
(r), (3)
and where the symbol ∗ stands for 
onvolution, i.e.,
ρν ∗ ω
dr′ρν(r
ν (r − r
′). The weights ω
are the 
hara
teristi
 fun
tions whose volume integrals

onstitute the fundamental measures of a single parti
le
(the edge lengths and surfa
e area). They are de�ned as
ω(0)ν (r) =
− |x|)δ(
− |y|), (4)
ω(1x)ν (r) =
− |x|)δ(
− |y|), (5)
ω(1y)ν (r) =
− |x|)Θ(
− |y|), (6)
ω(2)ν (r) = Θ(
− |x|)Θ(
− |y|), (7)
where σνµ = σ+(L− σ)δµν , with L and σ the length and
width of the re
tangle and δµν the Krone
ker fun
tion,
while δ(x) and Θ(x) are the Dira
 delta and Heaviside
fun
tions, respe
tively.
A. The bulk phases
To 
al
ulate the bulk phase diagram we need to mini-
mize the Helmholtz free energy fun
tional βF [{ρν(r)}] =
dr [Φid(r) + Φexc(r)] with respe
t to the density pro-
�les ρν(r). These density pro�les have the symmetries

orresponding to the equilibrium phases, whi
h 
an be
the isotropi
 or nemati
 �uids, the sme
ti
 phase (with
parti
les arranged in layers with their long axes pointing
perpendi
ular to the layers), the 
olumnar phase (with
long axes parallel to the layers), plasti
 solid (parti
les
lo
ated at the nodes of the square grid with averaged ori-
entational order parameter over the 
ell equal to zero),
and oriented solid (with both translational and orienta-
tional order). To take proper a

ount of all these possible
symmetries, we have used a Fourier-series expansion of
the density pro�les:
ρν(r) = ρ0xν
k=(0,0)
k1,k2
cos(q1x) cos(q2y), (8)
where we de�ned k ≡ (k1, k2) [with N = N(1, 1)℄,
q1 = 2πk1/dx, and q2 = 2πk2/dy are the wave ve
tor

omponents parallel to x and y axes respe
tively, and dx,
dy are the periods of the re
tangular grid along these di-
re
tions. α
k1,k2
are the Fourier amplitudes of the density
pro�le of the spe
ies ν with the 
onstraint α
0,0 = 1. ρ0 is
the average of the lo
al density over the 
ell, while xν is
the 
ell-averaged o

upan
y probability of spe
ies ν. The
Fourier series is trun
ated at that value N whi
h guar-
antees that α
N,N < 10
. With this parametrization the
weighted density 
an be 
al
ulated expli
ity as
nα(r) = ρ0
k1,k2
ω̂(α)ν (k) cos(q1x) cos(q2y), (9)
where ω̂
ν (k) are the Fourier transforms of the 
orre-
sponding weights, whi
h have the form
ω̂(0)ν (k) = χ0(q1σ
ν /2)χ0(q2σ
ν/2), (10)
ω̂(1x)ν (k) = σ
νχ1(q1σ
ν/2)χ0(q2σ
ν/2), (11)
ω̂(1y)ν (k) = σ
νχ0(q1σ
ν/2)χ1(q2σ
ν/2), (12)
ω̂(2)ν (k) = aχ1(q1σ
ν/2)χ1(q2σ
ν/2), (13)
Here a = Lσ is the surfa
e area of the parti
le, and
χ0(x) = cosx, χ1(x) = sin(x)/x. We have sele
ted the
orientational dire
tor parallel to y. Thus, the equilib-
rium sme
ti
 (
olumnar) phase should be found by min-
imizing the free energy with respe
t to the Fourier am-
plitudes α
0,k (α
k,0), the sme
ti
 (
olumnar) period dy
(dx) and the order parameter QN ∈ [−1, 1] [related to
the xν 's through the relations x‖,⊥ = (1 ±QN)/2 where
the symbols ‖,⊥ stand for parti
le alignment along y
and x respe
tively℄. For uniform phases [α
k1,k2
∀(k1, k2) 6= (0, 0)℄ QN 
oin
ides with the nemati
 order
parameter. The solid phase is to be found by minimiz-
ing the free energy with respe
t to all the Fourier ampli-
tudes α
k1,k2
, the 
rystal periods dx and dy, and the order
parameter QN in the 
ase of an orientationally ordered
solid. To measure the pa
king stru
ture and the orienta-
tional order of the bulk phases we use the lo
al density
and the order parameter pro�les, ρ(r) =
ν ρν(r), and
Q(r) = [ρy(r)− ρx(r)] /ρ(r) respe
tively.
B. The interfa
ial phases
As we want to study the hard wall-�uid interfa
e or the
HR �uid 
on�ned in a slit geometry, we have introdu
ed
the following external potential:
Vν(x) =
∞, x < σxν/2
0, x ≥ σxν/2,
for the semi-in�nite system, and
Vν(x) =
∞, x < σxν/2 and x > H − σ
0, σxν/2 ≤ x ≤ H − σ
ν /2,
for the slit geometry, where H is the slit width, and
the normal to the wall was sele
ted in the x dire
tion.
Note that this external potential represents a hard line
whi
h ex
ludes the 
enter of mass of parti
les at distan
es
less than their 
onta
t distan
es with the wall. In this
sense we 
an say that the external potential favors par-
allel alignment at the wall. This is in 
ontrast with the
favored homeotropi
 alignment usually 
onsidered in sev-
eral studies of three-dimensional liquid 
rystals 
on�ned
by a single or two walls (in parti
ular that of Ref. [8℄).
The one-dimensional equilibrium density pro�les ρν(x)
were found by minimizing the ex
ess surfa
e free energy
per unit length
+ P −
ρν(x) [µν − Vν(x)]
, (16)
where β = (kBT )
, Φ(x) = Φid(x) + Φexc(x), and µν
are the 
hemi
al potentials of spe
ies ν �xed at the bulk
�uid-phase value at in�nite distan
e from the wall, while
P is the �uid pressure. The 
hemi
al potential of the
bulk �uid phase is 
al
ulated, as usual, as µ =
xνµν ,
with xν the molar fra
tions of spe
ies ν. If the bulk phase
is an isotropi
 �uid then xν = 1/2, and µν = µ, ∀ν.
To measure the degree of interfa
ial order, we will
use the adsorption of the density pro�le, de�ned as
dx [ρ(x)− ρ(∞)], and the order parameter pro�le
Q(x).
The expression (16) 
oin
ides with the de�nition of the
surfa
e tension of the wall-�uid interfa
e for the semi-
in�nite 
ase, whi
h is approximately equal to half the
ex
ess surfa
e free-energy for the slit geometry when the
wall distan
e H is large enough to a

ommodate both
interfa
es.
To minimize the fun
tional given by (16), we have dis-

retized spa
e in the x dire
tion and minimize γ with
respe
t to ρν(xi) (xi ∈ [x0, xN ]) using the 
onjugate-
gradient algorithm.
III. RESULTS
In this se
tion we present the main results obtained
from the appli
ation of the theoreti
al model just de-
s
ribed to the study of surfa
e properties of a 2D HR
�uid. Parti
les were 
hosen to have aspe
t ratio κ ≡
L/σ = 3. This aspe
t ratio is 
hosen be
ause one of the
aims of the present work is the study of layered phases

on�ned by one or two walls. As we will show bellow for
κ = 3 the stable phase is the 
olumnar layered phase.
In the �rst subse
tion we will summarize the results
obtained in the 
al
ulation of the bulk phase diagram of
this system, while in the se
ond subse
tion we will fo
us
on the study of the surfa
e phase diagram.
A. Bulk phase diagram
We have minimized the free energy density of the HR
�uid, de�ned as Φ ≡ V −1
dr [Φid(r) + Φexc(r)], with
respe
t to the Fourier amplitudes, periods, and mean o
-

upan
y probability, as des
ribed in detail in Se
. II A.
The results are plotted in Fig. 1, where the free-energy
densities of all the stable and metastable phases found
are plotted as a fun
tion of the pa
king fra
tion η = ρ0a.
We have found, apart from the usual isotropi
 (I) and ne-
mati
 (N) phases, two di�erent sme
ti
 phases (Sm1, and
Sm2), a plasti
 solid (PS), perfe
tly oriented solid (OS),
and �nally the 
olumnar phase (C), whi
h is the stable
one in the whole range of pa
king fra
tions explored.
0.55 0.6 0.65 0.7 0.75
Figure 1: The res
aled free-energy density Φ∗ = Φ+2.9875−
5.8501η is plotted against the mean pa
king fra
tion for all
the stable and metastable phases found. These are: isotropi
(dashed line), nemati
 (dotted line), sme
ti
-1 and sme
ti
-
2 (dotted and dashed lines), plasti
 solid (dashed line la-
belled as PS), while the perfe
tly oriented solid and the

olumnar phases (labelled in the �gure as OS and C respe
-
tively) are plotted with solid lines. The open 
ir
le indi-

ates the isotropi
-nemati
 bifur
ation point; the open square,
the isotropi
-plasti
 solid bifur
ation point; and the solid 
ir-

les represent the 
oexisting pa
king fra
tions at isotropi
-

olumnar phase 
oexisten
e.
The 
oupling between the spatial and orientational de-
grees of freedom of the parti
les results in the presen
e of
phases (stable or metastable) with di�erent symmetries.
In Fig. (2) we have sket
hed some of the parti
le 
on-
�gurations 
orresponding to phases with 
olumnar (a),
sme
ti
-1 (b), sme
ti
-2 (
), and plasti
 solid (d) sym-
metries found from the numeri
al minimization of the
density fun
tional. The dire
tions of spatial periodi
ities
of ea
h phase have been depi
ted in the �gure.
In Fig. 3 (a) we have plotted the density and order-
parameter pro�les of the 
oexisting 
olumnar phase. The

olumnar phase is orientationally ordered in the y dire
-
tion with the long re
tangle axis pointing along the y axis,
while the periodi
ity of both density and order parame-
ter pro�les (whi
h are in phase) is along the x dire
tion
[see Fig. 3 (a)℄. The mean 
oexisten
e pa
king fra
tions
of the I and C phases are ηI = 0.57058 and ηC = 0.60310,
respe
tively while the period of the C phase, in units of
the HR width, was found to be dx/σ = 1.20102. In Fig.
3 (b) we have plotted the order parameter QN, and the
period of the 
olumnar phase as a fun
tion of the mean
pa
king fra
tion.
(c) (d)
Figure 2: Sket
h of parti
le 
on�gurations 
orresponding to
di�erent phases: 
olumnar (a), sme
ti
-1 (b), sme
ti
-2 (
),
and plasti
 solid (d) phases. The dire
tion of spatial period-
i
ities are labeled in the �gure.
To 
ompare the di�erent pa
kings of HR parti
les in
the metastable phases (found as the lo
al minima of the
free energy density) for a �xed mean pa
king fra
tion
η = 0.7, we have plotted the density and order-parameter
pro�les of the Sm1,2 [Fig. 4 (a) and (b)℄, and PS and OS
[Fig. 5 (a)�(
)℄ phases. As 
an be seen from Fig. 4
(a), the density pro�le of the Sm1 phase has two max-
ima per period. The less pronoun
ed maxima, lo
ated
at the interstitials, re�e
t the high population of parti-

les with long axes oriented parallel to the sme
ti
 layers
[see the sket
hed parti
le 
on�gurations in Fig. 2 (b)℄.
This alignment is also shown in the order-parameter pro-
�le, whi
h rea
hes high negative values at the intersti-
tial positions. This phase bears a strong resemblan
e
to the �ndings of Refs. [21℄ and [22℄ where the parti-

le equilibrium 
on�gurations in the 3D sme
ti
 phases
show the same pattern. As a 
onsequen
e of this (al-
ternating population of parti
les aligned perpendi
ular
�sharpest peak in the density pro�le� and parallel to the
layers), the sme
ti
 period in units of the parti
le length
is dy/L = 1.53025, higher than the sme
ti
 period of the
Sm2 phase (dy/L = 1.17935). The density and order-
parameter pro�les of the Sm2 are shown in Fig. 4 (b).
As 
an be seen from the �gure, these pro�les re�e
t the
usual pa
king in sme
ti
s, 
hara
terized by a single den-
sity peak with vanishingly small population of parti
les
in the interstitials, while the order parameter rea
hes its
maximum value at the position of the sme
ti
 layers [see
Fig. 2 (
) for the sket
hed parti
le 
on�gurations℄.
The density and order parameter pro�les of the PS
phase with mean pa
king fra
tion equal to 0.7 are plot-
-0.5 -0.25 0 0.25 0.5
0.5 0.55 0.6 0.65 0.7 0.75 0.8
Figure 3: (a): density ρ(x) (solid line) and order parameter
Q(x) (dashed line) pro�les of the 
olumnar phase at 
oex-
isten
e with the isotropi
 phase. (b): order parameter QN
and period of the 
olumnar phase against the mean pa
king
fra
tion.
ted in Fig. 5 (a) and (b). The plasti
 solid has the same
periodi
ity in the x and y dire
tion, i.e. dx = dy = d,
and the order parameter averaged over the unit 
ell is
stri
tly equal to zero. As we 
an see from Fig. 5 (b),
while the order parameter at the nodes of the square
latti
e is equal to zero, it rea
hes positive (negative) val-
ues at the (±0.5, 0) [(0,±0.5)℄ positions along the sides
of the 
ell (the same solution with the x and y dire
-
tions inter
hanged was found in the minimization of the
free energy). Finally, the density pro�le of the perfe
tly
aligned two-dimensional solid is plotted in Fig. 5 (
).
Although the phases des
ribed above are metastable
with respe
t to the 
olumnar phase, they 
an be stabi-
lized for di�erent values of the parti
le aspe
t ratio. A
detailed study of the 
omplete phase diagram, ne
essary
to elu
idate this point, is a work in progress.
We now pro
eed to make a 
omparison between the
results for the 2D Zwanzig model with κ = 3 obtained
-0.5 -0.25 0 0.25 0.5
-0.5 -0.25 0 0.25 0.5
Figure 4: Density (solid line) and order parameter (dashed
line) pro�les of the sme
ti
-1 (a) and sme
ti
-2 (b) metastable
phases for a value of mean pa
king fra
tion �xed at 0.7.
above and those for hard parallelepipeds with restri
ted
orientations and the same value of κ [23℄. This 
ompari-
son will show the di�eren
es in phase behaviour between
three and two dimensions as predi
ted by Fundamental-
Measure Theory (whi
h, as already pointed out, 
on-
forms with the dimensional 
rossover 
riterion). As
shown in Ref. [23℄, hard parallelepipeds exhibit a se
ond-
order phase transition between isotropi
 and plasti
 solid
phases. As density in
reases the system goes to a dis
oti
sme
ti
 phase (
on�rmed by simulations) via a �rst-order
phase transition, whi
h in turn dis
ontinuously 
hanges
to a 
olumnar phase and then to an oriented solid. By

ontrast, the present model shows that, in two dimen-
sions, the isotropi
 phase exhibits a �rst-order transition
to a 
olumnar phase that is stable until very high pa
k-
ing fra
tions (more stable that plasti
, oriented solid and
di�erent sme
ti
 phases). As a 
onsequen
e, one expe
ts
that the 
orresponding surfa
e phase diagrams be also
Figure 5: Density (a) and order-parameter (b) pro�les of the
plasti
 solid phase. (
): Density pro�le of the perfe
tly ori-
ented solid.
di�erent.
B. Surfa
e phase diagram
In this se
tion we deal with surfa
e phenomena. In the
�rst part we will 
on
entrate on the semi-in�nite wall-
isotropi
 interfa
e of a HR �uid, while in the se
ond part
we will fo
us on the slit geometry. We will demostrate
the presen
e of 
omplete wetting, 
apillary ordering and
layering transitions in the 
on�ned two dimensional hard
rod �uid. For a detailed dis
ussion on general grounds of
the phase behavior and 
riti
al phenomena of a 
on�ned
by a single wall �uid see Ref. [24℄.
1. The wall-�uid interfa
e
The intera
tion between the isotropi
 �uid phase and a
hard wall was studied by 
al
ulating the one-dimensional
equilibrium density ρ(x) and order-parameter Q(x) pro-
�les through the ex
ess surfa
e free-energy minimization
[see Eq. (16)℄. The 
hemi
al potential µ of the �uid phase
at in�nite distan
e from the wall was varied within the
range of isotropi
-phase stability, i.e. µ ∈ [−∞, µ0] (µ0
being the value at whi
h the I-C phase transition o

urs).
It is well known that the presen
e of a hard wall in a sys-
tem of elongated parti
les indu
es parallel alignment of
the parti
le axes with respe
t to the wall [25, 26℄. This
preferential alignment is a result of the entropi
 depletion
e�e
t. In the parallel 
on�guration, the 
enters of mass
of the parti
les are mu
h 
loser to the wall, so the gain
in volume per parti
le is larger and, as a 
onsequen
e,
the 
on�gurational entropy of the system is maximized.
This e�e
t is responsible for the o

urren
e of a biaxial
nemati
 phase whi
h breaks the orientational symmetry
in a three-dimensional nemati
 �uid [4℄. The same deple-
tion me
hanism is at work in 2D, as we will show below.
The results from the minimization are shown in Figs. 6
(a) and (b) for an undersaturation of β∆µ = −1.1×10−4.
As we 
an see from the �gure, the density and order-
parameter pro�les indi
ate 
olumnar order near the wall,
whi
h propagates several 
olumnar periods into the �uid
phase. The wall-�uid intera
tion enhan
es the orienta-
tional order near the surfa
e and the adsorption of par-
ti
les, 
reating a stru
tured layer with 
olumnar-phase
symmetry whi
h grows in width with in
reasing 
hem-
i
al potential and diverges at µ = µ0. Thus, 
omplete
wetting by a 
olumnar phase o

urs at the wall-isotropi
interfa
e. This result is shown in Fig. 7 (a) where the ex-

ess surfa
e free-energy γ and the adsorption 
oe�
ient Γ
are plotted against ∆µ = µ−µ0. As we 
an see, Γ grows

ontinuously, ultimately diverging logarithmi
ally with
∆µ (see inset of �gure). The ex
ess surfa
e free energy γ
has a maximum, and at this point the adsorption passes
through zero. This result is dire
tly related to the interfa-

ial Gibbs-Duhem equation, Γ = −dγ/dµ, whi
h relates
the adsorption 
oe�
ient with the �rst derivative of the
ex
ess surfa
e free-energy with respe
t to bulk 
hemi
al
potential. At µ0 the ex
ess surfa
e energy is equal to the
wall-isotropi
 surfa
e tension γWI, whi
h is in turn equal
to the sum of wall-
olumnar and 
olumnar-isotropi
 sur-
fa
e tensions, γ(µ0) = γWI = γWC + γCI (the Young's
equation for 
omplete wetting).
0 10 20 30 40 50 60
0 10 20 30 40 50 60
Figure 6: Density (a) and order parameter (b) pro�les of the
wall-isotropi
 �uid interfa
e. The undersaturation is �xed to
β∆µ = −1.1× 10−4.
We have 
arried out a logarithmi
 �t of the adsorp-
tion 
oe�
ient with respe
t to undersaturation β∆µ =
β (µ− µ0), and we �nd that Γσ = τ1+τ2 ln [β|∆µ|], with
τ1 = 0.02387 and τ2 = −0.03396. Then, integrating the
interfa
ial Gibbs-Duhem relation Γ = −dγ/dµ, we �nd
the expression
βγσ ≈ βγWIσ − [τ1 + τ2 (ln (β|∆µ|) − 1)]β∆µ, (17)
whi
h approximates the ex
ess surfa
e free energy near

omplete wetting. The above expression is plotted
against β∆µ in Fig. 7 (b), where the results from dire
t

al
ulation of βγσ, using the equilibrium density pro�les
obtained, are also plotted. As we 
an see the agreement
is ex
ellent even for relatively high values of undersatu-
ration.
0 2 4 6 8
σ 10-4 10-2 100
0.4(a)
0 0.01 0.02 0.03 0.04 0.05
0.136
0.138
0.142
0.144
Figure 7: (a): ex
ess surfa
e free energy (solid line) and ad-
sorption 
oe�
ient (dashed line), in redu
ed units, against
β∆µ. The inset shows Γσ vs. β∆µ in logarithmi
 s
ale. (b):
ex
ess surfa
e free energy vs. βµ in the neighborhood of zero
undersaturation. The open 
ir
les show the values obtained
from the numeri
al minimization, while the solid line repre-
sents the analyti
 
urve obtained by integrating the interfa
ial
Gibbs-Duhem relation with the �tted logarithmi
 dependen
e
of the adsorption 
oe�
ient (see text). The solid 
ir
le shows
the value of the W-I surfa
e tension βγWIσ = 0.13822
To 
al
ulate the stru
tural and thermodynami
 prop-
erties of the 
olumnar-isotropi
 interfa
e, we have imple-
mented a numeri
al s
heme already used in Ref. [27℄,

onsisting of minimizing the surfa
e ex
ess free energy γ
in a box of width h 
ontaining a stripe of a few 
olum-
nar layers surrounded by isotropi
 material with periodi
boundary 
onditions. h is 
hosen su
h that the density
pro�les 
an easily a

ommodate the two interfa
es and
go to the 
oexisten
e �uid density at the periodi
 bound-
ary. A typi
al result from this 
al
ulation is plotted in
Fig. 8 (a) and (b) for the density and order-parameter
pro�les, respe
tively. Thus, the I-C interfa
ial tension

an be 
al
ulated as half the ex
ess surfa
e free energy
resulting from the minimization. We have found a value
of βγICσ = 0.00672.
-60 -40 -20 0 20 40 60
-60 -40 -20 0 20 40 60
Figure 8: Density (a) and order-parameter (b) pro�les of a
numeri
al box 
ontaining two isotropi
-
olumnar interfa
es.
Finally, to verify that Young's law for 
omplete wet-
ting holds, we need to 
al
ulate the surfa
e tension of
the wall-
olumnar interfa
e. To 
onstru
t density pro-
�les 
ompatible with this semi-in�nite interfa
e, one has
to establish a boundary, at the side of the 
omputational
box opposite to the wall, and pla
e, beyond the bound-
ary and into the bulk, a periodi
ally stru
tured pro�le,

hoosing the phase (i.e. the value of the pro�le at the
boundary) arbitrarily within the bulk period. Although
this re
ipe 
an in prin
iple be implemented, we have 
ho-
sen to �x bulk I-C 
oexisten
e 
onditions in a 
on�ned

olumnar phase and 
al
ulate the density pro�le of the
system 
on�ned between two walls; the separation be-
tween the walls was 
hosen large enough so that the ef-
fe
ts of having a �nite interfa
e penetration length 
aused
by the presen
e of the 
on�ned external potential 
an be
negle
ted. Also, in order to ensure that 
ommensurabil-
ity e�e
ts 
an be ignored, the distan
e between the walls
was set to a (large) integer number of equilibrium periods
of the 
olumnar phase. The results from these 
al
ula-
tions are plotted in Fig. 9 (a) and (b). The W-C surfa
e
tension 
al
ulated as half the value of the ex
ess surfa
e
free energy results in βγWCσ = 0.13150, 
ompatible with
Young's law in 
onditions of 
omplete wetting of the W-I
interfa
e by the 
olumnar phase.
0 20 40 60 80 100 120
0 20 40 60 80 100 120
Figure 9: The density (a) and order parameter (b) pro�les
of two wall-
olumnar interfa
es
2. Capillary ordering
This se
tion is devoted to a study of the e�e
t of 
on-
�nement of a 2D HR �uid on the thermodynami
 and
stru
tural properties of the �uid. In parti
ular, we are
interested in the enhan
ement of the orientational and
layering ordering due to 
on�nement, and the 
ommen-
surability e�e
ts exhibited by a layered phase sandwi
hed
between two hard walls at a distan
e that may or may
not 
ommensurate with the period of the bulk 
olumnar
phase. It is well known that, under 
ertain 
ir
umstan
es
(related to the nature of the �uid-�uid and surfa
e-�uid
intera
tions), a �uid inside a pore 
an exhibit 
apillary
�rst-order phase transitions between two di�erent phases
at a 
hemi
al potential below the bulk 
oexisten
e value.
An example of this phenomenon is the re
ently studied

apillary nematization [4℄ and sme
tization [8℄ of a liquid

rystal �uid inside a pore. The bulk 
ondensed phase
may have uniform (nemati
) or nonuniform density pro-
�les. For the latter 
ase, 
apillary layering transitions be-
tween interfa
ial phases with di�erent number of sme
ti
layers [8℄ 
an also be found. Here we will show that these

apillary and layering phase transitions are not unique to
3D system. They are also present in 2D anisotropi
 �uids
whi
h 
an stabilize layered phases with di�erent spatial
symmetries, su
h as the 
olumnar phase.
With a view to �nding the e�e
ts of 
on�nement on

olumnar ordering in a HR �uid, we have minimized the
ex
ess surfa
e free energy with respe
t to the density pro-
�le for the parti
ular 
ase of HR's with κ = 3. The �uid
is 
on�ned by two hard walls at a distan
e H/σ = 30 (in
units of the parti
le width). As already pointed out, hard
walls favor alignment parallel to the wall, as well as ad-
sorption of parti
les at both surfa
es (density and order
parameters at 
onta
t are mu
h higher than their bulk
values). This 
oupled translational-orientational order-
ing near the surfa
es propagates into the �uid, 
reating

olumnar ordering. We have found that for low values
of the 
hemi
al potential of the bath the density pro�le
is stru
tureless (ex
ept just at the wall 
onta
t), similar
to the bulk isotropi
 phase. In
reasing the 
hemi
al po-
tential several damped 
olumnar peaks appear near the
wall in a 
ontinuous fashion, i.e. with their heights in-

reasing 
ontinuously. At some value of the 
hemi
al po-
tential, the system exhibits a �rst-order phase transition
between a phase with highly damped 
olumnar peaks to
a new phase with mu
h stronger 
olumnar ordering even
at the 
enter of the pore. The typi
al density and order-
parameter pro�les of both interfa
ial phases are shown in
Fig. 10 (a)-(d). Although the less-ordered phase exhibits
strong os
illations in both density and order-parameter
pro�les, the peak amplitudes are damped into the pore
faster than those of the higher ordered phase. We will
take the 
onvention to 
all the �rst `isotropi
', and the
se
ond `
olumnar' surfa
e phases. This 
onvention is jus-
ti�ed by the fa
t that, just before the transition des
ribed
above, 
olumnar ordering in
reases 
ontinuously, start-
ing from an isotropi
-like density pro�le, as the 
hemi-

al potential is in
reased. Thus we 
annot tra
e out a
de�nite boundary (a value for µ below that 
orrespond-
ing to �rst order phase transition) below or above whi
h
the pro�le inside the pore 
an be 
onsidered isotropi
 or

olumnar. Only the �rst order phase transition des
ribed
above 
an really distinguish two di�erent surfa
e phases,
one of them less ordered (following our 
onvention, the
isotropi
 phase) than the other (the 
olumnar phase). As
we 
an see in the �gure, the latter has 25 
olumnar peaks.
The transition point is 
al
ulated from the dis
ontinu-
ity in the �rst derivative of the ex
ess surfa
e free en-
0 5 10 15 20 25 30
0 5 10 15 20 25 30
0 5 10 15 20 25 30
0 5 10 15 20 25 30
Figure 10: Isotropi
 (a)-(b) and 
olumnar (
)-(d) phases that

oexist at the same 
hemi
al potential bellow µ0. (a), (
):
Density pro�les. (b), (d): Order-parameter pro�les.
ergy with respe
t to the bulk pa
king fra
tion η. The

orresponding plot is shown in Fig 11 (a). At this point
the adsorption 
oe�
ient jumps dis
ontinuously from the
less- (the damped 
olumnar) to the higher-ordered phase
[see Fig. 11 (b)℄.
This surfa
e transition point is lo
ated below the bulk
isotropi
-
olumnar phase transition [see Fig. 11 (a)℄,
0.54 0.55 0.56 0.57 0.58
0.54 0.55 0.56 0.57 0.58
Figure 11: Ex
ess surfa
e free energy (a) and adsorption 
oef-
�
ient (b) against pa
king fra
tion of the bulk isotropi
 �uid.
In the �gure at top, the solid 
ir
le represents the transition
point between both interfa
ial phases, while open square in-
di
ates the point 
orresponding to the bulk 
oexisten
e value
for isotropi
 and 
olumnar phases.
showing the presen
e of 
olumnar-order enhan
ement in
the pore. On further in
reasing the 
hemi
al potential
up to a su�
iently high value (above µ0), we �nd a �rst-
order layering transition between two interfa
ial 
olum-
nar phases whi
h di�er by just a single 
olumnar layer.
The behavior of the ex
ess surfa
e free energy and the
adsorption 
oe�
ient is similar to that shown in Fig. 11
(a) and (b). Alternatively we 
an �nd the transition from
n−1 to n 
olumnar layers by �xing the 
hemi
al potential
and in
reasing the pore width H .
The two surfa
e phase transitions des
ribed above,
namely �rst-order 
apillary I-C ordering and (n − 1)�
n layering transition, are 
onne
ted in the µ−H surfa
e
phase diagram through the pe
uliar stru
ture shown in
Fig. 12.
The parabola below the bulk transition line 
orre-
sponds to �rst-order transition lines separating regions
of stability of the isotropi
 and the 
olumnar interfa
ial
phases, while the straight lines indi
ate layering transi-
tions. In
reasing the 
hemi
al potential from low values
to those 
orresponding to the parabola, the density pro-
�les always 
hange 
ontinously from a stru
tureless to
damped 
olumnar density pro�le. Both types of transi-
28 28.5 29 29.5 30
24 C25
Figure 12: µ − H surfa
e phase diagram showing �rst-order

apillary 
olumnar ordering and layering transitions. The
pore width 
overs a range whi
h goes from 23 to 25 
olum-
nar layers, as labeled in the �gure. The open 
ir
les indi
ate
the 
ases 
hosen for 
al
ulations, while the solid line is a 
u-
bi
 spline interpolation. The horizontal dashed line shows the
value of the bulk 
hemi
al potential at the isotropi
-
olumnar
phase 
oexisten
e.
tions (the isotropi
-
olumnar and n−1-n layering transi-
tions) 
oales
e in triple points, two of whi
h are shown in
Fig. 12. At the triple points an isotropi
 and two 
olum-
nar interfa
ial phases with n− 1 and n layers 
oexist in
equilibrium. The set of 
onne
ted of Fig. 12 are similar
to those found in MC simulations of the 
on�ned hard-
sphere �uid [2℄. In this work the authors have shown the
existen
e of 
apillary freezing of the HS �uid, 
on�ned
in the slit geometry, for 
hemi
al potential values below
the bulk freezing transition. The transitions lines in the
µ−H surfa
e phase diagram follow the same topology of

onne
ted set of parabolas as found in our system.
Some of the topologi
al features of this surfa
e phase
diagram 
an be elu
idated from the Clausius-Clapeyron
equation as applied to the interfa
ial 
oexisten
e lines.
The ex
ess surfa
e free energy γ(µ,H) along 
oexisten
e
is a fun
tion of two variables, the 
hemi
al potential µ,
and the pore width H . Thus, in�nitesimal 
hanges in
these variables along the 
oexisten
e 
urve are related
through the equation
dγα − dγβ = ∆
dH = 0, (18)
where the 
oexisting 
ondition γα = γβ (for
α, β =I,Cn−1,Cn) was used, and ∆u = uα − uβ for any
fun
tion u. Using the interfa
ial Gibbs-Duhem equation
∂γ/∂µ = −Γ and the de�nition of the solvation for
e
f = −∂γ/∂H , we arrive at
, (19)
whi
h relates the �rst derivative of the 
hemi
al potential
with respe
t to the pore width with 
hanges in the sol-
vation for
e and in the adsorption 
oe�
ient at the tran-
sition point. The negative slope of the layering 
urves is
a dire
t result of Eq. (19), as the in
rement in the ad-
sorption is always positive for the (n − 1) → n layering
transition, while the 
hange in the solvation for
e is also
positive (the latter 
an be interpreted as an in
rement
with respe
t to the bulk of the ex
ess surfa
e pressure,
whi
h is obviously larger for the phase with n layers).
For values of the pore width that 
ommensurate with an
integer number of 
olumnar periods of the bulk 
olumnar
phase, the solvation for
e be
omes zero and we get a min-
imum in the I-C 
apillary transition 
urve (see Fig. 12).
At ea
h side of the minimum the solvation for
e 
hange
the sign to positive (left side) or negative (right side) as
we 
ompress or expand the �lm, respe
tively, while the

hange in adsorption remains positive.
The Kelvin equation for 
apillary 
ondensation in a slit
geometry relates the undersaturation in 
hemi
al poten-
tial with pore width H as
∆µ = µ(H)− µ0 = −
(ρα − ρβ)H
, (20)
where ρα and ρβ are the bulk 
oexisting densities of
phases α and β (α being the 
ondensed phase), while
γαβ is the surfa
e tension of the 
orresponding interfa
e.
It was assumed that 
omplete wetting by the α phase
o

urs at the W-β interfa
e. For a detailed dis
ussion of
the Kelvin equation in the 
ontext of liquid 
rystal phase
transitions see Ref. [28℄. Applying this equation using
H/σ = 28.88 (the lo
ation of the minimum in the µ−H
phase diagram of Fig. 12), we obtain an undersaturation
β∆µ = −0.0429, while its real value is β∆µ = −0.1255.
In the derivation of the Kelvin equation, deviations from
the bulk stru
ture of the density pro�le arising from
the 
on�nement by the external potential are negle
ted.
Also, the elasti
 energy resulting from the 
ompression or
expansion of a layered phase 
on�ned between two walls
is not taken into a

ount. These e�e
ts might be respon-
sible for the di�eren
es found between our 
al
ulations
and the estimation based on the Kelvin equation. We
have 
he
ked that the sequen
e of minima in the µ−H
phase diagram tends to µ0 as H → ∞, a result predi
ted
by Eq. (20).
Refs. [4℄ and [8℄ showed that the 
apillary nemati-
zation line of the 
on�ned liquid-
rystal �uid ends in a

riti
al point for small values of the pore width. In or-
der to study how the topology of the surfa
e phase dia-
gram 
hanges in the regime of small pore widths, we have

arried out the 
orresponding 
al
ulations of interfa
ial
stru
ture. We have found that the I-C 
apillary ordering
transition 
hanges at some parti
ular value of H (near
its maximum undersaturation represented by the mini-
mum in the I-C interfa
e 
oexisting 
urve) from �rst to
se
ond order. For lower values of H two 
riti
al points
emerge from this single point, the distan
e between them
in
reasing. In Fig. 13 one of these s
enarios is shown.
As we 
an see, there is a range of values of H (near
the triple points) where the �rst-order 
apillary order-
ing transitions are still present but, between the 
riti
al
points belonging to di�erent layering bran
hes, 
olumnar
ordering grows 
ontinously from the isotropi
 (damped

olumnar interfa
ial phase) to a highly-ordered 
olum-
nar phase. Layering transitions are always present even
for very small H , as will be shown below. An interest-
ing feature of this phase diagram is that the lo
ation of
the triple points moves above the bulk 
oexisten
e value
µ0. This indi
ates that the interfa
ial 
olumnar phase
just below the triple points 
an be unstable for values of

hemi
al potentials 
orresponding to those of 
olumnar-
phase stability at bulk (similar to the 
apillary evapora-
tion of the 
on�ned �uid). For wide enough slits (those
for whi
h the parabolas are 
onne
ted) the triple points
are pra
ti
ally lo
ated at µ0, as 
an be observed from
Fig. 12.
18 18.5 19 19.5 20 20.5
Figure 13: µ−H surfa
e phase diagram showing 
riti
al points
(�lled 
ir
les). Number of 
olumnar layers are indi
ated as
subs
ripts.
For even smaller values of H , only layering transitions
remain; these end in 
riti
al points lo
ated above µ0, as
Fig. 14 shows.
When the width H is su
h that the pore 
an only a
-

ommodate one parti
le with its long axis perpendi
u-
lar to the wall (or not more than four or three parti
les
aligned parallel to the wall) the system is near the one-
dimensional limit. It is known that hard-
ore systems in
this limit do not exhibit �rst-order phase transitions, but
even for very narrow slits we 
an still �nd �rst-order tran-
sitions at whi
h the density pro�le experien
es an abrupt

hange inside the pore. In Fig. 15 (a) and (b) we show
two 
oexisting density pro�les 
orresponding to oversat-
urations, β∆µ = 0.51760 and β∆µ = 0.72836, and pore
widths H/σ = 4.32 and H/σ = 3.14 in (a) and (b), re-
spe
tively. The �uid inside the pore undergoes a phase
transition, whi
h dramati
ally 
hanges the stru
ture of
the interfa
ial density pro�les by in
reasing the heigh of
11.4 11.45 11.5 11.55 11.6
7.8 7.85 7.9 7.95 8
Figure 14: µ −H surfa
e phase diagrams for small values of
four [Fig. 15(a)℄ or three [(b)℄ density peaks inside the
pore.
IV. CONCLUSIONS
In this arti
le we have shown that 2D �uids 
omposed
of anisotropi
 parti
les intera
ting via hard-
ore repul-
sion and 
on�ned in a slit geometry exhibit a 
omplex
and ri
h interfa
ial phase behavior. Apart from �rst-
order 
apillary 
olumnar ordering, we have also found
layering transitions in this system. These results are sim-
ilar to those found in 3D liquid-
rystal �uids 
on�ned in
a pore, where 
apillary sme
tization and layering phe-
nomena were also found [8℄. In view of these similarities,
we 
an extra
t the 
on
lusion that, independent of the
system dimensionality and the pe
uliarities of the layered
phases, either sme
ti
 or 
olumnar, if the �uid-wall inter-
a
tion enhan
es layered interfa
e ordering (homeotropi
in 
ase of sme
ti
 phases, and the entropi
ally favored
parallel alignment for the 
olumnar phase), 
ompatible
with the equilibrium bulk phase, then the 
on�ned �uid
exhibits the interfa
ial phase transitions des
ribed above.
In this study we have used as a model a hard-re
tangle
�uid, and the density and the order-parameter pro-
�les were 
al
ulated by minimizing the ex
ess surfa
e
free-energy fun
tional resulting from the Fundamental-
Measure Theory applied to the two-dimensional Zwanzig
model. The orientational degrees of freedom were
dis
retized, in order to take advantage of having a
free-energy fun
tional whi
h redu
es to the exa
t one-
dimensional fun
tional when the density pro�le is 
on-
strained to lie along a line. This property is 
ru
ial
to study strongly 
on�ned �uids (as is the 
ase in this
study), in parti
ular when the pore width has only a few
parti
le diameters in width.
As already pointed out in Se
. I, some experiments
had shown profound similarities between parti
le 
on-
�gurations obtained as stationary states of systems of
anisotropi
 grains and those 
orresponding to the equi-
0 1 2 3 4 5
0 0.8 1.6 2.4 3.2
Figure 15: (a): density pro�les of two 
oexisting phases
(shown with solid and dashed lines) at β∆µ = 0.5176. The
pore width is H/σ = 4.32. (b): same as in (a) but for a pore
with H/σ = 3.14 and for β∆µ = 0.72836.
librium states obtained by density fun
tional minimiza-
tion [12℄. These similarities 
an be explained by applying
a maximum-entropy prin
iple on granular 
olle
tions of
parti
les, i.e. for a �xed pa
king fra
tion, externally-
indu
ed vibrational motion for
es the system to explore
those stationary states whi
h maximize the 
on�gura-
tional entropy (sin
e the grains 
annot overlap). Of

ourse, equilibrium statisti
al me
hani
s is unable to pro-
pose an equation of state for granular matter, but it 
ould
be possible to predi
t that granular matter 
omposed of
anisotropi
 parti
les and 
on�ned between parallel walls
may support a stationary texture 
onsisting of layers of
parti
les oriented parallel to the wall. The manner in
whi
h the grain orientations propagate into the 
ontainer
would depend on the average pa
king fra
tion and on
the frequen
y of the external for
e. Only at this qualita-
tive level 
an we give some insight into possible 
omplete
wetting phenomena and 
apillary ordering in granular
rod �uids 
on�ned between two horizontal plates at a
distan
e slightly larger than the parti
le dimensions in
the verti
al dire
tion (thus simulating a two-dimensional
system), and also 
on�ned by one or two verti
al planes
(these playing the role of hard walls).
Some 
al
ulations (not shown here) on the 2D HR �uid
show that, for di�erent aspe
t ratios, 2D sme
ti
 and

rystal phases 
an be stable over some range of pa
k-
ing fra
tions. It would be interesting to explore whether

on�nement suppresses or enhan
es bulk ordering, and
to study the 
hanges in the surfa
e phase diagram when
phases of di�erent symmetries are in
luded. Work along
this dire
tion is 
urrently in progress.
A
knowledgments
I thank D. de las Heras, E. Velas
o, and L. Mederos for
useful dis
ussions, and E. Velas
o for a 
riti
al reading
of the manus
ript. The author gratefully a
knowledges
�nan
ial support from Ministerio de Edu
a
ión y Cien
ia
under grants No. BFM2003-0180 and from Comunidad
Autónoma de Madrid (S-0505/ESP-0299) and (UC3M-
FI-05-007). The author was supported by a Ramón y
Cajal resear
h 
ontra
t from the Ministerio de Edu
a
ión
y Cien
ia.
[1℄ M. S
hmidt, and H. Löwen, Phys. Rev. Lett. 76, 4552
(1996); Phys. Rev. E 55, 7228 (1997).
[2℄ M. Dijkstra, Phys. Rev. Lett. 93, 108303 (2004); A. For-
tini, and M. Dijkstra, J. Phys. Condes Matter 18, L371
(2006).
[3℄ L. Salama
ha, A. Patrykiejew, S. Sokolowski, and K.
Binder, Eur. Phys. J. E 13, 261 (2004); L. Salama
ha,
A. Patrykiejew, S. Sokolowski, Eur. Phys. J. E 18, 425
(2005).
[4℄ R. van Roij, M. Dijkstra, and R. Evans, Europhys. Lett.
49, 350 (2000); M. Dijkstra, R. van Roij, and R. Evans,
Phys.Rev. E 63, 051703 (2001)
[5℄ L. Harnau, and S. Dietri
h, Phys. Rev. E 66, 051702
(2002).
[6℄ I. Rodriguez-Pon
e, J. M. Romero-Enrique, E. Velas
o,
L. Mederos, and L. F. Rull, J. Phys.: Condens. Mat-
ter 12, A363 (2000); I. Rodriguez-Pon
e, J. M. Romero-
Enrique, and L. F. Rull, Phys. Rev. E 64, 051704 (2001)
[7℄ R. E. Webster, N. J. Mottram, and D. J. Cleaver, Phys.
Rev. E. 68, 021706 (2003); Z. Kutnjak, S. Kralj, G. La-
hajnar, and S. Zumer, Phys. Rev. E 70, 051703 (2004).
[8℄ D. de las Heras, E. Velas
o, and L. Mederos, Phys. Rev.
Lett. 94, 017801 (2005); Phys. Rev. E 74, 011709 (2006).
[9℄ V. M. Kaganer, H. Mohwald, and P. Dutta, Rev. Mod.
Phys. 71, 779 (1999).
[10℄ I. S. Aranson, and L. S. Tsimring, Rev. Mod. Phys. 78,
641 (2006).
[11℄ P. M. Reis, R. A. Ingale, and M. D. Shattu
k, Phys. Rev.
Lett. 96, 258001 (2006).
[12℄ V. Narayan, N. Menon, and S. Ramaswamy, J. Stat.
Me
h. P01005 (2006)
[13℄ Y. Martínez-Ratón, E. Velas
o, and L. Mederos, J. Chem.
Phys. 122, 064903 (2005); J. Chem. Phys. 125, 014501
(2006).
[14℄ A. Donev, J. Burton, F. H. Stillinger, and S. Torquato,
Phys. Rev. B 73, 054109 (2006).
[15℄ A. Fierro, M. Ni
odemi, and A. Coniglio, Eurphys. Lett.
59, 642 (2002); Phys. Rev. E 66, 061301 (2002); A.
Coniglio, A. Fierro, and N. Ni
odemi, Eur. Phys. J. E
9, 219 (2002).
[16℄ D. S. Dean, and A. Lefevre, Phys. Rev. Lett 90, 198301
(2003).
[17℄ J. Galanis, D. Harries, and D. L. Sa
kket, Phys. Rev.
Lett. 96, 028002 (2006).
[18℄ D. Chaudhuri, and S. Sengupta, Phys. Rev. Lett. 93,
115702 (2004).
[19℄ Y. Rosenfeld, M. S
hmidt, H. Löwen, and P. Tarazona,
J. Phys.: Condens. Matt. 8, L577 (1996); Phys. Rev. E
55, 4245 (1997); P. Tarazona, and Y. Rosenfeld, Phys.
Rev. E 55, R4873 (1997).
[20℄ J. A. Cuesta and Y. Martínez�Ratón, Phys. Rev. Lett.
78, 3681 (1997); J. Chem. Phys. 107, 6379 (1997).
[21℄ R. van Roij, P. Bolhuis, B. Mulder, and D. Frenkel, Phys.
Rev. E 52, R1277 (1995).
[22℄ J. S. van Duijneveldt, and M. P. Allen, Mole
. Phys. 90,
243 (1997).
[23℄ Y. Martínez-Ratón, Phys. Rev. E 69, 061712 (2004).
[24℄ R. Lipowsky, J. Appl. Phys. 55, 2485 (1984).
[25℄ A. Poniewierski, and R. Holyst, Phys. Rev. A 38, 3721
(1988).
[26℄ A. Chrzanowska, P. I. C. Teixeira, H. Ehrentraut, and D.
J. Cleaver, J. Phys.: Condens. Matter 13, 4715 (2001).
[27℄ Y. Martínez-Ratón, A. M. Somoza, L. Mederos, and D. E.
Sullivan, Faraday Dis
uss. 104, 111 (1996); Y. Martinez,
A. M. Somoza, L. Mederos, and D. E. Sullivan, Phys.
Rev. E 53 2466 (1996).
[28℄ T. J. Slu
kin and A. Poniewierski, Mole
. Cryst. Liq.
Cryst. 179, 349 (1990).
