Density profiles of Ar adsorbed in slits of CO_2: spontaneous symmetry breaking

Abstract

A recently reported symmetry breaking of density profiles of fluid argon confined by two parallel solid walls of carbon dioxide is studied. The calculations are performed in the framework of a nonlocal density functional theory. It is shown that the existence of such asymmetrical solutions is restricted to a special choice for the adsorption potential, where the attraction of the solid-fluid interaction is reduced by the introduction of a hard-wall repulsion. The behavior as a function of the slit's width is also discussed. All the results are placed in the context of the current knowledge on this matter. Comment: Text plus 8 figures

Full text

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arXiv:0710.5784v1 [cond-mat.soft] 31 Oct 2007
asymmetric-br.tex
Density profiles of Ar adsorbed in slits of CO2: spontaneous symmetry breaking
Leszek Szybisz
Laboratorio TANDAR, Departamento de F´ ısica, Comisi´ on Nacional de Energ´ ıa At´ omica,
Av. del Libertador 8250, RA–1429 Buenos Aires, Argentina
Departamento de F´ ısica, Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires, Ciudad Universitaria, RA–1428 Buenos Aires, Argentina and
Consejo Nacional de Investigaciones Cient´ ıficas y T´ ecnicas,
Av. Rivadavia 1917, RA–1033 Buenos Aires, Argentina
Salvador A. Sartarelli
Instituto de Desarrollo Humano, Universidad Nacional de General Sarmiento,
Gutierrez 1150, RA–1663 San Miguel, Argentina
(Dated: February 5, 2008)
A recently reported symmetry breaking of density profiles of fluid argon confined by two parallel
solid walls of carbon dioxide is studied. The calculations are performed in the framework of a
nonlocal density functional theory. It is shown that the existence of such asymmetrical solutions
is restricted to a special choice for the adsorption potential, where the attraction of the solid-fluid
interaction is reduced by the introduction of a hard-wall repulsion. The behavior as a function of
the slit’s width is also discussed. All the results are placed in the context of the current knowledge
on this matter.
PACS numbers: 61.20.-p, 61.25.Bi, 68.45.-v
I.INTRODUCTION
In a quite recent paper, Berim and Ruckenstein [1]
have reported symmetry breaking of the density profile
of fluid argon (Ar) confined in a planar slit with identical
walls of carbon dioxide (CO2). These authors claimed
that a completely symmetric integral equation provides
an asymmetric profile which has a lower free energy than
that of the lowest symmetric solution leading to a sym-
metry breaking phenomenon. It was assumed that the
Ar atoms interact via a standard Lennard-Jones (LJ) po-
tential characterized by the strength εff and the atomic
diameter σff. The presented results were obtained from
calculations carried out with the smoothed density ap-
proximation (SDA) version [2, 3] of the nonlocal density
functional theory (DFT) in the case of a closed planar
slit with an effective width of 15σff.
breaking was found for temperatures between the exper-
imental triple point for Ar, Tt= 83.8 K, and a critical
value Tsb = 106 K. At each temperature, it was deter-
mined a range of average densities ρsb1 ≤ ρav ≤ ρsb2
where the symmetry breaking occurs, outside this range
a symmetric profile has the lowest free energy.
The symmetry
As a matter of fact, the adsorption of fluid Ar on a
solid substrate of CO2 was intensively studied for sev-
eral decades (see, e.g., Refs. 4 and 5). In 1977 Ebner
and Saam [6] analyzed phase transitions by assuming
that atoms of the fluid interact with the solid wall via
a 9-3 van der Walls potential (from here on denoted
as ES potential) obtained from the assumption that Ar
atoms interact with CO2 molecules via a LJ interac-
tion with parameters εsf and σsf. After this pioneer-
ing work, a large amount of work has been devoted
to study this system with different numerical and ana-
lytical techniques. The attention was focused to ana-
lyze features like: the oscillatory behavior of the den-
sity profile which leads to a layered structure in the
neighborhood of the flat substrate; the thin- to thick-
film transitions; wetting properties and prewetting jumps
[7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22].
Berim and Ruckenstein [1] have also adopted the ES po-
tential, however, a hard-wall repulsion was introduced in
their calculations. In practice, such a hard wall dimin-
ishes the strength of the solid-fluid interaction.
The investigation of symmetry breaking in physical
systems is a very exciting issue. This is due to the fact
that such a feature may have fundamental theoretical
implications. In many fields of physics the discovery of
symmetry breaking lead to significant advances in the
theory. Therefore it is important to place the results of
Ref. [1] in the context of the current knowledge about
adsorption into planar slits.
Asymmetric solutions for fluids confined in slits have
been previously reported in the literature.
years ago a Dutch Collaboration has carried out cal-
culations on the Delft Molecular Dynamics Processor
(DMDP), which was specially designed for Molecular Dy-
namics (MD) simulations of simple fluids [23, 24]. The
results were published in a series of papers by Sikkenk
et al. [25, 26] and Nijmeijer et al. [27, 28]. The sim-
ulations were performed for a canonical ensemble with
two types of particles, 2904 of one type for building a
solid substrate and several thousand of the other type
for composing the fluid adsorbate. The temperature of
the system was kept at T∗= kBT/εff= 0.9 which is in
between the fluid’s triple-point temperature T∗
the critical temperature T∗
c≃ 1.3. The width of the slit
was taken as L = 29.1σff, supposing that this distance
be enough large to avoid any capillary effect. Such a sys-
About 20
t≃ 0.7 and

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tem can support solid-liquid (SL), solid-vapor(SV), and
liquid-vapor (LV) interfaces. These authors have stud-
ied wetting at LV coexistence by varying over a wide
range the relative strength of the solid-fluid and fluid-
fluid interactions defined by the ratio εr = εsf/εff of
the LJ parameters. The length scale of this interaction
was taken as σsf = 0.941σff. For increasing εr from
εr≃ 0.1 towards εr≃ 1.0 three cases were observed:
(i) at low εr, symmetric profiles consisting of two SV
interfaces and two LV interfaces are obtained, this sit-
uation corresponds to a complete wall drying as can be
seen in Fig. 1 of [28];
(ii) at intermediate εr, asymmetric profiles consisting
of a SL, a LV, and a SV interface are obtained, here the
wall attraction is sufficiently strong to produce a partial
wetting, i.e., to support a rather thick film on one wall
while a SV interface is present near the other wall, this
feature is shown in Fig. 2 of [28];
(iii) for the largest εr, symmetric profiles consisting of
two SL interfaces and two LV interfaces are obtained,
now the strength is enough to wet both walls as can be
seen in Fig. 3 of [28].
The structure of the profiles mentioned above depends
on the balance of the involved surface tensions γSL, γLV,
and γSV which are related by the Young’s law (see, e.g.,
Eq. (2.1) in Ref. [29])
γSV = γSL+ γLV cosθ , (1.1)
where θ is the contact angle. The latter quantity is de-
fined as the angle between the wall and the interface be-
tween the liquid and the vapor (see Fig. 1 in Ref. [29]).
The transition from (i) to (ii) takes place at the drying
point θ = π, whereas the transition from (ii) to (iii) takes
place at the wetting point θ = 0.
It is worth of notice that Velasco and Tarazona [30]
have carried out calculations in the frameworkof the SDA
obtaining density profiles with the same structure to that
reported in Refs. [25, 26, 27, 28]. The reader may look at
Ref. [31] for a further comparison between MD and DFT
results.
It is the aim of the present work to acquire a more
accurate picture of the symmetry breaking reported in
Ref.1. In so doing, we explore size effects by com-
paring the results obtained for slits of widths 15σff
and 30σff. Next, we investigate the existence of sta-
ble asymmetric solutions for the density profiles when
the position of the hard-wall repulsion introduced in Ref.
1 is changed.When the location of this hard wall is
moved, the strength of the adsorption potential is varied
allowing a connection to the studies described in Refs.
[25, 26, 27, 28, 30, 31]. Several properties of the obtained
solutions are discussed.
The paper is organized as follows. In Sec. II we pro-
vide a summary of the formalism underlying the present
calculations. Special attention is devoted to the adsorp-
tion potential. The results and its analysis are reported
in Sec. III. Final remarks are given in Sec. IV.
II. THE MODEL
The properties of a fluid adsorbed by an inert solid
substrate may be studied by analyzing the grand free
energy [32]
Ω = F − µN , (2.1)
where F is the Helmholtz free energy, µ the chemical
potential, and N the number of particles of the adsorbate
N =
?
ρ(r)dr . (2.2)
Quantity F contains the energy due to the interaction
between fluid atoms as well as the energy provided by the
confining potential. In a DFT it is expressed in terms of
the density profile ρ(r)
F[ρ(r)] = Fint[ρ(r)] +
?
drρ(r)Usf(r) .(2.3)
Here Fint[ρ(r)] is the intrinsic Helmholtz free energy func-
tional and Usf(r) is the external potential produced by
the slit’s walls.
This formulation is usually applied to systems de-
scribed by the grand canonical ensemble, i.e., at con-
stant volume V , temperature T, and chemical potential
µ. Such a situation corresponds to an open system in
contact with reservoir which fixes T and µ. A minimiza-
tion of Ω with respect to ρ(r) leads to the Euler-Lagrange
equation for the density profile and the number of parti-
cles may be evaluated with Eq. (2.2). For a closed system,
i.e., a canonical ensemble with fixed N, one should treat
µ as an unknown Lagrange multiplier to be determined
from the minimization procedure.
A.Density functional theory
Let us now summarize the DFT adopted for Fint[ρ(r)].
In the case of inhomogeneous classical fluids at tempera-
ture T the intrinsic free energy functional is decomposed
into two kind of contributions:
(i) the ideal gas term Fid[ρ(r)], which is given by the
exact expression
Fid[ρ(r)] = kBT
?
?
drρ(r)fid[(ρ)]
= kBTdrρ(r){ln[Λ3ρ(r)] − 1} , (2.4)
with Λ =
wavelength of a molecule of mass m;
(ii) the excess term Fex[ρ(r)], which accounts for the
interparticle interactions is a unique but unknown func-
tional of the local density. For fluids with attractive in-
teractions as the Lennard-Jones (LJ) one, the free energy
is decomposed into the repulsive and attractive contribu-
tions. The repulsive interactions are then approximated
?2π ?2/mkBT being the thermal de Broglie

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by a hard-sphere functional with a certain choice of the
hard-sphere diameter dHS
FHS[ρ(r)] =
?
drρ(r)fHS[¯ ρ(r);dHS] , (2.5)
whereas the attractive interactions are treated in most
cases in a mean field fashion
Fattr[ρ(r)] =1
2
? ?
drdr′ρ(r)ρ(r′)Φattr(| r − r′ |) .
(2.6)
Here Φattr(r =| r − r′ |) is the attractive part of the LJ
potential.
In summary, the intrinsic Helmholtz free energy func-
tional may be expressed as
Fint[ρ(r)] = Fid[ρ(r)] + Fex[ρ(r)]
= Fid[ρ(r)] + FHS[ρ(r)] + Fattr[ρ(r)] . (2.7)
The free energy functional for hard spheres plays a cen-
tral role in DFT. Expressions for fHS[¯ ρ(r);dHS] may be
taken from the Percus-Yevick [33] or Carnahan-Starling
(CS) [34] approximations for the equation of state (EOS)
of a uniform non-attractive hard-sphere fluid (see, e.g.
Ref. [35]). In a nonlocal DFT this quantity is evaluated
as a function of a conveniently averaged density ¯ ρ(r).
For the calculations performed in the present work we
used the same SDA formalism adopted in the paper of
Berim and Ruckenstein [1]. In this approach developed
by Tarazona [2, 3], the excess of free energy density of
hard spheres is written according to the semi-empirical
quasi-exact CS expression
fHS[¯ ρ(r);dHS]] = fCS[¯ η] = kBT
?¯ η (4 − 3¯ η)
(1 − ¯ η)2
?
. (2.8)
Here ¯ η = ¯ ρ(r)VHSis the packing fraction, where the fac-
tor VHS= π d3
HS/6 is the volume of a hard sphere. The
smoothed density ¯ ρ(r) is defined as
¯ ρ(r) =
?
drρ(r)w[| r − r′ |; ¯ ρ(r)] ,(2.9)
with the following weighting function:
w[| r − r′ |; ¯ ρ(r)] = w0[| r − r′ |] + w1[| r − r′ |] ¯ ρ(r)
+w2[| r − r′ |] ¯ ρ2(r) . (2.10)
The expansion coefficients w0(r), w1(r), and w2(r) are
density independent and its expressions as a function of
r =| r − r′ | are given in the Appendix of Ref. 3.
To account for the fluid-fluid interaction we adopted,
as in Ref. [1], the spherically symmetric L-J potential
given in Eq. (2) of Ref. 8
Φattr(r) =



4εff
??σff
r
?12−?σff
r
?6?
if r ≥ σff
0 if r < σff
. (2.11)
where σffis the hard-core diameter of the fluid. The au-
thors of Ref. 8 have used this L-J version just for studying
the adsorption of Ar on CO2and it has been also utilized
in several subsequent works on this system. The values of
the interaction parameters for Ar are εff/kB= 119.76 K
and σff= 3.405˚ A.
B.The Euler-Lagrange equation
The equilibrium density profile ρ(r) of the fluid ad-
sorbed in a closed slit is determined by a minimization
of the free energy with respect to density variations with
the constraint of a fixed number of particles N
δ
δρ(r)
?
Fint[ρ(r)] +
?
dr′ρ(r′)[Usf(r′) − µ]
?
= 0 .
(2.12)
Here, i.e., for an ensemble with fixed V , T, and N, the La-
grange multiplier µ is an unknown quantity which should
be determined from the constraint. It plays a role of a
chemical potential but off the liquid-vapor coexistence
conditions. Hence, it is not necessarily equal to µcoexof
an open slit in equilibrium with a reservoir at tempera-
ture T (see, e.g., Ref. 36).
In the case of a planar symmetry where the flat walls
exhibit an infinite extent in the x and y directions the
profile depends only of the coordinate z perpendicular
to the substrate. For this geometry the variation of Eq.
(2.12) yields the following Euler-Lagrange (EL) equation
δF
δρ(z)
+ Usf(z) =δ[Fid+ FHS]
δρ(z)
+
?
dz′ρ(z′)¯Φattr(| z − z′ |) + Usf(z) = µ .
(2.13)
For a slit of effective width ℓwthis EL equation may be
cast into the form
kBT ln[Λ3ρ(z)] + Q(z) = µ , (2.14)
where
Q(z) = kBT4 ¯ η(z) − 3¯ η2(z)
[1 − ¯ η(z)]2
+kBTπ d3
HS
6
?ℓw
0
dz′ρ(z′)4 − 2 ¯ η(z′)
[1 − ¯ η(z′)]3
δ¯ ρ(z′)
δρ(z)
+
?ℓw
0
dz′ρ(z′)¯Φattr(| z − z′ |) + Usf(z) .(2.15)
The number of particles per unit area of one wall of the
slit is
?ℓw
0
Ns=
ρ(z)dz . (2.16)
In order to get solutions for ρ(z) it is useful to rewrite
Eq. (2.14) as
ρ(z) = ρ0exp
?
−Q(z)
kBT
?
, (2.17)

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with
ρ0=
1
Λ3exp
?
µ
kBT
?
. (2.18)
The relation between µ and Nsis obtained by substitut-
ing Eq. (2.17) into the constraint given by Eq. (2.16)
µ = −kBT ln
?
1
NsΛ3
?ℓw
0
dz exp
?
−Q(z)
kBT
??
. (2.19)
For the calculations carried out in the present work we
set dHS= σff as it was done in Ref. 1.
The asymmetry of the density profiles is measured by
the parameter
∆N=
1
2Ns
?ℓw
0
dz | ρ(z) − ρ(ℓw− z) | . (2.20)
According to this definition, if the profile is completely
asymmetrical about the middle of the slit [ρ(z < ℓw/2) ?=
0 and ρ(z ≥ ℓw/2) = 0] this parameter becomes unity,
while for symmetric solutions it vanishes.
C. Adsorption potential
The model van der Waals (9-3) potential proposed by
Ebner and Saam [6], i.e. the ES potential, is
Usf(z) =



2π
3ǫeff
?
2
15
?σsf
z
?9−?σsf
z
?3?
if z > 0
0 if z ≤ 0
(2.21)
with ǫeff = εsfρsσ3
adopted for almost all the abovementioned studies of the
adsorption of Ar atoms on a flat wall of solid CO2. The
exception is the experimental and theoretical investiga-
tion performed by Mistura et al. [22], where a more re-
alistic adsorption potential calculated on the basis an
ab initio expansion of Marshall et al.
The ES expression is obtained when one assumes that
Ar atoms interact with CO2atoms via a Lennard-Jones
(12-6) potential and subsequently integrates this poten-
tial over a continuum of CO2substrate atoms with a re-
duced density ρ∗
sf= 0.988. The cross-parameters
of the potential are determined by using the Lorentz-
Berthelot rules. So that, the van der Waals strength εsf
is the square root of the product of the argon and CO2
van der Waals strengths, while the hard-core diameter
σsf is the mean of the argon and CO2hard-core diam-
eters, while The parameters evaluated in this way are
εsf/kB= 153 K and σsf= 3.727˚ A.
Berim and Ruckenstein [1] have investigated the Ar-
CO2 system utilizing, in principle, the ES potential.
However, by looking at their paper one realizes that ac-
cording to Eq. (A5) of the Appendix
sfbeing the effective strength, was
[37] was used.
s= ρsσ3
Usf1(z) =2π
3ǫeff
?2
15
?
σsf
z + σsf
?9
−
?
σsf
z + σsf
?3?
(2.22)
,
0123456789 10
−400
−300
−200
−100
0
100
200
ξ [A]
Usf(ξ) [K]
Ar−CO2
o minimum
σsf
σsf/2
< >
< >
FIG. 1: Adsorption potential as a function of the distance
from the real wall. The solid curve is the potential of Berim
and Ruckenstein close to the left wall given by Eq. (2.23),
while the dashed curve is that of Nilson and Griffiths given
by Eq. (2.25).
which accounts for the solid-fluid interaction at one of
the walls, a hard-wall repulsion was located at a distance
σsf from the real wall of the slit. In agreement with this
assumption, the total confining potential exerted on Ar
atoms by the two walls separated by a distance L was
expressed as
Usf(z) = Usf1(z + σsf) + Usf2(ℓw− z + σsf) .(2.23)
Here the effective width of the slit is
ℓw= L − 2σsf. (2.24)
This scenario is depicted in Fig. 2 of Ref. 1. In this con-
text, it is interesting to notice that Nilson and Griffiths
[38] in order to study the adsorption of a fluid in a pla-
nar slit have written in their Eq. (10) the total fluid-solid
potential as
Usf(z) = Usf1(z +σsf
2
) + Usf2(ℓw− z +σsf
2
) , (2.25)
i.e., locating a hard-wall repulsion at a distance σsf/2
from the substrate. In this case, the effective width is
ℓw= L − σsf, (2.26)
as it is shown in Fig. 2 of Ref. 38.
In Fig. 1 we compare the potentials outlined in the
previous paragraph. The comparison is restricted to the
region close to the substrate. The quantity ξ is the per-
pendicular distance from the real wall being
ξ =



z + σsf for Berim − Ruckenstein
z +σsf
2
for Nilson − Griffiths.
(2.27)
One may realize that Eq. (2.25) retains the “soft” re-
pulsion [Usf(z) ∝ (σsf/z)9], while Eq. (2.23) cuts the

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TABLE I: Values of the Helmholtz free energies Fsym, Fasym,
and Fcap of the symmetric, asymmetric and capillary solu-
tions, respectively, obtained with ν = 1 for the slit ℓ∗
at T = 87 K. The free energies are given for several average
densities in the range ρ∗
sb1≤ ρ∗
w= 15
av≤ ρ∗
sb2in units of kBT/σ2
ff.
Fsym
Fasym
Fcap
PWb
ρ∗
av
BRa
PWb
BRa
PWb
0.1546
0.2319
0.3092
0.3865
0.4638
−26.59
−39.66
−52.81
−66.01
−79.22
−26.62
−39.69
−52.85
−66.04
−79.28
−26.67
−39.86
−53.06
−66.26
−79.48
−26.70
−39.88
−53.08
−66.29
−79.51
−39.12
−52.32
−65.65
−79.26
aData taken from [1].
bCalculated in the present work.
0 0.1 0.2 0.3 0.40.5 0.60.7 0.8
−12.2
−12
−11.8
−11.6
−11.4
−11.2
−11
−10.8
ρ*
av
µ/kBT
T = 87 K lw = 15 σff
FIG. 2: Lagrange multiplier µ as a function of average density.
The solid curve are results for symmetric film solutions. The
dashed curve stands for values of asymmetric film solutions
which occur in the range ρ∗
curve corresponds to drying-CC like solutions.
sb1≤ ρ∗
av≤ ρ∗
sb2. The dot-dashed
potential before the minimum be reached. This feature
produces important effects on the behavior of the density
profiles. In fact, the calculations performed by Berim and
Ruckenstein [1] yielded density profiles with ρ(z = 0) and
ρ(z = ℓw) different from zero indicating that the fluid is
in contact with the hard walls, while in the case of Nilson
and Griffiths [38] the fluid forms a well defined first layer
separated from the wall.
In the present work we shall analyze the evolution of
asymmetric solutions when the total adsorption potential
is written as
Usf(z) = Usf1(z +σsf
ν
) + Usf2(ℓw− z +σsf
ν
) , (2.28)
and the parameter ν varies from 1 to 2. In doing so, one
goes from Eq. (2.23) towards Eq. (2.25) increasing the
strength of the solid-fluid attraction.
III.NUMERICAL RESULTS AND DISCUSSION
Let us now describe the obtained results. The EL equa-
tion (2.14) was solved at fixed ℓwand T for a given num-
ber of particles per unit area Ns. The latter quantity de-
termines an average fluid density ρav= Ns/ℓw. A widely
used computational algorithm consisting of a numerical
iteration of the coupled Eqs. (2.17)-(2.19) was applied.
This procedure yields the density profile ρ(z) and the
value of the Lagrange multiplier µ. The convergence of
the solutions are measured by the difference between two
consecutive profiles
δ1= σ5
ff
?ℓw
0
dz
?
ρi+1(z) − ρi(z)
?2
, (3.1)
where ρi(z) is the density profile after the i-th iteration,
and by the quantity
δ2= 1 −
1
Ns
?ℓw
0
dz ρ(z) ,(3.2)
accounting for the deviation from the required Ns.
In practice, for the calculations it is convenient to use
dimensionless variables: z∗= z/σff for the distance,
ρ∗= ρσ3
fffor the densities, and T∗= kBT/εff for the
temperature. In these units the average density becomes
ρ∗
ff/ℓ∗
region of integration [0, ℓ∗
w] was divided into a grid of
equal intervals ∆z∗= 0.02, i.e., a grid with 50 points per
atomic diameter σff. It is worthwhile to notice that in
the work of Berim and Ruckenstein the number of grid
points was taken equal to 10 per atomic diameters. If the
obtained profile did not change with increasing precision
from δ1≈ 10−8to δ1≈ 10−15, then it was accepted as a
solution of the coupled integral equations.
In a first step, we studied the same systems treated in
detail by Berim and Ruckenstein [1]. Hence, we set ν = 1
and solved the EL equation for a slit with an effective
width ℓ∗
w= 15 at T = 87 K (T∗= 0.73) for a series
of average fluid density ρ∗
av= N∗
solutions yield symmetric density profiles for ρ∗
ρ∗
sb1and for ρ∗
ρ∗
av≤ ρ∗
density profiles. This is due to the fact that in such a
regime the asymmetric solutions have lower free energy
than the symmetric ones. The free energies calculated for
some selected ρ∗
avare listed in Table I together with the
results obtained by Berim and Ruckenstein [1]. A glance
at this table indicates a good agreement between both
sets of values. In order to get symmetric solutions in the
range 0.1 ≤ ρ∗
av≤ 0.513 one must explicitly impose such
a condition to Eqs. (2.17)-(2.19).
The Lagrange multiplier µ (equivalent to the chemical
potential in the case of open slits) is displayed in Fig. 2
as a function of average density. We show the results for
a wider range of ρ∗
avthan it is done in Fig. 9 of Ref. 1.
Figure 2 clearly indicates that the asymmetric solutions
av= Nsσ2
w= N∗
s/ℓ∗
w. For the numerical task, the
s/ℓ∗
w. The ground state
av< 0.1 =
sb2, while in the range ρ∗
av> 0.514 = ρ∗
sb2the ground-state solutions provide asymmetric
sb1≤