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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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2

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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3

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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4

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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5

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≤