Journal of Colloid and Interface Science 313 (2007) 41–52www.elsevier.com/locate/jcis
Capillary condensation in pores with rough walls:A density functional approach
P. Bryk
a
, W. R˙zysko
a
, Al. Malijevsky
b
, S. Sokołowski
a
,
∗
a
Department for the Modeling of PhysicoChemical Processes, Maria CurieSkłodowska University, 20031 Lublin, Poland
b
E. Hála Laboratory of Thermodynamics, ICPF, Academy of Sciences, 165 02 Prague 6, Suchdol, Czech Republic
Received 18 January 2007; accepted 27 March 2007Available online 27 April 2007
Abstract
The effect of surface roughness of slitlike pore walls on the capillary condensation of a spherical particles and short chains is studied. The gasmolecules interact with the substrate by a LennardJones
(
9
,
3
)
potential. The rough layer at each pore wall has a variable thickness and densityand consists of a disordered quenched matrix of spherical particles. The system is described in the framework of a density functional approachand using computer simulations. The contribution due to attractive van der Waals interactions between adsorbate molecules is described by usingﬁrstorder mean spherical approximation and meanﬁeld approximation.
©
2007 Elsevier Inc. All rights reserved.
Keywords:
Adsorption; Pore; Capillary condensation; Density functional theory; Quenchedannealed systems
1. Introduction
Real adsorbing surfaces are usually rough, so a ﬂuid uponadsorption encounters geometrically and/or energetically heterogeneous walls. Recently, the effects of heterogeneity onadsorption have been considered theoretically and employingsimulation techniques for several model systems [1–19]. In par
ticular, it has been shown that the surface roughness can inﬂuence their wettability [20–24]. To study this effect some of
us proposed a model of a rough wall created by depositing alayer of molecules on an idealized substrate and next imposinga quenching procedure [23,24]. The quenched layer was then
exposed to the adsorbing gas. Surface roughness in this modelis a combined effect of the preparation procedure, the densityof the deposited layer and its thickness. To describe the adsorption of a ﬂuid a modiﬁed Fischer–Methfessel approach [25] thatis closely related to the socalled “MarkI” version of the Tarazona’s density functional theory (DFT) [26] was applied. Thismodel was used to study the effect of surface roughness on thewetting transitions [24], to investigate adsorption of ﬂuids in
*
Corresponding author.
Email address:
stefan.sokolowski@neostrada.pl (S. Sokołowski).
pores with rough walls and to examine the behavior of mixturesnear semipermeable membranes [27–29].Recently, Schmidt and coworkers proposed a DFT to describe quenchedannealed mixtures [30–33]. Their approach is
based on the replica trick and allows the treatment of situationswhere the quenched random matrix as well as the annealedﬂuid are inhomogeneous. Applications of the theory includedinvestigation of the adsorption properties of hard spheres andmodel colloid–polymer mixtures in bulk matrices and at matrix surfaces, and the inﬂuence of the quenched disorder onphase transitions like ﬂuid demixing, isotropicnematic ordering and freezing. Particularly rich wetting behavior was foundfor colloid–polymermixtures adsorbed against porous wall. Forreview of all these applications see Ref. [32].The theory of Schmidt and coworkers treats both the annealed and the quenched species on the level of their onebodydensity distributions. Hence the complicated external potentialthatthequenchedparticlesexertontheﬂuidneverentersexplicitly into the theoretical framework. This approach constitutesan enormous simpliﬁcation as far as practical computations areconcerned,butalsosuffersofseverallimitations,asitwasnotedby Lafuente and Cuesta [34]. These authors derived an alterna
tive DFT for quenchedannealed mixtures from ﬁrst principles
00219797/$ – see front matter
©
2007 Elsevier Inc. All rights reserved.doi:10.1016/j.jcis.2007.03.077
42
P. Bryk et al. / Journal of Colloid and Interface Science 313 (2007) 41–52
without resorting to the replica trick. They showed that thedisorderaveraged free energy of the ﬂuid is a functional of theaverage density proﬁle of the ﬂuid as well as the pair correlationof the ﬂuid and matrix particles. However, an implementationof this theory is difﬁcult and from this reason it is preferable touse the approach of Schmidt and coworkers.To employ theoretical methods for calculating adsorptionisotherms a model for ﬂuid–ﬂuid and ﬂuid–adsorbent interactions must be formulated ﬁrst. The molecular parameters forﬂuid–ﬂuid interaction are usually determined from bulk thermodynamic data, whereas solid–ﬂuid potential parameters areobtained by ﬁtting theoretical results against experimental adsorption isotherms. It appears, however, that in a vast majority of cases [18,19] the predicted isotherms are stepwise indicating the sequential formation of molecular layers, which,however, are not observed experimentally. In the case of someadsorbents, like graphitized carbon, improvement of the predictability of isotherms is achieved by modifying the ﬂuid–ﬂuidinteraction parameters [18,19,35]. The altering of their values is
justiﬁed by the importance of multibody effects [35–37]. Thesuccess is of this procedure is possible with graphitized carbonblack because of the fairly homogeneous nature of the surface,however, such an approach does not enjoy a similar success inthedescriptionofadsorptionisothermforotheradsorbents,e.g.,for silica gel [38–40]. One possible reason is energetic hetero
geneity of adsorbing surfaces (chemical heterogeneity) surfaceroughness (geometrical heterogeneity), which is considered atthe molecular level and attributed to a molecular feature of amorphous solid [18,19].In the case of heterogeneousadsorbents the adsorbing potential is a complicated function that varies in threedimensionalspace. Apart from the fact that usually only a little is knownabout the spatial dependence of the adsorbing potential forreal substrates, numerical calculations (including simulations)would be extremely timeconsuming. One possible route to describe effectively such systems is to replace the energy landscape by the density of “auxiliary matrix particles” landscapeand to employ the methods developed for quenchedannealedmixtures [41]. According to this treatment a complicated adsorbing potential does not enter explicitly theoretical equations.The model developed in our previous works [23,24,27] seemsto be appropriate to handle the effects of surface heterogeneity.Indeed, a similar model was used recently by Ravikovitch andNeimark [42], who applied DFT of Schmidt [30,32], and Reich
and Schmidt [31] to model adsorption on geometrically heterogeneous surfaces and porous solids with rough pore walls.In this work we present DFT of adsorption of chain molecules in pores with rough walls. The theory combines the approach of Schmidt [32] with the theory of Yu and Wu [43] and
Cao and Wu [44]. The latter theory can be successfully applied
to describe adsorption of such important from experimentalpoint of view [45–49] complex ﬂuids as carbon dioxide, hydrocarbons, polymers and many others [50]. Moreover, this theoryis numerically almost as simple as density functional theoriesof simple ﬂuids [26]. Obviously, the proposed approach can be
also used to study adsorption of spherically symmetric (atomic)molecules.Previous applications of the density functional theory havebeenprimarilyfocusedonvariousweighteddensityapproximations for the repulsive part of the Helmholtz energy functionaland on the mean ﬁeld approximation (MFA) for an attractivecontribution. The major problem of the MFA is that it signiﬁcantly underpredicts the critical temperature for uniform ﬂuids;hence the corresponding interfacial properties maybe unphysical. Recently Tang [51] proposed a ﬁrstorder meansphericalapproximation (FMSA) theory for a long ranged attraction.The FMSA theory, with the analytical expression of a radialdistribution function or a direct correlation function as input,signiﬁcantly improves the MFA for the long ranged attractionand is applicable to both bulk and inhomogeneous systems using a single set of molecular parameters. The DFT for simplenonuniform ﬂuids based on the Fundamental Measure Theory[52–54] for hardsphere freeenergy contribution and on theFMSA for the attractive contribution is probably the most accurate approach to the theory of simple conﬁned ﬂuids [55–58].Therefore the aim of this paper is also to extend the FMSAto the case of nonuniform systems with quenched component.In the case of adsorption of spherical symmetric molecules thetheoretical predictions are compared with the results of computer simulations. The phase diagrams for model systems havebeen evaluated using hyperparallel tempering technique [59].
2. Theory
The system under consideration is a ﬂuid composed of chainmolecules conﬁned in a slitlike pore of the width
H
. Thechains are built of
M
segments, each of diameter
σ
ss
. We stressthat setting
M
=
1 in all equations presented below the theoryreduces to that for adsorption of spherically symmetric (atomic)adsorbates.The chain connectivity is assured by imposing the bonding potential [43],
V
b
(
R
)
=
M
−
1
j
=
1
v
b
(

r
j
+
1
−
r
j

)
, where
R
≡
(
r
1
,
r
2
,...,
r
M
)
denotes a set of coordinates describing the positions of all segments and
v
b
is the bonding potential betweentwo adjacent segments. We assume that the segments are tangentially jointed, so that the binding potential is(1)exp
−
βV
b
(
R
)
=
M
−
1
j
=
1
δ

r
j
+
1
−
r
j
−
σ
ss
/
4
πσ
2ss
.
We also take into account attractive van der Waals potentialbetween the segments. This potential is the same for all the segments and is given by(2)
u
ss
(r)
=
4
ε
ss
(σ
ss
/r)
12
−
(σ
ss
/r)
6
,
for
r
r
cut
,
ss
,
0
,
for
r > r
cut
,
ss
.
In the above
r
cut
,
ss
is the cutoff distance, and
ε
ss
is the energyparameter.Each segment
j
interacts with a single pore wall viaLennardJones (9,3) potential(3)
v
j
(z
j
)
=
ε
gs
(z
0
/z
j
)
9
−
(z
0
/z
j
)
3
,
where
z
j
is the distance between the
j
th segment and the wall.The total external potential,
V
ext
(
R
)
, imposed on a chain molecule is the sum of the potentials acting on individual segments,
P. Bryk et al. / Journal of Colloid and Interface Science 313 (2007) 41–52
43
due to both pore walls
V
ext
(
R
)
=
M j
=
1
[
v
j
(z
j
)
+
v
j
(H
−
z
j
)
]
.The parameters
ε
gs
, and
z
0
are identical for all segments. Thelast assumption can be easily removed.In order to model pores with rough walls, a layer consisting of a quenched matrix is placed at each wall [23,24,27]. Weassume that the matrix is built of spherical molecules of diameter
σ
mm
. However, the presented theory can be also extended tothe case of matrices made of chain particles. The interaction between matrix species and a segment of the chain is given by thepotential
u
ms
(r)
, which can be either a hardsphere potential(4)
u
ms
(r)
=
∞
,
for
r
σ
ms
,
0
,
for
r > σ
ms
,
or a truncated LennardJones potential(5)
u
ms
(r)
=
4
ε
ms
(σ
ms
/r)
12
−
(σ
ms
/r)
6
,
for
r
r
cut
,
ms
,
0
,
for
r > r
cut
,
ms
.
Here
σ
ms
=
(σ
ss
+
σ
mm
)
and
r
cut
,
ms
is the cutoff distance forthe ﬂuid–matrix interaction.The local density of the matrix is described by the function
ρ
m
(z)
. At the moment we do not specify the form of
ρ
m
(z)
, northe process leading to the formation of the matrix distribution atthe pore walls. Of course, if the matrix inhomogeneity is generated in response to an external potential,
V
m
(z)
, that acts on thematrix particles before the quench, then the matrix conﬁguration is drawn from an equilibrium distribution according to theHamiltonian of the matrix particles, in particular an equilibriumDFT can be applied for this purpose [30–32]. In general, how
ever, in the DFT of quenchedannealed systems the distribution
ρ
m
(z)
is an input into the theory and the step of matrix development is completely separated from the second step, which relieson the adsorption of a ﬂuid.Following Schmidt [30–33] the replica trick is applied towrite down the grand potential functional for the conﬁned ﬂuid
Ω
ρ
m
(z),ρ(
R
)
=
F
id
ρ(
R
)
+
F
ex
ρ
m
(z),ρ(
R
)
(6)
+
d
R
ρ(
R
)
V
ext
ρ(
R
)
−
µ
,
where
ρ(
R
)
and
µ
are the local density of chain particles andtheir chemical potential, respectively. Moreover,
F
id
(ρ(
R
))
isthe conﬁgurational ideal free energy and the excess free energyfunctional,
F
ex
(ρ
m
(z),ρ(
R
))
, describes the interparticle interactions of adsorbate particles with adsorbate particles and withthe matrix particles. We are aware that this theory has a numberof weak points that should be pointed out. Although the replicatrick is a widely applied method of statistical physics, it introduces assumptions that are difﬁcult to justify concerning theanalytic continuation of the grand potential as a function of thenumber of replicas, and the replica symmetry or its breaking.Moreover,contrarytoDFTof ﬂuids,theformulationof thedensity functional theory of quenchedannealed systems by Eq. (6)makes difﬁcult to derive the set of Ornstein–Zernike equationsforthesystemfromfunctionalrelations.Forexample,atpresentit is not at all clear what the meaning of the second derivativesof
F
ex
(ρ
m
(z),ρ(
R
))
is [34]. Nevertheless, we have decided tofollow the ideas of Schmidt [30–32] because at the moment thisapproach is the only one that can be implemented to the considered case in a relatively easy manner.Similarly as in DFT of simple nonuniform ﬂuids we startwith writing down the system free energy. The ideal free energyfunctional is known exactly [43]
βF
id
ρ(
R
)
=
β
d
R
ρ(
R
)V
b
(
R
)
(7)
+
d
R
ρ(
R
)
ln
ρ(
R
)
−
1
.
The excess free energy,
F
ex
, can be split into the hardspherecontribution,
F
hs
, which results from the hardsphere repulsionbetween segments and between segments and matrix particles,the attractive contribution due to segment–segment interaction,
F
att
and, eventually, due to segment–matrix interaction, and thecontribution due to chain connectivity,
F
c
. According to theapproach of Yu and Wu, which is based on the FundamentalMeasure Theory [52–54] and on Wertheim [60,61] theory of
association, the contributions
F
hs
and
F
c
are functionals of theweighted densities. In order to write down the deﬁnitions of theweighted densities we ﬁrst introduce the average segment density proﬁles for chain molecules(8)
ρ
s
(
r
)
=
M
j
=
1
ρ
s
,j
(
r
)
=
M
j
=
1
d
R
δ(
r
−
r
j
)ρ(
R
),
where
ρ
s
,j
(
r
)
is the local density of the segment
j
of the chain.The weighted densities are then deﬁned as
n
α
(
r
)
=
n
α,
s
(
r
)
+
n
α,
m
(
r
)
=
d
r
ρ
s
(
r
)w
α

r
−
r

,d
ss
(9)
+
d
r
ρ
m
r
w
α

r
−
r

,d
mm
,
where scalar (
α
=
0
,
1
,
2
,
3) and vector (
α
=
V
1
,V
2) weightfunctions,
w
α
(r,σ
x
)
,
x
=
ss
,
ms are given in Ref. [52]. Thequantities
d
ss
and
d
mm
are the effective hardsphere diametersof the reference hardsphere ﬂuid.Each of the components,
F
α
,
α
=
hs or c is expressed as avolume integral
F
α
=
Φ
α
(
r
)
d
r
. The hardsphere contributionis evaluated from [52–54](10)
Φ
hs
(
r
)
= ˜
Φ
hs
r
;{
n
α,
s
}
,
{
n
α,
m
}
− ˜
Φ
hs
r
;{
n
α,
s
≡
0
}
,
{
n
α,
m
}
.
˜
Φ
hs
r
;{
n
α,
s
}
,
{
n
α,
m
}
= −
n
0
ln
(
1
−
n
3
)
+
n
1
n
2
−
n
V
1
·
n
V
2
1
−
n
3
(11)
+
n
32
1
−
ξ
2
3
n
3
+
(
1
−
n
3
)
2
ln
(
1
−
n
3
)
36
πn
23
(
1
−
n
3
)
2
,
where
ξ(
r
)
= 
n
V
2
(
r
)

/n
2
(
r
)
.The contribution
Φ
c
is obtained from Wertheim’s ﬁrstorderperturbation theory [60,61](12)
Φ
c
(
r
)
=
1
−
M M n
0
,
s
ζ
ln
y
hs
(d
ss
)
,
44
P. Bryk et al. / Journal of Colloid and Interface Science 313 (2007) 41–52
where
ζ
=
1
−
n
V
2
,
s
·
n
V
2
,
s
/(n
2
,
s
)
2
and
y
hs
is the contact valueof the hardsphere radial distribution function for the particlesoftheannealedcomponentinthemixturecontainingmatrixandannealed particles(13)
y
hs
(d
ss
)
=
11
−
n
3
+
n
2
d
ss
ζ
4
(
1
−
n
3
)
2
+
(n
2
d
ss
)
2
ζ
72
(
1
−
n
3
)
3
.
The simplest treatment of attractive forces follows from theassumption that
d
ss
=
σ
ss
and
d
mm
=
σ
mm
and from the application of a MFA. Introducing the Weeks–Chandler–Andersendivision [62] of the LennardJones potentials (Eqs. (2) and (5))
(14)
˜
u
x
(r)
=
−
ε
x
,
for
r
2
1
/
6
σ
x
, x
=
ss
,
ms
,u
x
(r),
for
r >
2
1
/
6
σ
x
, x
=
ss
,
ms
,
we have
F
att
=
12
d
r
d
r
˜
u
ss

r
−
r

ρ
s
(
r
)ρ
s
(
r
)
(15)
+
d
r
d
r
˜
u
ms

r
−
r

ρ
s
(
r
)ρ
m
(
r
).
If the matrix–ﬂuid interactions are of hardsphere type, the second term in Eq. (15) disappears.As we have already noted, the application of FMSA instead of MFA improves substantially accuracy of theoreticalpredictions, compared with MFA. However, it is possible onlywhen the ﬂuid–matrix interactions are given by the hardspherepotential. All the details concerning FMSA can be found in[55–58]. In the case of FMSA the effective hardsphere diameter
d
ss
is used. If the densitydependence of
d
ss
is neglected,its temperaturedependence can be approximated by the expression [63](16)
d
ss
=
σ
ss
0
d
r
1
−
exp
−
βu
ss
(r)
.
The diameter
d
ss
is used so often that it is very tempting to employ an approximation by a simple analytical expression. Oneof the most widely exploited expression is that due to Cotterman et al. [64](17)
d
ss
=
1
+
0
.
2977
T
∗
1
+
0
.
33163
T
∗
+
0
.
00104771
(T
∗
)
2
,
where
T
∗
=
kT/ε
ss
. A collection of other expressions proposedto approximate Eq. (16) can be found in Silva et al. [65] paper.
Sensitivity of thermodynamics of bulk ﬂuids to the Barker–Henderson diameter, which covers both its approximation incalculation and improvement in rationality was discussed byTang [66]. In the case of nonuniform systems, however, a sim
plifying assumption that
d
ss
is just
σ
ss
does not change theoverall picture of surface phase transition occurring in the system. Obviously,
d
mm
≡
σ
mm
, by deﬁnition. The Helmholtz freeenergy functional due to the long ranged attraction is then givenby(18)
F
att
= −
12
d
r
d
r
˜
c
ss

r
−
r

ρ
s
(
r
)ρ
s
(
r
),
where
c
ss
(

r
−
r

)
is the attractive part of the direct correlationfunction, which can be expressed as [55–58](19)
c
ss
(r)
=
c
Y
(T
∗
1
,z
1
,d
s
s,r)
−
c
Y
(T
∗
2
,z
2
,d
s
s,r), r < d
ss
,
0
, d
ss
< r < σ
ss
,u
ss
(r), r > σ
ss
,
where
z
1
=
2
.
9637
/σ
ss
,
z
2
=
14
.
0167
/σ
ss
,
T
∗
1
=
T
∗
d
ss
/k
1
,
T
∗
2
=
T
∗
d
ss
/k
2
,
k
1
=
k
0
exp
[
(σ
ss
−
d
ss
)z
1
]
k
2
=
k
0
exp
[
(σ
ss
−
d
ss
)z
2
]
and
k
0
=
2
.
1714
σ
ss
. The ﬁrstorder Yukawa direct correlation function,
c
Y
, in Eq. (19) is given in appendix of Ref. [67].TheequilibriumconﬁgurationofﬂuidmoleculessatisﬁestheEuler–Lagrange equation(20)
ρ(
R
)
=
exp
βµ
−
βV
b
(
R
)
−
βv(
R
)
−
βΛ(
R
)
,
where
µ
is the chemical potential and
Λ(
R
)
=
δF
ex
/δρ(
R
)
,where
F
ex
=
F
hs
+
F
att
+
F
c
is the effective potential ﬁeld dueto intra and intermolecular interactions. The excess free energyfunctional
F
ex
depends on the average segment density
ρ
s
, thus(21)
δF
ex
δρ(
R
)
=
M
i
=
1
λ
i
(
r
), λ
i
(
r
)
=
δF
ex
δρ
s
(
r
).
In this work we consider systems that are inhomogeneousin only one (say
z
) direction. Then one obtains (cf. Refs. [43,68,69])(22)
ρ
s
,i
(z)
=
exp
(βµ)
exp
−
βλ
i
(z)
G
i
(z)G
M
+
1
−
i
(z),
where the functions
G
i
(z)
are determined from the recurrencerelation(23)
G
i
(z)
=
d
z
exp
−
βλ
i
−
1
(z)
θ(σ
− 
z
−
z

)
2
σ G
i
−
1
(z
),
for
i
=
2
,
3
,...,M
and with
G
1
(z)
≡
1.The conﬁned system is in an equilibrium with the bulk ﬂuid.The bulk ﬂuid is characterized by the same value of the chemical potential
µ
as the conﬁned system and its properties aredescribed by the equations given above assuming that the external ﬁeld
V
ext
is zero and that the segment densities
ρ
s
,j
and thetotal segment density
ρ
s
are constant in the entire bulk systemand equal to
ρ
bs
,j
and
ρ
b
,
s
, respectively. Obviously, the matrixconcentration is zero in the bulk system.The numerical methods for the minimization of the grandpotential are identical with those described in our previous papers [68,69]. In particular, we introduce the appropriate propa
gator functions, as described in details in our previous works.For the sake of brevity these equations are not presented here,but we refer the reader to srcinal works [43,68,69].Finally, we introduce the reduced quantities. All the energy parameters are expressed in
ε
ss
units,
ε
∗
gs
=
ε
gs
/ε
ss
,
ε
∗
ms
=
ε
ms
/ε
ss
. Similarly, the diameter
σ
ss
is used as unit of length.Then
σ
∗
ms
=
σ
ms
/σ
ss
,
H
∗
=
H/σ
ss
,
z
∗
=
z/σ
ss
and the reduceddensities
ρ
∗
s
,j
(z)
are deﬁned as
ρ
∗
s
,j
(z)
=
ρ
s
,j
(z)σ
3ss
,
ρ
∗
bs
,j
=
ρ
bs
,j
(σ
ss
)
3
,
ρ
∗
bs
=
ρ
bs
(σ
ss
)
3
, etc.
P. Bryk et al. / Journal of Colloid and Interface Science 313 (2007) 41–52
45
3. Results and discussion
As we have noted, setting
M
=
1 in the relevant equationsgiven above, the theory reduces to that for atomic adsorbates.The aim of our model calculations performed for systems involving spherically symmetric molecules is two fold. First, wewould like to compare the results of MFA and FMSA with computer simulations. Secondly, we intend to compare two cases:the case of matrix molecules interacting with adsorbate atomsvia repulsive (hardsphere) interactions and via LennardJones
(
12
,
6
)
potential.TheMFAthatwasusedinavastmajorityofpapersaimingatthe application of DFT to experimental data, is perhaps the simplest possible that it is able to capture the behavior of nonuniform systems with repulsive and attractive forces at quantitativelevel. However, this approximation should not be treated asqualitatively correct. An attractive alternative for MFA providesFMSA. The latter theory, with analytical expression for a directcorrelation function as an input, which makes it numerically assimple as MFA, signiﬁcantly improves the MFA for the longranged attraction [55–58].At the beginning of our discussion we compare the densityfunctionalpredictionswiththeresultsofgrandcanonicalMonteCarlo simulations. Because the simulation method is quite standard, we omit all the technical details and only note that thephase diagrams for the conﬁned systems were obtained employing hyperparallel tempering technique [59]. The surface
roughness was modelled assuming that the matrix distributionis(24)
ρ
m
(z)
=
ρ
m0
,
for 0
z
z
m
and
H
−
z
m
z
H,
0
,
otherwise
,
where
z
m
is the matrix thickness at each pore wall. In the calculations below we have assumed that
z
m
=
σ
ss
. The diameterof matrix molecules was the same as the LennardJones diameter of ﬂuid particles,
σ
∗
mm
=
1. The matrix–ﬂuid interactionswere of hardsphere type and the cutoff of the ﬂuid–ﬂuid potential was
r
∗
cut
,
ss
=
2
.
5. The pore width was kept constant andequal
H
∗
=
5 and the ﬂuid–pore wall potential parameters were
ε
∗
gs
=
9 and
z
∗
0
=
0
.
5.Fig. 1 shows the density proﬁles obtained from computersimulations and from both theories at supercritical temperature,
T
∗
=
1
.
5. For all the systems in Fig. 1 the bulk (reference) system density was
ρ
∗
b
,
s
=
0
.
6958, which corresponds to the valueof the chemical potential from the bulk system Grand Ensemble Monte Carlo simulations equal to
µ/kT
= −
0
.
5. It is notsurprising that the FMSA improves the agreement with simulations in comparison with the data obtained from the meanﬁeldtheory. This is particularly well seen for the system without matrix. The improvement is also evident when one compares thedependence of the average adsorbate densities in the pore,(25)
ρ
=
H
0
ρ
s
(z)
d
z/H,
plotted versus the bulk density, see Fig. 2. The presence of
quenched molecules causes signiﬁcant changes in the structure
Fig. 1. Density proﬁles from simulations (symbols), from MFA (solid lines)and from FMSA (remaining lines) inside the pore of
H
∗
=
5 at
ρ
∗
b
,
s
=
0
.
6958,at
T
=
1
.
5. The consecutive curves from bottom are for
ρ
∗
m0
=
0
.
5, 0.2959,0.09694 and 0 (i.e., for the pore without the quenched particles). The values of all remaining parameters of the model are given in the text.Fig. 2. Average ﬂuid densities in the pore versus the bulk density. The consecutive curves from bottom are for
ρ
m0
=
0
.
5, 0.2959, 0.09694 and 0 (the porewithout the quenched particles). The results of computer simulations are givenas points, theoretical curves resulting from MFA and from FMSA are given as(
···
) and (—), respectively. Parameters as the proﬁles in Fig. 1 and the nomenclature is the same as in Fig. 1.
of adsorbed ﬂuid. Due to pore walls “screening” by the matrixmolecules, the height of the local density peak adjacent to thewall is lowered at higher matrix density
ρ
m0
.