What is liquid in random porous media: the Barker-Henderson perturbation theory
We apply the Barker-Henderson (BH) perturbation theory to the study of a Lennard-Jones (LJ) fluid confined in a random porous matrix formed by hard sphere (HS) particles. In order to describe the reference system needed in this perturbation scheme, the extension of the scaled particle theory (SPT) i...
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Інститут фізики конденсованих систем НАН України
2015
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| Цитувати: | What is liquid in random porous media: the Barker-Henderson perturbation theory / M.F. Holovko, T.M. Patsahan, V.I. Shmotolokha // Condensed Matter Physics. — 2015. — Т. 18, № 1. — С. 13607:1-17 — Бібліогр.: 37 назв. — англ. |
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irk-123456789-1534962019-08-29T22:46:20Z What is liquid in random porous media: the Barker-Henderson perturbation theory Holovko, M.F. Patsahan, T.M. Shmotolokha, V.I. We apply the Barker-Henderson (BH) perturbation theory to the study of a Lennard-Jones (LJ) fluid confined in a random porous matrix formed by hard sphere (HS) particles. In order to describe the reference system needed in this perturbation scheme, the extension of the scaled particle theory (SPT) is used. The recent progress in the development of SPT approach for a hard sphere fluid in a hard sphere matrix allows us to obtain very accurate results for thermodynamic properties in such a system. Hence, we combine the BH perturbation theory with the SPT approach to derive expressions for the chemical potential and the pressure of a confined fluid. Using the obtained expressions, the liquid--vapour phase diagrams of a LJ fluid in HS matrix are built from the phase equilibrium conditions. Therefore, the effect of matrix porosity and a size of matrix particles is considered. It is shown that a decrease of matrix porosity lowers both the critical temperature and the critical density, while the phase diagram becomes narrower. An increase of a size of matrix particles leads to an increase of the critical temperature. From the comparison it is observed that the results obtained from the theory are in agreement with computer simulations. The approach proposed in the present study can be extended to the case of anisotropic fluid particles in HS matrices. Застосовано теорiю збурень Баркера-Гендерсона (БГ) для вивчення плину Ленарда-Джонса в невпорядкованiй пористiй матрицi, сформованiй твердими сферичними частинками. З метою опису системи вiдлiку, яка необхiдна для теорiї збурень, було використано розвинення теорiї масштабної частинки (ТМЧ). Останнi досягнення у розвитку ТМЧ для твердокулькового плину в твердокульковiй матрицi дозволяють отримувати термодинамiчнi властивостi в такiй системi iз високою точнiстю. Таким чином, нами поєднано теорiю БГ з теорiєю ТМЧ та виведено вирази для хiмiчного потенцiалу i тиску плину в матрицi. Використовуючи отриманi вирази та умови фазової рiвноваги, побудовано фазовi дiаграми газ–рiдина плину Ленарда-Джонса в твердокульковiй матрицi. Дослiджено ефект пористостi матрицi i розмiру матричних частинок. Показано, що зменшення пористостi матрицi понижує значення критичної температури i критичної густини плину, разом з тим, фазова дiаграма звужується. Також спостережено, що збiльшення розмiру матричних частинок призводить до росту критичної температури. Зауважено, що результати теорiї узгоджуються iз даними комп’ютерного моделювання. Запропонований теоретичний пiдхiд може бути розвинутий до опису анiзотропних рiдин у твердокульковiй матрицi. 2015 Article What is liquid in random porous media: the Barker-Henderson perturbation theory / M.F. Holovko, T.M. Patsahan, V.I. Shmotolokha // Condensed Matter Physics. — 2015. — Т. 18, № 1. — С. 13607:1-17 — Бібліогр.: 37 назв. — англ. 1607-324X PACS: 61.20.Gy, 61.43.Gt DOI:10.5488/CMP.18.13607 arXiv:1504.01228 http://dspace.nbuv.gov.ua/handle/123456789/153496 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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| description |
We apply the Barker-Henderson (BH) perturbation theory to the study of a Lennard-Jones (LJ) fluid confined in a random porous matrix formed by hard sphere (HS) particles. In order to describe the reference system needed in this perturbation scheme, the extension of the scaled particle theory (SPT) is used. The recent progress in the development of SPT approach for a hard sphere fluid in a hard sphere matrix allows us to obtain very accurate results for thermodynamic properties in such a system. Hence, we combine the BH perturbation theory with the SPT approach to derive expressions for the chemical potential and the pressure of a confined fluid. Using the obtained expressions, the liquid--vapour phase diagrams of a LJ fluid in HS matrix are built from the phase equilibrium conditions. Therefore, the effect of matrix porosity and a size of matrix particles is considered. It is shown that a decrease of matrix porosity lowers both the critical temperature and the critical density, while the phase diagram becomes narrower. An increase of a size of matrix particles leads to an increase of the critical temperature. From the comparison it is observed that the results obtained from the theory are in agreement with computer simulations. The approach proposed in the present study can be extended to the case of anisotropic fluid particles in HS matrices. |
| format |
Article |
| author |
Holovko, M.F. Patsahan, T.M. Shmotolokha, V.I. |
| spellingShingle |
Holovko, M.F. Patsahan, T.M. Shmotolokha, V.I. What is liquid in random porous media: the Barker-Henderson perturbation theory Condensed Matter Physics |
| author_facet |
Holovko, M.F. Patsahan, T.M. Shmotolokha, V.I. |
| author_sort |
Holovko, M.F. |
| title |
What is liquid in random porous media: the Barker-Henderson perturbation theory |
| title_short |
What is liquid in random porous media: the Barker-Henderson perturbation theory |
| title_full |
What is liquid in random porous media: the Barker-Henderson perturbation theory |
| title_fullStr |
What is liquid in random porous media: the Barker-Henderson perturbation theory |
| title_full_unstemmed |
What is liquid in random porous media: the Barker-Henderson perturbation theory |
| title_sort |
what is liquid in random porous media: the barker-henderson perturbation theory |
| publisher |
Інститут фізики конденсованих систем НАН України |
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2015 |
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http://dspace.nbuv.gov.ua/handle/123456789/153496 |
| citation_txt |
What is liquid in random porous media: the Barker-Henderson perturbation theory / M.F. Holovko, T.M. Patsahan, V.I. Shmotolokha // Condensed Matter Physics. — 2015. — Т. 18, № 1. — С. 13607:1-17 — Бібліогр.: 37 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT holovkomf whatisliquidinrandomporousmediathebarkerhendersonperturbationtheory AT patsahantm whatisliquidinrandomporousmediathebarkerhendersonperturbationtheory AT shmotolokhavi whatisliquidinrandomporousmediathebarkerhendersonperturbationtheory |
| first_indexed |
2025-07-14T04:38:06Z |
| last_indexed |
2025-07-14T04:38:06Z |
| _version_ |
1837595767940841472 |
| fulltext |
Condensed Matter Physics, 2015, Vol. 18, No 1, 13607: 1–17
DOI: 10.5488/CMP.18.13607
http://www.icmp.lviv.ua/journal
What is liquid in random porous media: the
Barker-Henderson perturbation theory∗
M.F. Holovko, T.M. Patsahan, V.I. Shmotolokha
Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii St.,
79011 Lviv, Ukraine
Received March 3, 2015
We apply the Barker-Henderson (BH) perturbation theory to the study of a Lennard-Jones fluid confined in a
random porous matrix formed by hard sphere particles. In order to describe the reference system needed in
this perturbation scheme, the extension of the scaled particle theory (SPT) is used. The recent progress in the
development of SPT approach for a hard sphere fluid in a hard sphere matrix allows us to obtain very accurate
results for thermodynamic properties in such a system. Hence, we combine the BH perturbation theory with
the SPT approach to derive expressions for the chemical potential and the pressure of a confined fluid. Using
the obtained expressions, the liquid–vapour phase diagrams of a LJ fluid in HS matrix are built from the phase
equilibrium conditions. Therefore, the effect of matrix porosity and a size of matrix particles is considered.
It is shown that a decrease of matrix porosity lowers both the critical temperature and the critical density,
while the phase diagram becomes narrower. An increase of a size of matrix particles leads to an increase of
the critical temperature. From the comparison it is observed that the results obtained from the theory are in
agreement with computer simulations. The approach proposed in the present study can be extended to the
case of anisotropic fluid particles in HS matrices.
Key words: fluids in random porous media, Barker-Henderson perturbation theory, liquid–vapour
coexistence, scaled particle theory
PACS: 61.20.Gy, 61.43.Gt
1. Introduction
The original understanding of the nature of the liquid state of matter is connected with Van derWaals
equation of state formulated nearly 150 years ago [1]. The Van derWaals picture focuses on different roles
of the strong short-ranged repulsive and long-ranged attractive intermolecular interaction in forming the
equilibrium properties of dense fluids. According to this picture, the harsh repulsive interactions fix the
shape and size of molecules and essentially determine a high-density fluid structure. While the contribu-
tion of repulsive interaction is entropic, the contribution of attractive interaction is mainly energetic and
can be treated as the perturbation. The first theory of liquids based on the Van der Waals idea was pro-
posed by Barker and Henderson (BH) nearly fifty years ago [2–4]. Within this theory, the intermolecular
potential is separated into the repulsive and attractive parts. The short-ranged repulsive contribution is
described within the framework of the hard spheres model while the long-ranged attractive part is in-
cluded within the Zwanzig high-temperature perturbation theory [5]. A few years later, Andersen, Chan-
dler and Weeks (ACW) developed a somewhat different theory of liquid based on the Van der Waals ap-
proach [6–9]. In this theory, instead of intermolecular interactions, they separated intermolecular forces
into repulsive and attractive parts. They also used the optimized cluster expansion (OCE) instead of the
high-temperature perturbation theory for the treatment of attractive forces. However, the first term of
perturbation related to the high temperature approximation (HTA) in both theories is identical except
that the repulsive and attractive interactions are not exactly the same.
∗Dedicated to Prof. Douglas Henderson on the occasion of his 80th birthday.
© M.F. Holovko, T.M. Patsahan, V.I. Shmotolokha, 2015 13607-1
http://dx.doi.org/10.5488/CMP.18.13607
http://www.icmp.lviv.ua/journal
Holovko et al.
In this paper we make use of the Van der Waals ideas and expand BH theory for the description of
liquids adsorbed in random porous media. To this end, we use the Madden-Gland model [10]. According
to this model, a porous medium is presented as a quenched configuration of randomly distributed hard
spheres forming a so-called matrix. A specific description of a fluid in such porous media is connected
with double quenched-annealed averages: the annealed average is taken over all fluid configurations and
the additional quenched average should be taken over all realizations of a matrix. One of the popular ap-
proaches to the solution of this problem is based on the replica method, which allows for the extension
of many theoretical methods of liquid state physics to the case of a fluid confined in a random porous
medium. For instance, using the replica Ornstein-Zernike (ROZ) integral equation theory [11], the statis-
tical mechanics approach of liquid state was extended to the description of various fluids confined in
random porous media [12, 13], including the chemically reacting fluids [14, 15]. However, unlike bulk
fluids, no analytical result has been obtained from the ROZ integral equation approach even for such a
simple model as a hard sphere (HS) fluid in a HS matrix. At the same time, the model of HS fluid has
a peculiar importance, since similar to the case of a bulk fluid [16, 17], it can be used as the reference
system in the development of different perturbation schemes.
The first rather accurate analytical results for a HS fluid in a HS matrix were obtained quite recently
in [18–23] by extending the scaled particle theory (SPT) [24, 25] to the case of a HS fluid confined in
random porous media. The SPT approach is based on the combination of the exact treatment of a point
scaled particle in a HS fluid combined with the thermodynamical treatment of a macroscopic scaled par-
ticle. The exact result for a point scaled particle in a HS fluid in a random porous medium was obtained
in [18]. However, the approach proposed in [18] referred to as SPT1 contains a subtle inconsistency ap-
pearing when the size of matrix particles is much larger than the size of fluid particles. Later on, this
inconsistency was eliminated in a new approach referred to as SPT2 [20]. Starting from this formalism,
a series of new approximations were developed [20–23]. Among these approximations we select only
SPT2b and SPT2b1, which will be used in this paper for the description of the reference system in the BH
perturbation theory.
The liquid–vapour phase diagrams play a central role in understanding the nature of liquids. In con-
trast to the bulk case, the results of investigations of liquid–vapour phase equilibrium of a simple fluid in
random porous media are rather controversial. Computer simulations of a simple fluid confined in a HS
matrix [26–29] demonstrate a possibility of the existence of two phase transitions. One is analogous to the
bulk liquid-vapor transition, but with a narrower coexistence curve and lower values of the critical den-
sity and critical temperature. The second transition occurs at lower temperatures and at higher densities,
and it is interpreted as a phenomenon related to the wetting effects in the fluid located in more confined
regions of the matrix. On the other hand, in more thorough investigations [29], it was noticed that this
second transition is extremely sensitive to a particular matrix configuration. In the case of HS matrix,
it was shown that two phase transitions appear in some realizations of the matrix, while a single phase
transition was observed in the others. In order to explain this observation, optimized cluster expansions
were used in [30]. It was found that different approximations can lead to qualitatively different results.
For example, the mean spherical approximation (MSA) gives only a single liquid–vapour transition, but
the inclusion of the second and the third cluster coefficients result in two phase transitions. At the same
time, it was shown in [31] that the association theory leads to one phase transition. Our recent investiga-
tion, in which we generalized the Van der Waals equation for simple fluids in random porous media, also
demonstrates only one liquid–vapour transition [22].
In the present study we combine the BH theory with the previously developed SPT approach to de-
scribe a liquid–vapour phase behaviour of a Lennard-Jones fluid confined in a randommatrix formed by
hard spheres (HSmatrix). For a comparison, the results obtained in different approximations are consid-
ered as well. As it was mentioned above for the reference system, the SPT approach is applied using the
SPT2b approximation and its improved version SPT2b1. It should be noted that the SPT2b approximation
is our first really successful result for confined fluids [20], although it may have essential problems at
high fluid densities and/or for low matrix porosities. At the same time, the improved SPT2b1 approxima-
tion is based on the original SPT2b, but it is free of this shortcoming and gives an accurate description
of the thermodynamics for a hard sphere fluid in a hard sphere matrix up to the close packing condi-
tions [21–23, 32]. Apart from the BH theory, we present the results obtained in the HTA approximation.
Using the developed approach, the phase diagrams for a confined Lennard-Jones are built. Different ma-
13607-2
What is liquid in random porous media
trix porosities andmatrix particle sizes are considered. To check the accuracy of the approaches proposed
in this paper, a comparison with the MSA results as well as with the results of Monte Carlo simulations is
performed.
2. Theory
2.1. Reference system: HS fluid in HS matrix
We start our theoretical consideration with the description of the reference system. For this purpose
we briefly recapitulate themain ideas of the SPT theory and present here the expressions for the chemical
potential and pressure of a HS fluid in a HS matrix, which were obtained in our previous papers [20, 21],
and which are needed in the current study. The key point of the SPT theory consists in a derivation of the
excess chemical potential of an additional scaled particle of a variable size inserted in a fluid. This excess
chemical potential is equal to a work needed to create a cavity in a fluid which is free from any other
particles. For a small scaled particle in a HS fluid in the presence of a porous medium, the expression for
the excess chemical potential is equal to [20]:
βµex
s = ln p0(λs)− ln
[
1−η1
(1+λs)3
p0(λs)
]
, (2.1)
where β= 1/(kBT ), kB is the Boltzmann constant, T is the temperature, η1 = 1
6
πρ1σ
3
1 is the fluid packing
fraction, ρ1 is the fluid density, σ1 is the diameter of HS fluid particles. The term p0(λs) = exp(−βµ0
s ) is
defined by the excess chemical potential of the scaled particle confined in an empty matrix, µ0
s . It has the
meaning of probability to find a cavity created by the scaled particle in the matrix in the absence of fluid
particles. We should note that here we use conventional notations [11–15, 18–22], where the index “1” is
used to denote a fluid component, the index “0” denotes matrix particles, while for the scaled particles
the index “s” is used.
For a large scaled particle, the excess chemical potential is presented by the thermodynamic expres-
sion for the work needed to create a macroscopic cavity inside the fluid, which at the same time is con-
fined in a porous medium, and the corresponding expression can be presented as follows:
βµex
s = w(λs)+β
PVs
p0(λs)
, (2.2)
where P is the pressure of fluid, Vs is the volume of a scaled particle. The multiplier 1/p0(λs) appears
due to an excluded volume occupied by matrix particles. In this context, it should be mentioned that the
probability p0(λ) is directly related to two different types of porosity [20–22]. The first one corresponds
to the case of λs = 0 and provides the geometrical porosity
φ0 = p0(λs = 0), (2.3)
which depends only on the structure of a matrix and it is equal to the volume fraction of a void between
the matrix particles. For a HS fluid in a HS matrix, it is equal to
φ0 = 1−η0 , (2.4)
where η0 = 1
6πσ
3
0ρ0, ρ0 = N0
V , N0 is the number of matrix particles, σ0 is the diameter of the matrix
particles, V is the volume of the system.
The second type of porosity corresponds to the case λs = 1 and provides the probe particle poros-
ity [20–22]
φ= p0(λs = 1) = e−βµ
0
1 , (2.5)
which is defined by the excess chemical potential of fluid particles in the limit of infinite dilution µ0
1.
Using the SPT theory [24, 25] for the case of a HS fluid in a HS matrix, the following expression for φ can
be derived:
φ= (1−η0)exp
{
−
[
3η0τ
1−η0
+
3η0
(
1+ 1
2η0
)
τ2
(1−η0)2
+
βP0η0
ρ0
]
τ3
}
, (2.6)
13607-3
Holovko et al.
where τ=σ1/σ0, and P0 is the bulk pressure of the matrix particles.
βP0
ρ0
=
(1+η0 +η2
0)
(1−η0)3
. (2.7)
According to the ansatz of SPT [18–22, 24, 25], w(λs) can be presented in the form of an expansion:
w(λs) = w0 +w1λs+
1
2
w2λ
2
s . (2.8)
Coefficients of this expansion can be found from the continuity of µex
s and the corresponding derivatives
∂µs/∂λs and ∂2µs/∂λ2
s at λs = 0. After setting λs = 1, the expression (2.2) yields the relation between the
pressure P and the excess chemical potential µex
1 of a fluid:
β(µex
1 −µ0
1) =− ln
(
1−η1/φ0
)
+ A
η1/φ0
1−η1/φ0
+B
(η1/φ0)2
(1−η1/φ0)2
+
βP
φ
η1
ρ1
, (2.9)
where the coefficients A and B determine the porous medium structure, and for a HS fluid in a HSmatrix,
they are as follows:
A = 6+
3η0τ(τ+4)
1−η0
+
9η2
0τ
2
(1−η0)2
,
B =
9
2
(
1+
τη0
1−η0
)2
. (2.10)
Using the Gibbs-Duhem equation, which relates the pressure of a fluid with its total chemical potential
µ1 = ln(Λ3
1ρ1)+µex
1 as
(
∂P
∂ρ1
)
T
= ρ1
(
∂µ1
∂ρ1
)
T
(2.11)
one derives the fluid compressibility as
β
(
∂P
∂ρ1
)
T
=
1
(
1−η1/φ
) + (1+ A)
η1/φ0
(
1−η1/φ
)(
1−η1/φ0
)
+ (A+2B)
(
η1/φ0
)2
(
1−η1/φ
)(
1−η1/φ0
)2
+2B
(
η1/φ0
)3
(
1−η1/φ
)(
1−η1/φ0
)3
. (2.12)
After dividing the expression (2.12) by ρ1 and subsequently integrating it over ρ1, one obtains the chem-
ical potential:
βµSPT2
1 = ln(Λ3
1ρ1)− ln(φ)− ln(1−η1/φ)+ (A+1)
φ
φ−φ0
ln
1−η1/φ
1−η1/φ0
+ (A+2B)
φ
φ−φ0
(
η1/φ0
1−η1/φ0
−
φ
φ−φ0
ln
1−η1/φ
1−η1/φ0
)
+2B
φ
φ−φ0
[
1
2
(η1/φ0)2
(1−η1/φ0)2
−
φ
φ−φ0
η1/φ0
1−η1/φ0
+
φ2
(φ−φ0)2
ln
1−η1/φ
1−η1/φ0
]
. (2.13)
It is worth noting that the second term− ln(φ) in (2.13) follows from the relation (2.5) and the correspond-
ing substitution βµ0
1 = − ln(φ). Similarly, integration of the right-hand side of expression (2.12) over ρ1
13607-4
What is liquid in random porous media
leads to the pressure
(
βP
ρ1
)SPT2
=−
φ
η1
ln
1−η1/φ
1−η1/φ0
+ (1+ A)
φ
η1
φ
φ−φ0
ln
1−η1/φ
1−η1/φ0
+ (A+2B)
φ
φ−φ0
(
1
1−η1/φ0
−
φ
η1
φ
φ−φ0
ln
1−η1/φ
1−η1/φ0
)
+2B
φ
φ−φ0
[
1
2
η1/φ0
(1−η1/φ0)2
−
2φ−φ0
φ−φ0
1
1−η1/φ0
+
φ
η1
φ2
(φ−φ0)2
ln
1−η1/φ
1−η1/φ0
]
. (2.14)
The expressions (2.13) and (2.14) are considered as the expression derived within the framework of the
SPT2 approach [20]. A simple analysis of (2.13) and (2.14) shows that they have two divergences at η1 =
φ and η1 = φ0. Since φ < φ0, the divergence at η1 = φ occurs at lower densities. However, from the
geometrical point of view, this divergence should appear at higher densities near the maximum value
of the fluid packing fraction available for a fluid in a given matrix. Different corrections improving the
SPT2 approach were proposed in [20–22]. Here, we consider two of them, which provide rather accurate
results in comparison with computer simulations. The first of them known as SPT2b can be derived if φ
is replaced by φ0 everywhere in (2.12) except for the first term. In this case, the chemical potential and
the pressure of a confined fluid are as follows:
βµSPT2b
1 = ln(Λ3
1ρ1)− ln(φ)− ln(1−η1/φ)+ (1+ A)
η1/φ0
1−η1/φ0
+
1
2
(A+2B)
(η1/φ0)2
(1−η1/φ0)2
+
2
3
B
(η1/φ0)3
(1−η1/φ0)3
, (2.15)
(
βP
ρ1
)SPT2b
=−
φ
η1
ln
(
1−
η1
φ
)
+
φ0
η1
ln
(
1−
η1
φ0
)
+
1
1−η1/φ0
+
A
2
η1/φ0
(1−η1/φ0)2
+
2B
3
(η1/φ0)2
(1−η1/φ0)3
. (2.16)
The second approximation referred to as SPT2b1 can be derived from SPT2b by removing the divergence
at η1 =φ by an expansion of the logarithmic term in (2.15)
− ln
(
1−η1/φ
)
≈− ln
(
1−η1/φ0
)
+
η(φ0 −φ)
φ0φ(1−η1/φ0)
. (2.17)
As a consequence, one obtains the following expressions within the SPT2b1 approximation:
βµSPT2b1
1 = ln(Λ3
1ρ1)− ln(φ)− ln(1−η1/φ0)+ (1+ A)
η1/φ0
1−η1/φ0
+
η1(φ0 −φ)
φ0φ(1−η1/φ0)
+
1
2
(A+2B)
(η1/φ0)2
(1−η1/φ0)2
+
2
3
B
(η1/φ0)3
(1−η1/φ0)3
, (2.18)
(
βP
ρ1
)SPT2b1
=
1
1−η1/φ0
φ0
φ
+
(
φ0
φ
−1
)
φ0
η1
ln
(
1−
η1
φ0
)
+
A
2
η1/φ0
(1−η1/φ0)2
+
2B
3
(η1/φ0)2
(1−η1/φ0)3
. (2.19)
2.2. BH perturbation theory for simple fluid in random porous medium
The next step of theoretical treatment is connected with a consideration of an attractive part of inter-
action. We consider a simple fluid with an intermolecular interaction in the form
v11(r ) =
{
∞, r <σ1,
u11(r ), r >σ1,
(2.20)
13607-5
Holovko et al.
where u11(r )É 0 is a pure attractive part of interaction.
In order to take into account the attractive part of interaction, in this subsection we generalize the BH
perturbation theory for the case of a fluid in a randomHSmatrix. To this end, we use the replica trick [11]
according to which a system of a fluid in a matrix of unmovable (frozen) particles can be replaced by
an equilibrium mixture consisting of the movable (annealed) matrix particles and s identical copies (or
replicas) of a fluid. The condition is also set that the fluid replicas from different copies do not interact
with each other, but they interact with the matrix. Such a system can be described in a standard way
using the liquid state theories, and the properties of a fluid can be obtained by considering the limit
s → 0. Therefore, the Helmholtz free energy of a fluid in a matrix can be presented as [30]:
F = lim
s→0
d
ds
F (s), (2.21)
where F (s) is the free energy of the (s +1)-component equilibriummixture.
Within the framework of the Zwanzig high-temperature perturbation theory [5], the first term of the
free energy expansion corresponds to the high-temperature approximation:
β(F −F0)HTA
V
=
1
2
ρ2
1β
∫
dr̄ gHS
11 (r )u11(r ), (2.22)
which is of the same form as in the optimized cluster expansions [30]. F0 and gHS
11 (r ) are the free energy
and the pair distribution function of a HS fluid in a HS matrix, respectively.
The second correction term of Zwanzig expansion involves the three- and four-body distribution func-
tions, for which it is difficult to find simple satisfactory approximations. Therefore, instead of this, we
follow Barker and Henderson [3, 4] and we write the free energy in the form:
β(F −F0)BH
V
=
1
2
ρ2
1β
∫
dr̄ gHS
11 (r )u11(r )
−
1
4
ρ2
1β
(
∂ρ1
∂P
)HS
T
∫
dr̄ u2
11(r )
(
∂
[
ρ1gHS
11 (r )
]
∂ρ1
)
HS
, (2.23)
where we use the same semimacroscopic arguments as in [3, 4] and neglect the cross-correlation terms
between fluids from different replicas.
Differentiating the expressions (2.22) and (2.23) with respect to the fluid density, one derives the ex-
pression for the chemical potential of a fluid:
βµ1 =βµHS
1 +βµHTA
1 +βµBH
1 , (2.24)
where µHS
1 is the HS contribution of the reference system, which can be given by equation (2.15) within
the SPT2b approximation or by the equation (2.18) within the SPT2b1 approximation. The first term of
(2.23) corresponds to the HTA approximation, and it has the following form:
βµHTA
1 = 2πβρ1
[
2I (ρ1)+ρ1
∂
∂ρ1
I (ρ1)
]
. (2.25)
The contribution coming from the BH approximation as the second term of (2.23) is as follows:
βµBH
1 =−πβρ1
[
2J (ρ1)
(
∂ρ1
∂P
)HS
T
+ρ1 J (ρ1)
∂
∂ρ1
(
∂ρ1
∂P
)HS
T
+4ρ1
(
∂ρ1
∂P
)HS
T
∂
∂ρ1
J (ρ1)+ρ2
1
∂J (ρ1)
∂ρ1
∂
∂ρ1
(
∂ρ1
∂P
)HS
T
+ρ2
1
(
∂ρ1
∂P
)HS
T
∂2 J (ρ1)
∂ρ2
1
]
. (2.26)
The expressions for the first and second derivatives of the isothermal compressibility with respect to the
fluid density for the reference system,
(
∂ρ1
∂P
)HS
T
and ∂
∂ρ1
(
∂ρ1
∂P
)HS
T
, are found using equations (2.16) and (2.19)
13607-6
What is liquid in random porous media
within the framework of the SPT2b or SPT2b1 approximations, correspondingly, and they are presented
in Appendix. The functions I (ρ1) and J (ρ1) are the integrals from the first and second terms of (2.23)
I (ρ1)=
∞
∫
0
gHS
11 (r )u11(r )r 2dr,
J (ρ1) =
∞
∫
0
gHS
11 (r )u2
11(r )r 2dr. (2.27)
These integrals contain the pair distribution function gHS
11 (r ), which is unknown for this moment and will
be considered separately in the next subsection.
The pressure can be calculated by differentiating the expression (2.23) with respect to the volume of
a system or from the general thermodynamical relation:
βP =βρ1µ1 −β
F
V
. (2.28)
The general form of the pressure is as follows:
βP =βPHS+βPHTA+βPBH, (2.29)
where PHS is the HS contribution given by (2.16) within the framework of SPT2b approximation or by
(2.19) within the framework of SPT2b1 approximation. The HTA term for the pressure is as follows:
βPHTA
ρ1
= 2πβρ1
[
I (ρ1)+ρ1
∂
∂ρ1
I (ρ1)
]
. (2.30)
The contribution of BH term is as follows:
βPBH
ρ1
=−πβρ1
[
J (ρ1)
(
∂ρ1
∂P
)HS
T
+ρ1 J (ρ1)
∂
∂ρ1
(
∂ρ1
∂P
)HS
T
+3ρ1
(
∂ρ1
∂P
)HS
T
∂
∂ρ1
J (ρ1)+ρ2
1
∂J (ρ1)
∂ρ1
∂
∂ρ1
(
∂ρ1
∂P
)HS
T
+ρ2
1
(
∂ρ1
∂P
)HS
T
∂2 J (ρ1)
∂ρ2
1
]
. (2.31)
2.3. The replica Ornstein-Zernike equations
The pair distribution function gHS
11 (r ) needed to calculate the integrals (2.27) can be obtained from
a solution of the so-called replica Ornstein-Zernike (ROZ) equations, which were derived by Given and
Stell [11] using the replica trick:
h00 = c00 +ρ0c00 ⊗h00 ,
h10 = c10 +ρ0c10 ⊗h00 +ρ1cc⊗h10 ,
h11 = c11 +ρ0c10 ⊗h01 +ρ1cc⊗h11 +ρ1cb⊗hc ,
hc = cc+ρ1cc⊗hc , (2.32)
where the symbol ⊗ denotes a convolution. The pair and direct fluid-fluid correlation functions are sep-
arated into the connected and blocked parts
h11(r )= hc(r )+hb(r ),
c11(r )= cc(r )+cb(r ). (2.33)
As usual in the liquid state theory [16, 17], the ROZ equations need additional closure relations. The
Percus-Yevick (PY) approximation is used for a HS fluid in a HS matrix considered in this paper as the
13607-7
Holovko et al.
0 1 2 3 4
0
1
2
3
4
5
1.0
1.0
1.0
2/3
2/3
r/
g 1
1(r
)
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
r/
g 1
1(r
)
1.0
*
1 = 0.48 (GCMC)
*
1 = 0.10 (GCMC)
*
1 = 0.48 (ROZ)
*
1 = 0.10 (ROZ)
*
1 = 0.48 (SPT)
*
1 = 0.10 (SPT)
Figure 1. (Color online) Fluid-fluid pair distribution functions gHS
11
(r ) of a HS fluid in a HS matrix for
different parameters. Left-hand panel: the different matrix densities ρ∗0 = ρ0σ
3
1 and the size ratios of
fluid and matrix particles τ = σ1/σ0, but the fixed fluid density ρ∗1 = ρ1σ
3
1 = 0.5. Right-hand panel: the
different fluid densities ρ∗1 , but the fixed matrix density ρ∗0 = 0.3.
reference system. As it was shown in [33, 34], this approximation gives results for the pair distribution
functions which are in good agreement with computer simulations. In the PY approximation for aHSfluid
in a HS matrix, the blocking direct correlation function is zero cb(r ) = 0, thus cc(r ) = c11(r ). Therefore,
the closure conditions in our case are as follows:
h00(r )=−1 if r <σ0 , c00(r )= 0 if r >σ0 ,
h10(r )=−1 if r <σ01 , c10(r )= 0 if r >σ01 ,
h11(r )=−1 if r <σ1 , c11(r )= 0 if r >σ1 ,
(2.34)
where σ01 = 1
2
(σ0 +σ1).
The set of equations (2.32) in combination with the closure relations (2.34) is solved numerically using
the hybrid Newton-Raphson procedure [35]. Some of the results for gHS
11 (r ) = 1+h11(r ) are presented in
figure 1. As one can see, the function gHS
11 (r ) shows a typical behaviour for a HS fluid. In the left-hand
panel of figure 1 it is shown that for the fixed fluid density, an increase of the matrix density, i.e., lowering
thematrix porosity, leads to a short-range order increase. The same effect is observed if the fluid density is
increased, but the matrix density is kept constant (figure 1, right-hand panel). There is also a comparison
of the ROZ results with the Monte-Carlo simulations for the low and high fluid densities of a HS fluid in
a HS matrix. It is clearly seen that they fit very well, except the contact value g11(σ+
1 ) for the dense fluid,
which is somewhat lower in the case of ROZ equations. Besides, there are two points of the contact value
obtained from the SPT for a fluid in a matrix [32], which are a bit higher than the values of ROZ, thus
closer to the corresponding simulation results. In this paper, for a comparison we also consider the mean
spherical approximation (MSA) for the studiedmodel of a fluidwith the pair interaction between particles
in the form (2.20). Similarly to the HTA and BH approximations, to calculate the chemical potential and
the pressure within the MSA approximation, the correlation functions are needed. For this purpose, we
can also use the ROZ equations in combination with the corresponding closure relations:
h00(r )=−1 if r <σ0 , c00(r ) = 0 if r >σ0 ,
h10(r )=−1 if r <σ01 , c10(r ) = 0 if r >σ01 ,
h11(r )=−1 if r <σ1 , c11(r ) = cc(r )=−βu11(r ) if r >σ1 .
(2.35)
Again, using the replica procedure [11] and the general expressions of Hoye and Stell [36] for thermo-
dynamic properties of an equilibrium mixture of (s +1) components in the limit s → 0, one obtains the
chemical potential and pressure of a fluid within the framework of the MSA approximation:
βµ1 =βµHS
1 +βµMSA
1
βP =βPHS+βPMSA, (2.36)
13607-8
What is liquid in random porous media
where “HS” denotes the reference system, i.e., the HS system. For the perturbation part within the MSA
approximation, the expressions for the chemical potential is [30]:
βµMSA
1 = 2πβρ1
∞
∫
0
u11(r )g11(r )r 2dr
−2πρ1
∞
∫
0
[
c11(r )−cHS
11 (r )
]
r 2dr −2πρ0
∞
∫
0
[
c01(r )−cHS
01 (r )
]
r 2dr. (2.37)
In the same way the pressure of a fluid confined in a matrix can be found:
βPMSA/ρ1 =
1
3
πρ1σ
3
1
{
g 2
11(σ+
1 )−
[
g 2
11(σ+
1 )
]
HS
}
+
1
3
πρ0σ
3
10
{
g 2
10(σ+
10)−
[
g 2
10(σ+
10)
]
HS
}
+
2
3
πρ1
∞
∫
0
g11(r )
∂u11(r )
∂r
r 3dr, (2.38)
where cHS
11 (r ) and cHS
10 (r ) are the fluid-fluid and fluid-matrix direct correlation functions for a HS fluid in
a HS matrix, g11(σ+
1 ) and g10(σ+
10) are the contact values of pair distribution functions g11(r )= 1+h11(r )
and g10(r ) = 1+h10(r ).
2.4. Some calculation details
In the present study we consider a fluid of spherical molecules confined in a HS matrix. According
to (2.20), the fluid–fluid interaction is decomposed into a hard-sphere part and a Lennard-Jones (12-6) tail
following Weeks, Chandler and Andersen [6], i.e.,
u11(r ) =
{
−ǫ, σ1 < r <
6p
2σ1 ,
4ǫ
[
(σ1/r )12 − (σ1/r )6
]
, r >
6p
2σ1 .
(2.39)
Applying a hard core potential for the repulsive potential, we avoid difficulties connected with the tem-
perature dependence of the fluid diameter, which requires a special treatment in the case of soft repul-
sion [3]. The Lennard-Jones attractive tail is truncated at r = 2.5σ1 in order to compare our results with
the corresponding computer simulations [28].
The functions I (ρ1) and J (ρ1) used in the generalized BH theory should be calculated from the inte-
grals (2.27). The dependencies of I (ρ1) and J (ρ1) on the fluid density are illustrated in figure 2. As one
0.0 0.2 0.4 0.6 0.8
-1.20
-1.15
-1.10
-1.05
-1.00
-0.95
-0.90
-0.85
I
0.046
0.15
0.3
0.0 0.2 0.4 0.6 0.8
0.48
0.56
0.64
0.72
0.80
0.88
0.046
0.15
0.3
J
Figure 2. (Color online) The integrals I (ρ1) and J (ρ1) defined by (2.27) for different matrix densities.
13607-9
Holovko et al.
can see, these dependencies are rather smooth, that is why they can be interpolated as polynomials. It
is noticed that polynomials of the order of 9 provide a satisfactory fitting for the given functions. Since
finally the functions I (ρ1) and J (ρ1) are polynomials, all first and second derivatives of these functions
used in (2.25)–(2.26) and (2.30)–(2.31) are calculated analytically.
Having the chemical potential and pressure as functions of ρ1 (or η1) at different temperatures, one
can calculate the coexistence curves of the liquid–vapour phase transition. For this purpose, we solve a
set of two non-linear equations which follows from the conditions of thermodynamic equilibrium:
µ1(ρv
1,T ) =µ1(ρl
1,T ),
P (ρv
1,T ) = P (ρl
1,T ), (2.40)
where ρv
1 and ρl
1 are the fluid density of vapour and liquid phases, respectively. The numerical solution of
these equations is realized using the Newton-Raphson algorithm. Thus, the liquid–vapour phase diagrams
of a fluid confined in HS matrices are constructed.
3. Results and discussions
We apply the theory presented in the previous section for the description of liquid–vapour phase co-
existence of a simple fluid in a HS random porous medium. However, we first consider the bulk case,
i.e., when ρ∗
0 = 0. The fluid-fluid interaction v11(r ) is taken in the form (2.20) with the attractive poten-
tial u11(r ) (2.40). Infigure 3, one can see the liquid–vapour phase diagrams in coordinates T−ρ1 for a bulk
fluid obtained within different approximations. The simulation results obtained in [28] using the method
of grand-canonical Monte Carlo are shown for a comparison. As it is seen, the HTA approximation gives
a good description only for low temperature and leads to the overestimation at higher temperatures. The
BH approximation slightly improves the diagram, but still essentially overestimates the critical tempera-
ture. As expected, the MSA approximation provides the best description of the phase diagram among the
considered ones, although the critical temperature is still higher than in the simulations.
Considering a fluid in a matrix within the model applied in our study, one should take into account
that there is no attractive interaction between fluid and matrix particles. Thus, the only effect of the
matrix is to confine the fluid in the void volume formed between matrix particles. Therefore, the most
relevant parameters, which determine fluid properties, are the matrix density ρ∗
0 or the corresponding
matrix porosity φ0, and the size ratio of fluid and matrix particles τ= σ1/σ0. In figure 4, we present the
liquid–vapour phase diagrams obtained for a fluid in matrices of different densities ρ∗
0 = 0 (bulk), 0.046,
0.15 and 0.30. In this case, the fluid and matrix particles are of equal sizes, i.e., τ = 1. For this purpose,
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Bulk
BH(SPT)
HTA(SPT)
MSA
GEMC
T*
Figure 3. (Color online) Liquid–vapour phase diagram for the bulk fluid (ρ0 = 0) with the fluid-fluid in-
teraction (2.20) and (2.39), where T∗ = kBT /ǫ. Solid line corresponds to BH theory, dotted line — HTA,
dashed line — MSA, symbols — GEMC simulation results [28].
13607-10
What is liquid in random porous media
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
T*
HTA(SPT2b1)
HTA(SPT2b)
GCMC
bulk
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
T*
BH(SPT2b1)
BH(SPT2b)
MSA
GCMC
bulk
Figure 4. (Color online) Liquid–vapour phase diagrams for the same fluid as in figure 3, but confined in
a matrix of different densities ρ∗0 = ρ0σ
3
1 = 0.046, 0.15 and 0.30, where T∗ = kBT /ǫ. Left-hand panel:
dashed lines correspond to the HTA approximation combined with SPT2b, solid lines — HTA combined
with SPT2b1, symbols— GCMC results taken from [28]. Right-hand panel: dashed lines correspond to the
BH theory combined with SPT2b, solid lines — BH theory combined with SPT2b1, dotted lines — MSA
approximation, symbols— the GCMC results taken from [28].
we use the theoretical approaches considered in the previous section, i.e., the HTA and BH approxima-
tions in combination with the reference system obtained within the SPT2b and SPT2b1 approaches. The
computer simulation results taken from [28] are presented in figure 4 for comparison. Similarly to the
bulk case, one can see the MSA results for the considered systems [30]. It is observed that for a lowmatrix
density ρ∗
0 = 0.046, the coexistence curves calculated using the SPT2b and SPT2b1 approximations almost
coincide. With an increase of the matrix density up to ρ∗
0 = 0.15, the difference between the results ob-
tained with SPT2b and SPT2b1 becomes more distinguishable. And finally, for the high matrix density
ρ∗
0 = 0.3, the diagrams differ essentially in these approximations. Moreover, the results obtained with the
use of the SPT2b are rather anomalous, since they are far from any other approximation. This anomaly
can be explained by the divergence contained in the expressions for the chemical potential and pressure
of a fluid when η1 → φ [21, 22], and which become important at high matrix densities. To illustrate this
problem, the dependencies of the chemical potential of a HS fluid in a HS matrix of densities ρ∗
0 = 0.15
and ρ∗
0 = 0.3, in comparison with the results of grand-canonical Monte Carlo are shown in figure 5. It is
clearly seen that for the matrix density ρ∗
0 = 0.3 the chemical potential of a fluid becomes wrong at the
densities ρ∗
1 > 0.2 and tends to infinity around 0.33. The latter value corresponds to the η1 ≈ φ, where
the divergence is expected. However, this is not the case for ρ∗
0 = 0.15, where the SPT2b approximation
is close to the result of SPT2b1 approximation. On the other hand, it is observed that the SPT2b1 approxi-
mation perfectly fits the simulation results for the both matrix densities up to the highest values of fluid
densities. Therefore, STP2b1 should be considered as the best choice for the description of a HS fluid in
HS matrix. Consequently, hereafter we restrict ourselves only to this approximation for the reference
system.
All the approximations HTA, BH and MSA correctly reproduce the basic trends of the behaviour of
liquid–vapour coexistence curves of a fluid in a matrix, i.e., a decrease of matrix porosity (or an increase
of the matrix density) leads to a critical point shift toward lower fluid densities and lower temperatures,
simultaneously the phase diagram becomes narrower (figure 4). Furthermore, all of the considered ap-
proximations give only one critical point. A comparison of the diagrams obtained using the approxima-
tions presented in our paper with computer simulations (figure 4) shows that the inclusion of the second
term in the BH theory essentially improves the description of coexistence curves. The computer simula-
tions data are in semiquantitative agreement with the theoretical prediction based on the MSA and BH
approximation. It is worth mentioning that in contrast to MSA, which is mostly numerical approach, the
BH approximation is more an analytical theory.
The phase diagrams obtained for a fluid in the matrices of different densities are presented in the
13607-11
Holovko et al.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0
2
4
6
8
10
12
14
16
18
= 0.15
SPT2b
SPT2b1
MC
ex 1
1
= 0.30
Figure 5. The chemical potential for a HS fluid confined in a HS matrix of densities ρ∗0 = ρ0σ
3
1 =
0.15 and 0.30, and the ratio τ = σ1/σ0 = 1.0. A comparison between the results of SPT2b (dashed
lines), SPT2b1 (solid lines) and the grand-canonical Monte Carlo simulations performed in the present
study (symbols).
reduced units as a plot T /Tcr versus ρ1/ρ1,cr in figure 6. In such a way, it makes possible to check whether
the theory developed for a fluid confined in a HS matrix satisfies the law of the corresponding states. As
one can see in the figure, the considered diagrams are rather close to each other, except the case of high
matrix density ρ∗
0 = 0.30. A general trend of the phase diagrams becomes broader when the reduced
temperature decreases.
Now, we consider the confinement effect of amatrix on afluid by varying the size ratio of the fluid and
matrix particles τ=σ1/σ0 at a fixed porosity. At the same time, we compare the theoretical results with
computer simulations data obtained in [29] using the method of Gibbs-ensemble Monte Carlo (GEMC) for
the conventional Lennard-Jones (LJ) potential
v11(r ) = 4ǫ
[
(σ1
r
)12
−
(σ1
r
)6
]
, (3.1)
truncated at the distance rc = 2.5σ1 . Since in our study the reference system is taken as a HS system, the
0.6
0.7
0.8
0.9
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
/ 1,cr
Bulk
*
0=0.046
*
0=0.15
=0.3
T/
T c
r
Figure 6. Liquid–vapour coexistence diagrams in terms of reduced temperature T /Tcr and reduced den-
sity ρ1/ρ1,cr for different matrix densities ρ∗0 and for the same fluid model as in figures 3–4. All curves
are calculatedwithin the framework of the BH theory with the description of the reference systemwithin
SPT2b1 approximation.
13607-12
What is liquid in random porous media
0.6
0.7
0.8
0.9
1.0
1.1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
BH(SPT2b1)
BH(SPT2b1) corrected
GEMC
GEMC
T*
Figure 7. (Color online) Liquid–vapour phase diagram for a Lennard-Jones fluid in a HS matrix for dif-
ferent size ratios τ = σ1/σ0, but at fixed porosity φ0 = 0.95. Theoretical predictions are given using the
BH theory with the description of the reference system within framework of the SPT2b1 approach. Solid
lines — reference system with a hard core size d1(T ) = σ1. The dashed lines — corrected results with
d1(T ) defined according to the Barker-Henderson formula (3.4). Circles — GEMC results taken from [29].
repulsive part of LJ potential is a hard core potential. Thus, according to the form (2.20), we use
v11(r ) =
∞, r <σ1 ,
4ǫ
[
(σ1/r )12 − (σ1/r )6
]
, σ1 < r < 2.5σ1 ,
0, r > 2.5σ1 .
(3.2)
In figure 7 (solid lines) there is presented a comparison of theoretical results calculated for the fluid-
fluid potential (3.2) with computer simulations [29] (symbols) for the LJ potential (3.1) at τ= 1 and τ= 3/2,
but for the fixed matrix porosity φ0 = 0.95 (or the matrix packing fraction η0 = 0.05). As one can see,
the form of the phase diagrams obtained from the BH theory is rather close to the data of simulations,
although for the both cases of τ they are notably narrower. This deviation is systematic and anticipated,
since the simulations are performed for the conventional LJ potential, which has a soft core, while we use
the reference system as a hard core fluid. According to (3.1), the repulsive potential is as follows:
ϕ11(r )=
{
4ǫ
[
(σ1/r )12 − (σ1/r )6
]
, r <σ1 ,
0, r >σ1 .
(3.3)
It is acceptable to substitute this repulsive part of the LJ potential by a hard core, but instead of the
diameter of HS particles σ1, one should take an effective diameter d1 which is somewhat smaller than σ1
and depends on the fluid temperature. To take into account a correct value of the effective diameter d1
we use one of the successful and the most popular relations proposed by Barker and Henderson for a LJ
fluid [3]:
dBH
1 (T ) =
σ1
∫
0
{
1−exp
[
−βϕ11(r )
]}
dr. (3.4)
Simple calculations of dBH
1 (T ) depending on temperature show that it does not vary too heavily. For in-
stance, for the temperature T ∗ = 1.1, the effective diameter is dBH
1 = 0.9711 and for T ∗ = 0.6, the diameter
is dBH
1 = 0.9815. However, it still can have a strong effect on the thermodynamics of the system, hence on
the curves of the liquid–vapour coexistence. Therefore, we need to substitute the diameter of HS particles
σ1 by dBH
1 (T ) in every place where it is needed in the expressions used for the reference system, i.e., the
terms containing σ1 or depending on it should be modified. The corrected expressions for the chemical
potential and the pressure for a fluid in a HS matrix are used and calculated depending on the temper-
ature. Again using the BH theory in combination with the SPT2b1 approximation, the phase diagrams
of liquid–vapour transition are obtained. The results of this correction is presented as dashed lines in
figure 7, and, as one can see, the theoretical curves coincide very well with the computer simulation data.
13607-13
Holovko et al.
0.85 0.90 0.95 1.00
0.21
0.24
0.27
0.30
* 1
,c
r
0
0.85 0.90 0.95 1.00
0.7
0.8
0.9
1.0
1.1
1.2
1.3
T* cr
Figure 8. The dependencies of the critical density ρ∗
1,cr
and critical temperature T∗
cr on thematrix porosity
φ0 for the same model as in figures 3–4.
It is worth noting that there is a quicker and more efficient way to improve the present phase di-
agrams. Our preliminary calculations show that coexistence curves change negligibly along the T -axis,
and most deviations take place along the ρ1-axis. This is mainly related to the effect of an excluded vol-
ume, which depends on η1 = πd3
1ρ1/6 and is overestimated in the case of the reference system with a
hard sphere size d1 = σ1. Therefore, to improve our results for the conventional LJ fluid, the packing
fraction should be replaced by ηBH
1 = π[dBH
1 (T )]3ρ1/6, and this is equivalent to the rescaling of the fluid
density as:
ρBH
1 = ρ1
[
σ1
dBH
1 (T )
]3
. (3.5)
The correction made in such a way allows us to obtain the diagrams which are practically equivalent to
those shown in figure 7.
Finally, in figure 8 we present the critical temperature T ∗
cr and the critical density ρ∗
1,cr
as a function
of porosity φ0 for the fluid confined in a HS matrix within the framework of the model considered in
figure 4. One can see here the effects similar to those observed in figure 4 and figure 7, i.e., with a decrease
of porosity φ0, the critical point shifts toward lower temperatures and densities. In figure 8 we do not
present the results for the model discussed in figure 7, since the results in figure 4 and figure 7 are shown
for a differentmodel of the fluid. However, we should remark that the critical temperature decreases with
an increase of τ at the fixed porosity, and the change of the critical density is very small. For example,
from the results presented in figure 7 it is found that for φ0 = 0.95 and τ = 2/3, the critical temperature
T ∗
cr = 1.095 and the critical density is ρ∗
1,cr
= 0.273, while for τ= 1, the critical temperature is T ∗
cr = 1.031
and the critical density is ρ∗
1,cr
= 0.275. If we estimate this in the limit τ→ 0, one can see that the critical
temperature T ∗
cr shifts to T ∗(bulk)
cr = 1.209 and the critical density to ρ∗
1cr → ρ∗(bulk)
1cr φ0 = 0.305×0.95 = 0.29.
ρ∗(bulk)
1cr and T ∗(bulk)
cr correspond to the values of the critical parameters of a bulk fluid.
4. Conclusions
In this paper, the Barker-Henderson (BH) perturbation theory is generalized for the a Lennard-Jones
fluid confined in a random porous matrix. As the reference system, a hard sphere fluid in a hard sphere
matrix is chosen. To describe the reference system, the extension of the scaled particle theory (SPT) is
used, and two corresponding approximations are tested. It is shown that the SPT2b1 approximation,
which was developed recently, makes it possible to achieve a very accurate description of the thermody-
namics of confined hard sphere systems. Combining the SPT approach with the BH theory, the expres-
sions for the chemical potential and the pressure of a simple fluid in a hard sphere matrix are derived.
Based on the obtained expressions, the phase diagrams of liquid–vapour transition are calculated and
13607-14
What is liquid in random porous media
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1
P
SPT2b
SPT2b1
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
-15
-12
-9
-6
-3
0
SPT2b
SPT2b1
1
1 P
Figure 9. (Color online) Isothermal compressibility
(
∂ρ1
∂P
)
T
and the derivative ∂
∂ρ1
(
∂ρ1
∂P
)
T
for a HS fluid
in a HS matrix calculated within SPT2b and SPT2b1 approximations.
compared with other theoretical approaches such as the high-temperature approximation and the mean-
spherical approximation. A comparison of our results with computer simulation data found in the liter-
ature is made as well. Different matrix porosities as well as the size ratios of fluid and matrix particles
are considered in the present paper. The proposed extension of the BH theory for the case of a fluid in a
matrix provides a good qualitative agreement with the computer simulations, and in some situations it
provides a marvelous quantitative agreement, as it is observed in the case of low matrix porosities and
at temperatures which are not very close to the critical point. The theory correctly reproduces the basic
effects of porous media on the liquid–vapour phase coexistence of simple fluids, i.e., with a decrease of
porosity, the critical point shifts toward lower fluid densities and lower temperatures. It is also observed
that for a fixed matrix porosity, but for variable sizes of matrix particles, the critical temperature in-
creases if the size of matrix particles becomes larger and moves to the value of critical temperature of a
bulk fluid, while the critical density changes weakly, and in the limit τ→ 0, it moves to the bulk critical
value normalized by the porosity φ0.
The approach developed in this paper can be extended to the case of more complex fluid systems in
a confinement. In future, we plan to generalize the BH theory for anisotropic fluids in random porous
media. Our very recent investigation [37] connected with the extension of the Van der Waals theory to
the case of anisotropic fluids in random porous matrices shows that due to anisotropic interactions, the
orientational order causes a competition between isotropic and anisotropic interactions, and the effect
of a matrix can essentially modify the liquid–vapour phase diagram.
A. Isothermal compressibility for a HS fluid in HS matrix
Here, we present the expression for the isothermal compressibility
(
∂ρ1
∂P
)
T
and the derivative
∂
∂ρ1
(
∂ρ1
∂P
)
T
for a HS fluid in a HS matrix obtained within the SPT2b and SPT2b1 approximations. Using
simple differentiations of the expressions (2.16) and (2.19), one obtains
(
∂ρ1
dP
)SPT2b
=
[
−
φ0/φ−1
1−η1/φ0
+
φ0
φ
1
(
1−η1/φ0
)2
+ A
η1/φ0
(
1−η1/φ0
)3
+2B
(η1/φ0)2
(
1−η1/φ0
)4
]−1
, (A.1)
∂
∂ρ1
(
∂ρ1
∂P
)SPT2b
T
=−
[
−
φ0/φ−1
φ0(1−η1/φ0)2
+
2
φ(1−η1/φ0)3
+ A
1+2η1/φ0
φ0(1−η1/φ0)4
+4B
η1/φ0 + (η1/φ0)2
φ0(1−η1/φ0)5
]
×
−
φ0
φ
−1
1−η1/φ0
+
φ0
φ
1
(1−η1/φ0)2
+ A
η1/φ0
(1−η1/φ0)3
+2B
(η1/φ0)2
(1−η1/φ0)4
−2
, (A.2)
13607-15
Holovko et al.
(
∂ρ1
∂P
)SPT2b1
T
=
[
1
1−η1/φ
+
η1/φ0
(
1−η1/φ0
)2
+ A
η1/φ0
(
1−η1/φ0
)3
+2B
(η1/φ0)2
(
1−η1/φ0
)4
]−1
, (A.3)
∂
∂ρ1
(
∂ρ1
∂P
)SPT2b1
T
=−
[
−
1
φ(1−η1/φ0)2
+
1+η1/φ0
φ0(1−η1/φ0)3
+ A
1+2η1/φ0
φ0(1−η1/φ0)4
+4B
η1/φ0 + (η1/φ0)2
φ0(1−η1/φ0)5
]
×
[
1
1−η1/φ
+
η1/φ0
(1−η1/φ0)2
+ A
η1/φ0
(1−η1/φ0)3
+2B
(η1/φ0)2
(1−η1/φ0)4
]−2
, (A.4)
where A and B are given in equation (2.10).
The dependence of
(
∂ρ1
∂P
)
T
and ∂
∂ρ1
(
∂ρ1
∂P
)
T
on the fluid packing fraction η1 calculated within the SPT2b
and SPT2b1 approximations for different matrix densities ρ∗
0 is shown in figure 9. One can see that for
low densities of ρ∗
0 , the results in the both approximations are nearly identical, but for ρ∗
0 = 0.30, the
results within SPT2b show an odd behaviour for η1 larger than 0.15.
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http://dx.doi.org/10.1063/1.1712308
http://dx.doi.org/10.1063/1.1701689
http://dx.doi.org/10.1103/RevModPhys.48.587
http://dx.doi.org/10.1063/1.1740409
http://dx.doi.org/10.1063/1.1674820
http://dx.doi.org/10.1063/1.1677784
http://dx.doi.org/10.1063/1.1678512
http://dx.doi.org/10.1007/BF01028471
http://dx.doi.org/10.1063/1.463883
http://dx.doi.org/10.1007/978-94-011-4564-0_13
http://dx.doi.org/10.1021/jp961443l
http://dx.doi.org/10.1063/1.473042
http://dx.doi.org/10.1021/jp809706n
http://dx.doi.org/10.1021/jp9106603
http://dx.doi.org/10.1063/1.3532546
http://dx.doi.org/10.5488/CMP.15.23607
http://dx.doi.org/10.1351/PAC-CON-12-05-06
http://dx.doi.org/10.1016/j.molliq.2013.05.030
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http://dx.doi.org/10.1103/PhysRevE.54.6557
http://dx.doi.org/10.1103/PhysRevE.60.5495
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http://dx.doi.org/10.1063/1.474134
http://dx.doi.org/10.1016/S0167-7322(03)00058-8
http://dx.doi.org/10.1021/jz502135f
http://dx.doi.org/10.1063/1.465206
What is liquid in random porous media
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Що таке рiдина в невпорядкованому пористому
середовищi: теорiя збурень Баркера-Гендерсона
М.Ф. Головко, Т.М. Пацаган, В.I. Шмотолоха
Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
Застосовано теорiю збурень Баркера-Гендерсона (БГ) для вивчення плину Ленарда-Джонса в невпоряд-
кованiй пористiй матрицi, сформованiй твердими сферичними частинками. З метою опису системи вiд-
лiку, яка необхiдна для теорiї збурень, було використано розвинення теорiї масштабної частинки (ТМЧ).
Останнi досягнення у розвитку ТМЧ для твердокулькового плину в твердокульковiй матрицi дозволяють
отримувати термодинамiчнi властивостi в такiй системi iз високою точнiстю. Таким чином, нами поєд-
нано теорiю БГ з теорiєю ТМЧ та виведено вирази для хiмiчного потенцiалу i тиску плину в матрицi.
Використовуючи отриманi вирази та умови фазової рiвноваги, побудовано фазовi дiаграми газ–рiдина
плину Ленарда-Джонса в твердокульковiй матрицi. Дослiджено ефект пористостi матрицi i розмiру матри-
чних частинок. Показано, що зменшення пористостi матрицi понижує значення критичної температури
i критичної густини плину, разом з тим, фазова дiаграма звужується. Також спостережено, що збiльшен-
ня розмiру матричних частинок призводить до росту критичної температури. Зауважено, що результати
теорiї узгоджуються iз даними комп’ютерного моделювання. Запропонований теоретичний пiдхiд може
бути розвинутий до опису анiзотропних рiдин у твердокульковiй матрицi.
Ключовi слова: плини в невпорядкованих пористих середовищах, теорiя збурення
Баркера-Гендерсона, фазовий перехiд газ-рiдина, теорiя масштабної частинки
13607-17
http://dx.doi.org/10.1103/PhysRevE.48.233
http://dx.doi.org/10.1080/00268978500102651
http://dx.doi.org/10.1063/1.434887
Introduction
Theory
Reference system: HS fluid in HS matrix
BH perturbation theory for simple fluid in random porous medium
The replica Ornstein-Zernike equations
Some calculation details
Results and discussions
Conclusions
Isothermal compressibility for a HS fluid in HS matrix
|