On the improvement of SPT2 approach in the theory of a hard sphere fluid in disordered porous media
The SPT2 approach is based on the scaled particle theory and developed for the description of thermodynamic properties of hard sphere (HS) fluids in disordered porous media. Using this approach a porous medium is modelled as a quenched matrix of hard spheres (HS) or overlapping hard spheres (OHS)....
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Інститут фізики конденсованих систем НАН України
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irk-123456789-1570092019-06-20T01:28:44Z On the improvement of SPT2 approach in the theory of a hard sphere fluid in disordered porous media Holovko, M. Patsahan, T. Dong, W. The SPT2 approach is based on the scaled particle theory and developed for the description of thermodynamic properties of hard sphere (HS) fluids in disordered porous media. Using this approach a porous medium is modelled as a quenched matrix of hard spheres (HS) or overlapping hard spheres (OHS). A hard sphere fluid immersed in a matrix can move in a void between matrix particles. A number of approximations were previously proposed within the SPT2 approach. Among these approximations, the SPT2b1 has been considered as the most successful and accurate one in a large range of fluid densities and for different matrix parameters. However, at high densities, it can lack accuracy, since it does not take into account that the maximum packing fraction of a HS fluid in a matrix is limited, not by the geometrical porosity of a matrix φ0 and the probe particle porosity φ, but by another type of porosity φ ∗ introduced in our previous studies. The porosity φ ∗ is related to the maximal adsorption capacity of a matrix and it is lower than φ0 and larger than φ. This can be crucial for a fluid in matrices of low porosities and at high fluid density, especially in the region near close-packing conditions. Therefore, the approximations SPT2b2 and SPT2b3 taking into account this feature were suggested, although they still needed a correction because of their poor accuracy. In the present study, we improved the versions of these approximations, named as SPT2b2∗ and SPT2b3∗ . We compare these different approximations with the results of computer simulations performed in the Monte Carlo grand-canonical ensemble. We test the SPT2 approach both for the one- and three-dimensional cases. We show that the SPT2b3∗ provides a very good description of the chemical potential of a confined fluid, which is better than others. This extends the applicability of the SPT2 approach to the studies of very dense fluids confined in disordered matrices. Пiдхiд SPT2 базується на теорiї масштабної частинки i розроблений для опису термодинамiчних властивостей твердокулькового (HS) плину в невпорядкованих пористих середовищах. При використаннi цього пiдходу пористе середовище моделюється як заморожена матриця твердих кульок (HS) або твердих кульок, якi перетинаються (OHS). Твердокульковий плин, помiщений в матрицю, може рухатися у вiльному просторi помiж матричними частинками. Ранiше ряд наближень було запропоновано в рамках пiдходу SPT2. Серед цих наближень вважалося, що SPT2b1 є найбiльш успiшним i точним у широкiй областi густин плину i рiзних параметрiв матрицi. Проте, при високих густинах точнiсть може бути недостатньою, оскiльки цей пiдхiд не бере до уваги те, що максимальна упаковка твердокулькового плину в матрицi є обмеженою не тiльки геометричною пористiстю матрицi φ0 i пористiстю пробної частинки φ, але й ще одним типом пористостi φ ∗ , який був запропонований нами в попереднiх дослiдженнях. Пористiсть φ ∗ пов’язана iз максимальною адсорбцiйною ємнiстю матрицi, i вона є меншою нiж φ0 та бiльшою нiж φ. Вона може бути визначальною для плину в матрицях iз низькою пористiстю та при високих густинах плину, особливо в областi близькiй до щiльної упаковки. Тому було запропоновано наближення SPT2b2 i SPT2b3, що враховують цю властивiсть, хоча й вони все ще потребували удосконалення у зв’язку iз їх поганою точнiстю. В даному дослiдженнi ми покращуємо цi наближення i назвали їх SPT2b2∗ та SPT2b3∗ . Ми порiвнюємо цi наближення iз результатами комп’ютерного моделювання, яке проводилося за допомогою методу Монте-Карло у великому канонiчному ансамблi. Перевiрено пiдхiд SPT2 для одно- i тривимiрних випадкiв. Показано, що SPT2b3∗ забезпечує дуже добрий опис, який є кращий за iншi, для хiмiчного потенцiалу плину в просторовому обмеженнi. Це розширює область застосування пiдходу SPT2 на випадок вивчення дуже щiльних систем плину в невпорядкованих матрицях. 2017 Article On the improvement of SPT2 approach in the theory of a hard sphere fluid in disordered porous media / M. Holovko, T. Patsahan, W. Dong // Condensed Matter Physics. — 2017. — Т. 20, № 3. — С. 33602: 1–14 . — Бібліогр.: 31 назв. — англ. 1607-324X PACS: 61.20.Gy, 61.43.Gy DOI:10.5488/CMP.20.33602 arXiv:1710.01104 http://dspace.nbuv.gov.ua/handle/123456789/157009 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The SPT2 approach is based on the scaled particle theory and developed for the description of thermodynamic
properties of hard sphere (HS) fluids in disordered porous media. Using this approach a porous medium is
modelled as a quenched matrix of hard spheres (HS) or overlapping hard spheres (OHS). A hard sphere fluid
immersed in a matrix can move in a void between matrix particles. A number of approximations were previously
proposed within the SPT2 approach. Among these approximations, the SPT2b1 has been considered as the most
successful and accurate one in a large range of fluid densities and for different matrix parameters. However, at
high densities, it can lack accuracy, since it does not take into account that the maximum packing fraction of a
HS fluid in a matrix is limited, not by the geometrical porosity of a matrix φ0 and the probe particle porosity φ,
but by another type of porosity φ
∗
introduced in our previous studies. The porosity φ
∗
is related to the maximal
adsorption capacity of a matrix and it is lower than φ0 and larger than φ. This can be crucial for a fluid in matrices
of low porosities and at high fluid density, especially in the region near close-packing conditions. Therefore,
the approximations SPT2b2 and SPT2b3 taking into account this feature were suggested, although they still
needed a correction because of their poor accuracy. In the present study, we improved the versions of these
approximations, named as SPT2b2∗
and SPT2b3∗
. We compare these different approximations with the results
of computer simulations performed in the Monte Carlo grand-canonical ensemble. We test the SPT2 approach
both for the one- and three-dimensional cases. We show that the SPT2b3∗ provides a very good description of
the chemical potential of a confined fluid, which is better than others. This extends the applicability of the SPT2
approach to the studies of very dense fluids confined in disordered matrices. |
| format |
Article |
| author |
Holovko, M. Patsahan, T. Dong, W. |
| spellingShingle |
Holovko, M. Patsahan, T. Dong, W. On the improvement of SPT2 approach in the theory of a hard sphere fluid in disordered porous media Condensed Matter Physics |
| author_facet |
Holovko, M. Patsahan, T. Dong, W. |
| author_sort |
Holovko, M. |
| title |
On the improvement of SPT2 approach in the theory of a hard sphere fluid in disordered porous media |
| title_short |
On the improvement of SPT2 approach in the theory of a hard sphere fluid in disordered porous media |
| title_full |
On the improvement of SPT2 approach in the theory of a hard sphere fluid in disordered porous media |
| title_fullStr |
On the improvement of SPT2 approach in the theory of a hard sphere fluid in disordered porous media |
| title_full_unstemmed |
On the improvement of SPT2 approach in the theory of a hard sphere fluid in disordered porous media |
| title_sort |
on the improvement of spt2 approach in the theory of a hard sphere fluid in disordered porous media |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2017 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157009 |
| citation_txt |
On the improvement of SPT2 approach in the theory of a hard sphere fluid in disordered porous media / M. Holovko, T. Patsahan, W. Dong // Condensed Matter Physics. — 2017. — Т. 20, № 3. — С. 33602: 1–14
. — Бібліогр.: 31 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT holovkom ontheimprovementofspt2approachinthetheoryofahardspherefluidindisorderedporousmedia AT patsahant ontheimprovementofspt2approachinthetheoryofahardspherefluidindisorderedporousmedia AT dongw ontheimprovementofspt2approachinthetheoryofahardspherefluidindisorderedporousmedia |
| first_indexed |
2025-07-14T09:21:23Z |
| last_indexed |
2025-07-14T09:21:23Z |
| _version_ |
1837613589822701568 |
| fulltext |
Condensed Matter Physics, 2017, Vol. 20, No 3, 33602: 1–14
DOI: 10.5488/CMP.20.33602
http://www.icmp.lviv.ua/journal
On the improvement of SPT2 approach in the theory
of a hard sphere fluid in disordered porous media
M. Holovko1, T. Patsahan1, W. Dong2
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
2 École Normale Supérieure de Lyon, Laboratoire de Chimie, UMR 5182 CNRS,
46 Allée d’Italie, 69364 Lyon, Cedex 07, France
Received July 7, 2017, in final form August 5, 2017
The SPT2 approach is based on the scaled particle theory and developed for the description of thermodynamic
properties of hard sphere (HS) fluids in disordered porous media. Using this approach a porous medium is
modelled as a quenched matrix of hard spheres (HS) or overlapping hard spheres (OHS). A hard sphere fluid
immersed in amatrix canmove in a void betweenmatrix particles. A number of approximations were previously
proposed within the SPT2 approach. Among these approximations, the SPT2b1 has been considered as the most
successful and accurate one in a large range of fluid densities and for different matrix parameters. However, at
high densities, it can lack accuracy, since it does not take into account that the maximum packing fraction of a
HS fluid in a matrix is limited, not by the geometrical porosity of a matrix φ0 and the probe particle porosity φ,but by another type of porosity φ∗ introduced in our previous studies. The porosity φ∗ is related to the maximal
adsorption capacity of amatrix and it is lower than φ0 and larger than φ. This can be crucial for a fluid inmatricesof low porosities and at high fluid density, especially in the region near close-packing conditions. Therefore,
the approximations SPT2b2 and SPT2b3 taking into account this feature were suggested, although they still
needed a correction because of their poor accuracy. In the present study, we improved the versions of these
approximations, named as SPT2b2∗ and SPT2b3∗. We compare these different approximations with the results
of computer simulations performed in the Monte Carlo grand-canonical ensemble. We test the SPT2 approach
both for the one- and three-dimensional cases. We show that the SPT2b3∗ provides a very good description of
the chemical potential of a confined fluid, which is better than others. This extends the applicability of the SPT2
approach to the studies of very dense fluids confined in disordered matrices.
Key words: confined fluids, porous material, scaled particles theory, hard sphere fluid, chemical potential,
porosity
PACS: 61.20.Gy, 61.43.Gy
1. Introduction
We dedicate this paper to the memory of our good friend, colleague and teacher Jean-Pierre Badiali,
who passed away last year. Among different scientific works by Jean-Pierre, the investigation of phase
transition of fluids adsorbed in deformable porous matrices takes an important place [1–3]. In the present
paper we consider some improvements of the SPT2 approach which had been developed earlier by us [4]
for the description of thermodynamic properties of the simple hard sphere model of a fluid confined in a
disordered matrix.
Fluids confined in disordered porous materials undergo drastic modifications in their properties, the
understanding and prediction of which is highly required for many practical applications [5]. Starting
from the pioneering work of Madden and Gland [6], different theories have been developed for the
description of fluids confined in disordered porous media [7–13]. According to the model of Madden
and Glandt, a disordered porous medium is presented as a matrix of quenched configuration of randomly
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
33602-1
https://doi.org/10.5488/CMP.20.33602
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
M. Holovko, T. Patsahan, W. Dong
distributed hard spheres. Despite a comprehensive study of fluids in disordered matrices, the developed
approaches were totally numerical. The first rather accurate analytical expressions for chemical potential
and pressure of a hard sphere fluid confined in a hard sphere (HS) or an overlapping hard sphere (OHS)
matrix were obtained only in [4, 14, 15] by extending the scaled particle theory (SPT) [16–18]. The
developed approach is based on a combination of the exact treatment of a point scaled particle in a hard
sphere (HS) fluid with the thermodynamic consideration of a finite size scaled particle. The exact result
for a point scaled particle in HS fluid confined in a disordered matrix was obtained in [14]. However, the
approach proposed in [14] referred to as SPT1 contained a subtle inconsistency appearing when a size
of matrix particles is essentially larger than a size of fluid particles. This inconsistency was eliminated
in [4] within the framework of a new approach named SPT2. Later on, the SPT2 approach for a hard
sphere fluid in disordered porous media was generalized for one- and two-dimensional cases [19], and
more recently for a fluid of hard convex body particles [20] and for the mixture of hard spheres [21] in
disordered matrices.
The original expressions derived within the SPT2 approach include two parameters defining the
porosity of a matrix. The first parameter is related to a bare geometry of a matrix. It is the so-called
geometrical porosity, φ0, and characterizes the free volume, which is not occupied by matrix particles.
The second one is defined by the chemical potential of a fluid in the limit of infinite dilution, and it is
called a probe particle porosity, φ 6 φ0, characterizing the adsorption of a fluid particle in an empty
matrix. It was shown that the SPT2 agrees well with computer simulation data at low fluid densities,
while at medium and at high densities, an essential difference can be obtained. This difference becomes
especially essential when the packing fraction of a fluid η1 reaches the values close to the probe particle
porosity φ, since the obtained expressions diverge at η1 → φ. In order to improve the description in
the SPT2 approach, a series of approximations SPT2a, SPT2b, SPT2c and SPT2d were proposed [4] by
a substitution of φ by φ0 in different places during a derivation of the expressions for thermodynamic
properties of a HS fluid in HS and OHS matrices. Among these approximations only SPT2a has no
divergence at η1 = φ. However, it was shown that this approximation systematically underestimates the
chemical potential of HS fluid in a matrix even for rather low densities. Other approximations improve
the description, but at higher fluid densities, they lead to an overestimation of the chemical potential
due to the divergence at η1 = φ as in the original approximation SPT2. Nevertheless, for that time, the
approximation SPT2b was selected as the best one, since it gave results closest to computer simulation
data. It was noticed that SPT2b is capable of reproducing the simulation results with a good accuracy at
low and medium fluid densities in a wide range of matrix parameters. On the other hand, the problem of
divergence at η1 → φ still remained. Thus, to get accurate results at high fluid densities it had to be fixed.
It should be noted that the divergence in the SPT2b approximation is connected only with the
logarithmic term ln(1 − η1/φ), which appears in the cases of one and two-dimensional systems as well
[19]. In order to find the way to remove the divergence in the logarithm, a fluid confined in a disordered
matrix was considered in a lower space dimensionality, namely the one-dimensional case was studied
[22]. By the example of one-dimensional hard rod fluid in a hard rod matrix, the logarithmic term was
modified in such a way as to omit the divergence η1 → φ. Hence, three new approximations of SPT2b
were derived—SPT2b1, SPT2b2 and SPT2b3. In the last two variants, the new type of porosity parameter
φ∗ was introduced, and it corresponds to the maximum value of fluid packing fraction of a hard rod fluid
in a disordered hard rod matrix. This porosity parameter has a clear physical meaning, since it defines
the maximum adsorption capacity of a matrix for a given fluid, and since a matrix is frozen, the porosity
φ∗ should be less than φ0.
Due to a structure of the expressions obtained in the approximations SPT2b2 and SPT2b3, the
divergence appears at η1 → φ∗, but in the SPT2b1 it is located at η1 → φ0. Since φ 6 φ∗ 6 φ0, the
divergence η1 → φ∗ is in the correct range of densities. However, some disagreement between the theory
and computer simulation data was still observed [22]. This was also noticed after an application of this
concept to the case of three-dimensional system. Despite the more appropriate physical justification of
SPT2b2 and SPT2b3, in general, the variant SPT2b1 appeared to be better than the latter ones. Thus,
the SPT2b1 has been applied during the recent years as the most successful one [15, 22–28], while the
SPT2b2 and SPT2b3 needed an improvement. On the other hand, one should bear in mind that SPT2b1
has a wrong asymptotic for highly dense systems (close packing region), for which the divergence should
appear at φ∗ < φ0.
33602-2
On the improvement of SPT2 approach
In this paper, we revise the approximations SPT2b2 and SPT2b3 in order to get an appropriate
description of the thermodynamic properties of a HS fluid in a HS or an OHS matrix at nearly close
packing conditions. For this reason, we propose approximations SPT2b2* and SPT2b3* that are obtained
from the Taylor expansion of logarithmic term around (φ − φ∗). The corresponding expressions for the
chemical potential of a one-dimensional system of a hard rod fluid in a hard rod matrix are obtained.
We also apply the same concept to the case of three-dimensional system of a hard sphere fluid in hard
sphere and overlapping hard sphere matrices. A comparison with computer simulations has shown that
both the approximations SPT2b2∗ and SPT2b3∗ give an essential improvement of SPT2b2 and SPT2b3,
respectively.
The paper is arranged as follows. The theoretical part with a review of our previous results and a
formulation of new approximations is presented in section 2. Computer simulation details are described
in section 3. In section 4, a comparison of different approximations obtained from the SPT2 approach
with the computer simulation results is shown and discussed. And finally, we draw some conclusions in
the last section.
2. Theory
In this theoretical part, we start from the generalization of our previous results [4, 14, 15, 19, 22] of
the application of SPT2 theory and its different modifications for n-dimensional hard sphere (HS) fluid in
disordered matrices. The key point of the SPT theory consists in deriving the excess chemical potential
of an additional scale 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. For a small scaled particle inserted into a HS fluid in the
presence of a matrix, the expression for the excess chemical potential is equal to [14]
βµexs = ln p0(λs) − ln
(
1 − η1
(1 + λs)n
p0(λs)
)
, (2.1)
where β = 1/(kBT), kB is the Boltzmann constant, T is the temperature, λs = Rs/R1 is the scaling
parameter, Rs is the radius of a scaled particle, R1 = σ1/2 is the radius of a fluid particle, n is the
dimension of space, η1 = ρ1v1 is the fluid packing fraction, ρ1 is the fluid density, v1 is the volume of
a fluid particle. In the one-dimensional case (n = 1) v1 = 2R1, in the two-dimensional case (n = 2)
v1 = πR2
1 and in the three-dimensional case (n = 3) v1 = 4
3πR3
1 . We use here conventional notations
[4, 7, 11–15], where 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.
The term p0(λs) = exp[−βµ0
s (λs)] is defined by the excess chemical potential µ0
s (λs) of the scaled
particle confined in an empty matrix. It has also the meaning of probability to find a cavity created by
the scaled particle confined in the matrix in the absence of fluid particles. It is worth noting that in our
study we consider two types of matrices, namely a HS matrix and an OHS matrix. Both the matrices are
presented as quenched configurations of disordered hard spheres. The difference between them is that
the HS matrix is composed of an equilibrated one-component system of HS particles, while the OHS
matrix is a totally random configuration of non-interacting particles (ideal gas configuration). Therefore,
the particles of OHS matrix can freely overlap and compose a porous structure, which fundamentally
differs from that in the HS matrix. The p0(λs) is connected with two types of matrix porosity. The first
one is the geometrical porosity:
φ0 = p0(λs = 0), (2.2)
which defines the volume of a void between matrix particles. The second one is the so-called probe
particle porosity
φ = p0(λs = 1) = exp(−βµ0
1), (2.3)
which can be determined by the excess chemical potential of a fluid in infinite dilution µ0
1. In contrast to
the geometrical porosity φ0, it also depends on the nature of a fluid.
From (2.2) and (2.3), the geometrical porosity is obtained as follows:
φ0 = 1 − η0 (2.4)
33602-3
M. Holovko, T. Patsahan, W. Dong
for a HS matrix and
φ0 = exp(−η0) (2.5)
for an OHS matrix, where η0 = ρ0v0 is the packing fraction of matrix, ρ0 = N0/V is the matrix density,
N0 is the number of matrix particles and V is the total volume of a system. The volume of a single matrix
particle is defined as v0 = σ0 for n = 1, v0 = πσ2
0 /4 for n = 2 and v0 = πσ3
0 /6 for n = 3, where σ0 = 2R0
is the diameter of the matrix particle.
Using the probe particle porosity given by equation (2.3) and the conventional SPT theory [16–18]
for the pressure P0 of a bulk HS fluid and for the chemical potential of a fluid in the limit of infinite
dilution µ0
1, the following expression in the case of a HS matrix can be derived
φ = (1 − η0) exp
(
−
βP0η0
ρ0
τ
)
,
βP0
ρ0
=
1
1 − η0
(2.6)
for the one-dimensional system,
φ = (1 − η0) exp
[
−
(
2η0
1 − η0
+
βP0η0
ρ0
τ2
)]
,
βP0
ρ0
=
1
(1 − η0)2
(2.7)
for the two-dimensional system and
φ = (1 − η0) exp
{
−
[
3η0τ
1 − η0
+
3η0(1 + 1/2η0)τ
2
(1 − η0)2
+
βP0η0
ρ0
τ3
]}
,
βP0
ρ0
=
1 + η0 + η
2
0
(1 − η0)3
(2.8)
for the three-dimensional system, where τ = R1/R0. In the case of OHS matrix, one derives from (2.3)
the following expression for the probe particle porosity:
φ = exp[−η0(1 + τ)n]. (2.9)
For a large scaled particle (Rs � 0), the excess chemical potential is presented by a thermodynamic
expression for the work needed to create a macroscopic cavity inside a fluid and it can be written in the
following form
βµexs (λs) = βµ
0
s + ω(λs) + β
Pvs
p0(λs)
, (2.10)
where P is the pressure of the fluid, vs is the volume of a scaled particle. The multiplier 1/p0(λs) appears
due to an excluded volume occupied by matrix particles.
According to the ansatz of SPT theory [4, 15],ω(λs) can be presented in the form of Taylor expansion,
which is cut off at (n− 1)-th term in n-dimensional case. Therefore, in the one-dimensional case one gets
ω(λs) = ω0 , (2.11)
in the two-dimensional case
ω(λs) = ω0 + ω1λs , (2.12)
and in three dimensions
ω(λs) = ω0 + ω1λs +
1
2
ω2λ
2
s . (2.13)
The coefficients ω0, ω1 and ω2 in these expansions can be found from the continuity of µexs (λs) and
its corresponding derivatives ∂µexs (λs)/∂λs and ∂2µexs (λs)/∂λ
2
s for λs = 0. After setting λs = 1 the
expression (2.10) yields the relation between the pressure P and the chemical potential µex1 of a fluid
confined in a matrix
β(µex1 − µ
0
1) = ln(1 − η1/φ0) + A
η1/φ0
1 − η1/φ0
+ B
(η1/φ0)
2
(1 − η1/φ0)2
, (2.14)
where
A = n −
p′0
φ0
+
1
2
[
n(n − 1) − 2n
p′0
φ0
+ 2
( p′0
φ0
)2
−
p′′0
φ0
]
, (2.15)
33602-4
On the improvement of SPT2 approach
B =
1
2
(
n −
p′0
φ0
)2
, (2.16)
p′0 = ∂p0(λs)/∂λs and p′′0 = ∂
2p0(λs)/∂λ
2
s at λs = 0.
We note that in the one-dimensional case [19] the parameters A and B are
A = B = 0, (2.17)
in the two-dimensional case
A = n −
p′0
φ0
, B = 0, (2.18)
and for the three-dimensional case A and B are given by (2.15) and (2.16) when n = 3.
Using the Gibbs-Duhem equation we obtain the SPT2 result from the relation (2.14) for the isothermal
compressibility of a fluid:
β
∂P
∂ρ1
= 1 +
η1/φ
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
. (2.19)
A direct integration of this expression leads to the SPT2 result for the excess chemical potential and for
the pressure [4]
β(µex1 − µ
0
1)
SPT2 = − 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
]
, (2.20)
(
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
]
. (2.21)
As one can see the expressions (2.20)–(2.21) have two divergences, which appear at η1 = φ and
η1 = φ0. Since φ < φ0, the first divergence at η1 = φ occurs at densities lower than the second one.
This divergence can strongly worsen the prediction of thermodynamic properties, especially when the
difference between φ and φ0 increases. In order to improve the SPT2 approach, different approximations
were developed [4]. The most successful among them is the SPT2b approach, which was derived by
replacing φ with φ0 everywhere in (2.19) except the first term. Such a correction leads to the following
modification of the expression (2.20) for the excess chemical potential of a confined fluid:
β(µex1 − µ
0
1)
SPT2b = − 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
. (2.22)
We do not present here the expression for the pressure, but it can be found in [4]. This expression for the
excess chemical potential essentially improves the description of thermodynamic properties at low and
33602-5
M. Holovko, T. Patsahan, W. Dong
medium densities. However, the divergence at η1 = φ remains in the first logarithmic term. In a study of
a hard rod fluid (one-dimensional analog of HS fluid) [22] this term was presented in the form
ln
(
1 −
η1
φ
)
= ln
(
1 −
η
φ0
)
+ ln
[
1 −
(
η1
φ
−
η1
φ0
) (
1 −
η1
φ0
)−1
]
. (2.23)
After the expansion of the last logarithmic term in (2.23) it can be rewritten as follows:
− ln (1 − η1/φ) ≈ − ln (1 − η1/φ0) +
η1(φ0 − φ)
φ0φ(1 − η1/φ0)
. (2.24)
As a result, a new form of the excess chemical potential of a confined fluid is obtained, and it is referred
to as the SPT2b1 approximation:
β(µex1 − µ
0
1)
SPT2b1 = − 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
. (2.25)
Two other approximations called SPT2b2 and SPT2b3 contain the third type of porosity φ∗ defined
by the maximum value of packing fraction of a fluid in a random porous media. In order to introduce
φ∗ in the expression for the chemical potential (2.22) within the SPT2b2 approximation, the logarithmic
term is modified in the following way [22]
− ln (1 − η1/φ) ≈ − ln (1 − η1/φ
∗) +
η1(φ
∗ − φ)
φ∗φ (1 − η1/φ∗)
. (2.26)
Consequently, the chemical potential in the SPT2b2 approximation is derived as
β(µex1 − µ
0
1)
SPT2b2 = − ln(1 − η1/φ
∗) +
η1/φ0
1 − η1/φ0
(1 + A) +
η1(φ
∗ − φ)
φ∗φ(1 − η1/φ∗)
+
1
2
(A + 2B)
(η1/φ0)
2
(1 − η1/φ0)2
+
2
3
B
(η1/φ0)
3
(1 − η1/φ0)3
. (2.27)
The SPT2b3 approximation can be obtained similar to the SPT2b2 approximation through an ex-
pansion of the logarithmic term in the expression (2.26) for the chemical potential. As a result, one
derives
− ln(1 − η1/φ
∗) ≈ − ln(1 − η1/φ0) +
η1/φ
∗
1 − η1/φ0
−
η1/φ0
1 − η1/φ0
, (2.28)
β(µex1 − µ
0
1)
SPT2b3 = − ln(1 − η1/φ0) +
η1/φ
∗
1 − η1/φ0
+
η1(φ
∗ − φ)
φ∗φ(1 − η1/φ∗)
+ 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
. (2.29)
For the sake of brevity, we omit here the expressions for the pressure in the SPT2b1, SPT2b2 and SPT2b3
approximations, which can be found in [22, 30].
In the present study we also propose two other ways to modify the expression (2.22), which are based
on the expansion of the logarithmic term in (2.22) around (φ − φ∗). The resulting expressions can be
considered as alternative ones to the approximations SPT2b2 and SPT2b3. Therefore, taking into account
that
− ln(1 − η1/φ) ≈ − ln(1 − η1/φ
∗) +
η1
(1 − η1/φ∗)
(φ∗ − φ)
φ∗φ∗
, (2.30)
33602-6
On the improvement of SPT2 approach
instead of SPT2b2 we obtain the chemical potential and pressure of a HS fluid in a disordered matrix in
the approximation, denoted SPT2b2∗:
β(µex1 − µ
0
1)
SPT2b2∗ = − ln(1 − η1/φ
∗) +
η1(φ
∗ − φ)
φ∗φ∗(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
, (2.31)
(
βP
ρ1
)SPT2b2∗
= −
φ∗
η1
ln(1 − η1/φ
∗) +
(φ∗ − φ)
φ∗
[
ln(1 − η1/φ
∗) +
η1/φ
∗
1 − η1/φ∗
]
+
φ0
η1
ln(1 − η1/φ0)
+
1
1 − η1/φ0
+
A
2
η1/φ0
(1 − η1/φ0)2
+
2
3
B
(η1/φ0)
2
(1 − η1/φ0)3
. (2.32)
In the samemanner, we derive the expressions for the chemical potential and pressure in the approximation
referred to as SPT2b3∗ and it can substitute the SPT2b3 approximation developed earlier:
β(µex1 − µ
0
1)
SPT2b3∗ = − ln(1 − η1/φ0) +
η1/φ
∗
1 − η1/φ0
+
η1(φ
∗ − φ)
φ∗φ∗(1 − η1/φ∗)
+ 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
, (2.33)
(
βP
ρ1
)SPT2b3∗
=
(φ∗ − φ)
φ∗
[
ln(1 − η1/φ
∗) +
η1/φ
∗
(1 − η1/φ∗)
]
+
1
1 − η1/φ0
+
(φ0 − φ
∗)
φ∗
[
ln(1 − η1/φ0) +
η1/φ0
(1 − η1/φ0)
]
+
1
2
A
(η1/φ0)
(1 − η1/φ0)2
+
2
3
B
(η1/φ0)
2
(1 − η1/φ0)3
. (2.34)
3. Computer simulation details
The grand-canonical ensemble Monte Carlo (GCMC) simulations [31] were performed within this
study in order to verify an accuracy of different approximations of the SPT2 approach in one- and three-
dimensions. In the one-dimensional case, a system is presented as a hard rod (HR) fluid in a hard rod
(HR) or an overlapping hard rod (OHR) matrix. In thee dimensions, a system was considered as a hard
sphere fluid in a hard sphere (HS) matrix or an overlapping hard sphere (OHS) matrix. In the both cases
the systems consisted of two components. The first component was represented by fixed particles of a
matrix and the second one by particles of a fluid that could move in a space not occupied by the matrix
particles. The interactions between particles are defined by the hard-core pair potentials of matrix and
fluid particles with sizes σ0 = 2R0 and σ1 = 2R1, respectively.
Two types of a matrix were used in this study. One is composed of the particles distributed randomly
without overlapping (HR or HS matrix), and another one is built totally randomly with possible over-
lapping of matrix particles (OHR or OHS matrix). In all simulations, a number of matrix particles was
equal to N0 = 10000. Two sizes of matrix particles were considered, σ0 = σ1 and σ0 = 3σ1 in the one-
dimensional case and one size, σ0 = σ1, in the case of three dimensions. Also, for the one-dimensional
case a fluid in a matrix of point particles was studied (σ0 = 0). It should be noted that all sizes and lengths
in our study are presented in units of σ, which is equal to a diameter of fluid particles, σ1 = σ.
Another important parameter of a matrix is its porosity. For a one-dimensional matrix we fixed the
probe particle porosity equal to φ = 0.35. In the three-dimensional case, the geometrical porosity is set
equal to φ0 = 0.843, which depending on the type of a matrix corresponds to φ = 0.173 for a HS matrix
or φ = 0.255 for an OHS matrix. The parameters of matrix porosities (φ or φ0) as well as a number of
33602-7
M. Holovko, T. Patsahan, W. Dong
matrix particles N0 were used to calculate a size of simulation box from the relations (2.4), (2.5) and
(2.6), taking into account that V = N0/ρ0 and ρ0 = η0/v0, where v0 = σ0 or v0 = πσ3
0 /6 is a volume of a
single particle of a HR or a HS matrix, respectively. In our study a cubic simulation box was used with
the periodical boundary conditions.
In GCMC simulations, the chemical potential of a fluid is set, and after equilibration a corresponding
fluid density is obtained. Each simulation run starts from a system consisting of a matrix configuration
prepared preliminary and fluid particles of the number density ρ1 = N1/V = 0.3 placed randomly into
the void between matrix particles. In the case of one-dimensional systems, a number of steps for the
equilibration equal to 6 ·105 and for the production equal to 2 ·105 appeared to be sufficient to get reliable
results even for the highest fluid densities. In three dimensions, 2 · 106 steps were taken for equilibration
and 5 · 105 steps used for production. To speed up the simulations, the linked cell list algorithm was
applied [31].
It should be noted that the results being obtained from simulations are sensitive to a matrix con-
figuration. Thus, for each set of the parameters, 8 different matrix configurations were generated and
the obtained results have been averaged over these matrix realizations. This allowed us to obtain fluid
densities with a statistical error less than 0.5%.
Therefore, from the simulations we obtained numerical relations between the fluid density and the
chemical potential, which were compared with the results of the SPT2 approach.
4. Results and discussions
4.1. Hard rod fluid in random porous media
We use different approximations based on the SPT2 approach for the description of thermodynamic
properties of a HS fluid confined in disordered matrices. First we start with the application of the
presented theory to the one-dimensional case of a confined fluid, i.e., with the description of a hard
rod (HR) fluid in a hard rod (HR) or an overlapping hard rod (OHR) matrix. The hard rods is the one-
dimensional analog of hard spheres model and it should allow us to distinguish a role of the logarithmic
term more precisely. According to (2.17) in the case of n = 1 the parameters A = B = 0. Some results
obtained in the SPT2 approximation and its modifications like the SPT2b, SPT2b1, SPT2b2 and SPT2b3
approximations were already discussed in our previous paper [22]. It was noticed that the SPTb1, SPT2b2
and SPT2b3 approximations essentially improve the results for thermodynamic properties in comparison
with STP2b. They provide a rather accurate description up to high fluid densities both in the one- and
three-dimensional cases. However, the SPT2b1 approximation fails at the densities near the close-packing
region since it does not take into account the limitation on the maximum packing fraction of a fluid in a
matrix, ηmax
1 , which is directly related to the maximum adsorption capacity of this matrix. On the other
hand, the approximations SPT2b2 and SPT2b3 do not have such a defect since they contain the third type
of porosity, φ∗, which defines ηmax
1 . In [22] we proposed a generalized expression for φ∗:
φ∗ =
φ0φ
φ0 − φ
ln (φ0/φ) , (4.1)
which totally reproduces the analytical expressions derived from the quenched-annealed density func-
tional theory (QA DFT) by Reich and Schmidt for the partition coefficient K(ηr1 → 1) of a HR fluid
in a HR or OHR matrix (see [29], equations (38) and (41)). According to [29] this partition coefficient
corresponds to the maximal possible amount of fluid particles, which can be loaded into a matrix, i.e.,
K(ηr1 → 1) ≡ ηmax
1 = φ∗. It is shown in [22] that the expression (4.1) works well for one-dimensional
systems. However, in three dimensions it cannot be considered as an accurate one due to a sphericity of
particles, which cannot fill the whole space with their bodies, in the same way as hard spheres cannot
reach the packing fraction η1 = 1 even in the bulk system. Nevertheless, the expression (4.1) takes
into account an important feature related to the frozenness of matrix particles. Since, matrix particles
cannot move and they are not in equilibrium with fluid particles, the total packing fraction of matrix and
fluid in the system is always less than one (i.e., η0 + η1 < 1) in any space dimensionality, including the
one-dimensional space. To put it differently, the maximum packing fraction of a fluid confined in a matrix
33602-8
On the improvement of SPT2 approach
is always less than the geometrical porosity ηmax
1 ≡ φ∗ < φ0. This feature starts to play an essential role at
high densities of a confined fluid, especially near close-packing conditions. Therefore, in order to obtain
a correct description of thermodynamic properties of a fluid in disordered matrices at high densities, the
applied approximations should contain the porosity φ∗.
In the present study we introduce two new approximations which take into account φ∗, SPT2b2∗ and
SPT2b3∗, which are aimed at yielding better quantitative results than their forerunners, the SPT2b2 and
SPT2b3. In order to test an accuracy of all the approximations presented in section 2, first we consider
the one-dimensional system of a HR fluid in a HR or OHR matrix, and compare the excess chemical
potential obtained from the expressions given above and the computer simulations results taken from our
previous paper [22]. Similar as it was done in [22] we fix the probe particle porosity φ = 0.35 and show
the dependencies of the chemical potential on the packing fraction of a HR fluid in a HR matrix and in an
OHR matrix. We consider two sizes of matrix particles σ0 = σ1 (small matrix particles) and σ0 = 3σ1
(large matrix particles), which correspond to τ = 1 and τ = 1/3, respectively. In figures 1 and 2 one can
see a comparison of the approximations SPT2b, SPT2b1, SPT2b2, SPT2b3, SPT2b2∗ and SPT2b3∗ with
the results of GCMC simulations for the excess chemical potential of HR fluid in disordered HR and
OHR matrices. Also, on the same plots we present the results obtained from the analytical expressions
0.0 0.1 0.2 0.3 0.4 0.5
2
4
6
8
10
12
14
SPT2b
SPT2b1
SPT2b2
SPT2b3
SPT2b2*
SPT2b3*
RS
ex 1
1
HR matrix
= 0.630
0.35; = 0.463
0.0 0.1 0.2 0.3 0.4 0.5
2
4
6
8
10
12
14
SPT2b
SPT2b1
SPT2b2
SPT2b3
SPT2b2*
SPT2b3*
RS
ex 1
1
HR matrix
= 0.493
0.35; = 0.413
Figure 1. (Color online) The excess chemical potential of a HR fluid in a disordered HR matrix. A
comparison of different approximations (lines) with GCMC simulations data (symbols). The vertical
dotted line corresponds to the value of maximum packing fraction φ∗.
0.0 0.1 0.2 0.3 0.4 0.5
2
4
6
8
10
12
14
SPT2b
SPT2b1
SPT2b2
SPT2b3
SPT2b2*
SPT2b3*
RS
ex 1
1
OHR matrix
= ; 0.592
0.350; 0.450
0.0 0.1 0.2 0.3 0.4 0.5
2
4
6
8
10
12
14
SPT2b
SPT2b1
SPT2b2
SPT2b3
SPT2b2*
SPT2b3*
RS
ex 1
1
OHR matrix
= ; 0.455
0.350; 0.398
Figure 2. (Color online) The excess chemical potential of a HR fluid in a disordered OHR matrix. A
comparison of different approximations (lines) with GCMC simulations data (symbols). The vertical
dotted line corresponds to the value of maximum packing fraction φ∗.
33602-9
M. Holovko, T. Patsahan, W. Dong
for the chemical potential derived by Reich and Schmidt (RS) for the same systems with the use of the
QA DFT approach (see equations (42) and (43) in [29]). We have rewritten these expressions in terms of
φ0, φ and τ as follows:
β(µex1 − µ
0
1)
RS = − ln
(
1 −
η1
φ0
)
+
τη0q0 + η1/φ0
1 − η1/φ0
− τη0q0 , (4.2)
where the coefficient q0 = 1/φ0 for a HR matrix and q0 = 1 for an OHR matrix. As it is seen in
figure 1 (left-hand panel) the RS formula for a HR fluid in a HR matrix gives the results close to the
SPT2b1 approximation. However, it noticeably underestimates the chemical potential, as it is shown for
the equal sizes of fluid and matrix particles. In contrast to this, the SPT2b1 perfectly fits the GCMC data,
at least up to η1 = 0.3, for the both cases of matrix particle sizes, and then totally fails at the high densities
close to η1 = φ
∗ (figure 1, left-hand panel). As it was expected, the original SPT2b approximation leads to
overestimation of the chemical potential due to an improper divergence at η1 = φ = 0.35, thus at η = 0.3,
it gives the worst results. Having a correct location of the divergence at η1 = φ
∗, the chemical potentials
obtained in the SPT2b2 and SPT2b3 approximations start to deviate from the GCMC data too early, thus
we get the results in these approximations higher than they should be in the case σ0 = σ1. However, for
the size of matrix particles σ0 = 3σ1, they provide a rather good accuracy, which is much better than
in the SPT2b1. Moreover, for large matrix particles, the SPT2b2 and SPT2b3 almost coincide with their
improved versions, the SPT2b2∗ and SPT2b3∗ approximations. However, it is not the case when we have
small matrix particles (σ0 = σ1), where the SPT2b2∗ and SPT2b3∗ are closer to the simulations, and the
SPT2b3∗ fits the GCMC results even better than the SPT2b2∗ approximation.
For a HR fluid in OHR matrix one can see a very similar picture as it is in the case of a HR matrix. In
figure 2 (right-hand panel) we show a comparison for the same approximations as above. We also present
the results of the RS expression for the chemical potential of a HR fluid, but in a OHR matrix (4.2).
And again for this case we observe an underestimation given by the RS formula in comparison with the
GCMC results. All other conclusions are totally the same as for the HR matrix case.
It is worth noting that the strongest confinement effect is exhibited for small matrix particles. Also,
the smaller are the sizes of matrix particles the less accurate results can be predicted by the theory. On
the contrary, with an increase of the size of matrix particles, the prediction of thermodynamic properties
is getting better. Moreover, the best accuracy can be obtained in the limit σ → ∞ (or τ → 0), if the
probe particle porosity φ is fixed. In this limit all the porosities become equal and all the approximations
presented in this study lead to the same result equivalent to the bulk-like case with the effective density
η̂1 = η1/φ.
Tomake a severe test of an accuracy of the theory by showing itsmaximumdeviation from simulations,
0.0 0.1 0.2 0.3 0.4 0.5 0.6
2
4
6
8
10
12
14
SPT2b
SPT2b1
SPT2b2
SPT2b3
SPT2b2*
SPT2b3*
RS
ex 1
1
Point matrix
; = 1.0
= 0.35; = 0.565
Figure 3. (Color online) The excess chemical potential of a HR fluid in a disordered point matrix. A
comparison of different approximations (lines) with GCMC simulations data (symbols). The vertical
dotted line corresponds to the value of maximum packing fraction φ∗.
33602-10
On the improvement of SPT2 approach
the sizes of matrix particles should be taken as small as possible. For this purpose, we consider another
limit by considering the so-called point matrix, which is formed by random point particles (the limit
σ → 0). In this case, the geometrical porosity tends to one (φ0 → 1), since the packing fraction of
matrix particles tends to zero (η0 → 0). At the same time, we fix the probe particle porosity φ, which
after taking the limit assumes a finite value and depends only on the number density of matrix particles
ρ0 and the size of fluid particles σ1:
φ = exp[−η0(1 + τ)]
��
σ0→0 = exp(−ρ0σ1), (4.3)
where η0 = ρ0σ0 and τ = σ1/σ0. A set of dependencies of the chemical potential on the packing fraction
of a HR fluid confined in a point matrix is presented in figure 3. Again, the results obtained in different
approximations behave qualitatively in the same way as in the cases of HR and OHR matrices. However,
this time the deviations from the simulations are essentially larger. This also concerns the SPT2b3∗
approximation, which in the previous figures exhibited very accurate results. Nevertheless, even in this
case the SPT2b3∗ remains the best among all other approximations. It should also be noted that matrix
particles can hardly be smaller than fluid particles. Moreover, they cannot be considered as point ones.
However, for methodological reasons, this case is always worth testing.
4.2. Hard sphere fluid in random porous media
From the results obtained for one-dimensional systems, we conclude that the SPT2b3∗ approximation
provides the best and very good accuracy for the chemical potential of a HR fluid confined in disordered
HR and OHR matrices at all fluid densities, including the one quite near the close-packing condition.
However, this is not quite obvious that in the three-dimensional system we will come to the same
conclusion. Therefore, we use the approximations described in this paper for a hard sphere fluid in
disordered hard sphere (HS) and overlapping hard sphere (OHS) matrices. In this case, the parameters
A and B are given by the expressions (2.15) and (2.16) when n = 3. Since the relation (4.1) for φ∗ is
presented in a general form and does not depend on the space dimensionality, we extend its application
to the three-dimensional case. We restrict our calculations to the case of equal sizes of fluid and matrix
particles (σ0 = σ1 or τ = 1) and to the low value of matrix porosity in order to strengthen the confinement
effects on a fluid, and thus clarify all possible drawbacks of the theory. In figure 4 we show a comparison
of the excess chemical potential obtained in different approximations with the computer simulations. Two
types of matrices (HS and OHS) are taken at the same geometrical porosity φ0 = 0.843, although the
probe particle porosities φ and the maximum adsorption capacity φ∗ are different (see the corresponding
values in figure 4). Similar to the one-dimensional case, all the approximations correctly reproduce the
0.0 0.1 0.2 0.3 0.4
2
4
6
8
10
12
14
16
18
20
22
24
0 = 0.843
= 0.173; * = 0.344
SPT2b
SPT2b1
SPT2b2
SPT2b3
SPT2b2*
SPT2b3*
GCMC
ex 1
1
HS fluid in HS matrix
0.20 0.24 0.28
6
8
10
12
14
ex
1
0.0 0.1 0.2 0.3 0.4
2
4
6
8
10
12
14
16
18
20
22
24
0 = 0.843
= 0.255; * = 0.437
SPT2b
SPT2b1
SPT2b2
SPT2b3
SPT2b2*
SPT2b3*
GCMC
ex 1
1
HS fluid in OHS matrix
0.26 0.28 0.30 0.32
8
10
12
14
ex
1
Figure 4. (Color online) The excess chemical potential of a HS fluid in disordered HS and OHS matrices.
A comparison of different approximations (lines) with GCMC simulations data (symbols). The vertical
dotted line corresponds to the value of maximum packing fraction φ∗.
33602-11
M. Holovko, T. Patsahan, W. Dong
behaviour of the chemical potential at small fluid densities. At medium fluid densities, the approximation
SPT2b has a divergence at η1 = φ. Thus, it crucially overestimates the chemical potential in this region and
does not give any result at higher densities. The SPT2b1 approximation provides a very good accuracy in a
large range of fluid densities up to η1 = 0.25, at which it starts to deviate from the GCMC results, because
the SPT2b1 does not take into account the limitation defined by the maximum adsorption capacity of
the matrix, i.e., the porosity φ∗ is not included in this approximation. An analysis of the approximations
SPT2b2 and SPT2b3 shows that although they contain the porosity φ∗ and have a divergence in the
correct place, their accuracy is poor due to an essential overestimate of the chemical potential, which
they provide starting from the medium densities. This is clearly seen for the case of a HS fluid in a HS
matrix (figure 4). However, for a HS fluid in a OHS matrix they are surprisingly good. Moreover, they are
as good as the approximations SPT2b2∗ and SPT2b3∗, which also have a correct divergence at φ∗. We
should remind that we observed the same picture herein above for a HR fluid in a OHR matrix. Similar
to a HR fluid in a HR matrix, the SPT2b2∗ and SPT2b3∗ give a much better prediction for the case of
a HS fluid in a HS matrix, and again the SPT2b3∗ surpasses SPT2b2∗ by accuracy. At the same time, it
is seen that the SPT2b3∗ is somewhat worse than SPT2b2∗ in the case of OHS matrix at high densities
(figure 4). However, the difference in the relative errors of the both approximations with respect to the
simulation results is not so large (within 1.5% for SPT2b2∗ and 2.5% for SPT2b3∗) as to give an absolute
preference to the SPT2b2∗. Hence, in general, the SPT2b3∗ approximation can be selected as the best
one for a description of a chemical potential of a fluid confined in disordered matrices.
5. Conclusions
We continue our efforts on improving the SPT2 approach developed for the description of the
thermodynamic properties of a hard sphere fluid in disordered matrices [4, 15, 23]. This approach
is based on the scaled particles theory originally proposed by Reiss, Frisch and Lebowitz [16] and
successfully adapted by us for the case of quenched-annealed systems like fluids in disordered porous
media [4, 14, 15, 23]. These systems are modelled as a mixture of mobile hard sphere (HS) particles of a
fluid immersed in a matrix of frozen hard spheres. We consider two types of a matrix: (i) a matrix formed
by equilibrated hard spheres system (HS matrix); (ii) matrix particles are randomly distributed (ideal
gas-like distribution). Thus, they can overlap and compose the so-called overlapping hard sphere system
(OHS matrix). A void in these two types of matrices forms qualitatively different porous structures,
which are characterized by a specific surface area of pore walls as well as by their mean curvatures.
These factors lead to quantitatively different dependencies of thermodynamic properties of a confined
fluid even at the same matrix porosities[15].
For the SPT2 approach, a number of approximations have been proposed in our previous studies:
SPT2b, SPT2b1, SPT2b2 and SPT2b3. The most accurate was the SPT2b1 approximation, but it did
not take into account the fact that due to the frozenness of matrix particles, the fluid particles cannot be
suitably packed into a pore. Thus, they cannot occupy the whole accessible volume which is defined by
the geometrical porosity of matrix, φ0. Neglecting this fact leads to inappropriate results at high fluid
densities, especially near the region of close-packing conditions. Therefore, we additionally introduced
another porosity parameter, φ∗, which is equivalent to themaximum packing fraction of a fluid in amatrix,
ηmax
1 . The porosity φ∗ is less than φ0 and it corresponds to the maximal adsorption capacity of a porous
medium. To take this porosity into account, an expression was proposed for it and the approximations
SPT2b2 and SPT2b3 were derived [22]. However, as it was shown herein above, these approximations
lack accuracy.
In this paper we considered all the approximations mentioned above and introduced two new approx-
imations, SPT2b2∗ and SPT2b3∗, which were expected to substitute the SPT2b2 and SPT2b3 approxi-
mations as more accurate ones. We assessed the accuracy of these approximations by comparing them
with the GCMC simulations. Since, the proposed theory can be used for a space of any dimensionality
n 6 3, first a comparison was made for a one-dimensional case presented as a system of a hard rod
(HR) fluid confined in a hard rod (HR) or an overlapping hard rod (OHR) matrix. Then, we tested the
SPT2 approach with different approximations for three-dimensional systems of a hard sphere (HS) fluid
in a hard sphere (HS) or an overlapping hard sphere (OHS) matrix. It was shown that the SPT2b2∗ and
33602-12
On the improvement of SPT2 approach
SPT2b3∗ approximations are really better than the SPT2b2 and SPT2b3, and especially this is seen for
a HR fluid in a HR matrix and for a HS fluid in a HS matrix. Between the SPT2b2∗ and SPT2b3∗ one
can choose the SPT2b3∗ approximation since in most cases it provides the best results, although for a
HS fluid in an OHS matrix the SPT2b2∗ is somewhat better. It should be also noted that in the presented
comparisons, the SPT2b1 approximation showed a very good accuracy for the description of chemical
potential of a HS fluid in HS and OHS matrices in a wide range of fluid densities. Hence, the SPT2b1
can also be used for the problems where fluid densities are far enough from extreme ones. However, for
more general cases, the SPT2b3∗ approximation is preferable.
The results obtained within the SPT2 approach can be used as a reference system for the description
of thermodynamic properties of a fluid confined in disordered porous media. The application of this
approach has already been found in many studies of confined liquids such as simple [15, 23] and ionic
fluids [25–27], network-forming colloids [24] and binarymixtures [21]. The SPT2 approach has also been
developed and used for the description of nematic liquids in matrices [20, 30]. The new approximation
SPT2b3∗ presented in this paper extends the applicability of the SPT2 due to an essential improvement
of its accuracy at high fluid densities.
Acknowledgements
MH and TP acknowledge support from the European Union’s Horizon 2020 research and innovation
programme under the Marie Sklodowska-Curie (grant No 734276) and the State Fund For Fundamental
Research (project N F73/26-2017).
The computer simulations have been performed on the computing cluster of the Institute forCondensed
Matter Physics of NAS of Ukraine (Lviv, Ukraine).
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Удосконалення пiдходу SPT2 в рамках теорiї
твердокулькового плину в невпорядкованому пористому
середовищi
М. Головко1, Т. Пацаган1, В. Донг2
1 Iнститут фiзики конденсованих систем Нацiональної академiї наук України,
вул. Свєнцiцького, 1, 79011 Львiв, Україна
2 Вища нормальна школа Лiону, хiмiчна лабораторiя, UMR 5182 НЦНД,
алея Iталiї 46, 69364 Лiон, Францiя
Пiдхiд SPT2 базується на теорiї масштабної частинки i розроблений для опису термодинамiчних власти-
востей твердокулькового (HS) плину в невпорядкованих пористих середовищах. При використаннi цього
пiдходу пористе середовище моделюється як заморожена матриця твердих кульок (HS) або твердих ку-
льок, якi перетинаються (OHS). Твердокульковий плин, помiщений в матрицю, може рухатися у вiльному
просторi помiж матричними частинками. Ранiше ряд наближень було запропоновано в рамках пiдходу
SPT2. Серед цих наближень вважалося, що SPT2b1 є найбiльш успiшним i точним у широкiй областi гу-
стин плину i рiзних параметрiв матрицi. Проте, при високих густинах точнiсть може бути недостатньою,
оскiльки цей пiдхiд не бере до уваги те, що максимальна упаковка твердокулькового плину в матрицi є
обмеженою не тiльки геометричною пористiстю матрицi φ0 i пористiстю пробної частинки φ, але й ще
одним типом пористостi φ∗, який був запропонований нами в попереднiх дослiдженнях. Пористiсть φ∗
пов’язана iз максимальною адсорбцiйною ємнiстю матрицi, i вона є меншою нiж φ0 та бiльшою нiж φ. Во-
на може бути визначальною для плину в матрицях iз низькою пористiстю та при високих густинах плину,
особливо в областi близькiй дощiльної упаковки. Тому було запропоновано наближення SPT2b2 i SPT2b3,
що враховують цю властивiсть, хоча й вони все ще потребували удосконалення у зв’язку iз їх поганою
точнiстю. В даному дослiдженнi ми покращуємо цi наближення i назвали їх SPT2b2∗ та SPT2b3∗. Ми по-
рiвнюємо цi наближення iз результатами комп’ютерного моделювання, яке проводилося за допомогою
методу Монте-Карло у великому канонiчному ансамблi. Перевiрено пiдхiд SPT2 для одно- i тривимiрних
випадкiв. Показано, що SPT2b3∗ забезпечує дуже добрий опис, який є кращий за iншi, для хiмiчного по-
тенцiалу плину в просторовому обмеженнi. Це розширює область застосування пiдходу SPT2 на випадок
вивчення дуже щiльних систем плину в невпорядкованих матрицях.
Ключовi слова: плини в просторовому обмеженнi, пористi матерiали, теорiя масштабної частинки,
твердокульковий плин, хiмiчний потенцiал, пористiсть
33602-14
https://doi.org/10.1021/jz502135f
https://doi.org/10.1088/0953-8984/28/41/414003
https://doi.org/10.1016/j.molliq.2016.10.045
https://doi.org/10.1016/j.molliq.2016.11.030
https://doi.org/10.1063/1.4984773
https://doi.org/10.1023/B:JOSS.0000041752.55138.0a
https://doi.org/10.1007/978-3-319-20875-6_1
Introduction
Theory
Computer simulation details
Results and discussions
Hard rod fluid in random porous media
Hard sphere fluid in random porous media
Conclusions
|