Analytical theory of one- and two-dimensional hard sphere fluids in random porous media
The recently proposed scaled particle theory SPT2 approach to the description of three-dimensional hard sphere fluids in random porous media is extended for one- and two-dimensional cases. Analytical expressions for the chemical potential and pressure of one- and two-dimensional hard sphere fluids i...
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| Cite this: | Analytical theory of one- and two-dimensional hard sphere fluids in random porous media / M.F. Holovko, V.I. Shmotolokha, W. Dong // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23607: 1-7. — Бібліогр.: 12 назв. — англ. |
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| citation_txt | Analytical theory of one- and two-dimensional hard sphere fluids in random porous media / M.F. Holovko, V.I. Shmotolokha, W. Dong // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23607: 1-7. — Бібліогр.: 12 назв. — англ. |
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| description | The recently proposed scaled particle theory SPT2 approach to the description of three-dimensional hard sphere fluids in random porous media is extended for one- and two-dimensional cases. Analytical expressions for the chemical potential and pressure of one- and two-dimensional hard sphere fluids in hard sphere and overlapping hard sphere matrices are obtained and discussed. Some improvements and modifications of the obtained results are proposed.
Недавно запропоновану теорію масштабної частинки SPT2 для опису тривимірного плину твердих сфер у випадковому пористому середовищі узагальнено на одно- та двовимірні випадки. Одержано та розглянуто аналітичні вирази для хімічного потенціалу та тиску одно- та двовимірних твердих сфер у матрицях твердих сфер і твердих сфер, що перетинаються. Запропоновано деякі покращання та модифікації одержаних результатів.
We thank T. Patsahan for some interesting discussions. The support from the CNRS-NASU cooperation project is acknowledged.
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Condensed Matter Physics 2010, Vol. 13, No 2, 23607: 1–7
http://www.icmp.lviv.ua/journal
Analytical theory of one- and two-dimensional hard
sphere fluids in random porous media
M.F. Holovko1, V.I. Shmotolokha2, W. Dong3
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str.,
79011 Lviv, Ukraine
2 Physico-Mechanical Institute of the National Academy of Sciences of Ukraine, 5 Naukova Str., 79601 Lviv,
Ukraine
3 Ecole Normale Superiere de Lyon, 46 Allee d’ Italie, 69364 Lyon, Cedex 07, France
Received April 23, 2010, in final form April 28, 2010
The recently proposed scaled particle theory SPT2 approach to the description of three-dimensional hard
sphere fluids in random porous media is extended for one- and two-dimensional cases. Analytical expressions
for the chemical potential and pressure of one- and two-dimensional hard sphere fluids in hard sphere and
overlapping hard sphere matrices are obtained and discussed. Some improvements and modifications of the
obtained results are proposed.
Key words: confined fluids, porous materials, scaled particle theory
PACS: 61.20.Gy, 61.43.Gy
1. Introduction
Much theoretical effort has been devoted to the study of fluids in porous materials for the
last two decades starting with a pioneering work of Madden and Gland [1]. In that work, a very
simple model for fluid adsorption in random porous media was proposed. Within the framework of
this model, a porous medium forms a matrix of randomly distributed particles. With the replica
Ornstein-Zernike integral equation [2] for the description of this model, the statistical-mechanics
approach of liquid state physics was extended to a description of different fluids confined in a
random porous matrix including the chemical reacting fluids adsorbed in porous media [3]. However,
in sharp contrast with the bulk fluid, no analytical results have been obtained in the integral
equation approach even for a simple model like a hard sphere fluid in a hard sphere matrix. The
main difficulty in describing such a model is related to the absence of direct interaction between
hard spheres from different replicated copies of fluids. Due to this, the description of such a model
is equivalent to the study of a hard-sphere mixture with strongly nonadditive diameters for which
it is very difficult to find correct analytical results.
The first very accurate analytical results for a hard-sphere (HS) fluid in HS and overlapping
HS matrices were obtained quite recently [4–6] by extending the scaled particle theory (SPT) to a
HS fluid confined in random porous media. This approach is based on the combination of the exact
treatment of point scaled particle in HS fluid with the thermodynamical consideration of finite
size scaled particle [7, 8]. It should be noted that for the bulk thermodynamical properties of HS
fluid from SPT there was obtained the same result as for analytical solution of integral equation in
Percus-Yevick approximation [9]. The exact result for point scaled particle in HS fluid in random
porous media was obtained in [4]. However, the proposed approach, called SPT1, has got some
inconsistency. After some improvements to the SPT1 there was developed a SPT2 approach [6] free
from this inconsistency. The expressions obtained in SPT2 include two types of porosities. One of
them is defined by the geometry of porous media and the second one is defined by the chemical
potential of the fluid in infinite dilution. After an additional approximation there was developed a
SPT2b approach that reproduces the computer simulation data with a very high accuracy.
c© M.F. Holovko, V.I. Shmotolokha, W. Dong 23607-1
http://www.icmp.lviv.ua/journal
M.F. Holovko, V.I. Shmotolokha, W. Dong
All the above discussed results were obtained for three-dimensional systems. In this paper we
extend the SPT method to the treatment of one- and two-dimensional HS fluid in random porous
media. We should mention that SPT for the pure HS fluid in one-dimensional case [10] reproduces
the exact result of Tonks [11] and in two-dimensional case leads to an analytical result which very
well reproduces the computer simulation data. In contrast to three-dimensional case for pure HS
fluids, in the two-dimensional case, the integral equation theory does not provide analytical results.
2. SPT for one- and two-dimensional hard sphere fluid
The basic idea of SPT is to insert an additional scaled HS particle of a variable size into a
HS fluid. This is equivalent to creating a spherical cavity devoid of any other fluid particles. In
the original formulation of SPT [7–10], the contact value of the pair distribution function of fluid
particle around the scaled particle is the central function of the theory. However, as it was pointed
in [4] such formulation cannot be readily extended to fluids in random porous media because no
direct relation between pressure and the contact distribution function has been found. Due to this
in [4] there was presented a reformulation of SPT which is slightly different from the original one.
In this section we consider a generalization of this approach for the one- and two-dimensional HS
fluids.
The central point of reformulation of SPT presented in [4] is a direct calculation of chemical
potential of the scaled particle. After generalizing this approach to the excess of chemical potential
of small scaled particle in n-dimensional HS fluid we have
βµex
s = − ln
(
1 − η1
(
1 +
RS
R1
)n)
for 0 6 RS + R1 6 R1 , (1)
where β = 1/kT , k is Boltzmann constant, T – is temperature, RS is the radius of scaled particle,
R1 is the radius of HS fluid particle, η1 = ρ1v1 is the fluid packing fraction, ρ1 is fluid density, v1
is the volume of one fluid particle. For one-dimensional case v1 = 2R1, for two-dimensional case
v1 = 4πR2
1, for three-dimensional case v1 = 4
3πR3
1.
For a larger scaled particle (RS > 0), the excess of chemical potential in SPT is given by a
thermodynamic expression for the work needed to create a macroscopic cavity inside a fluid and
it can be presented in the following form:
µex
s = wS(RS) + PvS , (2)
where P is pressure, vS is the volume of scaled particle.
wS(RS) is the surface contribution which can be presented in the form of Taylor expansion
which we cut off on (n − 1)th term
wS(RS) = w0 + · · · +
1
(n − 1)!
wn−1R
n−1
S . (3)
The coefficients of this expansion can be found from the continuity of µex
s at RS = 0. As a result,
for n = 1 we will have only the first term
βw0 = − ln(1 − η1). (4)
For n = 2 we will also have the second term
βw1 =
nη1
1 − η1
1
R1
. (5)
For n = 3 we will also have the next term
βw2 =
[
n(n − 1)η1
1 − η1
+
η2
1n
2
(1 − η1)2
]
1
R2
1
. (6)
23607-2
Analytical theory of hard sphere fluids in random porous media
Now it we put RS = R1, the expression (2) gives us the relation between the pressure and the
chemical potential of HS fluid. Using Gibbs-Duhem equation
(
∂P
∂ρ1
)
T
= ρ1
(
∂µ1
∂ρ1
)
T
(7)
we will find that for one-dimensional HS fluid
βµex
1 = − ln(1 − η1) +
η1
1 − η1
,
βP
ρ1
=
1
1 − η1
, (8)
for two-dimensional HS fluid
βµex
1 = − ln(1 − η1) +
2η1
1 − η1
+
η1
(1 − η1)2
,
βP
ρ1
=
1
(1 − η1)2
, (9)
for three-dimensional HS fluid
βµex
1 = − ln(1 − η1) +
7η1
1 − η1
+
15η2
1
2(1− η1)2
+
6η3
1
(1 − η1)3
,
βP
ρ1
=
1
1 − η1
+
3η1
(1 − η1)2
+
3η2
1
(1 − η1)3
. (10)
All these results were obtained earlier by the standard SPT techniques [7–10].
3. SPT for n-dimensional hard sphere fluid in random porous media
In the presence of porous media the expression for the excess of chemical potential of small
scaled particle in n-dimensional HS fluids can be written in the form
βµex
s = − ln (p0(RS) − η1(1 + RS/R1)
n) (11)
= − ln p0(RS) − ln
(
1 −
η1(1 + RS/R1)
n
p0(RS)
)
, (12)
where p0(RS) = e−βµ0
S is defined by the excess of the chemical potential µ0
S of the scaled particle
in pure porous media and has the sense of probability of finding a cavity created by the particle of
radius RS in the porous media in the absence of fluid particles. p1/0(RS) = 1− η1(1+RS/R1)
n
p0(RS) is the
conditional probability of finding a cavity created by the particle of radius RS in the fluid-matrix
system under condition that the cavity is located entirely inside a pore created by the particle with
a radius equal to RS.
For a larger scaled particle (RS > 0) the excess of chemical potential µex
s is treated in ther-
modynamical way. To this end, there are two possibilities described by SPT1 [4] and SPT2 [6]
approaches. We will show hereafter that SPT1 can be considered as the special case of SPT2 and
due to this we will proceed now to SPT2 approach. According to SPT2 for (RS > 0)
µex
s − µ0
S = wS(RS) +
1
p0(RS)
PvS , (13)
where P is the pressure, which is similar to [6] and is modified here by the presence of porous
media, vS is the volume of the scaled particle. p0(RS) is the probability of finding a cavity created
by the particle of radius RS > 0 in the matrix in the absence of fluid particles.
For RS = R1
p0(RS = R1) = φ (14)
is thermodynamic porosity defined by the excess chemical potential at the infinite dilution.
For RS = 0
p0(RS = 0) = φ0 (15)
23607-3
M.F. Holovko, V.I. Shmotolokha, W. Dong
is the geometrical porosity of porous media.
Similar to the previous section, wS(RS) describes the surface contribution and can be presented
in the form of expansion (3). The coefficients of this expansion can be found from the continuity
of µex
S at RS = 0.
As a result, for n = 1
βw0 = −ln(1− η1/φ0). (16)
For n = 2 we have the second term
βw1 =
η1/φ0
1 − η1/φ0
[
n
R1
−
p′0
φ0
]
. (17)
For n = 3 we have the third term
βw2 =
η1/φ0
1 − η1/φ0
[
n(n − 1)
R2
1
− 2
n
R1
p′0
φ0
+ 2
(
p′0
φ0
)2
−
p′′0
φ0
]
+
(
η1/φ0
1 − η1/φ0
)2 [
n
R1
−
p′0
φ0
]2
, (18)
where p′0 = ∂p0(RS)
∂RS
, p′′0 = ∂2p0(RS)
∂R2
S
at RS = 0.
Now if we put RS = R1 the expression (13) gives us the following relation between pressure P
and the excess chemical potential of the fluid in porous media
β(µex
1 − µ0
1) = − ln(1 − η1/φ0) + A
η1/φ0
1 − η1/φ0
+ B
(η1/φ0)
2
(1 − η1/φ0)2
+
βP
ρ1
η1
φ
, (19)
where for one-dimensional case A = B = 0, for two-dimensional case
A = n −
p′0
φ0
R1, B = 0, (20)
for three-dimensional case
A = n −
p′0
φ0
R1 +
1
2
[
n(n − 1) − 2n
p′0
φ0
R1 + 2
(
p′0
φ0
)2
R2
1 −
p′′0
φ0
R2
1
]
,
B =
1
2
(
n −
p′0
φ0
R1
)2
. (21)
Using Gibbs-Duhem equation (7) from (19) we have
β
(
∂P
∂ρ1
)
= 1 +
η1/φ
1 − η1/φ
+ (A + 1)
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
(22)
which is of the same form as in [6]. The general result for the chemical potential and pressure for
three-dimensional case is presented in [6]. Here we present the results for one- and two-dimensional
cases. For one-dimensional case
β(µex
1 − µ0
1) = −ln(1− η1/φ) +
φ
φ − φ0
ln
1 − η1/φ
1− η1/φ0
, (23)
βP
ρ1
=
φ0
η1
φ
φ − φ0
ln
1 − η1/φ
1− η1/φ0
(24)
and for two-dimensional case
β(µex
1 − µ0
1) = −ln(1− η1/φ) + (A + 1)
φ
φ − φ0
ln
1 − η1/φ
1− η1/φ0
+ A
φ
φ − φ0
[
η1/φ0
1 − η1/φ0
−
φ
φ − φ0
ln
1 − η1/φ
1− η1/φ0
]
, (25)
23607-4
Analytical theory of hard sphere fluids in random porous media
βP
ρ1
= −
φ
η1
ln
1 − η1/φ
1− η1/φ0
+ (A + 1)
φ
η1
φ
φ − φ0
ln
1 − η1/φ
1− η1/φ0
+ A
φ
φ − φ0
[
1
1 − η1/φ0
−
φ
η1
φ
φ − φ0
ln
1 − η1/φ
1− η1/φ0
]
(26)
where A is given by (20).
4. Discussion and conclusions
The expressions (23)–(26) are the main results of this paper. At the beginning of this discussion
we will consider the application of the obtained results to HS and overlapping HS matrices.
For the HS matrix for RS 6 0
p0(RS) = 1 − η0(1 + RS/R0)
n, (27)
where η0 = ρ0v0, ρ0 = N0
V , N0 is the number of matrix particles, V is the volume of the system,
R0 is the radius of matrix particle, v0 is the volume of matrix particle. For the considered case
φ0 = p0(RS = 0) = 1 − η0 , −
p′0
φ0
R1 =
nη0
1 − η0
τ, (28)
where τ = R1
R0
.
The excess chemical potential in infinite dilution µ0
1 = µ0
S(RS = R1) related to the thermody-
namical porosity
φ = exp(−βµ0
1) (29)
can be found from the usual SPT theory discussed in the second section.
For one-dimensional case
βµ0
1 = −ln(1− η0) + β
η0
ρ0
P0τ,
βP0
ρ0
=
1
1 − η0
. (30)
For two-dimensional case
βµ0
1 = −ln(1− η0) +
2η0
1 − η0
τ +
βη0P0
ρ0
τ2,
βP0
ρ0
=
1
(1 − η0)2
. (31)
For the overlapping HS matrix for RS 6 0
p0(RS) = exp (−η0(1 + RS/R0)
n) (32)
and
φ0 = p0(RS = 0) = e−η0 , −
p′0
φ0
R1 = nη0τ, βµ0
1 = η0(1 + τ)n. (33)
For two-dimensional case in accordance with (20) and (28) for HS matrix
A = 2
(
1 +
η0
1− η0
τ
)
(34)
and for overlapping HS according to (33)
A = 2(1 + τη0). (35)
Now all expressions are defined for one- and two-dimensional HS fluids in HS and overlapping
HS matrices. Corresponding expressions for three-dimensional case are presented in [6].
From (22) it is easy to obtain virial expansions for pressure and for chemical potential. The
second virial coefficient was obtained in [6]
B2 = v1
(
1
φ
+
A + 1
φ0
)
(36)
23607-5
M.F. Holovko, V.I. Shmotolokha, W. Dong
and it was shown that in three-dimensional case it is very accurate. As was shown in [6] for the
three-dimensional case, SPT2 overestimates the role of porosity φ for the third and higher virial
coefficients. A better description can be provided by SPT2b approximation which can be obtained
by a slight modification of the expression (22)
β
(
∂P
∂ρ1
)
=
1
1 − η1/φ
+ (A + 1)
η1/φ0
(1 − η1/φ0)2
+ (2B + A)
(η1/φ0)
2
(1 − η1/φ0)3
+ 2B
(η1/φ0)
3
(1 − η1/φ0)4
. (37)
The SPT2b approximation does not change the value of the second virial coefficient but for
higher densities it is more accurate than SPT2. For one-dimensional case SPT2b gives the following
results
β(µex
1 − µ0
1) = −ln(1− η1/φ) +
η1/φ0
1 − η1/φ0
,
βP
ρ1
= −
φ
η1
ln(1 − η1/φ) +
φ0
η1
ln(1− η1/φ0) +
1
1 − η1/φ0
(38)
and for two-dimensional case
β(µex
1 − µ0
1) = −ln(1 − η1/φ) + (1 + A)
η1/φ0
1− η1/φ0
+
1
2
A
(η1/φ0)
2
(1 − η1/φ0)2
,
βP
ρ1
= −
φ
η1
ln(1 − η1/φ) +
φ0
η1
ln(1− η1/φ0) +
1
1 − η1/φ0
+
1
2
A
η1/φ0
(1 − η1/φ0)2
. (39)
The comparison of the obtained results with computer simulation data will be presented else-
where [12].
Finally, we discussed the possibility of applying the SPT1 approach. From (13) and (3) it is
possible to show that SPT2 reduces to SPT1 if we put in (13) p(RS) = 1 and subtract in (29) of
[4] the infinite dilution part µ0
1 from the excess chemical potential. As a result, from (23)–(26) it
is possible to obtain the corresponding expressions for one- and two-dimensional case by a simple
substitution φ = 1. Since usually φ < φ0, this approximation is not quite correct at least for small
densities.
Acknowledgements
We thank T. Patsahan for some interesting discussions. The support from the CNRS-NASU
cooperation project is acknowledged.
References
1. Madden W.G., Glandt E.D., J. Stat. Phys., 1988, 51, 537.
2. Given J.A., Stell G., J. Chem. Phys., 1992, 97, 4573.
3. Trokhymchuk A., Pizio O., Holovko M., Sokolowski S., J. Chem. Phys., 1997,106, 200.
4. Holovko M., Dong W., J. Phys. Chem. B, 2009,113, 6360, 16091.
5. Chen W., Dong W., Holovko M., Chen X.S., J. Phys. Chem. B, 2010, 114, 1225.
6. Patsahan T., Holovko M., Dong W., J. Chem. Phys., 2010. (in press)
7. Reiss H., Frisch H.L., Lebowitz J.L., J. Chem. Phys., 1959, 31, 369.
8. Reiss H., Frisch H.L., Helfand E., Lebowitz J.L., J. Chem. Phys., 1960, 32, 119.
9. Yukhnovskij I.R., Holovko M., Statistical Theory of Classical Equilibrium Systems. Naukova Dumka,
Kyiv, 1980.
10. Helfand E., Frisch H.L., Lebowitz J.L., J. Chem. Phys., 1961, 34, 1037.
11. Tonks L., Phys. Rev., 1936, 50, 955.
12. Patsahan T., Shmotolokha V., Holovko M., Dong W. (in preparation).
23607-6
Analytical theory of hard sphere fluids in random porous media
Аналiтична теорiя одно- та двовимiрних плинiв твердих сфер
у випадковому пористому середовищi
М.Ф. Головко1, В.I. Шмотолоха2, В. Донг 3
1 Iнститут фiзики конденсованих систем НАН України, 79011 Львiв, вул. Свєнцiцького, 1
2 Фiзико-механiчний iнститут iм. Г.В. Карпенка НАН України, 79061 Львiв, вул. Наукова, 5
3 Нормальна вища школа в Лiонi, 69364 Лiон, Алея Iталiї, 46, Францiя
Недавно запропонована теорiя масштабної частинки SPT2 для опису тривимiрного плину твердих
сфер у випадковому пористому середовищi узагальнена на одно- та двовимiрнi випадки. Аналiтичнi
вирази для хiмiчного потенцiалу та тиску одно- та двовимiрних твердих сфер у матрицях твердих
сфер та твердих сфер, що перетинаються, отриманi та обговорюються. Запропоновано деякi по-
кращення та модифiкацiї отриманих результатiв.
Ключовi слова: обмеженi плини, пористi матерiали, метод масштабної частинки
23607-7
Introduction
SPT for one- and two-dimensional hard sphere fluid
SPT for n-dimensional hard sphere fluid in random porous media
Discussion and conclusions
|
| id | nasplib_isofts_kiev_ua-123456789-32097 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T17:00:01Z |
| publishDate | 2010 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Holovko, M.F. Shmotolokha, V.I. Dong, W. 2012-04-08T16:16:39Z 2012-04-08T16:16:39Z 2010 Analytical theory of one- and two-dimensional hard sphere fluids in random porous media / M.F. Holovko, V.I. Shmotolokha, W. Dong // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23607: 1-7. — Бібліогр.: 12 назв. — англ. 1607-324X PACS: 61.20.Gy, 61.43.Gy https://nasplib.isofts.kiev.ua/handle/123456789/32097 The recently proposed scaled particle theory SPT2 approach to the description of three-dimensional hard sphere fluids in random porous media is extended for one- and two-dimensional cases. Analytical expressions for the chemical potential and pressure of one- and two-dimensional hard sphere fluids in hard sphere and overlapping hard sphere matrices are obtained and discussed. Some improvements and modifications of the obtained results are proposed. Недавно запропоновану теорію масштабної частинки SPT2 для опису тривимірного плину твердих сфер у випадковому пористому середовищі узагальнено на одно- та двовимірні випадки. Одержано та розглянуто аналітичні вирази для хімічного потенціалу та тиску одно- та двовимірних твердих сфер у матрицях твердих сфер і твердих сфер, що перетинаються. Запропоновано деякі покращання та модифікації одержаних результатів. We thank T. Patsahan for some interesting discussions. The support from the CNRS-NASU cooperation project is acknowledged. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Analytical theory of one- and two-dimensional hard sphere fluids in random porous media Аналітична теорія одно- та двовимірних плинів твердих сфер у випадковому пористому середовищі Article published earlier |
| spellingShingle | Analytical theory of one- and two-dimensional hard sphere fluids in random porous media Holovko, M.F. Shmotolokha, V.I. Dong, W. |
| title | Analytical theory of one- and two-dimensional hard sphere fluids in random porous media |
| title_alt | Аналітична теорія одно- та двовимірних плинів твердих сфер у випадковому пористому середовищі |
| title_full | Analytical theory of one- and two-dimensional hard sphere fluids in random porous media |
| title_fullStr | Analytical theory of one- and two-dimensional hard sphere fluids in random porous media |
| title_full_unstemmed | Analytical theory of one- and two-dimensional hard sphere fluids in random porous media |
| title_short | Analytical theory of one- and two-dimensional hard sphere fluids in random porous media |
| title_sort | analytical theory of one- and two-dimensional hard sphere fluids in random porous media |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/32097 |
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