Simple fluid in an attractive, disordered polydisperse matrix
The extension of the replica Ornstein-Zernike (ROZ) equations is applied to the study of the structural properties of a Lennard-Jones fluid confined in an attractive polydisperse disordered matrix. The ROZ equations in combination with the orthogonal polynomial expansions for the correlation func...
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nasplib_isofts_kiev_ua-123456789-1189592025-06-03T16:27:02Z Simple fluid in an attractive, disordered polydisperse matrix Проста рідина в невпорядкованій полідисперсній матриці із притягаючою взаємодією Patsahan, T. Holovko, M. The extension of the replica Ornstein-Zernike (ROZ) equations is applied to the study of the structural properties of a Lennard-Jones fluid confined in an attractive polydisperse disordered matrix. The ROZ equations in combination with the orthogonal polynomial expansions for the correlation functions are used. The radial distribution functions are calculated for the adsorbed fluid at different temperatures. The effect of matrix polydispersity on the excess internal energy is considered in our study as well. Використовуючи розвинення рівнянь реплік Орнштейна-Церніке (РОЦ), досліджувались структурні властивості Леннард-Джонсівського флюїду адсорбованого в полідисперсній матриці із притягальною взаємодією. Для цього було застосовано РОЦ рівняння в поєднанні із розкладами кореляційних функцій по ортогональних поліномах. Пораховано радіальні функції розподілу адсорбованого флюїда при різних температурах. Також, в даному дослідженні, розглядався ефект полідисперсності матриці на внутрішню енергію системи. 2004 Article Simple fluid in an attractive, disordered polydisperse matrix / T. Patsahan, M. Holovko // Condensed Matter Physics. — 2004. — Т. 7, № 2(38). — С. 321–330. — Бібліогр.: 13 назв. — англ. 1607-324X PACS: 05.20.Jj, 68.43.De, 82.70.Gg DOI:10.5488/CMP.7.2.321 https://nasplib.isofts.kiev.ua/handle/123456789/118959 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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English |
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The extension of the replica Ornstein-Zernike (ROZ) equations is applied to
the study of the structural properties of a Lennard-Jones fluid confined in an
attractive polydisperse disordered matrix. The ROZ equations in combination
with the orthogonal polynomial expansions for the correlation functions
are used. The radial distribution functions are calculated for the adsorbed
fluid at different temperatures. The effect of matrix polydispersity on the
excess internal energy is considered in our study as well. |
| format |
Article |
| author |
Patsahan, T. Holovko, M. |
| spellingShingle |
Patsahan, T. Holovko, M. Simple fluid in an attractive, disordered polydisperse matrix Condensed Matter Physics |
| author_facet |
Patsahan, T. Holovko, M. |
| author_sort |
Patsahan, T. |
| title |
Simple fluid in an attractive, disordered polydisperse matrix |
| title_short |
Simple fluid in an attractive, disordered polydisperse matrix |
| title_full |
Simple fluid in an attractive, disordered polydisperse matrix |
| title_fullStr |
Simple fluid in an attractive, disordered polydisperse matrix |
| title_full_unstemmed |
Simple fluid in an attractive, disordered polydisperse matrix |
| title_sort |
simple fluid in an attractive, disordered polydisperse matrix |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2004 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/118959 |
| citation_txt |
Simple fluid in an attractive, disordered
polydisperse matrix / T. Patsahan, M. Holovko // Condensed Matter Physics. — 2004. — Т. 7, № 2(38). — С. 321–330. — Бібліогр.: 13 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
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| first_indexed |
2025-11-24T02:23:52Z |
| last_indexed |
2025-11-24T02:23:52Z |
| _version_ |
1849636717759299584 |
| fulltext |
Condensed Matter Physics, 2004, Vol. 7, No. 2(38), pp. 321–330
Simple fluid in an attractive, disordered
polydisperse matrix
T.Patsahan, M.Holovko
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received May 6, 2004, in final form May 19, 2004
The extension of the replica Ornstein-Zernike (ROZ) equations is applied to
the study of the structural properties of a Lennard-Jones fluid confined in an
attractive polydisperse disordered matrix. The ROZ equations in combina-
tion with the orthogonal polynomial expansions for the correlation functions
are used. The radial distribution functions are calculated for the adsorbed
fluid at different temperatures. The effect of matrix polydispersity on the
excess internal energy is considered in our study as well.
Key words: fluid, integral equations, polydisperse matrix, molecular
dynamics, confinement, silica gel
PACS: 05.20.Jj, 68.43.De, 82.70.Gg
1. Introduction
Within the framework of the integral equation theory a porous medium involving
the fluid is usually treated as a quenched system of disordered particles and is called
a matrix. The method of replica Ornstein-Zernike (ROZ) equations of Given and
Stell [1] in combination with the corresponding approaches is a powerful theoretical
tool in investigating the structural and thermodynamic properties of such partly
quenched systems. A simple, but quite realistic model, which is reasonable from the
practical point of view, was proposed by Kaminsky and Monson [2] to describe the
system of the methane adsorbed into silica gel pores. Within the framework of the
model of Kaminsky and Monson (KM), the matrix (M) particles are characterized
by large sizes comparatively to the adsorbed fluid (F) molecule sizes (σM/σF ≈ 7).
The strong fluid-matrix attractive potential (≈ 6εF) causes an adsorption of fluid
molecules on the matrix particle surface that is treated as the surface of the silica
gel pores. The KM model is widely used in various theoretical studies of structural
[2–5], thermodynamic [6–8] and even dynamical [9] properties of the fluid confined
1In fond remembrance of Zynoviy Gurskii, excelent scientist and good friend.
c© T.Patsahan, M.Holovko 321
T.Patsahan, M.Holovko
in the silica gel. But this model is restricted to the monodisperse matrix particles as
constituent parts of a model gel material. Naturally, the structure of porous medium
has a more complicated geometry. One of the most essential features that should be
taken into account is polydispersity of matrix particle sizes.
In the present paper the ROZ equations [1] in combination with the orthogonal
polynomial expansions of size-dependent functions [10] are used in order to study
structural properties of fluids in disordered polydisperse porous media. Firstly, the
polydisperse replica Ornstein-Zernike (PROZ) equations were applied by Ilnytski et
al [11], where the structural properties of the hard-sphere fluid in the hard-sphere
polydisperse matrix were considered. The model of porous medium that was used in
[11] is similar to the KM model but without attractive potential. The authors of the
paper [11] did not observe significant effects caused by the matrix polydispersity.
However, their investigations were restricted only to the hard-sphere interactions,
while the structural properties of the system with the attractive fluid-fluid and
fluid-matrix potentials could be more sensitive to the polydispersity of the matrix
particle sizes. Thus, we propose the present study in order to continue the main
idea of previous investigations and to see how the polydispersity affects the struc-
tural properties of Lennard-Jones fluid in the attractive matrix. To see the effect of
the matrix attraction, the case of hard-core matrices is considered. The molecular
dynamics (MD) simulations are performed for the system of fluid in KM matrix
to compare radial distribution functions (RDFs) and the values of internal excess
energy with the results obtained from ROZ equations.
2. Model and theory
Following the polydisperse extension of the KM model being proposed in our
study, the confinement is modelled as an interconnected network of pores formed
by the space between unmovable spherical particles of different sizes presenting the
matrix. The fluid molecules fill up the space between the matrix particles. The sizes
of matrix particles are defined by some distribution function f(σM). The important
characteristics of the matrix is the porosity χ = 1− η, where η is a packing fraction
of matrix particles:
η =
π
6
ρM
∫
σ3
Mf(σM)dσM. (1)
In our study we consider the two cases of interaction between matrix particles and
fluid molecules: the first one is the hard-sphere (HS) interaction (the case of HS ma-
trix)
ϕFM(r) =
{ ∞, r < (σM + σF)/2,
0, r � (σM + σF)/2,
(2)
322
Fluid in polydisperse matrix
and the second one is the interaction by the Kaminsky-Monson (KM) potential [2]
(the case of KM matrix)
ϕFM(r) =
2
3
πρsσ
3
Mεgs
[
σ12
gs
(
r6 +
21
20
σ2
Mr
4 +
3
16
σ4
Mr
2 +
σ6
M
192
) (
r2 − σ2
M
4
)−9
− σ6
gs
(
r2 − σ2
M
4
)−3
]
, (3)
where σM and σF are the diameters of a matrix particle and a fluid molecule, respec-
tively. In both cases for the fluid-fluid interaction we use the Lennard-Jones (LJ)
potential (4) which is truncated and shifted with rc = 2.5σF. The KM potential is
truncated and shifted as well, but at Rc = 20.48σF.
ϕFF(r) = 4εLJ
{(σF
r
) 12
−
(σF
r
) 6
}
. (4)
The distribution function of the diameters of matrix particles σM is chosen to be
stepwise:
f(σM) =
{
1/(σU − σL), σL < σM < σU ,
0, otherwise;
σL = 6.055σF and σU = 8.055σF . (5)
In order to describe the system of the fluid adsorbed in the polydisperse matrix we
use the PROZ equations [11,12] which, in combination with the Percus-Yevick (PY)
approximation, represent the reduced form of those in the Madden-Glandt like ap-
proach (MGOZ) [13]:
h̃MM,nm(k) = c̃MM,nm(k) + ρM
∑
l
h̃MM,nl(k)c̃MM,lm(k),
h̃FM,n(k) = c̃FM,n(k) + ρM
∑
l
c̃FM,l(k)h̃MM,nl(k) + ρFc̃FF(k)h̃FM,n(k),
h̃FF(k) = c̃FF + ρM
∑
l
h̃FM,l(k)c̃FM,l(k) + +ρFc̃FF(k)h̃FF(k), (6)
where a tilde denotes the Fouriez transforms of the pair hαβ(r) and the direct cαβ(r)
correlation functions (α and β standing for M or F). The PY closure relations to
solve (6) can be written
cαβ(r, σα,i, σβ,j) = [γαβ(r, σα,i, σβ,j) + 1]Fαβ(r, σα,i, σβ,j), (7)
where the indirect correlation function γαβ(r) is defined as γαβ(r)=hαβ(r) − cαβ(r)
and Fαβ(r, σα,i, σβ,j) is the Mayer function
Fαβ(r, σα,i, σβ,j) = exp[−uαβ(r, σα,i, σβ,j)/kBT ] − 1. (8)
Any function ψ(r, σM,i, σM,j) and ψ(r, σM,i) can be represented by their expansions
ψ(r, σM,i, σM,j) =
∑
k,l
ψkl(r)Pk(σM,i)Pl(σM,j), (9)
323
T.Patsahan, M.Holovko
ψ(r, σM,i) =
∑
k
ψk(r)Pk(σM,i), (10)
while the expansion coefficients are given by
ψMM,kl(r) =
∫
dσM,idσM,jf(σM,i)f(σM,j)ψ(r, σM,i, σM,j)Pk(σM,i)Pl(σM,j), (11)
ψFM,k(r) =
∫
dσM,if(σM,i)ψ(r, σM,i)Pk(σM,i). (12)
We expand all σM-dependent functions by using a set of normalized orthogonal
polynomials Pi(σM) (i = 0, 1, 2, . . . , n) associated with the distribution function
f(σM) and defined as follows:∫
dσMf (σM)Pi(σM)Pj(σM) = δij. (13)
Figure 1. Expansion coefficients of the function γFM for fluid in KM polydisperse
matrix at T ∗ = 1.2.
According to the chosen distribution function (5) the polynomials are used as the
normalized, associated Legendre polynomials. Taking into account that an inclusion
of strong attraction between matrix and fluid can adversely affect the convergence
of the expansion coefficients, in our calculations we take n = 10 of the expansion
coefficients number to expect a reasonable accuracy. In figure 1 the first five expan-
sion coefficients of the fluid-matrix indirect correlation function γFM,i are presented
for the fluid in KM polydisperse matrix at fluid density ρ∗F = ρσ3
F = 0.2336 and the
packing fraction of matrix η = 0.386. As is seen these coefficients converge rapidly
324
Fluid in polydisperse matrix
and for i � 5 they are negligibly small as opposed to the pair correlation function,
where convergence of its coefficients is rather poor, especially in the case of the at-
tractive matrix. The extensive questions concerning an accuracy and the problem of
choosing the number of expansion coefficients for the considered system, need addi-
tional examination and will be published elsewhere. Now, we stay on the assumtion
that the choice n = 10 is rather satisfactory for us.
We solve the PROZ equations (6) with the PY closure (7) to obtain the expan-
sion coefficients for pair correlation functions hFM,i(r) = gFM,i(r) − 1 and hFF(r) =
gFF(r) − 1. Thus, the excess internal energy for a fluid confined in a polydisperse
matrix can be easily calculated in the following form [12]:
U ex
N
= 4πρM
∑
i
∫
drr2gFM,i(r)ϕFM,i(r) + 2πρF
∫
drr2gFF(r)ϕFF(r). (14)
3. Results and discussion
We performed our calculations for a LJ fluid confined in a matrix at the constant
fluid density ρ∗F = ρFσ
3
F = 0.2336 and the matrix packing fraction ηM = 0.386.
In order to see an effect of the matrix polydispersity we calculated the expansion
coefficients of fluid-fluid and fluid-matrix pair correlation functions for monodisperse
(f(σM) = δ(σM−[σU +σL]/2)) and polydisperse matrix with particle sizes defined by
distribution (5). The packing fraction, ηM, was fixed by the fitting of the density of
the matrix particles corresponding to the distribution function (5) and the equation
(1). In table 1 we present the values of excess internal energy of the adsorbed fluid
obtained from (14) at two temperatures T ∗ = kT/εF = 1.2 and 2.0 for the cases
of mono- and polydisperse KM matrices. The excess internal energy of the fluid
confined in the HS matrix is considered as well. The MD simulations are performed
to verify our results for the case of monodisperse matrix. It is seen from the PROZ
results that the excess internal energy is larger in the case of polydisperse matrix
than in the monodisperse one. Simple calculations that can be made for different
Table 1. Excess internal energy per fluid molecule uex = U ex/NεF.
T* HS KM HS KM bulk
mono mono poly poly
ROZ ROZ MD PROZ PROZ OZ MD
1.2 uex -2.56 -6.49 -7.35 -2.56 -7.30 -1.58 -1.62
uex
FF -2.56 -2.24 -2.28 -2.56 -2.24
uex
FM 0 -4.25 -5.07 0 -5.06
2.0 uex -2.30 -6.35 -6.27 -2.30 -6.81 -1.38 -1.39
uex
FF -2.30 -2.15 -2.02 -2.30 -2.15
uex
FM 0 -4.20 -4.25 0 -4.66
325
T.Patsahan, M.Holovko
matrices at a constant porosity show that the larger is the size dispersity, the smaller
is an effective area of the surface of matrix particles. At the same time, most part
of excess energy between fluid and matrix is caused by attractive interaction on
the contact with the matrix particle surface. The smaller is the surface area, the
smaller amount of fluid molecules may be located in the first coordination shell near
the surface. Thus, a decrease of potential energy can be expected, and this is the
opposite effect to what we have. Although, according to the (3) the minimum of
the KM potential for the matrix particle with the large diameter is deeper than
the small one. Besides, the matrix particles with large sizes can adsorb a greater
amount of the fluid molecules due to a larger surface area. Therefore, the increase
of excess internal energy is observed in the polydisperse case. And, in this way, our
last reasoning agrees well with the results presented in table 1.
σ σ
σ σ
Figure 2. Fluid-matrix radial distribution functions obtained from the PROZ
equations.
Any effect of polydispersity cannot be seen when there is no attraction between
fluid and matrix (HS matrix). We suspect that it must be a drawback of PY MGOZ
like approach which we use in our study. It is known that within the framework of
326
Fluid in polydisperse matrix
the PY ROZ approximation, the blocking part of the fluid-fluid direct correlation
function that was originally introduced in [1] vanishes making an additional equation
for the connected parts unnecessary. The resulting simplified version of the ROZ
equations (6) are the polydisperse version of MGOZ equations and they do not take
into account the interaction between fluid molecules belonging to different replicas.
Therefore, the results obtained from the ROZ equations, in combination with PY
and the ones obtained from MD simulations (table 1) for the fluid in monodisperse
matrix, essentially differ, especially at low temperatures (see T ∗ = 1.2).
ρ
σ
ρ
σ
ρ
σ
ρ
σ
Figure 3. Radial distribution functions for the system of fluid confined in
monodisperse matrix. Lines correspond to the results obtained from the ROZ
equations and points correspond to the MD simulation results.
Because of the vanishing effect of the matrix polydispersity on the fluid-fluid
structure in all considered cases, we do not present figures with the fluid-fluid pair
distribution functions. However, it should be noticed that the negligible effect of
polydispersity on fluid-fluid radial distribution functions is an evidence of the fact
that within the framework of the considered approximation the fluid-fluid structure
327
T.Patsahan, M.Holovko
is sensible to the matrix packing fraction only. Therefore, the contribution into excess
internal energy due to fluid-fluid interaction is practically permanent in the range
of the considered dispersity of matrix particle sizes. Also, the difference between
internal energy of the fluid confined in the HS matrix and the bulk one (see table 1)
can be most probably caused only by the excluded volume effect.
While the fluid-fluid structure is unaffected by matrix geometry, one can see
a strong effect of the matrix polydispersity on the fluid-matrix pair distribution
functions. In figure 2 the radial distribution functions gFM(r) for the cases mentioned
above are presented. The low smoothed first peak with a weird shape in the case
of the polydisperse matrix is quite expectable: due to the wide range of sizes of
the matrix particles (σM/σF = 6.055 − 8.055) the fluid molecules can localize at
different distances to the matrix center. One can observe that such localization is
additionally enforced by the attractive potential (figures 2c and 2d) and this is clearly
seen at low temperature T ∗ = 1.2. The large difference between the fluid-matrix pair
distribution functions in the case of the polydisperse and monodisperse matrices is
an evidence of the strong effect of the matrix polydispersity on the internal energy
of the system.
Finally, we would like to focus on the problem of the approach chosen in our
study. As it was mentioned above, the internal energy of the adsorbed fluid obtained
from the PY-MGOZ like approach does not agree with the one obtained from MD
simulations at a low temperature. A reasonable way to find out the cause of the
deviation is to calculate fluid-matrix RDFs for the fluid confined, for example, in the
monodisperse KM matrix. Earlier, in [4] it was shown that the PY-MGOZ approach
gives unphysical results for fluid-matrix RDF at fluid temperature T ∗ = 1.2, but for
a higher density ρ∗F = 0.36. Moreover, an adverse tendency was observed for other
parameters as well. The same problem was discussed in the previous paper [3], in
which the difference between values of the internal energy obtained from the GCMC
simulation and the MGOZ equations was observed for different densities of the fluid
adsorbed in the monodisperse KM matrix. Therefore, in order to check the RDFs for
our system the radial distribution functions calculated from ROZ equations in the
PY and the hypernetted chain (HNC) approximations are presented in figure 3. The
lack of coincidence of the PY-MGOZ with MD is clearly observed in the vicinity of
the gap between the first and the second peaks of the fluid-matrix RDF at T ∗ = 1.2
(see figure 3a). Also, the height of the first peak is noticeably higher than the one
obtained from MD simulation for both temperatures, while the RDFs obtained from
the HNC-ROZ approach almost completely fit the MD results (figures 3b and 3d).
The short review presented here is the evidence of the cruel necessity to introduce
the blocking parts of the correlation functions into the PROZ equations using at
least the HNC approach.
In our study, the effect of the matrix polydispersity on the excess internal en-
ergy has been shown. However, the chemical potential would be more important to
consider in order to see an effect of the matrix polydispersity on, for example, the
adsorption isotherms. Besides, if we compare the systems with the different distri-
butions of matrix particle sizes, but with the same thermodynamic conditions (say
328
Fluid in polydisperse matrix
with equal chemical potentials), the amount of fluid molecules in both cases can
differ. Thus, when we compare the same amount of the fluid adsorbed even in the
hard-sphere matrices, the physical conditions for this fluid are different and the ef-
fects that are (or not) observed could be also related to this fact. The application
of the HNC approximation in the PROZ equations to the study of the chemical po-
tential and other thermodynamic properties of the fluid confined in a polydisperse
matrix is under study and will be published in future.
References
1. Given J.A., Stell G. // J. Chem. Phys., 1992, vol. 97, p. 4573.
2. Kaminsky R.D., Monson P.A. // J. Chem. Phys., 1991, vol. 95, p. 2936.
3. Vega C., Kaminsky R.D., Monson P.A. // J. Chem. Phys., 1993, vol. 99, p. 3003.
4. Trokhymchuk A., Pizio O., Holovko M., Sokolowski S. //J. Chem. Phys., 1997,
vol. 106, p. 200.
5. Patsahan T., Trokhymchuk A., Holovko M. // J. Mol. Liq, 2001, vol. 92, p. 117.
6. Page K.S., Monson P.A. // Phys. Rev., 1996, vol. 54, p. 6557.
7. Sarkisov L., Monson P.A. // Phys. Rev. E, 2000, vol. 51, p. 7231.
8. Brennan J.K., Dong W. // J. Chem. Phys., 2002, vol. 116, p. 20.
9. Patsahan T., Trokhymchuk A., Holovko M. // Condens. Matter Phys., 2003, vol. 6,
p. 3.
10. Lado F. // Phys. Rev. E, 1996, vol. 54, p. 4411.
11. Ilnytsky Ja., Patrykiejew A., Sokolowski S., Pizio O. // J. Phys. Chem. B, 1999,
vol. 103, p. 868–871.
12. Jorge S., Scholl-Paschinger E., Kahl G., Fernaud M.-J. // Mol. Phys., 2003, vol. 101,
p. 1733–1740.
13. Madden W.G., Glandt E.D. // J. Stat. Phys., 1988, vol. 51, p. 537.
329
T.Patsahan, M.Holovko
Проста рідина в невпорядкованій полідисперсній
матриці із притягаючою взаємодією
Т.Пацаган, М.Головко
Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
Отримано 6 травня 2004 р., в остаточному вигляді –
19 травня 2004 р.
Використовуючи розвинення рівнянь реплік Орнштейна-Церніке
(РОЦ), досліджувались структурні властивості Леннард-Джонсівсько-
го флюїду адсорбованого в полідисперсній матриці із притягаль-
ною взаємодією. Для цього було застосовано РОЦ рівняння в поєд-
нанні із розкладами кореляційних функцій по ортогональних поліно-
мах. Пораховано радіальні функції розподілу адсорбованого флюїда
при різних температурах. Також, в даному дослідженні, розглядався
ефект полідисперсності матриці на внутрішню енергію системи.
Ключові слова: флюїд, полідисперсність, пористе середовище,
силікагель, інтегральні рівняння, матриця, молекулярна динаміка
PACS: 05.20.Jj, 68.43.De, 82.70.Gg
330
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